au pure · molecular response properties from a hermitian eigenvalue equation for a time-periodic...

Molecular response properties from a Hermitian eigenvalue equation for a time-periodic Hamiltonian

Filip Pawłowski, , Jeppe Olsen, and Poul Jørgensen

Citation: The Journal of Chemical Physics 142, 114109 (2015); doi: 10.1063/1.4913364View online: http://dx.doi.org/10.1063/1.4913364View Table of Contents: http://aip.scitation.org/toc/jcp/142/11Published by the American Institute of Physics

Articles you may be interested inMolecular response properties in equation of motion coupled cluster theory: A time-dependent perspectiveThe Journal of Chemical Physics 144, 024102 (2016); 10.1063/1.4939183

Perspective: Explicitly correlated electronic structure theory for complex systemsThe Journal of Chemical Physics 146, 080901 (2017); 10.1063/1.4976974

Linear and nonlinear response functions for an exact state and for an MCSCF stateThe Journal of Chemical Physics 82, 3235 (1998); 10.1063/1.448223

http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/20939943/x01/AIP-PT/JCP_ArticleDL_0117/PTBG_orange_1640x440.jpg/434f71374e315a556e61414141774c75?x

http://aip.scitation.org/author/Paw%C5%82owski%2C+Filip

http://aip.scitation.org/author/Olsen%2C+Jeppe

http://aip.scitation.org/author/J%C3%B8rgensen%2C+Poul

/loi/jcp

http://dx.doi.org/10.1063/1.4913364

http://aip.scitation.org/toc/jcp/142/11

http://aip.scitation.org/publisher/

http://aip.scitation.org/doi/abs/10.1063/1.4939183



THE JOURNAL OF CHEMICAL PHYSICS 142, 114109 (2015)

Molecular response properties from a Hermitian eigenvalue equationfor a time-periodic Hamiltonian

Filip Pawłowski,1,2,a) Jeppe Olsen,1 and Poul Jørgensen11qLEAP Center for Theoretical Chemistry, Department of Chemistry, Aarhus University, Langelandsgade 140,DK-8000 Aarhus C, Denmark2Institute of Physics, Kazimierz Wielki University, Plac Weyssenhoffa 11, 85-072 Bydgoszcz, Poland

(Received 23 December 2014; accepted 11 February 2015; published online 18 March 2015)

The time-dependent Schrödinger equation for a time-periodic perturbation is recasted into a Hermi-tian eigenvalue equation, where the quasi-energy is an eigenvalue and the time-periodic regular wavefunction an eigenstate. From this Hermitian eigenvalue equation, a rigorous and transparent formula-tion of response function theory is developed where (i) molecular properties are defined as derivativesof the quasi-energy with respect to perturbation strengths, (ii) the quasi-energy can be determinedfrom the time-periodic regular wave function using a variational principle or via projection, and (iii)the parametrization of the unperturbed state can differ from the parametrization of the time evolutionof this state. This development brings the definition of molecular properties and their determinationon par for static and time-periodic perturbations and removes inaccuracies and inconsistencies ofprevious response function theory formulations. The development where the parametrization of theunperturbed state and its time evolution may differ also extends the range of the wave function modelsfor which response functions can be determined. The simplicity and universality of the presentedformulation is illustrated by applying it to the configuration interaction (CI) and the coupled cluster(CC) wave function models and by introducing a new model—the coupled cluster configurationinteraction (CC-CI) model—where a coupled cluster exponential parametrization is used for theunperturbed state and a linear parametrization for its time evolution. For static perturbations, theCC-CI response functions are shown to be the analytical analogues of the static molecular propertiesobtained from finite field equation-of-motion coupled cluster (EOMCC) energy calculations. Thestructural similarities and differences between the CI, CC, and CC-CI response functions are alsodiscussed with emphasis on linear versus non-linear parametrizations and the size-extensivity of theobtained molecular properties. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4913364]

I. INTRODUCTION

Response functions provide a framework which allowsthe determination of molecular properties for a variety ofquantum chemical models. We present a transparent andrigorous formulation of response function theory, wherethe application range of response theory is extended andwhere the derivation follows a different and simpler routecompared to previous formulations.1,2 The cornerstone isthe recasting of the time-dependent Schrödinger equationfor a time-periodic perturbation into a Hermitian eigenvalueequation where the quasi-energy is an eigenvalue and the time-periodic regular wave function an eigenstate. In the staticlimit, this Hermitian eigenvalue equation becomes the Her-mitian eigenvalue equation for static perturbations. For staticperturbations, molecular properties are defined as derivativesof the eigenvalue with respect to perturbation strengths. Forperiodic perturbations, this definition of molecular propertiesthus extends to derivatives of the quasi-energy.

Recasting the time-dependent Schrödinger equation fora time-periodic perturbation into a Hermitian eigenvalue

a)Electronic mail: [email protected]

equation is by itself not new and has been carried out, forexample, in the context of Floquet theory3–5 and steady-state6

theory. In the steady-state formulation, the time-dependentwave function is written as an exponential phase-factorcontaining the quasi-energy times a time-dependent wavefunction, the steady-state wave function. Using simple time-translational arguments, it is easy to see that the steady-statewave function is time-periodic with the same period as theHamiltonian. However, the steady-state wave function doesnot provide the best starting point for the development ofapproximate time-dependent quantum mechanical methods. Inparticular, the steady-state wave function contains an overallphase, which is redundant and often leads to problems, inparticular in the time-independent limit. To avoid this problem,it is useful to introduce the regular wave function defined asthe steady-state wave function times a time-dependent phase-factor, which is chosen so that the time-dependent coefficientof a chosen reference state is real and positive at all times. Inthe development of Sambe,6 it is shown that the regular wavefunction has the same periodicity as the steady-state wavefunction, provided the reference state is a solution to a time-independent Schrödinger equation. In the present work, thisrestriction is removed, i.e., we allow an arbitrary normalizedstate with non-vanishing overlap with the steady-state wave

0021-9606/2015/142(11)/114109/51/$30.00 142, 114109-1 © 2015 AIP Publishing LLC

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

mailto:[email protected]


























http://crossmark.crossref.org/dialog/?doi=10.1063/1.4913364&domain=pdf&date_stamp=2015-03-18

114109-2 Pawłowski, Olsen, and Jørgensen J. Chem. Phys. 142, 114109 (2015)

function as the reference state and show that the regular andsteady-state wave functions have the same periodicity.

The regular wave function provides the starting pointfor developing approximate time-dependent methods freeof singularities for the wave functions that are normalizedat all times. For a number of standard approximate wavefunction models, in particular coupled cluster (CC) methods,an intermediate normalized rather than a unit-norm normalizedwave function is used. By eliminating the amplitude of thereference wave function from the regular wave function, weshow that the regular wave function is then also time-periodicand fulfills a slightly modified eigenvalue equation than for aunit-normalized state.

We have thus described a direct route where, usingthe time-translational invariance of the Hamiltonian, thetime-dependent Schrödinger equation can be recasted into aHermitian eigenvalue equation and where molecular responsefunctions are defined as derivatives of the quasi-energy withrespect to the perturbation strengths. The advantages ofdefining molecular properties in this way compared to thetraditional one, where response functions are defined in termsof the expansion coefficients of a perturbation expansion of anexpectation value for an observable, are also discussed.

The formulation we present allows for a different param-etrization of the unperturbed state and its time evolution. Thismakes it possible to derive response functions for new classesof states. For example, we introduce a new model with acoupled cluster unperturbed state and a linear parametrizationof the time evolution, which we denote the coupled clusterconfiguration interaction (CC-CI) model, and derive responsefunctions for this model. In the static limit, molecularproperties obtained with the CC-CI model become an analyticrepresentation of the static molecular properties obtainedfrom finite field equation-of-motion coupled cluster (EOMCC)energy calculations. The CC-CI molecular properties arenot size-extensive; however, we show that the lack of size-extensivity is weak as the unperturbed system is describedby a coupled cluster state. We also derive response functionsfor the standard configuration interaction (CI) and CC wavefunction models and discuss the size-extensivity of molecularproperties for these models for truncated excitation manifolds.

The formulation we present simplifies the derivationof response functions and allows us to solve inadequaciesand inconsistencies of previous formulations of responsefunction theory.1,2 For example, in the previous formulationsof response function theory, time evolution has only beenconsidered for states with a norm fixed in time and results wereassumed to carry over for states with time-evolving norms,as coupled cluster states. The formulation we develop hereis rigorously applicable to states with both a constant and atime-evolving norm.

Response function theory was introduced in the physicsliterature in the 1950s and 1960s. We refer to the books ofZubarev7 and Linderberg and Öhrn8 for more details. Responsefunction theory was introduced into chemistry in the 1960sand 1970s9–11 and in its contemporary form formulated in1985 by Olsen and Jørgensen1 for determining molecularproperties for both ground and excited states and transitionproperties between these states. Olsen and Jørgensen1 derived

response functions for exact and multi-configurational self-consistent field (MCSCF) reference states. The responsefunctions were identified from a perturbation expansion ofthe expectation value of an observable, and the permutationalsymmetry of response functions with respect to interchange ofoperator–frequency pairs was for that reason far from trivialto obtain within this formulation.12

The first-generation formulation of response functiontheory was followed by implementations of MCSCF linear,13

quadratic,14,15 and cubic16,17 response functions and theirresidues and numerous applications of MCSCF responsetheory in various fields of chemistry and physics have beenreported, see, for example, Refs. 18–31.

The derivation of response functions from a perturbationexpansion of the expectation value of an observable was in1990 generalized by Koch and Jørgensen32 to comprehendcoupled cluster wave functions and implementations werereported for a coupled cluster singles and doubles (CCSD)state for excitation energies,33 transition strengths,34 andpolarizabilities.35

In 1998, Christiansen, Hättig, and Jørgensen published thesecond generation formulation of response function theory2

that treated variational and non-variational states in a uniformmanner. Central for that formulation was the generalization ofthe energy of time-independent theory to the quasi-energy fortime-periodic perturbations and the recognition that the quasi-energy satisfies, as the energy, a variational principle. Usingthe generalized Hellmann-Feynman theorem, the responsefunctions were shown, in the spirit of an earlier work by Aiga,Sasagane, and Itoh,36,37 to be derivatives of the quasi-energywith respect to the perturbation strengths. For non-variationalstates, the quasi-energy was replaced by the quasi-energyLagrangian and the time evolution of a non-variational statewas determined from the stationary conditions for the quasi-energy Lagrangian. Response functions for non-variationalstates were determined as derivatives of the quasi-energyLagrangian. Expressing response functions as quasi-energyor quasi-energy Lagrangian derivatives automatically yieldedthe operator–frequency pair permutational symmetries of theresponse functions, which were cumbersome to identify in thefirst generation formulation of response function theory. Thesecond generation formulation also allowed the applicationof the 2n + 1 and 2n + 2 rules,38–41 thereby considerablysimplifying response functions expressions.

Identifying the response functions as quasi-energyLagrangian derivatives allowed also the derivation of responsefunctions for approximate coupled cluster models, wheresimplifications were introduced in the coupled cluster ampli-tude equations based, for example, on a Møller-Plessetpartitioning of the Hamiltonian. This led to the coupledcluster response hierarchy CCS, CC2,42 CCSD,43 CC3,44–46

CCSDT,47,48 . . . and also to response functions for other iter-ative approximate coupled cluster wave function models.49–54

The second generation formulation boosted efficientimplementations of response functions for coupled clusterand approximate coupled cluster wave function models thathave been widely used.55–66 Further, the second generationformulation of response function theory allowed for the devel-opment of linear-scaling implementations and applications of


density-matrix based quasi-energy Hartree-Fock and Kohn-Sham response theories.67–69 Damped response function the-ory was introduced by Norman et al.,70,71 where finite excited-state lifetimes were introduced as an alternative to a residueanalysis for addressing resonance spectroscopies.72–75 For acomprehensive review of the developments, implementations,and applications of response function theory until 2012, seealso Ref. 76. The second generation formulation thus advancedresponse function theory to become a commonly used blackbox tool for predicting molecular properties for ground andexcited states and for transitions between these states.

Despite its great success, the second generation formula-tion suffered from several formal deficiencies. One example isthe legitimacy of generalizing the determination of responsefunctions as derivatives of the quasi-energy to the quasi-energyLagrangian. The quasi-energy is obtained solving the time-dependent Schrödinger equation for a time-evolving state witha unit norm, an assumption that clearly cannot be appliedfor non-variational wave functions that have a time-evolvingnorm, for example, a coupled cluster wave function. Anotherexample is that for approximate non-variational states, theHermiticity of an expectation value of an observable isnot guaranteed. This deficiency was circumvented by an adhoc introduction of an operator that symmetrizes responsefunctions with respect to simultaneous sign inversion of thefrequencies and complex conjugation, thus superimposing thesymmetry relation of the response functions for an exact stateon the response functions for approximate non-variationalstates, for example, coupled cluster states.

The formulation presented in this article leads to molec-ular response properties expressed as derivatives of the quasi-energy or the quasi-energy Lagrangian, as did the secondgeneration formulation of 1998, but without the inadequaciesof the previous formulations. The way in which we arrive at thequasi-energy and quasi-energy Lagrangian is, however, verydifferent. In our formulation, the quasi-energy is an eigenvalueof a Hermitian eigenvalue equation and molecular propertiesmay therefore straightforwardly be defined by differentiationof the eigenvalue with respect to perturbation strengths.This direct way of defining molecular response propertiesdoes not require any recourse to the generalized Hellmann-Feynman theorem. The operator that symmetrizes responsefunctions with respect to simultaneous sign inversion of thefrequencies and complex conjugation arises automatically inthe formulation presented here.

Another advantage of the formulation presented here isthat the parametrization of the time evolution is separated fromthe parametrization of the unperturbed state. This allows usto eliminate the zero-order amplitude and multiplier equationsfrom the quasi-energy Lagrangian at an early stage of thederivation, which greatly simplifies the derivation of responsefunctions, compared to previous formulations. Additionalsimplifications result from the fact that we may use the 2n + 1and 2n + 2 rules, as in the formulation of 1998 in order toderive the response functions. However, we may also use thegeneralizations of these rules, as formulated by Kristensenet al.77

The simplicity in our derivation of response functions isillustrated by deriving linear, quadratic, and cubic response

functions for CI, CC-CI, and CC wave function models.The CI, CC-CI, and CC models constitute different param-etrizations of the same exact solution to the time-dependentSchrödinger equation and, therefore, provide full configura-tion interaction (FCI) results in the limit, where no truncationsare performed in the excitation manifold. In both the CI andthe CC-CI models, the time evolution is linearly parametrized,in CI combined with a time-independent CI reference state,while in CC-CI with a time-independent CC reference state.The linear parametrization of the time evolution makes itpossible to obtain an explicit representation of the groundand excited states, but this happens at the expense that theCI and CC-CI molecular properties are not size-extensive.In the CC model, both the time-independent reference stateand the time evolution are exponentially parametrized. Theexponential parametrization of the time evolution makes itimpossible to obtain an explicit representation of the groundand excited states. This, however, does not pose any problem,since explicit expressions for molecular properties may beobtained from the response functions. In turn, the exponentialparametrization ensures that the CC molecular properties aresize-extensive, also when truncations are performed in theexcitation manifold.

We also compare the CC-CI response functions with theEOMCC response functions. EOMCC response functions arenot derived by differentiation of a quasi-energy Lagrangianbut, as described by Stanton and Bartlett78 and Rozyczko andBartlett,79–81 from CI response functions by substituting theCI energies, states, and response vectors with their EOMCCcounterparts. In this article, we show that the EOMCC andCC-CI linear response functions are identical. We show alsothat the EOMCC and CC-CI quadratic (and higher-order)response functions differ and that the EOMCC quadratic,cubic, etc., response functions therefore cannot reproduce theFCI limit results.

In Sec. II, we recast the time-dependent Schrödingerequation for a time-periodic perturbation into a Hermitianeigenvalue equation. We eliminate the time-dependent phasefrom the eigenstate and show that in the time-independentlimit the phase-isolated eigenstate (the regular wave function)becomes the eigenstate and the quasi-energy the eigenvalueof the time-independent Hamiltonian. Molecular responseproperties for a time-periodic Hamiltonian may therefore bedefined as derivatives of the quasi-energy with respect to time-periodic perturbation strengths, in the same way the staticmolecular properties are defined as derivatives of the energywith respect to static perturbation strengths. In Sec. III weshow how molecular response functions may be obtained whenthe Hermitian eigenvalue equation is solved using a variationalprinciple. We also compare the definition of response functionsas derivatives of the quasi-energy to the standard definition asthe time-dependent expansion coefficients of the expectationvalue of a perturbation operator. In Sec. IV, we show howmolecular response functions may be obtained as derivatives ofthe quasi-energy Lagrangian when the Hermitian eigenvalueequation is solved via projection. We also parametrize thetime-dependent wave function such that the parametrizationand optimization of the unperturbed system may differ fromthe parametrization and optimization of its time evolution. We


apply the general development of Sec. IV in Secs. VI–VIIIto derive response functions for the CI, CC-CI, and CCmodels. The EOMCC response functions are given in Sec. IX.A comprehensive comparison between CI, CC-CI, and CCresponse functions is carried out in Sec. X with emphasison linear versus exponential parametrization and the size-extensivity of molecular response properties. We also showthat the CC-CI static limit response properties may be obtainedfrom EOMCC finite field energy calculations. Section XIcontains concluding remarks.

II. THE TIME-DEPENDENT SCHRÖDINGER EQUATIONFOR A PERIODIC PERTURBATION AS A HERMITIANEIGENVALUE EQUATION

In this section, we present the foundation for recastingthe time-dependent Schrödinger equation for a time-periodicHamiltonian into a Hermitian eigenvalue equation. We alsodescribe how molecular properties become the derivativesof the eigenvalue of this Hermitian eigenvalue equationwith respect to perturbation strengths and how molecularresponse functions therefore may be obtained solving theHermitian eigenvalue equation. This development brings thedetermination of molecular properties for time-periodic andstatic perturbations on par.

A. The steady-state wave function and its eigenvalueequation

Consider the time-dependent Schrödinger equation,which in atomic units becomes

H(t, ϵ)|0(t, ϵ)⟩ = i∂

∂t|0(t, ϵ)⟩, (1)

where H(t, ϵ) is a time-dependent bounded Hermitian Hamil-tonian

H(t, ϵ) = H0 + V (t, ϵ), (2)

where H0 is the time-independent Hamiltonian of the unper-turbed system and V (t, ϵ) is a time-dependent perturbation.V (t, ϵ) may be expanded in a sum over Fourier components,

V (t, ϵ) =j

X jϵXj

(ωXj

)exp

(−iωXj

t), (3)

where X j is a Hermitian time-independent operator and ϵXj(ωXj

)is the associated perturbation strength for the real fre-

quency ωXj. ϵ denotes a set of perturbation strengths ϵXj(

ωXj

). To obtain a Hermitian perturbation operator, the sum

in Eq. (3) includes for each term X jϵXj

(ωXj

)exp

(−iωXj

t)

also its adjoint X†j ϵXj

(−ωXj

)exp

(iωXj

t)

with

ϵXj

(−ωXj

)= ϵ∗Xj

(ωXj

). (4)

We will in the following consider solely perturbations thatare periodic in time with a period T ,

V (t + T, ϵ) = V (t, ϵ) . (5)

It then follows from Eq. (3) that associated with the period T ,there is a fundamental frequency ωT ,

ωT =2πT

(6)

such that the frequencies in Eq. (3) become an integer timesthe fundamental frequency,

ωXj= nXj

ωT . (7)

As the Hamiltonian, H(t, ϵ) of Eq. (2), is bounded, it hassolutions with finite norm, and we consider in the followingonly such solutions. From the Hermiticity of the Hamiltonian,it follows that the norm of |0(t, ϵ)⟩ is independent of time. Wechoose this norm to be one,

⟨0(t, ϵ)|0(t, ϵ)⟩ = 1. (8)

The time-periodicity of the perturbation operator inEq. (5) implies that the Hamiltonian H(t, ϵ) of Eq. (2) isinvariant with respect to time translations of the length kT ,where k is an integer,

Tt(kT),H(t, ϵ) = 0, k = 0,±1,±2, . . . , (9)

where Tt(∆t) is a time-translation operator that carries out atranslation of a state | f (t)⟩ from an initial time t to a final timet + ∆t,

Tt(∆t) | f (t)⟩ = | f (t + ∆t)⟩. (10)

Acting with the time-translation operator on theSchrödinger equation [Eq. (1)], we obtain

Tt(kT)H(t, ϵ)|0(t, ϵ)⟩ = Tt(kT)i ∂∂t

|0(t, ϵ)⟩. (11)

Using Eq. (9), and the fact that the time differentiation operatorcommutes with the time-translation operator, we obtain fromEq. (11)

H(t, ϵ) (Tt(kT)|0(t, ϵ)⟩) = i∂

∂t

(Tt(kT)|0(t, ϵ)⟩) , (12)

which, using Eq. (10), may be written as

H(t, ϵ)|0(t + kT, ϵ)⟩ = i∂

∂t|0(t + kT, ϵ)⟩. (13)

Comparing Eqs. (1) and (13) and assuming that |0(t, ϵ)⟩ is non-degenerate, we conclude that |0(t, ϵ)⟩ and |0(t + kT, ϵ)⟩ satisfythe same Schrödinger equation and they therefore differ atmost by a constant phase factor,

|0(t + kT, ϵ)⟩ = exp (−ikα) |0(t, ϵ)⟩, (14)

where α is the phase that describes a time-translation overone period T . Since |0(t, ϵ)⟩ is unit-normalized at each time[Eq. (8)], α is real.

Eq. (14) is satisfied for a wave function of the form

|0(t, ϵ)⟩ = exp(−iα

Tt) |u(t, ϵ)⟩, (15)

where |u(t, ϵ)⟩ is time-periodic with the same period T as theHamiltonian,

|u(t + T, ϵ)⟩ = |u(t, ϵ)⟩. (16)


This may be seen from the following derivation:

|0(t + kT, ϵ)⟩ = exp[−iα

T(t + kT)] |u(t + kT, ϵ)⟩

= exp(−ikα) exp(−iα

Tt) |u(t, ϵ)⟩

= exp(−ikα) |0(t, ϵ)⟩. (17)

From Eqs. (8) and (15), it follows that |u(t, ϵ)⟩ is unit-normalized,

⟨u(t, ϵ)|u(t, ϵ)⟩ = 1. (18)

Introducing the notation

E = α

T, (19)

|0(t, ϵ)⟩ of Eq. (15) becomes

|0(t, ϵ)⟩ = exp(−iE(ϵ)t) |u(t, ϵ)⟩. (20)

Inserting Eq. (20) into the Schrödinger equation [Eq. (1)]yields an eigenvalue equation from which E(ϵ) and |u(t, ϵ)⟩may be determined,(

H(t, ϵ) − i∂

∂t

)|u(t, ϵ)⟩ = E(ϵ) |u(t, ϵ)⟩. (21)

Note that E(ϵ) depends on the set of perturbation strengths ϵ .Equations (16) and (21) are the time-dependent Schrödingerequation within steady-state theory.6

B. The regular wave function and its eigenvalueequation

The steady-state wave function contains an undeterminedtime-dependent phase factor. To determine this phase factor,we introduce a reference state |s0⟩ which may be anynormalized time-independent state that has a non-vanishingcomponent in |u(t, ϵ)⟩,

⟨s0|u(t, ϵ)⟩ , 0. (22)

We then introduce the time-dependent regular wave function,|0R(t, ϵ)⟩, defined as

|u(t, ϵ)⟩ = ⟨s0|u(t, ϵ)⟩|⟨s0|u(t, ϵ)⟩| |0R(t, ϵ)⟩, (23)

where |⟨s0|u(t, ϵ)⟩| is the modulus of ⟨s0|u(t, ϵ)⟩. We now showthat |0R(t, ϵ)⟩ is connected to |u(t, ϵ)⟩ via a phase factor thathas a real time-periodic phase, that |0R(t, ϵ)⟩ is time-periodicand unit-normalized, and that it has a real positive componentin the |s0⟩ direction.

As ⟨s0|u(t,ϵ)⟩|⟨s0|u(t,ϵ)⟩| has modulus 1, it may be written in terms of

a real phase FP0(t, ϵ) as

⟨s0|u(t, ϵ)⟩|⟨s0|u(t, ϵ)⟩| = exp(−iFP0(t, ϵ)). (24)

Since |u(t, ϵ)⟩ is time-periodic, it follows from Eq. (24) thatFP0(t, ϵ) is also time-periodic. Substituting Eq. (24) in Eq. (23)gives as

|u(t, ϵ)⟩ = exp�−iFP0(t, ϵ)

� |0R(t, ϵ)⟩. (25)

From Eq. (25), we see that the time-periodicity of |u(t, ϵ)⟩ andFP0(t, ϵ) implies that the regular wave function is time-periodic

|0R(t + T, ϵ)⟩ = |0R(t, ϵ)⟩. (26)

Using that the steady-state wave function is unit-normalized[Eq. (18)] and that FP0(t, ϵ) is real, it follows from Eq. (25)that the regular wave function is unit-normalized,

⟨0R(t, ϵ)|0R(t, ϵ)⟩ = 1. (27)

Taking the scalar product of |u(t, ϵ)⟩ in Eq. (23) with ⟨s0| gives

⟨s0|0R(t, ϵ)⟩ = |⟨s0|u(t, ϵ)⟩| > 0. (28)

Inserting Eq. (25) in Eq. (20) shows the connectionbetween the full solution to the Schrödinger equation and theregular wave function,

|0(t, ϵ)⟩ = exp (−iF (t, ϵ)) |0R(t, ϵ)⟩, (29)

where

F (t, ϵ) = E(ϵ) t + FP0(t, ϵ). (30)

Substituting Eq. (25) in the Schrödinger equation for thesteady-state wave function [Eq. (21)] gives(

H(t, ϵ) − i∂

∂t

)exp

�−iFP0(t, ϵ)

� |0R(t, ϵ)⟩= E(ϵ) exp

�−iFP0(t, ϵ)

� |0R(t, ϵ)⟩, (31)

which may be expressed as(H(t, ϵ) − i

∂

∂t− FP0(t, ϵ)

)|0R(t, ϵ)⟩ = E(ϵ)|0R(t, ϵ)⟩. (32)

Eq. (32) is an eigenvalue equation with the eigenvalue E(ϵ)and the eigenstate |0R(t, ϵ)⟩. The regular wave function thus isa phase-isolated solution to the time-dependent Schrödingerequation.

The phase [Eq. (30)] is not uniquely defined as FP0(t, ϵ)[Eq. (24)] depends on the choice of the reference state |s0⟩.In Sec. II C, we introduce a composite Hilbert space6 andshow that in this space both Eqs. (21) and (32) are Hermitianeigenvalue equations and that the solution to Eq. (32) becomesindependent of FP0(t, ϵ).

C. The Hermiticity of the time-dependent Hamiltonian

We now introduce a composite Hilbert space in the timeand configuration space variables for solving Eqs. (21) and(32). First, we recognize that the functions exp (ipωTt), withωT defined in Eq. (6) and p = 0,±1,±2, . . ., satisfy

1T

T

0[exp (ipωTt)]∗ exp (iqωTt) dt = δpq (33)

and form a complete orthonormal basis for the Hilbert spacein the time variable, where the inner product for two time-periodic functions f (t) and g(t) is defined as

f ∗(t)g(t)

T=

1T

T

0f ∗(t)g(t)dt . (34)


Since |u(t, ϵ)⟩ is time-periodic with the period T , we mayexpand |u(t, ϵ)⟩ as

|u(t, ϵ)⟩ =mp

cmp exp (ipωTt) |m⟩, (35)

where {|m⟩} is an orthonormal basis in the Hilbert space overconfiguration space variables and the coefficients cmp are time-independent. The eigenstate |u(t, ϵ)⟩ of Eq. (35) therefore maybe expanded in the composite Hilbert space6 comprehendingboth the time and the configuration space variables.

The inner product in the composite Hilbert space is

⟨a(t)|b(t)⟩T=

1T

T

0⟨a(t)|b(t)⟩dt, (36)

where ⟨a(t)|b(t)⟩ denotes the standard inner product over theconfiguration space variables. The time average of a timederivative of a time-periodic function vanishes

f (t + T) = f (t)⇒

d f (t)dt

T

=1T

T

0

d f (t)dt

dt

=1T

f (t)T

0=

1T

(f (T) − f (0)) = 0. (37)

The operator i ∂∂t

is Hermitian in the composite Hilbertspace, since for two time-periodic states ⟨a(t)| and |b(t)⟩ withthe period T , we have from Eq. (37)

0 =

i∂

∂t⟨a(t) |b(t) ⟩

T

= −

i∂

∂ta(t)

��b(t)

T

+

a(t)

��i∂

∂tb(t)

T

(38)

and thus i∂

∂ta(t)

��b(t)

T

=

a(t)

��i∂

∂tb(t)

T

. (39)

Since the Hamiltonian H(t, ϵ) of Eq. (2) is Hermitian, it followsfrom Eq. (39) that the operator

�H − i ∂

∂t

�is Hermitian in the

composite Hilbert space and Eqs. (21) and (32) constitutestandard Hermitian eigenvalue equations in this space.6 Theregular wave-function is also periodic and may therefore alsobe expressed in the composite Hilbert space.

The time average of the time-differentiated overall phase-factor, Eq. (30), gives

�F (t, ϵ)

T= E(ϵ) + �FP0(t, ϵ)

T. (40)

Using that FP0(t, ϵ) is periodic, Eq. (37) may be invoked

�FP0(t, ϵ)

T= 0 (41)

and Eq. (40) becomes�F (t, ϵ)

T= E(ϵ). (42)

Whereas the phase F (t, ϵ) is not uniquely defined and dependson the choice of the state |s0⟩, the time average of the timederivative of the phase,

�F (t, ϵ)

T, is independent of that

choice of |s0⟩ and uniquely defined76 and equal to E(ϵ).

D. The Hermitian eigenvalue equation in the limitof a time-independent perturbation

In the limit, where the frequencies in Eq. (3) are zero,the perturbation operator of Eq. (3) and hence also theHamiltonian of Eq. (2) become time-independent

H(ϵ) = H0 + V (ϵ) (43)

and the time-dependent Schrödinger equation reads

H(ϵ)|00(t, ϵ)⟩ = i∂

∂t|00(t, ϵ)⟩. (44)

The solution to the time-dependent Schrödinger equation[Eq. (44)] may then be parametrized as

|00(t, ϵ)⟩ = exp(−iE0(ϵ)t)|00(ϵ)⟩, (45)

which, inserted in Eq. (44), gives the time-independentSchrödinger equation

H(ϵ)|00(ϵ)⟩ = E0(ϵ)|00(ϵ)⟩. (46)

The eigenstate |00(ϵ)⟩ contains an undetermined time-independent phase factor,

|00(ϵ)⟩ = exp(−iφ(ϵ))|00(ϵ)⟩. (47)

Inserting Eq. (47) in Eq. (46) gives

H(ϵ) exp(−iφ(ϵ))|00(ϵ)⟩ = E0(ϵ) exp(−iφ(ϵ))|00(ϵ)⟩. (48)

The phase factor is constant and may therefore straightfor-wardly be eliminated giving the time-independent Schrödingerequation for the phase-isolated state,

H(ϵ)|00(ϵ)⟩ = E0(ϵ)|00(ϵ)⟩. (49)

We now consider the time-independent limit of the time-dependent Schrödinger equation [Eq. (1)]. In this limit, thesteady-state eigenvalue equation [Eq. (21)] becomes the time-independent Schrödinger equation for the phase-includingstate [Eq. (46)]. Comparing Eqs. (25) and (47), we see thatin the time-independent limit the phase factor exp

�−iFP0(t, ϵ)

�

becomes time-independent and equal to exp (−iφ(ϵ)), whereasthe regular wave function |0R(t, ϵ)⟩ becomes equal to |00(ϵ)⟩.For the time-independent Schrödinger equation [Eq. (48)],the time-independent phase factor could be removed givingthe time-independent Schrödinger equation [Eq. (49)]. For thetime-dependent Schrödinger eigenvalue equation [Eq. (31)],the time-dependent phase factor introduces the time derivativeof the phase, FP0(t, ϵ), in the eigenvalue equation [Eq. (32)],which of course vanishes in the time-independent limit.

