Coupled-Cluster in Real Space. 1. CC2 Ground State Energies Using

Subscriber access provided by UNIVERSITY OF THE SUNSHINE COAST

Article

Coupled-Cluster in Real Space I: CC2 Ground State Energies using Multi-Resolution Analysis Jakob Siegfried Kottmann, and Florian Andreas Bischoff J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00694 • Publication Date (Web): 13 Sep 2017 Downloaded from http://pubs.acs.org on September 15, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Journal of Chemical Theory and Computation is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 39

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

Journal of Chemical Theory and Computation

Coupled-Cluster in Real Space I: CC2 Ground State Energies 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 A framework to calculate CC2 approximated coupled-cluster ground-state correlation energies in a multiresolution basis is derived and implemented into the MADNESS library. The CC2 working equations are formulated in first quantization which makes them suitable for real-space methods. The first-quantized equations can be interpreted diagrammatically using the usual diagrams from second quantization with adjusted interpretation rules. Singularities arising from the nuclear and electronic potentials are regularized by explicitly taking the nuclear and electronic cusps into account. The regularized three- and six-dimensional cluster functions are represented directly on an adaptive grid. The resulting equations are free of singularities and virtual orbitals, which results into a low intrinsic scaling. Correlation energies close to the basis set limit are computed for small molecules. This work is the first step towards CC2 excitation energies in a multiresolution basis. ∗ To

whom correspondence should be addressed

1

ACS Paragon Plus Environment


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

I Introduction The reliable computation of molecular properties is the main goal of quantum chemistry. Because the molecular wave function is not accessible analytically, a number of approximations has to be invoked. The most prominent approximations involve the use of finite basis sets, and the choice of the quantum chemical model, e.g. Hartree-Fock, density functional theory (DFT), or post-Hartree-Fock methods. The exact solution of the Schrödinger equation is approximated as the basis set goes to completeness and the model includes higher and higher excitation levels. The errors introduced by incomplete basis sets and excitation levels into the description of molecular properties can be of different magnitude, depending on the property of interest. For the atomization energy Hartree-Fock gives errors in the 100’s of kJ/mol for small molecules, 1 by far the largest errors, MP2 has only a few 10’s kJ/mol deviation, which is also the size of the basis set error even for the moderately large quadruple zeta set. 2,3 Excitation energies are off by 1-1.5eV for the uncorrelated CIS (configuration interaction singles) theory, and 0.2 - 0.5eV for CC2, 4 while the basis set error strongly depends on the kind of excitation: For valence excited states the error is small, 0.2eV, for Rydberg states it can become as large as 2-3eV. 5 In the latter case (very) diffuse functions must be included into the basis set, which might affect the numerical stability and convergence properties. Thus it can be said that both error sources, basis set incompleteness and model error, can be significant and it is often not clear which is the more severe one. In molecular calculations atom-centered Gaussian basis function (linear combination of atomic orbitals, LCAO) are in dominant use. In order to approach the exact solution of the Schrödinger equation both the quantum chemical model and the basis set incompleteness must be improved, keeping in mind that the computational costs increase steeply with the complexity of the model and the size of the basis set. While the LCAO basis functions offer many advantages, like fast initial convergence with the basis set size, efficient integral evaluation, or facile use of fast matrix techniques, they lack the ability to systematically 2


Page 2 of 39

Page 3 of 39

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


converge to the basis set limit and lead to artificially high scaling of the calculation with the system size. For instance, the CC2 ground and excited states scale as N 5 , where N is some measure of the system size. The reason for the high scaling is the use of the same set of virtual functions to express all singles and pair-functions of the correlated or excited wave function. Recently much progress has been made in employing local methods to reduce the scaling to lower orders, e.g. by using pair natural orbitals (PNOs). 6,7 Physi3 due the action of the exchange operator (O( N )) cally the scaling should only be Nocc occ 2 )). Furthermore, in most main group molecular on the correlated pair function (O( Nocc

systems with light atoms it is possible to converge to the basis set limit, but there are properties and elements where it is hard to systematically improve the accuracy of the wave function, because the systematic extension of the basis set to completeness is unclear, as witnessed by the multitude of basis set families, and additional specializations and re-parameterizations for various molecular properties. In solid state calculations it is common practice to use plane wave basis functions which are adapted to the periodic boundary conditions of the system, possibly by using special representations for the areas around the atoms. 8 These basis sets are primarily used for DFT calculations, and only rarely for correlated methods due to their increased computational costs. 9 In contrast to the simple and efficient, but high-scaling and possibly inaccurate LCAO expansion of the molecular wave function, the wave function can also be handled in the so-called real-space methods where the wave function is represented directly on a multidimensional grid. There exist different variants of real-space methods, e.g. the direct solution on a grid via finite differences 10,11 or finite elements. 12 Wavelets as basis functions can be used in a non-adaptive way 13 or in a fully adaptive way. 14 In this article we will focus on the formalism of multiresolution analysis (MRA), which employs wavelets as basis functions, placed on a fully adaptive grid around the molecule. The wave function is an n-dimensional object, with n = 3 for Hartree-Fock or DFT orbitals, and n = 6 for the pair functions of MP2 or CC2. The working equations are solved by inverting the shifted

3



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

kinetic energy operator (instead of diagonalizing a Hamilton matrix as in LCAO) by applying the bound-state Helmholtz operator on the wave function guess, and iterating until convergence. 15 MRA has successfully been applied in quantum chemistry for computing SCF energies and gradients, 16–18 CIS and TD-DFT excitation energies, 5,19,20 MP2 energies, 21,22 magnetic properties, 23 and also explicit time-dependent problems. 24,25 Its unique features include the computation of molecular properties at the limit of the complete basis and, despite having a high computational prefactor, MRA exhibits naturally 3 , similar to MP2. 22 Also the low-scaling algorithms, as e.g. CC2 formally scales as Nocc 3 scaling, as it actual scaling of MP2 and CC2 is expected to be close to the formal Nocc 2 formal scaling. 5 A possible volume factor was observed for HF and CIS with their Nocc

can be avoided by taking advantage of the adaptiveness of MRA, and by localizing the orbitals. 17 In this article the ground state CC2 formalism 26 for use in MRA will be introduced. Since MRA is formulated in terms of functions in the real space, the CC2 working equations are given in first quantization, as opposed to the usual second quantized form. A firstquantized reinterpretation of coupled-cluster diagrams is offered so that the existing literature and formalisms can be used without extensive re-derivations. 27 The resulting equations solve for the cluster functions directly and do not include virtual orbitals. A direct consequence of this is reduced formal scaling with system size. The CC2 equations are subsequently regularized to avoid the interelectronic cusp and the electron-nuclear cusps, which makes the numerical representation of wave functions and potentials tractable. 28,29 The implementation of the working equations and numerical results for small molecules are finally given. This article should be considered the first step to the computation of excitation energies using MRA-CC2, for which we lay the foundations here.

