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Excitation energies along a range-separated adiabatic connectionElisa Rebolini, Julien Toulouse, Andrew M. Teale, Trygve Helgaker, and Andreas Savin
Citation: The Journal of Chemical Physics 141, 044123 (2014); doi: 10.1063/1.4890652 View online: http://dx.doi.org/10.1063/1.4890652 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Multi-configuration time-dependent density-functional theory based on range separation J. Chem. Phys. 138, 084101 (2013); 10.1063/1.4792199 Excitation energies from range-separated time-dependent density and density matrix functional theory J. Chem. Phys. 136, 184105 (2012); 10.1063/1.4712019 Communications: Accurate description of atoms and molecules by natural orbital functional theory J. Chem. Phys. 132, 031103 (2010); 10.1063/1.3298694 Adiabatic connection forms in density functional theory: H 2 and the He isoelectronic series J. Chem. Phys. 129, 064105 (2008); 10.1063/1.2965531 Correlation energies for some two- and four-electron systems along the adiabatic connection in density functionaltheory J. Chem. Phys. 110, 2828 (1999); 10.1063/1.478234
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THE JOURNAL OF CHEMICAL PHYSICS 141, 044123 (2014)
Excitation energies along a range-separated adiabatic connectionElisa Rebolini,1,2,a) Julien Toulouse,1,2,b) Andrew M. Teale,3,4 Trygve Helgaker,4
and Andreas Savin1,2,c)
1Sorbonne Universités, UPMC Univ Paris 06, UMR 7616, Laboratoire de Chimie Théorique,F-75005 Paris, France2CNRS, UMR 7616, Laboratoire de Chimie Théorique, F-75005 Paris, France3School of Chemistry, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom4Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo,P.O. Box 1033 Blindern, N-0315 Oslo, Norway
(Received 14 April 2014; accepted 9 July 2014; published online 29 July 2014)
We present a study of the variation of total energies and excitation energies along a range-separatedadiabatic connection. This connection links the non-interacting Kohn–Sham electronic system to thephysical interacting system by progressively switching on the electron–electron interactions whilstsimultaneously adjusting a one-electron effective potential so as to keep the ground-state densityconstant. The interactions are introduced in a range-dependent manner, first introducing predomi-nantly long-range, and then all-range, interactions as the physical system is approached, as opposedto the conventional adiabatic connection where the interactions are introduced by globally scalingthe standard Coulomb interaction. Reference data are reported for the He and Be atoms and theH2 molecule, obtained by calculating the short-range effective potential at the full configuration-interaction level using Lieb’s Legendre-transform approach. As the strength of the electron–electroninteractions increases, the excitation energies, calculated for the partially interacting systems alongthe adiabatic connection, offer increasingly accurate approximations to the exact excitation ener-gies. Importantly, the excitation energies calculated at an intermediate point of the adiabatic con-nection are much better approximations to the exact excitation energies than are the correspondingKohn–Sham excitation energies. This is particularly evident in situations involving strong static cor-relation effects and states with multiple excitation character, such as the dissociating H2 molecule.These results highlight the utility of long-range interacting reference systems as a starting pointfor the calculation of excitation energies and are of interest for developing and analyzing practi-cal approximate range-separated density-functional methodologies. © 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4890652]
I. INTRODUCTION
Range-separated density-functional theory (DFT) (see,e.g., Ref. 1) constitutes an interesting alternative to standardKohn–Sham (KS) DFT.2, 3 In the standard KS approach, thephysical interacting electronic Hamiltonian is replaced by aneffective non-interacting Hamiltonian. By contrast, in range-separated DFT, the physical Hamiltonian is instead replacedby a partially interacting Hamiltonian that incorporates thelong-range part of the electron–electron interaction. This cor-responds to an intermediate point along a range-separated adi-abatic connection.1, 4–7 The KS Hamiltonian is linked to thephysical Hamiltonian by progressively switching on the long-range part of the two-electron interaction, whilst simultane-ously modifying the one-electron potential so as to maintain aconstant ground-state density. The ground-state energy of thephysical system can then be extracted from the ground state ofthe long-range interacting Hamiltonian by using a short-rangedensity functional describing the complementary short-rangepart of the electron–electron interaction. Be aware that this
a)Electronic mail: [email protected])Electronic mail: [email protected])Electronic mail: [email protected]
range-separated manner of introducing the interaction is notthe usual way of performing the adiabatic connection, wherethe Coulomb interaction is instead scaled by a multiplicativeconstant going from 0 to 1.
Several short-range density-functional approximationshave been developed1, 4, 8–13 and a diverse range of ap-proaches for calculating the ground state of the long-rangeinteracting Hamiltonian have been explored. To aid inthe description of static (or strong) correlation effects,which are poorly treated by standard density functionals,configuration-interaction,1, 4, 7, 14–17 multiconfiguration self-consistent-field (MCSCF),18–20 density-matrix functionaltheory (DMFT),21–23 and constrained-pairing mean-fieldtheory24, 25 descriptions of the long-range interacting systemshave been employed. To treat van der Waals interactions,second-order perturbation theory,26–37 coupled-clustertheory,11, 13, 38–40 and random-phase approximations41–51 havebeen used successfully.
