Time-dependent N-electron valence perturbation theory with matrix product statereference wavefunctions for large active spaces and basis sets: Applications to thechromium dimer and all-trans polyenesAlexander Yu. Sokolov, Sheng Guo, Enrico Ronca, and Garnet Kin-Lic Chan

Citation: The Journal of Chemical Physics 146, 244102 (2017); doi: 10.1063/1.4986975View online: http://dx.doi.org/10.1063/1.4986975View Table of Contents: http://aip.scitation.org/toc/jcp/146/24Published by the American Institute of Physics

THE JOURNAL OF CHEMICAL PHYSICS 146, 244102 (2017)

Time-dependent N-electron valence perturbation theory with matrixproduct state reference wavefunctions for large active spaces and basissets: Applications to the chromium dimer and all-trans polyenes

Alexander Yu. Sokolov,a) Sheng Guo, Enrico Ronca, and Garnet Kin-Lic Chanb)

Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena,California 91125, USA

(Received 8 April 2017; accepted 7 June 2017; published online 22 June 2017)

In earlier work [A. Y. Sokolov and G. K.-L. Chan, J. Chem. Phys. 144, 064102 (2016)], we introduced atime-dependent formulation of the second-order N-electron valence perturbation theory (t-NEVPT2)which (i) had a lower computational scaling than the usual internally contracted perturbation formu-lation and (ii) yielded the fully uncontracted NEVPT2 energy. Here, we present a combination oft-NEVPT2 with a matrix product state (MPS) reference wavefunction (t-MPS-NEVPT2) that allowsus to compute uncontracted dynamic correlation energies for large active spaces and basis sets, usingthe time-dependent density matrix renormalization group algorithm. In addition, we report a low-scaling MPS-based implementation of strongly contracted NEVPT2 (sc-MPS-NEVPT2) that avoidscomputation of the four-particle reduced density matrix. We use these new methods to computethe dissociation energy of the chromium dimer and to study the low-lying excited states in all-transpolyenes (C4H6 to C24H26), incorporating dynamic correlation for reference wavefunctions with up to24 active electrons and orbitals. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4986975]

I. INTRODUCTION

Recent advances in molecular electron correlation meth-ods have made it possible to describe dynamic correlationin weakly correlated systems with several hundreds of atomsby taking advantage of accurate local correlation approxima-tions.1–4 Similarly, efficient methods for static correlation5–11

now exist that allow us to simulate systems with a large num-ber of strongly correlated open-shell orbitals.12–15 Yet, despitethese efforts, the challenge still remains to efficiently describedynamic correlation in the presence of significant static cor-relation. This is a scenario one faces when treating somecomplicated chemical systems, such as transition metal com-pounds with multiple metals.12,15 Technically, the challengesare the high computational cost of the existing canonical algo-rithms, algebraic complexity of the underlying equations,16

and numerical instabilities in the simulations.17–20

The standard approach to electron correlation in multi-reference (strongly correlated) systems is to first carry out ahigh-level description of static correlation in a small subsetof near-degenerate (active) orbitals, followed by a lower-leveldescription of dynamic correlation in the remaining large setof core and external orbitals. Here, the main challenge is tocombine the high-level and low-level descriptions withoutsacrificing their accuracy, while retaining a low computa-tional cost. For this purpose, most multi-reference dynamiccorrelation theories use the internal contraction approxima-tion,21–23 which represents the complicated active-space wave-function in terms of simpler quantities, such as the reduceddensity matrices (RDMs). While this approximation has been

a)Electronic mail: alexander.y.sokolov@gmail.comb)Electronic mail: gkc1000@gmail.com

enormously useful in quantum chemistry,21–36 internally con-tracted methods require the computation of high-order (three-and four-particle) RDMs. These become prohibitively expen-sive for larger active spaces.

We have recently proposed a time-dependent formula-tion of multi-reference perturbation theory37 that efficientlydescribes dynamic correlation by representing the active-spacewavefunction in terms of compact time-dependent quanti-ties (active-space imaginary-time Green’s functions (GFs)).This does not introduce any additional approximations nordo high-order RDMs appear in the equations. The result-ing time-dependent theory is equivalent to the fully uncon-tracted perturbation theory but in fact displays a lower com-putational scaling than the internally contracted approxi-mations, particularly with respect to the number of activeorbitals.

Our previous work described the implementation of thetime-dependent second-order N-electron valence perturbationtheory (t-NEVPT2) for complete active-space self-consistentfield (CASSCF) reference wavefunctions. Here, we present anew implementation combining t-NEVPT2 with matrix prod-uct state (MPS) reference wavefunctions, thus allowing usto describe dynamic correlation in multi-reference systemswith much larger active spaces. As we will demonstrate,the resulting t-MPS-NEVPT2 approach requires computationof up to two-particle imaginary-time Green’s functions thatcan be evaluated using the time-dependent density matrixrenormalization group (td-DMRG) algorithm5,8–11 in a highlyparallel fashion. In addition, for comparison, we present alow-scaling MPS-based implementation of the strongly con-tracted NEVPT2 (sc-MPS-NEVPT2),22,23,35 which does notrequire computation of the four-particle density matrix. Todemonstrate the capabilities of these new methods, we apply

0021-9606/2017/146(24)/244102/14/$30.00 146, 244102-1 Published by AIP Publishing.

244102-2 Sokolov et al. J. Chem. Phys. 146, 244102 (2017)

them to compute the dissociation energy of the chromiumdimer and to study the low-lying excited states in the all-transpolyenes.

This paper is organized as follows. We begin with a briefoverview of t-NEVPT2 (Sec. II A) and matrix product statewavefunctions (Sec. II B). We then describe the details of ourt-MPS-NEVPT2 implementation (Sec. II C) and outline thecomputational details (Sec. III). In Sec. IV, we use t-MPS-NEVPT2 to compute energies along the N2 dissociation curve(Sec. IV A), the dissociation energy of the chromium dimer(Sec. IV B), and the vertical excitation energies in all-transpolyenes (Sec. IV C). In Sec. V, we present the conclusionsof our work. The details of the efficient sc-MPS-NEVPT2implementation are described in Appendix B.

II. THEORYA. Time-dependent formulation of N-electronvalence perturbation theory (t-NEVPT2)

We begin with a brief overview of multi-reference pertur-bation theory in its time-dependent form. Our starting point is azeroth-order electronic wavefunction |Ψ0〉 computed in a com-plete active space (CAS) of molecular orbitals. We require |Ψ0〉

to be an eigenfunction of a zeroth-order Hamiltonian H (0). Fol-lowing our previous work,37 we assume that H (0) is the DyallHamiltonian,38 H (0) = HD, defined as

HD =∑

i

εia†

i ai +∑

a

εaa†aaa + Hact , (1)

where we partition the spin-orbitals into three sets: (i) core(doubly occupied) with indices i, j, k, l; (ii) active with indicesu, v, w, x, y, z; and (iii) external (unoccupied) with indicesa, b, c, d. The orbital energies εi and εa are defined asthe core and external eigenvalues of the generalized Fockoperator,

f qp = hq

p +∑

rs

vqsprγ

rs , (2)

where hqp and vrs

pq are the usual one- and antisymmetrized two-

electron integrals, and γpq = 〈Ψ0 |a

†paq |Ψ0〉 is the one-particle

density matrix of |Ψ0〉. The Hact operator describes the two-electron interaction in the active space

Hact =∑xy

(hyx +

∑i

vyixi )a

†xay +

14

∑xywz

vzwxy a†xa†yawaz . (3)

Having specified |Ψ0〉 and H (0), we now consider anexpansion of the energy with respect to the perturbationλV = λ(H − HD). Rather than expressing the energy in atime-independent perturbation series, as is commonly donein the Rayleigh-Schrodinger perturbation theory, we considera time-dependent expansion37 with respect to the perturbationoperator V (τ) = e(HD−ED)τ V e−(HD−ED)τ . The second-orderenergy contribution can be written as

E(2) = −

∫ ∞0

dτ⟨Ψ0 |V

′†(τ)V ′(0)|Ψ0

⟩= −

∫ ∞0

dτ 〈Ψ0 |V′†e−(HD−ED)τV ′ |Ψ0〉 , (4)

where τ is the imaginary time and V ′ denotes the partof V that contributes to the first-order wavefunction |Ψ(1) 〉

(V ′ = QV , Q = 1 − |Ψ0〉 〈Ψ0 |). Equation (4) is the Laplacetransform of the second-order energy expression from theRayleigh-Schrodinger perturbation theory, which yields theexact (uncontracted) energy of the second-order N-electronvalence perturbation theory (NEVPT2).22,23 However, unlikethe standard uncontracted theory, the time-dependent energyexpression (4) does not require the costly representation of thefirst-order wavefunction in a large space of determinants andcan be evaluated very efficiently. Expanding the perturbationoperator V into classes of excitations, the second-order energycan be expressed as a sum of 8 terms related to the numbersof holes and particles created in the core and external spaces(see Refs. 22 and 39 for a complete definition)

E(2) = E[0] + E[+1] + E[−1] + E[+2] + E[−2]

+ E[+1′] + E[−1′] + E[0′] . (5)

The central quantities to compute in Eq. (5) are the one- andtwo-particle imaginary-time Green’s functions (1- and 2-GF)in the active space. Specifically, the E[1] and E[+1] termsrequire computation of 1-GFs (e.g., G(τ) =

⟨Ψ0 |a

†x(τ)ay |Ψ0

⟩),

while 2-GFs are necessary to compute the E[2], E[+2], andE[0′] contributions (e.g., G(τ) =

⟨Ψ0 |a

†w(τ)a†x(τ)ayaz |Ψ0

⟩).