4


Page 4 of 39

Page 5 of 39

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


II

Methodology

In this section we will derive the working equations for CC2 in their real-space representation and provide a simple scheme to switch between amplitude and real-space formalism. In the following we present a way to regularize the singular potentials which occur in the real-space representation using an explicitly correlated ansatz for the electronic pair functions. We will use the standard convention that occupied reference orbitals are labeled with i, j, k, l and virtual orbitals with a, b, c, d while general orbitals will be labeled p, q, r, s. In this work the reference orbitals are assumed to be canonical Hartree-Fock orbitals and the reference determinant is a closed-shell determinant. The extension to local orbitals is straight-forward. Furthermore we will use Einstein notation for index summations. For convenience the indexes i, j and a, b are never summed. An overview of the used notation is given in Tab. 1.

A Coupled-cluster in first and second quantization First we give a brief overview of the coupled-cluster equations in their amplitude representation. We will stick to the closed-shell formulation (see Refs. 30–32 ) of coupled-cluster singles and doubles (CCSD) and will give explicit equations for the singles and doubles of the CC2 approximation to CCSD. 26 For the formulation of explicitly correlated CC2-R12 we refer to Refs. 33,34 Coupled-cluster in second quantization The CCSD wave function is defined as |Ψi = exp Tˆ |0i,

Tˆ = Tˆ 1 + Tˆ 2

(1)

where |0i denotes the reference wave function (in this work this will always be the closedshell Hartree-Fock wave function) and Tˆ 1 , Tˆ 2 are the one- and two-electron excitation 5



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 6 of 39

operators. The CCSD Schrödinger equation is given by

H|0i = ω |0i,

ˆ N exp Tˆ , H = exp −Tˆ H

(2)

ˆ N is the normal ordered nonrelativistic electronic Hamiltonian and ω the correwhere H lation energy. If the eigenfunctions of the Fock operator are used as a basis the Tˆ 1 and Tˆ 2 operators for the closed-shell singlet can be written as 32 Tˆ 1 = tck Eˆ ck

(3)

1 ˆ ˆ Tˆ 2 = tcd E E , 2 kl ck dl

(4)

Eˆ ai = aˆ †aα aˆ iα + aˆ †a β aˆ i β

(5)

where the creation (annihilation) operator aˆ †pσ (ˆa pσ ) creates (annihilates) the spatial orbital φ p (r ) ≡ | pi with spin σ. The two sets of amplitudes tia and tijab are real numbers and can be determined by projecting the CCSD Schrödinger equation onto singly and doubly excited determinants Ωia = hia |H|0i =

∑ Sian = 0,

(6)

n

Ωijab = hijab |H|0i = 1 + Pˆ ij Pˆ ab

ijab

∑ Dn

=0

(7)

n

where the terms Sn and Dn depend on the amplitudes tia and tijab , and are given for CC2 in closed-shell form in Tabs. 2, 3 and 4. The operators Pˆ ij and Pˆ ab permute the corresponding indices. In the closed-shell case the doubles amplitudes follow the symmetry relation 32 tijab = tba ji . Coupled-cluster in first quantization

(8)

In the real-space representation the CCSD equa-

tions will not be solved for the set of amplitudes tia and tijab but for two sets of functions |τi i 6


Page 7 of 39

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


and |τij i defined on the virtual one-electron and two-electron spaces V1 and V2 . These functions are strongly orthogonal towards the occupied reference orbitals |i i. Following 35 and 27 the functions |τi i and |τij i will be referred to as one- and two-electron cluster functions. Orthogonality of the cluster functions to the occupied space can be ensured by the projection operators Q and Q12 which are given by

Q = |cihc| = 1 − |kihk|, Q12 = Q ⊗ Q .

(9) (10)

so that

|τi i = Q|τi i,

(11)

|τij i = Q12 |τij i.

(12)

A one-to-one correspondence between cluster functions |τi i and |τij i and cluster amplitudes tia and tijab may be established by expanding the cluster functions in the virtual basis, or, in the opposite direction, by projecting the cluster functions onto the virtual basis.

|τi i = tic |ci,

tia = h a|τi i,

(13)

|τij i = tijcd |cdi,

tijab = h ab|τij i.

(14)

The cluster functions obey the symmetry relation

|τij i = Pˆ 12 |τji i,

(15)

where Pˆ 12 permutes the coordinates of particles 1 and 2. To solve for the cluster functions the CCSD Schrödinger equation is projected onto the spaces V1 and V2 . For this we formally project against the sum of all singly (and doubly) excited determinants multiplied

7



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 8 of 39

with the corresponding (product of) virtuals

|Ωi i = |cihic |H|0i = Q ∑ |Sin i = 0,

(16)

n

|Ωij i = |cdihijcd |H|0i = Q12 1 + Pˆ ij Pˆ 12

ij

∑ |Dn i = 0.

(17)

n

Equations (16) and (17) together with the equations (13) and (14) provide a simple scheme to switch from amplitude to real-space formalism: 1. Write down amplitude equation for the singles (doubles). 2. Replace the amplitudes by using Eq. (13) and Eq. (14). 3. Multiply by |ci (or |cdi) from the left and sum over all such states. 4. Use Eq. (9) and Eq. (10) for the projectors to become independent of any virtuals. Rename integration variables if necessary. 5. Absorb the projectors Q and Q12 into the cluster functions of the singles and doubles if possible. The terms for the closed-shell singles and doubles equations of CC2 are given in amplitude and real-space formalism in Tabs. 2, 3 and 4 and an explicit example is given in the Appendix A. Diagrammatic formalism

The equations for the coupled-cluster singles (doubles) am-

plitudes (6) (and (7)) can be expanded diagrammatically (see 36 for a detailed introduction). Using the appropriate rules the terms which enter the singles and doubles potentials in their amplitude form can then be derived directly from these diagrams. These rules can be adjusted to give the terms in real-space representation from the same diagrams. It is only necessary to change the interpretation of particle (virtual) lines and horizontal bars which represent amplitudes in the second-quantized form. In the realspace representation particle lines will represent the coordinates in real-space and also 8


Page 9 of 39

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


carry the projector onto the virtual space which will act on the origin of the particle line (either a cluster function or an operator). Coordinates from internal particle lines are integrated over. The cluster functions will be represented by the horizontal bars, where the in- and outgoing lines will determine the index and the coordinate of the cluster function. The rules for the symmetry factors and signs do not change in the real-space representation. An overview and comparison of the interpretation of diagrams for the amplitude and real-space formalism is given in Tab. 5. We also provide a detailed example in the appendix. If the internal particle lines originate from a horizontal bar, which is always the case for the terms of Tabs. 2, 3 and 4, the projector can directly be absorbed into the corresponding cluster function. Otherwise the projector has to be taken into account As an example we give the diagram

= hkl | g12 Q12 f 12 |τk τl i,

(18)

which arises from the CC2 energy equation using the explicitly correlated ansatz of Eq. (37). Here we used the double lines to represent the correlation factor f 12 which will be introduced later.