Electronic excitation energies can also be calculatedin range-separated DFT by using the linear-response ap-proach with a time-dependent generalization of the staticground-state theory.52 In this case, the excitation energiesof the long-range interacting Hamiltonian act as starting
0021-9606/2014/141(4)/044123/9/$30.00 © 2014 AIP Publishing LLC141, 044123-1
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044123-2 Rebolini et al. J. Chem. Phys. 141, 044123 (2014)
approximations that are then corrected using a short-rangedensity-functional kernel, just as the KS excitation ener-gies act as starting approximations in linear-response time-dependent density-functional theory (TDDFT). Several suchrange-separated linear-response schemes have been devel-oped, in which the short-range part is described by an ap-proximate adiabatic semi-local density-functional kernel andthe long-range linear-response part is treated at the Hartree–Fock,52–55 MCSCF,52, 55 second-order polarization-propagatorapproximation (SOPPA),55 or DMFT56 levels. These schemesaim at overcoming the limitations of standard linear-responseTDDFT applied with usual adiabatic semi-local approxima-tions for describing systems with static correlation,57 dou-ble or multiple excitations,58 and Rydberg or charge-transferexcitations.59, 60
For the purpose of analyzing linear-response range-separated DFT approaches, it is desirable to have accu-rate reference values of the excitation energies of the long-range interacting Hamiltonian along the range-separatedadiabatic connection [cf. Eq. (5)]. In this work, we pro-vide and analyze reference data for the He and Be atomsand the H2 molecule. The short-range one-electron poten-tials required to keep the ground-density constant along arange-separated adiabatic connection [cf. Eq. (6)] are deter-mined at the full configuration-interaction (FCI) level us-ing Lieb’s Legendre-transform approach.61–63 The excited-state energies of the long-range interacting Hamiltonian alongthe adiabatic connection [cf. Eq. (10)] are then calculatedusing the FCI method. Several accurate ground-state cal-culations have been performed in the past along the stan-dard adiabatic connection62–67 and range-separated adia-batic connections1, 6, 67–69 for small atomic and molecularsystems, but accurate calculations of excited-state energiesalong adiabatic connections are very scarce—see, however,Refs. 62 and 70.
The remainder of this paper is organized as follows. InSec. II, range-separated DFT is briefly reviewed and the def-inition of the excited states along the range-separated adia-batic connection is introduced. In Sec. III, the behaviour ofthe excited-state energies near the two endpoints of the adi-abatic connection, the KS system and the physical system,is studied analytically. After giving computational details inSec. IV, results along the full adiabatic-connection path arepresented and discussed in Sec. V. Finally, some concludingremarks are made in Sec. VI.
II. RANGE-SEPARATED DENSITY-FUNCTIONALTHEORY
In range-separated DFT (see, e.g., Ref. 1), the ex-act ground-state energy of an N-electron system is ob-tained by the following minimization over normalized multi-determinantal wave functions �:
E0 = min�
{〈�|T + Vne + Wlr,μee |�〉 + E
sr,μHxc [n�]
}. (1)
This expression contains the kinetic-energy operator T , thenuclear–electron interaction operator Vne = ∫
vne(r)n(r)drexpressed in terms of the density operator n(r), and a long-
range (lr) electron–electron interaction operator
Wlr,μee = 1
2
∫∫w
lr,μee (r12)n2(r1, r2)dr1dr2, (2)
expressed in terms of the pair-density operator n2(r1, r2). Inthe present work, we use the error-function interaction
wlr,μee (r12) = erf(μr12)
r12
, (3)
where μ controls the range of the separation, with 1/μ act-ing as a smooth cut-off radius. The corresponding comple-mentary short-range (sr) Hartree–exchange–correlation den-sity functional E
sr,μHxc [n�] is evaluated at the density of �:
n� (r) = 〈�|n(r)|�〉.The Euler–Lagrange equation for the minimization of
Eq. (1) leads to the (self-consistent) eigenvalue equation
H lr,μ∣∣�μ
0
⟩ = Eμ
0
∣∣�μ
0
⟩, (4)
where �μ
0 and Eμ
0 are the ground-state wave function and as-sociated energy of the partially interacting Hamiltonian (withan explicit long-range electron–electron interaction)
H lr,μ = T + Vne + Wlr,μee + ˆV sr,μ
Hxc . (5)
It contains the short-range Hartree–exchange–correlation potential operator, evaluated at the densityn0(r) = 〈�μ
0 |n(r)|�μ
0 〉, which is equal to the ground-statedensity of the physical system for all μ,
ˆV sr,μHxc =
∫v
sr,μHxc [n0](r)n(r)dr, (6)
where
vsr,μHxc [n](r) = δE
sr,μHxc [n]
δn(r). (7)
For μ = 0, H lr,μ reduces to the standard non-interacting KSHamiltonian, H KS, while for μ → ∞ it reduces to the physi-cal Hamiltonian H :
H KS = H lr,μ=0 = T + Vne + VHxc, (8)
H = H lr,μ=∞ = T + Vne + Wee. (9)
Varying the parameter μ between these two limits, H lr,μ
defines a range-separated adiabatic connection, linking thenon-interacting KS system to the physical system with theground-state density kept constant (provided that the exactshort-range Hartree–exchange–correlation potential vsr,μ
Hxc (r) isused).