The E[−1′] and E[+1′] components formally involve the three-particle active-space Green’s function but can be efficientlyevaluated by forming intermediates. Defining

h†p =∑

x

hpxa†x +

12

∑xyz

vpzxy a†xa†yaz, (6)

hqp = hq

p +∑

i

vqipi , (7)

the E[−1′] and E[+1′] contributions can be expressed as expec-tation values of the single-index operators

E[+1′] = −

∫ ∞0

∑i

eεiτ 〈Ψ0 |hi(τ)h†i |Ψ0〉 dτ , (8)

E[−1′] = −

∫ ∞0

∑a

e−εaτ 〈Ψ0 |h†a(τ)ha |Ψ0〉 dτ . (9)

The bottleneck of the time-dependent NEVPT2 (t-NEVPT2)algorithm, when |Ψ0〉 is expanded in determinants, is theevaluation of the 2-GF that has O(Nτ × Ndet × N6

act) com-putational cost, where Nact is the number of active orbitals,Ndet is the dimension of the CAS Hilbert space, and Nτ is thenumber of time steps (Nτ ∼ 10-20). As a result, t-NEVPT2has a significantly lower scaling with Nact than the internallycontracted NEVPT2 approaches,22,23 which require compu-tation of up to the four-particle reduced density matrix (4-RDM) with O(Ndet × N8

act) cost. However, the computationalcost of these two types of NEVPT2 formulations still scalesexponentially with Nact , since the number of determinantsin the zeroth-order wavefunction Ndet ∼ O(eNact ), and solv-ing for |Ψ0〉 becomes a bottleneck for large active spaces. InSecs. II B and II C, we will describe how this limitation canbe ameliorated by expressing |Ψ0〉 in the matrix product state(MPS) representation.

244102-3 Sokolov et al. J. Chem. Phys. 146, 244102 (2017)

B. Matrix product state (MPS) wavefunctionsand the density matrix renormalization group(DMRG) algorithm

In this section, we very briefly introduce matrix productstate (MPS) wavefunctions, which will be used extensively inthis work. Further details about MPS algorithms can be foundin references such as Refs. 40 and 41. The MPS is a nonlinearwavefunction composed of a product of tensors where eachtensor corresponds to one or more orbitals in the basis. Themost common form of the MPS wavefunction is the one-siteMPS written in the following form:

|Ψ〉 =∑

n1...nk

An1 An2 . . .Anp . . .Ank |n1n2 . . . nk 〉, (10)

where |n1n2 . . . nk 〉 is a Slater determinant, Anp is a ten-sor that corresponds to only one orbital p with occupancynp ∈ | 〉 , | ↑ 〉 , | ↓ 〉 , | ↑↓ 〉 (also referred as the site p), and thetotal number of tensors Anp equals the number of orbitals kin the basis (for a CAS wavefunction, k = Nact). For each pin the range 2, . . . , k − 1, Anp is a matrix with fixed dimen-sions M ×M. The dimensions of An1 and Ank are 1 ×M andM × 1, respectively. As a result, for a specified set of occu-pation numbers n1n2 . . . nk , the product of tensors in Eq. (10)yields a scalar corresponding to the coefficient of the determi-nant |n1n2 . . . nk 〉 in the expansion of the wavefunction |Ψ〉.The parameter M, referred to as the bond dimension, controlsthe flexibility of the MPS, which increases as M increases.

The most common algorithm when working with MPS isthe sweep algorithm, where linear algebra operations are car-ried out on a single tensor at a time, sweeping successivelythrough the orbitals 1 . . . k. The density matrix renormaliza-tion group (DMRG) is the prototypical version of such analgorithm and corresponds to a variational energy minimiza-tion using a sweep through the tensors. At a given site in thesweep algorithm, the tensor that is being operated on yields thelinear expansion coefficients of the wavefunction in a many-body renormalized basis. For example, at site p, we define theleft and right renormalized bases as

|lαp 〉 =∑

n1...np−1

[An1 An2 . . .Anp−1 ]αp |n1n2 . . . np−1 〉 , (11)

|rαp+1 〉 =∑

np+1...nk

[Anp+1 Anp+2 . . .Ank ]αp+1|np+1np+2 . . . nk 〉 ,

(12)and the total wavefunction becomes

|Ψ〉 =∑

lαp nprαp+1

[Anp ]αpαp+1|lαp nprαp+1 〉 . (13)

In this context, we can think of Anp as representing a “sitewavefunction,” and operators projected into the renormalizedspace |lαp nprαp+1 〉 act on this site wavefunction as “site oper-ators.” General numerical algorithms for wavefunctions canbe converted into sweep algorithms for MPS by identifyingwavefunctions with site wavefunctions and operators with siteoperators. This idea is used in the time-dependent DMRG(td-DMRG) algorithm that we employ below.

C. t-NEVPT2 with MPS reference wavefunctions(t-MPS-NEVPT2)1. Overview of the algorithm

We now describe the implementation of t-NEVPT2 forthe MPS reference wavefunctions, which we denote as t-MPS-NEVPT2. To aid the discussion, we first rewrite the t-NEVPT2energy in a compact form

E(2) = E(2)ext +

∑[i]

E[i]act , (14)

where we combine terms involving the core and externalorbitals

E(2)ext =

14

∑ijab

vabij v

ijab

εi + εj − εa − εb+

∑ia

hai hi

a

εi − εa+ 2

∑ixya

hiav

ayix γ

xy

εi − εa

(15)and introduce a shorthand notation E[i]

act for the active-spacecomponents of the energy contributions E[i] in Eq. (5) (i ∈+1,−1, +2,−2, +1′,−1′, 0′). Each component E[i]

act can beexpressed in the general form

E[i]act = −

∫ ∞0

ε [i](τ)G[i](τ) dτ , (16)

where G[i](τ) is the Green’s function tensor and ε [i](τ) isthe integral prefactor tensor that can be computed using theone- and two-electron integrals and orbital energies (εi andεa). Explicit equations for the elements of tensors ε [i](τ) andG[i](τ) are given in Table I.

The general t-NEVPT2 algorithm consists of three steps:(i) Green’s function evaluation, (ii) computation of the energyas a function of imaginary time (τ), and (iii) integration inimaginary time. In step (i), the elements of G[i](τ) are com-puted for a set of τ values (time steps, τk). In step (ii), theprefactors ε [i](τ) are evaluated using the equations in Table I,and the energy contributions at each time step are com-puted as E[i]

act(τk) = ε [i](τk)G[i](τk). For example, E[−2]act (τk)

is computed as the dot product of two tensors: ε [−2](τk)= ε

xyzw(τk) = 1

8

∑ab v

abzwv

xyabe−(εa+εb)τk and G[−2](τk)=Gxy

zw(τk)

= 〈Ψ0 |a†x(τk)a†y(τk)awaz |Ψ0〉. Finally, in step (iii), each energy

contribution E[i]act is evaluated by fitting the computed values

E[i]act(τk) to an exponential function

∑i E(τ) = aie−biτ and inte-

grating the obtained result. We note that the above-outlinedalgorithm can be used to implement t-NEVPT2 for refer-ence wavefunctions in any representation (e.g., determinant-based37 or MPS-based). Only the details of step (i) depend onthe explicit form of |Ψ0〉. In Sec. II C 2, we will describe howimaginary-time Green’s functions G[i](τ) are evaluated in theMPS representation.

2. Green’s function evaluation

To organize our MPS implementation of the imaginary-time Green’s functions, we express the elements of G[i](τ)(Table I) in the general form G[i]

µν(τ) = 〈Φ[i]µ (τ)|Φ[i]

ν 〉, where|Φ

[i]ν 〉 is obtained by applying all creation and annihilation

operators at τ = 0 on |Ψ0〉 (e.g., ax |Ψ0〉, a†xay |Ψ0〉), whilethe application of operators at τ = τ′ yields |Φ[i]

µ (τ′)〉 (e.g.,

ax(τ′) |Ψ0〉, a†x(τ′)ay(τ′) |Ψ0〉). In our notation, indices µ and ν

244102-4 Sokolov et al. J. Chem. Phys. 146, 244102 (2017)

TABLE I. Equations for the elements of tensors ε[i](τ) and G[i](τ) used in the evaluation of the t-NEVPT2active-space energy contributions E[i]

act in Eq. (16).