Summarizing, in the time-independent limit, we have

E(ϵ) −→ E0(ϵ), (50)

|u(t, ϵ)⟩ −→ |00(ϵ)⟩, (51)|0R(t, ϵ)⟩ −→ |00(ϵ)⟩, (52)

and

FP0(t, ϵ) −→ φ(ϵ). (53)

The eigenvalue E(ϵ) is thus a generalization of the energy totime-periodic perturbations. E(ϵ) is therefore referred to as thequasi-energy.6


E. The variational principle for the quasi-energy

The eigenvalue E(ϵ) may be determined by projecting, inthe composite Hilbert space, either the Hermitian eigenvalueequation [Eq. (21)] against ⟨u(t, ϵ)|,

E(ϵ) =⟨u(t, ϵ)|

(H(t, ϵ) − i

∂

∂t

)|u(t, ϵ)⟩

T

, (54)

or the Hermitian eigenvalue equation [Eq. (32)] against⟨0R(t, ϵ)|,

E(ϵ) =⟨0R(t, ϵ)|

(H(t, ϵ) − i

∂

∂t

)|0R(t, ϵ)⟩

T

, (55)

where we have used the unit normalization of |u(t, ϵ)⟩[Eq. (18)] and |0R(t, ϵ)⟩ [Eq. (27)]. To obtain Eq. (55), wehave also used Eq. (41). In the following, we will determineE(ϵ) expressed in terms of |0R(t, ϵ)⟩ [Eq. (55)], rather than interms of |u(t, ϵ)⟩ [Eq. (54)], since |0R(t, ϵ)⟩, and not |u(t, ϵ)⟩,becomes equal to |00(ϵ)⟩ in the limit of static perturbation.E(ϵ) of Eq. (55) is a generalization of the expectation

value expression for the energy for a time-independent system

E0(ϵ) = ⟨00(ϵ)|H(ϵ)|00(ϵ)⟩. (56)

E(ϵ) is also an eigenvalue of the Hermitian eigenvalueequation [Eq. (32)] which is a generalization of the eigenvalueof the Hermitian eigenvalue equation for a time-independentHamiltonian [Eq. (49)]. For the time-independent perturbedsystem, the variational principle for the ground-state energyof Eq. (56) becomes

δE0(ϵ) = ⟨δ00(ϵ)|H(ϵ)|00(ϵ)⟩ + ⟨00(ϵ)|H(ϵ)|δ00(ϵ)⟩ = 0 (57)

and is equivalent to solving the time-independent Schrödingerequation [Eq. (49)]. Similarly, for a time-periodic system, thevariational principle for E(ϵ) in Eq. (55)

δE(ϵ) =⟨δ0R(t, ϵ)|

(H(t, ϵ) − i

∂

∂t

)|0R(t, ϵ)⟩

T

+

⟨0R(t, ϵ)|

(H(t, ϵ) − i

∂

∂t

)|δ0R(t, ϵ)⟩

T

= 0 (58)

is equivalent to solving the Hermitian eigenvalue equation[Eq. (32)]. This equivalence between a variational principleand a Hermitian eigenvalue equation is a general equivalenceand is discussed for time-independent theory, for example, inRef. 82. Alternatively, the equivalence may be shown treatingE(ϵ) of Eq. (55) as the action functional in the same manneras in Refs. 83–85.

When the variational principle is written in the formof Eq. (58), we have used that the time-periodic regularwave function |0R(t, ϵ)⟩ is unit-normalized for all perturbationstrengths and at each time [Eq. (27)]. The allowed variationsin Eq. (58) are therefore time-periodic variations whichaccording to Eq. (27) fulfill the normalization condition

⟨0R(t, ϵ) + δ0R(t, ϵ) |0R(t, ϵ) + δ0R(t, ϵ) ⟩ = 1. (59)

This implies that through first-order, the variations satisfy

⟨δ0R(t, ϵ) |0R(t, ϵ) ⟩ + ⟨0R(t, ϵ) |δ0R(t, ϵ) ⟩ = 0 (60)

corresponding to

⟨0R(t, ϵ) |δ0R(t, ϵ) ⟩ = −⟨0R(t, ϵ) |δ0R(t, ϵ) ⟩∗. (61)

Equation (61) shows that for an allowed variation satisfyingEq. (58), ⟨0R(t, ϵ) |δ0R(t, ϵ) ⟩ must either vanish or be purelyimaginary. Similar allowed variations may be applied to obtainEq. (57).

F. Molecular response properties as derivativesof the eigenvalue of the Hermitian eigenvalue equation

We will now discuss how molecular response propertiesmay be determined from the quasi-energy E(ϵ). Consider thelimit, where the frequencies of the perturbation of Eq. (3) arezero. The quasi-energy E(ϵ) then becomes the ground-stateenergy, E0(ϵ), of the perturbed time-independent system. Forthe perturbations described by the Hamiltonian of Eq. (43),the induced changes of the ground state energy may beexperimentally observed for general values of the perturbationstrengths. It is therefore experimentally possible to determinean order expansion of the ground state energy, which directlygives the static molecular properties. Carrying out a Taylorseries expansion of E0(ϵ) in the perturbation strengths aroundthe ground-state energy E0 of the unperturbed system

E0(ϵ) = E0 +j1

EXj10 ϵXj1

+12

j1 j2

EXj1

Xj20 ϵXj1

ϵXj2+ · · ·,

(62)

we may thus identify static molecular properties for the state|00⟩ of the energy E0 as the expansion coefficients of the Taylorseries

EXj1

. . .Xjn

0 = *,

∂nE0(ϵ)∂ϵXj1

. . . ∂ϵX jn

+-0

, (63)

where the index 0 denotes that the derivative is taken at zeroperturbation strength.

For a time-periodic Hamiltonian [Eq. (2)], the quasi-energy of Eq. (55) may also be determined in each orderof the perturbation strengths identifying molecular constantsdenoted the frequency-dependent molecular properties or themolecular response functions. Carrying out a Taylor seriesexpansion of E(ϵ) in the perturbation strengths around theground-state energy E0 of the unperturbed system

E(ϵ) = E0 +j1

EXj1(ωXj1)ϵXj1

(ωXj1)

+12

j1 j2

EXj1Xj2(ωX j1,ωX j2)

× ϵX j1(ωX j1)ϵX j2(ωX j2) + · · ·, (64)

we may thus identify the frequency-dependent molecularproperties for the state |00⟩ of the energy E0 as the expansioncoefficients of the Taylor series

EXj1. . .Xjn(ωXj1

, . . . ,ωX jn)

= *,

∂nE(ϵ)∂ϵXj1

(ωXj1) . . . ∂ϵX jn

(ωX jn)+-0

, (65)

where EXj1. . .Xjn(ωXj1

, . . . ,ωX jn) is symmetric with respect to

permutations of operator–frequency pairs(X jk,ωXjk

)↔

(X jl,ωXjl

), k, l = 1,2, . . . n. (66)


Eq. (65) thus is a generalization of Eq. (63) to time-periodic perturbations and determines the molecular responsefunctions.

The Hamiltonian H0 for the unperturbed system satisfiesthe eigenvalue equation

H0|0m⟩ = Em|0m⟩. (67)

Since the eigenstates of H0 form a complete set, the regularwave function, |0R(t, ϵ)⟩ in Eq. (32), may be expressed as

|0R(t, ϵ)⟩ = N(t, ϵ) *,|00⟩ +

m,0

bm(t, ϵ)|0m⟩+-, (68)

where the reference state |s0⟩ that has a real and positiveexpansion coefficient in |0R(t, ϵ)⟩ [Eq. (28)] has been chosenas |00⟩.

The regular wave function |0R(t, ϵ)⟩ describes the timeevolution of the perturbed ground state and thus determines themolecular response properties for this state. However, throughthe time-evolving coefficients bm(t, ϵ) [Eq. (68)], the regularwave function also describes how transitions may be inducedbetween the ground and excited states and between excitedstates and further probes the molecular properties of theexcited states. Consider, for example, the second-order quasi-energy EXj1

Xj2(ωXj1,ωXj2

). This derivative is non-vanishingonly for ωXj1

+ ωXj2= 0 [see Sec. III B 1, in particular

Eq. (142)] and may be expanded in the basis of eigenfunctions|0m⟩ as

EXj1Xj2(ωXj1

,ωXj2) =

m

*,

⟨00|X j1|0m⟩⟨0m|X j2|00⟩ωX j2

− (Em − E0)

+⟨00|X j2|0m⟩⟨0m|X j1|00⟩−ωX j2

− (Em − E0)+-. (69)

Equation (69) shows that the second-order quasi-energybecomes infinite whenever an experimental frequency, ωX j2

,matches an excitation energy, Em0 = Em − E0. Similarly,the general-order quasi-energy, EXj1

. . .Xjn(ωXj1, . . . ,ωXjn

) inEq. (65), becomes infinite whenever one of the frequenciesmatches an excitation energy. For this match, molecularproperties are determined by the residues of the second- andhigher-order energy corrections, as the energy corrections onlycontain simple poles. For example, when the experimentalfrequency ωXk

matches Em0 = Em − E0, the simple pole forωXk= Em0 can be removed by taking the limit

limωXk

→Em0(ωXk

− Em0) E(ϵ)

=12

limωXk

→Em0(ωXk

− Em0)

×j1 j2

EXj1Xj2ϵXj1

(ωXj1)ϵXj2

(ωXj2)

+16

limωXk

→Em0(ωXk

− Em0)j1 j2 j3

EXj1Xj2

Xj3

× ϵXj1(ωXj1

)ϵXj2(ωXj2

)ϵXj3(ωXj3

) + · · · (70)

and residues are determined at each order in the perturbation.These residues are molecular constants and determine the

molecular properties that describe the strength of the tran-sitions that are induced by the perturbation of the frequencyωXk

between the ground state |00⟩ and the excited state |0m⟩.For example, for the second-order quasi-energy in Eq. (69),we obtain

limωX2→ω

(ωX2 − ω) EXj1Xj2(ωXj1

,ωXj2)

=

m: Em−E0=ω

⟨00|X j1|0m⟩⟨0m|X j2|00⟩, (71)

where the summation is restricted to excited states withenergies fulfilling Em − E0 = ω. If ω matches an excitationenergy Em − E0 and the corresponding excited state is non-degenerate, we obtain

limωX2→ (Em−E0)

�ωX2 − (Em − E0)� EXj1

Xj2(ωXj1,ωXj2

)= ⟨00|X j1|0m⟩⟨0m|X j2|00⟩, (72)

showing that the ground to excited state transition strengthsmay be obtained from the residues of the frequency-dependentsecond-order quasi-energy.

Similarly, we have double residues as limωXk→Em0(ωXk

−Em0)

(limωXl

→En0(ωXl− En0) E(ϵ)

), which determine transi-

tion properties between two excited states |0m⟩ and |0n⟩ . Inparticular, if |0m⟩ = |0n⟩, this residue determines molecularproperties for the excited state |0m⟩. Transition propertiesand molecular properties of the excited states are inaccessiblein time-independent theory, as static perturbations cannot betargeted to excite to a single excited state.

In this article, we focus on determining the ground-stateresponse functions as derivatives of the quasi-energy (seeEq. (65)) and refer to Refs. 1, 2, and 76 for a detailed discussionhow the transition properties and molecular properties ofexcited states may be determined from the residues of theground-state response functions.

G. Size-extensivity of the quasi-energy and molecularproperties

In this subsection, we examine, for time-periodic pertur-bations, the size-extensivity of the exact wave-function andthe quasi-energy. Consider two molecular systems C,D thatare located infinitely apart. The two systems do not interact,so the Hamiltonian for the combined system may be written

HCD(t, ϵ) = HC(t, ϵ) + HD(t, ϵ), (73)

where HC(t, ϵ) and HD(t, ϵ) are the time-dependent Hamilto-nians for system C and D, respectively. The steady-state wavefunctions for the two sub-systems, |uC(t, ϵ)⟩ and |uD(t, ϵ)⟩,satisfy (

HC(t, ϵ) − i∂

∂t

)|uC(t, ϵ)⟩ = EC(ϵ)|uC(t, ϵ)⟩, (74a)(

HD(t, ϵ) − i∂

∂t

)|uD(t, ϵ)⟩ = ED(ϵ)|uD(t, ϵ)⟩. (74b)


From Eq. (74), it follows that the direct product wave function|uC(t, ϵ)uD(t, ϵ)⟩ satisfies(

HC(t, ϵ) + HD(t, ϵ) − i∂

∂t

)|uC(t, ϵ)uD(t, ϵ)⟩

=��

(HC(t, ϵ) − i

∂

∂t

)uC(t, ϵ)

uD(t, ϵ)

+��uC(t, ϵ)

(HD(t, ϵ) − i

∂

∂t

)uD(t, ϵ)

=�EC(ϵ) + ED(ϵ)�|uC(t, ϵ)uD(t, ϵ)⟩, (75)

showing that for a multiplicatively separable steady-state wavefunction

|uCD(t, ϵ)⟩ = |uC(t, ϵ)uD(t, ϵ)⟩, (76)

the quasi-energy is additively separable

ECD(ϵ) = EC(ϵ) + ED(ϵ). (77)

We now show that the regular wave function is alsomultiplicatively separable. To see this, we consider the phasefactor that connects the regular and steady-state wave functionin Eq. (23),

⟨s0,CD|uCD(t, ϵ)⟩|⟨s0,CD|uCD(t, ϵ)⟩| =

⟨s0,C |uC(t, ϵ)⟩|⟨s0,C |uC(t, ϵ)⟩|

⟨s0,D|uD(t, ϵ)⟩|⟨s0,D|uD(t, ϵ)⟩| , (78)

where the reference wave function for the compound system,|s0,CD⟩ = |s0,Cs0,D⟩, is assumed to be multiplicatively sepa-rable. Eq. (78) implies that the phase FP0(t, ϵ) of Eq. (24) isadditively separable,

FP0,CD(t, ϵ) = FP0,C

(t, ϵ) + FP0,D(t, ϵ), (79)

and the connection between the steady-state and the regularwave functions may therefore for two non-interacting sub-systems C and D be written as

|uC(t, ϵ)uD(t, ϵ)⟩ = exp(−iFP0,C

(t, ϵ))× exp

(−iFP0,D

(t, ϵ)) |0R,CD(t, ϵ)⟩. (80)

It follows from Eq. (80) that the regular wave function ismultiplicatively separable,

|0R,CD(t, ϵ)⟩ = |0R,C(t, ϵ)0R,D(t, ϵ)⟩. (81)

Equations (77) and (79) further show that the overall phase[Eq. (30)] is additively separable,

FCD(t, ϵ) = FP0,CD(t, ϵ) + ECD(ϵ) = FC(t, ϵ) + FD(t, ϵ). (82)

The full solution to the Schrödinger equation [Eq. (29)] ismultiplicatively separable and Eqs. (81) and (82) show that itmay be written in the form

|0CD(t, ϵ)⟩ = exp (−iFCD(t, ϵ)) |0R,CD(t, ϵ)⟩= exp (−iFC(t, ϵ)) exp (−iFD(t, ϵ))× |0R,C(t, ϵ)0R,D(t, ϵ)⟩ = |0C(t, ϵ)0D(t, ϵ)⟩. (83)

Molecular properties were in Eq. (65) identified asderivatives of the quasi-energy with respect to the pertur-

bation strengths. As the quasi-energy is additively separable[Eq. (77)] for all perturbation strengths, the molecularproperties are also additively separable

EXj1

. . .Xjn

CD(ωXj1

, . . . ,ωXjn) = *

,

∂nECD(ϵ)∂ϵXj1

(ωXj1) . . . ∂ϵX jn

(ωX jn)+-0

= EXj1

. . .Xjn

C(ωXj1

, . . . ,ωXjn)

+ EXj1

. . .Xjn

D (ωXj1, . . . ,ωXjn

).(84)

The frequency-dependent energy-corrections become in-finite when a frequency matches an excitation energy of thesystem [see Eq. (69)]. The excitation energies are propertiesof the individual molecules and are thus an intensive property.However, as the energy-corrections are additively separable[Eq. (84)], so are the corresponding residues. For example, forthe residue of second-order energy correction, we obtain fromEq. (71)

limωX2→ω

(ωX2 − ω) EXj1

Xj2CD

(ωXj1,ωXj2

)

=

m: Em,C−E0,C=ω

⟨00,C |X j1|0m,C⟩⟨0m,C |X j2|00,C⟩

+

m: Em,D−E0,D=ω

⟨00,D|X j1|0m,D⟩⟨0m,D|X j2|00,D⟩.

(85)

If the systems C and D are identical and the excited state mis non-degenerate for the individual systems, we obtain fromEq. (85)

limωX2→ω

(ωX2 − ω) EXj1

Xj2CC

(ωXj1,ωXj2

)= 2⟨00,C |X j1|0m,C⟩⟨0m,C |X j2|00,C⟩, (86)

showing the size-extensivity of the transition strengths.In the above, size-extensivity is defined as additive

separability of, for example, the energy for a set of non-interacting systems. In the nomenclature of Bartlett,86 thisproperty is referred to as size-consistency, whereas size-extensivity refers to the scaling of a set of weakly interactingsystems. For a thorough discussion of the relation betweenthese two concepts, we refer to Ref. 87. In the present context,only size-extensivity in our restricted meaning of the word canbe proven using the above general arguments. For the scalingof weakly interacting systems, an analysis of a specific methodis required and we will return to this when discussing specificmethods.

In Sec. III, we show in detail how response functionsmay be determined as derivatives of the quasi-energy, whenthe Hermitian eigenvalue equation [Eq. (32)] is solved byapplying the variational principle of Eq. (58). We also showhow the explicit time-dependence may be averaged out inthe composite Hilbert space and replaced by a frequencydependence. In Sec. IV, we carry out a similar development,where the Hermitian eigenvalue equation is solved viaprojection.


III. RESPONSE FUNCTIONS FROM THE HERMITIANEIGENVALUE EQUATION SOLVED USING THEVARIATIONAL PRINCIPLE

In this section, we determine molecular response prop-erties from the quasi-energy obtained using the variationalprinciple. In Sec. III A, we parametrize the regular wavefunction and transform the quasi-energy to the frequencydomain. In Sec. III B, molecular response properties aredetermined as derivatives of the quasi-energy with respect toperturbation strengths. In Sec. III C, we apply this developmentto the determination of molecular response functions for aconfiguration interaction wave function. In Sec. III D, weshow that the definition of molecular response properties asderivatives of the quasi-energy with respect to perturbationstrengths is identical to the standard definition based on aFourier component expansion of the expectation value of anoperator representing an observable.

To simplify the notation, we suppress in the remainder ofthis article the explicit time and perturbation argument, suchthat, for example, H(t, ϵ) and |0(t, ϵ)⟩ are denoted as H and|0⟩, respectively, and the perturbation-dependent quasi-energyE(ϵ) is denoted as E.

A. Transformation of the quasi-energy from the timedomain to the frequency domain

In Subsection III A 1, the quasi-energy is determined inthe time domain. In Subsection III A 2, the time-dependentwave function coefficients are transformed to the frequencydomain and an order expansion in the perturbation strength isintroduced for the wave function coefficients in the frequencydomain. The quasi-energy is determined in the frequencydomain in Subsection III A 3.

1. Parametrization of the quasi-energy in the timedomain

We now consider the parametrization of the regular wavefunction, |0R(t, ϵ)⟩ of Eq. (23), in the time domain, where thereference state |s0⟩ of Sec. II B is chosen as an eigenstate |00⟩of the unperturbed system with the energy E0,

|s0⟩ = |00⟩, (87a)H0|00⟩ = E0|00⟩. (87b)

The states |0m⟩, m , 0 form an orthogonal complement setto |00⟩ and are not necessarily eigenstates of H0. The regularwave function, |0R(t, ϵ)⟩ of Eq. (23), then becomes

|0R(t, ϵ)⟩ = N(t, ϵ)|0I(t, ϵ)⟩, (88)

where we have introduced the intermediate-normalized state

|0I(t, ϵ)⟩ = |00⟩ + |δ00⟩, (89a)

|δ00⟩ =m,0

bm(t, ϵ)|0m⟩ (89b)

and the normalization constant

N(t, ϵ) = ⟨0I(t, ϵ) |0I(t, ϵ) ⟩−1/2 =

(1 + ⟨δ00 |δ00 ⟩

)−1/2

=(1 +

m

|bm(t, ϵ)|2)−1/2

, (90)

since Eq. (60) is satisfied. The unperturbed ground state |00⟩may be determined using the time-independent variationalprinciple of Eq. (57). In the absence of the perturbation|0R(t, ϵ)⟩ becomes |00⟩, so |δ00⟩ and thereby bm(t, ϵ) vanish.

Inserting Eq. (88) into the quasi-energy expression ofEq. (55) gives

E =

N2(t, ϵ)0I(t, ϵ)��H(t, ϵ) − i∂

∂t��0I(t, ϵ)

T

− i

N(t, ϵ)N(t, ϵ) ⟨0I(t, ϵ) |0I(t, ϵ) ⟩T. (91)

Using Eq. (90), the second term on the right-hand side ofEq. (91) becomes

i�N(t, ϵ)N(t, ϵ) ⟨0I(t, ϵ) |0I(t, ϵ) ⟩T= i

�N−1(t, ϵ)N(t, ϵ)

T= i

ddt

ln N(t, ϵ)T

(92)

and vanishes

i

ddt

ln N(t, ϵ)T

= 0, (93)

as follows from Eq. (37) and the periodicity of N(t, ϵ). Thequasi-energy of Eq. (91) therefore becomes

E =

N2⟨0I |H0|0I⟩T

− i

N20I

��∂0I

∂t

T

+j1

N2⟨0I |X j1|0I⟩ exp

(−iωXj1

t)

T

ϵXj1

(ωXj1

),

(94)

where we have used Eqs. (2) and (3). The normalizationconstant N2 may be expressed as

N2 =(1 +

m

|bm(t, ϵ)|2)−1

=

∞k=0

(−1)k (m

|bm(t, ϵ)|2)k, (95)

where we have used

11 + x

=

∞k=0

(−1)kxk . (96)

The quasi-energy in Eq. (94) may be written in the form

E = EH0 + EF +j1

EX j1ϵXj1

(ωXj1), (97)

where

EH0 =

N2⟨0I |H0|0I⟩

T

, (98)

EF = −i

N20I

��∂0I

∂t

T

, (99)

EX j1= EX j1

(ωXj1) =

N2⟨0I |X j1|0I⟩ exp

(−iωXj1

t)

T

.

(100)

As we shall see later, the EF contribution introduces thefrequency dependence to the response functions and henceis labeled with F.


Using Eq. (89a), we may write EH0 [Eq. (98)] as

EH0 =

N2

(⟨00|H0|00⟩ + ⟨00|H0|δ00⟩

+ ⟨δ00|H0|00⟩ + ⟨δ00|H0|δ00⟩)

T

, (101)

which becomes

EH0 =

N2

(E0 + ⟨δ00|H0|δ00⟩

)T

, (102)

where we have used that Eq. (57) is satisfied in the limit ofvanishing perturbation. Using Eqs. (89b) and (95) and writingout explicitly terms through fourth order in bm(t, ϵ), Eq. (102)becomes

EH0 = E0 +mn

b∗m(t, ϵ)bn(t, ϵ)

−(

k

b∗k(t, ϵ)bk(t, ϵ))b∗m(t, ϵ)bn(t, ϵ)

×⟨0m|H0|0n⟩ − E0δmn

T

+ O(b6). (103)

Using Eq. (89) and the orthonormality of the|00⟩, |0i⟩

set, the EF term [Eq. (99)] may be simplified as

EF = −i

N2δ00

��∂δ00

∂t

T

= −i

N2(

m

b∗m(t, ϵ)⟨0m|)

k

bk(t, ϵ)|0k⟩T

= −i

N2m

b∗m(t, ϵ) bm(t, ϵ)T

, (104)

which, using Eq. (95) and keeping terms through fourth orderin bm(t, ϵ), becomes

EF = −im

b∗m(t, ϵ) bm(t, ϵ)

T

+ ikm

b∗k(t, ϵ) bk(t, ϵ)b∗m(t, ϵ) bm(t, ϵ)

T

+ O(b6).(105)

For EX j1[Eq. (100)], we obtain

EX j1=

N2

(⟨00|X j1|00⟩ + ⟨δ00|X j1|δ00⟩)

exp(−iωXj1

t)

T

+

N2⟨00|X j1|δ00⟩ exp

(−iωXj1

t)

T

+

N2⟨δ00|X j1|00⟩ exp

(−iωXj1

t)

T

, (106)

which through third order in bm(t, ϵ) may be expressed as

EX j1= ⟨00|X j1|00⟩

exp

(−iωXj1

t)

T

+mn

b∗m(t, ϵ)bn(t, ϵ) exp

(−iωXj1

t)

T

×(⟨0m|X j1|0n⟩ − ⟨00|X j1|00⟩δmn

)+

m

bm(t, ϵ) exp

(−iωXj1

t)

T

−k

b∗k(t, ϵ)bk(t, ϵ)bm(t, ϵ) exp

(−iωXj1

t)

T

× ⟨00|X j1|0m⟩ +m

b∗m(t, ϵ) exp

(−iωXj1

t)

T

−k

b∗k(t, ϵ)bk(t, ϵ)b∗m(t, ϵ) exp

(−iωXj1

t)

T

× ⟨0m|X j1|00⟩ + O(b5). (107)

In order to determine an order expansion of the quasi-energy in the perturbation, we now transform the quasi-energyfrom the time to the frequency domain. In Sec. III A 2,the expansion coefficients of Eq. (89) are transformed fromthe time to the frequency domain, and in Sec. III A 3, thisexpansion is used to express the quasi-energy in the frequencydomain.

2. Perturbation expansion of time-dependent wavefunction coefficients in the frequency domain

We will now consider a perturbation expansion of thetime-dependent expansion coefficients bk(t, ϵ) of Eq. (89b) infrequency domain with the frequencies satisfying Eq. (7). Tosimplify the notation, we suppress in the following the explicitreference to the perturbation strength ϵ for the coefficients,

bk(t) = bk(t, ϵ) (108)

and express the perturbation expansion of the time-dependentcoefficients bk(t) as

bk(t) = b(0)k+ b(1)

k(t) + b(2)

k(t) + · · · + b(n)

k(t) + · · ·, (109)

where

b(1)k(t) =

j1

bXj1k

(ωXj1)ϵXj1

(ωXj1) exp(−iωXj1

t), (110)

b(2)k(t) = 1

2

j1

j2

bXj1

Xj2k

(ωXj1,ωX j2

)ϵXj1(ωXj1

)

× ϵX j2(ωX j2

) exp(−i(ωXj1+ ωX j2

)t), (111)

...

b(n)k(t) = 1

n!

j1

. . .jn

bXj1

. . .Xjn

k(ωXj1

, . . . ,ωX jn)

×

nm=1

ϵXjm(ωXjm

)

exp *,−i

nm=1

ωXjmt+-,

(112)

and b(0)k

are the parameters that are determined in the absenceof the perturbation. The b(0)

kparameter vanishes since bk(t) is

zero in the absence of the perturbation. However, we will lateruse an expansion of the form of Eqs. (109)–(112) for otherquantities that have non-vanishing zero-order terms and wetherefore have kept the full expansion in Eq. (109).


The wave function parameter responses to a given orderin the perturbation are given as

bXj1···Xjn

k(ωXj1

, . . . ,ωX jn) = *

,

dnbk(t)dϵXj1

(ωXj1) · · · dϵX jn

(ωX jn)+-0

(113)

and are symmetric in the perturbation operator–frequencypairs.

Alternatively, the terms in the expansion of the time-dependent coefficients [Eq. (109)] may be collected in aperturbation series which reflects more directly the connectionto the frequency domain,

bk(t) =K

bk(ωK) exp(−iωKt), (114)

where

bk(ωK) = b(0)k∆(0 − ωK)

+j1

bXj1k

(ωXj1)ϵXj1

(ωXj1)∆(ωXj1

− ωK)

+12

j1 j2

bXj1

Xj2k

(ωXj1,ωX j2

)ϵXj1(ωXj1

)

× ϵX j2(ωX j2

)∆(ωXj1+ ωX j2

− ωK)+ . . .

+1n!

j1· · · jn

bXj1···Xjn

k(ωXj1

, . . . ,ωX jn)

×

nm=1

ϵXjm(ωXjm

)∆( nm=1

ωXjm− ωK

)+ . . . (115)

and

∆(x) =

0, x , 01, x = 0

. (116)

The frequency ωK thus only gives a non-vanishing contribu-tion, when it is equal to a sum of perturbation frequencies.

The b∗k(t) coefficients may be expressed by taking the

complex conjugate of Eq. (114),

b∗k(t) =K

b∗k(ωK) exp(iωKt), (117)

where the coefficients b∗k(ωK) are the complex conjugates of

bk(ωK) of Eq. (115),

b∗k(ωK) = b(0)k

∗∆(0 + ωK)

+j1

bXj1k

∗(−ωXj1)ϵXj1

(ωXj1)∆(ωXj1

+ ωK

)+

12!

j1 j2

bXj1

Xj2k

∗(−ωXj1,−ωX j2

)

× ϵXj1(ωXj1

)ϵXj2(ωXj2

)∆(ωXj1+ ωX j2

+ ωK

)+ · · ·+

1n!

j1· · · jn

bXj1···Xjn

k

∗(−ωXj1, . . . ,−ωX jn

)

× *,

nm=1

ϵXjm(ωXjm

)+-∆( nm=1

ωXjm+ ωK

)+ · · ·, (118)

where we have used Eq. (4) and that ∆(x) = ∆(−x) and that foreach summation index jp both ϵXjp

(ωXjp) and ϵXjp

(−ωXjp)

enter the expansion.The time derivative of the expansion coefficients, bk(t),

enters in Eq. (105) and may be expressed as

bk(t) = ddt

K

bk(ωK) exp(−iωKt)

= −iK

ωKbk(ωK) exp(−iωKt). (119)

3. The quasi-energy in frequency domain

In the following, we transform the individual terms, EH0[Eq. (103)], EF [Eq. (105)], and EX j1

[Eq. (107)], of the quasi-energy E to the frequency domain where the explicit referenceto time can be removed. This requires that the time averaging iscarried out in Eqs. (103), (105), and (107). In order to do this,we describe initially how the time averaging may be carriedout for an exponential operator containing the frequencies ofEq. (7). Then, we determine the quasi-energy in the frequencydomain.