In this work we also consider the excited-state wave func-tions and energies of the long-range interacting Hamiltonian
H lr,μ∣∣�μ
k
⟩ = Eμ
k
∣∣�μ
k
⟩, (10)
where H lr,μ is Hamiltonian in Eq. (5), with the short-range Hartree–exchange–correlation potential evaluated atthe ground-state density n0. In range-separated DFT, theseexcited-state wave functions and energies provide a naturalfirst approximation to the excited-state wave functions andenergies of the physical system. For μ = 0, they reduce to
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044123-3 Rebolini et al. J. Chem. Phys. 141, 044123 (2014)
the single-determinant eigenstates and associated energies ofthe non-interacting KS Hamiltonian,
H KS∣∣�KS
k
⟩ = EKSk
∣∣�KSk
⟩, (11)
while, for μ → ∞, they reduce to the excited-state wave func-tions and energies of the physical Hamiltonian
H |�k〉 = Ek|�k〉. (12)
We emphasize that, even with the exact (short-range) Hartree-exchange-correlation potential, the total energies EKS
k (Eμ
k ) arenot the exact energies of the physical system but the totalenergies of a non-interacting (partially interacting) fictitioussystem of electrons with Hamiltonian H KS (H lr,μ). Note alsothat, since the ionization energy is related to the asymptoticdecay of the ground-state density, the ionization energy of theHamiltonian in Eq. (10) is independent of μ and is equal tothe ionization energy of the physical system. This is an ap-pealing feature since it sets the correct energy window forbound excited states. Finally, note that the excitation energies�Eμ
k = Eμ
k − Eμ
0 calculated from Eq. (10) constitute a start-ing point for range-separated linear-response theory based onthe time-dependent generalization of Eq. (1).52
III. EXCITED-STATE ENERGIES NEARTHE KOHN–SHAM AND PHYSICAL SYSTEMS
In this section, we study analytically the behaviour of theexcited-state energies Eμ
k as a function of the range-separationparameter μ close to the endpoints of the adiabatic connec-tion: the KS system at μ = 0 and the physical system atμ → ∞. This study will aid in the understanding of the nu-merical results presented in Sec. V.
A. Excited-state energies near the Kohn–Shamsystem
We first derive the expansion of the excited-state energiesnear μ = 0, to see how the KS energies are affected by theintroduction of the long-range electron–electron interaction.We assume that the system is spatially finite.
We rewrite the long-range interacting Hamiltonian ofEq. (5) as
H lr,μ = H KS + Wlr,μee − V
lr,μHxc , (13)
with the long-range Hartree–exchange–correlation potentialoperator
Vlr,μ
Hxc = VHxc − ˆV sr,μHxc =
∫v
lr,μHxc(r)n(r)dr. (14)
The expansion of the long-range two-electron interaction isstraightforward1 (valid for μr12 � 1)
wlr,μee (r12) = erf(μr12)
r12
= 2μ√π
+ μ3wlr,(3)ee (r12) + O(μ5), (15)
with
wlr,(3)ee (r12) = − 2
3√
πr2
12. (16)
Note that the first term in the expansion of wlr,μee (r12) in
Eq. (15) is a spatial constant, 2μ/√
π , which shows that whatwe call the long-range interaction does in fact contain also acontribution at short range.1 Next, the expansion of the long-range Hartree–exchange–correlation potential
vlr,μHxc(r) = δE
lr,μHxc[n]
δn(r)(17)
can be determined from the expansion of the correspondingenergy functional Elr,μ
Hxc[n]. As derived in Ref. 1, the expansionof the Hartree–exchange part begins at first order and may bewritten as
Elr,μHx [n] = μ√
π
∫∫nKS
2 (r1, r2)dr1dr2
+μ3
2
∫∫nKS
2 (r1, r2)wlr,(3)ee (r12)dr1dr2
+O(μ5), (18)
where nKS2 (r1, r2) is the KS pair density, while the expansion
of the correlation part only begins at sixth order (assuming anon-degenerate KS ground state)
Elr,μc [n] = 0 + O(μ6). (19)
If the functional derivative of Elr,μHx [n] is taken with respect
to density variations that preserve the number of electrons,∫δn(r)dr = 0, then the first-order term in Eq. (18) does not
contribute due to the fixed normalization of the KS pair den-sity,
∫∫nKS
2 (r1, r2)dr1dr2 = N (N − 1). The derivative is thendefined up to an additive (μ-dependent) constant Cμ, whichcan be fixed by requiring that a distant electron experienceszero potential interaction in Eq. (13), amounting to settingthe zero-energy reference. The linear term in μ in the long-range Hartree–exchange–correlation potential can then be de-termined as follows.
To first order in μ, the long-range electron–electron in-teraction tends to a constant, 2μ/
√π . A distant electron
(with 1 � r12 � 1/μ) then experiences a constant interaction2(N − 1)μ/
√π with the remaining N − 1 other electrons.
This constant must be exactly compensated by the long-rangeHartree–exchange–correlation potential in Eq. (13), so that itsfirst-order term in μ must also be 2(N − 1)μ/
√π . The expan-
sion of vlr,μHxc(r) therefore takes the form
vlr,μHxc(r) = 2(N − 1)μ√
π+ μ3v
lr,(3)Hxc (r) + O(μ5),
(20)
where vlr,(3)Hxc (r) is the third-order contribution.
Combining Eqs. (15) and (20), we arrive at the follow-ing expansion of the long-range interacting Hamiltonian ofEq. (13):
H lr,μ = H KS + μH lr,(1) + μ3H lr,(3) + O(μ5), (21)
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044123-4 Rebolini et al. J. Chem. Phys. 141, 044123 (2014)
with a constant first-order correction
H lr,(1) = −N (N − 1)√π
(22)
and the following third-order correction
H lr,(3) = Wlr,(3)ee − V
lr,(3)Hxc , (23)
Wlr,(3)ee = 1
2
∫∫w
lr,(3)ee (r12)n2(r1, r2)dr1dr2, (24)
Vlr,(3)
Hxc =∫
vlr,(3)Hxc (r)n(r)dr. (25)
Since the first-order correction in the Hamiltonian is a con-stant, it does not affect the associated wave functions. The ex-pansion of the wave functions therefore begins at third orderin μ:
�μ
k = �KSk + μ3�
(3)k + O(μ5). (26)
Using normalized KS wave functions 〈�KSk |�KS
k 〉 = 1, the ex-pansion of the total energy for the state k is then
Eμ
k = EKSk − N (N − 1)√
πμ
+μ3⟨�KS
k
∣∣H lr,(3)∣∣�KS
k
⟩ + O(μ5). (27)
The first-order contribution is the same for all states, can-celling out in the differences between the energies of twostates. As a result, the corrections to the KS excitation en-ergies are third order in μ.
