Coupled-Cluster in Real Space. 2. CC2 Excited States Using

Article pubs.acs.org/JCTC

Cite This: J. Chem. Theory Comput. XXXX, XXX, XXX-XXX

Coupled-Cluster in Real Space. 2. CC2 Excited States Using Multiresolution Analysis Jakob S. Kottmann* and Florian A. Bischoff* Institut für Chemie, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany S Supporting Information *

ABSTRACT: We report a first quantized approach to calculate approximate coupled-cluster singles and doubles CC2 excitation energies in real space. The cluster functions are directly represented on an adaptive grid using multiresolution analysis. Virtual orbitals are neither calculated nor needed in this approach. The nuclear and electronic cusps are taken into account explicitly regularizing the corresponding equations exactly. First calculations on small molecules are in excellent agreement with the best available LCAO results.

correlated methods like MP228,29 and ground state CC2,30 as well as explicitly time dependent approaches which are not based on linear response were developed using MRA.31−33 MRA approaches usually implement algorithms which solve the bound-state Helmholtz equations in real space by inverting the shifted Laplacian with the corresponding Helmholtz Green’s function, contrasting LCAO based approaches where usually the Hamiltonian matrix is constructed and diagonalized. As a consequence the corresponding potentials have to be represented in real space, and there are no virtual orbitals calculated or needed.30 There are also grid-based approaches which construct the elements of the Hamiltonian matrix like Kim et al.34,35 where both approaches use Lagrange sinc functions or the BigDFT18,36 project which uses Daubechies wavelets.37 The first correlated method which was implemented using MRA was MP2.28,29 For MRA-MP2 the pair functions are represented directly in the six-dimensional two-electron space so that the usage of low-rank representations for the coefficient tensors is mandatory, unlike for one-electron models.38 This is even more true for the real-space representation of the doubles potential which includes singularities due to the unscreened Coulomb potential, making an explicitly correlated ansatz unavoidable.39 For MRA approaches the explicitly correlated ansatz has to regularize the singularities exactly in order to avoid representing the singular Coulomb potential on the adaptive grid. This is not the case in LCAO based approaches where approximations to the explicitly correlated ansatz are in wide use. Regularization of the nuclear potential was also shown to improve the memory demand of MRA based MP2.39 For this the

I. INTRODUCTION Numerical bases are an alternative to the well established and widely used Gaussian basis sets in the LCAO approach (linear combination of atomic orbitals). Especially for properties of excited states and excitation energies the error of the Gaussian basis set can become significantly large even for uncorrelated models like configuration interaction singles (CIS).1 For correlated models the basis set has to describe the correlation part (high angular momenta) and the excitation part (diffuse functions). To achieve accurate results with respect to the basis set error it is also crucial that the used basis set offers a balanced description between ground and excited states.2 Ground state correlation energies close to the basis set limit may be obtained by the explicitly correlated methods where the electronic cusp3 is taken into account explicitly.4,5 Also for excited states such as the coupled-cluster approximation CC26 explicitly correlated methods have been developed, e.g. CC2-R12 by Fliegl et al.7 The initial approach suffered from an imbalanced description between ground and excited states and was extended later by Neiss et al.8 and by Köhn.9 Numerical methods have emerged as alternatives to LCAO, e.g. finite elements (FEM),10−13 multiradial grids,14,15 or wavelet methods.16−18 Multiresolution analysis (MRA) is a wavelet based framework to represent a real-space function with desired accuracy on a series of adaptively refined and truncated (multidimensional) intervals (see Harrison et al.16 and the corresponding appendix or Jensen19 for an excellent introduction). MRA based approaches have been used for Hartree−Fock and density functional theory for a variety of molecular properties like total energies,16,20 atomization energies,21 first and second order ground state properties,22,23 magnetic properties,24 static polarizabilities,25 and excitation energies obtained with the CIS1 and TDHF/TDDFT26,27 models. Also © XXXX American Chemical Society

Received: July 6, 2017 Published: September 13, 2017 A

DOI: 10.1021/acs.jctc.7b00695 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

Article

Journal of Chemical Theory and Computation so-called Nemo40 (numerical exponential molecular orbital) approach is used for the reference orbitals.41 In the preceding work30 we generalized the first quantized approach from MP2 to coupled-cluster, implementing the CC2 model. In this article we will briefly recapture the CC2 ground state equations and will then derive the CC2 response equations in first quantization. The general scheme to switch between firstand second-quantized form is again connected by the diagrammatic representation where the same rules for interpretation as for the ground state are used. A regularization for the response equation will be introduced and compared to existing LCAO F12 approximations. Finally we will present numerical examples for some small molecules and put them into context of existing results.

sum of the reference orbitals and the corresponding one-electron cluster functions |ti⟩ = |i⟩ + |τi⟩

