THE JOURNAL OF CHEMICAL PHYSICS 128, 174103 共2008兲
Explicitly correlated coupled cluster F12 theory with single
and double excitations
Jozef Noga,1,2,a兲 Stanislav Kedžuch,2 Ján Šimunek,1 and Seiichiro Ten-no3,b兲
1
Department of Physical and Theoretical Chemistry, Faculty of Natural Sciences, Comenius University,
Mlynská Dolina CH1, SK-84215 Bratislava, Slovakia
2
Institute of Inorganic Chemistry, Slovak Academy of Sciences, SK-84536 Bratislava, Slovakia
3
Graduate School of Information Science, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan
and CREST, Japan Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi,
Saitama 332-0012, Japan
共Received 25 January 2008; accepted 18 March 2008; published online 2 May 2008兲
Full explicitly correlated F12 coupled cluster theory with single and double excitations and with
Slater-type geminal as a correlation factor is introduced and implemented within the standard
approximation. The variant “C” that does not require integrals over the commutator between the
kinetic operator and the correlation factor has been used. All the necessary integrals are analytically
calculated. With variant C also, first results are reported for the correlation factor being the
interelectronic distance coordinate, i.e., for original R12 method. Calculations have been performed
for a set of eight molecules including CH2共 1A1兲, CH4, NH3, H2O, HF, CO, N2, and F2, as well as
for the constituting atoms. Atomization energies are reported too. © 2008 American Institute of
Physics. 关DOI: 10.1063/1.2907741兴
I. INTRODUCTION
It is well-known that the traditional configuration interaction type expansions converge very slowly to the basis set
limits that are often assessed by extrapolation schemes.1 The
most popular extrapolation schemes are based on the convergence of principal expansion in the hierarchy of the correlation consistent basis sets1–3 or on the convergence of the pair
natural orbital expansions.4 This convergence goes as
⬀共L + 1兲−3, with L being the highest angular momentum
function in the one-particle basis set.5 It is also well-known
that acceleration of the convergence by several orders of
magnitude can be accomplished by the use of explicitly correlated wave function based ab initio methods.6–8 Although
still not in standard use among quantum chemists, such approaches become more and more accepted tools for achieving highly accurate result, since the total cost of equally accurate extrapolated quantities is comparable or even higher.
A family of R12/F12 methods based on the original ansatz
suggested by Kutzelnigg9 more than 20 years ago certainly
belongs to such competitive approaches. Essentially, the traditional configuration state function space is extended by
functions created by the action of a correlation factor 共operator兲 F̂ = 兺 p⬎qF共r pq兲 on a reference function 兩⌽典,
兩⌿典 = F̂兩⌽典 + 兩⌿CI
type典.
共1兲
F̂ should ensure the correct description of the correlation
cusp close to the coalescence of two electrons.10 Since there
is a substantial overlap between the two parts of the wave
functions, the part describable through the conventional configuration interaction 共CI兲 expansion has to be outprojected
a兲
Electronic mail: [email protected].
Electronic mail: [email protected].
b兲
0021-9606/2008/128共17兲/174103/10/$23.00
from F̂兩⌽典. Moreover, F̂ can be associated with certain
variational parameters.
In the original R12 ansatz, F̂ is the operator of interelectronic coordinates r̂ = 兺 p⬎qr pq. This operator, however, has an
inconvenient asymptotic behavior at large separations. Even
though variational parameters and, essentially, a finite impact
range of the basis functions can treat this problem to a great
degree,11 the linear r12 is not the best choice for the correlation factor. Although the exact solution of the Schrödinger
equation of a n-electron system interacting via Coulomb
forces at r pq → 0 behaves as10
⌿共rជ1, . . . rជp, . . . rជq . . . rជn兲
= 共1 + 21 r pq兲⌿共rជ1, . . . rជp, . . . rជp . . . rជn兲 + O共r2pq兲,
共2兲
explicit inclusion of the linear r12 only treats the leading
term. Of course, we have to recall that the “conventional”
part of the wave function expansion via the CI-type expansion also contributes to the proper description of the Coulomb hole related to what we call the dynamical electron
correlation. Within the R12 ansatz, the r2pq and higher power
terms are only described in a one-particle basis. This implies
that for a correct description, still relatively high angular
momenta must be included. From this point of view, various
alternative correlation factors that would preserve the correct
description of the cusp but could eventually perform better
than R12 in a wider region can be considered.12–14 Indeed, in
the analysis by May et al.,15 it was concluded that the main
source of the error in R12/F12 theory is the choice of the
correlation factor, and a thorough study by Tew and
Klopper14 confirms that the Slater-type geminal 共STG兲 suggested by Ten-no13 is an ideal candidate for the correlation
factor. The same authors have shown the superior performance of STG in weak intermolecular interactions.16 Very
128, 174103-1
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174103-2
J. Chem. Phys. 128, 174103 共2008兲
Noga et al.
recently, Tew et al. reported the implementation of the STG
into an approximate explicitly correlated coupled cluster
共CC兲 variant CCSD共T兲共F12兲 and demonstrated its very good
performance for a series of reaction enthalpies including
23 molecules.17
Many of the STG implementations are based on the expansions of STG in terms of a linear combination of contracted Gaussian-type geminals for the ease of the integral
evaluation. Nevertheless, such correlation factors do not exactly describe the correlation cusp. Ten-no has efficiently
implemented the analytic integrals over STG,13,18 and the
analytic STG has been successfully applied within the second order MP theory at different approximated levels with
very promising results.8,18,19
In this work, we introduce the STG correlation factor
into the framework of the explicitly correlated CC theory
that was originally based on the R12 ansatz 共CC-R12兲.11,20
The quoted formulation of CC-R12 was explicitly related to
constrains known as “the standard approximation” 共SA兲.21
To apply the theory for chemically interesting problems, one
needs to go beyond this approximation, which is our ultimate
aim. Such formulation of the CCSD共T兲-F12 theory is outlined in Sec. II. In this initial implementation of CCSD共T兲F12 with STG factor, we still used the SA closely following
the C variant of the matrix elements evaluation.22 Numerical
results including energies for a set of eight small molecules
and their atomization energies are presented in Sec. III. Conclusions are depicted in Sec. IV.