For closed shells, the expansion of the difference be-tween the singlet and triplet energies associated with the sin-gle excitation i → a can be obtained by applying Eq. (27)with the spin-adapted KS wave functions 1�KS = (�KS
i→a
+ �KSi→a
)/√
2, for the singlet state, and 3,1�KS = �KSi→a
, forthe triplet state with spin projection MS = 1. Only the two-electron term then contributes:
�Eμ,1−3i→a = 2μ3〈ia|wlr,(3)
ee |ai〉 + O(μ5)
= 8μ3
3√
π|〈i|r|a〉|2 + O(μ5), (28)
where we have used r212 = r2
1 + r22 − 2r1 · r2. The appearance
of the transition dipole moment integral in Eq. (28) meansthat, for an atomic system, the singlet–triplet energy splittingappears at third order in μ if the difference between the an-gular moment of the orbitals ϕi and ϕa is �� = +1 or −1.Otherwise, the splitting appears at a higher order in μ.
B. Excited-state energies near the physical system
We now derive the asymptotic expansion of the excited-state energies when μ → ∞, which shows how the exactexcited-state energies are affected by the removal of the veryshort-range part of the electron–electron interaction.
For this purpose, we rewrite the long-range interactingHamiltonian of Eq. (5) as
H lr,μ = H − Wsr,μee + ˆV sr,μ
Hxc , (29)
where H is the Hamiltonian of the physical system,
Wsr,μee = 1
2
∫∫w
sr,μee (r12)n2(r1, r2)dr1dr2 (30)
is the short-range electron–electron interaction operator de-fined with the complementary error-function interaction
wsr,μee (r12) = erfc(μr12)
r12
, (31)
and ˆV sr,μHxc is the short-range Hartree–exchange–correlation
potential operator in Eq. (6). The first term in the asymp-totic expansion of w
sr,μee (r12) can be written in terms of a delta
function1 (valid for μr12 � 1)
wsr,μee (r12) = π
μ2δ(r12) + O
(1
μ3
), (32)
while the expansion of vsr,μHxc (r) = δE
sr,μHxc [n]/δn(r) can be ob-
tained from that of Esr,μHxc [n]. As derived in Ref. 1, the expan-
sion of the long-range Hartree–exchange energy is
Esr,μHx [n] = π
2μ2
∫nKS
2 (r, r)dr + O(
1
μ4
), (33)
where nKS2 (r, r) is the KS on-top pair density, while the ex-
pansion of the long-range correlation energy is
Esr,μc [n] = π
2μ2
∫n2,c(r, r)dr + O
(1
μ3
), (34)
where n2,c(r, r) is the on-top correlation pair density of thephysical system. Therefore, the expansion of the short-rangeHartree–exchange–correlation potential takes the form
vsr,μHxc (r) = 1
μ2v
sr,(−2)Hxc (r) + O
(1
μ3
), (35)
where vsr,(−2)Hxc (r) is the μ−2 contribution formally obtained by
taking the functional derivative of Eqs. (33) and (34).Substituting Eqs. (32) and (35) into Eq. (29), we obtain
the asymptotic expansion of the long-range interacting Hamil-tonian as
H lr,μ = H + 1
μ2H lr,(−2) + O
(1
μ3
), (36)
where H lr,(−2) = −Wsr,(−2)ee + ˆV sr,(−2)
Hxc is composed of an on-top two-electron term and a one-electron term:
Wsr,(−2)ee = π
2
∫n2(r, r)dr, (37)
ˆV sr,(−2)Hxc =
∫v
sr,(−2)Hxc (r)n(r)dr. (38)
The expansion of the Hamiltonian in Eq. (36) suggests asimilar expansion for the excited-state wave functions, �
μ
k
= �k + μ−2�(−2)k + O(μ−3). However, as shown in Ref. 71,
this expansion is not valid for r12 � 1/μ. The contribution ofthe wave function for small r12 to the integral for the total en-ergy Eμ
k = 〈�μ
k |H lr,μ|�μ
k 〉 nevertheless vanishes in the limitμ → ∞, and the asymptotic expansion of the total energy ofthe state k is
Eμ
k = Ek + 1
μ2〈�k|H lr,(−2)|�k〉 + O
(1
μ3
), (39)
where the wave function �k is normalized to unity.
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044123-5 Rebolini et al. J. Chem. Phys. 141, 044123 (2014)
IV. COMPUTATIONAL DETAILS
Calculations have been performed for the He and Beatoms and for the H2 molecule with a development versionof the DALTON program,72, 73 using the implementation de-scribed in Refs. 63 and 69. First, a FCI calculation was per-formed to determine the exact ground-state density withinthe basis set considered, followed by a Lieb optimization62
of the short-range potential vsr,μ(r) = vne(r) + vsr,μHxc (r) also
at the FCI level to reproduce the FCI ground-state densityin the presence of the long-range electron–electron interac-tion w
lr,μee (r12). The FCI excited-state energies were then cal-
culated using the partially interacting Hamiltonian with theinteraction w
lr,μee (r12) and effective potential vsr,μ(r).