The Coulomb potential which enters the equation for the ground state doubles is hard to represent numerically. In the preceding work30 we regularized the doubles equations using the ansatz t |τij⟩ = |uij⟩ + 812 f12 |tit j⟩

t ij ̂ − ϵij)|uij⟩ + 812 |Ωij⟩ = (F12 g12 ̃ |tit j⟩

t ̂ − ϵij) + Û12 − [K̂ 12, f ] + [F12 ̂ , 812 g12 ]f12 ̃ ij = f12 (F12 12

(7)

II.B. CC2 Excitation Energies. Excitation energies from the linear response theory for the CC2 model are the eigenvalues of the CC2 Jacobian6 Ax n = ωn x n

(8)

where ωn is the nth excitation energy, and xn is the corresponding eigenvector. For clarity of notation we will drop the index n labeling the excitation number. In a basis of occupied and virtual orbitals eq 8 can be written as two coupled equations ∂Ωia c ∂Ωia cd x + xkl = ωxia k ∂tkc ∂tklcd

symbol |i⟩ ≡ |ϕi⟩ ϵi

∂Ωijab

xkc ∂tkc

Δ ̂ + 2Ĵ − K̂ F̂ = − 2 + Vnuc

F̂ 12 = F̂ (r1) + F̂(r2)

+

∂Ωijab ∂tklcd

(9)

xklcd = ωxijab

(10)

with Ωai and Ωab ij being the CC2 singles and doubles equations for the ground state in amplitude-formalism. In real-space representation eqs 9 and 10 become

ϵij = ϵi + ϵj |xy⟩ = |x⟩ ⊗ |y⟩ xαβ γκ = ⟨αβ|x12|γκ⟩, x12 ∈ {g12,f12} xαγ = ⟨α|x12|γ⟩(r), x12 ∈ {g12,f12} ⟨r1i|g12|τkl⟩ ≡ ⟨i|g12|τkl⟩2 = (∫ d3r2 ϕi(r2)g12τkl(r1,r2))(r1)

δ Ω (r) k

kl 1

2

= ωxi(r)

∫ d3r′

6 = ∑i |i⟩⟨i|

8 t = 1 − ∑i |ti⟩⟨i|

(11)

δ Ωij(r1, r2) δτk(r′)

xk(r′) +

δ Ω (r , r )

∫ d3r1′ ∫ d3r2′ δτ (ijr ′1, r 2′) xkl(r′1, r2′) kl 1

2

= ωxij(r1, r2)

8 = 1 − ∑i |i⟩⟨i|

|Ωi⟩ = (F̂ − ϵi)|τi⟩ + |Vτi⟩

(1)

t ̂ − ϵij)|τij⟩ + 812 |Ωij⟩ = (F12 g12|tit j⟩

(2)

|xi⟩ = 8|xi⟩

(13)

|xij⟩ = 812|xij⟩

(14)

and they can be expanded in virtual orbitals and vice versa |xi⟩ = xic|c⟩, |xij⟩ = xijcd|cd⟩,

with the applied singles potential + |Si2b⟩)

(12)

Similar to the ground state cluster functions |τi⟩ and |τij⟩, the excited state cluster functions |xi⟩ and |xij⟩ are orthogonal to the occupied space

with i, j, k, l, m, and virtuals are labeled with a, b, c, d; furthermore we use Einstein notation for index summation. For clarity the indices i, j as well as a, b are never summed. II.A. CC2 Ground State. The CC2 singles and doubles for the ground state in real-space representation are given by

8 t (|St5bi ⟩

δ Ω (r)

∫ d3r′ δτ (i r′) xk(r′) + ∫ d3r1′ ∫ d3r2′ δτ (r i′, r ′) xkl(r′1, r2′)

6 t = ∑i |ti⟩⟨i|

|Vτi⟩ = 8(|St4ai ⟩ + |Si4c ⟩) +

(6)

with the regularized Coulomb operator

Table 1. Overview of the Notation and the Abbreviations Used in This Work Hartree−Fock orbitals Hartree−Fock orbital energies closed-shell Fock operator pair Fock operator pair orbital energies tensor product integrals convolutions 6D to 3D convolution projectors

(5)

which resulted in regularized equations for the cusp free pair functions |uij⟩

II. METHODOLOGY First we will give a brief summary of the regularized CC2 ground state equations in real-space representation derived in the preceding paper.30 Afterward the real-space representation for the linear response of CC2 is derived and regularized with an appropriate explicitly correlated ansatz. Similar to the ground state the resulting equations are iteratively solved by inverting with the Helmholtz Green’s function.30 An overview over the used notation is given in Table 1. Reference orbitals are labeled

name

(4)

xia = ⟨a|xi⟩

(15)

xijab = ⟨ab|xij⟩

xai

(16)

xab ij

where the amplitudes and of eqs 9 and 10 act as expansion coefficients. If we use the equations for the ground state singles and doubles (eqs 1 and 2) for eqs 11 and 12 we arrive at (details provided in the Appendix)

(3)

Explicit expressions for the projectors (8 , 8 t ) and the potential terms are given in Table 1 and Table 2. The |ti⟩ functions are the B


Article

Journal of Chemical Theory and Computation Table 2. Diagrams for the Ground State CC2 Singles and the Corresponding Interpretation in Real-Space Formalisma

a Open hole lines are labeled with i, while open particle lines carry the functions coordinates. Horizontal bars represent the cluster functions |τij⟩ and |τi⟩. Dashed horizontal bars represent the |ti⟩ functions.