We start out showing that the time average of theexponential satisfies

exp(− i

(K

ωK −L

ωL

)+

(m

ωXjm−

n

ωX jn

)t)T

= ∆ *,

(K

ωK −L

ωL

)+

(m

ωXjm−

n

ωX jn

)+-,

(120)

where ωK and ωL refer to the frequencies of the wave functionparameters in Eq. (114) (where ωK = nKωT and nK is aninteger) and where ωXjm

and ωX jnare defined by Eq. (7). To

prove Eq. (120), we note that the exponent on the left-handside of Eq. (120) may be expressed as(

K

ωK −L

ωL

)+

(m

ωXjm−

n

ωX jn

)=

(K

nKωT −L

nLωT

)+

(m

nXjmωT −

n

nX jnωT

)=

q

nqωT , (121)

where nq is an integer that may be positive or negative.Inserting Eq. (121) in the left-hand side of Eq. (120) andassuming

q nqωT , 0 gives

exp(− i

q

nqωTt)

T

=1

−i

q nqωT

ddt

exp(− i

q

nqωTt)T

= 0, (122)


where we have used Eq. (37). For

q nqωT = 0, we obtain

exp *.,−i

q

nqωTt+/-

T

= 1,q

nqωT = 0. (123)

Combining Eqs. (121)–(123) gives Eq. (120).We now express the individual terms, EH0 [Eq. (103)],

EF [Eq. (105)], and EX j1[Eq. (107)], of the quasi-energy

E [Eq. (97)] in the frequency domain removing the explicitreference to time. Inserting Eq. (114) in Eq. (103) and keepingterms through fourth order in the perturbation strength, EH0becomes

EH0 = E0 +mn

N

(M

b∗m(ωM)bn(ωN)

×

exp(− i(ωN − ωM)t

)T

−k

KLM

b∗k(ωK)× bk(ωL)b∗m(ωM)bn(ωN)×

exp

(− i(ωL + ωN − ωK − ωM)t

)T

)×

(⟨0m|H0|0n⟩ − E0δmn

)+ O(ϵ (6)). (124)

Using Eq. (120), we remove any reference to time from EH0,

EH0 = E0 +mn

MN

b∗m(ωM)bn(ωN)∆(ωN − ωM

)×⟨0m|H0|0n⟩ − E0δmn

−kmn

KLMN

b∗k(ωK)bk(ωL)b∗m(ωM)bn(ωN)

×∆(ωL + ωN − ωK − ωM

) ⟨0m|H0|0n⟩ − E0δmn

+O(ϵ (6)). (125)

The wave function coefficients bn(ωN) and b∗m(ωM) maybe viewed as independent parameters. Differentiating EH0[Eq. (125)] with respect to these independent parameters andtaking the limit of the zero perturbation strength gives theJacobian

Jmn = *,

∂2EH0

∂b∗m(ωM)∂bn(ωN)+-0

, (126)

which plays a central role in response function theory. Notethat the Jacobian is defined in terms of the quasi-energydifferentiated with respect to the time-dependent parameters,rather than in terms of the energy differentiated with respect

to the time-independent parameters. The eigenvalues of theJacobian occur at the excitation energies of the unperturbedmolecular system, which may be seen by substituting Eq. (125)in Eq. (126) and using that the coefficients bn(ωN) and b∗m(ωM)vanish at the zero perturbation strength, where the frequenciesare equal to zero, giving

Jmn = ⟨0m |H0|0n⟩ − E0δmn. (127)

In the basis where |0n⟩ is an eigenstate of H0 with theeigenvalue En, the Jacobian matrix is therefore diagonal,

Jmn = (En − E0) δmn, (128)with the eigenvalues equal to the excitation energies of theunperturbed system. Inserting Eq. (127) in Eq. (125) gives

EH0 = E0 +mn

MN

b∗m(ωM)bn(ωN)Jmn∆

(ωN − ωM

)−

kmn

KLMN

b∗k(ωK)bk(ωL)b∗m(ωM)bn(ωN)Jmn


)O(ϵ (6)). (129)

We consider now EF of Eq. (105). Inserting Eqs. (114)and (119) into Eq. (105) and keeping terms through fourthorder in the perturbation strength gives

EF = −ωN

m

MN

b∗m(ωM)bm(ωN)∆(ωN − ωM

)+ωN

km

KLMN

b∗k(ωK)bk(ωL)b∗m(ωM)bm(ωN)


)O(ϵ (6)). (130)

For the derivation of response function in Sec. III C, it isconvenient to combine the EH0 [Eq. (129)] and EF [Eq. (130)]contributions in EH ,

EH = EH0 + EF

= E0 +mn

MN

b∗m(ωM)bn(ωN)Jmn − ωNδmn

×∆(ωN − ωM

)−

kmn

KLMN

b∗k(ωK)bk(ωL)

× b∗m(ωM)bn(ωN)Jmn − ωNδmn


)+ O(ϵ (6)). (131)

Proceeding in a similar way, EX j1[Eq. (107)] may be

expressed in the frequency domain as

EX j1= ⟨00|X j1|00⟩∆

(ωXj1

)+

mn

MN

b∗m(ωM)bn(ωN)⟨0m|X j1|0n⟩ − ⟨00|X j1|00⟩δmn

∆

(ωXj1

+ ωN − ωM

)+

m

M

bm(ωM)∆(ωXj1

+ ωM

)−

k

KLM

b∗k(ωK)bk(ωL)bm(ωM)∆(ωXj1

+ ωL − ωK + ωM

)⟨00|X j1|0m⟩

+m

M

b∗m(ωM)∆(ωXj1

− ωM

)−

k

KLM

b∗k(ωK)bk(ωL)b∗m(ωM)∆(ωXj1

+ ωL − ωK − ωM

)⟨0m |X j1|00⟩

+O(ϵ (5)). (132)


The quasi-energy in Eq. (97) may thus be written as

E = EH +j1

EX j1ϵXj1

(ωXj1), (133)

where EH and EX j1are given in Eqs. (131) and (132),

respectively.

B. Molecular response properties from an orderexpansion of the quasi-energy in the frequencydomain

In Sec. II F, molecular response functions were identifiedas expansion coefficients EXj1

. . .Xjn(ωXj1, . . . ,ωXjn

) [Eq. (65)]of a Taylor series expansion of the quasi-energy E in theperturbation strengths [Eq. (64)], where the expansion coeffi-cients were symmetric with respect to the operator–frequencypair permutations in Eq. (66). In Sec. III A, we derived anexpansion of the quasi-energy [Eq. (133)] expressed in termsof two contributions: EH [Eq. (131)] and EX j1

[Eq. (132)].From the expansion in Eq. (133), it is not obvious how torearrange the terms to end up with a Taylor series, wherethe expansion coefficients satisfy the operator–frequency pairpermutational symmetry. In Sec. III B 1, we describe howthis rearrangement can be performed. As a byproduct ofthis derivation, we also show that the frequencies of theexpansion coefficients EXj1

. . .Xjn(ωXj1, . . . ,ωXjn

) must satisfynm=1ωXjm

= 0. In Sec. III B 2, the variational conditions areestablished for the expansion coefficients.

1. Order expansion of the quasi-energy in theperturbation strengths

To establish the rearrangement of the terms in Eq. (133)leading to a Taylor series, we first express Eq. (64) in terms ofsummation over distinct (i.e., not related by the permutationalsymmetry) n-fold multiplets of the operator–frequency pairs,

E =∞n=0

E(n), (134)

where

E(n) =1n!

j1≤···≤ jn

NXj1

. . .XjnωXj1

...ωXjnEXj1

. . .Xjn(ωXj1, . . . ,ωXjn

)

× ϵXj1(ωXj1

) . . . ϵXjn(ωXjn

) (135)

and where

NXj1

. . .XjnωXj1

...ωXjn=

n!k nk!

(136)

and nk is the number of times the perturbation–frequency pair(X jk,ωXjk

) occurs in the chain X j1 · · · X jn. NXj1

. . .XjnωXj1

...ωXjnis

thus the number of distinct permutations of a given multiplet.For example, for a set of two operator–frequency pairs{(Y,ωY), (Z,ωZ)}, the second-order term (n = 2) in Eq. (64)becomes

12!

(d2E

dϵYdϵY

)0ϵYϵY +

(d2E

dϵYdϵZ

)0ϵYϵZ

+

(d2E

dϵZdϵY

)0ϵZϵY +

(d2E

dϵZdϵZ

)0ϵZϵZ

=12!

(d2E

dϵYdϵY

)0ϵYϵY + 2

(d2E

dϵYdϵZ

)0ϵYϵZ

+

(d2E

dϵZdϵZ

)0ϵZϵZ

(137)

since the ZY and Y Z components are related by permutationalsymmetry and NYZ

ωYωZ= 2 and NYY

ωYωY= N ZZ

ωZωZ= 1.

We now consider the expansion in Eq. (133). By insertingEqs. (131) and (132) into Eq. (133) and collecting termsof the same order in the perturbation strength and usingEqs. (114) and (117), the nth-order quasi-energy correctionmay be written as

E(n) =j1· · · jn

fXj1

. . .XjnωXj1

...ωXjnϵXj1

(ωXj1) . . . ϵXjn

(ωXjn)

×∆( nm=1

ωXjm

). (138)

E(n) contains n perturbation strength parameters, an expansioncoefficient f

Xj1. . .Xjn

ωXj1...ωXjn

, and a factor∆(nm=1ωXjm

). The latteroriginates from the presence of ∆(K ωK) in Eq. (131) and∆(ωXj1

+

K ωK) in Eq. (132), where the frequenciesωK frombk(ωK) in Eq. (115) are resolved according to the frequencyconstraint ∆(n

m=1ωXjm− ωK) in Eq. (115), and similar for

b∗k(ωK) in Eq. (118).

The expansion coefficients fXj1

. . .XjnωXj1

...ωXjncontain matrix

elements of the unperturbed Hamiltonian and the perturbationoperators multiplied by a sum of products of coefficientsbXj1···Xjl

k(where l runs through all orders up to n). For example,

one of the third-order contributions from EH of Eq. (131),

E(3) ←−mn

MN

b∗m(ωM)bn(ωN)Jmn − ωNδmn

×∆(ωN − ωM

), (139)

has the form

E(3) ←− 12

j1 j2 j3

mn

bXj1m

∗(−ωXj1)bXj2

Xj3n (ωXj2

,ωXj3)

×Jmn − (ωXj2

+ ωXj3)δmn

× ϵXj1(ωXj1

)ϵXj2(ωXj2

)ϵXj3(ωXj3

)×∆

(ωXj1

+ ωX j2+ ωXj3

), (140)

where the factor12

arises from the second-order coeffi-

cients bXj2

Xj3m (ωXj2

,ωXj3) in Eq. (115). The contribution to

fXj1

Xj2Xj3

ωXj1ωXj2

ωXj3, therefore, becomes

fXj1

Xj2Xj3

ωXj1ωXj2

ωXj3←− 1

2

mn

bXj1m

∗(−ωXj1)bXj2

Xj3n (ωXj2

,ωXj3)

×Jmn − (ωXj2

+ ωXj3)δmn

. (141)

It follows from Eq. (138) that


E(n) =j1· · · jn

fXj1

. . .XjnωXj1

...ωXjnϵXj1

(ωXj1) . . . ϵXjn

(ωXjn),

nm=1

ωXjm= 0, (142)

where we have introduced the frequency constraint explicitly.We now introduce the permutation operator P

Xj1. . .Xjn

ωXj1...ωXjn

thatgenerates all n! (distinct and non-distinct) permutations of theoperator–frequency pairs

PXj1

. . .XjnωXj1

...ωXjnfXj1

. . .XjnωXj1

...ωXjn=

n!i=1

Pi

(fXj1

. . .XjnωXj1

...ωXjn

), (143)

where Pi is one of the permutations. Applying PXj1

. . .XjnωXj1

...ωXjn

on the right-hand side of Eq. (142) only describes a relabelingof the terms in the summation and we may therefore rewriteEq. (142) as

E(n) =j1· · · jn

ϵXj1(ωXj1

) . . . ϵXjn(ωXjn

)

× 1n!

PXj1

. . .XjnωXj1

...ωXjnfXj1

. . .XjnωXj1

...ωXjn,

nm=1

ωXjm= 0,

(144)

where the factor 1n! has been introduced because the permu-

tation operator generates n! copies of each term. To iden-tify EXj1

. . .Xjn(ωXj1, . . . ,ωXjn

) of Eq. (135), we have torearrange Eq. (144) to contain only distinct multiplets ofoperator–frequency pairs. Since P

Xj1. . .Xjn

ωXj1...ωXjn

fXj1

. . .XjnωXj1

...ωXjnis

symmetric with respect to operator–frequency pair permuta-tions as is the case for EXj1

. . .Xjn(ωXj1, . . . ,ωXjn

), we mayin Eq. (144) restrict the summation range introducing thefactor N

Xj1. . .Xjn

ωXj1...ωXjn

, giving

E(n) =

j1≤···≤ jn

ϵXj1(ωXj1

) . . . ϵXjn(ωXjn

)N

Xj1. . .Xjn

ωXj1...ωXjn

n!

× PXj1

. . .XjnωXj1

...ωXjnfXj1

. . .XjnωXj1

...ωXjn,

nm=1

ωXjm= 0. (145)

Equating Eqs. (135) and (145), we obtain

EXj1. . .Xjn(ωXj1

, . . . ,ωXjn) = P

Xj1. . .Xjn

ωXj1...ωXjn

fXj1

. . .XjnωXj1

...ωXjn,

nm=1

ωXjm= 0. (146)

Equation (146) shows that EXj1. . .Xjn(ωXj1

, . . . ,ωXjn) may

be obtained identifying the symmetry distinct contributionsin Eq. (133) and then applying the permutation operatoron these contributions. EXj1

. . .Xjn(ωXj1, . . . ,ωXjn

) will thenbe symmetric with respect to the operator–frequency pairpermutations in Eq. (66) and satisfy the frequency constraintin Eq. (142).

2. Stationary conditions for the quasi-energyand its perturbation components

Let us now consider the stationary conditions for thequasi-energy E of Eq. (133), where EH and EX j1

are defined inEqs. (131) and (132), respectively, in terms of the frequency-dependent coefficients bi(ωK). The quasi-energy is variationalwith respect to all periodic variations of the wave function(see Eq. (58)) and in particular stationary in the coefficientsbi(ωK) of Eq. (115),

∂E∂bi(ωK) = 0, ωK = nKωT , (147)

and in the coefficients bXj1···Xjn

i (ωXj1, . . . ,ωX jn

) of Eq. (113),

∂E

∂bXj1···Xjk

i (ωXj1, . . . ,ωXjk

)= 0,

km=1

ωXjm= nωT ,

(148)

where nK and n are integers. Equations (147) and (148) holdfor all values of the perturbation strengths ϵ . DifferentiatingEq. (148) with respect to perturbation strengths and settingϵ = 0 gives

0 = *,

dn

dϵXj1(ωXj1

) · · · dϵX jn(ωX jn

)

× ∂E

∂bXj1···Xjk


)+//-0

=∂

∂bXj1···Xjk


)

× *,

dnEdϵXj1


(ωX jn)+-0

. (149)

Using Eqs. (65) and (146), we obtain

∂EXj1···Xjn(ωXj1

, . . . ,ωX jn)

∂bXj1···Xjk


)= 0,

nm=1

ωXjm= 0. (150)

Since E is also stationary in the coefficients bXj1···Xjk

i

∗

(−ωXj1, . . . ,−ωX jk

), we obtain in a similar way that

∂EXj1···Xjn(ωXj1

, . . . ,ωX jn)

∂bXj1···Xjk

i

∗(−ωXj1, . . . ,−ωX jk

)= 0,

nm=1

ωXjm= 0. (151)

We thus conclude that the response functions EX j1...X jn(ωX j1,

. . . ,ωX jn) are variational in the perturbation components

bXj1···Xjk


) and bXj1···Xjk

i

∗(−ωXj1, . . . ,−ωXjk

).

C. Response functions through fourth order

We now derive explicit expressions for molecularresponse properties through fourth order. We initially identifyEXj1

. . .Xjn(ωXj1, . . . ,ωXjn

) by writing E(n) in terms of theindividual contributions from EH [Eq. (131)] and EX j1


[Eq. (132)],

E(n) = E(n)H +

j1

E(n−1)X j1

(ωXj1)ϵXj1

(ωXj1), (152)

identifying, according to Eq. (146), the individual distinctcontributions to EXj1

. . .Xjn(ωXj1, . . . ,ωXjn

). We then obtainresponse equations by differentiating EX j1...X jn(ωX j1

, . . . ,

ωX jn) with respect to b

Xj1···Xjm

k

∗(−ωXj1, . . ., −ωXjm) using the

stationary condition of Eq. (151). Finally, we determine molec-ular property expression by simplifying EX j1...X jn(ωX j1

, . . . ,ωX jn

) using the 2n + 1 rule, see, for example, Ref. 77.

1. Energy and first-order molecular properties

From Eq. (152), we determine E(0) as

E(0) = E(0)H = E

(0)H0= E0 (153)

since the expansion coefficients bi(ωK) do not contain zero-order contributions.

Using Eq. (146), the first-order molecular propertyexpression is obtained as

EXj1(ωXj1) = ⟨00|X j1|00⟩, ωXj1

= 0. (154)

The first-order molecular properties are thus frequency-independent and expressed in terms of the ground stateexpectation value of a perturbation operator, as in the time-independent perturbation theory.

2. Second-order molecular properties

Using Eq. (146), the second-order quasi-energy becomes

EXj1Xj2(ωXj1

,ωXj2)

= PXj1

Xj2ωXj1

ωXj2

m

bXj1m

∗(−ωXj1)n

bXj2n (ωXj2

)

×(Jmn − ωX j2

δmn

)+ b

Xj2m (ωXj2

)⟨00|X j1|0m⟩+ b

Xj2m

∗(−ωXj2)⟨0m|X j1|00⟩

, ωXj1

+ ωXj2= 0,

(155)

where we have used Eqs. (131) and (132).Differentiating EXj1

Xj2(ωXj1,ωXj2

) of Eq. (155) with

respect to bXj2k

∗(−ωXj2) and using the stationary condition of

Eq. (151)

∂EXj1Xj2(ωXj1

,ωXj2)

∂bXj2k

∗(−ωXj2)

= 0, (156)

we obtain the first-order amplitude equationsn

(Jkn − ωXj1

δkn)

bXj1n (ωXj1

) = −⟨0k |X j1|00⟩, (157)

from which bXj1n (ωXj1

) may be calculated.

Using Eq. (157), second-order molecular properties maybe simplified as

EXj1Xj2(ωXj1

,ωXj2) = P

Xj1Xj2

ωXj1ωXj2

m

bXj2m (ωXj2

)× ⟨00|X j1|0m⟩, ωXj1

+ ωXj2= 0,

(158)

where bXj2m (ωX j2

) are obtained from Eq. (157). This is equiv-alent to applying the 2n + 1 rule for even-order corrections toquasi-energy and using that the complex conjugate coefficientssatisfy for even-order a 2n + 2 rule.

3. Third-order molecular properties

Following the same route as for EXj1Xj2(ωXj1

,ωXj2), we

obtain EXj1Xj2

Xj3(ωXj1,ωXj2

,ωXj3) as

EXj1Xj2

Xj3(ωXj1,ωXj2

,ωXj3)

= PXj1

Xj2Xj3

ωXj1ωXj2

ωXj3

m

12

bXj1

Xj2m

∗(−ωXj1,−ωXj2

)n

bXj3n (ωXj3

) (Jmn − ωXj3δmn

)+

12

bXj1m

∗(−ωXj1)n

bXj2

Xj3n (ωXj2

,ωXj3) (Jmn − (ωXj2

+ ωXj3)δmn

)+ b

Xj2m

∗(−ωXj2)n

bXj3n (ωXj3

)(⟨0m|X j1|0n⟩ − ⟨00|X j1|00⟩δmn

)+

12

(bXj2

Xj3m (ωXj2

,ωXj3)⟨00|X j1|0m⟩ + b

Xj2Xj3

m


)⟨0m|X j1|00⟩)

, ωXj1+ ωXj2

+ ωXj3= 0. (159)

Differentiating EXj1Xj2

Xj3(ωXj1,ωXj2

,ωXj3) of Eq. (159) with

respect to bXj2

Xj3k


) and using the stationarity

of EXj1Xj2

Xj3(ωXj1,ωXj2

,ωXj3), we recover the first-order

response equation of Eq. (157). Differentiating with respect to


bXj3k

∗(−ωXj3), we obtain the second-order response equation

n

(Jkn − (ωXj1

+ ωX j2)δkn

)bXj1

Xj2n (ωXj1

,ωX j2)

= −PXj1

Xj2ωXj1

ωXj2

n

bXj2n (ωX j2

)

×(⟨0k |X j1|0n⟩ − ⟨00|X j1|00⟩δkn

), (160)

from which bXj1

Xj2n (ωXj1

,ωX j2), needed in Subsection III C 4

for the fourth-order properties, may be calculated.

Using the 2n + 1 rule, third-order molecular propertiesbecome

EXj1Xj2

Xj3(ωXj1,ωXj2

,ωXj3)

= PXj1

Xj2Xj3

ωXj1ωXj2

ωXj3

mn

bXj2m

∗(−ωXj2)bXj3

n (ωXj3)

×(⟨0m|X j1|0n⟩ − ⟨00|X j1|00⟩δmn

),

ωXj1+ ωXj2

+ ωXj3= 0, (161)

where bXj3n (ωXj3

) are obtained from Eq. (157) and bXj2m

∗(−ωXj2)

from the complex conjugate of Eq. (157).

4. Fourth-order molecular properties

Proceeding in the same way as for EXj1Xj2

Xj3(ωXj1,ωXj2

,ωXj3) and applying the 2n + 1 rule for the even-order energy

corrections and using that the complex conjugate coefficients satisfy for the even-order corrections a 2n + 2 rule, and in additionapplying the first-order equation [Eq. (157)], the fourth-order molecular property expression becomes

EXj1Xj2

Xj3Xj4(ωXj1

,ωXj2,ωXj3

,ωXj4) = P

Xj1Xj2

Xj3Xj4

ωXj1ωXj2

ωXj3ωXj4

12

mn

bXj2m

∗(−ωXj2)bXj3

Xj4n (ωXj3

,ωXj4)(⟨0m|X j1|0n⟩

−⟨00|X j1|00⟩δmn

)−

k

bXj2k

∗(−ωXj2)bXj3

k(ωXj3

)m

bXj4m (ωXj4

)⟨00|X j1|0m⟩,

ωXj1+ ωXj2

+ ωXj3+ ωXj4

= 0. (162)

where bXj4m (ωXj4

) are obtained from Eq. (157), bXj2m

∗(−ωXj2)

from the complex conjugate of Eq. (157), and bXj2

Xj3n

(ωXj2,ωXj3

) from Eq. (160).The derivation presented in this section may be extended

to any order.

D. Molecular properties from the quasi-energycompared to the standard definition

We have in this article identified molecular responseproperties as derivatives of the quasi-energy with respectto perturbation strengths. Traditionally, molecular responseproperties have been defined as the expansion coefficientsof a perturbation expansion of the expectation value of anobservable, ⟨0|Xi |0⟩, for the time-dependent wave function|0⟩. Since |0⟩ is non-periodic, it cannot be expanded in thecomposite Hilbert space. To arrive at an expression that canbe expanded in the composite Hilbert space, we use Eq. (29) toexpress the expectation value in terms of the periodic regularwave function,

⟨0|Xi |0⟩ = ⟨0R| exp(iF

)Xi exp

(− iF

) |0R⟩ = ⟨0R|Xi |0R⟩.(163)

For the perturbations of the Hamiltonian in Eq. (3), theexpectation value ⟨0|Xi |0⟩ may be experimentally determinedfor all perturbation strengths. All terms in the expansion of⟨0R|Xi |0R⟩ in the perturbation strengths may therefore be

obtained from experiments identifying at each order in theperturbation strength the molecular constants, i.e., molec-ular properties of the molecular system. The expansion of⟨0R|Xi |0R⟩ in the perturbation strength may be expressed as

⟨0R|Xi |0R⟩= ⟨⟨Xi⟩⟩0 +

j1

Xi; X j1

��ωXj1

ϵXj1

(ωXj1

)× exp

(−iωXj1

t)+

12

j1

j2

Xi; X j1,X j2

��ωXj1

,ωXj2

× ϵXj1

(ωXj1

)ϵX j2

(ωX j2

)exp

−i

(ωXj1

+ ωX j2

)t

+ · · ·+

1n!

j1

· · ·jn

Xi; X j1, . . . ,X jn

��ωXj1

, ...,ωXjn

×n

m=1

ϵXjm

(ωXjm

)exp *

,−i

nm=1

ωXjmt+-, (164)

where the expansion coefficients are symmetric with respectto the operator–frequency pair permutations (X jk,ωXjk

)↔ (X jl,ωXjl

), for k, l = 1,2, . . .. The zero-order term, ⟨⟨Xi⟩⟩0

= ⟨00|Xi |00⟩, is the static first-order molecular property. Theexpansion coefficient of the first-order term,

Xi; X j1

��ωXj1

,

is the linear response function and measures how themolecular property described by the operator Xi is alteredlinearly by applying a perturbation field X j1 oscillatingwith a frequency ωXj1

.

Xi; X j1

��ωXj1

is a second-order

molecular property. In general, the expansion coefficients,


Xi; X j1, . . . ,X jn

��ωXj1

, ...,ωXjn

, are the nth-order response

functions and determine the response of the molecularproperty described by the operator Xi to n perturbations.

To see that the two definitions of molecular properties areidentical, we first have to derive the generalized Hellmann-Feynman theorem for the quasi-energy. We therefore takethe derivative of E in Eq. (55) with respect to a perturbationstrength ϵXi

�ωXi

�,

dEdϵXi

�ωXi

� =d�

0R

�H − i ∂

∂t

�0R

�T

dϵXi

�ωXi

�

=�⟨0R |Xi | 0R⟩ exp

�−iωXi

t�

T

+

∂0R

∂ϵXi

�ωXi

��H − i

∂

∂t

��0R

+

0R

��H − i

∂

∂t

��∂0R

∂ϵXi

�ωXi

�T

, (165)

where we have used Eqs. (2) and (3). Using the variationalprinciple in Eq. (58) and the fact that | ∂0R

∂ϵXi⟩ is an allowed

variation, since it preserves periodicity of |0R⟩ and satisfiesEq. (61), we obtain from Eq. (165) the generalized Hellmann-Feynman theorem

dEdϵXi

�ωXi

� =⟨0R|Xi |0R⟩ exp

�−iωXi

t�

T. (166)

Inserting ⟨0R|Xi |0R⟩ of Eq. (164) into Eq. (166), we obtain

dEdϵXi

�ωXi

� = ⟨⟨Xi⟩⟩0

exp

�−iωXi

t�

T

+j1

Xi; X j1

��ωXj1

ϵXj1

(ωXj1

) exp

−i

(ωXi+ ωXj1

)t

T

+12

j1

j2

Xi; X j1,X j2

��ωXj1

,ωXj2ϵXj1

(ωXj1

)ϵX j2

(ωX j2

) exp

−i

(ωXi+ ωXj1

+ ωX j2

)t

T

+ · · · +1n!

j1

· · ·jn

Xi; X j1, . . . ,X jn

��ωXj1

, ...,ωXjn

nm=1

ϵXjm

(ωXjm

)

exp−i *

,ωXi+

nm=1

ωXjm+-

t

T

.

(167)

Using Eq. (120) gives

dEdϵXi

�ωXi

� = ⟨⟨Xi⟩⟩0∆�ωXi

�+

j1

Xi; X j1

��ωXj1

ϵXj1

(ωXj1

)∆

(ωXi+ ωXj1

)+

12

j1

j2

Xi; X j1,X j2

��ωXj1

,ωXj2

× ϵXj1

(ωXj1

)ϵX j2

(ωX j2

)∆

(ωXi+ ωXj1

+ ωX j2

)+ · · · + 1

n!

j1

· · ·jn

Xi; X j1, . . . ,X jn

��ωXj1

, ...,ωXjn

×n

m=1

ϵXjm

(ωXjm

)∆ *,ωXi+

nm=1

ωXjm+-. (168)

Differentiating Eq. (168) with respect toϵXj1

(ωXj1) . . . ϵXjn

(ωXjn) at the zero perturbation strength

gives

*,

dn+1EdϵXi

(ωXi)dϵXj1


(ωX jn)+-0

=

Xi; X j1, . . . ,X jn

��ωXj1

, ...,ωXjn

,

ωXi+

nm=1

ωXjm= 0, (169)

where we have used that response functionsXi; X j1, . . . ,X jn

��ωXj1

, ...,ωXjn

are symmetric with respect

to permutations of operator–frequency pairs (X jk,ωXjk)↔

(X jl,ωXjl), k, l = 1,2, . . .. Using Eq. (65), we obtain from

Eq. (169)

Xi; X j1, . . . ,X jn

��ωXj1

, ...,ωXjn

= EXiX j1···Xjn(ωXi

,ωXj1, . . . ,ωX jn

), ωXi+

nm=1

ωXjm= 0

(170)

and conclude that the traditional definitions of the responsefunctions as the coefficients of the expansion in Eq. (164) areidentical to the definitions from the quasi-energy expansion inEq. (65).

When molecular properties are defined in terms of theexpansion coefficients of the Taylor series in Eq. (164), heredenoted ⟨⟨X j1; X j2, . . . ,X jn⟩⟩ωXj2

, ...,ωXjn, they are symmetric

with respect to the operator–frequency pair permutations

(X jk,ωXjk)↔ (X jl,ωXjl

), k, l = 2,3, . . . . (171)


Further, there is no constraint on the frequencies. Thefrequency constraint arises from the integration over timewhen the periodicity of |0R⟩ is introduced. In the abovederivation, this is done when the expansion of ⟨0R|Xi |0R⟩in Eq. (164) is inserted into the generalized Hellmann-Feynman theorem expression in Eq. (166) and the timeaveraging is carried out. Since from Eq. (170) the responsefunctions are derivatives of the quasi-energy with respect toperturbation strengths, they are symmetric with respect topermutations of all the operator–frequency pairs, includingthe first operator–frequency pair (X j1,ωXj1

),

(X jk,ωXjk)↔ (X jl,ωXjl

), k, l = 1,2, . . . . (172)

The extension of the symmetry relation to include also the firstoperator–frequency pair (X j1,ωXj1

) is far from trivial to provewhen molecular properties are defined through the expansionin Eq. (164).12

For a variational wave function, the generalized Hellmann-Feynman theorem is satisfied and molecular properties areidentical whether they are defined in terms of derivatives ofthe quasi-energy or from an average value of a perturbationoperator. However, for a non-variational wave function, wherethe generalized Hellmann-Feynman theorem is not satisfied,molecular properties are expressed in terms of quasi-energyderivatives. The obtained expressions then deviate from thesimple average value expression due to the non-variationalnature of the wave function. From a theoretical point ofview, the quasi-energy expressed as an eigenvalue of theHermitian eigenvalue equation is the fundamental quantity fora molecular system when a periodic perturbation is imposed, asthe energy is the fundamental quantity when the perturbation isstatic. This should be reflected in the way molecular responseproperties are defined.

IV. RESPONSE FUNCTIONS FROM THE HERMITIANEIGENVALUE EQUATION SOLVED VIA PROJECTION

In this section, we consider how response functions maybe obtained when the Hermitian eigenvalue equation [Eq. (32)]is solved via projection in the composite Hilbert space. Thedevelopment we present allows for determination of responsefunctions where the parametrization of the ground state for theunperturbed system differs from the parametrization of its timeevolution. This allows response functions to be determined fora new general class of approximate wave function models,extending the range of wave function models for whichresponse functions can be determined.