The Lieb maximization was performed using the short-range analogue of the algorithm of Wu and Yang,74 in whichthe potential is expanded as
vsr,μ(r) = vne(r) + vsr,μref (r) +
∑t
btgt (r), (40)
where the reference potential is the short-range analogue ofthe Fermi–Amaldi potential
vsr,μref (r) = N − 1
N
∫n0(r′)wsr
ee(|r − r′|)dr′, (41)
calculated for a fixed N-electron density n0, to ensure the cor-rect asymptotic behaviour. The same Gaussian basis set {gt}is used for the expansion of the potential and the molecu-lar orbitals. The coefficients bt are optimized by the New-ton method, using a regularized Hessian with a truncatedsingular-value-decomposition cutoff of 10−7 for He and 10−6
for Be and H2.Even-tempered Kaufmann basis sets75 and uncontracted
correlation consistent Dunning basis sets76 augmented withdiffuse functions were tested extensively for the He atom, es-pecially to converge the lowest P state. No significant differ-ences were observed using the two basis sets and only theDunning basis sets are used in the following. The basis setsused are: uncontracted t-aug-cc-pV5Z for He, uncontracted d-aug-cc-pVDZ for Be, and uncontracted d-aug-cc-pVTZ Dun-ning basis sets for H2.
Calculations were performed for about 30 values of μ be-tween 0 to 10 bohrs−1 (with about half the points between 0and 1 where the energies vary the most). Cubic spline interpo-lation has been used on this calculated data when plotting thetotal and excitation energies as a function of μ. For later use,analytical expressions were also fitted to the calculated totalenergies and excitation energies. The forms used in the fittingwere chosen to satisfy the expansions at small and large μ
values as presented in Eqs. (27) and (39). The details of thesefits are given in the supplementary material.77
V. RESULTS AND DISCUSSION
A. Helium atom
The total energies of the ground state 11S and of the firstRydberg-like singlet and triplet S and P excited states of theHe atom are plotted as a function of the range-separation pa-rameter μ in Figure 1. At μ = 0, the KS non-interacting total
-3
-2.5
-2
-1.5
-1
0 1 2 3 4 5 6 7 8 9 10µ in bohr−1
Tot
alen
ergi
esin
hart
ree
E2S − 2√πµ
11S23S21S13P11P
FIG. 1. Ground- and excited-state total energies Eμk (in hartree) of the He
atom as a function of μ (in bohr−1). The total energies of the physical systemE
k= Eμ→∞
k are plotted as horizontal dotted lines. The slope at μ = 0 isshown by the black dashed line for the first excited state.
energies are obtained. Being sums of orbital energies with aresulting double counting of electron repulsion, these quanti-ties are well above the total energies of the physical system(higher by about 1 hartree). When the long-range electron–electron interaction is added by increasing μ from μ = 0,the total energies decrease linearly with μ with a slope of−2/
√π , in accordance with the linear term in the expansion
of Eq. (27) for N = 2. For larger μ values, the total energycurves flatten and approach the energies of the physical sys-tem asymptotically as 1/μ2 as μ → ∞, in accordance withEq. (39). The total energies along the adiabatic connectionare poor approximations to the total energies of the physi-cal system unless the range-separation parameter μ is large.Specifically, μ � 6 is required to be within 10 mhartree of theexact total energies.
The lowest singlet and triplet excitation energies are plot-ted in Figure 2. The KS singlet and triplet excitation energiesare degenerate and, as already observed for a few atomic sys-tems in Refs. 78–80, are bracketed by the singlet and tripletexcitation energies of the physical system. As μ increases
0.72
0.73
0.74
0.75
0.76
0.77
0.78
0.79
0 1 2 3 4 5µ in bohr−1
Exc
itat
ion
ener
gies
inha
rtre
e
11S → 23S11S → 21S11S → 13P11S → 11P
FIG. 2. Excitation energies �Eμk = Eμ
k − Eμ0 (in hartree) of the He atom as
a function of μ (in bohr−1). The excitation energies of the physical system�E
k= �Eμ→∞
k are plotted as horizontal dotted lines.
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044123-6 Rebolini et al. J. Chem. Phys. 141, 044123 (2014)
0
0.005
0.01
0.015
0 0.2 0.4 0.6 0.8 1µ in bohr−1Si
ngle
t-tr
iple
ten
ergy
split
ting
inha
rtre
e
2S1P
FIG. 3. Singlet–triplet energy splittings (in hartree) for the He atom as afunction of μ (in bohr−1).
from μ = 0, the excitation energies vary as μ3 since the lin-ear term in Eq. (27) cancels out for energy differences. Thesinglet–triplet degeneracy is lifted and the excitation energiesconverge to the exact singlet and triplet excitation energieswhen μ → ∞. Whereas a monotonic variation of the ex-citation energy with μ can be observed for the singlet andtriplet 1S→ 2S excitations and for the triplet 11S → 13P ex-citation, a non-monotonic variation is observed for the singlet11S → 11P excitation. This behaviour could be an artefactof the basis-set expansions (either orbital or potential), notingthat a similar behaviour was observed for other excitations ina smaller basis set and was removed by enlarging the basisset (the basis set dependence of the singlet 11S → 11P excita-tion energy is given in the supplementary material77). In linewith previous observations in Refs. 78 and 80 for the KS sys-tem, the excitation energies for Rydberg-type states along theadiabatic connection are rather good approximations to theexcitation energies of the physical system (the maximal erroris about 0.02 hartree at μ = 0 for the triplet 11S → 23S ex-citation), becoming better and better for high-lying states asthey must eventually converge to the exact ionization energy.