Table 3. Diagrams for the Variation of the CC2 Singles the Corresponding Interpretation in Real-Space Formalisma

a Open hole lines are labeled with i, while open particle lines carry the functions coordinates. Horizontal bars represent the cluster functions |τij⟩ and |τi⟩. Dashed horizontal bars represent the |ti⟩ functions, and the marked horizontal bars represent the response cluster functions |xi⟩ and |xij⟩.

0 = (F̂ − ϵi − ω)|xi⟩ + |Vxi⟩

(17)

t ̂ − ϵij − ω)|xij⟩ + δ(812 0 = (F12 g12|tit j⟩)

(18)

II.C. Regularization of the CC2 Response. For the MRA representation the regularization of the potential in eq 18 results in the removal of the cusp from the pair functions |xij⟩. The singular Coulomb potentials occur in the first term on the righthand side of eq 20, for which we can choose in analogy to the ground state the ansatz

with the variation of the applied singles potential and the doubles potential

t t |xij⟩ ≡ |δτij⟩ = |vij⟩ + 812 f12 (|xit j⟩ + |tixj⟩) + δ 812 f12 |tit j⟩

|Vxi⟩ ≡ |δVτi⟩ = 8(|δ Si4a ⟩ + |δ Si4c⟩) + 8 t (|δ Si5b⟩ + |δ Si2b⟩) + δ 8 t (|St5bi ⟩ + |Si2b⟩) t δ(812 g12|tit j⟩)

=

t 812 g12(|xit j⟩

+ |tixj⟩) + t

The variation of the projectors 8 and t

δ 8 = −6 t δ 812

(23)

(19)

x

t 812

t δ 812 g12|tit j⟩

With the chosen ansatz the regularized equations for the cuspfree |vij⟩ functions are

(20)

̂ − ϵij − ω)|vij⟩ + V̂ rxij (|xit j⟩ + |tixj⟩) + δ V̂ rxij |tit j⟩ = 0 (F12

is given by

(24) (21)

t

x

x

t

= − (8 ⊗ 6 + 6 ⊗ 8 )

where the regularized potentials are given by x t V̂ r ij = 812 (g12 ̃ ij − ωf12 )

(22)

and the explicit expressions of the CC2 singles response are given in Table 3.

(25)

x

t t δ V̂ r ij = δ 812 (g12 δg12 ̃ ij − ωf12 ) + 812 ̃ ij

C

(26)


Article

Journal of Chemical Theory and Computation t ̂ , δ 812 δg12 ]f12 ̃ ij = [F12

Table 4. Diagrams for the CIS(D) Excitation Energy and the Corresponding Interpretation in Real-Space Formalisma

(27)

with g̃ij12 defined in eq 7. We note at this point that (see ref 30) t t t ̂ , δ 812 ̂ , δ 812 812 [F12 ] = [F12 ]

(28)

so that we also can rewrite eq 26 as x t ij t t ̂ , δ 812 δ V̂ r ij = δ 812 g12 ] − ωδ 812 )f12 ̃ + ([F12

(29)

which will be useful when the commutator gets evaluated. II.D. Evaluation of Commutators. The commutators of the t Fock operator with the projector 812 and the variation of the t projector δ 812 can be evaluated using the singles equations for the ground state, eqs 1 and 3, and the excited state, eqs 17 and 19. Here we give the result which was already derived in the preceding work30 t ̂ , 812 [F12 ] = 6 Vτ ⊗ 8 t + 8 t ⊗ 6 Vτ

̂ )6 Vτ ⊗ 8 t = (1 + P12

(30)

where P̂ 12 permutes the two electrons. The commutator for the variation of the projector is calculated in a similar way t ̂ , δ 812 ̂ )([F,̂ 6 x] ⊗ 8t − 6 x ⊗ [F,̂ 6 τ]) [F12 ] = −(1 + P12

̂ )(−6 Vx ⊗ 8t + ω 6 x ⊗ 8t + 6 x ⊗ 6 Vτ) = −(1 + P12 t ̂ )(6 Vx ⊗ 8t − 6 x ⊗ 6 Vτ) + ωδ 812 = (1 + P12

(31)

where we have used [F,̂ 6 x] = (F̂ − ϵk )|xk⟩⟨k|

a

Open hole lines are labeled with i, while open particle lines carry the functions coordinates. Horizontal bars represent the MP2 pairs |τij⟩, and the marked horizontal bars represent the response pairs |xij⟩ or the CIS singles |xi⟩.