9



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

B 1

Page 10 of 39

Implementation details Energy equation

The working equation for the correlation energy ω is obtained by projecting the CC2 or CCSD Schrödinger equation onto the reference state ω = h0|H|0i

=

+

c d kl lk tcd + t t = 2gcd − gcd kl k l , kl lk kl lk = 2gτkl − gτkl + 2gτk τl − gτk τl .

(19)

Once the cluster functions |τi i and |τij i are determined, the energy is a simple integral of the cluster functions with the ground state orbitals over the electron repulsion operator g12 . 2

Singles equations

The singles equations Eq. (16) contains all terms of Tab. 3. The number of terms can be reduced by introducing relaxed orbitals {|ti i} defined by

|ti i = |i i + |τi i.

(20)

This transformation is similar to the T1-transformed orbitals in LCAO, see the Appendix for the detailed relations. The |ti i functions are defined in the occupied and the virtual space, however the projector Qt , which we define as (see also Tab. 1)

Qt = 1 − |ti ihi |,

10


(21)

Page 11 of 39

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


projects still onto the virtual space which can be seen by

Qt Qt = Qt ,

(22)

QQt = Qt ,

(23)

OQt = 0.

(24)

With this we can make the following simplifications to the terms which enter the singles equation (note that the symbols abbreviating the terms do not contain the projector onto the virtual space) ti i i |S4a i, i = |S2c i + |S4a

(25)

ti i i |S5b i = |S3c i + |S5b i,

(26)

i |S6ti i = |S5c i + |S6i i, ti ti Qt |S5b i = Q |S5b i + |S6ti i , i i i Qt |S2b i = Q |S2b i + |S4b i ,

(27) (28) (29)

so that the singles equation can be written as

|Ωi i = ( F − ǫi ) |τi i + Q

ti i |S4a i + |S4c i

+Q

t

ti i i |S5b i + |S2b

.

(30)

The first term on the right hand side corresponds to the terms S3a and S3b , where the strong orthogonality projector Q has been absorbed into the cluster function |τi i. Note t

that this short-hand notation is not unique, since the term S4ai could also be written as t

S2ci . The Fock operator and the Q projector commute as the HF reference has been solved numerically exactly. From Eq. (30) and Tab. 3 it may be seen that the singles formally 3 in the real-space representation. For later use we name the second and third scale as Nocc

11



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 12 of 39

term of Eq. (30) "applied singles potential" Vτi ti ti i i |Vτi i = Q |S4a i + |S4c i + Qt |S5b i + |S2b i 3

(31)

Doubles equations

For the CC2 doubles equations (17) we can do similar simplifications as for the singles and write tt

ij

tt

ij

tt

ij

ij

ij

(32)

ij

ij

(33)

ij

ij

(34)

|D6ai j i = |D1 i + |D4a i + |D6a i, |D8ai j i = |D4b i + |D6c i + |D8a i, |D8ci j i = |D6b i + |D8b i + |D8c i, tt 1 t tt tt tt t g12 |ti t j i. Q12 |D6ai j i = Q |D6ai j i + |D8ai j i + |D8ci j i = Q12 2

(35)

ti t j

Again this short-hand notation is not unique, since the term D6a could also be written as ti t j

D1 . The doubles equation in real-space now read as

|Ωij i = Q12 1 + Pˆ ij Pˆ 12

ti t j ij ij t 1 + Pˆ ij Pˆ 12 |D6a i |D2a i + |D2b i + Q12

t g12 |ti t j i, = Fˆ 12 − ǫij |τij i + Q12 where the factor of

1 2

(36)

in the D6a term cancels due to the Pˆ ij Pˆ 12 permutation. Since the

t are separable the CC2 doubles equations also formally scale as projectors Q12 and Q12 3 . 22 Nocc

C

Regularization of CC2 in real-space formalism

The CC2 doubles equations Eq. (36) contain the singular Coulomb potential which is hard to represent numerically. Since the exact cusp condition following from the Coulomb potential is well-known, 37 the Coulomb singularity can be factored out by using an appro12


Page 13 of 39

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


priately constructed wave function containing the correlation cusp. 38,39 For the MRA representation of the correlated wave function the removal of the cusp is mandatory. 21,22,29 t g | t t i term of the doubles working equation Eq. (36) contains the Coulomb sinThe Q12 12 i j

gularity. To remove this singularity equation Eq. (36) we choose a corresponding cluster function ansatz as t |τij i = |uij i + Q12 f 12 |ti t j i,

(37)

introducing the cusp free pair function |uij i and the correlation factor f 12 . As for MP2 21 we use a slater-type 40 correlation factor given by r f 12 = 1 − exp − 12 , 2

(38)

with the electronic distance r12 = |~r1 −~r2 |. Its leading term is 1 2 ), f 12 = r12 + O(r12 2

(39)

and it cancels the singularities of Eq. (36) exactly. For a discussion of singlet and triplet pairs we refer to Ref. 22 We note that there is some freedom in choosing the strong orthogonality projector Q or Qt , with the current choice of Qt being the most convenient. The commutator between the correlation factor and the Fock operator

ˆ 12 − K, ˆ f 12 , Fˆ 12 , f 12 = − g12 + U

(40)

ˆ 12 being Kutzelnigg’s regularized potential cancels the electronic singularity g12 , with U

13



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 14 of 39

operator. 38 With this ansatz and Eq. (40) the CC2 doubles equations (36) become t |Ωij i = Fˆ 12 − ǫij |τij i + Q12 g12 |ti t j i t t f 12 |ti t j i + Q12 g12 |ti t j i = Fˆ 12 − ǫij |uij i + Fˆ 12 − ǫij Q12 t = Fˆ 12 − ǫij |uij i + Q12 g˜ij |ti t j i,

(41)

t ˆ Qt = F, ˆ Qt (see Appendix B) and define the regularF, where we use the identity Q12 ized Coulomb operator g˜ij as

ˆ 12 − K ˆ 12 , f 12 + Fˆ 12 , Qt f 12 g˜ij = f 12 Fˆ 12 − ǫij + U 12

(42)

Note that there is no summation over the indices i and j in Eq. (41). The second and the third term of Eq. (42) are computed in analogy to MP2. 21,22,29 The commutator which appears in the last term of the modified Coulomb operator acts on each particle separately

t ˆ Qt ⊗ Qt + Qt ⊗ F, ˆ Qt = F, Fˆ 12 , Q12

= O Vτ ⊗ Qt + Qt ⊗ O Vτ .

(43)

Once the singles equations are solved, the relation |Ω j i = 0 of Eq. (30) holds, and the commutator which acts on only one particle can be evaluated with the applied singles potential |Vτk i of Eq. (31)

ˆ Qt = − F, ˆ Oτ F, = − Fˆ − ǫk |τk ihk|

= |Vτk ihk| ≡ −O Vτk .