[i] ε[i](τ) G[i](τ)

[+1] εyx (τ)=

12

∑ijav

ayij v

ijaxe(εi + εj − εa)τ Gy

x(τ)= 〈Ψ0 |ax(τ)a†y |Ψ0〉

[1] ε xy (τ)=

12

∑iabvab

iy vixabe(εi − εa − εb)τ Gx

y(τ)= 〈Ψ0 |a†x (τ)ay |Ψ0〉

[+2] ε zwxy (τ)=

18

∑ijvzw

ij vijxye(εi + εj )τ Gzw

xy (τ)= 〈Ψ0 |ax(τ)ay(τ)a†wa†z |Ψ0〉

[2] εxyzw (τ)=

18

∑abvab

zw vxyabe−(εa + εb)τ Gxy

zw (τ)= 〈Ψ0 |a†x (τ)a†y (τ)awaz |Ψ0〉

[0′] ε xwyz (τ)=

∑iavaw

iz vixaye(εi − εa)τ Gxw

yz (τ)= 〈Ψ0 |a†x (τ)ay(τ)a†waz |Ψ0〉

[+1′] ε ii (τ)= eεiτ Gi

i(τ)= 〈Ψ0 |hi(τ)h†i |Ψ0〉

[−1′] εaa (τ)= e−εaτ Ga

a(τ)= 〈Ψ0 |h†a(τ)ha |Ψ0〉

run over the total number of states |Φ[i]µ 〉 (i.e., Nact for ax |Ψ0〉

or N2act for a†xay |Ψ0〉).We begin the computation of G[i](τ) by optimizing the

zeroth-order MPS wavefunction |Ψ0〉 using the DMRG algo-rithm with bond dimension M0. To evaluate the elements ofG[i](τ), we employ the algorithm below.

1. Compute initial wavefunctions |Φ[i]µ 〉 for every µ. Each

wavefunction |Φ[i]µ 〉 is computed as an individual MPS

and stored on disk. Evaluation of |Φ[i]µ 〉 ([i] ∈ [+1],

[1], [+2], [2], [0′]) requires applying the operatora†x or ax on |Ψ0〉 one or more times. The resultingMPS is of exactly the same bond dimension as theoriginal MPS. For [i] ∈ [+1′], [−1′], the wavefunc-tions |Φ[+1′]

µ 〉 and |Φ[−1′]µ 〉 are computed by compressing

the MPS form of h†i |Ψ0 〉 and ha |Ψ0〉, where h†i and

ha are defined as in Eq. (6). Since applying h†i or ha

involves a summation over the active-space labels x, theresulting MPS is of larger bond dimension than |Ψ0〉.We use variational compression to obtain |Φ[+1′]

µ 〉 and

|Φ[−1′]µ 〉 MPS by minimizing | | |Φ[+1′]

i 〉 − h†i |Ψ0 〉 | | and

| | |Φ[−1′]a 〉 − ha |Ψ0〉 | |, where |Φ[+1′]

i 〉 and |Φ[−1′]a 〉 have

bond dimension M1 > M0. To maximize the efficiencyof computing the overlaps appearing in the compres-sion (e.g., 〈Φ[+1′]

i | h†i |Ψ0 〉), we use the standard DMRGformalism of “normal” and “complementary” operators,which requires building O(k) such operators from theleft and right blocks. In practice, we find that M1 ≈ 2M0

usually provides a sufficiently accurate compression.Overall, this step of the t-MPS-NEVPT2 algorithm hasO(N2

actM30 ) + O(NextN2

actM21 M0) scaling for computing

all |Φ[i]µ 〉 wavefunctions with [i] ∈ [+2], [−2], [0′] and

[i] ∈ [+1′], [−1′], respectively (Next is the number ofexternal orbitals).

2. Propagate wavefunctions |Φ[i]µ 〉 in imaginary time τ

according to the time-dependent Schrodinger equation|Φ

[i]µ (τ)〉 = e−(HD−ED)τ |Φ

[i]µ 〉. In this step, the algorithm

we use is based on the time step targeting time-dependent

DMRG algorithm of Feiguin and White described inRef. 42, with some small modifications. Specifically, weuse the embedded Runge-Kutta [ERK (4,5)] time stepalgorithm,43 which allows us to automatically control thetime step δτ based on the error estimate of the fifth-orderapproximation of the propagator e−(HD−ED)τ . In the ERK(4,5) method, the wavefunction at time τ′ = τ + δτ isapproximated to O((δτ)5) as

|Φ[i]µ (τ′)〉 ≈ |Φ[i]

µ (τ)〉 +6∑

n=1

cn |kn(δτ)〉 , (17)

where states |kn(δτ)〉 are obtained by successively apply-ing the zeroth-order Hamiltonian six times

|kn(δτ)〉 = −δτ(HD−ED)|Φ[i]µ (τ)〉 +

n− 1∑m=1

bnm |km(δτ)〉.

(18)

The coefficients bnm and cn are given in Ref. 43. In addi-tion to the fifth-order approximation, Eq. (17) can be usedto compute the fourth-order estimate of |Φ[i]

µ (τ′)〉, whichwe denote as |Φ[i]

µ (τ′)[4]〉. This can be done by setting cn

= cn[4], where the fourth-order coefficients cn[4] can befound in Ref. 43. This gives an estimate of the time step

error, ∆ =√

|Φ[i]µ (τ′)〉 − |Φ[i]

µ (τ′)[4]〉 .

As mentioned in Sec. II B, we can adapt the above generalwavefunction propagation to the propagation of a MPSwithin a sweep algorithm. This is the idea behind thetime step targeting td-DMRG. The quantities in the aboveequations are then to be interpreted as applying to eachsite, i.e., the wavefunction |Φ[i]

µ 〉 is now the site wave-function, the states |kn 〉 are vectors in the renormalizedsite basis, and the Hamiltonian HD is projected into therenormalized site basis. The site error in the propagatorapproximation is estimated from the site wavefunction as∆p (where p is the site index). As in Ref. 42, the site wave-functions |Φ[i]

µ (τ)〉, |Φ[i]µ (τ + δτ/3)〉, |Φ[i]

µ (τ + 2δτ/3)〉,and |Φ[i]

µ (τ + δτ)〉 are averaged in the density matrix to

244102-5 Sokolov et al. J. Chem. Phys. 146, 244102 (2017)

construct density matrix eigenvectors to update the sitewavefunction, before proceeding to the next step in thesweep.

We begin the time propagation with a small value ofδτ ∼ 10−3 and determine the new time step aftereach propagation as δτ′ = min(2 × δτ, δτemb), where

δτemb = δτ |∆Econv

maxp(∆p )|1/5

and∆Econv is the specified accu-

racy threshold. Note that if maxp(∆p)>∆Econv , thetime step is decreased. In this case, we repeat thetime propagation using the smaller time step. The com-putational cost of a single time step for all states|Φ

[i]µ 〉 has O(N5

actM30 ) + O(N6

actM20 ) + O(NextN3

actM31 )

+ O(NextN4actM

21 ) scaling.

3. Compute Green’s function elements at every timestep τk as the overlap between two MPSs G[i]

µν(τk)= 〈Φ

[i]µ (τk)|Φ[i]

ν 〉. The computational cost of computingall G[i]

µν(τk) elements has O(N5actM

30 ) scaling.

The cost of the t-MPS-NEVPT2 algorithm outlined above isdominated by the computation of the 2-GF, which requirestime-propagation of O(N2

act) MPS wavefunctions |Φ[i]µ 〉 and

evaluation of O(N4act) matrix elements G[i]

µν(τk), leading toa total O(NτN5

actM30 ) + O(NτN6

actM20 ) computational scaling,

where Nτ ∼ 10-20 is the number of time steps. While this isless than the cost of computing the 4-RDM in DMRG, comput-ing the 3-RDM, which has O(N4

actM30 ) +O(N6

actM20 ) scaling, is

of lower cost.35 The higher scaling of our 2-GF implementa-tion is because each MPS |Φ[i]

µ 〉 is propagated in imaginarytime independently, while in the efficient 3-RDM implemen-tation, a common set of renormalized operators is reused tocompute all elements of the density matrix together. How-ever, the higher scaling of this particular implementation isnot an intrinsic property of the time-dependent theory. In thedeterminant basis, the computation of the 2-GF and 3-RDMhas the same computational scaling. For MPS, it is possibleto similarly devise an algorithm to compute the 2-GF withthe same cost as the 3-RDM by propagating multiple |Φ[i]

µ 〉

wavefunctions using the same renormalized basis (similar toperforming a state-averaged DMRG optimization, and relatedto the algorithm used in Ref. 44). An advantage of our currentt-MPS-NEVPT2 implementation, however, is that it is easilyparallelized by separating the correlation energy contributionsinto O(N2

act) components,

E[i]act = −

∑µ

∫ ∞0

∑ν

ε [i]µν(τ)G[i]

µν(τ) dτ =∑µ

E[i]µ , (19)

where each component E[i]µ can be evaluated independently

with O(NτN3actM

30 ) + O(NτN4

actM20 ) cost. In addition, as we

will demonstrate in Sec. IV, the energy terms that depend onthe 2-GF converge very quickly with increasing M0 and do notrequire a large bond dimension.

3. Spin-adaptation

In Sec. II C 2, we discussed the evaluation of the G[i]µν(τ)

matrix elements in terms of creation and annihilation opera-tors a†p(τ) and ap(τ). In a non-spin-adapted DMRG algorithm,a† and a are the spin-orbital operators, and the index p corre-sponds to a spatial orbital with a particular spin. For example,

for [i] = [2], the spin-orbital 2-GF has the following form:G

xρyσzλwκ (τ) = 〈Ψ0 |a

†xρ (τ)a†yσ (τ)awκ aaλ |Ψ0〉, where we use the

Roman characters x, y, w, and z to denote spatial orbitals andGreek characters ρ,σ, κ, λ to denote the spin of these orbitals.Here, awκ and azλ are simple operators, and application of a pairof operators awκ azλ |Ψ0〉 results in a single MPS wavefunction|Φ

[−2]wκ zλ 〉.

