11 February 2000
Chemical Physics Letters 317 Ž2000. 535–544
www.elsevier.nlrlocatercplett
Size-extensive quadratic CI methods including quadruple
excitations: QCISDTQ and QCISDTQ ž6 / – On the importance of
four-electron correlation effects
Yuan He, Zhi He, Dieter Cremer
)
Department of Theoretical Chemistry, Goteborg
UniÕersity, Reutersgatan 2, S-41320 Goteborg,
Sweden
¨
¨
Received 23 November 1999; in final form 9 December 1999
Abstract
Size-extensive Quadratic CI with single ŽS., double ŽD., triple ŽT., and quadruple ŽQ. excitations ŽQCISDTQ. and its
non-iterative extension to a method correct at sixth-order perturbation theory, QCISDTQŽ6., are developed and applied to
electronic systems, for which full CI ŽFCI. results are available. It is shown that QCISDTQ results are more accurate than
either QCISDT or CCSDT. In particular, QCISDTQŽ6. correlation energies differ from FCI correlation energies on the
average by just 0.04 mhartree. The improvement is caused preferentially by the balanced addition of connected and
disconnected four-electron correlation effects, which confirms observations that four-electron correlation effects are
important for a balanced description of electron correlation. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction
Among the electron correlation effects that are
included into correlated ab initio methods to improve
the results of Hartree–Fock ŽHF. theory, two- Žpair.
and three-electron correlation effects are the most
important Žfor a recent review, see Ref. w1x.. It is
generally assumed that four-electron correlation effects are not important and, accordingly, can be
neglected. In recent work by Cremer and He w2,3x, it
was shown that this assumption seems to be correct
for atoms and molecules with well-separated Žlocalized. electron pairs Žclass-A systems: saturated hydrocarbons, boranes, derivatives of Li, Be, etc. w2x..
Using sixth-order many-body perturbation theory
)
Corresponding author. Fax: q46-31-7735590, e-mail:
[email protected]
with the Møller–Plesset ŽMP. perturbation operator
ŽMBPT-6 ' MP6. w1,2x, which considers correlation contributions from single ŽS., double ŽD., triple
ŽT., quadruple ŽQ., pentuple ŽP., and hextuple ŽH.
excitations, these authors could show that all connected Q contributions that describe the correlated
movement of four electrons at the same time are
negligible. However, in the case of electronic systems with strong clustering of electrons in a confined
region of atomic or molecular space Žclass-B systems: electronegative atoms, molecules with multiple
bonds, hypervalent or nonclassical bonding, etc. w2x.,
Q contributions became significant. As a simple
example, correlation contributions at MP6 calculated
with an cc-pVTZ basis set w4x for the Fy anion are
shown in Fig. 1.
While at fifth-order MBPT ŽMBPT-5 ' MP5.
the Q contributions DQ, TQ, and QQ are all disconnected describing the independent, but simultaneous
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 4 0 9 - 8
536
Y. He et al.r Chemical Physics Letters 317 (2000) 535–544
Fig. 1. MP6 spectrum for the Fy anion calculated with a pVTZ
basis set. The square root of the the sum of squared correlation
Ž6.
energy contributions EAB
C was used as scaling factor to establish
a basis for the comparison with other MP6 results. The MP6 terms
are given in the way they are calculated. Terms in bold ŽQQQŽII.,
TQQŽII., and TQT. give the correlation contributions arising from
connected Q excitations that correspond to four-electron correlation effects.
pair correlations of two electron pairs, at MP6 there
are three important contributions ŽQQQŽII., TQQŽII.,
TQT., which involve connected four-electron correlation effects ŽQQQŽI., TQQŽI., DQQ, DQQ, TTQ,
TQD cover the corresponding disconnected fourelectron contributions. w2,3x. Fig. 1, which gives the
relative magnitude of MP6 correlation contributions
and which is typical of the MP6 results for a class-B
system, reveals that the connected Q correlation
contributions are predominantly negative Žstabilizing. and amount to ; 10% of the total correlation
contribution at MP6, which if neglected would lead
to ; 0.3 kcalrmol error per electron pair in energy
comparisons involving both class-A and -B systems.