The singlet–triplet energy splittings for the 2S and 1Pstates are plotted in Figure 3. The expansion at small μ ofEq. (28) predicts the singlet–triplet splitting to increase as μ3
for the 1P state since it corresponds to the 1s → 2p excitationin the KS system, so that �� = 1. By contrast, the singlet–triplet splitting should increase at most as μ5 for the 2S statesince it corresponds to the 1s → 2s excitation in the KS sys-tem, so that �� = 0. This difference is clearly visible in Figure3, where the 2S curve for the singlet–triplet splitting initiallyincreases more slowly than the 1P curve.
B. Beryllium atom
The total energies of the ground state 11S and of the va-lence singlet and triplet 1P excited states of the Be atom areplotted in Figure 4. The KS total energies are approximately6 hartree above the physical energies. At small μ, an initialslope of −12/
√π is observed for all states, in accordance
with Eq. (27) with N = 4. However, convergence to the phys-
-15
-14
-13
-12
-11
-10
-9
-8
0 2 4 6 8 10µ in bohr−1
Tot
alen
ergi
esin
hart
ree
E1S − 12√πµ
11S13P11P
FIG. 4. Ground- and excited-state total energies Eμk (in hartree) of the Be
atom as a function of μ (in bohr−1). The total energies of the physical systemE
k= Eμ→∞
k are plotted as horizontal dotted lines. The slope at μ = 0 isshown in dashed line.
ical energies with increasing μ is much slower than for theHe atom, owing to the short inter-electronic distances in theBe 1s core region, which are consequently probed at larger μ
values.The singlet and triplet excitation energies are plotted in
Figure 5. As for He, the KS excitation energies are brack-eted by the singlet and triplet excitation energies of the physi-cal system. Not surprisingly, the KS excitation energies arepoorer approximations to the exact excitation energies forthese valence excitations in Be than for the Rydberg excita-tions in He. As μ increases, the KS excitation energies rapidlyconverge to the physical excitation energies. Clearly, the slowconvergence of the core energies does not affect the conver-gence of the valence excitation energies.
Close to the KS system, at μ = 0, the excitation energiesare quite sensitive to the introduction of a small portion ofelectron–electron interaction in the Hamiltonian, which maybe interpreted as a sign of static correlation. For μ ≈ 0.4–0.5,a typical μ value in range-separated DFT calculations,18, 81
the calculated excitation energies are significantly better
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0 1 2 3 4 5µ in bohr−1
Exc
itat
ion
ener
gies
inha
rtre
e
11S → 13P11S → 11P
FIG. 5. Excitation energies �Eμk = Eμ
k − Eμ0 (in hartree) of the Be atom as
a function of μ (in bohr−1). The excitation energies of the physical system�E
k= �Eμ→∞
k are plotted as horizontal dotted lines.
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044123-7 Rebolini et al. J. Chem. Phys. 141, 044123 (2014)
0.38
0.4
0.42
0.44
0.46
0.48
0.5
0 1 2 3 4 5µ in bohr−1
Exc
itat
ion
ener
gies
inha
rtre
e11Σ+
g → 13Σ+u
11Σ+g → 11Σ+
u
11Σ+g → 23Σ+
g
11Σ+g → 21Σ+
g
11Σ+g → 13Πu
11Σ+g → 11Πu
FIG. 6. Excitation energies �Eμk = Eμ
k − Eμ0 (in hartree) of the H2 molecule
at the equilibrium internuclear distance as a function of μ (in bohr−1). Theexcitation energies of the physical system �E
k= �Eμ→∞
k are plotted ashorizontal dotted lines.
approximations to the exact excitation energies than arethe KS excitation energies. This observation justifies range-separated multi-determinantal linear-response DFT calcula-tions, which take these excitation energies as a starting point.
C. Hydrogen molecule
The first few excitation energies of H2 at the equilibriumbond distance are plotted against μ in Figure 6. As for theatoms, the valence excitations energies vary much more alongthe adiabatic connection than do the Rydberg-like excitationenergies. Note also that the energetic ordering of the stateschanges along the adiabatic connection. With our choice ofbasis set, we also observe that the higher singlet excitationenergies do not depend monotonically on μ, approaching thephysical limits from above, as observed for He. Again, the ex-citation energies around μ ≈ 0.4–0.5 represent better approxi-mations to the exact excitation energies than the KS excitationenergies.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5µ in bohr−1
Exc
itat
ion
ener
gies
inha
rtre
e
11Σ+g → 13Σ+
u
11Σ+g → 11Σ+
u
11Σ+g → 21Σ+
g
FIG. 7. Excitation energies �Eμk = Eμ
k − Eμ0 (in hartree) of the H2 molecule
at 3 times the equilibrium internuclear distance as a function of μ (in bohr−1).The excitation energies of the physical system �E
k= �Eμ→∞
k are plottedas horizontal dotted lines.
Finally, we consider the interesting case of the dissocia-tion of the H2 molecule. The first excitation energies at threetimes the equilibrium distance are shown in Figure 7. Withincreasing bond distance, the 1σg and 1σu molecular orbitalsbecome degenerate. Consequently, the KS excitation energyfor the single excitation 1σg → 1σu goes to zero. Moreover,the KS excitation energy for the double excitation (1σg)2
→ (1σu)2 also goes to zero (albeit more slowly). This be-haviour is in contrast to that of the physical system, whereonly the excitation energy to the triplet 13+
u state goes tozero, whilst those to the singlet 11+
u state and the 21+g state
(the latter connected to the double excitation in the KS sys-tem) go to finite values.