In a spin-adapted DMRG implementation,45 a† and a referto spin tensor creation and annihilation operators, and stringsof these operators generate multiple eigenstates of the S2 oper-ator when acting on a reference state. As a result, expectationvalues of spin tensor operators depend on the spin quantumnumbers, e.g., 〈Ψ0 |[a

†x(τ)a†y(τ)]S1

M1[awaz]

S2M2S3M3|Ψ0〉, where

we use brackets to denote the coupling of spins S1, S2, andS3 for different tensor products. As an example, we considerthe evaluation of G[−2]

µν (τ) for a reference state |Ψ0〉with S = 0.(Generalization to reference states with S , 0 is straightfor-ward and is simplified using singlet embedding.)45,46 Applyinga pair of tensor operators on the reference state [awaz]

S1M1|Ψ0 〉

results in two sets of eigenstates |Φ[−2]wz

S1

M1〉 with S1 = 0, 1.

The wavefunctions |Φ[−2]wz

S1

M1〉 are propagated in imaginary time

for each value of S1, and the matrix elements are computedusing Clebsch-Gordan coefficients cS1,S2,S3

M1,M2,M3as (Gxy

zw)S1 (τ)

=∑

M1(−1)S1+M1 cS1,S1,0

M1,−M1,0 〈Ψ0 |[a†x(τ)a†y(τ)]S1

M1[awaz]

S1M1|Ψ0〉,

where we used the fact that S1 = S2 and S3 = 0 for a closed-shell |Ψ0〉. Expanding the spin tensor operators [awaz]S1 asa combination of spin-orbital operators awρ and azσ leads toa system of linear equations that can be solved to obtain thespin-orbital 2-GF elements G

xρyσzλwκ (τ) from the matrix elements

(Gxyzw)S1 (τ) (see Appendix A for explicit equations).

III. COMPUTATIONAL DETAILS

All DMRG computations for the reference wavefunctions,reduced density matrices, and Green’s functions were per-formed using the Block program.45 The t-MPS-NEVPT2 andsc-MPS-NEVPT2 correlation energies were computed usingPyscf,47 through its interface with Block. For all NEVPT2computations, the DMRG reference wavefunctions were fullyoptimized with respect to active-external orbital rotationsusing the DMRG-SCF algorithm implemented in Pyscf. Wedenote active spaces used in DMRG-SCF as (ne, mo), wheren is the number of active electrons and m is the number oforbitals. We used tight convergence parameters for the energyin the DMRG sweeps (109 Eh) and orbital optimization iter-ations (106 Eh). In t-MPS-NEVPT2, the accuracy of timepropagation was controlled by setting ∆Econv = 10−4 Eh forall systems but the N2 molecule, where the tighter threshold∆Econv = 10−6 Eh was used (see Sec. II C 2 for details). Spe-cific details pertaining to our computations of the all-transpolyenes are described in Sec. IV C.

As described in Sec. II C, the bond dimensions necessaryto represent the reference MPS (M0) and the compressed MPS(M1) are usually different (M0 < M1). However, to simplifythe discussion, in our computations, we set M0 and M1 to thesame value, and whenever we refer to bond dimension M, weimply that M0 = M1 = M.

244102-6 Sokolov et al. J. Chem. Phys. 146, 244102 (2017)

IV. RESULTSA. Analysis of numerical accuracy: N2 dissociation

We first investigate the numerical accuracy of our algo-rithm by computing the errors of t-MPS-NEVPT2 as a func-tion of the bond dimension relative to the uncontractedNEVPT2 energies computed using our t-NEVPT2/CASSCFimplementation.37 For this purpose, we evaluate the t-MPS-NEVPT2 energies for a range of geometries along the ground-state potential energy curve (PEC) of the N2 molecule. Ateach geometry, we optimize the molecular orbitals using theCASSCF method in the (10e, 10o) active space and performa single DMRG computation using these orbitals to obtain aMPS wavefunction with a large bond dimension M ′ = 1000.We subsequently compress this reference wavefunction downto an MPS with a smaller bond dimension (M = 50, 100, 200)to be used in the NEVPT2 computation. In this section, we useDunning’s cc-pVQZ basis.48

Figure 1 plots the t-MPS-NEVPT2 correlation energyerrors for different values of the bond dimension M, as well asthe error in sc-NEVPT2/CASSCF due to the strong contrac-tion approximation. Notably, even with a small M = 50, theerror from finite bond dimension in t-MPS-NEVPT2 is signif-icantly smaller than the contraction error of sc-NEVPT2 forall geometries along the N2 potential energy curve (PEC). Asthe bond dimension M increases, the t-MPS-NEVPT2 errorsdecrease, becoming smaller than 0.1 mEh at M = 200 forall energy points. As can be seen from Fig. 1, the conver-gence of t-MPS-NEVPT2 energies is not monotonic along thedissociation curve. In particular, in the range of short bonddistances (r < 2.1 Å), the correlation energies converge sig-nificantly faster to the exact (uncontracted) NEVPT2 energiesthan the energies at the dissociation limit (r ≥ 2.1 Å). Thisresult is consistent with the fact that both the zeroth-orderand first-order wavefunctions at longer bond distances containcontributions from a larger number of determinants than at

FIG. 1. Error in the t-MPS-NEVPT2 correlation energy relative to that of theuncontracted NEVPT2 theory for N2 as a function of the N–N bond distanceand the DMRG bond dimension M. Computations employed the (10e, 10o)active space and the cc-pVQZ basis set. Also shown are the strong contractionerrors computed as the difference between sc-NEVPT2 and t-NEVPT2 basedon the CASSCF reference.

the shorter bond distances. To illustrate this, we computedthe t-MPS-NEVPT2 energies using a large bond dimension M= 1000 (Fig. 1). For all geometries with r ≤ 2.1 Å, our t-MPS-NEVPT2 algorithm achieves an accuracy of better than 106

Eh, while errors in the range of∼10−5 Eh still remain in the dis-sociation limit. The slow convergence of correlation energieswith bond dimension near the dissociation limit has also beenobserved in Sharma and Chan’s MPS-PT2 study of N2 disso-ciation using the cc-pVDZ basis set and the (10e, 8o) activespace.49

We now analyze the errors in the t-MPS-NEVPT2correlation energy from two different contributions: (i) thesum of active-space energy components that depend on the1- and 2-GF

∑[i] E[i]

act ([i] ∈ [+1], [−1], [+2], [−2], [0′]) and(ii) the sum of energy terms that depend on the contracted 3-GF (E[+1′]

act + E[−1′]act ). These errors are plotted in Figs. 2(a) and

2(b), for different values of r and M. Between the two energycontributions, the contribution (i) from the 1- and 2-GF exhibits

FIG. 2. Error in the t-MPS-NEVPT2 correlation energy contributions for N2 as a function of the N–N bond distance and the DMRG bond dimen-sion M. Plot (a) shows the errors for E[i]

act ([i] ∈ [+1], [−1], [+2], [−2], [0′]) components that depend on the 1- and 2-GF, while plot (b) presents the

errors for E[i]act ([i] ∈ [+1′], [−1′]) that depend on the contracted 3-GF. Computations employed the (10e, 10o) active space and the cc-pVQZ basis

set.

244102-7 Sokolov et al. J. Chem. Phys. 146, 244102 (2017)

significantly faster convergence with increasing M than thecontribution (ii) from the contracted 3-GF. At M = 50, con-tribution (i) shows errors of ∼10−4 Eh for a large range ofgeometries, while the errors in contribution (ii) are an order ofmagnitude larger. Comparing Figs. 2(a) and 2(b), we observethat the energy components (i) and (ii) achieve similar accu-racy when computed with bond dimensions of M and 2M,respectively. In fact, for a specified M, the errors in thetotal t-MPS-NEVPT2 correlation energy are dominated bythe errors in contribution (ii) [cf. Figs. 1 and 2(b)]. As dis-cussed in Sec. II C 2, the observed slower convergenceof contribution (ii) is the result of the MPS compressionused to evaluate E[+1′]

act and E[−1′]act , which requires a larger

bond dimension than the one used in the optimization ofthe reference MPS for an accurate representation. Since thetwo contributions can be evaluated separately, we recom-mend to compute contribution (i) using a small bond dimen-sion M and to use a larger bond dimension ∼2M to obtaincontribution (ii).

B. Chromium dimer

The chromium dimer (Cr2) is a notoriously challeng-ing system for multi-reference methods. Computational stud-ies29,35,37,50–60 have demonstrated that including the dynamiccorrelation in combination with the multi-configurationaldescription of static correlation and large basis sets is cru-cial to properly describe the ground-state potential energycurve (PEC) of this molecule. In 2011, Kurashige and Yanaishowed that the Cr2 dissociation curve can be accuratelydescribed by combining the CASPT2 variant of the inter-nally contracted perturbation theory with DMRG in a large(12e, 28o) active space,29 which included the 3d, 4d, 4s,and 4p orbitals of the Cr atoms. Very recently (2016),Guo et al. computed Cr2 PEC using a combination of sc-NEVPT2 and DMRG (sc-DMRG-NEVPT2) with the (12e,22o) active space (3d, 4d, and 4s orbitals of Cr).35 Atthe complete basis set limit (CBS), the Cr2 PEC fromsc-DMRG-NEVPT2 agrees well with the dissociation curvefrom the experiment, yielding spectroscopic parameters for theequilibrium bond length re = 1.656 Å and a binding energy

De = 1.432 eV that are close to the experimental rexpe = 1.679 Å

and Dexpe = 1.47 ± 0.1 eV.61 However, the Cr2 binding energy

in sc-NEVPT2 can be significantly affected by the strong con-traction approximation, as we demonstrated in our t-NEVPT2study of the Cr2 PEC using the (12e, 12o) active space.37 Inthis work, we recompute the Cr2 equilibrium binding energyin the large (12e, 22o) active space using our t-MPS-NEVPT2algorithm.