In Sec. IV A, we parametrize the regular wave functionof Eq. (68) where the parametrization of the unperturbedground state may be different from the parametrization ofthe time evolution. In Sec. IV B, we describe how theHermitian eigenvalue equation [Eq. (32)] may be solvedvia projection. In Secs. IV C and IV D, we introduce thequasi-energy Lagrangian for the solution of Eq. (32), and inSec. IV E, response functions are derived from the quasi-energy Lagrangian.

A. Parametrization of the regular wave function

In this section, we consider wave function models whereprojection is used to solve the Hermitian eigenvalue equation.In Subsection IV A 1, we consider the parametrization ofthe ground state for the unperturbed system, whereas inSubsection IV A 2, the parametrization of the time evolutionof this state is described.

1. Parametrization of the unperturbed state

The eigenstate |00⟩ of the unperturbed Hamiltonian H0may be parametrized in terms of an orthonormal basis con-sisting of a reference state |R⟩ and its orthogonal complement{|k⟩},

|B⟩ = |R⟩, |k⟩. (173)

With this parametrization, |00⟩ may be written as

|00⟩ = N0|0I0⟩, (174)

where

|0I0⟩ = exp(B0)|R⟩ (175)

and N0 is the normalization constant,

N0 = ⟨R| exp(B0)† exp(B0)|R⟩−1/2. (176)

We discuss below the parametrization of the exponentialoperator exp(B0). To comprehend coupled cluster theory, weconsider |0I0⟩ states that are intermediate normalized,

R

�0I0

�= 1, (177)

hence the subscript I in |0I0⟩. The intermediate normalizationcondition may be satisfied by requiring that the action of B0 onany state |P⟩ leads to a state that is orthogonal to the referencestate |R⟩,

B0|P⟩ =k

b(0)k|k⟩. (178)

We express the operators B0 satisfying Eq. (178) in the genericform

B0 =k

b(0)kβk . (179)

The operator βk may take on the form of a state-transferoperator

βk = |k⟩⟨R| (180)

and Eq. (175) then describes a linear parametrization of |0I0⟩in the basis |B⟩ of Eq. (173),

|0I0⟩ = exp(B0)|R⟩ = |R⟩ +k

b(0)k|k⟩. (181)

The advantage of expressing a linear parametrization of awave function in terms of an exponential operator is that theinverse of the wave function may then straightforwardly bedetermined,

⟨0I0|0I0⟩ = ⟨R| exp(−B0) exp(B0)|R⟩ = 1. (182)

The operator βk may alternatively be defined as a many-body orbital excitation operator carrying out excitations from


orbitals occupied in the reference state to its orthogonalcomplement

βk = τk . (183)

Equation (175) then describes an exponential parametrizationof |0I0⟩, as in coupled cluster theory. Using the exponentialoperator form of a linear parametrization allows us to developa generic expression for the quasi-energy Lagrangian thatencompasses both a linear and exponential parametrizationof |0I0⟩.

For a given choice of the operator manifold, the genericexcitation operators commute

�βk, β j

�= 0. (184)

The operators βk satisfy

βk |R⟩ = |k⟩ (185)

as follows from Eq. (178). Using Eq. (178) and the orthonor-mality of the basis in Eq. (173), we have for any state |P⟩

⟨R| exp(−B0)|P⟩ = ⟨R|(1−B0 +

12

B20 + · · ·

)|P⟩ = ⟨R |P ⟩ .

(186)

Since Eq. (186) is fulfilled for any state |P⟩, it follows that

⟨R|B0 = 0 (187)

and

⟨R| = ⟨R| exp(−B0). (188)

Inserting Eq. (174) in Eq. (87b) and using Eq. (175), weobtain

H0 exp(B0)|R⟩ = E0 exp(B0)|R⟩. (189)

The reference to the normalization constant thus vanishestrivially, when the Schrödinger equation is solved for a time-independent unperturbed system. Multiplying Eq. (189) byexp(−B0), we obtain the similarity-transformed Schrödingerequation

exp(−B0)H0 exp(B0)|R⟩ = E0|R⟩. (190)

E0 is thus the eigenvalue of a similarity-transformed Hamilto-nian

HB00 = exp(−B0)H0 exp(B0). (191)

Equation (190) emphasizes that the eigenvalues of theSchrödinger equation do not change when the Hamiltonian issubjected to a similarity transformation. Equation (190) maybe solved by projecting it against the basis

⟨B | = ⟨R|,⟨k |, (192)

giving

⟨R| exp(−B0)H0 exp(B0)|R⟩ = E0, (193a)⟨k | exp(−B0)H0 exp(B0)|R⟩ = 0. (193b)

For the determination of response functions, it turns outto be convenient to express the solution to the Schrödingerequation for the unperturbed system in Eqs. (193a) and (193b)in terms of a Lagrangian, where Eq. (193b) is added asa constraint to Eq. (193a) using undetermined Lagrangianmultipliers, b

(0)k ,

L0 = ⟨R|HB00 |R⟩ +

k

b(0)k ⟨k |HB0

0 |R⟩, (194)

where we have used Eq. (191). The Lagrangian, L0, inEq. (194) is stationary with respect to the Lagrangianmultipliers

∂L0

∂b(0)k

= ⟨k |HB00 |R⟩ = 0. (195)

The multipliers, b(0)k , are undetermined and may be chosen

such that the Lagrangian in Eq. (194) is also stationary withrespect to the coefficients b(0)

kin Eq. (179),

∂L0

∂b(0)k

= ⟨R| HB00 , βk

|R⟩

+m

b(0)m ⟨m| HB0

0 , βk |R⟩ = 0. (196)

2. Parametrization of the time evolution

The formal separation of the description of the unper-turbed ground state and the time evolution of this state may beachieved by parametrizing the unit normalized regular wavefunction |0R(t, ϵ)⟩ of Eq. (68) as

|0R(t, ϵ)⟩ = N(t, ϵ)|0I(t, ϵ)⟩, (197)

where

|0I(t, ϵ)⟩ = exp(B0) exp(B(t))|R⟩ (198)

and N(t, ϵ) is the time-dependent normalization constant

N(t, ϵ) = ⟨R| exp(B(t))† exp(B0)† exp(B0) exp(B(t))|R⟩−1/2.

(199)

The operator B0 parametrizes the unperturbed ground state (asdiscussed in Sec. IV A 1) and the operator B(t) parametrizesthe perturbation-induced time evolution of this state. Similarlyto B0, the operator B(t) is defined such that |0I(t, ϵ)⟩ isintermediate normalized against ⟨R|,

⟨R |0I(t, ϵ) ⟩ = 1, (200)

and therefore the action of B(t) on any state leads to a statethat is orthogonal to the reference state |R⟩. Analogous toEq. (179), B(t) may be written as

B(t) =k

bk(t)βk . (201)

A time-dependent analogue of Eq. (188) also holds for B(t),⟨R| = ⟨R| exp(−B(t)). (202)


The operators B0 and B(t) in general do not commute,for example, if B0 is expressed in terms of the many-bodyoperators in Eq. (183), while B(t) is expressed in terms of thestate-transfer operators in Eq. (180). The order that is chosenin Eq. (198) allows to describe the time evolution in termsof a similarity-transformed Hamiltonian HB0 containing thetime-independent operator exp(B0).

Since the normalization constant N(t, ϵ) in Eq. (199) istime-dependent, it is not trivially eliminated when |0R(t, ϵ)⟩ ofEq. (197) is inserted in the Hermitian eigenvalue equation[Eq. (32)] in contrast to the time-independent case wherethe normalization constant can be removed straightforwardlygiving Eq. (189). In Sec. IV B, we show how the referenceto the normalization constant can be eliminated when theHermitian eigenvalue equation in Eq. (32) is solved viaprojection in the composite Hilbert space and that the solutionto the Hermitian eigenvalue equation then may be expressedas a straightforward generalization of Eqs. (193a) and (193b).

B. Projected Hermitian eigenvalue equation

We now describe how the solution to the time-dependentSchrödinger equation [Eq. (32)] may be determined usingprojection. Inserting Eq. (197) in the Hermitian eigenvalueequation [Eq. (32)] and using Eq. (198) gives

(H − i

∂

∂t− FP0 − iN−1N

)exp(B0) exp(B(t))|R⟩

= E exp(B0) exp(B(t))|R⟩. (203)

Multiplying Eq. (203) by exp(−B(t)) exp(−B0), we obtain

exp(−B(t))(HB0 − i

∂

∂t

)exp(B(t))|R⟩

=

(E + FP0 + i

ddt

ln N)|R⟩, (204)

where we have used that B0 is time-independent and wherethe similarity transformed Hamiltonian HB0 is defined analo-gously to HB0

0 of Eq. (191). Projecting Eq. (204) against thebasis ⟨B | of Eq. (192), we obtain the set of equations

ReR�� exp(−B(t))

(HB0 − i

∂

∂t

)exp(B(t))��R

= E + FP0,

(205a)

ImR�� exp(−B(t))

(HB0 − i

∂

∂t

)exp(B(t))��R

=

ddt

ln N,

(205b)k �� exp(−B(t))

(HB0 − i

∂

∂t

)exp(B(t))��R

= 0, (205c)

where we have used that E and FP0 are real and i ddt ln N

is purely imaginary (since N is real by construction—seeEq. (199)). Time-averaging the set of equations (205) andusing Eqs. (41) and (93), we obtain

E = Re

R�� exp(−B(t))(HB0 − i

∂

∂t

)exp(B(t))��R

T

,

(206a)

0 = Im


∂

∂t

)exp(B(t))��R

T

,

(206b)k �� exp(−B(t))

(HB0 − i

∂

∂t

)exp(B(t))��R

T

= 0. (206c)

The explicit reference to the normalization constant thereforevanishes when the Hermitian eigenvalue equation [Eq. (32)] issolved via projection in the composite Hilbert space. Equation(206) shows that the solution to the Hermitian eigenvalueequation [Eq. (32)] is obtained by first solving Eq. (193b) toobtain HB0, then solving Eq. (206c) and inserting this solutionto Eq. (206a) yielding E. Knowledge of the time-periodicphase FP0 is thus not required for determining |0R⟩ and E.

The set of equations (206) may alternatively be obtained ina single step by projecting Eq. (204) directly in the compositeHilbert space and using the definition of the inner productin this space [Eq. (36)]. The right-hand side of Eq. (206a)would then not be ensured to be real. This does not posea problem for an exact representation of the regular wavefunction |0R(t, ϵ)⟩, where the quasi-energy E is real beingan eigenvalue of the Hermitian operator. For approximatewave function models, the time averages of d

dt ln N and FP0

still vanish. However, ⟨R| �HB0 − i ∂∂t

�exp(B(t))|R⟩ may be

complex. Using the two-step approach above to derive the setof equations (206) ensures that the quasi-energy, and hencealso molecular properties, is real also for approximate wavefunction models.

C. Stationary conditions for the quasi-energyLagrangian

Molecular response properties may be determined bysolving Eq. (206c) explicitly and then determining E fromEq. (206a). However, to get variational flexibility to simplifythe determination of response functions, we introduce, as forthe static case, a Lagrangian, where Eq. (206c) is addedas a constraint to Eq. (206a) using frequency-dependentundetermined Lagrangian multipliers, bk(ωK), which aredefined analogously to bk(ωK) of Eq. (114),

bk(t) =K

bk(ωK) exp(−iωKt). (207)

The Lagrangian then becomes

L = Re


∂

∂t

)exp(B(t))��R

T

+kK

bk(ωK)

k �� exp(−B(t))(HB0 − i

∂

∂t

)

× exp(B(t))��R

exp(−iωKt)T

. (208)

The Lagrangian multipliers, bk(ωK), contain the zero-ordercontribution, b

(0)k . In the limit of a time-independent unper-

turbed system, the Lagrangian L therefore reduces to L0[Eq. (194)].


The Lagrangian of Eq. (208) is per construction stationarywith respect to the multipliers

∂L∂bk(ωK)

=

k �� exp(−B(t))

(HB0 − i

∂

∂t

)× exp(B(t))��R

exp(−iωKt)

T

= 0. (209)

The Lagrangian multipliers are undetermined and are chosensuch that the Lagrangian of Eq. (208) is stationary also withrespect to the coefficients bk(ωK) (entering B(t) of Eq. (201)in the frequency domain),

∂L∂bk(ωK) = 0. (210)

We now consider a Lagrangian, where the real part is takenon both terms in Eq. (208),

L = Re(


∂

∂t

)exp(B(t))��R

T

+kK

bk(ωK)

k �� exp(−B(t))(HB0 − i

∂

∂t

)exp(B(t))��R

× exp(−iωKt)

T

). (211)

To show that the two forms of the Lagrangian, Eqs. (208) and(211), lead to the same stationary points, we consider bk(ωK)and b

∗k(ωK) as independent parameters. Then, noting that the

first term in Eq. (211) does not depend on multipliers, weobtain

∂L∂bm(ωM) =

12

∂

∂bm(ωM)(

kK

bk(ωK)

k �� exp(−B(t))(HB0 − i∂

∂t

)exp(B(t))��R

exp(−iωKt)

T

+kK

b∗k(ωK)

k �� exp(−B(t))(HB0 − i

∂

∂t

)exp(B(t))��R

∗exp(iωKt)

T

)

=12

m�� exp(−B(t))

(HB0 − i

∂

∂t

)exp(B(t))��R

exp(−iωMt)

T

. (212)

A similar equation also holds for b∗m(ωM). The Lagrangian

of Eq. (211) is thus stationary with respect to variations inbm(ωM) and b

∗m(ωM) if and only if Eq. (206c) is satisfied.

The Lagrangian expression in Eq. (208) and in Eq. (211) thusboth have stationary points at the solution to the projectedSchrödinger equation [Eq. (206c)]. Further, it follows fromEqs. (208) and (211) that the two forms of the Lagrangian areequal at the stationary points.

The set of equations (206) implies that at the stationarypoints the Lagrangian of Eq. (211) is equal to the quasi-energyof Eq. (55),

L = E for δE = 0,∂L

∂bk(ωK)= 0,

∂L∂bk(ωK) = 0. (213)

We therefore denote the Lagrangian L of Eq. (211) as thequasi-energy Lagrangian. It follows from Eq. (213) that at thestationary points the derivatives of E and L with respect tothe perturbation strength are equal, in particular

dEdϵXi

�ωXi

� = dLdϵXi

�ωXi

�

for δE = 0,∂L

∂bk(ωK)= 0,

∂L∂bk(ωK) = 0. (214)

When the Hermitian eigenvalue equation is solved viaprojection, molecular response properties may therefore beobtained as derivatives of the real quasi-energy Lagrangian Lof Eq. (211), in the same way as molecular response propertiesare obtained as derivatives of the real quasi-energy E of

Eq. (55), when the Hermitian eigenvalue equation is solved us-ing the variational principle. For exact wave functions, the twoapproaches lead to the same molecular response properties.

D. Complex quasi-energy Lagrangian and itsstationary conditions

In Subsection IV D 1, we derive an explicit expressionfor the complex quasi-energy Lagrangian, and in SubsectionIV D 2, the stationary condition is derived for this Lagrangian.

1. Complex quasi-energy Lagrangian

To determine molecular properties, it is convenient toexpress the real quasi-energy Lagrangian, L of Eq. (211), interms of the complex quasi-energy Lagrangian, cL,

L = Re cL = 12( cL + cL∗) , (215)

where

cL =


∂

∂t

)exp(B(t))��R

T

+kK

bk(ωK)

k �� exp(−B(t))(HB0 − i

∂

∂t

)

× exp(B(t))��R

exp(−iωKt)T

. (216)


Further, it is convenient to split the complex quasi-energy Lagrangian, cL of Eq. (216), into the contributions similar to theones obtained for the quasi-energy in Eq. (133). Using Eqs. (2) and (3), we may write the complex quasi-energy Lagrangian as

cL =

R�� exp(−B(t))HB00 exp(B(t))��R

T

+kK

bk(ωK)

k �� exp(−B(t))(HB0

0 − i∂

∂t

)exp(B(t))��R

exp(−iωKt)

T

+j1

R�� exp(−B(t))XB0

j1exp(B(t))��R

exp(−iωXj1

t)T

ϵXj1(ωXj1

)

+j1

kK

bk(ωK)

k �� exp(−B(t))XB0j1

exp(B(t))��R

exp(−i(ωK + ωXj1)t)

T

ϵXj1(ωXj1

), (217)

where we have used that

⟨R| exp(−B(t)) ∂∂t

exp(B(t))|R⟩ = ⟨R| ∂∂t

exp(B(t))|R⟩=

ddt⟨R| exp(B(t))|R⟩

=ddt

⟨R |R ⟩ = 0, (218)

as follows from Eq. (202) and the time independence of |R⟩.The complex quasi-energy Lagrangian cL in Eq. (217) maybe written in the form

cL = cLH0 +cLF +

j1

cLX j1ϵXj1

(ωXj1), (219)

where

cLH0 =

R�� exp(−B(t))HB0

0 exp(B(t))��R

T

+kK

bk(ωK)

k �� exp(−B(t))HB00 exp(B(t))��R

× exp(−iωKt)

T

, (220)

cLF = −ikK

bk(ωK)

k �� exp(−B(t)) ∂∂t

exp(B(t))��R

× exp(−iωKt)T

= −ikmK

bk(ωK)�bm(t) exp(−iωKt)T⟨k |βm|R⟩, (221)

andcLX j1

=

R�� exp(−B(t))XB0

j1exp(B(t))��R

exp(−iωXj1

t)T

+kK

bk(ωK)

k �� exp(−B(t))XB0j1

exp(B(t))��R

× exp(−i(ωK + ωXj1)t)

T

. (222)

To obtain the second equality in Eq. (221), we have usedEq. (201).

2. Stationary conditions for the complex quasi-energyLagrangian

We have shown that the real quasi-energy Lagrangian Lof Eq. (211) is stationary with respect to the wave-function

coefficients bk(ωK) and Lagrangian multipliers bk(ωK) andwill now show that this implies that the complex quasi-energyLagrangian cL is also stationary with respect to bk(ωK) andbk(ωK). To do this, we first note from Eqs. (219)–(222) thatcL depends on a set of wave-function coefficients, b(ωK),entering B(t) of Eq. (201) in the frequency domain, and a setof Lagrangian multipliers, b(ωK), in Eq. (207),

cL = cL(b(ωK),b(ωK)

). (223)

Similarly, cL∗ depends on b∗(ωK) and b∗(ωK),

cL∗ = cL∗(b∗(ωK),b∗(ωK)

). (224)

Using that cL∗ does not depend on b(ωK) and b(ωK) and thestationarity of L, we obtain that the complex quasi-energyLagrangian, cL of Eq. (219), is stationary with respect tobk(ωK) and bk(ωK),

∂ cL∂bk(ωK) =

∂

∂bk(ωK) (2L −cL∗) = 2

∂L∂bk(ωK) = 0, (225)

∂ cL∂bk(ωK)

=∂

∂bk(ωK)(2L − cL∗) = 2

∂L∂bk(ωK)

= 0. (226)

Similarly, cL∗ is stationary with respect to b∗k(ωK) and b

∗k(ωK).

E. Molecular properties from the quasi-energyLagrangian

In Subsection IV E 1, the quasi-energy Lagrangian isexpanded in orders of the perturbation strength, and inSubsection IV E 2, stationary conditions are established foreach term in the order expansion.

1. Molecular properties from an order expansion ofthe quasi-energy Lagrangian

Following the developments in Sec. III B, we expand thequasi-energy Lagrangian as in Eqs. (134) and (135),

L =∞n=0

L(n), (227)


where

L(n) =1n!

j1≤···≤ jn

NXj1

. . .XjnωXj1

...ωXjnLXj1

. . .Xjn(ωXj1, . . . ,ωXjn

)× ϵXj1

(ωXj1) . . . ϵXjn

(ωXjn) (228)

and

LXj1. . .Xjn(ωXj1

, . . . ,ωXjn) = *

,

dnLdϵ Xj1

(ωXj1) . . . dϵ Xjn

(ωXjn)+-0

.

(229)

The complex quasi-energy Lagrangian, cL, and its complexconjugate cL∗, may be expanded in the same way,

cL =n

cL(n), (230)

cL∗ =n

cL(n)∗, (231)

where

cL(n) = 1n!

j1≤···≤ jn

NXj1

. . .XjnωXj1

...ωXjn

cLXj1. . .Xjn(ωXj1

, . . . ,ωXjn)

× ϵ Xj1(ωXj1

) . . . ϵ Xjn(ωXjn

), (232)

cL(n)∗ = 1n!

j1≤···≤ jn

×NXj1

. . .XjnωXj1

...ωXjn

cLXj1. . .Xjn

∗(−ωXj1, . . . ,−ωXjn

)× ϵ Xj1

(ωXj1) . . . ϵ Xjn

(ωXjn), (233)

andcLXj1

. . .Xjn(ωXj1, . . . ,ωXjn

)= *,

dn cLdϵXj1

(ωXj1) . . . dϵXjn

(ωXjn)+-0

, (234)

cLXj1. . .Xjn

∗(−ωXj1, . . . ,−ωXjn

)= *,

dn cL∗

dϵXj1(ωXj1

) . . . dϵXjn(ωXjn

)+-0

. (235)

Note thatcL(0) = L0 (236)

is given in Eq. (194). To obtain Eq. (233), we take the complexconjugate of Eq. (232) with no restriction on the summationindices,

cL(n)∗ =1n!

j1· · · jn

cLXj1

. . .Xjn(ωXj1, . . . ,ωXjn

)∗ϵ∗Xj1(ωXj1

) . . . ϵ∗Xjn(ωXjn

) = 1n!

j1· · · jn

cLXj1. . .Xjn

∗(ωXj1, . . . ,ωXjn

)

× ϵXj1(−ωXj1

) . . . ϵXjn(−ωXjn

) = 1n!

j1· · · jn

cLXj1. . .Xjn

∗(−ωXj1, . . . ,−ωXjn

)ϵXj1(ωXj1

) . . . ϵXjn(ωXjn

), (237)

where to obtain the second equality we have used Eq. (4)and to obtain the last equality we have used that for eachsummation index jk both ϵXjk

(ωXjk) and ϵXjk

(−ωXjk) enter

the expansion. From Eq. (237), we straightforwardly obtainEq. (233).

We will now express response functions in terms ofcLXj1

. . .Xjn(ωXj1, . . . ,ωXjn

). We initially insert Eq. (215) inEq. (214) giving

dEdϵXj1

�ωXj1

� = 12*,

d cLdϵXj1

�ωXj1

� + d cL∗

dϵXj1

�ωXj1

� +-. (238)

Differentiating both sides of Eq. (238) with respect to theperturbation strengths ϵXj2

(ωX j2), . . . , ϵXjn

(ωX jn) and setting

all perturbation strengths to zero gives

*,

dnEdϵXj1


(ωXjn)+-0

=12

*,

dn cLdϵXj1


(ωXjn)+-0

+ *,

dn cL∗

dϵXj1(ωXj1


)+-0

. (239)

Using Eqs. (65) and (170), we obtain

⟨⟨X j1; X j2, . . . ,X jn⟩⟩ωXj2, ...,ωXjn

=12

*,

dn cLdϵXj1


(ωXjn)+-0

+ *,

dn cL∗

dϵXj1(ωXj1


)+-0

,

nm=1

ωXjm= 0. (240)

Inserting Eqs. (234) and (235) in Eq. (240) gives

⟨⟨X j1; X j2, . . . ,X jn⟩⟩ωXj2, ...,ωXjn

=12

(cLXj1

. . .Xjn(ωXj1, . . . ,ωXjn

)+ cLXj1

. . .Xjn∗(−ωXj1

, . . . ,−ωXjn)),

nm=1

ωXjm= 0. (241)

Introducing the complex conjugation and frequency signinversion operator C±ω defined as

C±ωhXj1···Xjn(ωXj1

, . . . ,ωX jn)

= hXj1···Xjn(ωXj1

, . . . ,ωX jn)

+ hXj1···Xjn

∗(−ωXj1, . . . ,−ωX jn

), (242)

where hXj1···Xjn(ωXj1

, . . . ,ωX jn) is an arbitrary complex func-

tion of the set of operator–frequency pairs, we may write


Eq. (241) in a compact manner as

⟨⟨X j1; X j2, . . . ,X jn⟩⟩ωXj2, ...,ωXjn

=12

C±ω cLXj1. . .Xjn(ωXj1

, . . . ,ωXjn),

nm=1

ωXjm= 0.

(243)

Following the same route as for deriving EX j1...X jn

(ωX j1, . . . ,ωX jn

) of Eq. (146), we may identify cLX j1...X jn

(ωX j1, . . . ,ωX jn

) from Eq. (219) in terms of the symmetry

distinct contributions, c f Xj1. . .Xjn

ωXj1...ωXjn

,

cLXj1. . .Xjn(ωXj1

, . . . ,ωXjn) = P

Xj1. . .Xjn

ωXj1...ωXjn

c f Xj1. . .Xjn

ωXj1...ωXjn

,n

m=1

ωXjm= 0. (244)

When the Hermitian eigenvalue equation is solved viaprojection, molecular response functions are thus determinedfrom Eq. (243), where the complex quasi-energy Lagrangianis obtained from Eq. (244).

2. Stationary conditions for the perturbationcomponents of the quasi-energy Lagrangian

The individual components of the complex quasi-energyLagrangian, cLXj1

···Xjn(ωXj1, . . . ,ωX jn

), are stationary withrespect to the components of the wave-function param-eters, b

Xj1···Xjm

i (ωXj1, . . . ,ωXjm

), and multipliers, bXj1···Xjm

i(ωXj1, . . . ,ωXjm

), when the conditionn

m=1ωXjm= 0 is satis-

fied,

∂ cLXj1. . .Xjn(ωXj1

, . . . ,ωXjn)

∂bXj1···Xjm


)= 0,

nk=1

ωXjk= 0, (245)


, . . . ,ωXjn)

∂bXj1···Xjm


)= 0,

nk=1

ωXjk= 0. (246)

To show this, we consider the derivative ofcLXj1


) in Eq. (234),


, . . . ,ωXjn)

∂bXj1···Xjm


)=

∂

∂bXj1···Xjm


)*,

dn cLdϵXj1

(ωXj1) . . . ϵXjn

(ωXjn)+-0

= *,

dn

dϵXj1(ωXj1

) . . . ϵXjn(ωXjn

)∂ cL

∂bXj1···Xjm


)+/-0

= *,

dn

dϵXj1(ωXj1

) . . . ϵXjn(ωXjn

)k

L

∂ cL∂bk(ωL)

∂bk(ωL)∂b

Xj1···Xjm


)+/-0

, (247)

where in the last equality we have used the chain rule.Applying the stationary condition of Eq. (225) in Eq. (247), weconclude that the individual components of the quasi-energyLagrangian, cLXj1


), are stationary withrespect to variations in the individual components of the coeffi-cients, b

Xj1···Xjm


), as expressed by Eq. (245).A similar derivation shows that the stationary condition inEq. (246) is fulfilled for multipliers. Equations (245) and(246) may be used to obtain response equations that determinethe Lagrangian multipliers, b

Xj1···Xjm

i , and the wave-functioncoefficients, b

Xj1···Xjm

i , respectively.

V. QUASI-ENERGY LAGRANGIANS FOR THE CI,CC-CI AND CC WAVE FUNCTION-MODELS

In this section, we introduce the CI, CC-CI, and CCwave-function models and determine the response func-tions for these models using projection. For the CI model,we use a linear parametrization for both the unperturbedand the perturbed system. For the CC model, we use anexponential parametrization for both the unperturbed andthe perturbed system, whereas for the CC-CI model, anexponential parametrization is used for the unperturbed systemand a linear parametrization for the perturbed system. In Sec.V A, the CI, CC-CI, and CC models are introduced, and in

Sec. V B, we show how simplifications may be introduced incLH0 [Eq. (220)], cLF [Eq. (221)], and cLX j1

[Eq. (222)]for these wave function models. In particular, we use that thezero-order amplitude and multiplier equations [Eqs. (195) and(196)] are satisfied to simplify the expression for cLH0 andidentify a generic expression for the quasi-energy Lagrangianfor the CI, CC-CI, and CC models. Response functions for theCI, CC-CI, and CC models are given in Secs. VI–VIII.

A. Wave function models in terms of differentparametrizations of the unperturbed stateand the time evolution

We start out by introducing the notation we will use forB0 [Eq. (179)] and B(t) [Eq. (201)] when these operators referto either a linear or an exponential parametrization. For thelinear parametrization [Eq. (180)], we use the operator label Sand the amplitudes bk and the multipliers bk will be denotedsk and sk, respectively,

B = S =k

sk |k⟩⟨R|. (248)

Equation (248) implies that

S2(t) = 0. (249)


For the exponential parametrization [Eq. (183)], we assumethat |R⟩ = |HF⟩ is the Hartree-Fock state and relabel the many-body orbital excitation operators of Eq. (183) as τµk

, wherek denotes an excitation level and µk is an excitation at thisexcitation level. The orthogonal complement set of states thenbecomes (cf. Eq. (185))

|µk⟩ = τµk|HF⟩. (250)

We use the T , tµk, and tµk

symbols for the exponentialparametrization,

B = T =µk

tµkτµk

. (251)

Four wave function models may be obtained fromEq. (198),

|0CII ⟩ = exp(S0) exp(S(t))|R⟩, (252)

|0CCI ⟩ = exp(T0) exp(T(t))|R⟩, (253)

|0CC−CII ⟩ = exp(T0) exp(S(t))|R⟩, (254)

|0CI−CCI ⟩ = exp(S0) exp(T(t))|R⟩. (255)

When the excitation manifolds, |k⟩⟨R| and τµk, are

complete, the four wave function models [Eqs. (252)–(255)]all describe an exact time evolution and give the same exactresult. When the excitation manifolds are truncated, the fourmodels describe different but well-defined time evolutions. Inthe CC model [Eq. (253)], both the unperturbed state and thetime evolution are described using an exponential expansionleading to the well-known advantages of the rapid convergencewith the excitation rank and a size-extensive energy for theunperturbed systems. By contrast, the CI model [Eq. (252)],where both the unperturbed state and the time evolutionare linearly parametrized, exhibits a slow convergence withthe excitation rank and a nonsize-extensive energy for theunperturbed systems. In the CC-CI model [Eq. (254)], theunperturbed state is described with a coupled cluster state andthe time evolution of this state by a simple linear CI expansion.

As the time-dependent perturbation is small, it may suffice todescribe it using a linear parametrization. In the fourth model[Eq. (255)], the unperturbed state is described using the CImodel. The exponential parametrization of the time evolutioncannot remedy the deficiencies of the CI model for describingthe unperturbed system and this model will not be consideredfurther.

B. Simplifications in the complex quasi-energyLagrangian in the frequency domain

In this subsection, we simplify the complex quasi-energyLagrangian of Eq. (219) to obtain a generic expression that canbe applied for the three wave function models CI, CC-CI, andCC. We first transform cLH0 [Eq. (220)], cLF [Eq. (221)], andcLX j1

[Eq. (222)] to the frequency domain and then simplifythe resulting expressions.

