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Coupled-Cluster in Real Space II: CC2 Excited States using Multi-Resolution Analysis Jakob Siegfried Kottmann, and Florian Andreas Bischoff J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00695 • Publication Date (Web): 13 Sep 2017 Downloaded from http://pubs.acs.org on September 15, 2017

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Journal of Chemical Theory and Computation

Coupled-Cluster in Real Space II: 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 E-mail: [email protected]; [email protected]

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.

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 ∗ To

whom correspondence should be addressed

1

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(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 cusp 3 is taken into account explicitly. 4,5 Also for excited states such as the coupled-cluster approximation CC2 6 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 state 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 Jensen 19 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 CIS 1 and TDHF/TDDFT 26,27 models. Also correlated methods like MP2 28,29 and ground-state CC2, 30 as well as explicitly time dependent approaches which are not based on linear response where 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 func-

2


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tions or the BigDFT 18,36 project which uses Daubechies wavelets. 37 The first correlated method which was implemented using MRA was MP2. 28,29 For MRAMP2 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 so called Nemo 40 (numerical exponential molecular orbital) approach is used for the reference orbitals. 41 In the preceding work 30 we generalized the first quantized approach from MP2 to coupledcluster, 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 first- and 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.

II

Methodology

First we will give a brief summary of the regularized CC2 ground state equations in realspace representation derived in the preceding paper. 30 Afterwards the real-space repre-

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sentation 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 Tab. 1. Reference orbitals are labeled with i, j, k, l, m and virtuals 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.

A CC2 ground state The CC2 singles and doubles for the ground state in real-space representation are given by |Ωi i = Fˆ − ǫi |τi i + |Vτi i, t |Ωij i = Fˆ 12 − ǫij |τij i + Q12 g12 |ti t j i,

(1) (2)

with the applied singles potential ti ti i i |Vτi i = Q |S4a i + |S4c i + Qt |S5b i + |S2b i .

(3)

Explicit expressions for the projectors (Q ,Qt ) and the potential terms are given in Tab. 1 and Tab. 2. The |ti i functions are the sum of the reference orbitals and the corresponding one-electron cluster functions

|ti i = |i i + |τi i.

(4)

The Coulomb potential which enters the equation for the ground state doubles is hard to represent numerically. In the preceding work 30 we regularized the doubles equations

4


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using the ansatz t |τij i = |uij i + Q12 f 12 |ti t j i,

(5)

which resulted in regularized equations for the cusp free pair-functions |uij i t |Ωij i = Fˆ 12 − ǫij |uij i + Q12 g˜ij |ti t j i,

(6)

with the regularized Coulomb operator ˆ 12 − K ˆ 12 , f 12 + Fˆ 12 , Qt f 12 . g˜ij = f 12 Fˆ 12 − ǫij + U 12

B

(7)

CC2 Excitation Energies

Excitation energies from the linear response theory for the CC2 model are the eigenvalues of the CC2 Jacobian 6 Axn = ωn xn ,

(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 , ∂tck k ∂tcd kl ∂Ωijab

xc ∂tck k

+

∂Ωijab ∂tcd kl

5

cd xkl = ωxijab ,


(9) (10)


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with Ωia and Ωijab being the CC2 singles and doubles equations for the ground state in amplitude-formalism. In real-space representation equations (9) and (10) become Z 3 ′ Z 3 ′ δΩi (r) δΩi (r) ′ ′ ′ = ωxi (r) , r r x d r r + d r x kl k 2 2 1 1 δτk (r′ ) δτkl r1′ , r2′ Z Z 3 ′ Z 3 ′ δΩij (r1 , r2 ) δΩij (r1 , r2 ) ′ xkl r1′ , r2′ = ωxij (r1 , r2 ) . r + d r1 d r2 x d3 r′ k ′ ′ ′ δτk (r ) δτkl r1 , r2 Z

d3 r ′

(11) (12)

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

| xi i = Q| xi i,

(13)

| xij i = Q12 | xij i,

(14)

and they can be expanded in virtual orbitals and vice versa

| xi i = xic |ci,

xia = h a| xi i,

(15)

| xij i = xijcd |cdi,

xijab = h ab| xij i,

(16)

where the amplitudes xia and xijab of Eq. (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) 0 = Fˆ − ǫi − ω | xi i + |Vxi i, t t 0 = Fˆ 12 − ǫij − ω | xij i + Q12 g12 | xi t j i + |ti x j i + δ Q12 g12 |ti t j i ,

6


(17) (18)

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with the variation of the applied singles potential and the doubles potential ti i i i i i |Vxi i ≡ |δVτi i = Q |δS4a i + |δS4c i + Qt |δS5b i + |δS2b i + δQt |S5b i + |S2b i ,

(19)

t t t δ Q12 g12 |ti t j i = Q12 g12 | xi t j i + |ti x j i + δQ12 g12 |ti t j i.