We obtained the MPS reference wavefunction asdescribed in the sc-DMRG-NEVPT2 study by Guo et al.35

In short, we used the (12e, 22o) active space and optimizedthe molecular orbitals using the DMRG-SCF algorithm withM ′ = 1000 (no frozen core). We then performed a DMRGcomputation with M ′ = 6000 using the optimized orbitalsto obtain an accurate reference wavefunction. To computeimaginary-time Green’s functions, the reference MPS with thelarge bond dimension M ′ was compressed down to an MPSwith a smaller bond dimension M. We computed each t-MPS-NEVPT2 energy contribution E[i]

act by starting with M = 500and increasing M until convergence. To account for the scalarrelativistic effects, we used the cc-pwCV5Z-DK basis set62

combined with the spin-free one-electron variant of the X2CHamiltonian.63–65 Since strong contraction reduces the bind-ing energy37 in Cr2, the CBS-extrapolated equilibrium bondlength re computed using the uncontracted t-MPS-NEVPT2method is expected to be somewhat shorter than re = 1.656 Åobtained from sc-DMRG-NEVPT2. Based on the results of ourpreliminary study,37 we estimate re obtained from the uncon-tracted theory to be∼0.005 Å shorter and use the bond distancer(Cr–Cr) = 1.650 Å in our computations.

Table II presents the t-MPS-NEVPT2 correlation energycontributions computed for different values of the bond dimen-sion M. Increasing M from 500 to 800 does not significantlyaffect the energy components E[i]

act that depend on the 1-GF([i] ∈ [+1], [−1]) and 2-GF ([i] ∈ [+2], [−2], [0′]). Out offive of these contributions, the largest change of +0.3 mEh isobserved for E[0′]

act , while each of the other four energy termschanges by less than 0.1 mEh. These changes largely canceleach other, leading to a total energy difference of ∼0.1 mEh.Thus, we estimate the total error in the sum of the 1- and 2-GFenergy contributions to be less than 0.1 mEh. However, the

TABLE II. t-MPS-NEVPT2 correlation energy contributions E[i]act for Cr2 at 1.650 Å computed for different

values of the bond dimension M. Computations employed the (12e, 22o) active space and the cc-pwCV5Z-DKbasis set. Also shown are strong contraction errors ∆Econtr in the correlation energy for Cr2, the Cr atom, andthe Cr2 binding energy (De), evaluated as the difference between the sc-DMRG-NEVPT2 and t-MPS-NEVPT2energies. The error of strong contraction in the binding energy is shown in bold.

M E[+1]act E[−1]

act E[+2]act E[−2]

act E[0′]act E[+1′]

act E[−1′]act

500 0.1632 0.2357 0.0939 0.1140 1.6425 0.0282 0.0416800 0.1633 0.2357 0.0940 0.1140 1.6422 0.0286 0.04311200 0.0288 0.04411600 0.0290 0.04462000 0.0291 0.0449

∆Econtr (Cr2) = E(2)sc−NEVPT2 − E(2)

ext −∑

[i]E[i]

act = 1.6351 0.6801 + 2.3232 = 0.0080 Eh

∆Econtr (Cr) = E(2)sc−NEVPT2(Cr) − E(2)

t −NEVPT2(Cr) = 0.7758 + 0.7781 = 0.0023 Eh

∆Econtr (De) = ∆Econtr (Cr2) 2 × ∆Econtr (Cr) = 0.0034 Eh = 0.09 eV

244102-8 Sokolov et al. J. Chem. Phys. 146, 244102 (2017)

remaining energy components E[+1′]act and E[−1′]

act that dependon the contracted 3-GF exhibit slower convergence with M,changing by 1.9 mEh for M = 500→ 800 and 1.2 mEh forM = 800→ 1200 (Table II). We increased the bond dimensionup to M = 2000, where the remaining error in E[+1′]

act and E[−1′]act

is estimated to be less than 0.4 mEh (<0.01 eV). The resultingactive-space energy contributions E[i]

act can be used to computethe Cr2 binding energy for the incomplete cc-pwCV5Z-DKbasis set as De = E(2)

ext +∑

[i] E[i]act 2×E(2)

t −NEVPT2(Cr), where

E(2)ext is the Cr2 correlation energy from the external space

[Eq. (15)] and E(2)t −NEVPT2(Cr) is the correlation energy for

an isolated Cr atom. To estimate De at the CBS limit, we firstcompute the error of strong contraction in the binding energy∆Econtr(De) as shown in Table II, where E(2)

sc−NEVPT2 is thesc-DMRG-NEVPT2 correlation energy for Cr2 reported byGuo et al.,35 while E(2)

sc−NEVPT2(Cr) and E(2)t −NEVPT2(Cr) are

the correlation energies for a Cr atom computed using the(6e, 11o) CASSCF reference. Assuming that the contractionerror does not change significantly from the incomplete (cc-pwCV5Z-DK) to the complete basis set, the CBS-extrapolatedCr2 binding energy at the t-MPS-NEVPT2 level of theory canbe estimated as (De)CBS

t −NEVPT2 ≈ (De)CBSsc−NEVPT2 +∆Econtr(De),

where we use (De)CBSsc−NEVPT2 = 1.43 eV from Ref. 35. Although

the resulting binding energy (De)CBSt −NEVPT2 = 1.52 eV is 0.09 eV

larger than (De)CBSsc−NEVPT2, it is still in a good agreement with

the experimental binding energy of 1.47 ± 0.1 eV reported byCasey and Leopold.61

C. Singlet excited states in all-trans polyenes1. Background

Conjugated polyenes are important model systems forunderstanding properties of organic materials (such as poly-acetylene), as well as pigments involved in photosynthesis andvision (e.g., carotenoids or retinals). The key to the function-ality of these molecules is the ability to absorb and efficientlytransfer energy via the low-lying π–π∗ excited states of theconjugated backbone. One of the simplest examples of polyenesystems is the unsubstituted all-trans polyenes that consist ofalternating single and double carbon bonds arranged in a lin-ear chain (CnHn+2). The excited states of these molecules arelabeled by the symmetry of the C2h point group (Ag, Au, Bg, andBu), as well as an additional +/ label due to their approximateparticle-hole symmetry, which is typically used to distinguishexcited states with mainly ionic (+ states) or mainly covalent( states) excitation character.66,67

The electronic spectra of all-trans polyenes have beenthe subject of many experimental and computational stud-ies.39,67–85 Of particular interest are the excitation energiesand ordering of the low-lying singlet electronic states. Whilethe symmetry of the singlet ground state is known to alwaysbe A−g , the experimental and theoretical characterization ofthe low-lying singlet excited states has been very challeng-ing. In shorter polyenes (CnHn+2, n ≤ 8), it has been estab-lished that the two lowest-energy singlet transitions are thedipole-allowed excitation to the ionic 1B+

u state and the dipole-forbidden excitation to the 2A−g state. Despite the fact thatthe allowed transition 1A−g → 1B+

u has primarily HOMO →

LUMO excitation character, fluorescence studies suggest that,starting from octatetraene (C8H10),75 the lowest-energy sin-glet excitation corresponds to a forbidden transition, such asthe 1A−g → 2A−g transition. Recent high-level theoretical studiesof butadiene (C4H6) predict the bright 1B+

u state vertical exci-tation to be ∼0.2 eV lower in energy than the excitation to thedark 2A−g state,73 whereas the two vertical transitions are pre-dicted to be almost degenerate in C8H10.72 In longer polyenes,the low-energy electronic spectrum is more complicated,involving additional dark states 1B−u and 3A−g with energiesclose to 1B+

u and 2A−g . These dark states have been observedexperimentally in all-trans-carotenoids with n = 18–26π-electrons.76–78

Early computational studies of long polyenes have beenprimarily based on model Hamiltonians, such as the Pariser-Parr-Pople (PPP) and Hubbard models.67,68 In 1986, Tavanand Schulten67 performed multi-reference configuration inter-action (MRCI) computations using the PPP model, whichsuggested that at n = 10, the 1B−u state becomes the secondexcited state, leading to the 1A−g < 2A−g < 1B−u < 1B+

u < 3A−gordering of states for longer polyenes. These computationalresults were recently reassessed using a combination of MRCIand the semi-empirical OM2 method,86 which predicted theinversion of the 1B−u and 1B+

u transitions at n = 14.In 1998, Hirao and co-workers performed the first ab

initio computations of the vertical excitation energies in smalland medium-size polyenes (n ≤ 10) using multi-reference per-turbation theory (MRMP) based on the CASSCF referencewavefunctions.69 In a later study, Kurashige et al. used a com-bination of MRMP and CASCI with the (10e, 10o) active space(incomplete π-valence space) for polyene series up to n = 28.70

Their computations also suggested that the 1B−u and 1B+u verti-

cal transitions cross at n = 14 and predicted a crossing of 3A−gand 1B+

u vertical excitations at n = 22. In 2008, Ghosh et al. per-formed a DMRG-SCF study of polyenes with 8 ≤ n ≤ 24 usingthe complete π-valence active space (ne, no).71 Although inthis study the ionic 1B+

u state was not investigated, the authorsdemonstrated that the self-consistent optimization of orbitalsin the DMRG computations of these systems is very importantin achieving reasonable agreement with the experiment for thecovalent 2A−g , 1B−u , and 3A−g states. The work by Ghosh et al.,however, did not incorporate dynamic correlation. Althoughit is generally believed that the dynamic correlation mainlyaffects the ionic 1B+

u state, excitation energies of the cova-lent states computed using DMRG-SCF overestimate thoseobtained from the experiment.71 Using our sc-MPS-NEVPT2and t-MPS-NEVPT2 methods, we are now in the position toinvestigate the importance of dynamic correlation effects onthe electronic excitations in all-trans polyenes in combina-tion with the complete π-valence space description of staticcorrelation.