Consider first cLH0. Applying the Baker-Campbell-Hausdorff (BCH) expansion to cLH0 [Eq. (220)] gives

cLH0 =cLL

H0+ cLNL

H0, (256)

where cLLH0

and cLNLH0

denote terms that are linear and non-linear in the bm(t) coefficients, respectively. cLL

H0reads

cLLH0= ⟨R|HB0

0 |R⟩ +⟨R| HB0

0 ,B(t) |R⟩T

+kK

bk(ωK)⟨k |HB0

0 |R⟩ exp(−iωKt)T

+kK

bk(ωK)⟨k | HB0

0 ,B(t) |R⟩ exp(−iωKt)T

,

(257)

whereas the explicit form of cLNLH0

depends on whetherthe time evolution is linearly (B(t) = S(t)) or exponentially(B(t) = T(t)) parametrized,

cLNLH0=

−kK

sk(ωK)

k ��S(t)HB00 S(t)��R

exp(−iωKt)

T

, for B(t) = S(t), (258a)

12

⟨R| HT0

0 ,T(t) ,T(t) |R⟩T

+kK

tk(ωK)(

12⟨k | HT0

0 ,T(t) ,T(t) |R⟩

+16⟨k | HT0

0 ,T(t) ,T(t) ,T(t) |R⟩

+1

24⟨k | HT0

0 ,T(t) ,T(t) ,T(t) ,T(t) |R⟩)

exp(−iωKt)T

, for B(t) = T(t). (258b)

To obtain Eq. (258a), we have used Eq. (249) to truncatethe BCH expansion. In Eq. (258b), B0 has been replacedby T0 because for an exponential parametrization of thetime evolution, the only unperturbed state we consider isexponentially parametrized, as discussed in Subsection V ATo obtain Eq. (258b), we have used that H0 and HT0

0 are rank-

two operators, and therefore, the BCH expansion truncatesafter the quadratic term for ⟨R| exp(−T(t))HT0

0 exp(T(t))|R⟩and after the quartic term for ⟨k | exp(−T(t))HT0

0 exp(T(t))|R⟩.We now transform cLL

H0[Eq. (257)] to the frequency

domain. To do that we substitute Eq. (201) in Eq. (257) andthen insert Eq. (114) for bm(t), giving


cLLH0= ⟨R|HB0

0 |R⟩ +mM

bm(ωM)⟨R| HB00 , βm

|R⟩

×

exp(−iωMt)T

+kK

bk(ωK)⟨k |HB00 |R⟩

exp(−iωKt)

T

+

kmKM

bk(ωK)bm(ωM)⟨k | HB00 , βm

|R⟩

×

exp�− i(ωK + ωM)t�

T

. (259)

Using Eq. (120), the explicit time dependence may be removedfrom Eq. (259),

cLLH0= E0 +

mM


|R⟩∆(ωM)

+kK

bk(ωK)⟨k |HB00 |R⟩∆(ωK)

+

kmKM

bk(ωK)bm(ωM)⟨k | HB00 , βm

|R⟩

×∆(ωK + ωM), (260)

where we have also used Eq. (193a).We now introduce the Jacobian in Eq. (260). cLH0

contains bm(ωM) and bk(ωK) as independent parameters. TheJacobian therefore becomes [cf. Eq. (126)]

PJkm = *,

∂2 cLH0

∂bk(ωK)∂bm(ωM)+-0

= *,

∂2 cLLH0

∂bk(ωK)∂bm(ωM)+-0

= ⟨k |�HB00 , βm

�|R⟩, (261)

where we have used that at zero perturbation strength thebm(ωM) coefficients vanish and the frequencies are equal tozero. Inserting Eq. (261) in Eq. (260) gives

cLLH0= E0 +

mM


|R⟩∆(ωM)

+kK


+

kmKM

bk(ωK)bm(ωM) PJkm∆(ωK + ωM). (262)

We now separate out the zero-order contribution in theLagrangian multipliers bk(ωK),

bk(ωK) = b(0)k + b

(\0)k (ωK), (263)

where b(0)k denotes the zero-order multipliers in Eq. (194) and

b(\0)k (ωK) denotes multipliers that do not contain the zero-order

contribution. Inserting Eq. (263) in the last term of Eq. (262)gives

cLLH0= E0 +

mM

bm(ωM)(⟨R| HB0

0 , βm |R⟩

+k

b(0)k

PJkm

)∆(ωM)

+kK


+

kmKM

b(\0)k (ωK)bm(ωM) PJkm∆(ωK + ωM). (264)

We now show that cLLH0

may be simplified to

cLLH0= E0 +

kmKM

b(\0)k (ωK)bm(ωM) PJkm

×∆(ωK + ωM) (265)

using the amplitudes equation [Eq. (195)] and the Lagrangianmultipliers equation [Eq. (196)] for the unperturbed system.

To obtain Eq. (265), we initially note that the thirdterm in Eq. (264) contains the zero-order amplitude equation[Eq. (195)] and vanishes for all the three models: CI, CC,and CC-CI. The second term of Eq. (264) contains theJacobian of Eq. (261). For the CI and CC models, theoperators βn describing the unperturbed system are identicalto the operators describing the time evolution. The zero-orderLagrangian multipliers equation [Eq. (196)] may therefore beinvoked to show that the second term in Eq. (264) vanishes forthe CI and CC models.

To see that the second term vanishes also for the CC-CI model, we may write for this model the terms in theparentheses as

HF��HT0

0 , |νm⟩⟨HF| ��HF+

µk

t(0)µk

CC−CIJµkνm

= ⟨HF|HT00 |νm⟩ +

µk

t(0)µk

CC−CIJµkνm, (266)

where we have used that in the CC-CI model B0 = T0 andthe |B⟩ basis [Eq. (173)] reads

�|HF⟩, |µk⟩, and where[cf. Eq. (261)]

CC−CIJµkνm = ⟨µk |HT0

0 , |νm⟩⟨HF| |HF⟩= ⟨µk |HT0

0 |νm⟩ − δµkνm⟨HF|HT00 |HF⟩. (267)

In the CC-CI model, we use the CC state as the unperturbedstate and the cluster amplitudes therefore satisfy the zero-ordermultiplier equation [see Eq. (196)]

0 = ⟨HF| HT00 , τνm

|HF⟩ +µk

t(0)µk⟨µk |

HT0

0 , τνm |HF⟩

= ⟨HF|HT00 |νm⟩ +

µk

t(0)µk

CC−CIJµkνm, (268)

where we have used

⟨µk |HT0

0 , τνm |HF⟩ = ⟨µk |HT0

0 |νm⟩ − δµkνm⟨HF|HT00 |HF⟩

= CC−CIJµkνm. (269)

To obtain the second equality in Eq. (269), we have usedthat ⟨µk |τνmHT0

0 |HF⟩ vanishes for k < m due to Eq. (187)and for k > m it becomes ⟨µn |HT0

0 |HF⟩, where n < k, and alsovanishes since the zero-order stationary condition of Eq. (195)


is satisfied. The last equality in Eq. (269) is obtained usingEq. (267). It follows from Eqs. (266) and (268) that the secondterm in Eq. (264) vanishes also for the CC-CI wave function.

Turning our attention to cLNLH0

[Eq. (258)], we maytransform it to the frequency domain, using Eqs. (114), (120),and (201), as

cLNLH0=

−KLM

k

sk(ωK)sk(ωL)m

sm(ωM)R��HB00��m

∆(ωK + ωL + ωM), for B(t) = S(t), (270a)

12

νmλnMN

tνm(ωM)tλn(ωN)⟨R| HT00 , τνm

, τλn

|R⟩∆(ωM + ωN)

+12

kνmλnKMN

tk(ωK)tνm(ωM)tλn(ωN)⟨k |

HT00 , τνm

, τλn

|R⟩∆(ωK + ωM + ωN)

+16

kνmλnσpKMNP

tk(ωK)tνm(ωM)tλn(ωN)tσp(ωP)⟨k |

HT00 , τνm

, τλn

, τσp

|R⟩

×∆(ωK + ωM + ωN + ωP)+

124

kνmλnσpρqKMNPQ

tk(ωK)tνm(ωM)tλn(ωN)tσp(ωP)tρq(ωQ)⟨k |

HT00 , τνm

, τλn

, τσp

, τρq

|R⟩

×∆(ωK + ωM + ωN + ωP + ωQ), for B(t) = T(t). (270b)

cLF may be transformed to the frequency domainsubstituting Eq. (119) in Eq. (221),

cLF = −

kmKM

ωMbk(ωK)bm(ωM)⟨k |βm|R⟩

×

exp(− i(ωK + ωM)t)

T

= −

kmKM

ωMbk(ωK)bm(ωM)⟨k |βm|R⟩∆(ωK + ωM

)= −

kKM

ωMbk(ωK)bk(ωM)∆(ωK + ωM

), (271)

where to obtain the last equality we have used Eq. (185) andthe orthonormality of the |B⟩ set [Eq. (173)].

Considering the cLX j1contribution [Eq. (222)], we note

that as long as X jk are assumed to be one-electron operators,the BCH expansion truncates at the same order for both linearand exponential parametrization of the time evolution, giving

cLX j1= ⟨R|XB0

j1|R⟩∆(ωXj1

) +mM

bm(ωM)⟨R|

×XB0

j1, βm

|R⟩∆(ωXj1+ ωM)

+kK

bk(ωK)⟨k |XB0j1|R⟩∆(ωXj1

+ ωK)

+kmKM

bk(ωK)bm(ωM)⟨k | XB0j1, βm

|R⟩

×∆(ωXj1+ ωK + ωM) + 1

2

kmnKMN

bk(ωK)bm(ωM)

× bn(ωN)⟨k |

XB0j1, βm

, βn

|R⟩×∆(ωXj1

+ ωK + ωM + ωN). (272)

For convenience in derivation of response functions, wecombine cLH0 and cLF in cLH ,

cLH = cLH0 +cLF

= E0 +

kmKM

b(\0)k (ωK)bm(ωM)

(PJkm − ωMδkm

)×∆(ωK + ωM) + cLNL

H0, (273)

where in the second term cLF [Eq. (271)] is combined withthe second term of cLL

H0in Eq. (265) and where cLNL

H0is

given in Eq. (270). The quasi-energy Lagrangian in Eq. (219)may thus be written as

cL = cLH +j1

cLX j1ϵXj1

(ωXj1), (274)

where cLH and cLX j1are given in Eqs. (273) and (272),

respectively.

VI. MOLECULAR PROPERTIES FOR THE CI MODEL

We consider in this section how molecular responsefunctions are derived for the CI model where projectionis applied to the time-dependent Schrödinger equation todetermine the time evolution. Projection is also used todetermine the unperturbed ground state.

A. Parametrization of the CI model

To derive molecular response properties for the CImodel we use, as in Sec. III (see Eq. (89)), the groundstate for the unperturbed system as the expansion point forthe perturbation-induced time development. The basis |B⟩[Eq. (173)], in which we describe the time development,


therefore becomes

|B⟩ = |0CI0 ⟩, |µCI

k ⟩, (275)

where |0CI0 ⟩ is the unit-normalized unperturbed ground state

satisfying

H0|0CI0 ⟩ = ECI

0 |0CI0 ⟩ (276)

and�|µCI

k⟩ denotes the orthogonal complement set of states

to |0CI0 ⟩. The orthogonal complement states |µCI

k⟩ may or may

not be the exact solution to the Schrödinger equation for theunperturbed system [Eq. (67)]. Since the unperturbed groundstate, |0CI

0 ⟩, is used as the reference state, the intermediatenormalized state in Eq. (181) may be expressed as

|0CII0⟩ = exp

(µk

s(0)µk|µCI

k ⟩⟨0CI0 |

)|0CI

0 ⟩

= |0CI0 ⟩ +

µk

s(0)µk|µCI

k ⟩. (277)

As we consider the time evolution with |0CI0 ⟩ as an expansion

point, the s(0)µkcoefficients, and hence B0, become zero. We

may, therefore, replace the similarity-transformed Hamilto-nian, HB0

0 [Eq. (191)], by the bare Hamiltonian, H0, in thecomplex quasi-energy Lagrangian expression [Eqs. (272) and(273)].

Projecting Eq. (276) against the basis

⟨B | = ⟨0CI0 |,⟨µCI

k |

(278)

gives

⟨0CI0 |H0|0CI

0 ⟩ = ECI0 , (279a)

⟨µCIk |H0|0CI

0 ⟩ = 0. (279b)

It follows from Eq. (279b) and the Hermiticity of theHamiltonian that

⟨0CI0 |H0|µCI

k ⟩ = 0 (280)

is also satisfied.Equations (279) and (280) describe the optimization

conditions obtained when the variational principle is invokedto solve the time-independent Schrödinger equation [Eq. (87b)].The zero-order Lagrangian multipliers in Eq. (194) satisfyEq. (196), which in the CI model become

⟨0CI0 | H0, |µCI

k ⟩⟨0CI0 | |0CI

0 ⟩+

νm

s(0)νm⟨νCI

m | H0, |µCIk ⟩⟨0CI

0 | |0CI0 ⟩ = 0, (281)

which may be written as

⟨0CI0 |H0|µCI

k ⟩ +νm

s(0)νm

(⟨νCIm |H0|µCI

k ⟩ − CIE0δµkνm

)= 0. (282)

Applying Eq. (280) in Eq. (282), we thus see that the zero-order multipliers vanish in CI theory,

s(0)µk= 0. (283)

As S0 = B0 is zero and

S(t) =µk

sµk(t)|µCI

k ⟩⟨0CI0 |, (284)

the time-dependent CI wave function [Eq. (252)] is linearlyexpanded around the unperturbed ground state |0CI

0 ⟩,|0CI

I ⟩ = exp(S(t))|0CI0 ⟩ = |0CI

0 ⟩ +µk

sµk(t)|µCI

k ⟩. (285)

We now derive explicit expressions for the CI molecularresponse functions through fourth order. To do this, we initiallynote that the expressions for cLH [Eq. (273)] and cLX j1[Eq. (272)] in the CI model may be further simplified byreplacing HB0

0 and XB0j1

with the bare operators and settingthe zero-order multipliers to zero [Eq. (283)]. Using thesesimplifications and introducing the relabeling |R⟩ = |0CI

0 ⟩,|k⟩ = |µCI

k⟩, bk = sµk

, b(\0)k = s(\0)µk

, the expression for cLH[Eq. (273)] becomes in the CI model

cLCIH =

CIE0 +µkνmKM

s(\0)µk(ωK)sνm(ωM)

×(

CIJµkνm − ωMδµkνm

)∆(ωK + ωM

), (286)

where the CI Jacobian is given by [see Eq. (261)]CIJµkνm = ⟨µCI

k |H0, |νCI

m ⟩⟨0CI0 | |0CI

0 ⟩= ⟨µCI

k |H0|νCIm ⟩ − CIE0δµkνm. (287)

We have used in Eq. (286) that cLNLH0

[Eq. (270a)] vanishessince Eq. (280) is satisfied. cLX j1

[Eq. (272)] becomes

cLCIX j1= ⟨0CI

0 |X j1|0CI0 ⟩∆(ωXj1

) +νmM

sνm(ωM)⟨0CI0 |

×X j1, |νCI

m ⟩⟨0CI0 | |0CI

0 ⟩∆(ωXj1+ ωM)

+µkK

s(\0)µk(ωK)⟨µCI

k |X j1|0CI0 ⟩∆(ωXj1

+ ωK)

+µkνmKM

s(\0)µk(ωK)sνm(ωM)⟨µCI

k |X j1, |νCI

m ⟩⟨0CI0 |

× |0CI0 ⟩∆(ωXj1

+ ωK + ωM)

+12

µkνmλnKMN

s(\0)µk(ωK)sνm(ωM)sλn(ωN)⟨µCI

k |

×

X j1, |νCIm ⟩⟨0CI

0 | , |λCIn ⟩⟨0CI

0 | |0CI0 ⟩

×∆(ωXj1+ ωK + ωM + ωN). (288)

Expanding the commutators in Eq. (288), we obtain

cLCIX j1= ⟨0CI

0 |X j1|0CI0 ⟩∆(ωXj1

) +νmM

sνm(ωM)

× ⟨0CI0 |X j1|νCI

m ⟩∆(ωXj1+ ωM)

+µkK

s(\0)µk(ωK)⟨µCI

k |X j1|0CI0 ⟩∆(ωXj1

+ ωK)

+µkνmKM

s(\0)µk(ωK)sνm(ωM)

×(⟨µCI

k |X j1|νCIm ⟩ − δµkνm⟨0CI

0 |X j1|0CI0 ⟩

)


×∆(ωXj1+ ωK + ωM) − (

µkKL

s(\0)µk(ωK)sµk

(ωL))

×νmM

sνm(ωM)⟨0CI0 |X j1|νCI

m ⟩

×∆(ωXj1+ ωK + ωL + ωM). (289)

Using cLH [Eq. (286)] and cLX j1[Eq. (289)], we follow

the same route as in Sec. III C to derive response functionexpressions. We thus initially identify cLX j1...X jn(ωX j1

, . . . ,

ωX jn) by writing cL(n) in terms of the individual contributions

from cLCIH [Eq. (286)] and cLCI

X j1[Eq. (289)],

cL(n) = cL(n)H +

j1

cL(n−1)X j1

ϵXj1(ωXj1

), (290)

identifying, according to Eq. (244), the individual distinctcontributions to cLX j1...X jn(ωX j1

, . . . ,ωX jn). We then obtain

response equations by differentiating cLX j1...X jn(ωX j1,

. . . ,ωX jn) with respect to s

Xj1···Xjm

µk(ωXj1

, . . . ,

ωXjm) and s

Xj1···Xjm

µk(ωXj1

, . . . ,ωXjm) using the stationary

conditions of Eq. (245) and Eq. (246), respectively. Finally,we simplify cLXj1

. . .Xjn(ωXj1, . . . ,ωXjn

) using the 2n + 1

and 2n + 2 rules (see, for example, Ref. 77) and obtainmolecular response functions ⟨⟨X j1; X j2, . . . ,X jn⟩⟩ωXj2

, ...,ωXjn

by inserting cLXj1. . .Xjn(ωXj1

, . . . ,ωXjn) in Eq. (243).

B. Energy and first-order molecular properties

From Eqs. (290) and (286), we determine cL(0) as

cL(0) = cL(0)H =

cL(0)H0= CIE0 = ⟨0CI

0 |H0|0CI0 ⟩. (291)

The first-order complex quasi-energy Lagrangian isobtained using Eqs. (286) and (289) as

cLXj1(0) = ⟨0CI0 |X j1|0CI

0 ⟩, (292)

which, inserted in Eq. (243), gives the first-order molecularproperties

X j1

��0 =

12

C±ω⟨0CI0 |X j1|0CI

0 ⟩ = ⟨0CI0 |X j1|0CI

0 ⟩. (293)

C. Second-order molecular properties

The second-order complex quasi-energy Lagrangian isobtained from Eq. (244) using Eqs. (286) and (289),

cLXj1Xj2(ωXj1

,ωXj2) = P

Xj1Xj2

ωXj1ωXj2

µkνm

sXj1µk

(ωXj1)sXj2

νm (ωX j2)(CIJµkνm − ωX j2

δµkνm

)

+νm

sXj2νm (ωX j2

)⟨0CI0 |X j1|νCI

m ⟩ +µk

sXj2µk

(ωX j2)⟨µCI

k |X j1|0CI0 ⟩

, ωXj1

+ ωX j2= 0. (294)

Applying the stationary condition of Eq. (246) tocLXj1

Xj2(ωXj1,ωXj2

) of Eq. (294),

0 =∂ cLXj1

Xj2(ωXj1,ωXj2

)∂s

Xj2µk

(ωXj2)

(295)

givesνm

(CIJµkνm − ωXj1

δµkνm

)sXj1νm (ωXj1

) = −⟨µCIk |X j1|0CI

0 ⟩.

(296)

Equation (296) is the first-order response equation for the first-order coefficients s

Xj1νm (ωXj1

). Similarly, applying the stationarycondition of Eq. (245) gives a response equation for the first-order multipliers s

Xj1νm (ωXj1

),νm

sXj1νm (ωXj1

) (CIJνmµk+ ωXj1

δνmµk

)= −⟨0CI

0 |X j1|µCIk ⟩,

(297)

where for the left-hand side, we have used the conditionωXj1

+ ωX j2= 0. It follows from Eqs. (296) and (297) that

the first-order amplitudes and the first-order multipliers are

the adjoints of each other,

sXj1µk

(ωXj1) = s

Xj1µk

∗(−ωXj1). (298)

Simplifying cLXj1Xj2(ωXj1

,ωXj2) [Eq. (294)] using the

2n + 1 and 2n + 2 rules77 and inserting the simplified formof cLXj1

Xj2(ωXj1,ωXj2

) in Eq. (243), we obtain the linearresponse function

X j1; X j2

��ωXj2=

12

C±ωPXj1

Xj2ωXj1

ωXj2

νm

sXj2νm (ωX j2

)

× ⟨0CI0 |X j1|νCI

m ⟩, ωXj1+ ωX j2

= 0, (299)

where sXj2νm (ωX j2

) are obtained by solving Eq. (296).

D. Third-order molecular properties

Following the same route as for cLXj1Xj2(ωXj1

,ωXj2), we

obtain cLXj1Xj2

Xj3(ωXj1,ωXj2

,ωXj3) as


cLXj1Xj2

Xj3(ωXj1,ωXj2

,ωXj3) = P

Xj1Xj2

Xj3ωXj1

ωXj2ωXj3

12

µkνm

sXj1µk

(ωXj1)sXj2

Xj3νm (ωX j2

,ωXj3)(CIJµkνm − (ωX j2

+ ωXj3)δµkνm

)

+12

µkνm

sXj1

Xj2µk

(ωXj1,ωX j2

)sXj3νm (ωXj3

)(CIJµkνm − ωXj3δµkνm

)

+12

νm

sXj2

Xj3νm (ωX j2

,ωXj3)⟨0CI

0 |X j1|νCIm ⟩ + 1

2

µk

sXj2

Xj3µk

(ωX j2,ωXj3

)⟨µCIk |X j1|0CI

0 ⟩

+µkνm

sXj2µk

(ωX j2)sXj3

νm (ωXj3)(⟨µCI

k |X j1|νCIm ⟩ − ⟨0CI

0 |X j1|0CI0 ⟩δµkνm

), ωXj1

+ ωX j2+ ωXj3

= 0,

(300)

where the factor 12 in the first four terms arises from Eq. (115).

Differentiating cLXj1Xj2

Xj3(ωXj1,ωXj2

,ωXj3) of Eq. (300)

with respect to sXj2

Xj3µk

(ωXj2,ωXj3

) and using the station-ary condition of Eq. (246), we recover the first-orderresponse equation of Eq. (296). Similarly, differentiatingcLXj1

Xj2Xj3(ωXj1

,ωXj2,ωXj3

) with respect to sXj2

Xj3µk

(ωXj2,

ωXj3) and using the stationary condition of Eq. (245), we

recover the first-order multipliers equation of Eq. (297).Differentiating cLXj1

Xj2Xj3(ωXj1

,ωXj2,ωXj3

) with respect to

sXj3µk

(ωXj3) and using the stationary condition of Eq. (246)

gives the response equation for the second-order coefficients,νm

(CIJµkνm − (ωXj1

+ ωXj2)δµkνm

)sXj1

Xj2νm (ωXj1

,ωX j2)

= −PXj1

Xj2ωXj1

ωXj2

νm

(⟨µCIk |X j1|νCI

m ⟩ − ⟨0CI0 |X j1|0CI

0 ⟩δµkνm

)× s

Xj2νm (ωX j2

), (301)

from which sXj1

Xj2νm (ωXj1

,ωX j2), needed in Subsection VI E

for the fourth-order properties, may be calculated.

Using the 2n + 1 and 2n + 2 rules, the third-order molecular property expression becomes

X j1; X j2,X j3

��ωXj2

,ωXj3

=12

C±ωPXj1

Xj2Xj3

ωXj1ωXj2

ωXj3

µkνm

sXj2µk

(ωX j2)sXj3

νm (ωXj3)(⟨µCI



),

ωXj1+ ωX j2

+ ωXj3= 0, (302)

where sXj3µk

(ωXj3) and s

Xj2µk

(ωX j2) are obtained by solving Eqs. (296) and (297), respectively.

E. Fourth-order molecular properties

Using the 2n + 1 and 2n + 2 rules, we obtain fourth-order molecular properties in terms of the first- and second-ordercoefficients and first-order multipliers as

X j1; X j2,X j3,X j4

��ωXj2

,ωXj3,ωXj4

=12

C±ωPXj1

Xj2Xj3

Xj4ωXj1

ωXj2ωXj3

ωXj4

12

µkνm

sXj2µk

(ωX j2)sXj3

Xj4νm (ωXj3

,ωXj4)(⟨µCI



)

−(

µk

sXj2µk

(ωX j2)sXj3

µk(ωXj3

)) νm

sXj4νm (ωXj4

)⟨0CI0 |X j1|νCI

m ⟩, ωXj1

+ ωX j2+ ωXj3

+ ωXj4= 0, (303)


where sXj3µk

(ωXj3), s

Xj2µk

(ωXj2), and s

Xj3Xj4

µk(ωXj3

,ωXj4) are

obtained by solving Eqs. (296), (297), and (301), respectively.

F. The CI Jacobian as a part of the CI eigenvalueequation

We now consider a matrix representation of the Hermitianeigenvalue equation for the unperturbed Hamiltonian in thebasis of Eq. (275). We then show that the Jacobian matrixcontains a sub-matrix of the Hamiltonian matrix.

1. The CI eigenvalue equation

The time development of the CI wave function inEq. (285) is described in terms of a linear expansion in the basis|B⟩ of Eq. (275). We now consider a matrix representationof the Hermitian eigenvalue equation for the unperturbedHamiltonian H0 in this basis

CIH0CIC = CIC CIE. (304)

The unperturbed Hamiltonian matrix, CIH0, has the blockstructure

CIH0 = *,

CIE0 00 CIH⊥0

+-, (305)

where CIH⊥0 denotes the sub-block of the Hamiltonian matrixin the orthogonal complement subspace {|µCI

k⟩},

�CIH⊥0�µkνm

=�CIH0

�µkνm

= ⟨µCIk |H0|νCI

m ⟩. (306)

To obtain the block structure in Eq. (305), we have usedEqs. (279) and (280),

�CIH0�00 = ⟨0CI

0 |H0|0CI0 ⟩ = CIE0, (307)

�CIH0�0µk= ⟨0CI

0 |H0|µCIk ⟩ = 0, (308)

�CIH0�µk0 = ⟨µCI

k |H0|0CI0 ⟩ = 0. (309)

The CIE matrix is diagonal and contains the energies of theunperturbed Hamiltonian H0,

CIE =

*......,

CIE0CIE1

CIE2

. . .

+//////-

. (310)

The block structure of Eq. (305) implies that

CIC = *,

1 00 CIC

+-. (311)

The ground state vector therefore becomes

CIC0 = *,

10+-, (312)

whereas excited state vectors for energies CIEn, n = 1,2, . . .,become

CICn = *,

0CICn

+-, (313)

where CICn is the nth column of the matrix CIC [Eq. (311)]. Asthe matrix CIH0 is Hermitian, the eigenvectors may be chosento constitute an orthonormal set

CIC† CIC = I. (314)

For later convenience, we express the eigenstates in thebracket form as

|0CIR ⟩ =

|0CI

0 ⟩, |0CI1 ⟩, |0CI

2 ⟩, · · ·, (315)

⟨0CIL | =

⟨0CI

0 |,⟨0CI1 |,⟨0CI

2 |, · · ·, (316)

where, for n , 0,

|0CIn ⟩ =

µk

CICµkn |µCIk ⟩ (317)

and where CICµkn are elements of the CICn vector [Eq. (313)].

2. The CI Jacobian eigenvalue equation

We now consider the CI Jacobian in more detail. It followsfrom Eqs. (287) and (306) that the Jacobian matrix may beexpressed in terms of a matrix representation of H0 in theorthogonal complement subspace {|µCI

k⟩} minus a diagonal

matrix containing CIE0,

CIJ = CIH⊥0 −CIE0I. (318)

Using the block structure of CIH0 [Eq. (305)] and CIC[Eq. (311)] and denoting the sub-block of CIE [Eq. (310)] inthe orthogonal complement subspace as CIE⊥, the eigenvalueequation [Eq. (304)] in the orthogonal complement subspacemay be written as(

CIH⊥0 −CIE0I

)CIC = CIC

(CIE⊥ − CIE0I

), (319)

or, equivalently, as

CIJ CIC = CIC ∆CIE, (320)

where

∆CIE = CIE⊥ − CIE0I =

*......,

CIE1 0CIE2 0

CIE3 0

. . .

+//////-

(321)

andCIEn0 =

CIEn − CIE0, n , 0. (322)

Equations (320)–(322) show that the eigenvalues of CIJ areequal to the excitation energies of the unperturbed system.

From Eqs. (319) and (311), we see that the eigenvectorsof the CI Jacobian, CIJ, are equal to the excited stateeigenvectors of the CI Hamiltonian matrix, CIH⊥0 , in Eq. (305)


in accordance with Eq. (128). The excited states are thuslinearly parametrized in the basis |B⟩ of Eq. (275) as is thetime evolution of the ground state [Eq. (285)]. As there isno interaction between the ground state and the orthogonalcomplement set of states, neither for the unperturbed systemnor for the time-evolving system, we therefore have anexplicit representation of the excited states in CI responsetheory.

G. Comparison of response functions obtained fromthe quasi-energy and from the quasi-energyLagrangian

In Secs. III and VI, molecular response functions weredetermined using a linearly parametrized unperturbed groundstate and employing a linear parametrization of the time evolu-tion of the perturbed wave function. In Sec. III, the Hermitianeigenvalue equation for the time evolution [Eq. (32)] wassolved using the variational principle, while in Sec. VI, itwas solved via projection.

When the Hermitian eigenvalue equation was solvedusing the variational principle, the molecular response prop-erties were determined as derivatives of the quasi-energyE [Eq. (133)] given in terms of EH [Eq. (131)] and EX j1[Eq. (132)], where each of these contributions contains termsto infinite order introduced by a normalization constant Nthrough Eq. (95) [cf. Eqs. (98)–(100)].

By contrast, when the Hermitian eigenvalue equationwas solved via projection, the molecular response prop-erties were obtained as derivatives of the complex quasi-energy Lagrangian cLCI [Eq. (274)] given in terms of cLCI

H[Eq. (286)] and cLCI

X j1[Eq. (289)]. The two definitions lead

to the same response properties, see Eq. (239). However, thecLCIH [Eq. (286)] and cLCI

X j1[Eq. (289)] contributions do not

contain a normalization constant. cLCI therefore has a simplerstructure than E. Although cLCI contains an extra set ofparameters compared to E, the Lagrangian multipliers, theseextra parameters are through the nth order easily obtainedfrom the nth-order wave function amplitudes. In particular, thefirst-order multipliers are simply the adjoints of the first-orderwave function amplitudes [Eq. (298)].

The complex quasi-energy Lagrangian [Eq. (274)] doesnot contain the norm since the time development is expressedin terms of an intermediate normalized state [Eq. (89a)] ratherthan in terms of the unit-normalized regular wave function[Eq. (88)]. This simplification is similar to the one obtainedin time-independent perturbation theory, where the use ofintermediate normalization also simplifies the determinationof an order expansion for the energy and the wave function,removing the reference to the normalization constant.

We have seen in this section the advantages of solvingthe Hermitian eigenvalue equation via projection for a linearparametrization of the time evolution of the perturbed wavefunction. However, for non-variational wave function models,the Hermitian eigenvalue equation [Eq. (32)] can only besolved via projection and molecular response properties thenhave to be determined using a quasi-energy Lagrangian asdescribed in Sec. IV. In Secs. VII and VIII, we describe the

determination of response functions, where the unperturbedground state is a coupled cluster state. In Sec. VII, the timeevolution is described in terms of a linear parametrization,while in Sec. VIII, the time evolution is described in terms ofan exponential parametrization.