Clearly, the excitation energies of KS theory are poor ap-proximations to the exact excitation energies, making it dif-ficult to recover from these poor starting values in practicallinear-response TDDFT calculations. As μ increases from μ
= 0, the excitation energies to the singlet 11+u and 21+
g
states vary abruptly, rapidly approaching the physical values.This sensitivity to the inclusion of the electron–electron inter-action is a clear signature of strong static correlation effects,emphasizing the importance of a multi-determinantal descrip-tion in such situations. At μ ≈ 0.4–0.5, the 11+
u and 21+g
excitation energies, although still too low, are much better ap-proximations than the KS excitation energies, constituting astrong motivation for range-separated multi-determinantal ap-proaches in linear-response theory.
VI. CONCLUSIONS
We have studied the variation of total energies andexcitation energies along a range-separated adiabatic con-nection, linking the non-interacting KS system (μ = 0) tothe physical system (μ → ∞) by progressively switching onthe long-range part of the electron–electron interaction withthe range-separation parameter μ, whilst keeping the ground-state density constant. This behaviour is of interest for thedevelopment and analysis of range-separated DFT schemesfor the calculation of excitation energies, such as the linear-response range-separated schemes of Refs. 52, 53, and 55.
Reference calculations were performed for the He andBe atoms and the H2 molecule. Except when μ is large, theground- and excited-state total energies along the adiabaticconnection are poor approximations to the correspondingenergies of the physical system. On the other hand, theexcitation energies are good approximations to the excitationenergies of the physical system for most of the adiabaticconnection curve, except close to the KS system (μ = 0). Inparticular, the excitation energies obtained at μ ≈ 0.4–0.5,typically used in range-separated DFT calculations, aresignificantly better approximations to the exact excitationenergies than are the KS excitation energies. This behaviourappears to be particularly evident for situations involvingstrong static correlation effects and double excitations, asobserved for the dissociating H2 molecule.
These observations suggest that the excitation energiesof the long-range interacting Hamiltonian in range-separatedDFT may be useful as first estimates of the excitation energies
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044123-8 Rebolini et al. J. Chem. Phys. 141, 044123 (2014)
of the physical system. However, if one cannot afford to uselarge μ values (μ > 2–3), these excitation energies should beconsidered only as starting approximations, suitable for cor-rection by, for example, linear-response range-separated the-ory.
In future work, we will utilize the present referencedata to assess the approximations made in practical linear-response range-separated schemes, where the long-range con-tribution is treated, for example, at the Hartree–Fock, MC-SCF, or SOPPA levels of theory, while the short-range partis described by semi-local density-functional approximations.We will also use the results of this work to guide the de-velopment of time-independent range-separated DFT meth-ods for the calculation of excitation energies as alternatives tolinear-response schemes—in particular, for methods based onperturbation theories79, 82 or extrapolations83, 84 along the adi-abatic connection.
ACKNOWLEDGMENTS
E.R. and J.T. gratefully acknowledge the hospitality ofthe Centre for Theoretical and Computational Chemistry(CTCC), University of Oslo, where part of this research wasdone. E.R. also thanks A. Borgoo and S. Kvaal for help-ful discussions. T.H. acknowledges support from the Nor-wegian Research Council through the CoE Centre for The-oretical and Computational Chemistry (CTCC) Grant Nos.179568/V30 and 171185/V30 and through the European Re-search Council under the European Union Seventh Frame-work Program through the Advanced Grant ABACUS, ERCGrant No. 267683. A.M.T. is grateful for support from theRoyal Society University Research Fellowship scheme.
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Supplementary Material for “Excitation energies along a range-separatedadiabatic connection”
Elisa Rebolini1,2,∗ Julien Toulouse1,2,† Andrew M. Teale3,4, Trygve Helgaker4, and Andreas Savin1,2‡1Sorbonne Universites, UPMC Univ Paris 06, UMR 7616,Laboratoire de Chimie Theorique, F-75005 Paris, France
2CNRS, UMR 7616, Laboratoire de Chimie Theorique, F-75005 Paris, France3School of Chemistry, University of Nottingham,
University Park, Nottingham NG7 2RD, United Kingdom4Centre for Theoretical and Computational Chemistry,
Department of Chemistry, University of Oslo,P.O. Box 1033 Blindern, N-0315 Oslo, Norway
(Dated: July 3, 2014)
I. FIT FORM AND PARAMETERS
The total energies Eµk and excitation energies ∆Eµk = Eµk −Eµ0 of the partially interacting Hamiltonian
given in Eq. (5) of the main article have been calculated with the DALTON program as a function of therange-separation parameter µ for the helium and beryllium atoms and for the dihydrogen molecule. Thecomputational details can be found in Section IV. The total ground-state energies were then fitted tothe following analytical expression which satisfies the form of the expansions at small µ and large µ givenin Eqs. (27) and (39)
Eµ0 = E0 +EKS0 − E0 + c1µ+ c2µ
2 + c3µ3
1 + d1µ+ d2µ2 + d3µ3 + d4µ4 + d5µ5,
where c1 = −N(N − 1)/√π+ (EKS
0 −E0)d1 and c2 = −N(N − 1)d1/√π+ (EKS
0 −E0)d2 are fixed by thesmall-µ expansion, E0 and EKS
0 give the ground-state total energies of the physical system and of theKohn-Sham (KS) system, and N is the number of electrons. The excitation energies were fitted to theexpression
∆Eµk = ∆Ek +∆EKS
k −∆Ek + c1µ+ c2µ2 + c3µ
3
1 + d1µ+ d2µ2 + d3µ3 + d4µ4 + d5µ5,
where c1 = d1(∆EKSk −∆Ek) and c2 = d2(∆EKS
k −∆Ek) to ensure the correct behavior at small µ, and∆Ek and ∆EKS
k give the excitation energies of the physical system and of the KS system.The fits were performed on about 30 points for a range of µ going from 0 to 10 bohr−1. The parameters
of the fit can be found in Tables I, II, III and IV, and reproduce the calculated curves shown in thearticle with a maximum error of about 0.1 mhartree. All the energies are in hartree and µ is in bohr−1.