2. Details of computations

The ground-state molecular geometries of the all-transpolyenes CnHn+2 (n = 4, 8, 16, and 24) were optimizedusing the B3LYP method87,88 implemented in the Psi4program.89 Geometry optimizations were constrained tohave C2h symmetry. For all polyenes, we used the cc-pVDZ basis set.48 In addition, to study the effect of the

244102-9 Sokolov et al. J. Chem. Phys. 146, 244102 (2017)

basis set on the excitation energies, we also employed thelarger aug-cc-pVTZ basis for the smaller polyenes (n = 4and 8).

At each optimized geometry, the reference MPS wave-functions were computed using the DMRG-SCF algorithmfor the complete π-valence active space (ne, no). To con-struct this active space, we first performed a self-consistentfield (SCF) computation to obtain a set of canonical Hartree-Fock orbitals. The occupied subset of the SCF orbitals wasused to generate an orthonormal set of intrinsic atomic orbitals(IAOs),90 from which the 2pz orbitals of carbon atoms (z beingthe C2 axis) were easily identified and used as the initialactive space orbitals. The initial core and external orbitals wereobtained by projecting the active-space orbitals out of the occu-pied and virtual subspaces of the SCF orbitals, respectively,followed by their symmetric orthogonalization. These initialorbitals were subsequently optimized using the state-averagedDMRG-SCF algorithm with M ′ = 250, averaging over the fivelowest-energy eigenstates with equal weights. At the end of theDMRG-SCF computations, the active orbitals were relocalizedusing the Pipek-Mezey procedure,91 and the core and externalorbitals were canonicalized as the eigenvalues of the state-specific generalized Fock operator [Eq. (2)]. To compute thet-MPS-NEVPT2 and sc-MPS-NEVPT2 energies, referenceMPS for each eigenstate was first computed with M ′ = 500and then compressed down to M = 250. For CnHn+2 with n = 4,8, and 16, the remaining error in the reference and correlation

energies was estimated to be less than 0.1 mEh. To achievea similar accuracy for C24H26, we used M = 500 in sc-MPS-NEVPT2 and to compute the E[+1′]

act and E[−1′]act contributions in

t-MPS-NEVPT2.Finally, to assign the spatial symmetries of the excited

states, we computed transition dipole moments between thestates. Forbidden transitions were assigned to have Ag sym-metry, while the allowed transitions were labeled as Bu. TheBu states corresponding to a large transition dipole momentwere assigned as B+

u , indicating that the electronic excita-tion involves a change of particle-hole character, whereas theremaining Bu states were labeled as B−u . For polyenes withn = 4, 8, 16, and 24, the 1B+

u state was found to be the 3rd, 5th,7th, and 9th singlet eigenstate of the zeroth-order Hamiltonian,respectively. Thus, to compute the 1B+

u excitation energies, weincreased the number of states evaluated in the state-averagedDMRG computation with M ′ = 500 to 9 and 11 for n = 16 and24, respectively.

3. Discussion

Table III presents energies, symmetries, and oscilla-tor strengths for the low-lying singlet states of polyenesCnHn+2 (n = 4, 8, 16, and 24) computed using the DMRG-SCF, sc-MPS-NEVPT2, and t-MPS-NEVPT2 methods withthe cc-pVDZ basis set. In addition, Fig. 3 plots the relativeenergies of the excited states obtained from DMRG-SCF and

TABLE III. Energies, symmetries, and oscillator strengths for the low-lying singlet states in the all-trans conju-gated polyenes CnHn+2. Entries for the 1A−g ground states give the total energies in Eh computed using DMRG-SCF,sc-MPS-NEVPT2, and t-MPS-NEVPT2 with the cc-pVDZ basis set. Entries for the excited states give verticalexcitation energies from the ground state in eV. The notation (ne, mo) denotes the active spaces used in the DMRGand NEVPT2 computations. Oscillator strengths in a.u. were computed using DMRG-SCF. The reference numbersin brackets are experimental measurements for substituted polyenes.

Polyene State DMRG-SCF sc-MPS-NEVPT2 t-MPS-NEVPT2 Oscillator strength Reference

C4H6 1A−g −154.9772 −155.4862 −155.4866(4e, 4o) 1B+

u 8.29 6.27 6.17 2.251 5.92a, 6.21b

2A−g 6.67 6.96 6.94 Forbidden 6.41b

C8H10 1A−g −308.8259 −309.8154 −309.8168(8e, 8o) 1B+

u 6.74 4.24 4.12 3.345 4.41c, 4.76d

2A−g 4.71 4.77 4.75 Forbidden 3.59e, 4.81d

1B−u 5.91 6.03 6.00 0.056 5.96d

3A−g 6.64 6.80 6.76 ForbiddenC16H18 1A−g −616.5142 −618.4719 −618.4754(16e, 16o) 1B+

u 5.51 2.82 2.64 4.964 (2.82)f

2A−g 3.24 3.14 3.09 Forbidden (2.21)f

1B−u 4.02 3.95 3.89 0.0313A−g 4.77 4.73 4.65 Forbidden

C24H26 1A−g −924.2014 −927.1284 −927.1354(24e, 24o) 1B+

u 5.04 2.25 2.00 6.167 (2.25)g

2A−g 2.72 2.54 2.45 Forbidden (1.51)g

1B−u 3.24 3.08 2.94 0.025 (1.80)g

3A−g 3.77 3.63 3.49 Forbidden (2.04)g

aExperimental adiabatic excitation energy.79,80

bBest theoretical vertical excitation energy.73

cExperimental adiabatic excitation energy.81–83

dBest theoretical vertical excitation energy.72

eExperimental adiabatic excitation energy.84

fExperimental adiabatic excitation energy for a substituted polyene.85

gExperimental adiabatic excitation energy for a substituted polyene.78

244102-10 Sokolov et al. J. Chem. Phys. 146, 244102 (2017)

FIG. 3. Excitation energies in eV for the 1B+u , 2A−g , 1B−u , and 3A−g singlet states

in the all-trans conjugated polyenes CnHn+2 computed using DMRG-SCF andt-MPS-NEVPT2 with the cc-pVDZ basis set. Computations employed (ne, no)active spaces, where the number of active electrons and orbitals n is equal tothe number of C atoms.

t-MPS-NEVPT2 as a function of the chain length n. Forpolyenes with n = 8, 16, and 24, our DMRG-SCF computationspredict the energies and ordering of the covalent excited states2A−g < 1B−u < 3A−g in close agreement with the results obtainedby Ghosh et al.71 In particular, for all polyenes (includingC4H6), DMRG-SCF predicts the 2A−g state to be the lowest-energy singlet excited state, while the energy of the ionic1B+

u state dominated by the HOMO → LUMO transition iscomputed to be 1.6 eV to 2.3 eV higher.

Including the dynamic correlation lowers the relativeenergy of the ionic 1B+

u state drastically. This is seen in thedifference of the 1B+

u excitation energies computed using sc-MPS-NEVPT2 and DMRG-SCF, which ranges from 2 eV inC4H6 to 2.7 eV in C24H26, reducing the excitation energy forthe longer polyene by more than a factor of two. Similar resultsare obtained using the uncontracted t-MPS-NEVPT2 method,which lowers the 1B+

u energy by an additional 0.10 to 0.25 eVrelative to sc-MPS-NEVPT2, due to the removal of the strongcontraction approximation. Both methods predict 1B+

u to bethe lowest-energy singlet excited state. The effect of dynamiccorrelation on the relative energies of the covalent states 2A−g ,1B−u , and 3A−g is much smaller (Fig. 3). Nevertheless, the cor-relation contributions to the relative energies of these statesincrease with chain length and become significant for longerpolyenes, reaching ∼0.3 eV in C24H26 at the t-MPS-NEVPT2level of theory. Interestingly, in shorter polyenes (C4H6 andC8H10), including the dynamic correlation actually leads toan increase of the excitation energies for the covalent statescomputed using sc-MPS-NEVPT2 and t-MPS-NEVPT2 rel-ative to DMRG-SCF, indicating that the correlation is moreimportant in the ground rather than the excited electronic statesfor these molecules. As we describe below, a similar trend isobserved even when the larger aug-cc-pVTZ basis set is usedin the computations. For longer polyenes, dynamic correla-tion lowers the vertical excitation energies for the covalenttransitions. As a result, the sc-MPS-NEVPT2 and t-MPS-NEVPT2 excitation energies decrease with increasing chainlength n faster than compared to DMRG-SCF, as shown inFig. 3.