(20)

t is given by The variation of the projectors Qt and Q12

δQt = −O x

(21)

t δQ12 = − Qt ⊗ O x + O x ⊗ Qt ,

(22)

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

C

Regularization of the CC2 response

For the MRA representation the regularization of the potential in Eq. (18) result in the removal of the cusp from the pair functions | xij i. The singular Coulomb potentials occurs in the first term on the right hand side of Eq. (20), for which we can choose in analogy to the ground state the ansatz t t | xij i ≡ |δτij i = |vij i + Q12 f 12 | xi t j i + |ti x j i + δQ12 f 12 |ti t j i.

(23)

With the chosen ansatz the regularized equations for the cusp-free |vij i functions are ˆ rxij |ti t j i = 0 ˆ rxij | xi t j i + |ti x j i + δV Fˆ 12 − ǫij − ω |vij i + V

7


(24)


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where the regularized potentials are given by ˆ rxij =Qt g˜ij − ω f 12 , V 12 ˆ rxij =δQt g˜ij − ω f 12 + Qt δ g˜ij , δV 12 12 t f 12 , δ g˜ij = Fˆ 12 , δQ12

(25) (26) (27)

with g˜ij defined in Eq. (7). We note at this point that (see Ref. 30 ) t t t Q12 Fˆ 12 , δQ12 = Fˆ 12 , δQ12 ,

(28)

so that we also can rewrite Eq. (26) as x

ˆ r ij = δQt g˜ij + δV 12

t t Fˆ 12 , δQ12 − ωδQ12 f 12 ,

(29)

which will be useful when the commutator gets evaluated.

D Evaluation of commutators t and the variation of the The commutators of the Fock operator with the projector Q12 t can be evaluated using the singles equations for the ground state, Eqs. (1) projector δQ12

and (3), and the excited state, Eqs. (17) and (19). Here we give the result which was already derived in the preceding work 30

t Fˆ 12 , Q12 = O Vτ ⊗ Qt + Qt ⊗ O Vτ = 1 + Pˆ 12 O Vτ ⊗ Qt ,

8


(30)

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where Pˆ 12 permutes the two electrons. The commutator for the variation of the projector is calculated in a similar way

t ˆ O x ⊗ Qt − O x ⊗ F, ˆ Oτ F, Fˆ 12 , δQ12 = − 1 + Pˆ 12 = − 1 + Pˆ 12 −O Vx ⊗ Qt + ω O x ⊗ Qt + O x ⊗ O Vτ t , = 1 + Pˆ 12 O Vx ⊗ Qt − O x ⊗ O Vτ + ωδQ12

(31)

where we have used

ˆ O x = Fˆ − ǫk | xk ihk | F, = Fˆ − ǫk − ω | xk ihk| + ω | xk ihk|

= −|Vxk ihk| + ω | xk ihk|

= −O Vx + ω O x .

(32)

t term from the commutator evaluation will cancel in Eq. (29). In a Note that the ωδQ12

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 ) Fˆ 12 − ǫij |ti t j i = − |Vτi τj i + |τi Vτj i , Fˆ 12 − ǫij − ω | xi t j i = − |Vxi τj i + | xi Vτj i ,

(33) (34)

which occur in the terms which include the modified Coulomb operator g˜ij and g˜ij − ω f 12 .

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 9



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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 i functions and projectors in the following way | ti i

τ =0

= | i i,

t Q12

τ =0

t δQ12

= Q12 ,

τ =0

= − 1 + Pˆ 12 Q ⊗ O x .

(35)

The CIS(D) excitation energy is computed by the summation of all diagrams of Tab. 4 where the three S3 terms are the CIS energy m m m m m i + h xm |δS4a i + h xm |δS4b i + h xm |δS4c i, ωCIS(D) = ωCIS + h xm |δS2b i + h xm |δS2c m m m ωCIS = h xm |δS3a i + h xm |δS3b i + h xm |δS3c i.