(44)

In a similar way we can evaluate the first term of the regularized Coulomb operator when

14


Page 15 of 39

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


it is applied to the pair state |ti t j i f 12 Fˆ 12 − ǫij |ti t j i = − f 12 |Vτi τj i + |τi Vτj i

(45)

where we have used Fˆ 12 − ǫij |ti t j i = Fˆ 12 − ǫij |τi τj i

= −|Vτi τj i − |τi Vτj i.

(46)

D Solving the residual equations To solve for the cluster functions |τi i and the regularized pair functions |uij i we use the Green’s function approach of Ref. 41 for the singles and doubles equations ˆ i |Vτ i + V ˆ nuc |τi i + 2ˆJ|τi i − K ˆ |τi i , |τi i = −2G i

ˆ ij Qt g12 |ti t j i + V ˆ nuc |uij i + 2ˆJ12 |uij i − K ˆ 12 |uij i , |uij i = −2G 12

(47) (48)

where we used the bound-state Helmholtz Green’s operators (BSH) ˆ i = (−∆ − 2ǫi )−1 , G ˆ ij = −∆12 − 2ǫij −1 . G

(49) (50)

A complete iteration consists of iterating the singles equations until convergence and subsequently applying the BSH operator on the doubles equations once, followed by converging the singles equations again. The singles can be solved similarly to CIS 5 while the doubles are solved similarly to MP2. 21 In contrast to the CIS equations the CC2 singles equations have an inhomogeneity arising from the CC2 doubles so that we can take the initial guess for the singles to be zero-functions. It should also be noted that the CC2 singles are not normalized while the CIS singles in Ref. 5 are. 15



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

III

Page 16 of 39

Computational Details

The regularized CC2 working equations (47) and (48) were implemented into the MADNESS library. 14,42 All reported calculations where performed using the regularized nuclear potential of Ref., 28 were we used the Slater correlation factor (parameter a = 1.5, see Eq. (28) in Ref. 28 ) for the nuclear cusps. For an improved convergence of the reference orbitals, singles and doubles we used the KAIN 43 approach. The polynomial orders for the multiwavelets were k = 5 and k = 6. Thresholds for the MRA accuracy were ǫ = 10−3 and ǫ = 10−4 for 6D functions (electron pairs |uij i) as given in the text, and 0.01ǫ for 3D functions (singles |τi i, and reference orbitals |i i). For all molecules the 1s core orbitals were kept frozen and the size of the simulation box was 60 bohrs in each dimension. LCAO calculation for MP2, CC2 and MP2-F12 where calculated with the ricc2 44 program of the TURBOMOLE program package. 45 MP2-F12 uses Ansatz 2B with F+K approximation and the fixed-amplitudes (sp) Ansatz. 46 As LCAO basis sets we used the aug-cc-pVXZ 47–50 basis sets of Dunning et al. and the corresponding auxiliary basis sets from the TURBOMOLE library. 51,52 The two LCAO calculations with the highest cardinal number in their basis set are extrapolated by the formula 53

CBS( X, Y ) =

EX X 3 − EY Y 3 , X3 − Y3

(51)

where EX and EY are the correlation energies obtained with the two basis sets with cardinal number X and Y. The molecular structures are MP2/aug-cc-pVTZ minimum structures except for the BH molecule for which the coordinates were taken from reference. 34 Coordinates to all used structures are given in the Supporting Information. The CC2-F12 results were computed with the KOALA 54,55 program using Ansatz 3B (equals Ansatz 2B in Turbomole) with F+K approximation and the fixed-amplitudes (sp) Ansatz.

16


Page 17 of 39

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


IV

Results and Discussion

Calculations on the MP2 and CC2 ground state of BeH2 , BH, BH3 , H2 O, and CH4 were performed using MRA and LCAO.

BeH2 .

The results for the BeH2 molecule listed in Tab. 6 show that the MRA calcula-

tions with polynomial order k=6 is nearly identical to the extrapolated LCAO calculations while the MRA calculations with k=5 is in magnitude slightly below the extrapolated values. The same holds true for the excitation energies obtained with explicitly correlated methods. BH. The results for the BH molecule are given in Tabs. 7 and 8. The MP2 pair correlation energies (see Tab. 8) calculated with MRA agree to the given precision with the best obtained LCAO result which is MP2-F12/aug-cc-pCV6Z. The MP2-F12 correlation energies obtained with the core-polarized 5Z basis sets differ only slightly from the core-polarized 6Z result. For CC2-F12 we expect a similar convergence behaviour with respect to the basis set than for MP2-F12 so that the CC2-F12/aug-cc-pwC5Z result can be assumed to be very close to the basis set limit. MRA reproduces this results up to the given accuracy. The CC2-R12/aug-cc-pV6Z calculation of Ref. 34 slightly overestimates the result.

BH3 , CH4 and H2 O.

The total correlation energies of BH3 , H2 O, and, CH4 given in

Tabs 9, 10, and 11 are up to 3.5 mEh below the absolute extrapolated values for MP2 and CC2 but still in very good agreement with the correlation energies obtained with the very large aug-cc-pV6Z basis set. The extrapolated values are again in very good agreement with the explicitly correlated results. Like for MP2 22 precision in CC2 is only guaranteed for the individual pair correlation energies so that an accumulation of small errors in magnitude below the given accuracy has to be expected for total energies. We expect that local reformulations and the use of

17



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

tighter thresholds solve this problem. Note that relative quantities like excitation energies from linear-response equations are more suitable for calculations with MRA than total energies. The computational performance of the CC2 implementation is similar to MP2 22 where the CPU times and memory are dominated by the application of the exchange operator on 6D functions. Both in MP2 and in CC2 the regularized Coulomb operator g˜ij (Eq. 42) contains the Fock operator. Unlike in MP2, where the Fock operator cancels with the orbital energies after being applied on the ground state orbitals, in CC2 the Fock operator does not cancel. Explicit application of the Fock operator is unfavorable for MRA representation since the kinetic energy operator and therefore the Laplacian is included. This can be prevented by taking the equations for the singles into account so that the terms which initially contained the Fock operator are replaced with terms containing the CC2 singles potential. We note at this point that the explicit application of the Fock operator is still possible in our approach since the used precision for the 3D functions is two orders of magnitudes higher. The used approach with the singles potential is however cheaper and more precise. Also the additional memory cost resulting from the storage of the singles potential can be neglected since the singles potentials are 3D functions and demand much less space than the 6D pair functions.

V

Conclusions

The CC2 equations have been reformulated in first-quantized form suited for real-space calculations. The diagrammatic formalism may be used with only minor reinterpretation, avoiding extensive rederivations and giving access to the standard literature on coupledcluster. An explicitly correlated ansatz has been used to regularize the singularities arising from the nuclear and electronic coulomb potentials. In LCAO based calculations approximations to the explicitly correlated ansatz are possible and in wide use while the explicitly correlated ansatz in MRA has to regularize all terms of the working equations

18


Page 18 of 39

Page 19 of 39

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


to avoid the representation of the singular potentials on the grid. The resulting regularized working equations solve for singles and doubles free of nuclear and electronic cusps 3 , where N and scale formally as Nocc occ is the number of occupied orbitals, two orders less

than LCAO which scales as N 5 . The reduced scaling is due to the absence of a fixed set of virtual orbitals in the working equations and the adaptiveness of the MRA grid, which avoids a possible volume factor. Local reformulations are expected to reduce the scaling further. The implementation of the regularized CC2 equations using multiresolution analysis is part of the MADNESS library and available on Github. 42 This work lays the foundation towards the computation of excitation energies in a multiresolution framework. An extension of this approach to linear-response is under development and will be reported in the second part of this series. 56

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.