II. THEORY
After the pilot calculations,23 the essence of the CC-R12
theory together with working equations including singles,
doubles, and approximate treatment of triples 关CCSD共T兲R12兴 using the standard approximation has been described in
several subsequent articles.11,20,24
In the following, indices i , j , . . . denote occupied spin
orbitals, a , b , . . . denote the virtual ones, and greek letters
␣ , ␤ , . . . will refer to 共virtual兲 spin orbitals within the orthogonal complement to the computational set of spin orbitals. Of course, this complement can only be used as an auxiliary notation in some intermediate steps, since the pertinent
spin orbitals belong to an unrealistic complete basis set. Referring to any spin orbital from the complete set is denoted
by ␬ , ␭ , . . .. We shall use tensor notation for the matrix elements and the Einstein summation convention as in the
aforementioned works. Second quantized hole-particle formalism will be employed throughout by using normal ordered operators expressed in terms of matrix elements and
normal ordered n-body replacement operators,
ã ␬␭1␬␭2.. .. ..␭␬n = 兵a␭† a␭† ¯ a␭† a␬n ¯ a␬2a␬1其.
1 2
n
1
2
n
共3兲
In this formalism, the normal ordered form of F̂ reads
F̂N = 41 F̄␬␮␭␯ã␮␬␭␯ + F̄␬␮iiã␮␬ ,
and we shall use
共4兲
ĤN = Ĥ1 + Ĥ2 = f ␬␮ã␮␬ + 41 g̃␬␮␭␯ã␮␬␭␯
共5兲
for the normal ordered Hamiltonian.
The CC-R12 ansatz follows the traditional CC method
with the exponential type of the wave operator ⍀̂ that transforms the independent particle model for a many electron
共reference兲 wave function 兩⌽典 into an 共more兲 exact wave
function 兩⌿典,
ˆ
兩⌿典 = ⍀̂兩⌽典 = eS兩⌽典.
共6兲
Unlike in traditional CC, Ŝ is not only represented by a global excitation operator
ij ab
T̂ = tiaã ai + 41 tab
ã ij +
1 ijk
abc
36 tabcã ijk ¯
共7兲
or its truncated form in an approximated variant but, in addition, Ŝ contains an operator R̂ related to the correlation
factor of Eq. 共1兲, i.e.,
Ŝ = T̂ + R̂.
共8兲
By using a final computational basis, R̂ comprises one- and
two-particle interactions.11 Thus, with an arbitrary correlation factor 共F̂兲, the operator R̂ is defined as
R̂ = R̂1 + R̂2 = cikR̃ki + 41 cijklR̃kl
ij ,
共9兲
␣
R̃ki = F̄kj
␣ jã i ,
共10兲
1 kl
kl ␣b
␣␤
R̃kl
ij = 2 F̄␣␤ã ij + F̄␣bã ij .
共11兲
From here, we shall use F␣k ⬅ F̄kj
␣ j. This choice of R̂ guarantees that all operators within Ŝ do commute. Further, if we
consider operators
Ĉ1 = cikã ki ,
共12兲
ij kl
Ĉ2 = 41 ckl
ã ij ,
共13兲
F̂1 = F̄␣k ã k␣ ,
共14兲
kl
␣␤ 1 kl ␣b
ã kl
+ 2 F̄␣bã kl ,
F̂2 = 41 F̄␣␤
共15兲
R̂1 and R̂2 can be understood as connected and doubly connected products of Ĉ1 with F̂1 and Ĉ2 with F̂2, respectively.
F̂1 and F̂2 originate from normal ordered form of F̂ after
eliminating terms that give zero contribution when acting on
兩⌽典 together with parts describable within the computational
basis set 共cf. Ref. 25兲. Ĉ1 and Ĉ2 ensure the orbital invariance
and relate the variational parameters 共c兲 to configuration
functions created by the action of F̂1 and F̂2 on 兩⌽典.11,20
R̂1 can be important if the primary computational basis
sets are far from the Hartree–Fock limit.25 Even more saturated basis sets up to 3Locc are required to satisfy the applicability requirement within the standard approximation.21
Locc is the highest angular momentum function involved in
the occupied orbitals. We have to recall that the argumenta-
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174103-3
J. Chem. Phys. 128, 174103 共2008兲
Explicitly correlated coupled cluster F12 theory
tion behind the SA holds exactly for atoms, for which fulfilling the aforementioned conditions leads to an exact method.
With basis set saturated up to Locc, the generalized Brillouin
condition 共GBC兲 holds, that is, the Fock matrix elements
f i␣ = 0. Since F̂ should be a totally symmetric operator, R̂1
also disappears under the latter condition. A stronger extended BC 共EBC兲 adopted in SA assumes even f a␣ = 0. Such
condition is only fulfilled if the basis set is saturated for each
angular momentum involved in this basis set. In CC-R12, SA
further implies that matrix elements whose truncation errors
go as 共L + 1兲−7 are disregarded. Three and four electron integrals are factorized via insertions of the resolution of identity
共RI兲, while in SA, RI is expressed in terms of the computational one-particle basis set. Alternatively, RI can be expressed in an auxiliary set 共ABS兲, enabling such calculations
with smaller main computational sets and even abandoning
the SA. Such approach has been used in the latest developments within MP2-F12 共R12兲 theory,26–31 the simplified variant of CCSD-R12 denoted as CCSD共R12兲,32 and even more
simplified CC2-R12 model for excitation energies.33 Nevertheless, all these approaches still at least implicitly assumed
the GBC via the wave function ansatz. We have addressed
this problem, especially in the context of MP2-R12 theory.25
In this pilot work that uses a genuine Slater-type geminal
correlation factor,
F共r12兲 = exp共− ␥r12兲
共16兲
within the CC framework, we employ the SA. R̂1 has not
been considered in the final implementation. However, a
more general theory outlined below has to consider this contribution. Parameter ␥ can be externally adjusted and kept
fixed, or taken as a variational parameter within the theory.
Such an approach would, however, require some additional
types of integrals and has not been done in the course of this
work.
The CCSD共T兲-F12 theory within the SA is fully compatible with CCSD共T兲-R12 published earlier,11,20 while the
implementation closely follows the 共semi兲direct atomic orbital integral driven and matrix operations based algorithm,34
which has been effectively parallelized within the DIRCCR1235
OS program. The latter also includes treatment of open shell
24
systems and a novel algorithm for perturbative contributions due to triples36 within the CCSD共T兲.37–39 The input
matrices involving the correlation factor are, of course,
changed. Modifications are more complex than only, replacing the integrals over r12 by integrals over the factor of Eq.