(36) (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

ωCIS(D) = ωCIS − ω MP2 + R, R = h x |δS2b i + h x |δS2c i.

(38) (39)

The R term depends on the CIS singles | xi i 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 MADNESS 17,43 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 10


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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 ǫ = 10−3 for all six-dimensional functions and ǫ = 10−5 for the three-dimensional functions. For the BH and H2 molecule we also used ǫ = 10−4 for six- and ǫ = 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 sets 48,49 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. 7,8 were optimized with CC2/aug-cc-pVTZ. The coordinates of the structures are given in the Supporting Information. We used the KAIN 54 approach to improve the convergence of the singles and doubles.

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

A Numerical results H2 .

The CIS, CIS(D), and CC2 energies for the first three excited states of H2 , are given

in Tab. 5. The Σ states are valence states and agree with our MRA results already for medium sized singly augmented basis sets while the more diffuse Πu state needs double 11



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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 Tab. 6 we give the difference of those three terms from an MRA calculation with tight threshold (ǫ = 10−4 ) for the Σu+ and Πu states of H2 . For the valence state Σu+ 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 is depends on the poorly described CIS vectors. The MRA calculations for ǫ = 10−3 meet the required accuracy so that all differences to the ǫ = 10−4 calculations stay in the sub milihartree domain. BH. For the BH molecule we calculated the first two excitation energies Π and Σ+ and give the results in Tab 7. MRA calculations with ǫ = 10−3 and ǫ = 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 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

12


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which could result from the not fully converged one-electron contribution. 8

BeH2 .

In Tab. 8 we give excitation energies for the first four states labeled as Πg , Πu ,

Σu+ , and Σg+ for the BeH2 molecule. The 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. H2 O. For the H2 O molecule we calculated the 1B1 valence and the 2A1 Rydberg excited state energies and give the results in Tabs. 9 and 10. The valence state shows good agreement between MRA and CC2. For the Rydberg state calculations the 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 CC2F12-SP, because the latter tends to be imbalanced between ground and excited state. 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 H2 O 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 reqested 1 mEh . To overcome this problem for ground state correlation energies, local formulations are necessary, and 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 intensive quantities. 13



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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 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 ansätze, see Tab. 11 for an overview. In the first implementation of CC2 excitation energies using explicit correlation the CC2-R12 ansatz 7 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 Tab. 11). The CC2-R12+ model 8 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 ansatz 9 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 jus14


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tified 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 state 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 Q 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).

V

Conclusion

We have derived and implemented the fully regularized first quantized working equations for CC2 excitation energies in real-space form. Like for the ground state 30 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 Köhn 9 but also takes

15



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an additional term into account. Regularization of the nuclear cusp is straightforward following preceeding works of one of the authors 41 and using the so called numerical exponential molecular orbitals (nemos 40 ). 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 preceeding work 30 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 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 description of ground and excited state 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 six dimensional 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 molecule we suggest an extension of the current equations to local schemes and use improved lowrank tensor representations.

VI

Acknowledgments

This work was financial supported by the Deutsche Forschungsgemeinschaft DFG (BI1432/2-1) and the Fonds der Chemischen Industrie (FCI). The authors thank Sebastian Höfener for helpful discussions and access to the KOALA code.

16


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VII

Supporting Information

We give all excitation energies in mEh and the coordinates of all used molecular structures in atomic units. This information is available free of charge via the Internet at http://pubs.acs.org

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(52) Höfener, S. Coupled-cluster frozen-density embedding using resolution of the identity methods. J. Comp. Chem. 2014, 35, 1716–1724. (53) Koala, an ab-initio electronic structure program, written by S. Höfener with contributions from A. S. Hehn, J. Heuser, and N. Schieschke. (54) Harrison, R. J. Krylov subspace accelerated inexact Newton method for linear and nonlinear equations. J. Comp. Chem. 2002, 25, 328–334.