Clearly, with the number of electrons clustering in
a confined area of atomic or molecular space, fourelectron correlation effects are no longer negligible
and it is desirable to have some theory to estimate
their contributions. This is of particular importance
in configuration interaction ŽCI. theory where the Q
contributions represent the most important contribution to correct CI for its size-extensivity error Žfor a
recent review, see Ref. w5x..
MP6 theory is probably the easiest way to get a
description of four-electron correlation effects w2x.
However, MP results for class-B systems are known
to oscillate with the order of perturbation applied
w2,7x. At lower orders this is known to result from an
exaggeration of pair correlation effects but also
three-electron correlation effects can be exaggerated
so that one cannot exclude a similar exaggeration for
connected Q contributions. Alternatively, one could
carry out CI calculations with S, D, T, Q excitations
ŽCISDTQ. where the possibility exists to reduce the
calculational work by deleting many of the Ž A, T .
and Ž A, Q . off-diagonal elements of the CI matrix
Ž A s S, D, T, Q. as was recently demonstrated by
ˇ´
w6x. In this way rather accuSychrovsky´ and Carsky
rate energies are obtained, which do not suffer any
longer so much from the size-extensivity error of CI
theory w5x.
However, the best account of correlation effects is
provided by Coupled Cluster ŽCC. theory since CC
theory is size-extensive and includes, contrary to
MBPT and CI, infinite order effects w8x. CCSD,
CCSDŽT., and CCSDT are well-established in the
repertoire of correlation corrected ab initio methods
w8x. Even CCSDTQ w9–11x was developed and exploratory calculations for few small molecules were
reported. The price for the high accuracy of the
CCSDTQ approach is its enormous calculational cost
which scale with Niter M 10 Ž M: number of basis
functions; M 8 to calculate the Q matrix elements;
M 2 to calculate CC amplitudes a. where Niter gives
the number of CC iterations needed for getting selfconsistent CC amplitudes. Any time, Q excitations
are fully included in a self-consistent way, the M 10
working-load cannot be avoided and, therefore, one
has to focus on reducing the number of M 10 terms
or keeping the number of iterations to a minimum.
Pople and co-workers w12x developed quadratic CI
theory with S and D ŽQCISD. and a perturbative
inclusion of T excitations ŽQCISDŽT.. as a size-extensive improvement of CI theory and, accordingly,
as an approximation to CCSD and CCSDŽT.. Due to
some error cancellation and Žsome minor. reduction
of calculational cost w13x, QCISD and QCISDŽT. are
frequently used as substitutes for the more correct
CC methods. The non-perturbative inclusion of T
excitations into QCI theory as suggested by Pople
and co-workers turned out to be no longer size-extensive w14x and, therefore, Cremer and He w15x
developed a connected QCI theory in which the
Y. He et al.r Chemical Physics Letters 317 (2000) 535–544
original QCI concept is systematically extended to
size-extensive QCISDT, QCISDTQ, etc., no longer
considering this theory as an improvement of the
corresponding CI methods, but simply using QCI as
a simple approximation to the more correct CC
method.
QCISDT was implemented by He and co-workers
w16,17x and it was found that QCISDT energies
favorably approach the corresponding CCSDT values offering at the same time two important advantages: Ži. QCISDT is easier to program since it
contains less cluster terms; and Žii. calculation of the
QCI amplitudes, although a O Ž M 8 . procedure, is
cheaper than a CCSDT calculation because of the
smaller number of cluster terms and a somewhat
faster convergence of the cluster amplitudes.
Because of the positive experience with QCISDT,
we extend in this work QCI theory to the QCISDTQ
level where three goals should be accomplished: Ža.
a faster converging and easier to program alternative
to CCSDTQ should be established; Žb. available
MP6, QCISDT and CCSDT results should be checked
with a CC method containing Q excitations; and Žc.
the importance of connected four-electron correlation
effects established with the help of MP6 calculations
should be verified with a Q-containing method that
also includes infinite order effects.