共16兲 in the final formulas for the matrix elements, and we
shall address these changes in the context of simplified recapitulation. First, however, we shall show the working equations by using ansatz of Eq. 共8兲 with full inclusion of R̂
defined by Eq. 共9兲 and then restrict our consideration to approximated variants.
With the wave function ansatz defined by Eqs. 共6兲 and
共8兲 and after multiplying from the left by exp共−Ŝ兲, one can
rewrite the Schrödinger equation as:
exp共− T̂兲H̄R exp共T̂兲兩⌽典 = E兩⌽典,
共17兲
where E is directly the correlation energy. H̄R is a similarity
transformed effective 关F共r12兲 dependent兴 Hamiltonian,
H̄R = exp共− R̂兲ĤN exp共R̂兲.
共18兲
Unlike the transcorrelated Hamiltonian of Boys and Handy,40
H̄R formally contains up to six particle terms. However, neither six- nor five-particle terms can contribute even in the
full CC-F12 with the ansatz of Eq. 共8兲. Four-particle terms
contribute in CC variants including double, triple, or quadruple excitations in T̂, and three-particle terms formally
contribute also in CCSD共T兲-F12.
Analogically, we shall use H̄T for
H̄T = exp共− T̂兲ĤN exp共T̂兲,
共19兲
which is an effective Hamiltonian formally corresponding to
the conventional CC theory. The energy and the system of
nonlinear equations to determine t and c amplitudes are obtained in a standard way by projecting Eq. 共17兲 onto the
reference and the pertinent configuration spaces created by
the action of Ŝ on 兩⌽典, respectively.
A. F12-transformed effective Hamiltonian
Since the involved second quantized operators can be
formally expressed in terms of matrix elements over the
complete basis, explicit formulas of these equations can be
effectively derived by using diagrammatic techniques.20,41,42
From R, only the interactions of F contain formal summations over the complete basis. F is a variation parameter free
operator and so is the Hamiltonian. Hence, it is convenient to
pre-evaluate the aforementioned interactions and then only
work in terms of the computational orbital basis. In H̄R, one
can identify the following one-, two-, three- and four-particle
intermediates of such type,
V ip = f ␣p F␣i ,
共20兲
ij
␣b ij
V ijpq = 21 ḡ␣␤
pq F̄␣␤ + ḡ pq F̄␣b ,
共21兲
Uirpq = ḡ␣pqrF␣i ,
共22兲
i j
Gijpq = ḡ␣␤
pq F␣F␤ ,
共23兲
ijr
= ḡ␤pqrF̄aij␤ ,
V apq
共24兲
ij
= f ␣p F̄aij␣ ,
Y ap
共25兲
ijk
ij k
= ḡ␤␣
Y apq
pq F̄a␤F␣ ,
共26兲
ijkl
ij kl
V apqb
= 21 ḡ␤␣
pq F̄a␤F̄␣b .
共27兲
Exact evaluation of these quantities would require calculation of many electron integrals over different operators in the
atomic orbital basis, which is certainly a nontrivial task. Obviously, partial wave expansion of Eqs. 共20兲 and 共22兲–共27兲
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174103-4
J. Chem. Phys. 128, 174103 共2008兲
Noga et al.
TABLE I. Explicit formulas for the matrix elements of one-, two- and threeparticle parts of H̄R
⬘ projected onto the space of the main computational
spin-orbital basis. See the text for definitions of the contributing matrices.
Element
Explicit form
1 kl in
k i
ki m
kl m i
2 V pnckl − V pck − U mpck + Gmpck cl
1 kli
a mn 1 kl a in
+ 4 V amn␦ pckl − 2 Y an␦ pckl
1
a in o 1 on i
− 2 Y klm
ano␦ p ckl cm − 2 ckl cm
kc m 1 klc
mn
−U mp
ck + 2 V amn␦ apckl
1 kl ij
km n 1 kl m n ¯ ij
2 V pqckl + U pq ck + 2 G pqck cl ␦ mn
1 kl ij ¯ ar 1
kll1 n mo ¯ ar ¯ ij
kln
+ 2 Y ar
ckl␦ pq + 2 共V aor
− Y aor
cl 兲ckl ␦ pq␦ mn
1
1
1 klm n ij ¯ ar
kk1l1l mk2 l2n ¯ ab ¯ ij
+ 16 V ak l bckk cl l ␦ pq␦ mn − 2 Y arn
cmckl␦ pq
22
1 1
1 klc io ¯ ar
kc i
V
c
␦
−
U
c
pq k
2 aor kl pq
1
m1n1o
ijk ¯ atu
− Y matu1n1lcol 兲cmmnn ␦¯ mno
␦ pqr
8 共V atu
1 1
1
m1n1m2n2 lo
mn ¯ ijk ¯ atc
+ 16 V atlc
cm n cm n ␦ mno␦ pqr
2 2
1 1
1 mnc ij ¯ atu
4 V atu cmn␦ pqr
共H̄R
⬘ 兲ip
共
共H̄R
⬘ 兲cp
共H̄R
⬘ 兲ijpq
兲
共
共H̄R
⬘ 兲icpq
共H̄R
⬘ 兲ijk
pqr
共H̄R
⬘ 兲ijc
pqr
兲
breaks off after a finite number of terms so that these can be
evaluated by using auxiliary basis sets for ␣ and ␤, more
precisely an auxiliary set that is orthogonalized to the computational spin-orbital basis—denoted by Valeev as CABS
共complementary ABS兲.28 When explicitly used in the formulas, we shall use double primed indices to denote the CABS
共p⬙ , q⬙ , . . . 兲, whereas p⬘ , q⬘ , . . . are reserved for the union of
the spin-orbital basis and CABS 共p⬘ ⬅ p 艛 p⬙兲. Using the
identity
兺␣ = 兺␬ − 兺p ,
lap = ␦ ap − ␦ iptia .
共36兲
Within the SA, only two terms survive from Table I, namely,
the first term of 共H̄R
⬘ 兲ijpq and the first term of 共H̄R
⬘ 兲ip.
In fact, R1 recovers the deficiency of the Hartree–Fock
solution rather than the genuine short range correlation. The
Hartree–Fock relaxation can be alternatively treated by extending the T̂1 operator in Eq. 共7兲 by excitations to the
complementary space, in practice, replaced by the CABS
space. However, without further approximations, such an approach would make the CCSD-F12 solution much more expensive because two-electron repulsion integrals involving
two CABS functions would be involved in each iteration.