VII. MOLECULAR PROPERTIES FOR THE CC-CIMODEL

In Sec. VI, we have derived molecular response func-tions using the CI model, where both the time-independentunperturbed state and the time development are linearlyparametrized. In this and Sec. VIII, we consider an approachwhere the time-independent unperturbed state is an exponen-tially parametrized coupled cluster state. In this section, weconsider the CC-CI model [Eq. (254)], where the exponentialparametrization of the unperturbed state is followed by a linearparametrization of the time development, whereas in Sec. VIII,we turn our attention to the CC model [Eq. (253)], where thetime development is also exponentially parametrized.

A. Parametrization of the CC-CI model

In the CC-CI model [Eq. (254)], the set |B⟩ becomes

|B⟩ = |HF⟩, |µCCk ⟩, (323)

where |HF⟩ is the Hartree-Fock reference state and theorthogonal complement set of states |µCC

k⟩ is defined via

Eq. (250).For the unperturbed system, the coupled cluster energy

reads [Eq. (193a)]

CCE0 = ⟨HF|HT00 |HF⟩, (324)

where the cluster amplitudes, t(0)µk, satisfy the coupled cluster

amplitudes equation (cf. Eq. (193b))

⟨µCCk |HT0

0 |HF⟩ = 0. (325)

The coupled cluster zero-order Lagrangian (cf. Eq. (194))therefore reads

L0 = ⟨HF|HT00 |HF⟩ +

µk

t(0)µk⟨µCC

k |HT00 |HF⟩. (326)

The zero-order Lagrangian multipliers satisfy (cf. Eq. (196))µk

t(0)µk

CCJµkνm = −ηνm, (327)

where, using Eq. (261), we have introduced the CC JacobianCCJµkνm = ⟨µCC

k | HT00 , τνm

|HF⟩

= ⟨µCCk |HT0

0 |νCCm ⟩ − δµkνm⟨HF|HT0

0 |HF⟩ (328)

and where

ηνm = ⟨HF| HT00 , τνm

|HF⟩ = ⟨HF|HT00 |νCC

m ⟩. (329)

Using that in the CC-CI model, the B(t) operator has theform

S(t) =µk

sµk(t)|µCC

k ⟩⟨HF|, (330)


the time-dependent CC-CI wave function |0CC−CII ⟩ [Eq. (254)]

may be written as

|0CC−CII ⟩ = exp(T0) exp(S(t))|HF⟩

= exp(T0)|HF⟩ +µk

sµk(t) exp(T0)|µCC

k ⟩. (331)

Equation (331) shows that the coupled cluster state exp(T0)|HF⟩is obtained for the unperturbed system in the CC-CI model andthat the perturbation expansion of the quasi-energy Lagrangiantherefore may be carried out with the unperturbed coupledcluster state as expansion point. Eq. (331) further shows thatin the CC-CI model the perturbation-induced time evolutionis expressed in terms of a linear CI expansion in the basis ofthe many-electron functions

|BT0⟩ = exp(T0)|HF⟩,exp(T0)|µCC

k ⟩. (332)

The CC-CI molecular response functions may be obtainedfollowing the same route as for the CI model in Sec. VIand using the following relabeling in cLH [Eq. (273)]and cLX j1

[Eq. (272)]: |R⟩ = |HF⟩, |k⟩ = |µCCk⟩, B0 = T0, βk

= |µCCk⟩⟨HF|, b(0)

k= t(0)µk

, b(0)k = t(0)µk

, bk = sµk, and bk = sµk

. Inthe following, we determine the CC-CI response functionsthrough fourth order.


From Eq. (273), we determine cL(0) as

cL(0) = cL(0)H =

cL(0)H0= CCE0 = ⟨HF|HT0

0 |HF⟩ (333)

since the expansion coefficients sk(ωK) do not contain zero-order contributions.

The first-order complex quasi-energy Lagrangian isobtained using Eqs. (272) and (273) as

cLXj1(0) = ⟨HF|XT0j1|HF⟩ +

µk

t(0)µk⟨µCC

k |XT0j1|HF⟩,

(334)

which, inserted in Eq. (243), gives the first-order molecularproperties

X j1

��0 =

12

C±ω *.,⟨HF|XT0

j1|HF⟩ +

µk

t(0)µk⟨µCC

k |XT0j1|HF⟩+/

-,

(335)

where the zero-order amplitudes, t(0)µk, and the zero-order

multipliers, t(0)µk, are obtained by solving Eqs. (325) and (327),

respectively.


The second-order complex quasi-energy Lagrangian isobtained from Eqs. (272) and (273),

cLXj1Xj2(ωXj1

,ωXj2) = P

Xj1Xj2

ωXj1ωXj2

µkνm

sXj1µk

(ωXj1)sXj2

νm (ωXj2)(

CCJµkνm − ωXj2δµkνm

)

−µk

t(0)µksXj1µk

(ωXj1)νm

sXj2νm (ωXj2

)⟨HF|HT00 |νCC

m ⟩ +νm

sXj2νm (ωXj2

)⟨HF| XT0j1, |νCC

m ⟩⟨HF| |HF⟩

+µk

sXj2µk

(ωXj2)⟨µCC

k |XT0j1|HF⟩ +

µkνm

t(0)µksXj2νm (ωXj2

)⟨µCCk | XT0

j1, |νCC

m ⟩⟨HF| |HF⟩, ωXj1

+ ωX j2= 0,

(336)

where we have used that

CCJµkνm =CC−CIJµkνm, (337)

as seen from Eqs. (269) and (328).Applying the stationary condition of Eq. (246) to

cLXj1Xj2(ωXj1

,ωXj2) of Eq. (336) gives the first-order ampli-

tudes equationνm

(CCJµkνm − ωXj1

δµkνm

)sXj1νm (ωXj1

) = −⟨µCCk |XT0

j1|HF⟩.

(338)

Similarly, applying the stationary condition of Eq. (245) givesthe first-order multipliers equation

νm

sXj1νm (ωXj1

)(

CCJνmµk+ ωXj1

δνmµk

)

=(νm

t(0)νmsXj1νm (ωXj1

))⟨HF|HT00 |µCC

k ⟩

+ t(0)µk

νm

sXj1νm (ωXj1

)⟨HF|HT00 |νCC

m ⟩

− ⟨HF| XT0j1, |µCC

k ⟩⟨HF| |HF⟩

−νm

t(0)νm⟨νm|

XT0

j1, |µCC

k ⟩⟨HF| |HF⟩, (339)


where for the left-hand side, we have used the condition ωXj1+ ωX j2

= 0.Simplifying cLXj1

Xj2(ωXj1,ωXj2

) [Eq. (336)] using the 2n + 1 and 2n + 2 rules77 and inserting the simplified form ofcLXj1

Xj2(ωXj1,ωXj2

) in Eq. (243), we obtain the linear response function

X j1; X j2

��ωXj2

=12

C±ωPXj1

Xj2ωXj1

ωXj2

(−

µk

t(0)µksXj1µk

(ωXj1)νm

sXj2νm

(ωXj2)⟨HF|HT0

0 |νCCm ⟩

+νm

sXj2νm

(ωXj2)⟨HF| XT0

j1, |νCC

m ⟩⟨HF| |HF⟩ +µkνm

t(0)µksXj2νm

(ωXj2)⟨µCC

k | XT0j1, |νCC

m ⟩⟨HF| |HF⟩), ωXj1

+ ωX j2= 0,

(340)

where t(0)µk, t(0)µk

, and sXj2µk

(ωXj2) are obtained by solving Eqs. (325), (327), and (338), respectively.



,ωXj2), we obtain cLXj1

Xj2Xj3(ωXj1

,ωXj2,ωXj3

) as

cLXj1Xj2

Xj3(ωXj1,ωXj2

,ωXj3)

= PXj1

Xj2Xj3

ωXj1ωXj2

ωXj3

12

µkνm

sXj1µk

(ωXj1)sXj2

Xj3νm (ωXj2

,ωXj3)(

CCJµkνm −(ωX j2

+ ωXj3

)δµkνm

)

+12

µkνm

sXj1

Xj2µk

(ωXj1,ωXj2

)sXj3νm (ωXj3

)(


)

− 12

µk

t(0)µksXj1µk

(ωXj1)νm

sXj2

Xj3νm (ωXj2

,ωXj3)⟨HF|HT0

0 |νCCm ⟩

− 12

µk

t(0)µksXj1

Xj2µk

(ωXj1,ωXj2

)νm

sXj3νm (ωXj3

)⟨HF|HT00 |νCC

m ⟩

−µk

sXj1µk

(ωXj1)sXj2

µk(ωXj2

)νm

sXj3νm (ωXj3

)⟨HF|HT00 |νCC

m ⟩

+12

νm

sXj2

Xj3νm (ωXj2

,ωXj3)⟨HF| XT0

j1, |νCC

m ⟩⟨HF| |HF⟩

+12

µk

sXj2

Xj3µk

(ωXj2,ωXj3

)⟨µCCk |XT0

j1|HF⟩

+12

µkνm

t(0)µksXj2

Xj3νm (ωXj2

,ωXj3)⟨µCC

k | XT0j1, |νCC

m ⟩⟨HF| |HF⟩

+µkνm

sXj2µk

(ωXj2)sXj3

νm (ωXj3)⟨µCC

k | XT0j1, |νCC

m ⟩⟨HF| |HF⟩

−µk

t(0)µksXj2µk

(ωXj2)νm

sXj3νm (ωXj3

)⟨HF|XT0j1|νCC

m ⟩, ωXj1

+ ωX j2+ ωXj3

= 0. (341)

Differentiating cLXj1Xj2

Xj3(ωXj1,ωXj2

,ωXj3) with respect to s

Xj3µk

(ωXj3) and using the stationary condition of Eq. (246) gives

the response equation for the second-order coefficients,


νm

(CCJµkνm −

(ωXj1

+ ωXj2

)δµkνm

)sXj1

Xj2νm (ωXj1

,ωXj2) = −P

Xj1Xj2

ωXj1ωXj2

(− s

Xj1µk

(ωXj1)νm

⟨HF|HT00 |νCC

m ⟩sXj2νm (ωXj2

)

+νm

⟨µCCk | XT0

j1, |νCC

m ⟩⟨HF| |HF⟩sXj2νm (ωXj2

)), (342)

from which sXj1

Xj2νm (ωXj1

,ωX j2), needed in Subsection VII E for the fourth-order properties, may be calculated.


X j1; X j2,X j3

��ωXj2

,ωXj3=

12

C±ωPXj1

Xj2Xj3

ωXj1ωXj2

ωXj3

−

µk

sXj1µk

(ωXj1)sXj2

µk(ωXj2

)νm

sXj3νm (ωXj3

)⟨HF|HT00 |νCC

m ⟩

+µkνm

sXj2µk

(ωXj2)sXj3

νm (ωXj3)⟨µCC

k | XT0j1, |νCC

m ⟩⟨HF| |HF⟩

−µk

t(0)µksXj2µk

(ωXj2)νm

sXj3νm (ωXj3

)⟨HF|XT0j1|νCC

m ⟩, ωXj1

+ ωX j2+ ωXj3

= 0, (343)


, sXj2µk

(ωXj2), and s

Xj2µk

(ωXj2) are obtained by solving Eqs. (325), (327), (338), and (339), respectively.


Using the 2n + 1 and 2n + 2 rules, fourth-order molecular properties may be expressed in terms of the first- and second-ordercoefficients and first-order multipliers as


��ωXj2

,ωXj3,ωXj4

=12

C±ωPXj1

Xj2Xj3

Xj4ωXj1

ωXj2ωXj3

ωXj4

− 1

4

µk

t(0)µksXj1

Xj2µk

(ωXj1,ωXj2

)νm

sXj3

Xj4νm (ωXj3

,ωXj4)⟨HF|HT0

0 |νCCm ⟩

− 12

µk

sXj1µk

(ωXj1)sXj2

µk(ωXj2

)νm

sXj3

Xj4νm (ωXj3

,ωXj4)⟨HF|HT0

0 |νCCm ⟩

− 12

µk

sXj1µk

(ωXj1)sXj2

Xj3µk

(ωXj2,ωXj3

)νm

sXj4νm (ωXj4

)⟨HF|HT00 |νCC

m ⟩

+12

µkνm

sXj2µk

(ωXj2)sXj3

Xj4νm (ωXj3

,ωXj4)⟨µCC

k | XT0j1, |νCC

m ⟩⟨HF| |HF⟩

− 12

µk

t(0)µksXj2µk

(ωXj2)νm

sXj3

Xj4νm (ωXj3

,ωXj4)⟨HF|XT0

j1|νCC

m ⟩

− 12

µk

t(0)µksXj2

Xj3µk

(ωXj2,ωXj3

)νm

sXj4νm (ωXj4

)⟨HF|XT0j1|νCC

m ⟩

−µk

sXj2µk

(ωXj2)sXj3

µk(ωXj3

)νm

sXj4νm (ωXj4

)⟨HF|XT0j1|νCC

m ⟩, ωXj1

+ ωX j2+ ωXj3

+ ωXj4= 0, (344)


, sXj1µk

(ωXj1), s

Xj1µk

(ωXj1), and s

Xj2Xj3

µk(ωXj2

,ωXj2)

are obtained by solving Eqs. (325), (327), (338), (339), and(342), respectively.

F. The CC Jacobian as a part of the CC-CI eigenvalueequation

We now consider a matrix representation of the eigenvalueequation for the unperturbed Hamiltonian in the basis of

Eq. (332) in which the CC-CI time-dependent wave functionis expanded [Eq. (331)]. We then express the Jacobian matrixand its eigenvalue equation in this basis.

1. The CC-CI eigenvalue equation

The time development of the CC-CI wave function inEq. (331) is described in terms of a linear expansion in the basis|BT0⟩ of Eq. (332). We now determine a matrix representation


of the unperturbed Hamiltonian H0 considering |BT0⟩ as theright basis and using its biorthonormal counterpart

⟨BT0| = ⟨HF| exp(−T0),⟨µCCk | exp(−T0)

(345)

as the left basis. In this biorthonormal representation, theeigenvalue equation for the unperturbed Hamiltonian becomes

CCHT00

CCCR = CCCR CCE, (346)

CCCL CCHT00 =

CCE CCCL, (347)

whereCCCL CCCR = I. (348)

The unperturbed Hamiltonian matrix, CCHT00 , has the block

structure

CCHT00 =

*,

CCE0 η

0(

CCHT00

)⊥+-, (349)

where(

CCHT00

)⊥denotes the sub-block of the Hamiltonian

matrix in the orthogonal complement subspace(CCHT0

0

)⊥µkνm

=

CCHT00

µkνm

= ⟨µCCk | exp(−T0)H0 exp(T0)|νCC

m ⟩

= ⟨µCCk |HT0

0 |νCCm ⟩, (350)

where we have used Eq. (191). To obtain the block structurein Eq. (349), we have used Eqs. (324), (325), and (329),

CCHT0

0

HF HF

= ⟨HF| exp(−T0)H0 exp(T0)|HF⟩

= ⟨HF|HT00 |HF⟩ = CCE0, (351)

CCHT0

0

µk HF

= ⟨µCCk | exp(−T0)H0 exp(T0)|HF⟩

= ⟨µCCk |HT0

0 |HF⟩ = 0, (352)

CCHT00

HF µk

= ⟨HF| exp(−T0)H0 exp(T0)|µCCk ⟩

= ⟨HF|HT00 |µCC

k ⟩ = ηµk. (353)

The CCE matrix is diagonal and contains the energies of theunperturbed Hamiltonian H0,

CCE =

*......,

CCE0CCE1

CCE2

. . .

+//////-

. (354)

Equations (350)–(353) imply that Eq. (349) may beviewed as a matrix representation of the similarity-transformedHamiltonian, HT0

0 [Eq. (191)], in the basis |B⟩ [Eq. (323)].As the similarity-transformed Hamiltonian, HT0

0 , is non-Hermitian, so is its matrix representation and therefore theleft and right eigenvectors, contained in CCCL [Eq. (346)] andCCCR [Eq. (347)] matrices, respectively, are not adjoints ofeach other. To obtain explicit expressions for the left and right

eigenvectors, we use the block structure of CCHT00 [Eq. (349)]

to write the CCCR and CCCL matrices as

CCCR = *,

CCD0CCD

CCC0CCC

+-

(355)

and

CCCL = *,

CCD0CCC0

CCD CCC+-. (356)

Inserting Eqs. (355) and (356) in Eqs. (346)–(348), we obtain,after some algebra (see, for example, Ref. 88), the ground stateeigenvectors as

CCCR0 =

*,

10+-, (357)

CCCL0 =

(1 t(0)

), (358)

whereas the excited state eigenvectors for the energies CIEn,n = 1,2, . . ., become

CCCRn =

*,

−t(0) CCCn

CCCn

+-, (359)

CCCLn =

(0 CCCn

), (360)

where t(0) is obtained from Eq. (327). Further, CCCn and CCCn

are the right and left eigenvectors of(

CCHT00

)⊥,(

CCHT00

)⊥ CCC = CCC CCE⊥, (361)

CCC(

CCHT00

)⊥= CCE⊥ CCC, (362)

whereCCC CCC = I (363)

and CCE⊥ is the sub-block of CCE [Eq. (354)] in the orthogonalcomplement subspace.

Using Eqs. (357)–(360), we may express the right and lefteigenstates of the standard Hamiltonian H0 (without similaritytransformation) in the bracket form as

|0CCR ⟩ =

|0CC

0 ⟩, |0CC1 ⟩, |0CC

2 ⟩, · · ·, (364)

⟨0CCL | =

⟨0CC

0 |,⟨0CC1 |,⟨0CC

2 |, · · ·, (365)

where

|0CC0 ⟩ = exp(T0)|HF⟩, (366)

⟨0CC0 | = ⟨HF| exp(−T0) +

µk

t(0)µk⟨µCC

k | exp(−T0), (367)

and, for n , 0,

|0CCn ⟩ = −

µk

t(0)µk

CCCµkn exp(T0)|HF⟩

+µk

CCCµkn exp(T0)|µCCk ⟩, (368)


⟨0CCn | =

µk

CC Cnµk⟨µCC

k | exp(−T0), (369)

and where CCCµkn are elements of the CCCn vector [Eq. (359)],whereas CC Cnµk

are elements of CCCn [Eq. (360)].The eigenvalue equations defined by Eqs. (346)–(354)

are often referred to as the EOMCC eigenvalue equation88 andform the foundation for the EOMCC model, which will bediscussed in Sec. IX.

2. The CC Jacobian eigenvalue equation

We now consider the CC-CI Jacobian or, equivalently, theCC Jacobian [see Eq. (337)]. Using Eqs. (328) and (350), wemay express the CC Jacobian matrix as

CCJ =(

CCHT00

)⊥− CCE0I. (370)

Using Eq. (370), we may write Eqs. (361) and (362) as

CCJ CCC = CCC ∆CCE, (371)

CCC CCJ = ∆CCE CCC, (372)

where CCC and CCC are orthonormal [Eq. (363)] and where

∆CCE = CCE⊥ − CCE0I

=

*......,

CCE1 0CCE2 0

CCE3 0

. . .

+//////-

, (373)

with

CCEn0 =CCEn − CCE0. (374)

Equations (371)–(374) show that the eigenvalues of CCJ areequal to the excitation energies of the unperturbed system.

Comparing Eqs. (371) and (372) with Eqs. (361) and(362), we see that the eigenvectors of the CC Jacobian, CCJ,are equal to the eigenvectors of the CC Hamiltonian matrix,(

CCHT00

)⊥, in Eq. (349). The excited states are thus linearly

parametrized in the biorthonormal basis |BT0⟩ [Eq. (332)] and⟨BT0| [Eq. (345)] as is the time evolution of the CC groundstate [Eq. (331)]. As there is no interaction between the groundstate and the orthogonal complement set of states, neither forthe unperturbed system nor for the time-evolving system, wetherefore have an explicit representation of the excited statesin CC-CI response theory.

VIII. MOLECULAR PROPERTIES FOR THE CC MODEL

A. Parametrization of the CC model

In the CC model, the unperturbed ground state, exp(T0)|HF⟩,satisfies Eqs. (323)–(327), and the time evolution is expo-

nentially parametrized in terms of the time-dependent clusteroperator

T(t) =µk

tµk(t)τµk

, (375)

where tµk(t) are time-dependent cluster amplitudes. The CC

wave function |0CCI ⟩ [Eq. (253)] becomes

|0CCI ⟩ = exp(T0) exp(T(t))|HF⟩= exp(T0)|HF⟩ + exp(T0)T(t)|HF⟩

+12

exp(T0)T2(t)|HF⟩ + · · ·


tµk(t) exp(T0)|µCC

k ⟩

+12

µkνm

tµk(t)tνm(t) exp(T0)|µkνCC

m ⟩ + · · ·. (376)

Equation (376) shows that for a complete basis�|HF⟩, |µCC

k⟩

[Eq. (323)], the wave function |0CCI ⟩ describes a non-linear

parametrization of the time evolution in this basis. If atruncation is carried out in the basis

�|HF⟩, |µCCk⟩, then the

time evolution in the CC model is expanded in a basis that islarger than the one of the CC-CI model [Eqs. (331) and (332)]which contains only a linearized form of Eq. (376).

We now follow the same route as in Sec. VII to deriveexplicit expressions for the CC molecular response functionsthrough fourth order and use the relabeling |R⟩ = |HF⟩, |k⟩= |µCC

k⟩, B0 = T0, bk = tµk

, and bk = tµkin cLH [Eq. (273)]

and cLX j1[Eq. (272)].


From Eq. (273), we see that cL(0) is equal to the standardground-state CC energy,

cL(0) = cL(0)H =

cL(0)H0= CCE0 = ⟨HF|HT0

0 |HF⟩, (377)

since the amplitudes tµk(ωK) do not contain zero-order

contributions.The first-order complex quasi-energy Lagrangian is

obtained using Eqs. (272) and (273) as

cLXj1(0) = ⟨HF|XT0j1|HF⟩ +

µk

t(0)µk⟨µCC

k |XT0j1|HF⟩, (378)

which, inserted in Eq. (243), gives the first-order molecularproperties as

X j1

��0 =

12

C±ω *.,⟨HF|XT0

j1|HF⟩

+µk

t(0)µk⟨µCC

k |XT0j1|HF⟩+/

-, (379)

where the zero-order amplitudes, t(0)µk, and the zero-order

multipliers, t(0)µk, are obtained by solving Eqs. (325) and (327),

respectively.



The second-order complex quasi-energy Lagrangian is obtained using Eqs. (272) and (273),

cLXj1Xj2(ωXj1

,ωXj2) = P

Xj1Xj2

ωXj1ωXj2

µkνm

tXj1µk

(ωXj1)tXj2νm (ωX j2

)(

CCJµkνm − ωX j2δµkνm

)

+12

νmλn

tXj1νm (ωXj1

)tXj2λn

(ωX j2)⟨HF| HT0

0 , τνm, τλn

|HF⟩

+12

µkνmλn

t(0)µktXj1νm (ωXj1

)tXj2λn

(ωX j2)⟨µCC

k | HT00 , τνm

, τλn

|HF⟩

+νm

tXj2νm (ωX j2

)⟨HF| XT0j1, τνm

|HF⟩

+µk

tXj2µk

(ωX j2)⟨µCC

k |XT0j1|HF⟩ +

µkνm

t(0)µktXj2νm (ωX j2

)⟨µCCk | XT0

j1, τνm

|HF⟩, ωXj1

+ ωX j2= 0,

(380)where the CC Jacobian, CCJµkνm, has been given in Eq. (328).

Applying the stationary condition of Eq. (246) to cLXj1Xj2(ωXj1

,ωXj2) of Eq. (380) gives the first-order amplitudes

equation

νm


δµkνm

)tXj1νm (ωXj1

) = −⟨µCCk |XT0

j1|HF⟩. (381)

Similarly, applying the stationary condition of Eq. (245) gives the first-order multipliers equation

νm

tXj1νm (ωXj1

)(

CCJνmµk+ ωXj1

δνmµk

)= −

νm

tXj1νm (ωXj1

)⟨HF| HT00 , τνm

, τµk

|HF⟩

−νmλn

t(0)νmtXj1λn

(ωXj1)⟨νCC

m | HT00 , τλn

, τµk

|HF⟩ − ⟨HF| XT0j1, τµk

|HF⟩

−νm

t(0)νm⟨νCC

m | XT0j1, τµk

|HF⟩, (382)

where for the left-hand side, we have used the condition ωXj1+ ωX j2

= 0.Simplifying cLXj1

Xj2(ωXj1,ωXj2

) [Eq. (380)] using the 2n + 1 and 2n + 2 rules77 and inserting the simplified form ofcLXj1

Xj2(ωXj1,ωXj2

) in Eq. (243), we obtain the linear response function

X j1; X j2

��ωXj2=

12

C±ωPXj1

Xj2ωXj1

ωXj2

12

νmλn

tXj1νm (ωXj1

)tXj2λn

(ωXj2)⟨HF| HT0

0 , τνm, τλn

|HF⟩

+12

µkνmλn


)tXj2λn

(ωXj2)⟨µCC

k | HT00 , τνm

, τλn

|HF⟩

+νm

tXj2νm (ωXj2

)⟨HF| XT0j1, τνm

|HF⟩ +µkνm


)⟨µCCk | XT0

j1, τνm

|HF⟩, ωXj1

+ ωX j2= 0,

(383)


, and tXj2µk

(ωXj2) are obtained by solving Eqs. (325), (327), and (381), respectively.




,ωXj2), we obtain cLXj1

Xj2Xj3(ωXj1

,ωXj2,ωXj3

) as

cLXj1Xj2

Xj3(ωXj1,ωXj2

,ωXj3)

= PXj1

Xj2Xj3

ωXj1ωXj2

ωXj3

12

µkνm

tXj1µk

(ωXj1)tXj2

Xj3νm (ωX j2

,ωXj3)(

CCJµkνm −(ωX j2

+ ωXj3

)δµkνm

)

+12

µkνm

tXj1

Xj2µk

(ωXj1,ωX j2

)tXj3νm (ωXj3

)(


)

+12

νmλn

tXj1νm (ωXj1

)tXj2Xj3

λn(ωX j2

,ωXj3)⟨HF| HT0

0 , τνm, τλn

|HF⟩

+12

µkνmλn


)tXj2Xj3

λn(ωX j2

,ωXj3)⟨µCC

k | HT00 , τνm

, τλn

|HF⟩

+12

µkνmλn

tXj1µk


)tXj3λn

(ωXj3)⟨µCC

k | HT00 , τνm

, τλn

|HF⟩

+16

µkνmλnσp


)tXj2λn

(ωX j2)tXj3σp (ωXj3

)⟨µCCk | HT0

0 , τνm, τλn

, τσp

|HF⟩

+12

νm

tXj2

Xj3νm (ωX j2

,ωXj3)⟨HF| XT0

j1, τνm

|HF⟩ + 12

µk

tXj2

Xj3µk

(ωX j2,ωXj3

)⟨µCCk |XT0

j1|HF⟩

+12

µkνm

t(0)µktXj2

Xj3νm (ωX j2

,ωXj3)⟨µCC

k | XT0j1, τνm

|HF⟩ +µkνm

tXj2µk

(ωX j2)tXj3νm (ωXj3

)⟨µCCk | XT0

j1, τνm

|HF⟩

+12

µkνmλn


)tXj3λn

(ωXj3)⟨µCC

k | XT0j1, τνm

, τλn

|HF⟩, ωXj1

+ ωX j2+ ωXj3

= 0. (384)

Using the stationary condition of Eq. (246), we obtain the second-order amplitudes equation,νm

(CCJµkνm −

(ωXj1

+ ωXj2

)δµkνm

)tXj1

Xj2νm (ωXj1

,ωXj2)

= −νmλn

⟨µCCk | HT0

0 , τνm, τλn

|HF⟩tXj1νm (ωXj1

)tXj2λn

(ωX j2)

− PXj1

Xj2ωXj1

ωXj2

νm

⟨µCCk | XT0

j1, τνm

|HF⟩tXj2νm (ωX j2

), (385)

from which tXj1

Xj2νm (ωXj1

,ωX j2), needed in Subsection VIII E for fourth-order properties, may be calculated.


X j1; X j2,X j3

��ωXj2

,ωXj3

=12

C±ωPXj1

Xj2Xj3

ωXj1ωXj2

ωXj3

12

µkνmλn

tXj1µk


)tXj3λn

(ωXj3)⟨µCC

k | HT00 , τνm

, τλn

|HF⟩

+16

µkνmλnσp


)tXj2λn


)⟨µCCk | HT0

0 , τνm, τλn

, τσp

|HF⟩

+µkνm

tXj2µk


)⟨µCCk | XT0

j1, τνm

|HF⟩

+12

µkνmλn


)tXj3λn

(ωXj3)⟨µCC

k | XT0j1, τνm

, τλn

|HF⟩

ωXj1+ ωX j2

+ ωXj3= 0, (386)



, tXj2µk

(ωXj2), and t

Xj2µk

(ωXj2) are obtained by solving Eqs. (325), (327), (381), and (382), respectively.


Using the 2n + 1 and 2n + 2 rules, we obtain fourth-order molecular properties in terms of the first- and second-orderamplitudes and first-order multipliers as


��ωXj2

,ωXj3,ωXj4

=12

C±ωPXj1

Xj2Xj3

Xj4ωXj1

ωXj2ωXj3

ωXj4

(18

νmλn

tXj1

Xj2νm (ωXj1

,ωX j2)tXj3

Xj4λn

(ωXj3,ωXj4

)⟨HF| HT00 , τνm

, τλn

|HF⟩

+18

µkνmλn

t(0)µktXj1

Xj2νm (ωXj1

,ωX j2)tXj3

Xj4λn

(ωXj3,ωXj4

)⟨µCCk | HT0

0 , τνm, τλn

|HF⟩

+12

µkνmλn

tXj1µk


)tXj3Xj4

λn(ωXj3

,ωXj4)⟨µCC

k | HT00 , τνm

, τλn

|HF⟩

+14

µkνmλnσp


)tXj2λn

(ωX j2)tXj3

Xj4σp (ωXj3

,ωXj4)⟨µCC

k | HT00 , τνm

, τλn

, τσp

|HF⟩

+16

µkνmλnσp

tXj1µk


)tXj3λn

(ωXj3)tXj4σp (ωXj4

)⟨µCCk | HT0

0 , τνm, τλn

, τσp

|HF⟩

+124

µkνmλnσpρq


)tXj2λn


)tXj4ρq (ωXj4

)⟨µCCk | HT0

0 , τνm, τλn

, τσp

, τρq

|HF⟩

+12

µkνm

tXj2µk

(ωX j2)tXj3

Xj4νm (ωXj3

,ωXj4)⟨µCC

k | XT0j1, τνm

|HF⟩

+12

µkνmλn


)tXj3Xj4

λn(ωXj3

,ωXj4)⟨µCC

k | XT0j1, τνm

, τλn

|HF⟩

+12

µkνmλn

tXj2µk


)tXj4λn

(ωXj4)⟨µCC

k | XT0j1, τνm

, τλn

|HF⟩),ωXj1

+ ωX j2+ ωXj3

+ ωXj4= 0, (387)


, tXj2µk

(ωX j2), t

Xj2µk

(ωX j2), and t

Xj2Xj3

µk(ωX j2

,ωXj3)

are obtained by solving Eqs. (325), (327), (381), (382), and(385), respectively.