= (F̂ − ϵk − ω)|xk⟩⟨k| + ω|xk⟩⟨k| = −|Vxk⟩⟨k| + ω|xk⟩⟨k| = −6 Vx + ω 6 x

(32)

ωCIS(D) = ωCIS + ⟨xm|δ Sm2b⟩ + ⟨xm|δ Sm2c ⟩ + ⟨xm|δ Sm4a⟩

t ωδ 812

Note that the term from the commutator evaluation will cancel in eq 29. In a similar way the terms which include the Fock operator and act on pair states created from simple tensor products can be evaluated. This leads to the following intermediates (see also ref 30) ̂ − ϵij)|tit j⟩ = −(|Vτ τj⟩ + |τiVτ ⟩) (F12 i j

(33)

̂ − ϵij − ω)|xit j⟩ = −(|Vx τj⟩ + |xiVτ ⟩) (F12 i j

(34)

+ ⟨xm|δ Sm4b⟩ + ⟨xm|δ Sm4c ⟩ (36)

ωCIS =

t 812 τ = 0 = 812,

+

⟨xm|δ S3mb⟩

+

⟨xm|δ S3mc ⟩

(37)

For systems with only one reference orbital the S4a and S4c terms cancel, and the S4b term becomes the MP2 energy with inverted sign. The CIS(D) excitation energy is in this case

which occur in the terms which include the modified Coulomb operator g̃ij12 and g̃ij12 − ωf12. II.E. The CIS(D) Model. The CIS(D) model was developed by Head-Gordon et al.42 as a perturbative correction to CIS excitation energies and may be seen as an approximation to CC2. By definition, in the CIS(D) model the ground state is the MP2 ground state, and the singles of the excited state are the CIS singles. We arrive at the regularized equation for the CIS(D) doubles by taking eq 24 for the CC2 doubles and setting the ground state singles to zero. This affects the |ti⟩ functions and projectors in the following way ti⟩ τ = 0 = i⟩,

⟨xm|δ S3ma⟩

ωCIS(D) = ωCIS − ωMP2 + R

(38)

R = ⟨x|δ S2b⟩ + ⟨x|δ S2c ⟩

(39)

The R term depends on the CIS singles |xi⟩ in a similar way as the MP2 correlation energy depends on the ground state orbitals and may be interpreted as the correlation energy of the excited state.

III. COMPUTATIONAL DETAILS The MRA-CC2 response equations were implemented into the MADNESS17,43 library. Solutions for the CC2 ground state were obtained with the implementation and parameters of ref 30. Similar to the preceding work on the CC2 ground state the regularized nuclear potential of ref 41 was used so that all orbitals and pair functions which are represented with MRA are free of nuclear cusps. We used the Slater correlation factor of ref 41 (parameter a = 1.5) as ansatz for the nuclear cusps. For the MRA representation we used precision thresholds of ϵ = 10−3 for all sixdimensional functions and ϵ = 10−5 for the three-dimensional

t x ̂ δ 812 τ = 0 = − (1 + P12)8 ⊗ 6 (35)

The CIS(D) excitation energy is computed by the summation of all diagrams of Table 4 where the three S3 terms are the CIS energy D


Article

Journal of Chemical Theory and Computation Table 5. Excitation Energies in eV for the H2 Molecule (1.394a0) A1u = Σ+u

A1g = Σ+g

E1u = Πu

basis

CIS

CIS(D)

CC2

CIS

CIS(D)

CC2

CIS

CIS(D)

CC2

aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z aug-cc-pV6Z d-aug-cc-pVDZ d-aug-cc-pVTZ d-aug-cc-pVQZ d-aug-cc-pV5Z d-aug-cc-pV6Z MRA(k = 5) MRA(k = 6)

12.730 12.765 12.766 12.768 12.766 12.723 12.759 12.764 12.764 12.764 12.764 12.764

12.717 12.828 12.853 12.865 12.867 12.710 12.821 12.851 12.861 12.864 12.870 12.870

12.708 12.802 12.822 12.832 12.833 12.701 12.794 12.819 12.828 12.830 12.839 12.846

13.109 13.071 13.058 13.038 13.032 12.989 13.025 13.029 13.028 13.028 13.027 13.027

13.140 13.203 13.224 13.220 13.220 13.035 13.165 13.201 13.212 13.217 13.227 13.224

13.129 13.176 13.193 13.187 13.186 13.023 13.138 13.169 13.179 13.183 13.190 13.195

15.793 14.534 14.041 13.783 13.622 13.102 13.127 13.129 13.128 13.126 13.123 13.123

15.839 14.550 14.105 13.878 13.736 13.197 13.269 13.295 13.304 13.307 13.314 13.311

15.827 14.527 14.078 13.848 13.705 13.187 13.242 13.264 13.271 13.274 13.283 13.285

Table 6. CIS(D) Convergence in mEh for the H2 Molecule (1.394a0)a A1u = Σ+u

E1u = Πu

basis

MP2

CIS

CIS(D)

R

CIS

CIS(D)