Using CABS in evaluation of matrix elements involving R1
is similar to using a T̂1 extended by externally contracted
single excitations over the CABS. Approximately, one can
just correct the total energy by a single step and cheap second order contribution from the single excitations over the
CABS.43
B. Energy and the T̂ amplitudes
At this moment, let us consider H̄R
⬘ as resolved. Projecting Eq. 共17兲 onto the conventional configuration space defines the energy and the working equations to determine the
conventional amplitudes, which in CCSD-F12 read
共37兲
共28兲
one can relatively easily evaluate Eq. 共21兲 共and similar
products22,26兲, giving rise to CABS and SA expressions, respectively,
共38兲
cabs
1 rs kl
kl
mq⬙
V kl
pq = 共F12/r12兲 pq − ḡ pq F̄mqv − 2 ḡ pqF̄rs ,
kl
共29兲
1 rs kl
kl
V kl
pq = 共F12/r12兲 pq − 2 ḡ pqF̄rs .
共30兲
s.a.
By having matrix elements of Eqs. 共20兲–共27兲, H̄R projected onto the space of computational spin-orbital basis can
be expressed in the standard way. For our purpose, it was
convenient to shift H̄R as
H̄R
⬘ = H̄R − ĤN .
共31兲
Explicit forms of its matrix elements up to the three-body
terms are given in Table I. To facilitate the formulas in this
table and in the following, we introduce
¯␦ pq = ␦ p␦ q − ␦ q␦ p ,
共32兲
pqr
␦ stu
= ␦ sp␦ qt ␦ ru + ␦ sq␦ rt ␦ up + ␦ sr␦ tp␦ qu ,
共33兲
¯␦ pqr = ¯␦ pq␦ r + ¯␦ qr␦ p + ¯␦ rp␦ q ,
共34兲
kip = ␦ ip + ␦ aptia ,
共35兲
rs
stu
r
s
st
r
u
s
st
u
st
u
共39兲
Formally, the first terms on the right hand side are always
fully compatible with the conventional CC theory. Underbraced terms contribute within SA and all the other terms
remain if CABS approach is adopted.
The amplitudes of R̂ 关Eq. 共9兲兴 need not necessarily be
varied in the F12 case. These can be explicitly fixed, e.g., at
the theoretical values to assure the cusp conditions for singlet
and triplet pairs, respectively, as it has been done for
MP2-F12.18,44 In that case only the system of Eqs. 共38兲 and
共39兲 have to be solved 共in CCSD-F12兲, and the matrix elements of H̄R
⬘ given by Table I are obtained in a one step
calculation, while “c” matrices are diagonal with fixed values. Results with such an approach will be published
elsewhere.45 A similar ansatz has been employed in approximate explicitly correlated CCSD共T兲 methods.43,46
Alternatively, simple models to correct for the electronic
cusp perturbatively in combination with the CC ansatz can
be considered and have been proposed very recently.47
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174103-5
J. Chem. Phys. 128, 174103 共2008兲
Explicitly correlated coupled cluster F12 theory
C. Correction due to triple excitations
Being by orders of magnitude cheaper than full CC with
singles, doubles, and triples,48,49 the quasiperturbative a posteriori corrections for triple excitations became very popular
due to their high reliability and accuracy proved in numerous
applications. Namely, the well-known CCSD共T兲,37 which
evolved from the former
共40兲
CCSD关T兴 = CCSD + T共CCSD兲
38
approximation, has found the broadest use. Nevertheless,
these approximations naturally fail for stretched geometries
when the systems become quasidegenerate. In that case, alternative corrections should be superior, e.g., the asymmetric
formula using the components of the left eigenvector of H̄T
as pioneered by Stanton and Crawford,50 corrections based
on the perturbation expansion of H̄T,51 or those derived from
the connected moments known as renormalized
corrections.52–54 Here, we shortly comment on CCSD共T兲F12. For a general non-Hartree–Fock reference, the correction due to triples reads55
ET =
具⌽兩T̂†3Ĥ1T̂3兩⌽典
+
具⌽兩T̂†1Ĥ2T̂3兩⌽典
+
具⌽兩T†2Ĥ1T̂3兩⌽典.
共41兲
While T̂1 and T̂2 are taken from the CCSD solution, amplitudes of approximate T̂3 are determined from
ijk
共Ĥ1T̂3 + Ĥ2T̂2兲兩⌽典 = 0.
具⌽兩ã abc
Element
Explicit form
Fk␣Fi␣
␹ ki
Bki
Fk␣ f ␤␣F̄i␤
Fk␣ḡ␤␣qpFi␤
Fk␣ḡ␣␤␥l Fi␤F␥j
ip
Zkq
ij
Dkl
ij
Qkl
ijk
Dlmn
ijp
Ekqr
ib
X kl
ijb
X kla
ij
W kl
ijp
J klm
ijk
I lmn
i1 j 1i2 j 2
Zkmnl
ij
Fk␣共f l␥F̄␣␥
+ f cl F̄␣ijc兲
␤␥ ij
b␥ ij
Fl␣共ḡmn
F̄␣␤ + ḡmn
F̄␣b兲F␥k
␤ p ij
bp ij
Fk␣共ḡqr
F̄␣␤ + ḡqr
F̄␣b兲
␣b i
F̄kl
F␣
␣b ij
F̄kl
F̄␣a
␣␤ i j
F̄kl
F ␣F ␤
␣␥ ␤ p i j
a␥ ␤ p i j
F̄kl
ḡ␣mF␤F␥ + F̄kl
ḡamF␤F␥
␣␤ ␥␦ i j k
␣b ␥␦ i j k
F̄mn
ḡl␤ F␥F␦F␣ + F̄mn
ḡlb F␥F␦F␣
a␥ ␤␦ i1 j 1 i2 j 2
␣␥ ␤␦ i1 j1 i2 j2
F̄kl
ḡmnF̄␣␤ F̄␦␥ + F̄kl
ḡmnF̄a␤ F̄␦␥
␣c ␤␦ i1 j 1 i2 j 2
a␥ ␤d i1 j 1 i2 j 2
+F̄kl
ḡmnF̄a␤ F̄d␥ + F̄kl
ḡmnF̄␣␤ F̄␦c
␣c b␦ i1 j 1 i2 j 2
+F̄kl
ḡmnF̄␣b F̄␦c
aip
Kklm
aij
Mklm
aj 1i2 j2
Mkmnl
ijab
Kkmnl
i1 j 1i2 j 2
J kmnl
a␣ ␤ p i
F̄kl
ḡ␣mF␤
a␣ ␤␥ i j
F̄kl
ḡ␣mF␤F␥
a␥ ␤d j 1 i2 j 2
a␥ ␤␦ j 1 i2 j 2
F̄kl
ḡmnF␤ F̄␦␥ + F̄kl
ḡmnF␤ F̄d␥
␣b ␤a ij
␣b ca ij
F̄kl
ḡmnF̄␣␤ + F̄kl
ḡmnF̄␣c
␣␥ ␤␦ i1 j1 i2 j2
␣c ␤␦ i1 j 1 i2 j 2
F̄kl
ḡmnF␣ F␤ F̄␦␥ + F̄kl
ḡmnF␣ F␤ F̄␦c
␣␥ ␤d i1 j1 i2 j2
+F̄kl
ḡmnF␣ F␤ F̄d␥
共42兲
The argumentation behind says that T2 is of the first order in
the sense of generalized perturbation theory,39,56 whereas the
other product terms that would contribute to T̂3 are of higher
order. In fact, within the SA, there is no nonzero threeparticle part of ĤR
⬘ projected onto the main basis and there is
no direct contribution from the first order R̂ to T̂3.20 Nevertheless, the correction itself is influenced via coupling terms
in Eqs. 共38兲 and 共39兲. The situation is different from the
CABS approach, when the three-particle part of H̄R
⬘ that contains the elements of Eq. 共24兲 is of second order 共first order
in the amplitudes of R̂兲. Consequently, in CCSD共T兲-F12, Eq.