The detailed formulas for the second-, third-, andfourth-order molecular properties in Eqs. (383), (386), and(387), respectively, only include terms where all indices areconnected and all matrix elements are fully expressed ascommutators. Such an expansion is called connected and issize-extensive also in the more restricted meaning of this termadvocated by Bartlett.86

F. The CC Jacobian

The CC and CC-CI Jacobians are equal [Eq. (337)].However, in the CC model, where the time evolution isexponentially parametrized, the time-evolving state cannotbe expanded in the time-independent complete basis|HF⟩, |µCC

k⟩, as was the case for the linear expansion of

the time evolution in the CC-CI model, and we, therefore,

cannot obtain an explicit representation for the excited states.However, it is not important that the wave function for theindividual states cannot be given an explicit representation,because expressions for experimental observables may beidentified from the response functions and their residueswithout a recourse to an explicit representation of the states.

IX. MOLECULAR PROPERTIESFOR THE EOMCC MODEL

EOMCC response functions were introduced by Stantonand Bartlett78 and Rozyczko and Bartlett79–81 and weredefined in terms of CI response functions, where the CIenergies of Eq. (310) and states of Eqs. (315) and (316)were replaced by the CC energies of Eq. (354) and states ofEqs. (364) and (365). The replacement must be performedfor CI response functions that are derived by solving theHermitian eigenvalue equation via projection, as in Sec. VI,in order to get the operator 1

2C±ω introduced in the EOMCCresponse functions.


In the following, we derive the first-, second-, and third-order molecular property expressions for the EOMCC model.First-order properties are straightforwardly obtained fromEq. (293) as

X j1

��EOMCC0 =

12

C±ω⟨0CC0 |X j1|0CC

0 ⟩, (388)

where the |0CC0 ⟩ and ⟨0CC

0 | are given in Eqs. (366) and (367),respectively, and therefore,

EOMCCb(0)µk= t(0)µk

. (389)

The CI second- and third-order molecular propertyexpressions may be expressed in the diagonal representationas

X j1; X j2

��CIωXj2=

12

C±ωPXj1

Xj2ωXj1

ωXj2

k

⟨0CI0 |X j1|0CI

k ⟩

× CIsXj2k

(ωX j2), ωXj1

+ ωX j2= 0, (390)

X j1; X j2,X j3

��CIωXj2

,ωXj3

=12

C±ωPXj1

Xj2Xj3

ωXj1ωXj2

ωXj3

km

CIsXj2k

(ωX j2)CIs

Xj3m (ωXj3

)

×(⟨0CI

k |X j1|0CIm⟩ − ⟨0CI

0 |X j1|0CI0 ⟩δkm

),

ωXj1+ ωX j2

+ ωXj3= 0, (391)

where CIsXj3k

(ωXj3) and CIs

Xj2k

(ωXj2) denote the coefficients

and multipliers in the basis of the eigenstates of H0,

CIsXj1n (ωXj1

) =µk

CC Cnµk

CIsXj1µk

(ωXj1), (392)

CIsXj1n (ωXj1

) =µk

CIsXj1µk

(ωXj1)CCCµkn (393)

and may be obtained from the first-order response equations(CIEk0 − ωXj1

)CIs

Xj1k

(ωXj1) = −⟨0CI

k |X j1|0CI0 ⟩, (394)

CIsXj1k

(ωXj1) (CIEk0 + ωXj1

)= −⟨0CI

0 |X j1|0CIk ⟩. (395)

The EOMCC second- and third-order molecular prop-erties are obtained by replacing in Eqs. (390) and (391)and Eqs. (394) and (395) the CI eigenstates and eigenvaluesof Eqs. (315), (316), and (321) by the CC eigenstates andeigenvalues of Eqs. (364), (365), and (373),

X j1; X j2

��EOMCCωXj2

=12

C±ωPXj1

Xj2ωXj1

ωXj2

k

⟨0CC0 |X j1|0CC

k ⟩ EOMCCbXj2k

(ωX j2),

ωXj1+ ωX j2

= 0, (396)

X j1; X j2,X j3

��EOMCCωXj2

,ωXj3

=12

C±ωPXj1

Xj2Xj3

ωXj1ωXj2

ωXj3

km

EOMCCbXj2k

(ωX j2)

×(⟨0CC

k |X j1|0CCm ⟩ − ⟨0CC

0 |X j1|0CC0 ⟩δkm

)EOMCCb

Xj3m (ωXj3

),

ωXj1+ ωX j2

+ ωXj3= 0, (397)

where EOMCCbXj3k

(ωXj3) and EOMCCb

Xj2k

(ωXj2) are the solu-

tions to the first-order EOMCC right and left responseequations in the diagonal representation,(

CCEk0 − ωXj1

)EOMCCb

Xj1k

(ωXj1) = −⟨0CC

k |X j1|0CC0 ⟩, (398)

EOMCCbXj1k

(ωXj1) (CCEk0 + ωXj1

)= −⟨0CC

0 |X j1|0CCk ⟩. (399)

The response functions may be obtained in the non-diagonal representation by transforming Eqs. (396) and (397)back to the basis of Eqs. (332) and (345), using Eqs. (363) and(366)–(369). The back transformation may be expressed as

EOMCCbXj1µk

(ωXj1) =

n

CCCµknEOMCCb

Xj1n (ωXj1

), (400)

EOMCCbXj1µk

(ωXj1) =

n

EOMCCbXj1n (ωXj1

)CC Cnµk(401)

and gives the first-order amplitude equations [Eq. (398)] inthe non-diagonal representation as

νm


δµkνm

)EOMCCb

Xj1νm (ωXj1

)

= −⟨µCCk |XT0

j1|HF⟩, (402)

which is identical to Eqs. (338) and (381), and therefore,the first-order wave function coefficients in the CC-CI andEOMCC models and the first-order amplitudes in the CCmodel are equal,

EOMCCbXj1µk

(ωXj1) = CC−CIs

Xj1µk

(ωXj1) = t

Xj1µk

(ωXj1). (403)

For the first-order Lagrangian multipliers of Eq. (399), theback transformation in Eq. (401) gives

µk

EOMCCbXj1µk

(ωXj1)(

CCJµkνm + ωXj1δµkνm

)

= −⟨HF|XT0j1|νCC

m ⟩ + t(0)νm⟨HF|XT0

j1|HF⟩

−µk

t(0)µk⟨µCC

k |XT0j1|νCC

m ⟩

+ t(0)νm

µk

t(0)µk⟨µCC

k |XT0j1|HF⟩ (404)

which is clearly different from Eqs. (339) and (382),

EOMCCbXj1µk

(ωXj1) , CC−CIs

Xj1µk

(ωXj1), (405)


EOMCCbXj1µk

(ωXj1) , t

Xj1µk

(ωXj1). (406)

The EOMCC second- and third-order molecular property expressions become

X j1; X j2

��EOMCCωXj2

=12

C±ωPXj1

Xj2ωXj1

ωXj2

(νm

EOMCCbXj2νm (ωXj2

)⟨HF|XT0j1|νCC

m ⟩

−(νm

t(0)νmEOMCCb

Xj2νm (ωXj2

))⟨HF|XT0j1|HF⟩ +

µkνm

t(0)µk

EOMCCbXj2νm (ωXj2

)⟨µCCk |XT0

j1|νCC

m ⟩

−νm

t(0)νmEOMCCb

Xj2νm (ωXj2

)µk

t(0)µk⟨µCC

k |XT0j1|HF⟩

), ωXj1

+ ωX j2= 0, (407)

X j1; X j2,X j3

��EOMCCωXj2

,ωXj3=

12

C±ωPXj1

Xj2Xj3

ωXj1ωXj2

ωXj3

( µkνm

EOMCCbXj2µk

(ωXj2) EOMCCb

Xj3νm (ωXj3

)⟨µCCk |XT0

j1|νCC

m ⟩

−(νm

t(0)νmEOMCCb

Xj3νm (ωXj3

)) µk

EOMCCbXj2µk

(ωXj2)⟨µCC

k |XT0j1|HF⟩

−(νm

EOMCCbXj2νm

(ωXj2) EOMCCb

Xj3νm (ωXj3

))⟨HF|XT0j1|HF⟩

−(νm

EOMCCbXj2νm

(ωXj2) EOMCCb

Xj3νm (ωXj3

)) µk

t(0)µk⟨µCC

k |XT0j1|HF⟩

),

ωXj1+ ωX j2

+ ωXj3= 0, (408)

where t(0)µkand t(0)µk

are obtained by solving Eqs. (325) and

(327), respectively, while EOMCCbXj3µk

(ωXj3) and EOMCCb

Xj2µk(ωXj2

) are obtained from Eqs. (402) and (404).

X. COMPARISON OF THE CI, CC-CI, CC, AND EOMCCMOLECULAR RESPONSE PROPERTIES

In Sec. X A, we discuss how static molecular propertiesmay be obtained from finite field energy calculations. Inparticular, we describe how the finite field method can begeneralized to be applicable for the CC-CI model. In Sec.X B, the size-extensivity of molecular properties is discussedfor the CI, CC-CI, and CC models. In Sec. X C, the focusis on the different characteristics and properties of responsefunctions when a linear versus an exponential parametrizationis considered for the unperturbed state and its time evolution.In Sec. X D, a comparison is performed of CC-CI andEOMCC response functions.

A. Static molecular properties from finite field energycalculations in the CI, CC-CI and CC models

We have seen in Sec. II D that in the limit of a time-independent perturbation, where the frequencies in Eq. (3) arezero, the perturbed Hamiltonian becomes time-independent

H(ϵ) = H0 + V (ϵ). (409)

The quasi-energy, E(ϵ), then becomes the energy, E0(ϵ)[Eq. (50)], of the eigenvalue equation for H(ϵ) [Eq. (49)],

E(ϵ) −→ E0(ϵ), (410)

and the regular wave function, |0R(t, ϵ)⟩, becomes the eigen-state |00(ϵ)⟩,

|0R(t, ϵ)⟩ −→ |00(ϵ)⟩. (411)

For any wave function model, in which a time-dependentphase is isolated giving the regular wave function [see Sec.II B], the perturbed energy, E0(ϵ), may therefore be usedin finite field energy calculations to obtain static molecularproperties that are equal to static limit response functionresults.

1. Standard finite field energy calculations of staticmolecular properties

The standard way of performing finite field energycalculations for determination of static molecular propertiesis to include the static perturbation operator in the time-independent Hamiltonian H(ϵ) [Eq. (409)] and optimize theenergy for this perturbed Hamiltonian. The optimization may,for example, be done using the variational principle [Eq. (57)]

⟨δ00(ϵ)|H(ϵ)|00(ϵ)⟩ + ⟨00(ϵ)|H(ϵ)|δ00(ϵ)⟩ = 0 (412)


and the optimized perturbed energy E0(ϵ) [Eq. (56)] thenbecomes

E0(ϵ) = ⟨00(ϵ)|H(ϵ)|00(ϵ)⟩. (413)

Differentiation of E0(ϵ) with respect to the perturbationstrengths, ϵ , then yields static molecular properties. When thefinite field method defined by Eqs. (412) and (413) is usedto determine static molecular properties, we obtain propertiesthat are equal to the static limit response function propertiesof Sec. III, as in Sec. III, we have assumed that both theunperturbed state and its time evolution are determined usingthe variational principle.

The standard finite field energy method may, of course,also be applied when the variational principle is replacedby projection. This is, for example, done in CC theory[Sec. VIII] where projection is used to obtain an optimizedCC wave function both for the unperturbed and the perturbedsystem.

In this article, we have considered more general wavefunction models, where the unperturbed state and its timeevolution are determined using different parametrizations andwhere different optimization conditions have been appliedfor the unperturbed and perturbed system. To obtain staticmolecular properties from finite field calculations for suchmodels, the above described finite field method has to begeneralized. In Sec. X A 2, we describe a finite field approachfor obtaining static limit CC-CI response function results, andin Sec. X A 3, we show that these static limit properties mayalso be obtained from standard finite field EOMCC energycalculations.

2. Finite field energy calculations for the CC-CI model

We now describe a finite field energy method fordetermining static CC-CI molecular properties. In the staticlimit, the CC-CI intermediate-normalized wave function[Eq. (331)] becomes

|0CC−CII (ϵ)⟩ = exp(T0) exp(S(ϵ))|HF⟩


sµk(ϵ) exp(T0)|µCC

k ⟩, (414)

where the cluster amplitudes, t(0)µk, entering the operator T0, are

determined from the CC amplitudes equations [Eq. (325)].Within the CC-CI framework, the energy of the CC-CI wavefunction in Eq. (414) is obtained as the static limit equivalentof Eq. (206a), whereas the amplitudes sµk

(ϵ) are obtainedfrom the static limit equivalent of Eq. (206c).

The CC-CI perturbed energy thus may be obtained fromthe quasi-energy expression in Eq. (206a) as

CC−CIE(ϵ) = ⟨HF| exp(−S(ϵ))(HT00 + VT0(ϵ)) exp(S(ϵ))|HF⟩

= ⟨HF|(1 − S(ϵ)) (HT00 + VT0(ϵ)) (1 + S(ϵ)) |HF⟩

= CCE0 +µk

⟨HF|HT00 |µCC

k ⟩sµk(ϵ) + ⟨HF|VT0(ϵ)

× |HF⟩ +µk

⟨HF|VT0(ϵ)|µCCk ⟩sµk

(ϵ), (415)

where the set of perturbed amplitudes, sµk(ϵ), is determined

from the static limit of the amplitude equations in Eq. (206c),µCCk�� exp(−S(ϵ))(HT0

0 + VT0(ϵ)) exp(S(ϵ))��HF= 0. (416)

To simplify the perturbed amplitude equations, consider thefirst term on the left-hand side of Eq. (416),

⟨µCCk | exp(−S(ϵ))HT0

0 exp(S(ϵ))|HF⟩= ⟨µCC

k |(1 − S(ϵ))HT00

(1 + S(ϵ)) |HF⟩

= ⟨µCCk |HT0

0 |HF⟩+

νm

(⟨µCCk |HT0

0 |νCCm ⟩ − δµkνm⟨HF|HT0

0 |HF⟩)sνm(ϵ)

− sµk(ϵ)

νm

⟨HF|HT00 |νCC

m ⟩sνm(ϵ)

=νm

CCJµkνmsνm(ϵ) − sµk(ϵ)

νm

⟨HF|HT00 |νCC

m ⟩sνm(ϵ),

(417)

where to obtain the last equality we have used the definitionof the CC Jacobian [Eq. (328)]. Using Eq. (417), we maywrite Eq. (416) asνm

CCJµkνmsνm(ϵ) − sµk(ϵ)

νm

⟨HF|HT00 |νCC

m ⟩sνm(ϵ)

+νm

(⟨µCCk |VT0(ϵ)|νCC

m ⟩ − δµkνm⟨HF|VT0(ϵ)|HF⟩)sνm(ϵ)

− sµk(ϵ)

νm

⟨HF|VT0(ϵ)|νCCm ⟩sνm(ϵ)

+ ⟨µCCk |VT0(ϵ)|HF⟩ = 0. (418)

Differentiating the energy of Eq. (415) with respect toperturbation strengths, ϵ , yields the static CC-CI molecularproperties.

3. Static CC-CI molecular properties from finite fieldEOMCC energy calculations

In Subsection X A 2, we have described how CC-CI static molecular properties may be determined fromfinite field energy calculations, where the perturbed systemis described by the CC-CI intermediate-normalized wavefunction |0CC−CI

I (ϵ)⟩ of Eq. (414). The determination of|0CC−CI

I (ϵ)⟩ in Eq. (414) is performed in the biorthonormalbasis |BT0⟩ [Eq. (332)] and ⟨BT0| [Eq. (345)], which isthe basis that is used to obtain the EOMCC eigenvalueequation in Eqs. (346)–(354). We now show that static CC-CI molecular properties may also be obtained from finite fieldEOMCC energy calculations.

The EOMCC right ground state eigenvalue equation inEq. (346) for the perturbed Hamiltonian, H(ϵ) [Eq. (409)],may be written as

HT0(ϵ) CR0 (ϵ) = CR

0 (ϵ) E0(ϵ), (419)


where the matrix elements of the perturbed Hamiltonianmatrix, HT0(ϵ), are given byHT0(ϵ)

HF HF= ⟨HF| exp(−T0)

(H0 + V (ϵ)) exp(T0)|HF⟩

= CCE0 + ⟨HF|VT0(ϵ)|HF⟩, (420)HT0(ϵ)

HF νm= ⟨HF| exp(−T0)

(H0 + V (ϵ)) exp(T0)|νCC

m ⟩

= ⟨HF|HT00 |νCC

m ⟩ + ⟨HF|VT0(ϵ)|νCCm ⟩, (421)

HT0(ϵ)

µk HF= ⟨µCC

k | exp(−T0)(H0 + V (ϵ)) exp(T0)|HF⟩

= ⟨µCCk |VT0(ϵ)|HF⟩, (422)

HT0(ϵ)

µkνm= ⟨µCC

k | exp(−T0)(H0 + V (ϵ)) exp(T0)|νCC

m ⟩

= ⟨µCCk |HT0

0 |νCCm ⟩ + ⟨µCC

k |VT0(ϵ)|νCCm ⟩. (423)

To obtain Eq. (422), we have used that the cluster amplitudesequation [Eq. (325)] is satisfied for the unperturbed system.We will assume that the right ground state eigenvector, CR

0 (ϵ),is intermediate normalized

CR0 (ϵ) = *

,

1cµk

(ϵ)+-, (424)

as is the CC-CI ground state wave function [Eq. (414)].Inserting Eqs. (420)–(424) in Eq. (419), we obtain

*,

CCE0 + ⟨HF|VT0(ϵ)|HF⟩ ⟨HF|HT00 |νCC

m ⟩ + ⟨HF|VT0(ϵ)|νCCm ⟩

⟨µCCk |VT0(ϵ)|HF⟩ ⟨µCC

k |HT00 |νCC

m ⟩ + ⟨µCCk |VT0(ϵ)|νCC

m ⟩+-*,

1cνm(ϵ)

+-= *,

1cµk

(ϵ)+-

E0(ϵ), (425)

which gives the perturbed energy and amplitude equations

E0(ϵ) = CCE0 + ⟨HF|VT0(ϵ)|HF⟩ +νm

⟨HF|HT00 |νCC

m ⟩cνm(ϵ)

+νm

⟨HF|VT0(ϵ)|νCCm ⟩cνm(ϵ), (426)

⟨µCCk |VT0(ϵ)|HF⟩ +

νm

⟨µCCk |HT0

0 |νCCm ⟩cνm(ϵ)

+νm

⟨µCCk |VT0(ϵ)|νCC

m ⟩cνm(ϵ) − cµk(ϵ) E0(ϵ) = 0.

(427)

Substituting Eq. (426) in Eq. (427) and using Eq. (328) gives,after some rearrangements,

⟨µCCk |VT0(ϵ)|HF⟩ +

νm

CCJµkνmcνm(ϵ)

+νm

(⟨µCCk |VT0(ϵ)|νCC

m ⟩ − δµkνm

× ⟨HF|VT0(ϵ)|HF⟩)cνm(ϵ)

− cµk(ϵ)

νm

⟨HF|HT00 |νCC

m ⟩cνm(ϵ)

− cµk(ϵ)

νm

⟨HF|VT0(ϵ)|νCCm ⟩cνm(ϵ) = 0. (428)

It follows from Eqs. (418) and (428) that

cµk(ϵ) = sµk

(ϵ), (429)

for all µk, and therefore, the EOMCC perturbed energyEOMCCE0(ϵ) [Eq. (426)] is equal to the CC-CI perturbedenergy [Eq. (415)],

EOMCCE0(ϵ) = CC−CIE(ϵ). (430)

CC-CI static molecular properties may therefore be obtainedby differentiation of the EOMCC energy in Eq. (426). If theEOMCC eigenvalue equation [Eq. (419)] is solved for excitedstate energies, we obtain static molecular properties for theexcited states that are equal to the CC-CI excited state staticmolecular properties.

B. Size-extensivity of molecular properties in the CI,CC-CI, and CC models

In Sec. II G, we have seen that a multiplicativelyseparable steady-state wave function [Eq. (76)] leads to size-extensive molecular properties. In Sec. V A, we have definedthe CI, CC-CI, and CC models [Eqs. (252)–(254)] in termsof the intermediate normalized wave function in Eq. (198),which in turn may be expressed in terms of the steady-state wave function using Eqs. (25) and (197). Therefore,to investigate the size-extensivity of molecular propertiesin the approximate models, CI, CC-CI, and CC, we willin the following examine the multiplicative separability ofthe intermediate normalized wave function for these models,both with respect to the parametrization of the unperturbedstate and the time evolution of this state. For the CI, CC-CI, and CC models, we initially examine the multiplicativeseparability of the reference state |R⟩ and of the exponentialoperators, exp(B0) and exp(B(t)), entering the parametriza-tion of the unperturbed state and the time evolution of thisstate. We will then examine the separability of the Jacobianmatrix and molecular properties.


1. Separability of the reference and time-dependentstates

Let us initially assume that the reference state |R⟩ ismultiplicatively separable,

|RCD⟩ = |RC⟩ ⊗ |RD⟩, (431)

where ⊗ denotes that |RCD⟩ is expressed in the direct productspace referencing sub-system C and D. We now considerthe exponential operator exp(B). To obtain multiplicativeseparability,

exp(BCD) = exp(BC + BD) = exp(BC) exp(BD), (432)

the operator B must be additively separable,

BCD = BC + BD. (433)

The last equality of Eq. (432) follows from Eq. (184).When the operator B is expressed in terms of the many-

body excitation operators, τµkof Eq. (250), it is additively

separable for two non-interacting sub-systems C and D,

TCD = TC + TD, (434)

even for truncated excitation manifolds. This is well-established for the time-independent Hamiltonians, see, forexample, Ref. 82, and the extension to the time-dependentHamiltonians is straightforward. Eq. (434) is thus satisfied forboth complete and truncated manifolds.

Turning our attention to the exponential operator ex-pressed in terms of the state transfer operators, we find thatthe operator S of Eq. (248) is not additively separable,

SCD , SC + SD. (435)

To see this, we express the exact operator S for sub-systems Cand D as

SC =kC

skC |kC⟩⟨RC |, (436)

SD =kD

skD|kD⟩⟨RD|, (437)

whereas for the compound system CD, the SCD operator inthe direct product space becomes

SCD =

kCkD

skCskD(|kC⟩ ⊗ |kD⟩) (⟨RC | ⊗ ⟨RD|)

+kC

skC(|kC⟩ ⊗ |RD⟩) (⟨RC | ⊗ ⟨RD|)

+kD

skD(|RC⟩ ⊗ |kD⟩) (⟨RC | ⊗ ⟨RD|) . (438)

Using Eqs. (436) and (437), we may write Eq. (438) as

SCD = SC ⊗ SD + SC ⊗ |RD⟩⟨RD| + |RC⟩⟨RC | ⊗ SD,

(439)

showing that the operator SCD is not additively separable,as expressed by Eq. (435). The exponential operator exp(S)is thus not multiplicatively separable, even for a completeexcitation manifold.

We now examine how the lack of multiplicative sepa-rability of exp(S) influences the multiplicative separabilityof the intermediate normalized wave function exp(S)|R⟩. Todo that we use Eqs. (436)–(438) to write the intermediatenormalized wave functions for sub-systems C and D and forthe compound system CD as

exp(SC)|RC⟩ = |RC⟩ +kC

skC |kC⟩, (440)

exp(SD)|RD⟩ = |RD⟩ +kD

skD|kD⟩, (441)

exp(SCD)|RC⟩ ⊗ |RD⟩

= |RC⟩ ⊗ |RD⟩ +kC

skC |kC⟩ ⊗ |RD⟩

+ |RC⟩ ⊗kD

skD|kD⟩ +

kCkD

skCskD|kC⟩ ⊗ |kD⟩.

(442)

For complete excitation manifolds, the compound wavefunction of Eq. (442) may be written as


=

(|RC⟩ +

kC

skC |kC⟩)⊗

(|RD⟩ +

kD

skD|kD⟩)

= exp(SC)|RC⟩ ⊗ exp(SD)|RD⟩, complete manifold,

(443)

where to obtain the last equality we have used Eqs. (440)and (441). For a complete manifold, the compound wavefunction exp(SCD)|RC⟩ ⊗ |RD⟩ is therefore multiplicativelyseparable, even though Eq. (432) is not satisfied. For trun-cated manifolds, however, the term

kCkD skCskD|kC⟩ ⊗

|kD⟩ arising from the direct product of the sub-system wavefunctions of Eqs. (440) and (441) contains contributionsthat are beyond the excitation manifold of the compoundsystem wave function of Eq. (442). The direct product andthe compound wave functions therefore differ for truncatedmanifolds, even if the reference state is separable,


, exp(SC)|RC⟩ ⊗ exp(SD)|RD⟩, truncated manifolds.

(444)

2. Separability of the Jacobian

We will now consider the Jacobian of the compoundsystem CD and study under which conditions this matrix isseparable and thereby gives excitation energies that are size-intensive, i.e., independent of system size. Consider first thegeneral form of the Jacobian given in Eq. (261), which for the


compound system reads

JCDkm = ⟨k | (HC

0 + HD0 )BCD

0 , βm |RCD⟩. (445)

We will first show that the Jacobian of Eq. (445) is additivelyseparable provided BCD

0 and the excitation manifold {β}used for the time-development are additively separable, andthe reference state is multiplicatively separable,

BCD0 = BC

0 + BD0 , (446a)

|RCD⟩ = |RC⟩ ⊗ |RD⟩, (446b)

{βCD} = {βC} ⊕ {βD}. (446c)

To show that the three conditions of Eq. (446) are sufficientto ensure separability of the Jacobian, we first note thatthe separability of the operator B0 for the time-independentdevelopment ensures that the similarity-transformed Hamilto-nian is extensive,

(HC0 + HD

0 )BCD0

= exp(−(BC0 + BD

0 ))(HC0 + HD

0 ) exp(BC0 + BD

0 )

= exp(−BC0 )HC exp(BC

0 ) + exp(−BD0 )HD exp(BD

0 )

= (HC0 )B

C0 + (HD

0 )BD0 . (447)

Furthermore, the blocks of the Jacobian coupling thedifferent sub-systems are then vanishing

JCD

µCkνDm= ⟨βµC

kRD| (HC

0 )BC0 + (HD

0 )BD0 , βνDm

|RCRD⟩

= ⟨βµCk|RC⟩⟨RD| (HD

0 )BD0 , βνDm

|RD⟩ = 0, (448)

where we have used that |βµCk⟩ is orthogonal to |RC⟩. For

two operators referring to the same sub-system, the Jacobianreduces trivially to the standard sub-system Jacobian

JCD

µCkνCm= ⟨βµC

kRD| (HC

0 )BC0 , βνCm

|RCRD⟩

= ⟨βµCk| (HC

0 )BC0 , βνCm

|RC⟩ = JCµCkνCm

. (449)

From Eqs. (448) and (449), we therefore have the additiveseparability of the Jacobian,

JCD = JC ⊕ JD. (450)

Any model, exact or approximate, fulfilling the conditionsof Eq. (446) will therefore provide an additively separableJacobian.

It is important to note that there are size-extensivemethods that do not fulfill Eq. (446) but still provide size-intensive excitation energies. Consider, for example, themodels, including the full CC-CI and CI models, where theexcitation manifold for the time-development have the formof Eq. (439), rather than that of Eq. (446c). For this form ofthe excitation manifold, the blocks of the Jacobian couplingany two of the three types of excitations vanish and the blockscontaining excitations only in one of the systems reduce asEq. (449). For two excitations containing excitations in bothsub-systems, one obtains for the CI-model

JCD

µCkµDk,νCmνDm

= ⟨µCk µDk |HC

0 + HD0 , |νCm⟩⟨0C0 | ⊗ |νDm⟩⟨0D

0 | |0C0 0D

0 ⟩

= ⟨µCk |HC0 |νCm⟩δµD

kνDm+ δµC

kνCm

⟨µDk |HD0 |νDm⟩

− δµCkνCm

δµDkνDm

(EC0 + ED

0 )

= JCµCkνCm

δµDkνDm+ δµC

kνCm

JD

µDkνDm

. (451)

Using a full linear expansion for the time-development, oneobtains therefore the following separation of the Jacobian:

JCD = JC ⊕ JD ⊕ (JC ⊗ 1D + 1C ⊗ JD). (452)

The eigenvalues of the first two parts of the Jacobian arethe excitation energies of the individual systems, whereasthe eigenvalues of the last term are the sum of excitationenergies of the two sub-systems. As the excitation energies ofthe individual systems are correctly reproduced, these modelsyield size-intensive excitation energies.

Armed with the above general developments, we are nowable todiscuss the separabilityof the Jacobian for the threewavefunction models, CC, CI, and CC-CI. The results are summa-rized in Table I, where we also summarize our later conclusionsconcerning the separability of molecular properties.

For the CC model, where both manifolds are addi-tively separable, the Jacobian is also additively separable,independent of whether a complete or truncated expansionis used. The excitation energies obtained as eigenvalues ofthe compound Jacobian become the excitation energies ofC and D, and the CC model therefore yields size-intensive

TABLE I. Summary of separability of excitation manifolds, Jacobians, excitation energies, and molecular properties for the standard wave function models.

CC CI CC-CI

Manifold, TI {τC} ⊕ {τD} SC ⊗�RD

�RD

�⊕�RC

�RC

�⊗ SD ⊕ SC ⊗ SD {τC} ⊕ {τD}

Manifold, TD {τC} ⊕ {τD} SC ⊗�RD

�RD

�⊕�RC

�RC

�⊗ SD ⊕ SC ⊗ SD SC ⊗

�RD

�RD

�⊕�RC

�RC

�⊗ SD ⊕ SC ⊗ SD

Jacobian, full expansion JC ⊕ JD JC ⊕ JD ⊕ (JC ⊗1D+1C ⊗ JD) JC ⊕ JD ⊕ (JC ⊗1D+1C ⊗ JD)Jacobian, truncated

expansionJC ⊕ JD Not separable JC ⊕ JD⊕ non-separable part

Excitation energies Size-intensive Not size-intensive Size-intensiveMolecular properties Size-extensive Not size-extensive Not size-extensive


excitation energies. For the CI-model, we obtain a Jacobianof the form of Eq. (452) when no truncations are performed.The eigenvalues of the first two parts of the Jacobian arethe excitation energies of the individual systems, whereas theeigenvalues of the last term are the sum of excitation energiesof the two sub-systems. For truncated CI-expansions, wherethe manifolds for both the time-independent and time-dependent parts are truncated, the reference state and therebythe Jacobian are not separable.

Consider finally the CC-CI model, where differentexcitation manifolds are used for the time-independent andthe time-dependent parts. Without truncations, the CC-CIJacobian behaves as the CI Jacobian and gives the individualplus the combined excitation energies. However, when theexcitation manifold is truncated, a difference between theCI and CC-CI models occurs. As the first two conditions ofEq. (446) are fulfilled for the CC-CI model, the Jacobianbecomes a direct sum of the sub-system Jacobians and apart containing excitations in both systems. The parts ofthe Jacobian referring to the individual sub-systems areindependent of whether one considers this sub-system orthe compound system, so this model correctly reproducesthe excitations of the individual systems. The part of theJacobian containing the combined excitations in the two sub-systems does not separate into direct products of the sub-system Jacobians and gives therefore a non-separable blockof the Jacobian, resulting in excitation energies that are notsize-intensive and less accurate than those obtained from thesub-system Jacobians.