In the following, we will present QCISDTQ by
first developing the basic theory ŽSection 2.. Then,
we will extend QCISDTQ to a new method that is
correct at sixth order perturbation theory ŽSection 3..
Finally, in Section 4, we will present results of
benchmark calculations that make it possible to test
the usefulness of QCISDTQ and to discuss the importance of four-electron correlation effects.
537
or, in general,
Tˆn s
1
Ž n! .
2
Ý a a b c . . . i jk . . . bˆ aq bˆ i bˆ qb bˆ j bˆcq bˆ k . . . . Ž 3.
ˆ creation and annihilation operators; inŽ bˆ q and b:
dices i, j, k, l denote occupied spin orbitals, indices
a, b, c, d virtual spin orbitals. and the products Tˆ2 Tˆn
are considered thus drastically simplifying the CC
projection equations w15x. This simplification is based
on the fact that pair correlation effects are the most
important effects and that higher-order correlation
effects introduced by cluster operator Tˆn have to be
considered with regard to their coupling to electron
pair correlation w1,15x. The projection equations for
the hierarchy of size-extensive QCI methods Žtruncated at level n. are given in Eqs. Ž4. and Ž7. in their
connected ŽC. form w15x:
D EQC I s ²F 0 < HTˆ2 <F 0 : ,
Ž 4.
²Fs < H Tˆ1 q Tˆ2 q Tˆ3 q Tˆ1Tˆ2 <F 0 :C s 0 ,
Ž 5.
ž
/
²F < H ž 1 q Tˆ q Tˆ q Tˆ q Tˆ q Tˆ / <F : s 0 ,
d
1
2
3
1
2
4
The QCI methods can be described as CC methods, for which only the cluster operators Tˆn :
Tˆ1 s Ý
Ý a ia bˆ aq bˆ i ,
min w pq2, n x
²Fp < H
ž
Ý
ispy1
/
Tˆi q Tˆ2 Tˆp <F 0 :C s 0
Ž n G p G 3. ,
Ž 7.
where H denotes the normal-order Hamiltonian,
H s Hˆ y E Ž HF . s H0 q Vs Ý bˆrq bˆ s ² r < Fˆ < s :
Tˆ2 s 14 Ý
ij
Ý a iajb bˆ aq bˆ i bˆ qb bˆ j ,
ab
q 14
½
Ý ½ bˆrq bˆsq bˆ t bˆu 5 ² rs < < ut : ,
5
Ž 8.
rstu
<F 0 : the HF reference wavefunction, and s, d, and p
are S, D, and general excitation indices. D EQC I
corresponds to the QCI correlation energy
Ž 1.
D EQC I s E Ž QCI . y E Ž HF . .
a
i
0 C
Ž 6.
rs
2. Quadratic CI theory with S, D, T, and Q
(QCISDTQ)
2
2
Ž 2.
Ž 9.
The corresponding QCISDTQ projection equations
are obtained by truncating the expansion for the
Y. He et al.r Chemical Physics Letters 317 (2000) 535–544
538
operator Tˆ at n s 4, i.e. to S, D, T, and Q excitations:
D EQC ISDTQ s ²F 0 < HTˆ2 <F 0 : ,
Ž 10 .
²F ia < H Ž Tˆ1 q Tˆ2 q Tˆ3 q Tˆ1Tˆ2 <F 0 :C s 0 ,
Ž 11 .
P
q Ý Ž y1 . P Ž irjk .
ž
1 q Tˆ1 q Tˆ2 q Tˆ3 q Tˆ4 q ˆ
1
2
T22
/
P
q Ý Y2 Ž n,i . a na bjkc y Ý Ž y1 . P Ž irjk .
n
<F 0 :C s 0 ,
Ž 12 .
Ý X4 Ž m,n, j,k . a iamb cn
mn
P
=
²F iaj b < H
1
2
P
1
2
Ý ² lm < < id : a lajkb cmd
lmd
P
y Ý Ž y1 . P Ž irjk < arbc .