共42兲 should be modified to
ijk
共Ĥ1T̂3 + Ĥ2T̂2 + Ĥ2R̂2兲兩⌽典 = 0.
具⌽兩ãabc
共43兲
If the Fock matrix is diagonal at least separately in the
occupied-occupied and virtual-virtual blocks, which can be
generally accomplished by using semicanonical orbitals,57
explicit form of Eq. 共43兲 reads
ijk
共f ii + f jj + f kk − f aa − f bb − f cc兲
tabc
共
TABLE II. Amplitudeless intermediates that can be identified in
具⌽兩共R†兲ki H̄R
⬘ 兩⌽典 and 具⌽兩共R†兲ijklH̄R
⬘ 兩⌽典 and that can be directly evaluated by
using CABS instead of the complete complementary basis.
兲
1 m1n1o mn
gm no
ol
ijk
def
t fg − ḡ mn
= ḡde
ld tef + 2 V def cm n ␦ mno␦ abc .
1 1
共44兲
D. R amplitudes
To simplify the notation for matrix elements in this section, we shall use a shorthand form for elements of Hermitian conjugate matrices as
+ ..
† rs. . .
Ors.
pq. . . ⬅ 共Ō 兲 pq. . . .
共45兲
As compared to Eqs. 共37兲–共39兲, many more amplitudeless
intermediates can be identified if the Eq. 共17兲 is projected
onto 具⌽兩共R†兲ki and 具⌽兩共R†兲kl
ij . However, most of these intermediate matrix elements have partial wave expansions that
break off after a finite l value and can be simply computed
by replacing ␣ , ␤ , . . . by CABS. These particular elements
are collected in Table II. In addition, in this table, we have
also included the four-particle Z whose partial wave expansion does not break off after a finite number of terms, but the
partial wave increments decrease very rapidly, at least as 共l
−12
+ 21 兲 共see Appendix G of Ref. 20兲. In Table III, we list the
remaining amplitudeless intermediates together with their
explicit forms within the CABS approach. These can be
evaluated by using derivations similar to those in Ref. 22,
whose further background can be found in Refs. 21 and 26.
To obtain the formulas for SA, one simply has to disregard summations over CABS and replace p⬘ by p in the latter
table. In fact, only X, B, two-particle P, and Z survive
within the SA. Explicit computational formulas for all of
these elements for the genuine R12 approach within SA approximation “B” have been given in the original work on
CC-R12.20 Unlike in former works, in Ref. 22, we adopted a
derivation when the SA restrictions were applied a posteriori
after an exact evaluation, while the exact one-particle RI has
been replaced by a full projector onto the computational oneparticle basis. This has been denoted as approximation C.
The main advantage follows from the fact that no integrals
over the commutator of the kinetic operator and the inter-
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174103-6
J. Chem. Phys. 128, 174103 共2008兲
Noga et al.
TABLE III. Amplitudeless intermediates that can be identified in 具⌽兩共R†兲ki H̄R
⬘ 兩⌽典 and 具⌽兩共R†兲ijklH̄R
⬘ 兩⌽典 and are
not included in Table II. For definition of ⌳ see the text. K and h are the exchange and the Hartree parts of the
Fock matrix, respectively.
Element
ij
Lkp
ij
X kl
ij
Bkl
ij
P kl
Exact form
Within CABS approach
ij
␤c ij 兲
Fk␣共 2 ḡ␤␥
␣ p F̄␤␥ + ḡ␣ pF̄␤c
1
1 ␣␤ ij
␣b ij
2 F̄kl F̄␣␤ + F̄kl F̄␣b
␣␤ ␥ ij
␣b ␥ ij
␣b c ij
F̄kl f ␣F̄␥␤ + F̄kl
f ␣F̄␥b + F̄kl
f bF̄␣c
␣␤ c ij
␣␤ ␥ ij
+F̄kl f ␣F̄c␤ + F̄kl f ␣F̄␥␤
1 ␣␤ ␥␦ ij
1 ␣b ␥␦ ij
4 F̄kl ḡ␣␤F̄␥␦ + 2 F̄kl ḡ␣bF̄␥␦
␣␤ c␦ ij
␣b c␦ ij
+ 2 F̄kl
ḡ␣␤F̄c␦ + F̄kl
ḡ␣bF̄c␦
1
P
ijk
lnm
␣␤共
c␦ ij 兲 k
␥␦ ij
F̄mn
2 ḡl␤ F̄␥␦ + ḡl␤ F̄c␦ F␣
1
共
兲
ij
kl
c␦ ij
␣b 1 ␥␦ ij
k
+F̄mn
2 ḡlb F̄␥␦ + ḡlb F̄c␦ F␣
␣␤ ␥ ij
c ij
k
␣b ␥ ij k
+F̄mn共f l F̄␥␤ + f l F̄c␤兲F␣ + F̄mn
f l F̄␥bF␣
a␤ 1 ␥␦ ij
c␦ ij
␥ ij
F̄mn 2 ḡl␤ F̄␥␦ + ḡl␤ F̄c␦ + f l F̄␥␤ + f lcF̄cij␤
1 ␣␤ ␥␦ i j
␣b ␥␦ i j
2 F̄kl ḡ␣␤F␥F␦ + F̄kl ḡ␣bF␥F␦
ijp
Zklm
a␥ ␤ p ij
a␥ bp ij
␣␥ ␤ p ij
ḡamF̄b␥
F̄kl
ḡ␣mF̄␤␥ + F̄kl
ḡamF̄␤␥ + F̄kl
ija
P lnm
I
共
兲
␣␥ bp ij
␣c ␤ p ij
+F̄kl
g␣mF̄b␥ + F̄kl
ḡ␣mF̄␤c
ijk
Llmn
a␥ ␤␦ ij
a␥ b␦ ij
␣␥ ␤␦ ij
共F̄lm
ḡ␣nF̄␤␥ + F̄lm
ḡan F̄␤␥ + F̄lm
ḡanF̄b␥兲
␣␥ b␦ ij
␣c ␤␦ ij
共 + F̄lm
ḡ␣nF̄b␥ + F̄lm
ḡ␣nF̄␤c兲Fk␦
electronic distance appeared in the final expression. With respect to B though, the matrix elements of X and P remained
unchanged, whereas B and Z are different. In B,
ˆ =
⌳
=
1
2
关F12, 关− 21 ⵜ21 − 21 ⵜ22,F12兴兴
1
2
2 共ⵜ1F12兲
+
1
2
2 共ⵜ2F12兲 ,
共46兲
which for the two different correlation factors considered in
this work means
ˆ 共r 兲 = 1,
⌳
12
共47兲
ˆ 共exp共− ␥r 兲兲 = ␥2 exp共− 2␥r 兲.