VII

Supporting Information

The Supporting Information for this work contains 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

References (1) Tajti, A.; Szalay, P. G.; Csaszar, A. G.; Kállay, M.; Gauss, J.; Valeev, E. F.; Flowers, B. A.; Vázquez, J.; Stanton, J. F. HEAT: High accuracy extrapolated ab initio thermochem19



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

istry. J. Chem. Phys. 2004, 121, 11599–11613. (2) Klopper, W.; Ruscic, B.; Tew, D. P.; Bischoff, F. A.; Wolfsegger, S. I. Chem. Phys. 2009, 356, 14–24. (3) Jensen, S. R.; Saha, S.; Flores-Livas, J. A.; Huhn, W.; Blum, V.; Goedecker, S.; Frediani, L. The Elephant in the Room of Density Functional Theory Calculations. J. Phys. Chem. Lett. 2017, 8, 1449–1457. (4) Send, R.; Kühn, M.; Furche, F. Assessing Excited State Methods by Adiabatic Excitation Energies. J. Chem. Theory Comput. 2011, 7, 2376–2386. (5) Kottmann, J. S.; Höfener, S.; Bischoff, F. A. Numerically accurate linear responseproperties in the configuration-interaction singles (CIS) approximation. Phys. Chem. Chem. Phys. 2015, 17, 31453–31462. (6) Neese, F.; Wennmohs, F.; Hansen, A. Efficient and accurate local approximations to coupled-electron pair approaches: An attempt to revive the pair natural orbital method. J. Chem. Phys. 2009, 130, 114108. (7) Schütz, M.; Werner, H.-J. Low-order scaling local electron correlation methods. IV. Linear scaling local coupled-cluster (LCCSD). J. Chem. Phys. 2001, 114, 661. (8) Draxl, A. G.; Kontur, S.; Meisenbichler, C.; Nabok, D.; Pavone, P.; Rigamonti, S.; Sagmeister, S.; Werner, U.; Claudia, exciting: a full-potential all-electron package implementing density-functional theory and many-body perturbation theory. Journal of Physics: Condensed Matter 2014, 26, 363202. (9) Grüneis, A.; Shepherd, J. J.; Alavi, A.; Tew, D. P.; Booth, G. H. Explicitly correlated plane waves: Accelerating convergence in periodic wavefunction expansions. J. Chem. Phys. 2013, 139, 084112.

20


Page 20 of 39

Page 21 of 39

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


(10) Shiozaki, T.; Hirata, S. Grid-based numerical Hartree-Fock solutions of polyatomic molecules. Phys. Rev. A 2007, 76, 040503. (11) Hirata, S.; Shiozaki, T.; Johnson, C. M.; Talman, J. D. Numerical solution of the ˇ equation using a multicentre radial-angular grid. Mol. Phys. 2017, 115, Sinanoglu 510–525. (12) Flores, J. R.; Kolb, D. Atomic MP2 correlation energies fast and accurately calculated by FEM extrapolations. J. Phys. B: At. Mol. Opt. Phys. 1999, 32, 779–790. (13) Genovese, L.; Neelov, A.; Goedecker, S.; Deutsch, T.; Ghasemi, S. A.; Willand, A.; Caliste, D.; Zilberberg, O.; Rayson, M.; Bergman, A.; Schneider, R. Daubechies wavelets as a basis set for density functional pseudopotential calculations. J. Chem. Phys. 2008, 129, 014109. (14) Harrison, R. J.; Beylkin, G.; Bischoff, F. A.; Calvin, J. A.; Fann, G. I.; FossoTande, J.; Galindo, D.; Hammond, J. R.; Hartman-Baker, R.; Hill, J. C.; Jia, J.; Kottmann, J. S.; Yvonne Ou, M.-J.; Pei, J.; Ratcliff, L. E.; Reuter, M. G.; RichieHalford, A. C.; Romero, N. A.; Sekino, H.; Shelton, W. A.; Sundahl, B. E.; Thornton, W. S.; Valeev, E. F.; Vázquez-Mayagoitia, Á.; Vence, N.; Yanai, T.; Yokoi, Y. MADNESS: A Multiresolution, Adaptive Numerical Environment for Scientific Simulation. Siam J. Sci. Comput. 2016, 38, S123–S142. (15) Beylkin, G.; Coifman, R.; Rokhlin, V. Fast wavelet transforms and numerical algorithms I. Comm. Pure Appl. Math. 1991, 44, 141–183. (16) Harrison, R. J.; Fann, G. I.; Yanai, T.; Gan, Z.; Beylkin, G. Multiresolution quantum chemistry: basic theory and initial applications. J. Chem. Phys. 2004, 121, 11587– 11598. (17) Yanai, T.; Fann, G. I.; Gan, Z.; Harrison, R. J.; Beylkin, G. Multiresolution quantum

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

chemistry in multiwavelet bases: Hartree-Fock exchange. J. Chem. Phys. 2004, 121, 6680–6688. (18) Yanai, T.; Fann, G. I.; Gan, Z.; Harrison, R. J.; Beylkin, G. Multiresolution quantum chemistry in multiwavelet bases: Analytic derivatives for Hartree-Fock and density functional theory. J. Chem. Phys. 2004, 121, 2866–2876. (19) Yanai, T.; Harrison, R. J.; Handy, N. Multiresolution quantum chemistry in multiwavelet bases: time-dependent density functional theory with asymptotically corrected potentials in local density and generalized gradient approximations. Mol. Phys. 2005, 103, 413–424. (20) Yanai, T.; Fann, G. I.; Beylkin, G.; Harrison, R. J. Multiresolution quantum chemistry in multiwavelet bases: excited states from time-dependent Hartree-Fock and density functional theory via linear response. Phys. Chem. Chem. Phys. 2015, 17, 31405–31416. (21) Bischoff, F. A.; Harrison, R. J.; Valeev, E. F. Computing many-body wave functions with guaranteed precision: The first- order Møller-Plesset wave function for the ground state of helium atom. J. Chem. Phys. 2012, 137, 104103. (22) Bischoff, F. A.; Valeev, E. F. Computing molecular correlation energies with guaranteed precision. J. Chem. Phys. 2013, 139, 114106. (23) Jensen, S. R.; Flå, T.; Jonsson, D.; Monstad, R. S.; Ruud, K.; Frediani, L. Magnetic properties with multiwavelets and DFT: the complete basis set limit achieved. Phys. Chem. Chem. Phys. 2016, 18, 21145–21161. (24) Vence, N.; Harrison, R.; Krsti´c, P. Attosecond electron dynamics: A multiresolution approach. Phys. Rev. A 2012, 85, 033403. (25) Domínguez-Gutiérrez, F. J.; Krstic, P. S. Charge transfer in collisions of H + with Li