P
²F iajkb c < H Tˆ2 q Tˆ3 q Tˆ4 q Tˆ2 Tˆ3 <F 0 :C s 0 ,
ž
/
Ž 14 .
/
Eqs. Ž10. and Ž14. have to be transformed into
two-electron integral Eqs. Ž15. and Ž19. to obtain
QCISDTQ in a form that can be programmed for a
computer.
D EQC ISDTQ s
1
4
Ý
² ij < < ab : a iajb
,
P
=
Ý ² ab < < ie : a ecjk ld
e
P
Ý Ý ² lm < < de : a ialdme ,
Ž 16 .
e i q e j y e a y e b a iajb
.
P
s Ý Ž y1 . P Ž arb .
P
P
=
Ý ² ma< < ij : a mb ckdl
m
P
q Ý Ž y1 . P Ž irjkl < arbcd .
P
² bl < < de : a iajld e
Ý
=
l, d-e
P
P
q Ý Ž y1 . P Ž abrcd .
l-m , d
q ² ab < < ij : q u iajb q Õiajb q 14
Ý X5 Ž m,a,i ,e . a mebjkc dl
me
Ý Ž ² lm < < dj : a ialbmd
q Ý Ž y1 . P Ž irj .
P
P
s Ý Ž y1 . P Ž irjkl < abrcd .
q Ý Ž y1 . P Ž ijrkl < arbcd .
l-m d-e
Ž
Ž e i q e j q e k q e l y e a y e b y e c y e d . a iajkb cl d
Ž 15 .
ij, ab
Ž e i y e a . a ia s u ia q Õia q
= Ý X5 Ž m,a,i ,e . a bjkcme ,
me
²F iajkb lc d < H Tˆ3 q Tˆ4 q Tˆ2 Tˆ4 <F 0 :C s 0 .
ž
Ž 13 .
P
Ý ² kl < < cd : a iajkb cl d ,
cd, kl
=
Ž 17 .
Ž e i q e j q e k y e a y e b y e c . a iajkb c
1
2
Ý X 3 Ž a,b,e, f . a iejkf cld
ef
P
q Ý Ž y1 . P Ž ijrkl .
P
P
s y Ý Ž y1 . P Ž irjk < arbc .
Ý X1Ž i ,d,b,c . a ajkd
P
d
=
1
2
Ý X4 Ž m,n,i , j . a ma bnckdl
mn
P
q Ý X 2 Ž j,k ,l,a . a iblc q Ý Ž y1 . P Ž arbc .
l
P
P
q Ý Ž y1 . P Ž ijrkl < arbcd .
P
=
1
2
Ý X 3 Ž b,c,e, f . a iajke f q Ý Y1Ž f ,a . a if jkb c
ef
f
=
Ý Z1Ž k ,l,b,c,d,e . a iaje
e
P
q Ý Ž y1 . P Ž arbc .
P
1
2
Ý
lde
² al < < de : a idjkb cl e
P
q Ý Ž y1 . P Ž irjkl < abrcd .
P
Ž 18 .
Y. He et al.r Chemical Physics Letters 317 (2000) 535–544
=
Ý Z2 Ž j,k ,l,c,d,n . a ianb
n
P
q Ý Ž y1 . P Ž arbcd .
P
=
Ý Y1Ž f ,a . a if jkb cl d
f
P
q Ý Ž y1 . P Ž irjkl .
P
=
Ý Y2 Ž n,i . a na bjkcld
.
Ž 19 .
f
The intermediate arrays u ia, Õia, u iajb ,Õiajb , X i , Yi and
Zi appearing in these equations are given in Appendix A.
3. QCISDTQ theory correct at sixth-order MBPT:
QCISDTQ(6)
The performance of a particular CC method can
be predicted by analyzing which MBPT contributions are covered at a given order w18,19x. The results
of such an analysis are given for QCISDT and
QCISDTQ as well as the corresponding CC methods
at fifth and sixth order in Fig. 2.