⌳
12
12
共48兲
Consequently, with the STG correlation factor in CC-F12,
one needs integrals over exp共−␥r12兲, exp共−2␥r12兲,
exp共−␥r12兲 / r12, and exp共−2␥r12兲 / r12, whose implementation
has been previously reported13,18 and need not be repeated
here.
冉
冊
Fkp⬙V ijp p
⬙
1
ij
mq⬙ ij
共F212兲kl
− F̄kl
F̄mq − 2 F̄klpqF̄ijpq
⬙
1
ij
ij
ij
⌳kl
+ 2 关共F̄2兲 p lhkp⬘ + 共F̄2兲kp
h p⬘ + H.c.兴 − F̄klp⬙q⬙Krp⬙ F̄rij q
⬘
⬘ l
⬙ ⬙⬙
ij
aq⬙ b ij
+ H.c.兲
−F̄klp⬙bKrp⬙ F̄rij b − F̄kl
KaF̄bq − 共F̄klp⬙q⬙Kap F̄aq
⬙ ⬙
⬙
⬙ ⬙
ij
ij
ij
−共F̄klp⬙qhrp F̄rq
+ H.c.兲 − F̄klpqhrpF̄rq
+ H.c.兲 − 共F̄klp⬙q⬙hnp F̄nq
⬙ ⬙
⬙
aq⬙ n ij
mq⬙ n ij
p⬙m r⬙ ij
−共F̄kl haF̄nq + H.c.兲 − F̄kl hmF̄nq − F̄kl h p F̄r m
⬙ ⬙
⬙
⬙
1
ij 1
kl
共F212 / r12兲kl
− 2 关共F12 / r12兲klpqF̄ijpq + H.c.兴 + F̄ijpqḡrs
pqF̄rs
4
n
ij
r
ij
−共F̄klp⬙mV p m + H.c.兲 − F̄klp⬙mḡ p⬙ mF̄r n
⬙
⬙ ⬙
p⬙q⬙ ij
p⬙b ij k
F̄mn
V lq Fkp + F̄mn
V lbF p
⬙ ⬙
⬙
ij
p⬙q⬙ r⬙ ij
+F̄mn
共f l F̄r q + f cl F̄cq
兲Fk + F̄ p⬙b f r⬙F̄ij Fk
⬙⬙
⬙ p⬙ mn l r⬙b p⬙
ij
aq⬙
F̄mn
共V lq
+ f r⬙F̄ij + f cF̄ij 兲
⬙ l r⬙q⬙ l cq⬙
⫹ p⬙q⬙ i j
V kl F p Fq
⬙ ⬙
1
p⬙q⬙ pr⬙ ij
p⬘ p
2 ij p⬘ p
2 ij
2 关共F̄ 兲 p⬘lgkm + 共F̄ 兲kp⬘glm + H.c.兴 − F̄kl g p⬙mF̄r⬙q⬙
ij
aq⬙ pb ij
−F̄klp⬙bg ppr⬙mF̄rij b − F̄kl
gamF̄bq − 共F̄klp⬙q⬙g ppamF̄aq
+ H.c.兲
⬙ ⬙
⬙
⬙
⬙
ij
p⬙q⬙ pn
qs rp ij
p⬙q pr
ij
−共F̄kl g p mF̄rq + H.c.兲 − 共F̄kl g p mFnq + H.c.兲 − F̄kl
gsmF̄qr
⬙
⬙
⬙
ijp⬙ k
Zlmn F p
⬙
To facilitate the equations wherever appropriate, we
shall introduce “tilded” notation for elements of Tables II and
III as soon as they are associated with the amplitudes, i.e.,
when we replace
ij
F̄␬ij␭ → 21 F̄␬mn␭ cmn
and
i
F␬i → − F␬mcm
.
共49兲
This means that, e.g.,
ij
X̃ ijkl = 21 X mn
kl cmn,
i1 j1k1 ij
ijk
P̃ lmn
= − 21 P lmn
ci j ckk , . . . ,
1 1
1
and so on 共but not for Hermitian conjugates兲. We shall also
use
i l
ma l i
共H̄N兲ik = 共H̄T兲ik + 共H̄R
⬘ 兲ik − Uma
kl cmta − Ulk cmta ,
m n¯ ij
共H̄N兲ijkl = 共H̄T兲ijkl + 共H̄R
⬘ 兲ijkl − Uoa
kl co ta␦mn .