Figure 4 plots the error of the strong contraction approx-imation in the excitation energies of polyenes with the num-ber of carbon atoms computed as the difference between the

FIG. 4. Strong contraction errors in excitation energies (eV) for the lowest-lying singlet states of the all-trans conjugated polyenes CnHn+2 com-puted as the difference between the sc-MPS-NEVPT2 and t-MPS-NEVPT2results.

sc-MPS-NEVPT2 and t-MPS-NEVPT2 results. Although inthe shorter polyenes (C4H6 and C8H10), strong contractionmainly affects the energy of the ionic 1B+

u state, in the longerpolyenes, the errors of strong contraction become significanteven for the covalent 2A−g , 1B−u , and 3A−g states. In particular, forC24H26, the strong contraction errors amount to 0.09–0.14 eVin the excitation energy for the covalent transitions, which iscomparable to the ∼0.15 eV lowering of the energies due tothe dynamic correlation computed by sc-MPS-NEVPT2.

To assess the relative performance of sc-MPS-NEVPT2and t-MPS-NEVPT2 for the description of excited stateswith ionic and covalent character, we computed the C4H6 andC8H10 vertical excitation energies using sc-NEVPT2 and t-NEVPT2 with the large aug-cc-pVTZ basis set and CASSCFreference wavefunctions (Table IV). Increasing the basis setfrom cc-pVDZ to aug-cc-pVTZ lowers the t-NEVPT2 relativeenergies of the 1B+

u and 2A−g states in C4H6 by 0.34 and 0.32eV, respectively. For the covalent 2A−g state, both sc-NEVPT2and t-NEVPT2 predict excitation energies of ∼6.6 eV,in a reasonable agreement with a reference value of 6.41 eVfrom highly accurate coupled cluster computations.73 Consid-ering the strong basis set dependence of the 2A−g excitationenergy, we expect that this agreement will improve at the CBSlimit. On the other hand, the sc-NEVPT2 and t-NEVPT2 ver-tical excitation energies for the 1B+

u state (6.06 and 5.83 eV)are too low relative to the best available theoretical value of6.21 eV (Table IV), indicating that the two methods will sig-nificantly underestimate the excitation energy of the ionic stateat the CBS limit. A similar performance of sc-NEVPT2 andt-NEVPT2 is observed for C8H10. In this case, both methodsyield excitation energies for the covalent states 2A−g and 1B−uthat are in the excellent agreement with the best available theo-retical reference values,72 while the relative energy of the ionic1B+

u state is underestimated by more than 0.7 eV. Similar errorsfor the 1B+

u state have been observed by Angeli and Pastorein the study of C8H10 using the second-order quasi-degenerateperturbation theory (QD-PT2) with the (8e, 8o) active space.72

They demonstrated that increasing the active space from (8e,8o) to (8e, 16o) results in ∼0.7 eV increase in the 1B+

u exci-tation energy computed using QD-PT2, leading to a closeragreement with the experiment.

The underestimation of the 1B+u relative energies due to

insufficiently large active spaces may explain the dependence

244102-11 Sokolov et al. J. Chem. Phys. 146, 244102 (2017)

TABLE IV. Energies for the lowest-lying singlet states in the all-trans conjugated polyenes C4H6 and C8H10.Entries for the 1A−g ground states give the total energies in Eh computed using CASSCF, sc-NEVPT2, and t-NEVPT2 with the aug-cc-pVTZ basis set. Entries for the excited states give vertical excitation energies from theground state in eV. The notation (ne, mo) denotes the active spaces used in the CASSCF computations.

Polyene State CASSCF sc-NEVPT2 t-NEVPT2 Reference

C4H6 1A−g 154.9845 −155.7199 −155.7211(4e, 4o) 1B+

u 7.33 6.06 5.83 5.92a, 6.21b

2A−g 5.40 6.61 6.62 6.41b

C8H10 1A−g 308.9070 −310.2677 −310.2694(8e, 8o) 1B+

u 6.65 4.02 3.79 4.41c, 4.76d

2A−g 4.79 4.81 4.78 3.59e, 4.81d

1B−u 6.00 6.07 6.02 5.96d

3A−g 6.74 6.82 6.75

aExperimental adiabatic excitation energy.79,80

bBest theoretical vertical excitation energy.73

cExperimental adiabatic excitation energy.81–83

dBest theoretical vertical excitation energy.72

eExperimental adiabatic excitation energy.84

of the excitation energies with the chain length n computedusing t-MPS-NEVPT2 (Fig. 3), where we observe no crossingof 1B+

u with the covalent 2A−g and 1B−u transitions, contraryto the experimental results on substituted polyenes78,85 andsome of the calculations.70,86 Re-investigating the trends inthe excitation energies of the long polyenes with a double π-active space, as employed by Angeli and Pastore, will be thesubject of future work.

V. CONCLUSIONS

In this work, we developed an efficient algorithmto describe dynamic correlation in strongly correlatedsystems with large active spaces and basis sets based on acombination of the time-dependent second-order N-electronvalence perturbation theory (t-NEVPT2) with matrix productstate (MPS) reference wavefunctions. The resulting t-MPS-NEVPT2 approach is equivalent to the fully uncontractedN-electron valence perturbation theory but can efficientlycompute correlation energies using the time-dependent den-sity matrix renormalization group algorithm. In addition, wepresented a new MPS-based implementation of strongly con-tracted NEVPT2 (sc-MPS-NEVPT2) that has a lower compu-tational scaling than the commonly used internally contractedNEVPT2 variants. Using these new methods, we computedthe dissociation energy of the chromium dimer and inves-tigated the importance of dynamic correlation in the low-lying excited states of all-trans polyenes (C4H6 to C24H26).For the chromium dimer, the active space used included the“double d” shell of the chromium atoms. Our Cr2 dissocia-tion energy computed using the uncontracted t-MPS-NEVPT2method is in a good agreement with the experimental dataand the previously reported results from strongly contractedNEVPT2.35 In a study of polyenes, using a complete π-valence active space, our results demonstrate that includingdynamic correlation is very important for the ionic π–π∗

transitions, while it is less important for excitations withcovalent character. In particular, for the C24H26 polyene,incorporating dynamic correlation at the t-MPS-NEVPT2

level of theory lowers the energy of the ionic transition by∼3 eV, whereas only ∼0.3 eV change is observed for the cova-lent electronic states. Our results suggest that both sc-MPS-NEVPT2 and t-MPS-NEVPT2 significantly underestimate therelative energy of the ionic state when combined with the com-plete π-valence active space. In this case, using the “double”π-active space is expected to improve the description of theexcitation energies and will be the subject of our future work.Overall, our results demonstrate that t-MPS-NEVPT2 andsc-MPS-NEVPT2 are promising methods for the study ofstrongly correlated systems that can be applied to a varietyof challenging problems.

ACKNOWLEDGMENTS

This work was supported by the U.S. Department ofEnergy (DOE), Office of Science through Award No. DE-SC0008624 (primary support for A.Y.S.) and Award No.DE-SC0010530 (additional support for G.K.-L.C.), and usedresources of the National Energy Research Scientific Comput-ing Center, a DOE Office of Science User Facility supported bythe Office of Science of the U.S. Department of Energy underContract No. DE-AC02-05CH11231. The authors would liketo thank Dr. Zhendong Li for insightful discussions.

APPENDIX A: SPIN-ORBITAL TWO-BODY GREEN’SFUNCTIONS IN SPIN-ADAPTED DMRG

Here, we describe how to compute the spin-orbital2-GF using the spin-adapted DMRG algorithm. First,we evaluate the elements of G[−2](τ), which in thespin-orbital basis have the following form: G

xρyσzκwλ (τ)

=⟨Ψ0 |a

†xρ(τ)a†yσ(τ)awλazκ |Ψ0

⟩≡

⟨a†xρ(τ)a†yσ(τ)awλazκ

⟩, wh-

ere x, y, w, z denote the spatial orbitals and ρ,σ, κ, λ denotethe spin of these orbitals (e.g., ρ = α, β). As discussed in Sec.II C 3, in spin-adapted DMRG, a†x and ax are the spin tensorcreation and annihilation operators. We use a shorthand nota-tion for the product of two tensor operators AS,M

xy ≡ [a†xa†y]SM ,

where square brackets denote spin coupling (S = 0, 1). Theadjoint of a spin tensor operator is defined with an additional