According to this analysis, both QCISDT
ŽCCSDT. and QCISDTQ are not correct at fifth
order. QCISDTQ corrects QCISDT at fifth order just
Ž5. .
by the QT term Ž EQT
, which actually represents a
disconnected Q contribution that is added in the first
iteration of the Q projection ŽEq. Ž14.. when all Tˆ4
amplitudes are still zero. According to Fig. 2, the
Fig. 2. Analysis of energy contributions at: Ža. fifth-order and Žb.
sixth-order many-body perturbation theory covered by QCISDT,
CCSDT, QCISDTQ, and QCISDTQŽ6. correlation energies. Yes
or y denote that the particular termŽs. shown at the left-hand side
of the diagram is Žare. fully contained in the correlation energy
while Žyes. or Žy. indicate that the termŽs. is Žare. only partially
covered.
539
Y. He et al.r Chemical Physics Letters 317 (2000) 535–544
540
relationship between QCISDT and QCISDTQ is
given by Eq. Ž20.:
Q
ž
q
Ž5.
Ž6.
Ž6.
D EQC ISDTQ s D EQCISDT q l5 EQT
q l6 EQTD
q EQTT
Ž6.
Ž6.
Ž6.
qEQQ
T q E TQT q E DQT
7
qOŽ l . .
Ž 20 .
One can predict that the improvements in going from
QCISDT to QCISDTQ are dominated by the QT
term, which is known to have normally relatively
large positive values Žsee also Fig. 3. w2x. Hence,
absolute QCISDTQ correlation energies could be
smaller than the corresponding QCISDT correlation
energies. Also, it is certainly not useful to compare
MP6 results for four-electron correlations with a
method that is not correct at both fifth and sixth
order. Therefore, we added to the QCISDTQ method
additional terms in a non-iterative manner that make
the method correct up to sixth order thus yielding
QCISDTQŽ6..
2
D E5 Ž 6 . s Ý ²F 0 < Tˆ3† q 12 Ž Tˆ2† . V
= Ž E0 y E q .
y1
/
C
<Fq :
²Fq < Ž V 12 Tˆ22 . C <F 0 :
Ž6.
Ž6.
s l6 ETQQ
Ž II . q EQQQ
Ž II . b q O Ž l7 . ,
Ž 26 .
D E6 Ž 6 . s ²F 0 < Tˆ3† VTˆ1Tˆ2
ž
/
C
Ž6.
<F 0 :s l6 E TTS
Ž II .
Ž6.
qETTQ
Ž I . b q O Ž l7 . ,
T
D E7 Ž 6 . s Ý ²F 0 <
t
ž
1
2
Ž Tˆ2† .
= Ž E 0 y Et .
y1
2
V
/
C
Ž 27 .
<F t :
²F t < Ž V 12 Tˆ22 . C <F 0 :
Ž6.
s l6 EQTQ
Ž II . b q O Ž l7 . ,
2
D E8 Ž 6 . s ²F 0 < 12 Ž Tˆ2† . V Tˆ2 Tˆ3 q
ž
Ž 28 .
1
3!
Tˆ23
/
<F 0 :
C
9
D EQC ISDTQŽ6 . s D EQCISDTQ q Ý D Eu Ž 6 . ,
Ž 21 .
u
q O Ž l7 . ,
with
Ž5.
Ž5.
D E1 Ž 6 . s ²F 0 < Tˆ2† VTˆ1Tˆ2 <F 0 :C s l5 Ž ETS
q E TQ
Ž I. .
Ž6.
Ž6.
Ž6.
Ž6.
qETSS
q ETSD
q ETPT
Ž I . q ETPQ
Ž I.
Ž 22 .
Ž5.
D E2 Ž 6 . s ²F 0 < Tˆ3† Ž V 12 Tˆ22 . C <F 0 :s l5 Ž E TQ
Ž II . .
Ž6.
Ž6.
Ž6.
q l6 EDTQ
Ž II . q ETQD
Ž II . q ETQQ
Ž I. b
Ž 23 .
2
Ž5.
D E3 Ž 6 . s ²F 0 < 12 Ž Tˆ2† . Ž V 12 Tˆ22 . C <F 0 :s l5 Ž EQQ
Ž II . .