22


Page 22 of 39

Page 23 of 39

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


(1s 2 2s, 2p z ): TD-MADNESS approach. J. Phys. B: At. Mol. Opt. Phys. 2016, 49, 195206. (26) Christiansen, O.; Koch, H.; Jorgensen, P. The second-order approximate coupled cluster singles and doubles model CC2. Chem. Phys. Lett. 1995, 243, 409–418. (27) Bukowski, R.; Jeziorski, B.; Szalewicz, K. Gaussian geminals in explicitly correlated coupled cluster theory including single and double excitations. J. Chem. Phys. 1999, 110, 4165–4183. (28) Bischoff, F. A. Regularizing the molecular potential in electronic structure calculations. I. SCF methods. J. Chem. Phys. 2014, 141, 184105. (29) Bischoff, F. A. Regularizing the molecular potential in electronic structure calculations. II. Many-body methods. J. Chem. Phys. 2014, 141, 184106. (30) Paldus,

J.

Correlation

problems

in

atomic

and

molecular

systems.

V.

˘ Radapted ˘ Relectron ˇ ˇ SpinâA coupled cluster manyâA theory. J. Chem. Phys. 1977, 67, 303–318. ˘ Rshell ˇ (31) Scuseria, G. E.; Scheiner, A. C.; Lee, T. J.; Rice, J. E.; Schaefer, H. F. The closedâA coupled cluster single and double excitation (CCSD) model for the description of electron correlation. A comparison with configuration interaction (CISD) results. J. Chem. Phys. 1987, 86, 2881–2890. (32) Helgaker, T.; Jorgensen, P.; Olsen, J. Molecular electronic-structure theory; John Wiley & Sons Inc, 2000. (33) 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.

23



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 24 of 39

(34) Neiss, C.; Hättig, C.; Klopper, W. Extensions of r12 corrections to CC2-R12 for excited states. J. Chem. Phys. 2006, 125, 64111. (35) Jeziorski, B.; Monkhorst, H. J.; Szalewicz, K.; Zabolitzky, J. G. Atomic and molecular correlation energies with explicitly correlated Gaussian geminals. III. Coupled cluster treatment for He, Be, H2, and LiH. J. Chem. Phys. 1984, 81, 368–388. (36) Shavitt, I.; Bartlett, R. J. Many-body methods in chemistry and physics: MBPT and coupled-cluster theory. 2009. (37) Kato, T. On the eigenfunctions of many-particle systems in quantum mechanics. Comm. Pure Appl. Math. 1957, 10, 151. (38) Kutzelnigg, W. R12-Dependent terms in the wave-function as closed sums of partialwave amplitudes for large-l. Theoret. Chim. Acta 1985, 68, 445–469. (39) Kong, L.; Bischoff, F. A.; Valeev, E. F. Explicitly Correlated R12/F12 Methods for Electronic Structure. Chem. Rev. 2012, 112, 75–107. (40) Ten-no, S. Initiation of explicitly correlated Slater-type germnal theory. Chem. Phys. Lett. 2004, 398, 56–61. (41) Kalos, M. Monte Carlo Calculations of the Ground State of Three- and Four-Body Nuclei. Phys. Rev. 1962, 128, 1791–1795. (42) MADNESS: tific

Multiresolution

Simulation,

Adaptive

Numerical

Environment

https://github.com/m-a-d-n-e-s-s/madness,

git

tag

for

Sciencc2

(or

ba6f8fbd864c97fc71af7f828d47f0fa3e5186b9). (43) Harrison, R. J. Krylov subspace accelerated inexact Newton method for linear and nonlinear equations. J. Comp. Chem. 2002, 25, 328–334. (44) Hättig, C.; Weigend, F. CC2 excitation energy calculations on large molecules using the resolution of the identity approximation. J. Chem. Phys. 2000, 113, 5154–5161. 24


Page 25 of 39

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


(45) TURBOMOLE V7.0 2015, a development of University of Karlsruhe and Forschungszentrum Karlsruhe GmbH, 1989-2007, TURBOMOLE GmbH, since 2007; available from http://www.turbomole.com. (46) Bachorz, R. A.; Bischoff, F. A.; Glöß, A.; Hättig, C.; Höfener, S.; Klopper, W.; Tew, D. P. The MP2-F12 method in the TURBOMOLE program package. Journal of Computational Chemistry 2011, 32, 2492–2513. (47) Dunning, T. H. Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron through Neon and Hydrogen. J. Chem. Phys. 1989, 90, 1007–1023. (48) Wilson, A. K.; van Mourik, T.; Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. VI. Sextuple zeta correlation consistent basis sets for boron through neon. J. Mol. Struc.-Theochem. 1996, 388, 339–349. (49) Peterson, K. A.; Woon, D. E.; Dunning, T. H. Benchmark calculations with correlated molecular wave functions. IV. The classical barrier height of the H+H2 - H2+H reaction. J. Chem. Phys. 1994, 100, 7410–7415. (50) Kendall, R. A.; Dunning, T. H.; Harrison, R. J. Electron Affinities of the First-Row Atoms revisited. Systematic Basis Sets and Wave Functions. J. Chem. Phys. 1992, 96. (51) Weigend, F.; Köhn, A.; Hättig, C. Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations. J. Chem. Phys. 2002, 116, 3175–3183. (52) Hättig, C. Optimization of auxiliary basis sets for RI-MP2 and RI-CC2 calculations: Core–valence and quintuple-ζ basis sets for H to Ar and QZVPP basis sets for Li to Kr. Phys. Chem. Chem. Phys. 2004, (53) Halkier, A.; Helgaker, T.; Jorgensen, P.; Klopper, W.; Koch, H.; Olsen, J.; Wilson, A. K. Basis-set convergence in correlated calculations on Ne, N2, and H2O. Chem. Phys. Lett. 1998, 286, 243–252. 25



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

(54) Koala, an ab-initio electronic structure program, written by S. Höfener with contributions from A. S. Hehn and J. Heuser. (55) Höfener, S. Coupled-cluster frozen-density embedding using resolution of the identity methods. J. Comp. Chem. 2014, 35, 1716–1724. (56) Kottmann, J. S.; Bischoff, F. A. Coupled-Cluster in Real Space II: CC2 Excited States using Multiresolution Analysis. J. Chem. Theory Comput. 2017, submitted.