3. Size-extensivity of molecular properties

Applying the above analysis to the CI, CC-CI, and CCmodels, we find that for the CI model [Eq. (252)] it followsfrom Eq. (444) that already the unperturbed state is notmultiplicatively separable when truncations are introduced inthe excitation manifold and the truncated CI model thereforecannot yield size-extensive molecular properties.

In the CC-CI and CC models, the CC state is used as theunperturbed state and this state is multiplicatively separable,as follows from Eq. (434). The time evolution in the CC-CI model is, however, linearly parametrized in terms of theexp(S(t)) operator, and therefore, as follows from Eq. (444),the time-dependent wave function is not multiplicativelyseparable for this model, when truncations are introducedin the excitation manifold. Consequently, molecular responseproperties are not size-extensive in truncated CC-CI models.In the CC model, the time evolution is parametrized in termsof the exp(T(t)) operator and, as follows from Eq. (434), thetime-dependent wave function is multiplicatively separable,provided that the reference state is multiplicatively separable.Molecular response properties are, therefore, size-extensivein the CC model, even if truncations are introduced in theexcitation manifold. For the CC-CI model, the non-size-extensivity is weak because, compared to the CC model, weare neglecting only non-linear terms for the time-evolvingstate and these non-linear terms only enter when excitationsare simultaneously considered for the two sub-systems in thecompound system.

Let us finally consider size-extensivity of molecularproperties obtained as residues of response functions. Whena perturbation frequency matches an excitation energy of theunperturbed system, the frequency-dependent quasi-energycorrection of Eq. (65) becomes infinite and a number ofmolecular properties are determined as residues of theresponse function, for example [Eq. (71)],

limωX2→ω

(ωX2 − ω) EXj1Xj2(ωXj1

,ωXj2)

=

m: Em−E0=ω

⟨00|X j1|0m⟩⟨0m|X j2|00⟩. (453)

For an exact wave function, the quasi-energy corrections aresize-extensive [Eq. (84)] and excitation energies are size-intensive and therefore it follows from Eq. (453) that theresidues are also size-extensive. For truncated CI models,transition properties are not size-extensive, because neitherexcitation energies are size intensive nor quasi-energy correc-tions are size-extensive. For the CC-CI model, excitationenergies are size intensive, but the quasi-energy corrections arenot size-extensive for truncated manifolds, and therefore, theCC-CI transition properties are not size-extensive for the trun-cated manifolds. In the CC model, even when truncations areintroduced in the excitation manifold, excitation energies aresize-intensive and quasi-energy corrections are size-extensive,and therefore, transition properties are size-extensive.

C. Linear versus exponential parametrizationin the CI, CC-CI and CC models

We now compare the CI, CC-CI, and CC molecularresponse properties in the context of linear versus exponentialparametrization of the unperturbed state and its time evolution.For the CI, CC-CI, and CC models, different parametrizationsare used for the wave function and the response functions arestructurally very different [see, for example, the linear responsefunctions in Eqs. (299), (340), and (383)]. However, in the limitwhere no truncation is performed in the excitation manifold(the FCI limit), the results obtained with the three modelsare identical, as in this limit, the Schrödinger equation for allthree models is solved with no approximations, just in termsof different parametrizations. However, when truncations areperformed in the excitation manifold, different results areobtained for the three models.

For the CI model, the unperturbed state satisfies thevariational condition. This gives large simplifications in theresponse functions. In particular, the cLNL

H0term in Eq. (270a)

vanishes in the CI model. The linear response function forthe CI model in Eq. (299) therefore only contains one singleterm. For both the CC-CI and CC models, a coupled clusterunperturbed state is used. The coupled cluster unperturbedstate satisfies the cluster amplitudes and multipliers equations[Eqs. (325) and (327)]. These equations also introducesimplifications in the response functions, but to less extent thanthe variational condition. As a result, the CC-CI [Eq. (340)] andCC [Eq. (383)] linear response functions contain additionalterms compared to the CI linear response function [Eq. (299)].


Comparing the expressions for the CC-CI and CCresponse functions, we see that the non-linear terms containingthe HT0

0 operator are greatly simplified for the CC-CI modelcompared to the CC model. The CC-CI response functions thusonly contain one non-linear HT0

0 term [Eq. (270a)], whereasthe CC response functions have, in general, four non-linearHT0

0 terms [Eq. (270b)]. Comparing the CC-CI linear responsefunction [Eq. (340)] with the CC linear response function[Eq. (383)], we therefore also see that the CC linear responsefunctions contain an additional term (the first term in Eq. (383))compared to the CC-CI linear response function and also thatthe second term in Eq. (383) has a more involved structuralform than its counterpart in the first term of Eq. (340). For thequadratic and higher-order response functions, this differencebetween the number and the complexity of the terms enteringthe CC and CC-CI response functions becomes even morepronounced, resulting in much simpler expressions for theCC-CI than for the CC response functions.

For the CI and CC-CI models, the time evolution isdescribed in terms of a linear parametrization and an explicitrepresentation is therefore obtained for the ground and excitedstates. However, this is obtained at the expense that these mod-els do not yield molecular properties that are size-extensive.This may, for example, be seen explicitly in the CC-CI linearresponse function that contains a term (the first term inEq. (340)) that increases quadratically with system size. Con-trary, for the CC model, where the time evolution is exponen-tially parametrized, molecular properties are size-extensive,but we cannot obtain an explicit representation of the groundand excited states. However, this is not important as we havea procedure for determining molecular properties withoutrecourse to an explicit representation of individual states.

D. Comparison of CC-CI and EOMCC molecularresponse properties

We now compare the CC-CI and EOMCC molecularresponse properties. For first-order molecular properties, weobtain from Eqs. (335) and (388)

X j1

��CC−CI0 =

X j1

��EOMCC0 . (454)

In the supplementary material,89 we have shown explicitly thatthe CC-CI and EOMCC linear response functions are equal,

X j1; X j2

��CC−CIωXj2

=

X j1; X j2

��EOMCCωXj2

, (455)

and that the CC-CI and EOMCC quadratic response functionsdiffer,

X j1; X j2,X j3

��CC−CIωXj2

,ωXj3,

X j1; X j2,X j3

��EOMCCωXj2

,ωXj3. (456)

We now show that this difference holds also for cubic andhigher-order response functions and that only the CC-CImodel reproduces FCI results.

The EOMCC model is defined by the non-Hermitianeigenvalue equation [Eqs. (346)–(349)]. The optimizationcondition for the EOMCC ground state is defined by van-ishing Hamiltonian matrix elements between the ground stateand the EOMCC excited states,

⟨0CCn |H0|0CC

0 ⟩ = 0, n > 0, (457)

⟨0CC0 |H0|0CC

n ⟩ = 0, n > 0. (458)

Further, the excited states in the EOMCC model are obtainedfrom a diagonalization of the similarity transformed Hamilto-nian in Eq. (350) and the excited states are therefore linearlyexpanded in the |BT0⟩ [Eq. (332)] and ⟨BT0| [Eq. (345)] basis.

We now consider the optimization conditions in moredetail. Substituting Eqs. (366) and (369) in Eq. (457), weobtain

⟨0CCn |H0|0CC

0 ⟩ =µk

CC Cnµk⟨µCC

k |HT00 |HF⟩ = 0. (459)

The optimization condition in Eq. (457) is thus equivalent torequiring that the cluster amplitudes equation [Eq. (325)] issatisfied for the unperturbed system. Inserting Eqs. (367) and(368) in Eq. (458) gives

⟨0CC0 |H0|0CC

n ⟩ =µk

CCCµkn

⟨HF|HT0

0 |µCCk ⟩

+νm

t(0)νm

(⟨νCC

m |HT00 |µCC

k ⟩

− ⟨HF|HT00 |HF⟩δνmµk

− t(0)µk⟨νCC

m |HT00 |HF⟩

)= 0. (460)

The CC amplitudes equation [Eq. (325)] is satisfied and thelast term in Eq. (460) therefore vanishes, giving

µk

CCCµkn

ηµk+

νm

t(0)νmCCJνmµk

= 0, (461)

where we have used Eqs. (328) and (329). The optimizationcondition in Eq. (458) is thus equivalent to requiring that theCC multipliers satisfy the multipliers equation [Eq. (327)] forthe unperturbed system. The excited states in the EOMCCmodel are further linearly expanded in the |BT0⟩ and ⟨BT0|basis. Such linear expansions can only be obtained when thetime evolution is described in terms of a linear expansion.EOMCC response functions therefore should be defined bya model where the CC amplitude and multiplier equationsare satisfied for the unperturbed system and where the timeevolution is described in terms of a linear expansion. Theseare precisely the premises for the CC-CI model [Eq. (254)],in accordance with the fact that static CC-CI molecularproperties are obtained from finite field EOMCC energycalculations, as discussed in Sec. X A 3.

In Sec. IX, we have discussed how the EOMCC responsefunctions are obtained expressing the CI response functionsof Sec. VI in the basis of the eigenstates and energies|0CI

n ⟩,⟨0CIn |,CIEn

,n = 0,1,2, . . . of the CI eigenvalue equa-

tion in Eq. (304) and replacing the CI eigenstates and ener-gies with the eigenstates and energies

|0CCn ⟩,⟨0CC

n |,CCEn

,n

= 0,1,2, . . . of the EOMCC eigenvalue equation in Eqs. (346)and (347). However, when the CI response functions aredetermined, it is done using a stronger optimization conditionthan is satisfied for the EOMCC model. For the EOMCCmodel, the CC amplitude and multiplier equations are


satisfied for the unperturbed system. In CI theory, theamplitude and multiplier equations are also satisfied, but inaddition we have

⟨0CIn |H0|0CI

0 ⟩† = ⟨0CI0 |H0|0CI

n ⟩ = 0, (462)

which is not satisfied for EOMCC, because HT00 is not

Hermitian. The stronger optimization condition [Eq. (462)]implies that the cLNL

H0term [Eq. (270a)],

cLNLH0=

KLM

k

sk(ωK)sk(ωL)m

sm(ωM)

×0CI

0��H0

��0CIm

∆(ωK + ωL + ωM), (463)

vanishes in CI. This term is therefore not present in theEOMCC response functions in Sec. IX. The CC-CI andEOMCC response functions therefore differ when this termcontributes to the CC-CI response functions. When thisterm is neglected, FCI results cannot be obtained using theEOMCC response functions of Sec. IX. Contrary, the CC-CImodel gives FCI limit results.

The cLNLH0

term [Eq. (463)] does not contribute whenfirst-order amplitude equations are determined [see Eq. (338)]and EOMCC and CC-CI first-order amplitudes are thereforeidentical. When the linear response function is calculatedusing the 2n + 1 and 2n + 2 rules, only the first-orderamplitudes are used and the EOMCC and CC-CI linearresponse functions therefore become identical. However, ifthe linear response function is determined in the asymmetricform, where first-order multipliers are used to determinethe linear response function, the EOMCC linear responsefunction is not equal to the one obtained when the 2n + 1and 2n + 2 rules are applied and is not able to reproduce FCIresult. EOMCC response functions in general are not capableof reproducing FCI results.

XI. SUMMARY, CONCLUSION, AND OUTLOOK

For a time-periodic perturbation, we have recastedthe time-dependent Schrödinger equation into a Hermitianeigenvalue equation by carrying out time averaging over oneperiod of the perturbation in the composite Hilbert space. Thesolution to the time-dependent Schrödinger equation for atime-periodic perturbation thereby becomes a straightforwardgeneralization of the solution to the Hermitian eigenvalueequation for a static perturbation. The Hermitian eigenvalueequation may be solved using the same techniques as forthe static perturbation. The Hermitian eigenvalue equationhas the quasi-energy as an eigenvalue and the time-periodicregular wave function as the corresponding eigenstate. Differ-entiation of the quasi-energy with respect to the perturbationstrengths determines molecular response functions. We havethus arrived at a rigorous and transparent formulation ofresponse function theory applicable to both variational andnon-variational wave functions where both a linear and anon-linear parametrization of the time evolution can beused and where the optimization of the wave function for

the unperturbed and perturbed system may be carried outdifferently.

We have illustrated the generality and simplicity of thenew formulation of response function theory by derivingresponse functions for a CI model, where both the unper-turbed state and its time evolution are linearly parametrized,and for a coupled cluster unperturbed state, where we haveconsidered both a linear (CC-CI model) and a non-linear(CC model) parametrization of its time evolution. We haveshown that the CC-CI molecular properties in the time-independent limit become an analytic representation of thestatic properties obtained from EOMCC finite field energycalculations. We have carried out a detailed comparison oftraits of response functions obtained for CI, CC, and CC-CImodels with emphasis on linear versus non-linear parametri-zations of the time evolution. The linear parametrization ofthe time-evolution in the CI and CC-CI models allows foran explicit representation of the ground and excited states,but at the expense of having molecular properties that arenot size-extensive. For an exponential parametrization ofthe time-evolution of the CC state, an explicit representa-tion of the ground and excited states cannot be obtained.However, this is not important as explicit expressions canbe obtained for molecular properties that in turn are size-extensive. A detailed analysis has further shown that thenon-size-extensivity of CC-CI molecular properties is weak.Taking into consideration the simplifications that arise whenmolecular properties are calculated using the CC-CI ratherthan the CC model, the CC-CI model may provide analternative and practical vehicle for determining molecularproperties.

The development we have presented has thus brought thedefinition of molecular response properties and their determi-nation on par for static and time-periodic perturbations, andit has removed inadequacies and inconsistencies of previousresponse function theory formulations. The development hasalso allowed response functions to be determined for newwave function models, e.g., the CC-CI model. In a laterpublication, we will also show how the Hermitian eigenvalueequation may be solved using perturbation theory and howresponse functions may then be determined for, for example,ΛCCSD[T]90–92 and CCSD(T-2)93 reference states.

ACKNOWLEDGMENTS

F.P. and P.J. acknowledge support from The EuropeanResearch Council under the European Union’s (EU) SeventhFramework Programme (FP/2007-2013)/ERC Grant Agree-ment No. 291371 and P.J. and J.O. acknowledge supportfrom The Danish Council for Independent Research—NaturalSciences. F.P. also acknowledges support form the PolishNational Science Centre (Project No. 3714/B/H03/2011/40).

1J. Olsen and P. Jørgensen, J. Chem. Phys. 82, 3235 (1985).2O. Christiansen, P. Jørgensen, and C. Hättig, Int. J. Quantum Chem. 68, 1(1998).

3G. Floquet, Ann. Ecole Norm. Sup. 2 12, 47 (1883), available online athttp://www.numdam.org/item?id=ASENS_1883_2_12__47_0.

4D. J. Tannor, Introduction to Quantum Mechanics: A Time-dependentPerspective (University Science Books, 2007).

http://dx.doi.org/10.1063/1.448223

http://dx.doi.org/10.1002/(SICI)1097-461X(1998)68:1%3C1::AID-QUA1%3E3.0.CO;2-Z

http://www.numdam.org/item?id=ASENS_1883_2_12__47_0




















































5S. Blanes, F. Casas, J. A. Oteo, and J. Ros, Phys. Rep. 470, 151 (2009).6H. Sambe, Phys. Rev. A 7, 2203 (1973).7D. N. Zubarev, Nonlinear Statistical Thermodynamics (Consultant Bureau,Plenum, New York, 1974).

8J. Linderberg and Y. Öhrn, Propagators in Quantum Chemistry (AcademicPress, New York, 1973).

9A. D. McLachlan and M. A. Ball, Rev. Mod. Phys. 36, 844 (1964).10P. Jørgensen, Annu. Rev. Phys. Chem. 26, 359 (1975).11J. Oddershede, P. Jørgensen, and D. L. Yeager, Comput. Phys. Rep. 2, 33

(1984).12J. Olsen and P. Jørgensen, in Modern Electronic Structure Theory II, edited

by D. Yarkoni (VCH, New York, 1995).13P. Jørgensen, H. J. A. Jensen, and J. Olsen, J. Chem. Phys. 89, 3654 (1988).14H. Hettema, H. J. A. Jensen, P. Jørgensen, and J. Olsen, J. Chem. Phys. 97,

1174 (1992).15O. Vahtras, H. Ågren, P. Jørgensen, H. J. A. Jensen, T. Helgaker, and J. Olsen,

J. Chem. Phys. 97, 9178 (1992).16P. Norman, D. Jonsson, O. Vahtras, and H. Ågren, Chem. Phys. Lett. 242, 7

(1995).17D. Jonsson, P. Norman, and H. Ågren, J. Chem. Phys. 105, 6401 (1996).18M. Jaszunski and A. Rizzo, Mol. Phys. 96, 855 (1999).19V. Carravetta, H. Ågren, H. J. A. Jensen, P. Jørgensen, and J. Olsen, J. Phys.

B: At., Mol. Opt. Phys. 22, 2133 (1989).20V. Spirko, H. J. A. Jensen, and P. Jørgensen, Chem. Phys. 144, 343 (1990).21J. M. Schins, L. D. A. Siebbeles, J. Los, M. Kristensen, and H. Koch, J. Chem.

Phys. 93, 3887 (1990).22D. Nordfors, H. Ågren, and H. J. A. Jensen, Int. J. Quantum Chem. 40, 475

(1991).23H. J. A. Jensen, P. Jørgensen, H. Hettema, and J. Olsen, Chem. Phys. Lett.

187, 387 (1991).24M. Jaszunski, P. Jørgensen, H. Koch, H. Ågren, and T. Helgaker, J. Chem.

Phys. 98, 7229 (1993).25H. Ågren, V. Carravetta, H. J. A. Jensen, P. Jørgensen, and J. Olsen, Phys.

Rev. A 47, 3810 (1993).26B. Minaev, O. Vahtras, and H. Ågren, Chem. Phys. 208, 299 (1996).27B. F. Minaev, N. A. Murugan, and H. Ågren, Int. J. Quantum Chem. 113,

1847 (2013).28P. Norman, Y. Luo, D. Jonsson, and H. Ågren, J. Chem. Phys. 106, 1827

(1997).29P. Norman, D. Jonsson, and H. Ågren, Chem. Phys. Lett. 268, 337 (1997).30D. Jonsson, P. Norman, O. Vahtras, H. Ågren, and A. Rizzo, J. Chem. Phys.

106, 8552 (1997).31D. Jonsson and P. Norman, J. Chem. Phys. 109, 572 (1998).32H. Koch and P. Jørgensen, J. Chem. Phys. 93, 3333 (1990).33H. Koch, H. J. A. Jensen, P. Jørgensen, and T. Helgaker, J. Chem. Phys. 93,

3345 (1990).34H. Koch, R. Kobayashi, A. Sanchez de Meras, and P. Jørgensen, J. Chem.

Phys. 100, 4393 (1994).35H. Koch, O. Christiansen, R. Kobayashi, P. Jørgensen, and T. Helgaker,

Chem. Phys. Lett. 228, 233 (1994).36F. Aiga, K. Sasagane, and R. Itoh, J. Chem. Phys. 99, 3779 (1993).37K. Sasagane, F. Aiga, and R. Itoh, J. Chem. Phys. 99, 3738 (1993).38E. A. Hylleraas, Z. Phys. 65, 209 (1930).39E. Wigner, Math. Natur. Anz. (Budapest) 53, 477 (1935).40T. Helgaker and P. Jørgensen, in Methods in Computational Molecular

Physics, edited by S. Wilson and G. H. F. Diercksen (Plenum Press, NewYork, 1992), Vol. 293, pp. 353–421.

41T. Helgaker and P. Jørgensen, Theor. Chim. Acta 75, 111 (1989).42O. Christiansen, H. Koch, and P. Jørgensen, Chem. Phys. Lett. 243, 409

(1995).43G. D. Purvis and R. J. Bartlett, J. Chem. Phys. 76, 1910 (1982).44H. Koch, O. Christiansen, P. Jørgensen, A. M. Sanchez de Merás, and T.

Helgaker, J. Chem. Phys. 106, 1808 (1997).45O. Christiansen, H. Koch, and P. Jørgensen, J. Chem. Phys. 103, 7429 (1995).46C. E. Smith, R. A. King, and T. D. Crawford, J. Chem. Phys. 122, 054110

(2005).47J. Noga and R. J. Bartlett, J. Chem. Phys. 86, 7041 (1987); erratum, 89, 3401

(1988).48G. E. Scuseria and H. F. Schaefer, Chem. Phys. Lett. 152, 382 (1988).49G. Trucks, J. Noga, and R. Bartlett, Chem. Phys. Lett. 145, 548 (1988).50J. Noga, R. Bartlett, and M. Urban, Chem. Phys. Lett. 134, 126 (1987).51M. Urban, J. Noga, S. Cole, and R. Bartlett, J. Chem. Phys. 83, 4041

(1985); erratum, 85, 5383 (1986).52Y. Lee, S. Kucharski, and R. Bartlett, J. Chem. Phys. 81, 5906

(1984); erratum, 82, 5761 (1985).

53Y. Lee and R. Bartlett, J. Chem. Phys. 80, 4371 (1984).54D. H. Magers, R. J. Harrison, and R. J. Bartlett, J. Chem. Phys. 84, 3284

(1986).55O. Christiansen, H. Koch, A. Halkier, P. Jørgensen, T. Helgaker, and A.

Sanchez de Meras, J. Chem. Phys. 105, 6921 (1996).56O. Christiansen, A. Halkier, H. Koch, P. Jørgensen, and T. Helgaker, J. Chem.

Phys. 108, 2801 (1998).57F. Pawłowski, P. Jørgensen, and C. Hättig, Chem. Phys. Lett. 389, 413

(2004).58R. Kobayashi, H. Koch, and P. Jørgensen, Chem. Phys. Lett. 219, 30 (1994).59O. Christiansen, J. Gauss, and J. F. Stanton, Chem. Phys. Lett. 292, 437

(1998).60K. Hald, F. Pawłowski, P. Jørgensen, and C. Hättig, J. Chem. Phys. 118, 1292

(2003).61C. Hättig, O. Christiansen, H. Koch, and P. Jørgensen, Chem. Phys. Lett.

269, 428 (1997).62C. Hättig, O. Christiansen, and P. Jørgensen, Chem. Phys. Lett. 282, 139

(1998).63J. Gauss, O. Christiansen, and J. F. Stanton, Chem. Phys. Lett. 296, 117

(1998).64F. Pawłowski, P. Jørgensen, and C. Hättig, Chem. Phys. Lett. 391, 27

(2004).65F. Pawłowski, P. Jørgensen, and C. Hättig, Chem. Phys. Lett. 413, 272

(2005).66M. Pecul, F. Pawłowski, P. Jørgensen, A. Köhn, and C. Hättig, J. Chem. Phys.

124, 114101 (2006).67A. J. Thorvaldsen, K. Ruud, K. Kristensen, P. Jørgensen, and S. Coriani, J.

Chem. Phys. 129, 214108 (2008).68T. Kjærgaard, P. Jørgensen, A. J. Thorvaldsen, P. Sałek, and S. Coriani, J.

Chem. Theory Comput. 5, 1997 (2009).69A. J. Thorvaldsen, L. Ferrighi, K. Ruud, H. Ågren, S. Coriani, and P.

Jørgensen, Phys. Chem. Chem. Phys. 11, 2293 (2009).70P. Norman, D. M. Bishop, H. J. A. Jensen, and J. Oddershede, J. Chem. Phys.

115, 10323 (2001).71P. Norman, D. M. Bishop, H. J. A. Jensen, and J. Oddershede, J. Chem. Phys.

123, 194103 (2005).72L. Jensen, J. Autschbach, and G. C. Schatz, J. Chem. Phys. 122, 224115

(2005).73D. Rocca, R. Gebauer, Y. Saad, and S. Baroni, J. Chem. Phys. 128, 154105

(2008).74S. Coriani, T. Fransson, O. Christiansen, and P. Norman, J. Chem. Theory

Comput. 8, 1616 (2012).75K. Kristensen, J. Kauczor, T. Kjærgaard, and P. Jørgensen, J. Chem. Phys.

131, 044112 (2009).76T. Helgaker, S. Coriani, P. Jørgensen, K. Kristensen, J. Olsen, and K. Ruud,

Chem. Rev. 112, 543 (2012).77K. Kristensen, P. Jørgensen, A. J. Thorvaldsen, and T. Helgaker, J. Chem.

Phys. 129, 214103 (2008).78J. F. Stanton and R. J. Bartlett, J. Chem. Phys. 98, 7029 (1993).79P. B. Rozyczko, S. A. Perera, M. Nooijen, and R. J. Bartlett, J. Chem. Phys.

107, 6736 (1997).80P. Rozyczko and R. J. Bartlett, J. Chem. Phys. 107, 10823 (1997).81P. B. Rozyczko and R. J. Bartlett, J. Chem. Phys. 108, 7988 (1998).82T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic-Structure

Theory (Wiley, Chichester, 2000).83S. Kvaal, J. Chem. Phys. 136, 194109 (2012).84J. Arponen, Ann. Phys. 151, 311 (1983).85P. Kramer and M. Saraceno, Geometry of the Time-Dependent Variational

Principle (Springer, 1981).86R. J. Bartlett, Annu. Rev. Phys. Chem. 32, 359 (1981).87M. Nooijen, K. R. Shamasundar, and D. Mukherjee, Mol. Phys. 103, 2277

(2005).88J. Geertsen, M. Rittby, and R. J. Bartlett, Chem. Phys. Lett. 164, 57 (1989);

H. Sekino and R. J. Bartlett, Int. J. Quantum Chem. Symp. 18, 255 (1984);I. Shavitt and R. J. Bartlett, Many-Body Methods in Chemistry and Physics:MBPT and Coupled-Cluster Theory (Cambridge University Press, 2009).

89See supplementary material at http://dx.doi.org/10.1063/1.4913364 forappendices showing relations between CC-CI and EOMCC responsefunctions.

90S. A. Kucharski and R. J. Bartlett, J. Chem. Phys. 108, 5243 (1998).91S. A. Kucharski and R. J. Bartlett, J. Chem. Phys. 108, 9221 (1998).92T. D. Crawford and J. F. Stanton, Int. J. Quantum Chem. 70, 601

(1998).93J. J. Eriksen, K. Kristensen, T. Kjærgaard, P. Jørgensen, and J. Gauss, J.

Chem. Phys. 140, 064108 (2014).

http://dx.doi.org/10.1016/j.physrep.2008.11.001

http://dx.doi.org/10.1103/PhysRevA.7.2203

http://dx.doi.org/10.1103/RevModPhys.36.844

http://dx.doi.org/10.1146/annurev.pc.26.100175.002043

http://dx.doi.org/10.1016/0167-7977(84)90003-0

http://dx.doi.org/10.1063/1.454885

http://dx.doi.org/10.1063/1.463245

http://dx.doi.org/10.1063/1.463344

http://dx.doi.org/10.1016/0009-2614(95)00716-H

http://dx.doi.org/10.1063/1.472493

http://dx.doi.org/10.1080/00268979909483023

http://dx.doi.org/10.1088/0953-4075/22/13/017

http://dx.doi.org/10.1088/0953-4075/22/13/017

http://dx.doi.org/10.1016/0301-0104(90)80099-J

http://dx.doi.org/10.1063/1.458774

http://dx.doi.org/10.1063/1.458774

http://dx.doi.org/10.1002/qua.560400404

http://dx.doi.org/10.1016/0009-2614(91)80269-4

http://dx.doi.org/10.1063/1.464714

http://dx.doi.org/10.1063/1.464714



http://dx.doi.org/10.1016/0301-0104(96)00126-7


http://dx.doi.org/10.1063/1.473338

http://dx.doi.org/10.1016/S0009-2614(97)00213-3

http://dx.doi.org/10.1063/1.473910

http://dx.doi.org/10.1063/1.476593

http://dx.doi.org/10.1063/1.458814

http://dx.doi.org/10.1063/1.458815

http://dx.doi.org/10.1063/1.466321

http://dx.doi.org/10.1063/1.466321

http://dx.doi.org/10.1016/0009-2614(94)00898-1

http://dx.doi.org/10.1063/1.466124

http://dx.doi.org/10.1063/1.466123

http://dx.doi.org/10.1007/BF01397032

http://dx.doi.org/10.1007/BF00527713

http://dx.doi.org/10.1016/0009-2614(95)00841-Q

http://dx.doi.org/10.1063/1.443164

http://dx.doi.org/10.1063/1.473322

http://dx.doi.org/10.1063/1.470315

http://dx.doi.org/10.1063/1.1835953

http://dx.doi.org/10.1063/1.452353

http://dx.doi.org/10.1063/1.455742

http://dx.doi.org/10.1016/0009-2614(88)80110-6

http://dx.doi.org/10.1016/0009-2614(88)87418-9

http://dx.doi.org/10.1016/0009-2614(87)87107-5

http://dx.doi.org/10.1063/1.449067

http://dx.doi.org/10.1063/1.451873

http://dx.doi.org/10.1063/1.447591

http://dx.doi.org/10.1063/1.448990

http://dx.doi.org/10.1063/1.447214

http://dx.doi.org/10.1063/1.450259

http://dx.doi.org/10.1063/1.471985

http://dx.doi.org/10.1063/1.475671

http://dx.doi.org/10.1063/1.475671

http://dx.doi.org/10.1016/j.cplett.2004.03.126

http://dx.doi.org/10.1016/0009-2614(94)00051-4

http://dx.doi.org/10.1016/S0009-2614(98)00701-5

http://dx.doi.org/10.1063/1.1523905

http://dx.doi.org/10.1016/S0009-2614(97)00311-4

http://dx.doi.org/10.1016/S0009-2614(97)01227-X

http://dx.doi.org/10.1016/S0009-2614(98)01013-6



http://dx.doi.org/10.1063/1.2173253

http://dx.doi.org/10.1063/1.2996351

http://dx.doi.org/10.1063/1.2996351

http://dx.doi.org/10.1021/ct9001625

http://dx.doi.org/10.1021/ct9001625

http://dx.doi.org/10.1039/b812045e

http://dx.doi.org/10.1063/1.1415081

http://dx.doi.org/10.1063/1.2107627

http://dx.doi.org/10.1063/1.1929740

http://dx.doi.org/10.1063/1.2899649

http://dx.doi.org/10.1021/ct200919e

http://dx.doi.org/10.1021/ct200919e

http://dx.doi.org/10.1063/1.3173828

http://dx.doi.org/10.1021/cr2002239

http://dx.doi.org/10.1063/1.3023123

http://dx.doi.org/10.1063/1.3023123

http://dx.doi.org/10.1063/1.464746

http://dx.doi.org/10.1063/1.474917

http://dx.doi.org/10.1063/1.474225

http://dx.doi.org/10.1063/1.476238

http://dx.doi.org/10.1063/1.4718427

http://dx.doi.org/10.1016/0003-4916(83)90284-1

http://dx.doi.org/10.1146/annurev.pc.32.100181.002043

http://dx.doi.org/10.1080/00268970500083952

http://dx.doi.org/10.1016/0009-2614(89)85202-9


http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.4913364

http://dx.doi.org/10.1063/1.475961

http://dx.doi.org/10.1063/1.476376

http://dx.doi.org/10.1002/(SICI)1097-461X(1998)70:4/5%3C601::AID-QUA6%3E3.0.CO;2-Z

http://dx.doi.org/10.1063/1.4862501

http://dx.doi.org/10.1063/1.4862501

au pure · molecular response properties from a hermitian eigenvalue equation for a time-periodic...

Documents