Ž6.
Ž6.
Ž6.
q l6 E DQQ
Ž II . q EQQQ
Ž II . a q EQQD
Ž II .
Ž6.
qEQQQ
Ž I . a q O Ž l7 . ,
Ž 24 .
Ž6.
D E4 Ž 6 . s ²F 0 < Tˆ2† Tˆ1† Ž V 12 Tˆ22 . C <F 0 :s l6 ESTQ
Ž II .
Ž6.
qEQTQ
Ž II . a q O Ž l7 . ,
ž
/
C
Ž6.
<F 0 :s l6 EQPT
Ž I. b
q O Ž l7 . .
Ž6.
Ž6.
Ž6.
qETTQ
Ž I . a q ETQD
Ž I . q ETQQ
Ž I. a
Ž6.
qETTQ
Ž II . q O Ž l7 . ,
D E9 Ž 6 . s ²F 0 < Tˆ2† VTˆ1Tˆ3
Ž 29 .
Ž6.
Ž6.
Ž6.
qEQPQ
Ž I . b q EQTS
Ž I . b q EQTQ
Ž I. b
Ž6.
Ž6.
Ž6.
q l6 E DTS
q EDTQ
Ž I . q ETTS
Ž I.
Ž6.
qETST
q O Ž l7 . ,
Ž6.
Ž6.
Ž6.
s l6 EQH
Q Ž II . q EQPT Ž II . q EQPQ Ž II .
Ž 25 .
Ž 30 .
The computational cost for these additional energy
terms scale just with O Ž M 7 ., which is negligible in
view of the O Ž M 10 . dependence of QCISDTQ.
QCISDTQŽ6. should be considerably better than
QCISDTQ since it covers all terms up to MP6:
D EQCISDTQŽ6 . s D EQCISDTQ q l2 E Ž2. q l3 E Ž3.
q l4 E Ž4. q l5 E Ž5. q l6 E Ž6. q O Ž l7 . .
Ž 31 .
QCISDTQ was programmed within the ab initio
package COLOGNE 99 w20x by adding to an existing
QCISDT program w15x the Q equation. QCISDTQ
was debugged by checking in the first iteration the
QT term obtained from the Q equations with the
corresponding result from an MP5 calculation. In a
similar test, the TQT term obtained in the second
iteration was compared with the TQT term from an
Y. He et al.r Chemical Physics Letters 317 (2000) 535–544
Fig. 3. Average MP5 spectra for 10 class-A and 14 class-B
electronic systems w26x.
MP6 calculation w2x. Differences between the QCISDTQ terms and the corresponding MP energies were
smaller than 10y8 which was considered to be a
sufficient indication that QCISDTQ results are correct.
The QCISDTQ projection equations are solved in
an iterative procedure Žsteps Ž1. – Ž4..:
Ž1. initialize a ia, a iajb amplitudes while setting a iajkb c
s 0, a iajkb cl d s 0;
Ž2. calculate new a ia, a iajb , a iajkb c and a iajkb cl d amplitudes according to Eqs. Ž16. and Ž19.;
Ž3. check the convergence of a ia and a iajb amplitudes; if they are converged, go to step Ž4.,
otherwise back to step Ž2.;
Ž4. calculate the energy D EQC ISDTQ and the additional terms D Eu Ž6..
QCISDTQ converges in about the same number of
iterations as QCISDT does. According to theory, the
computational cost of QCISDTQ should be larger
than those of QCISDT by a factor of M 2 . However,
in reality the increase in computer time is just 30%
of the expected cost increase.
4. Results and discussion
In this Letter, we present QCISDTQ and QCISDTQŽ6. correlation energies for atoms and molecules,
for which full CI ŽFCI. energies are available Žw21x;
for Ne, see Ref. w22x; for NH 2 , see Ref. w23x; for BH,
CH 2 , see Ref. w24x; for H 2 O, FH, see Ref. w25x.. In
the previous study on QCISDT, basis sets and ge-
541
ometries used in the FCI calculations are described
in detail and, therefore, these details are not repeated
here w16x. In Table 1, results are compared with
QCISD, QCISDT, CCSD, CCSDŽT., CCSDT, and
FCI energies where for reasons of simplification
deviations from FCI correlation energies are listed.