26


Page 26 of 39

Page 27 of 39

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 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

xαβ = hαβ| x12 |γκ i,

Convolutions

xα = hα| x12 |γi (r) ,

Closed-shell Fock operator

6D to 3D convolution Projectors

γκ

x12 ∈ { g12 , f 12 }

γ

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 |



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 2: Diagrams and the corresponding interpretation in amplitude and real-space formalism for the Brillouin terms (which vanish for canonical orbitals). Open hole lines are labeled with i while open particle lines for the amplitude formalism are labeled with a. All other indices are summed over. label S1i

diagram F

Q|S1i i i S2a i i Q|S2a i S5a i i Q|S5a

expression h a|Fˆ |i i Q Fˆ |i i

ac − tca hk|Fˆ |ci 2tik ik Q 2hk|Fˆ 2 |τik i2 − hk|Fˆ 1 |τik i1

F

F

−hk|Fˆ |citic tka −Q |τk ihk|Fˆ |τi i


Page 28 of 39

Page 29 of 39

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 3: Diagrams for the CC2 singles and the corresponding interpretation in amplitude and real-space formalism. Open hole lines are labeled with i while open particle lines for the amplitude formalism are labeled with a. All other indices are summed over. label

diagram

ia S2b i Q|S2b i

Q (2hk| g12 |τik i2 − hk| g12 |τik i1 ) kl 2t ac − tca − gic kl kl

ia S2c i i Q|S2c ia S3a i Q|S3b i ia S3b i i Q|S3a ia S3c

expression ak − g ak tcd 2gcd dc ik

−Q 2hl | gik (2)|τkl i2 − hl | gik (1)|τkl i1 F

F

h a|Fˆ |citic Q Fˆ |τi i

−hk|Fˆ |i itka −Q hk|Fˆ |i i|τk i ak − g ak tc 2gic ci k

i Q|S3c i

Q 2gτkk |i i − gik |τk i

ia S5b

ak − g ak tc td 2gcd dc i k

i Q|S5b i

Q 2gτkk |τi i − gτki |τk i

ia S5c

kl − gkl t a tc − 2gic ci k l kl kl −Q 2giτl − gτl i |τk i

i i Q|S5c

S6ia

Q|S6i i ia S4a i Q|S4a i ia S4b i Q|S4b i ia S4c i Q|S4c i

kl − gkl t a tc td − 2gcd dc k i l kl −Q gτi τl − gτkll τi |τk i kl − gkl t ad tc − 2gcd dc kl i

−Q 2hl | gτki (2)|τkl i2 − hl | gτki (1)|τkl i1 kl − gkl t a tcd − 2gcd dc k il −Q 2gτklil − gτklli |τk i kl − 2gkl − 2glk + glk t ad tc 4gcd dc cd dc il k

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 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 39

Table 4: Diagrams for the CC2 doubles and the corresponding interpretation in amplitude and real-space formalism. The open hole lines are labeled with i, j from left to right while the internal hole lines are labeled with k, l. Open particle lines for the amplitude represenation are labeled with a, b from left to right. All indices which are not i, j, a, b are summed over. label

diagram

expression

ijab

1 ab gij

D1

ij

Q12 |D1 i

ij

Q12 |D2a i

1 2 g12 |ij i

Q12 F

Q12 Fˆ (1) |τij i

ijab

D2b ij Q12 |D2b i

−Q12 ǫi |τij i

ijab ij

Q12 ( g12 |τi ji)

ijab ij

ijab

D6a

ij

Q12 |D6a i

− gijkb tka −Q12 |τk i ⊗ gik | ji 1 ab c d gcd ti t j Q12 12 g12 |τi τj i

ijab ij

ijab

D6c

ij

Q12 |D6c i

1 kl ab gij tkl

Q12

ijab

D8b

ij

Q12 |D8b i ijab

D8c

ij

Q12 |D8c i

1 kl 2 gij | τk τl i

−Q12 |τk i ⊗ gτki | ji + gik |τj i

ijab ij

2

kb tc + gkb tc t a − gcj i ic j k kb t a tc td − gcd k i j

D8a

Q12 |D8a i

2

D6b

Q12 |D6b i

ab tc gcj i

D4b

Q12 |D4b i

−ǫi tijab

F

D4a

Q12 |D4a i

h a|Fˆ |citijcb

ijab

D2a

2

−Q12 |τk i ⊗ gτki |τj i kl tc t a tb gcj i k l Q12 gτkli j |τk τl i 1 kl c d a b 2 gcd ti t j tk tl

Q12


1 kl 2 gτi τj | τk τl i

Page 31 of 39

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 5: Interpretation of diagrams in the amplitude and real-space formalism. name

symbol

amplitude-formalism

real-space formalism

particle lines

particle index sum over internal

coordinate integrate over internal

hole lines

hole index sum over internal

hole index sum over internal

singles

tia

hr1 |τi i ≡ τi (r1 )

doubles

tijab

hr1 r2 |τij i ≡ τij (r1 , r2 ) hout|Fˆ |ini

F

one-electron operator two-electron operator

hleft out, right out| g12 |left in, right ini

Table 6: Total correlation energies in mEh for the BeH2 molecule (2.513 a0 ). Basis

MP2

CC2

aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z CBS(5,6)

−52.60 −61.60 −64.75 −65.90 −67.10

−52.68 −61.70 −64.88 −66.03 −67.24

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

−66.63 −66.77 −67.03 −67.17

MP2-F12

CC2-F12

−64.35 −66.66 −67.10 −67.19

−64.46 −66.79 −67.24 −67.33



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 32 of 39

Table 7: Total correlation energies in mEh for the BH molecule (2.329 a0 ). CC2-R12 (Ansatz 1B) calculations are taken from the supporting information of Ref. 34 Basis

MP2

CC2

MP2-F12

CC2-F12

CC2-R12

aug-cc-pVQZ aug-cc-pV5Z aug-cc-pV6Z CBS(5,6)

−78.63 −80.21 −80.92 −81.91

−79.02 −80.62 −81.34 −82.33

−81.77 −81.91 −82.00

−82.19 −82.36

−81.24 −82.02 −82.63

aug-cc-pwCVQZ aug-cc-pwCV5Z aug-cc-pCV5Z aug-cc-pCV6Z CBS(5,6)

−79.06 −80.41 −80.33 −81.01 −81.94

−79.47 −80.83 −80.75 −81.43 −82.37

−81.88 −82.01 −82.00 −82.04

−82.31 −82.43

MRA(ǫ=10−3 ,k=5) MRA(ǫ=10−3 ,k=6) MRA(ǫ=10−4 ,k=5) MRA(ǫ=10−4 ,k=6)

−81.40 −81.69 −82.00 −81.98

−81.70 −81.98 −82.47 −82.47

Table 8: Pair correlation energies in mEh of the BH molecule. Comparison of MRA-MP2 and MP2-F12/aug-cc-pCV6Z. MRA-MP2 (ǫ=10−3 ) Pair 22 23 33

MP2-F12

MRA-MP2 (ǫ=10−4 )

(k=5)

(k=6)

(k=5)

(k=6)