R

aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z aug-cc-pV6Z d-aug-cc-pVDZ d-aug-cc-pVTZ d-aug-cc-pVQZ d-aug-cc-pV5Z d-aug-cc-pV6Z MRA (ϵ = 10−3, (k = 5)) MRA (ϵ = 10−3, (k = 6)) MRA (ϵ = 10−4, (k = 6))

−6.9937 −2.2755 −1.0084 −0.5173 −0.3039 −6.9395 −2.2623 −1.0010 −0.5131 −0.3012 0.0166 −0.0291 −34.2462

1.2535 −0.0362 −0.0721 −0.1517 −0.0803 1.5050 0.1954 0.0000 −0.0178 −0.0034 0.0022 0.0023 469.0644

5.7073 1.6200 0.7107 0.2692 0.2053 5.9702 1.8990 0.8005 0.4333 0.3014 0.0747 0.0937 473.0529

−2.5399 −0.6193 −0.2256 −0.0964 −0.0183 −2.4743 −0.5587 −0.2005 −0.0620 0.0036 0.0892 0.0624 −30.2577

−98.0934 −51.8504 −33.7058 −24.2202 −18.3239 0.7746 −0.1388 −0.2089 −0.1485 −0.1030 0.0090 0.0022 482.2788

−92.8052 −45.4373 −29.0925 −20.7263 −15.5319 4.2930 1.6503 0.6861 0.3567 0.2457 −0.0080 0.0976 489.2645

−1.7055 4.1376 3.6049 2.9765 2.4881 −3.4212 −0.4732 −0.1060 −0.0079 0.0475 −0.0004 0.0663 −27.2605

All numbers are given as deviations from the MRA (k = 6) values. The quantity R is given by the relation ωCIS(D) = ωCIS − ωMP2 + R. For a twoelectron system: R = ⟨x|δS2b⟩ + ⟨x|δS2c⟩. a

functions. For the BH and H2 molecule we also used ϵ = 10−4 for six- and ϵ = 10−6 for three-dimensional functions. The polynomial orders of the scaling functions were k = 5 or k = 6, and the size of the simulation box was 60 bohrs in each dimension. LCAO calculations were performed with the aug- and d-aug-cc-pVXZ basis sets of Dunning et al.44−47 and their corresponding density fitting basis sets48,49 in the ricc2 programm of TURBOMOLE.50,51 RI-CC2-F12/3B-sp (abbreviated as CC2-F12-SP) results were calculated with the KOALA program.52,53 All structures except for BH where we used the same coordinates as in refs 7 and 8 were optimized with CC2/ aug-cc-pVTZ. The coordinates of the structures are given in the Supporting Information. We used the KAIN54 approach to improve the convergence of the singles and doubles.

already for medium sized singly augmented basis sets, while the more diffuse Πu state needs double augmentations. Since H2 only has one occupied orbital we can use the CIS(D) calculation to gain insight into the different origins of basis set errors. For a two-electron system the CIS(D) excitation energy consists of the MP2 correlation energy, the CIS excitation energy, and an additional term R (see eq 38) which does not depend on the MP2 pair functions and which we interpret as the CIS(D) correlation energy. In Table 6 we give the difference of those three terms from an MRA calculation with tight threshold (ϵ = 10−4) for the Σ+u and Πu states of H2. For the valence state Σ+u both the CIS energy and the R term are small and converge quickly, so that the leading error is the MP2 correlation energy. This is also true for the Rydberg state Πu if the augmentation is sufficient, otherwise the CIS error will dominate the total error. In the latter case the R term behaves erratic, since it depends on the poorly described CIS vectors. The MRA calculations for ϵ = 10−3 meet the required accuracy so that all differences to the ϵ = 10−4 calculations stay in the submilihartree domain. BH. For the BH molecule we calculated the first two excitation energies Π and Σ+ and give the results in Table 7. MRA calculations with ϵ = 10−3 and ϵ = 10−4 thresholds for the Π state result in the same excitation energy in the given accuracy and are in very good agreement with the LCAO calculations, both conventional and with explicit correlation. For the Σ+ state the

IV. RESULTS AND DISCUSSION We performed calculations on small molecules and compared with conventional LCAO results. The excited states are labeled according to their irreducible representation, but the MRA implementation does not take advantage of symmetry. Most excitation energies are given in electronvolts; the corresponding numbers in milihartree are provided in the Supporting Information. IV.A. Numerical Results. H2. The CIS, CIS(D), and CC2 energies for the first three excited states of H2 are given in Table 5. The Σ states are valence states and agree with our MRA results E


Article

Journal of Chemical Theory and Computation

Tables 9 and 10. The valence state shows good agreement between MRA and CC2. For the Rydberg state calculations the

Table 7. Excitation Energies in eV for the BH Molecule (2.329a0) A1 = Π basis aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z aug-cc-pV6Z d-aug-cc-pVQZ d-aug-cc-pV5Z d-aug-cc-pV6Z MRA (ϵ = 10−3, (k = 5)) MRA (ϵ = 10−3, (k = 6)) MRA (ϵ = 10−4, (k = 6)) a