23



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Page 24 of 36

Table 1: Overview of the notation and the abbreviations used in this work. Hartree-Fock orbitals

|i i ≡ |φi i

Hartree-Fock orbital energies Pair Fock operator

ǫi ˆ nuc + 2ˆJ − K ˆ Fˆ = − ∆2 + V Fˆ 12 = Fˆ (r1 ) + Fˆ (r2 )

Pair orbital energies

ǫij = ǫi + ǫ j

Tensor product

| xyi = | x i ⊗ |yi

Integrals

xγκ = hαβ| x12 |γκ i,

Convolutions

xγα = hα| x12 |γi (r) ,

Closed-shell Fock operator

6D to 3D convolution Projectors

αβ

x12 ∈ { g12 , f 12 }

x12 ∈ { g12 , f 12 } R d3 r2 φi (r2 ) g12 τkl (r1 , r2 ) (r1 ) hr1 i | g12 |τkl i ≡ hi | g12 |τkl i2 =

O t = ∑i |ti ihi |

O = ∑i |i ihi | Qt = 1 − ∑i |ti ihi | Q = 1 − ∑i |i ihi |


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Table 2: Diagrams for the ground state CC2 singles and the corresponding interpretation in real-space formalism. Open hole lines are labeled with i while open particle lines carry the functions coordinates. Horizontal bars represent the cluster functions |τij i and |τi i. Dashed horizontal bars represent the |ti i functions. label

i Q|S2b i

diagram

expression

Q (2hk| g12 |τik i2 − hk| g12 |τik i1 )

ti Q|S5b i

ti Q|S4a i

i Q|S4c i

Q

−Q

2gτkk |ti i −

2hl | gtki (2)|τkl i2

gtki |τk i

− hl | gtl i (1)|τkl i1

Q 4hl | gτkk |τil i2 − 2hl | gτkk |τil i1 − 2hk| gτl k |τil i2 + hk| gτl k |τil i1



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

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Table 3: Diagrams for the variation of the CC2 singles the corresponding interpretation in real-space formalism. Open hole lines are labeled with i while open particle lines carry the functions coordinates. Horizontal bars represent the cluster functions |τij i and |τi i. Dashed horizontal bars represent the |ti i functions. And the marked horizontal bars represent the response cluster functions | xi i and | xij i. label

diagram

expression

i Q|δS2b i

ti Q|δS5b i

ti Q|δS4a i

i Q|δS4c i

Q (2hk| g12 | xik i2 − hk| g12 |τik i1 )

+

+

+

Q

2gkxk |ti i −

gtki | xk i

+

2gτkk | xi i −

gkxi |τk i

−Q 2hl | gtki (2)| xkl i2 − hl | gtl i (1) xτkl 1 −Q 2hl | gkxi (2)|τkl i2 − hl | glxi (1)|τkl i1

Q 4hl | gτkk |τil i2 − 2hl | gkxk |τil i1 − 2hk| glxk |τil i2 + hk| glxk |τil i1 k k l l +Q 4hl | gτk | xil i2 − 2hl | gτk | xil i1 − 2hk| gτk | xil i2 + hk| gτk | xil i1


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Table 4: Diagrams for the CIS(D) excitation energy and the corresponding interpretation in real-space formalism. Open hole lines are labeled with i while open particle lines carry the functions coordinates. Horizontal bars represent the MP2 pairs |τij i and the marked horizontal bars represent the response pairs | xij i or the CIS singles | xi i. label

diagram

m h xm |δS2b i

expression

(2h xm k| g12 | xmk i − hkxm | g12 |τmk i)

m h xm |δS2c i

k (2)| τ i − h lx | gk (1)| τ i − 2h x m l | gm m m kl kl

m h xm |δS3a i

F

h xm |Fˆ | xm i

m h xm |δS3b i

F

hk|Fˆ |mih xm | xk i

m h xm |δS3c i

m h xm |δS4a i

m h xm |δS4b i

m h xm |δS4c i

(2h xm k| g12 |ixk i − hkxm | g12 |ixk i)

− 2h xm l | gkxm (2)|τkl i − hlxm | gkxm (1)|τkl i

−

2gτklml

−

gτkllm

h xm | xk i

4h xm l | gkxk (2)|τml i − 2hlxm | gkxk (1)|τml i

−2h xm k| glxk (2)|τml i + hkxm | glxk (1)|τml i



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Page 28 of 36

Table 5: Excitation energies in eV for the H2 molecule (1.394a0 ). A1u = Σ+ u Basis