For the given T̂, we can express the equations determining
the amplitudes of R̂,
+
+pc m
pm
ilm
ilc m
mnp i
ci m
+ Ẽklm
tc ,
0 = − Blkcil + V kp + Ukm
tc − Z̃km
+ 21 Ẽkmn
k p + D̃ilkl + Z̃km
tc − X̃ nk 共H̄N兲in + Q̃ilkl + L̃ilkl + D̃klm
共50兲
共51兲
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174103-7
J. Chem. Phys. 128, 174103 共2008兲
Explicitly correlated coupled cluster F12 theory
TABLE IV. Comparison of CCSD共T兲-F12 approaches using different correlation factors and/or approximations in the matrix elements evaluations. Energies
for atoms that are contained in the set of the investigated molecules. 19s14p8d6f4g3h basis set and its subsets are used 共see the text兲. Highest angular
momentum included in the subset is denoted in the pertinent columns. Total energies are in Eh, differences 共⌬兲 in mEh.
exp共−r12兲 / C
r12 / C
h
⌬共f − h兲
⌬共g − h兲
C
N
O
F
−37.844 334
−54.588 631
−75.066 678
−99.733 454
0.220
0.341
0.646
0.864
0.041
0.065
0.098
0.129
C
N
O
F
−37.789 314
−54.529 880
−75.004 582
−99.668 081
0.211
0.324
0.630
0.850
0.041
0.064
0.098
0.128
⌬共g − h兲
f
g
h
All electrons correlated
−37.844 331
0.240
−54.588 628
0.370
−75.066 673
0.561
−99.733 443
0.414
0.050
0.076
0.115
0.140
−0.041
−0.106
−0.663
−1.422
−0.004
−0.024
−0.157
−0.237
0.003
−0.009
−0.091
−0.107
Valence electrons correlated
−37.789 313
0.226
−54.529 879
0.347
−75.004 580
0.534
−99.668 074
0.391
0.049
0.075
0.114
0.139
−0.088
−0.148
−0.711
−1.467
−0.053
−0.069
−0.211
−0.290
−0.046
−0.055
−0.144
−0.160
h
Again, the part that contributes within SA is underbraced. Let us note that with the big basis sets used in SA,
the contributions from Z̃ and T̂1 couplings are very small and
can be neglected for most applications.24 This means that
5
instead of having an ⬀nocc
N2 scaling for constructing the Z,
6
one only needs ⬀noccN steps. For the purpose of the present
work, such approximation was appropriate, and we have
used it.
We note that explicit equations given in this section are
not optimized for the practical implementation in which the
many particle terms would be most probably factorized in
terms of the auxiliary basis. Also, one has to keep in mind
that in most of the terms, the summation over CABS can be
effectively restricted to summations over the Hartree–Fock
limit complement to the computational basis. Finally, the dimensions of matrices and the complexity are drastically reduced in the diagonal approximation.
III. RESULTS AND DISCUSSION
These pilot CCSD共T兲-F12 calculations have been carried
out for a test set of eight molecules including CH2共 1A1兲,
CH4, NH3, H2O, HF, CO, N2, and F2 as well as for the
constituting atoms. The molecular calculations were carried
out by using the geometries optimized at the CCSD共T兲/ccpCVQZ level.58
We employed a hierarchy of R12 suited basis sets
that for a nonhydrogen atom include subsets of
19s14p8d6f4g3h,59 from which the whole subsets of the
highest angular momentum functions are skipped step by
step down to 19s14p8d6f. Similarly, for hydrogen, we have
used a series that originates from 9s6p4d3f going down to
9s6p.60
These fairly extended sets fulfill the SA requirements to
a great degree. For hydrogen, the 9s set is identical to an
uncontracted s set from aug-cc-pV5Z,61 while for nonhydrogen atoms, their sp sets are derived from the Hartree–Fock
limit sets of Partridge.62
An extensive study on the optimal STG exponent at
MP2-F12 and CCSD共F12兲 levels of theory has been given in
Refs. 14 and 17 for the basis set hierarchy of aug-cc-pVXZ.
⌬共f − h兲
r12 / ⌬共C − B兲
As pointed out by the authors, the optimum ␥ is related to the
size of the average Coulomb hole and depends on the effective nuclear charge experienced by the valence electrons.14 It
is also mentioned that the optimal ␥ is more dependent on
the chosen basis set than on the method. Nevertheless, their
results also indicate that the ␥ dependence is much less pronounced if they used ansatz 1, which is not the method of
choice with smaller basis sets. On the other hand, this ansatz
is essentially compatible with the use of standard approximation and becomes equivalent with our approach in SA and for
basis sets that fulfill the SA requirements. Indeed, with our
basis sets and applied SA, the energy versus ␥ dependence
was very flat; hence, in the next subsections, we report results that correspond to a fixed value of ␥ = 1.0, which is a
compromise exponent resulting from MP2-F12. The inaccuracies introduced due to this setting are fairly small, and the
atomization energies 共Sec. III B兲 are only affected at the
level of a tiny fraction of a kJ/mol.
Although the differences in energy with the changes in ␥
are very small along a fairly wide range, our incomplete
preliminary results indicate that CCSD-F12 optimal exponents can be sometimes quite different from those of MP2F12, at least if SA and our basis set hierarchy were used.
With CCSD共F12兲,17,46 by using aug-cc-pVXZ hierarchy and
using auxiliary basis set for the RI, these differences were
much less pronounced. More investigation related to this issue is necessary.
A. Absolute energies of atoms and molecules
The SA requirements are more clearly defined and also
more easily fulfilled for atoms than for molecules. Hence, it
is appropriate to discuss the trends separately. In Table IV,
we compare 共i兲 the performance of the STG correlation factor versus r12 and 共ii兲 the formerly introduced approximation
B with our more recent suggestion denoted as C.
The overall picture is the same both for valence correlation and if the core orbitals are included. With the largest
basis sets, both correlation factors provided almost identical
results. Comparison of ⌬共f − h兲 columns suggests somewhat
faster convergence toward the basis set limit if the original
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174103-8
J. Chem. Phys. 128, 174103 共2008兲
Noga et al.
TABLE V. Comparison of CCSD共T兲-F12 molecular energies using different correlation factors and/or approximations in the matrix elements evaluations.