CIS

CC2

2.850 2.861 2.862 2.863 2.863 2.855 2.863 2.863 2.863

2.865 2.834 2.823 2.819 2.817 2.806 2.819 2.817 2.816

2.863

2.816

2.863

2.816

B1 = Σ+ CC2R12+a 2.918 2.857 2.829 2.822 2.819

CIS

CC2

6.487 6.434 6.420 6.406 6.401 6.394 6.395 6.395 6.394

6.392 6.444 6.465 6.469 6.472 6.457 6.460 6.467 6.484

6.394

6.486

Table 9. Ground State Correlation Energies in mEh and Excitation Energies in eV for the 1B2 State of the H2O Molecule (1.820a0, 103.9°)

CC2R12+ 6.492 6.490 6.490 6.484 6.485

GS CC2-F12-SP

CIS

CC2

CC2-F12-SP

aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z aug-cc-pV6Z d-aug-cc-pV6Z MRA (k = 5) MRA (k = 6)

−295.546 −300.848 −302.370 −302.937

8.640 8.658 8.659 8.660 8.659 8.658 8.658 8.658

7.061 7.204 7.265 7.290 7.302 7.301 7.336 7.331

7.407 7.361 7.340 7.332

−299.648 −300.824

Table 10. Excitation Energies in eV for the 2A1 State of the H2O Molecule (1.820 a0, 103.9°)

Results are taken from the appendix of ref 8

CIS excitation energies only converge fully for a doubly augmented basis, but the difference to the MRA result of the CIS excitation energies is below the milihartree domain for the quintuple and hextuple zeta basis sets. Still the difference between the CIS results for the singly augmented basis sets and the MRA results is already larger than the difference between the best explicitly correlated results and MRA. For the doubly augmented basis sets the CIS results are equal to MRA. Results for the largest LCAO basis sets are in good agreement with the MRA calculations. Other than for the Π state the convergence of the CC2-R12+ calculations for the Σ+ state is not monotonous which could result from the not fully converged one-electron contribution.8 BeH2. In Table 8 we give excitation energies for the first four states labeled as Πg, Πu, Σ+u , and Σ+g for the BeH2 molecule. The

E1u = Πu

basis

CIS

CIS(D)

CC2

CIS

CIS(D)

CC2

aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z MRA (k = 5) MRA (k = 6)

7.052 7.071 7.071 7.071 7.071 7.071

6.770 6.798 6.808 6.811 6.811 6.815 A1u = Σ+u

6.765 6.789 6.797 6.799 6.803 6.808

9.266 9.235 9.221 9.212 9.208 9.208

8.921 8.907 8.908 8.909 8.908 8.914 A1g = Σ+g

8.928 8.899 8.892 8.889 8.898 8.904

basis

CIS

CIS(D)

CC2

CIS

CIS(D)

CC2

aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z MRA (k = 5) MRA (k = 6)

9.395 9.399 9.400 9.397 9.395 9.395

9.272 9.376 9.413 9.424 9.440 9.448

9.263 9.352 9.382 9.392 9.405 9.415

9.972 9.980 9.974 9.964 9.954 9.954

9.852 9.968 10.001 10.006 10.011 10.019

9.844 9.941 9.966 9.969 9.973 9.982

basis

CIS

CC2

aug-cc-pV6Z d-aug-cc-pVDZ d-aug-cc-pVTZ d-aug-cc-pVQZ d-aug-cc-pV5Z d-aug-cc-pV6Z MRA (k = 5) MRA (k = 6)

11.658 11.454 11.466 11.466 11.464 11.458 11.456 11.455

9.932 9.522 9.646 9.707 9.730 9.736 9.732 9.725

CC2-F12-SP 9.839 9.788 9.775

conventional CC2 calculations agree well with the MRA numbers, but the CC2-F12-SP are off by 0.05 eV. We believe that in this case the conventional CC2 numbers are more reliable than CC2-F12-SP, because the latter tends to be imbalanced between ground and excited states. This is especially the case if the excitation has significant contributions from virtual orbitals with different symmetry than any ground state orbitals.8 This effect might also occur if the occupied orbitals are of very different shape than the relevant virtual orbitals, as it might be the case here, with very diffuse virtual orbitals. It is interesting to note that the CC2 excitation energies maintain their accuracy independently of the system size. MRA grants guaranteed precision for all functions, but in pair theories like MP2 or CC2 the number of pair functions increases quadratically with the system size. This leads to a size-consistency error with fading accuracy for larger systems since many small error contributions accumulate, as correlation energies are extensive quantities. In the case of a H2O molecule this leads to an accumulated error up to 3 mEh for the ground state correlation energy compared to CC2-F12-SP, where the error of each of the 10 pairs is below the requested 1 mEh. To overcome this problem for ground state correlation energies, local formulations are necessary along with an increased initial accuracy threshold, similar to other local (LCAO) methods. In contrast, the excitation energies maintain the accuracy with system size, as they may be regarded as intensive quantities. IV.B. Comparison to CC2-F12. The explicitly correlated wave function in MRA is related but not equal to its counterpart in LCAO. In LCAO the F12-contribution is designed to capture the parts of the Coulomb cusp which is not described by the finite basis set. The singularities in the electron repulsion potential are no numerical problem, since only its matrix elements need to be