Also, an error analysis in terms of mean and standard
deviation of calculated QCI and CC energies from
the corresponding FCI results is given in Table 1.
While this first comparison is limited to ground and
some excited states in their equilibrium geometry, in
Table 2, similar data are collected for electronic
systems with considerable multi-reference character
such as C 2 or BH, H 2 O, FH, and C 2 with symmetrically stretched AH bonds Ž‘‘stretched geometries’’..
Calculations of the latter represent critical tests on
the performance of QCISDTQ and QCISDTQŽ6.
since single determinant theories are known to fail
for systems with considerable multireference character.
All energies obtained at the QCISDTQ level of
theory are less negative than the corresponding
QCISDT energies obtained with the same basis set at
the same geometry. By this, the agreement with FCI
correlation energies is significantly improved relative
to that found for QCISDT w16,17x, which is documented by the calculated mean Žfrom 0.35 to 0.25.,
mean absolute Žfrom y0.31 to 0.01., and standard
deviation Žfrom 0.87 to 0.48 mhartree, see Table 1..
For QCISD and CCSD, the corresponding values for
the mean absolute deviation are ; 3 mhartree, for
CCSDŽT. and CCSDT 0.49 and 0.30 mhartree, respectively.
In Fig. 3, the MP5 correlation energy contributions for the same set of electronic systems are
shown in form of an energy spectrum w2x for both
class-A and -B electronic systems. Obviously, the
TQ energy term represents always a positive correction to other negative fifth-order Žor lower. order
energy contributions, which in the case of class-B
systems is actually the dominant term at MP5. Hence,
the inclusion of the TQ term at QCISDTQ represents
the most important correction relative to QCISDT
and is responsible for the better agreement with FCI
results. Similar observations can be made for the
stretched geometries although deviations in theses
cases are larger by a factor 3–10 because of the
strong multi-reference character of these systems.
Y. He et al.r Chemical Physics Letters 317 (2000) 535–544
542
Table 1
Comparison of QCISDTQ energies with FCI and other CC energies for atoms and molecules in their equilibrium geometry
Molecule
y
F
Ne
BH
CH 2 Ž3 B1 .
CH 2 Ž1A 1 .
NH 2 Ž2 B1 .
NH 2 Ž2A 1 .
H 2O
FH
N2
Basis
cc-pVDZ
aug-cc-pVDZ
cc-pVTZŽ-f.
4s2p1d
6s4p1d
cc-pVDZ
aug-cc-pVDZ
cc-pVTZŽ-f.
DZP
cc-pVDZ
aug-cc-pVDZ
DZP
DZP
DZP
DZP
DZ
DZP
cc-pVDZ
cc-pVDZŽq.
DZP
cc-pVDZ
aug-cc-pVDZ
cc-pVDZ
Mean abs. deviation
Mean deviation
Standard derivation
FCI
y99.558917
y99.669369
y99.675158
y128.702462
y128.767889
y128.679025
y128.709476
y128.777048
y25.227627
y25.215126
y25.218227
y39.046260
y39.027183
y55.742620
y55.688762
y76.157866
y76.256624
y76.241650
y76.258208
y100.250969
y100.228640
y100.264113
y109.276527
a
EŽapprox.. y EŽFCI.
CCSD
CCSDŽT.
CCSDT
MP6
QCISD
QCISDT
QCISDTQ
QCISDTQŽ6.