A1g = Σ+ g

E1u = Πu

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

12.730 12.765 12.766 12.768 12.766

12.717 12.828 12.853 12.865 12.867

12.708 12.802 12.822 12.832 12.833

13.109 13.071 13.058 13.038 13.032

13.140 13.203 13.224 13.220 13.220

13.129 13.176 13.193 13.187 13.186

15.793 14.534 14.041 13.783 13.622

15.839 14.550 14.105 13.878 13.736

15.827 14.527 14.078 13.848 13.705

d-aug-cc-pVDZ d-aug-cc-pVTZ d-aug-cc-pVQZ d-aug-cc-pV5Z d-aug-cc-pV6Z

12.723 12.759 12.764 12.764 12.764

12.710 12.821 12.851 12.861 12.864

12.701 12.794 12.819 12.828 12.830

12.989 13.025 13.029 13.028 13.028

13.035 13.165 13.201 13.212 13.217

13.023 13.138 13.169 13.179 13.183

13.102 13.127 13.129 13.128 13.126

13.197 13.269 13.295 13.304 13.307

13.187 13.242 13.264 13.271 13.274

MRA(k=5) MRA(k=6)

12.764 12.764

12.870 12.870

12.839 12.846

13.027 13.027

13.227 13.224

13.190 13.195

13.123 13.123

13.314 13.311

13.283 13.285


Page 29 of 36

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Table 6: CIS(D) convergence in mEh for the H2 molecule (1.394a0 ). All numbers are given as deviations from the MRA(k=6) values. The quantity R is given by the relation ωCIS(D) = ωCIS − ω MP2 + R which for a two electron system is R = h x |δS2b i + h x |δS2c i A1u = Σ+ u Basis

E1u = Πu

MP2

CIS

CIS(D)

R

CIS

CIS(D)

R


−6.9937 −2.2755 −1.0084 −0.5173 −0.3039

1.2535 −0.0362 −0.0721 −0.1517 −0.0803

5.7073 1.6200 0.7107 0.2692 0.2053

−2.5399 −0.6193 −0.2256 −0.0964 −0.0183

−98.0934 −51.8504 −33.7058 −24.2202 −18.3239

−92.8052 −45.4373 −29.0925 −20.7263 −15.5319

−1.7055 4.1376 3.6049 2.9765 2.4881


−6.9395 −2.2623 −1.0010 −0.5131 −0.3012

1.5050 0.1954 0.0000 −0.0178 −0.0034

5.9702 1.8990 0.8005 0.4333 0.3014

−2.4743 −0.5587 −0.2005 −0.0620 0.0036

0.7746 −0.1388 −0.2089 −0.1485 −0.1030

4.2930 1.6503 0.6861 0.3567 0.2457

−3.4212 −0.4732 −0.1060 −0.0079 0.0475

MRA(ǫ = 10−3 , k=5) MRA(ǫ = 10−3 , k=6)

0.0166 −0.0291

0.0022 0.0023

0.0747 0.0937

0.0892 0.0624

0.0090 0.0022

−0.0080 0.0976

−0.0004 0.0663

−34.2462 469.0644 473.0529 −30.2577

482.2788

489.2645

−27.2605

MRA(ǫ = 10−4 , k=6)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 36

Table 7: Excitation energies in eV for the BH molecule (2.329a0 ). B1 = Σ +

A1 = Π Basis

CC2


2.850 2.861 2.862 2.863 2.863

2.865 2.834 2.823 2.819 2.817

d-aug-cc-pVQZ d-aug-cc-pV5Z d-aug-cc-pV6Z

2.855 2.806 2.863 2.819 2.863 2.817

6.394 6.457 6.395 6.460 6.395 6.467

MRA(ǫ = 10−3 , k=5) 2.863 2.816 MRA(ǫ = 10−3 , k=6) 2.863 2.816

6.394 6.484 6.394 6.486

MRA(ǫ = 10−4 , k=6) †

CC2-R12+

†

CIS

2.918 2.857 2.829 2.822 2.819

2.863 2.816

results taken from the appendix of. 8


CIS

CC2

CC2-R12+

6.487 6.434 6.420 6.406 6.401

6.392 6.444 6.465 6.469 6.472

6.492 6.490 6.490 6.484 6.485

Page 31 of 36

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Table 8: Excitation energies in eV for the linear BeH2 molecule (2.513a0 ). E1g = Π g Basis

E1u = Πu

CIS

CIS(D)

CC2

CIS

CIS(D)

CC2

aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z

7.052 7.071 7.071 7.071

6.770 6.798 6.808 6.811

6.765 6.789 6.797 6.799

9.266 9.235 9.221 9.212

8.921 8.907 8.908 8.909

8.928 8.899 8.892 8.889

MRA(k=5) MRA(k=6)