244102-12 Sokolov et al. J. Chem. Phys. 146, 244102 (2017)

sign factor to preserve the Condon-Shortley phase convention:AS,M‡

xy = (−1)S+M AS,−M†xy . In addition, we define the spin-

adapted tensor product ⊗S as AS11 ⊗S AS2

2 = [AS11 AS2

2 ]S , where

[AS11 AS2

2 ]SM =

∑M1M2

cS1,S2,SM1,M2,M (A1)S1

M1(A2)S2

M2and cS1,S2,S

M1,M2,M are

the Clebsch-Gordan coefficients. Thus, the elements Gxρyσzκwλ (τ)

can be computed by solving the linear system of equations

*................,

0 12

12 − 1

2 − 12 0

1√3

1√12

1√12

1√12

1√12

1√3

0 − 12

12

12 − 1

2 0

0 12 − 1

212 − 1

2 0

1√2

0 0 0 0 − 1√2

1√6− 1√

6− 1√

6− 1√

6− 1√

61√6

+////////////////-

*................,

〈a†xα(τ)a†yα(τ)awαazα〉

〈a†xα(τ)a†yβ(τ)awβazα〉

〈a†xβ(τ)a†yα(τ)awαazβ〉

〈a†xβ(τ)a†yα(τ)awβazα〉

〈a†xα(τ)a†yβ(τ)awαazβ〉

〈a†xβ(τ)a†yβ(τ)awβazβ〉

+////////////////-

=

*................,

〈A0xy(τ) ⊗0 A0‡

zw〉

〈A1xy(τ) ⊗0 A1‡

zw〉

〈A0xy(τ) ⊗1 A1‡

zw〉

〈A1xy(τ) ⊗1 A0‡

zw〉

〈A1xy(τ) ⊗1 A1‡

zw〉

〈A1xy(τ) ⊗2 A1‡

zw〉

+////////////////-

. (A1)

Equation (A1) can also be used to compute elements of G[+2](τ) by replacing creation operators with annihilation operators andvice versa, e.g., 〈a†xα(τ)a†yβ(τ)awβazα〉 → 〈axα(τ)ayβ(τ)a†wβa†zα〉 and 〈A1

xy(τ) ⊗0 A1‡zw〉 → 〈A

1‡yx (τ) ⊗0 A1

wz〉. To compute elements

of G[0′](τ), we define a product of tensor operators BS,Mxy ≡ [a†xay]S

M . In this case, the system of linear equations has the form

*................,

12

12 0 0 1

212

1√12− 1√

121√3

1√3− 1√

121√12

− 12

12 0 0 − 1

212

12

12 0 0 − 1

2 − 12

0 0 1√2− 1√

20 0

− 1√6

1√6

1√6

1√6

1√6− 1√

6

+////////////////-

*................,

〈a†xα(τ)ayα(τ)a†wαazα〉

〈a†xα(τ)ayα(τ)a†wβazβ〉

〈a†xα(τ)ayβ(τ)a†wβazα〉

〈a†xβ(τ)ayα(τ)a†wαazβ〉

〈a†xβ(τ)ayβ(τ)a†wαazα〉

〈a†xβ(τ)ayβ(τ)a†wβazβ〉

+////////////////-

=

*................,

〈B0xy(τ) ⊗0 B0‡

zw〉

〈B1xy(τ) ⊗0 B1‡

zw〉

〈B0xy(τ) ⊗1 B1‡

zw〉

〈B1xy(τ) ⊗1 B0‡

zw〉

〈B1xy(τ) ⊗1 B1‡

zw〉

〈B1xy(τ) ⊗2 B1‡

zw〉

+////////////////-

. (A2)

Equations (A1) and (A2) can be simplified for a closed-shell reference wavefunction |Ψ0〉 to obtain compact expressions

〈a†xα(τ)a†yα(τ)awαazα〉 =1√

3〈A1

xy(τ) ⊗0 A1‡zw〉, (A3)

〈a†xα(τ)a†yβ(τ)awβazα〉 =1√

12〈A1

xy(τ) ⊗0 A1‡zw〉 +

12〈A0

xy(τ) ⊗0 A0‡zw〉, (A4)

〈a†xβ(τ)a†yα(τ)awβazα〉 =1√

12〈A1

xy(τ) ⊗0 A1‡zw〉 −

12〈A0

xy(τ) ⊗0 A0‡zw〉, (A5)

〈a†xα(τ)ayβ(τ)a†wβazα〉 =1√

3〈B1

xy(τ) ⊗0 B1‡zw〉, (A6)

〈a†xα(τ)ayα(τ)a†wαazα〉 =12〈B0

xy(τ) ⊗0 B0‡zw〉 +

1√

12〈B1

xy(τ) ⊗0 B1‡zw〉, (A7)

〈a†xα(τ)ayα(τ)a†wβazβ〉 =12〈B0

xy(τ) ⊗0 B0‡zw〉 −

1√

12〈B1

xy(τ) ⊗0 B1‡zw〉. (A8)

The remaining spin-orbital elements can be obtained using the following symmetry relations:〈a†xα(τ)a†yα(τ)awαazα〉 = 〈a†xβ(τ)a†yβ(τ)awβazβ〉, 〈a

†xα(τ)a†yβ(τ)awβazα〉 = 〈a†xβ(τ)a†yα(τ)awαazβ〉, 〈a

†

xβ(τ)a†yα(τ)awβazα〉

= 〈a†xα(τ)a†yβ(τ)awαazβ〉, 〈a†xα(τ)ayα(τ)a†wαazα〉 = 〈a

†

xβ(τ)ayβ(τ)a†wβazβ〉, 〈a†xα(τ)ayβ(τ)a†wβazα〉 = 〈a

†

xβ(τ)ayα(τ)a†wαazβ〉,

〈a†xα(τ)ayα(τ)a†wβazβ〉 = 〈a†

xβ(τ)ayβ(τ)a†wαazα〉.

APPENDIX B: STRONGLY CONTRACTED NEVPT2WITH MPS COMPRESSION (sc-MPS-NEVPT2)

In this section, we describe an efficient implementationof strongly contracted NEVPT2 with MPS reference wave-functions (sc-MPS-NEVPT2). Although a sc-NEVPT2 imple-mentation on top of MPS wavefunctions was reported in

Ref. 35, here we additionally introduce MPS compression,which greatly reduces the computational cost and allows forlarger active spaces to be treated. We will not describe the gen-eral theory of sc-NEVPT2 in detail but instead refer the readerto Refs. 22 and 23.

In sc-NEVPT2, for each of the 7 subspaces (i∈ +1,−1, +2,−2, +1′,−1′, 0′) described in Sec. II A, the

244102-13 Sokolov et al. J. Chem. Phys. 146, 244102 (2017)

first-order wavefunction is expanded in a basis consisting ofa single perturber function for each core or external index.The perturber function is constructed by fixing (a set of) ofcore or external indices in V and acting on |Ψ0〉. For exam-ple, in the [+1′] subclass, the perturber functions are |Ψ[+1′]

i 〉

= h†i ai |Ψ0〉. Because the perturber functions are all orthogo-nal to each other, they lead to very simple expressions for thesecond-order energy. For example, the perturbation contribu-tion in the [+1′] subclass is

E[+1′] =∑

i

|〈Ψ0 |a†

i hi |Ψ[+1′]i 〉|2

EΨ

[+1′]i− E0

=∑

i

| | |Ψ[+1′]i

⟩| |2

EΨ

[+1′]i− E0

, (B1)

where |Ψ[+1′]i 〉 =

|Ψ[+1′]i 〉

| | |Ψ[+1′]i 〉 | |

and EΨ

[+1′]i=〈Ψ

[+1′]i |HD |Ψ

[+1′]i 〉

〈Ψ[+1′]i |Ψ

[+1′]i 〉

.

The main bottleneck in obtaining the sc-NEVPT2 energyis computing the term 〈Ψ[+1′]

i |HD |Ψ[+1′]i 〉 and the analogous

contribution for |Ψ[−1′]a 〉. This corresponds to

〈Ψ[+1′]i |HD |Ψ

[+1′]i 〉 = 〈Ψ0 |a

†

i hiHDh†i ai |Ψ0〉

= 〈Ψ0 |hiHDh†i |Ψ0〉 − ε i 〈Ψ0 |hih†

i |Ψ0〉

= 〈Ψ0 |hi

[HD, h†i

]|Ψ0〉

+ (E0 − ε i) 〈Ψ0 |hih†

i |Ψ0〉 . (B2)

The above object involves 8 active indices, and the expectationvalue can be computed from the contraction of the 4-RDM andtwo-electron integrals.22,23,92

However, similarly to how we avoid calculating the 3-GFin Eqs. (8) and (9) in t-NEVPT2, we can avoid calculatingthe 4-RDM in sc-NEVPT2. We can represent |Ψ[+1′]

i 〉 and

|Ψ[−1′]a 〉 as MPS of bond dimension M1, by carrying out a vari-

ational compression as described in Sec. II C 2 to minimize| | |Ψ

[+1′]i 〉 − h†i ai |Ψ0〉 | | and | | |Ψ[−1′]

a 〉 − a†aha |Ψ0〉 | |. Assum-ing M1 ∼ M0, the cost of carrying out compressions for allthe perturbers in class [+1′] and [−1′] is O(NextN2

actM30 ). The

computational cost for the subsequent HD expectation value isO(NextN3

actM30 ); however, as no sweeps are required, this has

a very low prefactor, and in our calculations, the amount oftime in this step is negligible. Thus, the computational costof sc-NEVPT2 with MPS compression (sc-MPS-NEVPT2)scales in practice as O(N4

actM30 +N6

actM20 +NextN2

actM30 ). This is

significantly lower than the O(N5actM

30 + N8

actM20 ) cost of com-

puting the 4-RDM, which dominates the standard sc-NEVPT2algorithm presented in Ref. 35.

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