19s14p8d6f4g3h / 9s6p4d3f basis sets for nonhydrogen/hydrogen and their subsets are used 共see the text兲. Highest angular momenta included in the subset are
denoted in the pertinent columns. Total energies are in Eh and differences 共⌬兲 in mEh.
exp共−r12兲 / C
r12 / C
h/ f
⌬共f / d − h / f兲
⌬共g / d − h / f兲
CH2共 1A1兲
CH4
NH3
H 2O
HF
CO
N2
F2
−39.132 840
−40.514 091
−56.563 336
−76.437 890
−100.459 222
−113.324 680
−109.540 638
−199.528 015
0.365
0.408
0.562
0.684
0.844
0.589
0.489
1.276
0.105
0.131
0.147
0.155
0.156
0.128
0.212
0.268
CH2共 1A1兲
CH4
NH3
H 2O
HF
CO
N2
F2
−39.077 163
−40.457 042
−56.503 476
−76.375 146
−100.393 542
−113.205 956
−109.421 702
−199.397 394
0.361
0.404
0.552
0.672
0.828
0.628
0.527
1.285
0.106
0.131
0.147
0.155
0.154
0.134
0.160
0.261
h/ f
⌬共f / d − h / f兲
r12 / ⌬共C − B兲
⌬共g / d − h / f兲
f /d
g/d
h/ f
0.226
0.135
0.215
0.093
−0.065
−0.602
−0.656
−1.023
0.097
0.098
0.098
0.132
0.133
0.025
0.010
0.128
−0.491
−0.844
−1.054
−1.635
−2.139
−2.624
−2.360
−4.298
−0.181
−0.328
−0.396
−0.460
−0.407
−0.648
−0.639
−0.799
−0.057
−0.118
−0.154
−0.211
−0.175
−0.207
−0.174
−0.310
Valence electrons correlated
−39.077 168
0.211
−40.457 044
0.126
−56.503 475
0.190
−76.375 165
0.066
−100.393 545
−0.070
−113.206 030
−0.588
−109.421 737
−0.674
−199.397 462
−0.971
0.090
0.087
0.087
0.120
0.128
0.022
0.023
0.166
−0.534
−0.875
−1.090
−1.677
−2.161
−2.683
−2.447
−4.458
−0.230
−0.373
−0.437
−0.510
−0.456
−0.731
−0.709
−0.874
−0.103
−0.158
−0.190
−0.255
−0.222
−0.294
−0.269
−0.416
All electrons correlated
−39.132 831
−40.514 071
−56.563 321
−76.437 901
−100.459 222
−113.324 756
−109.540 656
−199.528 068
r12 correlation factor was used. On the other hand, results
with STG factor are clearly more regular, e.g., with respect
to increasing number of electrons. This comparison indicates
that r12 values with smaller basis sets might be 共and most
probably are兲 overestimated. Comparison between approximations B and C, on the other hand, suggests that the values
with approximation B were underestimated. Although, as expected, the two values converge to each other with the increasing basis set.
Table V collects the same data for molecules. Here, the
differences between the performance of the correlation factors are much more pronounced, although the values with the
largest basis sets are almost identical. With STG, the trends
are physical and the convergence toward the basis set limit is
fairly clear. This is not so with r12 correlation factor, when
with the smallest basis sets, some of the CCSD共T兲-R12/C
values were evidently overestimated or even below the most
accurate h / f ones 共CO, N2, F2兲. Having this in mind, the
seemingly better 共closer to the limit兲 g / d results with r12 / C
than with STG may still be little overestimated due to an
unphysical reason. On the contrary, the trends with STG
seem to be more reliable in this sense.
Certainly, it is much more difficult to fulfill the SA assumption for molecules than for atoms. This is one of the
factors that has a major impact in the differences between
approximations B and C. With SA and r12 factor, these differences are significant at the MP2-R12 level22 as well. On
the other hand, using the two approximations B and C in
MP2-F12 with an STG factor 共fitted through Gaussian geminals兲 and auxiliary basis set in matrix element evaluation
provided almost identical results.31 However, this result
might be preferably attributed to abandoning the SA, rather
than the use of STG.
FIG. 1. 共Color online兲 Errors of the CCSD共T兲-F12/C atomization energies for the two investigated correlation factors, exp共−r12兲 共solid lines兲 and r12 共dashed
lines兲. Values are with respect to the experimental ones. Basis sets as in Table V.
Downloaded 03 Aug 2009 to 133.6.32.19. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
174103-9
J. Chem. Phys. 128, 174103 共2008兲
Explicitly correlated coupled cluster F12 theory
B. Atomization energies
Calculation of atomization energies can be considered a
much harder test case for new methods than that of reaction
energies. It is not the purpose of the present work to address
all the aspects related to the accurate calculation of atomization energies. Related to R12 based explicitly correlated approaches, these aspects are mentioned in our former
works.63,64 Here we can restrict ourselves to a short comment
regarding Fig. 1. In this figure, the errors of the atomization
energies are depicted with respect to experimental values estimated in Ref. 65. Although, in this context, comparison
with experiment might not be the best choice, it is also difficult to define reliable CCSD共T兲 basis set limits.
This figure clearly confirms the superiority of STG with
spdf / spd basis that provides results very close to the limiting atomization energies obtained by using larger sets. For
r12 with the aforementioned set, some of the atomization
energies are impacted by the overestimation of 共mainly兲 the
molecular energies. With g / d and h / f basis sets, the r12 and
STG results are practically equivalent. This figure also confirms that even if the exponent of the STG is not optimized,
the accuracy of the atomization energy is much less influenced due to a certain compensation of errors on atoms and
the molecule.
IV. CONCLUSIONS
To introduce an R12 ansatz based explicitly correlated
coupled cluster theory for an arbitrary correlation factor, in
general, all the basic assumptions of the standard approximation have to be abandoned. We have illustrated the complexity of such an approach in the context of the recapitulation of
the CCSD-F12 theory that we have implemented for the first
time by using Slater-type geminal correlation factor. We have
derived the basic working equations, although further factorization of the matrix elements is possible and desired in further developments.
Current implementation still used the standard approximation. In matrix element evaluations, we have followed the
variant C.22 Two types of two-electron integrals are needed
beside the usual coulomb repulsion, namely, integrals over
exp共−␰r12兲 and exp共−␰r12兲 / r12, where ␰ is either ␥ or 2␥.
Our sample calculations including eight small molecules
and their atomization energies confirmed the superiority of
the STG factor over the r12 if smaller basis sets are used.
With the most extended sets in this study, the two factors
provide almost identical results. However, with smaller sets,
the CCSD-R12/C energies tend to be overestimated and STG
provides more stable and more reliable values.
ACKNOWLEDGMENTS
This work has been supported by the Grant Agency of
the Ministry of Education of the Slovak Republic and Slovak
Academy of Sciences 共VEGA Project No. 2/6182兲 as well as
by the Slovak Research and Development Agency 共APVV
No. 20-018405兲. Computations have been carried out by using computer facilities of COMCHEM, a virtual Center for
Advanced Computational Chemistry sponsored by SAS.
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