∗Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]
Table I: Fitted parameters of the ground-state and excitation energies along the range-separated adiabatic con-nection for the helium atom using an uncontracted triple-augmented quintuple zeta basis set and a truncatedsingular-value decomposition cutoff of 10−7.
Ground-state EKS0 E0 c3 d1 d2 d3 d4 d5
11S -1.813977 -2.902589 0.2886122 -0.5256672 1.267965 0.7302989 1.729618 0.6215862
Transition ∆EKSk ∆Ek c3 d1 d2 d3 d4 d5
11S → 23S 0.7476677 0.7281453 0.06186663 -1.148460 0.7875350 3.601280 -0.8453350 3.279870
11S → 21S 0.7476670 0.7576321 -0.09598863 -1.799520 3.139774 9.153716 6.331953 8.164700
11S → 13P 0.7787323 0.7701976 0.2572886 -4.517757 11.06152 37.53672 -27.73374 90.37671
11S → 11P 0.7787322 0.7795772 0.05426567 11.97447 -47.11893 114.6076 -82.48554 51.72106
Table II: Fitted parameters of the ground-state and excitation energies along the range-separated adiabaticconnection for the beryllium atom using an uncontracted double-augmented double zeta basis set and a truncatedsingular-value decomposition cutoff of 10−6.
Ground-state EKS0 E0 c3 d1 d2 d3 d4 d5
11S -9.124165 -14.65438 46.83671 -0.2090221 -1.923411 3.658671 10.96260 5.215731
Transition ∆EKSk ∆Ek c3 d1 d2 d3 d4 d5
11S → 13P 0.1336714 0.1009080 -0.02498641 2.675899 -1.103249 66.59735 -39.94845 24.42414
11S → 11P 0.1336461 0.1974410 -0.3142418 2.670149 -5.243878 40.77140 -44.04497 36.69480
Table III: Fitted parameters of the excitation energies along the range-separated adiabatic connection for thedihydrogen molecule at the equilibrium distance using an uncontracted double-augmented triple zeta basis setand a truncated singular-value decomposition cutoff of 10−6.
Transition ∆EKSk ∆Ek c3 d1 d2 d3 d4 d5
11Σ+g → 13Σ+
u 0.4359619 0.3890173 0.2799389 1.767023 13.40149 19.23359 24.79910 19.29466
11Σ+g → 11Σ+
u 0.4359571 0.4677408 -0.01241781 1.264479 -0.4431237 21.85013 -16.49858 12.33904
11Σ+g → 23Σ+
g 0.4740336 0.4598110 0.2481387 0.9229492 -10.10530 21.52073 -16.87971 25.28795
11Σ+g → 21Σ+
g 0.4740150 0.4814739 0.04156341 -5.018161 10.11410 -3.755105 -2.033443 8.145262
11Σ+g → 13Πu 0.480003 0.4670848 0.1376852 30.80894 -13.52176 121.4909 -67.14926 32.50543
11Σ+g → 11Πu 0.4799835 0.4852236 0.01995048 0.5766261 -5.036269 37.87564 -20.46073 10.75388
Table IV: Fitted parameters of the excitation energies along the range-separated adiabatic connection for thedihydrogen molecule at three times the equilibrium distance using an uncontracted double-augmented triple zetabasis set and a truncated singular-value decomposition cutoff of 10−6.
Transition ∆EKSk ∆Ek c3 d1 d2 d3 d4 d5
11Σ+g → 13Σ+
u 0.05176212 0.01700837 0.6515038 -1.415220 4.923966 118.7312 -176.1438 667.7109
11Σ+g → 11Σ+
u 0.05174852 0.2813186 -9.042050 -9.080417 27.82670 57.35565 -188.3668 718.6815
11Σ+g → 23Σ+
g (σ+u )2 0.1034820 0.2988327 -8.364428 -6.876502 37.68192 51.86662 -151.0807 783.8883
II. BASIS SET CONVERGENCE
The short-range potential changes regime at about r ' 1/µ. If the basis set is not large enough, thisregion of the potential is not well described. The higher in energy, the more affected the excited statesare, which might result in a bump as observed in the the singlet 11S → 11P excitation energy of thehelium atom . The evolution of this excitation energy is shown in Figure 1 as a function of µ, for different
2
0.778
0.78
0.782
0.784
0.786
0.788
0.79
0 0.5 1 1.5 2
µ in bohr−1
Exci
tati
on
ener
gie
sin
hart
ree
DACPVTZDACPVQZTACPVTZTACPVQZTACPV5Z
Figure 1: Singlet 11S → 11P excitation energy ∆Eµk = Eµk −Eµ0 (in hartree) of the He atom as a function of µ (in
bohr−1) for different basis sets.
basis sets, going from a triple to a quintuple zeta, and with double or triple augmentation. As the sizeof the basis set increases, and especially, its diffuse nature, the amplitude of the bump and the positionof its maximum decreases. Similar observations were obtained for singlet 11S → 21S excitation energybut as the 21S is lower in energy, smaller basis set were soon sufficient to recover what we believe to bethe correct behavior. In this case, as the excited state is more diffuse, larger basis set are required, andmemory issues prevented us to increase the basis set further more.
3