7.071 7.071

6.811 6.815

6.803 6.808

9.208 9.208

8.908 8.914

8.898 8.904

A1u = Σu+

A1g = Σg+

CIS

CIS(D)

CC2

CIS

CIS(D)

CC2

aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z

9.395 9.399 9.400 9.397

9.272 9.376 9.413 9.424

9.263 9.352 9.382 9.392

9.972 9.980 9.974 9.964

9.852 9.968 10.001 10.006

9.844 9.941 9.966 9.969

MRA(k=5) MRA(k=6)

9.395 9.395

9.440 9.448

9.405 9.415

9.954 9.954

10.011 10.019

9.973 9.982

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◦ ). GS Basis aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z aug-cc-pV6Z

CC2-F12-SP

CIS

CC2

CC2-F12-SP

−295.546 −300.848 −302.370 −302.937

8.640 8.658 8.659 8.660 8.659

7.061 7.204 7.265 7.290 7.302

7.407 7.361 7.340 7.332

d-aug-cc-pV6Z MRA(k=5) MRA(k=6)

B2

8.658 7.301

−299.648 −300.824

8.658 7.336 8.658 7.331



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table 10: Excitation energies in eV for the 2A1 state of the H2O molecule (1.820 a0 , 103.9◦ ). Basis

CIS

CC2

aug-cc-pV6Z

11.658

9.932


11.454 11.466 11.466 11.464 11.458

9.522 9.646 9.707 9.730 9.736

MRA(k=5) MRA(k=6)

11.456 11.455

9.732 9.725

CC2-F12-SP

9.839 9.788 9.775


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Table 11: Overview over explicit correlated methods based on LCAO. 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 according to Ref. 9 Acronym

RI

DF

CC2-R12 yes no CC2-R12+ yes no CC2-F12-SP yes yes CCSD-F12-XSP yes no MRA-CC2 — — label

F12-pairs Program c c,(d+e+f) c c,d c,d,f

Reference 7

Dalton Dalton Koala GECCO MADNESS

diagram

8 53 9

this work

expression

c

Q12 f 12 |iji

d

Q12 f 12 |τi ji

e

Q12 f 12 |τij i

f

Q12 f 12 |τi τj i

δd

Q12 f 12 | xi ji

δe

Q12 f 12 |τij i

δf

Q12 f 12 | xi τj i

model

CC2-R12 ansätze R12 ansatz R12 response ansatz

CC2-R12

Q12 f 12 |kl ickl

CC2-R12+

Q12 f 12 | pqic pq

CC2-XSP

Q12 f 12 |iait a 21

ij

ij

j

ij

Q12 f 12 |kl iδckl ij

Q12 f 12 | pqiδc pq j

Q12 f 12 |iaiδt a 21



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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VIII Appendix A Derivation of the CC2 linear response working equations The CC2 linear response working equations (17) and (18) are derived through the functional derivatives of the corresponding ground state equations (1) and (2). We will show this in detail for some examples. First we note that the following identities hold δτ (r) δtk (r) = k ′ , ′ δτl (r ) δτl (r ) δτk (r) ′ = δ δ r − r kl δτl (r′ )

(40) (41)

which results in Z

dr′

δti (r) xk r′ = ′ δτk (r )

Z

dr′

δτi (r) x k r ′ = xi ( r ) . ′ δτk (r )

(42)

Or in shortened notation

|δti i = |δτi i = | xi i.

(43)

The diagrammatic representation of Eq. (43) is

δ

!

!

=δ

,

=

(44)

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

δ

!

=

.


(45)

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For terms which are quadratic in the singles the product rule for functional derivatives t

i has to be taken into account. For the S5b term we receive then the following variation

ti |δS5b i

!

=δ

=

.

+

(46)

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

(r) =

Z

dr′′ φi r′′ g r − r′′ τi r′′ .

(47)

Using the identity (41) we get the following functional derivative δgτi j (r)

δ = δτk (r′ ) δτk (r′ )

=

Z

Z

′′

dr φi r

′′

g r−r

′′

τj r

dr′′ φi r′′ g r − r′′ δ r′′ − r′ δjk

= φi r′ g r − r′ δjk

′′

(48)

which results into the variation δgτi j

=

Z

dr

′

δgτi j (r) δτk (r′ )

xk r′ = gix j (r) .


(49)


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Coupled-Cluster in Real Space. 2. CC2 Excited ... - ACS Publications

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