Table 8. Excitation Energies in eV for the Linear BeH2 Molecule (2.513a0) E1g = Πg

B2

basis

MRA results are up to the given accuracy in good agreement with the largest CC2 calculations. For all four states singly augmented basis sets are sufficient to get results comparable to MRA. H2O. For the H2O molecule we calculated the 1B1 valence and the 2A1 Rydberg excited state energies and give the results in F


Article

Journal of Chemical Theory and Computation calculated, not the potential itself. It is therefore not necessary to regularize all singular terms in LCAO calculations, and often some of the singularities are not considered in the explicitly correlated ansatz. In MRA, in contrast, the singular Coulomb potential has to be removed from all terms of the working equations, because the potential itself must be represented on a grid. The explicitly correlated part of the wave function captures the major part of electron correlation, and only the regular remains are handled through MRA. The F12 ansatz therefore depends on the quantum chemical model and is different e.g. for MP2 and CC2, since in CC2 singular terms arise in the singles contribution, as shown in eq 5. In LCAO, the choice of the virtual space obviously determines the quality of the wave function. For the explicit correlation there are several possible wave function ansatzes, see Table 11 for an overview. In the first implementation of CC2 excitation energies using explicit correlation the CC2-R12 ansatz7 was used. With this ansatz the ground and excited states suffer from imbalances resulting in large errors (larger than conventional CC2), since only hole states are included in the explicitly correlated wave function ansatz (structurally term c of Table 11). The CC2-R12+ model8 tries to alleviate these imbalances by including all combinations of particle and hole states (structurally term c and a combination of terms d, e, and f) in the explicitly correlated wave function, but the truncation scheme based on natural orbitals would still give relatively large errors (see the discussion of the truncation of the virtual space in ref 2). The XSP ansatz9 includes hole states and particle/hole states, namely terms c and d, and gives excellent results, despite being formally less complete than the CC2-R12+ ansatz. Neglecting terms f and δf can be justified in LCAO, since their contribution to the (excitation) energies is usually small due to the small norms of the ground state singles. It should be noted that the CC2-R12 and CC2-R12+ models can only structurally be cast into the c, d, e, and f diagrams by virtue of their particle/hole structure, but there is no one-to-one correspondence. The main difference is that these two models perform a variation of the R12 amplitudes, while in XSP these amplitudes are kept fixed and contracted with the conventional singles. Meeting the exact cusp condition via the fixed-amplitudes ansatz is crucial for the complete removal of the singularities in MRA. Physically, for ground state MP2 and CC2 only term c is significant, while terms d and f have only negligible contributions. The variation of term d in particular balances the ground and excited states and must be included in the ansatz for the excited state. For a complete handling of the singularities in the working equations also term f must be included in the wave function ansatz, for both the ground and the excited state. This is important for MRA, since in this case all singularities have to be regularized, even if they have only small contributions to the (excitation) energies. Therefore, terms c, d, and f are included but not term e, which would only arise in CCSD. Thus, while in LCAO several approximations may be discussed, in MRA it is only possible to completely regularize the working equations, and only the 8 projector may be modified within constraint of full regularization. Such a complete regularization of the ground state (eq (37) of ref 30) then directly generates a complete regularization of the excited state in eq 23.

Table 11. Overview over Explicit Correlated Methods Based on LCAOa

a

Diagrammatic expressions of the significant terms of the regularization residual for the ground state (labeled c, d, e, f) and the resulting terms for the excited state (labeled δd, δf). The accronyms RI and DF stand for resolution of the identity and density fitting. Labeling of the diagrams is according to ref 9.

space form. Like for the ground state30 we used an explicitly correlated ansatz for the pair functions which cancels the singularities in the doubles equations. Our ansatz is comparable to the XSP ansatz of Köhn9 but also takes an additional term into account. Regularization of the nuclear cusp is straightforward following the preceding works of one of the authors41 and using the so-called numerical exponential molecular orbitals (nemos40). Like for the ground state we can use the same reinterpretation of diagrams to arrive at the first quantized form of the equations (see the preceding work30 for details). The first quantized equations do not include any sort of virtual orbitals but solve for the singles and doubles cluster-functions directly. Calculations on small molecules are in very good agreement with the best explicitly correlated LCAO methods available and with conventional LCAO calculations with large basis sets. Unlike for the ground state we do not observe error accumulation for the

V. CONCLUSION We have derived and implemented the fully regularized first quantized working equations for CC2 excitation energies in realG


Article

Journal of Chemical Theory and Computation δg τi (r)

excitation energies. We believe this is because the MRA representation represents ground and excited state functions with the same accuracy so that the overall description of the excitation energies is balanced. Imbalanced descriptions of ground and excited states resulting from the chosen F12 ansatz are known problems for different approaches in LCAO based algorithms but do not occur within this work. The limitations and bottlenecks of MRA based CC2 excitation energies are the same as for ground state CC2 and MP2 which is primarily the memory requirements arising from the MRA representation of the sixdimensional pair functions. The usage of low-rank representations for the coefficient tensors is again crucial;29 also the usage of nemos for the reference function improves the memory requirement significantly.39 In order to apply the current algorithms to larger molecules we suggest an extension of the current equations to local schemes and use improved low-rank tensor representations.