−27.02 −26.81 −25.93 −25.66 −29.09 −28.93

−26.89 −25.81 −28.99

−27.01 −25.92 −29.08

−27.00 −25.95 −29.03


Page 33 of 39

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 9: Total correlation energies in mEh for the BH3 molecule (2.244 a0 ). Basis

MP2

CC2

MP2-F12

CC2-F12

aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z aug-cc-pV6Z CBS(5,6)

−95.22 −114.68 −121.00 −123.26 −124.28 −125.68

−95.45 −115.01 −121.40 −123.67 −124.71 −126.13

−120.91 −124.90 −125.55 −125.69 −125.73

−121.20 −125.31 −125.99

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

−123.94 −124.34 −124.47 −124.88

Table 10: Total correlation energies in mEh for the water molecule (1.820 a0 , 103.9◦ ). Basis

MP2

CC2

MP2-F12

CC2-F12


−219.91 −268.86 −286.41 −293.40 −296.46 −300.65

−222.22 −271.34 −288.79 −295.77 −298.81 −302.98

−293.72 −298.46 −300.02 −300.60 −300.83

−295.55 −300.85 −302.37 −302.94

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

−297.52 −299.65 −298.70 −300.82

Table 11: Total correlation energies in mEh for the methane molecule (2.054 a0 ). Basis

MP2

CC2

MP2-F12

CC2-F12


−167.73 −200.79 −211.10 −214.90 −216.61 −218.97

−168.55 −201.87 −212.30 −216.13 −217.87 −220.26

−212.58 −217.91 −218.71 −218.91 −218.99

−213.59 −219.17 −220.00

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

−215.44 −216.53 −216.93 −218.19



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 34 of 39

VIII Appendix A Switching from amplitude formalism to real-space formalism: Example Here we use the S3c term (see Tab. 3) to give an explicit example of how to switch between amplitude and real-space form of the coupled-cluster equations. For this we use the Eqs. (9), (13), and (16). Note that we still use the sum convention for repeated indices. ia | aiS3c

= | ai

ak 2gic

= | ai

ak 2gic

−

ak gci

tck

−

ak gci

hc|τk i

= | ai (2h ak| g12 |ici − h ak| g12 |ci i) hc|τk i = | ai (2h ak| g12 |iτk i − h ak| g12 |τk i i) = Q (2hk| g12 |τk i|i i − hk| g12 |i i|τk i) = Q 2gτkk |i i − gik |τk i i = Q|S3c i.

(52)

The same result can be derived directly from the diagram which represents the S3c term using the rules given in Tab. 5. We start from the diagram S3c and label the open hole line with i, and the open particle line with 1. The internal hole and particle lines are labeled with k and 2 i

(k)

1

(2)


(53)

Page 35 of 39

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


We put the labels on the internal lines in brackets to indicate that they will be summed respectively integrated over. The horizontal bar is now interpreted as hr2 |τk i ≡ τk (r2 ). The wavy line which represents the coulomb term gives the term hkr1 | g12 |r2 i i. In the end we have to sum over the internal index k and integrate over the internal coordinate r2 i hr1 |S3c i=∑ k

Z

d r2 hkr1 | g12 |r2 i ihr2 |τk i

= ∑hkr1 | g12 |τk i i, k

= ∑hr1 | gkτk |i i,

(54)

Z

(55)

k

where we have used 1=

d r2 |r2 ihr2 |.

In the paper we drop the formal projection onto hr1 | and just write i |S3c i = gkτk |i i.

(56)

The factor of 2 comes from the closed loop and the second part from the evaluation of the corresponding exchange diagram (see 36 for more information), but those rules stay the same in the real-space formalism. The exchange diagram for S3c can be obtained by reconnecting the hole lines (or particle lines) to the operator i

(2)

(k)

1

≡


.

(57)


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 36 of 39

If we apply the rules of Tab. 5 again we obtain

−∑ k

Z

d r2 φk (r2 ) g12 τk (r1 ) φi (r2 ) = − gik (r1 ) τk (r1 ) = − gik |τk i,

(58)

where the minus sign has again to be derived from the structure of the diagram (see again 36 ).

B

Properties of the Q and Qt projectors

t has no effect on its commutator with the Fock operator. It holds that Q12

t t t . = Fˆ 12 , Q12 Fˆ 12 , Q12 Q12

(59)

From the definitions in Tab. 1 and the orthogonality between the |k i and |τk i states we get the identities

Qt Qt = Qt , O τ O τ = 0, O τ Fˆ O τ = 0, ˆ Qt = F, ˆ Oτ F,

(60) (61) (62) (63)

and also the identity ˆ Qt = Qt F, ˆ Oτ Qt F, ˆ O τ + O τ F, ˆ Oτ = F, ˆ Oτ = F, ˆ Qt . = F,


(64)

Page 37 of 39

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


With the given definitions and identities we get t t t ˆ Qt + F, ˆ Qt ⊗ Qt Qt ⊗ F, = Q12 Fˆ 12 , Q12 Q12 ˆ Qt + Qt F, ˆ Qt ⊗ Qt = Qt ⊗ Qt F, t = Fˆ 12 , Q12

C

(65)

Comparison to Tˆ 1 -transformation

In the amplitude form it is convenient to use so called Tˆ 1 -transformed integrals which for an LCAO basis can be defined as (see chapter 13.7.3 in 32 ) pq

tu . g˘rs = x pt yqu xrm ysn gmn

(66)

The transformation matrices are defined over the t1 matrix which holds the singles amplitudes x = (1 − t1 )

(67)

y = (1 + t1 ) † .

(68)

The real-space form of those transformations is given by the operators

Qτ = 1 − |τk ihk|,

(69)

Q˜ τ = 1 + |τk ihk|

(70)

With these operators we define the transformed coulomb operator τ τ g12 Q˜ 12 g˘ 12 = Q12


(71)


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 38 of 39

and see that for the second part of the real-space doubles (36) it holds that t Q12 g12 |ti t j i = Q12 g˘ 12 |iji.

(72)

So for the CC2 doubles the approach using the |ti i states and the Qt projectors is equivalent to the Tˆ 1 -transformation in real-space formalism. For the singles this is not the case since we did not transform the terms depending on the one electron operator. The realspace version of the Tˆ 1 -transformed coupled-cluster singles is |Ωi i =Q F˘ |ti i + 2hk|F˘ 2 |τik i2 − hk|F˘ 1 |τik i1 + Q 2hk|g˘ 12 |τik i2 − hk|g˘ 12 |τik i1 − 2hl |g˘ ik |τkl i2 − hl |g˘ ik |τkl i1

(73)

where we have used the transformed one-electron operators ˘ Q˜ τ , F˘ = Qτ Fˆ + 2˘J − K

(74)

˘J = gτk , k

(75)

˘ | f i = gk |τk i. K f

(76)

For this work we will use the singles potential given in Eq. (31) since it is more suitable for our solving scheme using the Helmholtz Green’s function.


Page 39 of 39


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 ACS Paragon Plus Environment

Coupled-Cluster in Real Space. 1. CC2 Ground State Energies Using

Recommend Documents