■

j

δτk(r′)

δg τi j

■

(41)

δt (r) k

■

j

(49)

ASSOCIATED CONTENT

AUTHOR INFORMATION

Funding

This work was financially supported by the Deutsche Forschungsgemeinschaft DFG (BI-1432/2-1) and the Fonds der Chemischen Industrie (FCI). Notes

(42)

The authors declare no competing financial interest.

(43)

ACKNOWLEDGMENTS The authors thank Sebastian Höfener for helpful discussions and access to the KOALA code.

■ ■

REFERENCES

(1) Kottmann, J. S.; Höfener, S.; Bischoff, F. A. Numerically accurate linear response-properties in the configuration-interaction singles (CIS) approximation. Phys. Chem. Chem. Phys. 2015, 17, 31453−31462. (2) Höfener, S.; Klopper, W. Natural transition orbitals for the calculation of correlation and excitation energies. Chem. Phys. Lett. 2017, 679, 52−59. (3) Kato, T. On the eigenfunctions of many-particle systems in quantum mechanics. Comm. Pure Appl. Math. 1957, 10, 151. (4) Kong, L.; Bischoff, F. A.; Valeev, E. F. Explicitly Correlated R12/ F12 Methods for Electronic Structure. Chem. Rev. 2012, 112, 75−107. (5) Hättig, C.; Klopper, W.; Köhn, A.; Tew, D. P. Explicitly correlated electrons in molecules. Chem. Rev. 2012, 112, 4−74. (6) Christiansen, O.; Koch, H.; Jorgensen, P. The second-order approximate coupled cluster singles and doubles model CC2. Chem. Phys. Lett. 1995, 243, 409−418. (7) Fliegl, H.; Hättig, C.; Klopper, W. Coupled-cluster response theory with linear-r12 corrections: The CC2-R12 model for excitation energies. J. Chem. Phys. 2006, 124, 044112. (8) Neiss, C.; Hättig, C.; Klopper, W. Extensions of r12 corrections to CC2-R12 for excited states. J. Chem. Phys. 2006, 125, 064111. (9) Köhn, A. A modified ansatz for explicitly correlated coupled-cluster wave functions that is suitable for response theory. J. Chem. Phys. 2009, 130, 104104.

For terms which are quadratic in the singles the product rule for functional derivatives has to be taken into account. For the St5bi term we receive then the following variation

We give an explicit example which is not in diagrammatic form. Take the term

∫ dr″ϕi(r″)g(r − r″)τi(r″)

k

Jakob S. Kottmann: 0000-0002-4156-2048 Florian A. Bischoff: 0000-0002-7717-3183

where the dashed bar represents the |ti⟩ functions, and the marked thin bar represents the response cluster functions |xi⟩. In diagrammatic representation the variations are then easily derived like for example the d and δd terms from Table 11

i

j

ORCID

The diagrammatic representation of eq 43 is

g τi (r) =

∫ dr′ δτ (r′) xk(r′) = gxi (r)

*E-mail: [email protected]. *E-mail: florian.bischoff@hu-berlin.de.

Or in shortened notation |δti⟩ = |δτi⟩ = |xi⟩

=

δg τi (r)

Corresponding Authors

δτ (r) k

(48)

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.7b00695. Excitation energies in mEh and the coordinates of all used molecular structures in atomic units (PDF)

which results in

∫ dr′ δτ i(r′) xk(r′) = ∫ dr′ δτ i(r′) xk(r′) = xi(r)

∫ dr″ϕi(r″)g(r − r″)δ(r″ − r′)δjk

S Supporting Information *

The CC2 linear response working eqs 17 and 18 are derived through the functional derivatives of the corresponding ground state eqs 1 and 2. We will show this in detail for some examples. First we note that the following identities hold

δτk(r) = δklδ(r − r′) δτl(r′)

=

∫ dr″ϕi(r″)g(r − r″)τj(r″))

which results into the variation

Derivation of the CC2 Linear Response Working Equations

(40)

δ ( δτk(r′)

= ϕi(r′)g (r − r′)δjk

APPENDIX

δtk(r) δτ (r) = k δτl(r′) δτl(r′)

=

(47)

Using the identity (41) we get the following functional derivative H


Article

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Coupled-Cluster in Real Space. 2. CC2 Excited States Using

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