1.071
6.679
5.109
2.142
2.875
1.233
2.972
3.756
1.792
1.834
1.986
2.090
3.544
3.212
2.993
1.790
4.123
3.666
4.421
3.006
2.414
4.667
13.465
0.464
0.735
0.208
y0.001
0.114
0.189
0.181
0.066
0.412
0.479
0.474
0.359
0.873
0.548
0.534
0.574
0.718
0.634
0.684
0.397
0.493
0.528
1.708
0.446
0.397
0.232
y0.066
0.119
0.160
0.061
0.059
0.068
0.067
0.027
0.017
0.207
0.216
0.223
0.434
0.531
0.472
0.494
0.266
0.404
0.329
1.626
0.044
y8.662
y0.194
y0.227
y0.368
y0.027
y0.550
y0.176
1.305
1.328
1.263
0.363
1.977
0.336
1.263
0.087
0.077
0.016
y0.204
y0.229
y0.008
y0.972
y0.321
1.057
4.177
4.734
1.708
2.511
1.097
2.343
3.459
1.760
1.815
1.973
2.054
3.522
3.134
2.908
1.482
3.880
3.527
3.995
2.567
2.232
3.861
12.510
0.398
y2.719
y0.607
y0.549
y0.493
y0.008
y0.634
y0.523
y0.019
y0.020
y0.061
y0.107
y0.024
y0.046
y0.023
0.072
0.030
0.090
y0.274
y0.268
0.117
y0.727
y0.736
0.452
y1.675
0.002
y0.202
y0.183
0.139
y0.294
y0.155
0.014
0.025
y0.020
y0.037
0.113
0.121
0.139
0.297
0.348
0.335
0.108
0.079
0.291
y0.263
0.683
y0.002
0.330
0.007
0.052
0.000
0.020
0.060
0.002
0.021
y0.004
y0.059
y0.049
0.014
y0.025
y0.016
0.036
0.015
0.000
0.016
0.056
0.012
0.070
0.064
3.351
3.515
2.496
0.492
0.494
0.329
0.298
0.295
0.328
0.862
y0.169
2.189
2.993
3.144
2.243
0.349
y0.310
0.870
0.253
0.014
0.483
0.040
0.027
0.073
a
Absolute energies in hartree, deviations in mhartree. FCI energies are taken from Refs. w21–25x where details on basis sets and geometries
are published.
Contrary to CCSDT correlation energies, QCISDT
correlation energies are typically below FCI energies
since they are lacking the important QT term. Since
QCISDTQ energies contain the QT term, their value
is mostly above or in some cases slightly below the
FCI value. QCISDTQŽ6. covers in addition MP5 and
MP6 terms Žsee Eqs. Ž22. and Ž30.., which can be
either positive or negative, but in total correct the
QCISDTQ energies in such a way that the mean
absolute deviation from FCI values decreases by a
factor 6 to 0.04 mhartree and the standard deviation
by a factor of almost 7 to 0.07 mhartree. With a few
exceptions, QCISDTQŽ6. correlation energies approach FCI values from above. For basis sets without
diffuse functions the reproduction of FCI results is
clearly better than for basis sets including diffuse
functions. Cremer and co-workers showed recently
that the addition of diffuse functions to a basis set
not fully saturated with s, p functions leads to an
exaggeration of higher order correlation effects,
which diminishes the accuracy of a given MBPT or
CC method w7x.
It is interesting to note that QCISDTQŽ6. performs poorly in the cases of FH and H 2 O with bond
lengths of 2 R e ŽTable 2.. In these cases orbital
relaxation and state mixing are most important and
an inclusion of correction terms with frozen amplitudes seems to drastically overcorrect QCISDTQ
results.
Analysis of MP5 and MP6 contributions that are
added because of the Q projection equation at QCISDTQ reveals that the former are predominantly positive thus correcting the too negative QCISDT values
back to a level relatively high over FCI correlation
Y. He et al.r Chemical Physics Letters 317 (2000) 535–544
543
Table 2
Comparison of QCISDTQ energies with FCI and other CC energies for electronic systems with multi-reference character
Molecule Geom. Basis
FCI
EŽapprox.. y EŽFCI.
CCSD
C2
BH
BH
FH
FH
H 2O
H 2O
H 2O
H 2O
1.0
1.5
2.0
1.5
2.0
1.5
2.0
1.5
2.0
Re
Re
Re
Re
Re
Re
Re
Re
Re
CCSDŽT. CCSDT MP6
cc-pVDZ y75.721843 22.713 y4.853
DZP
y25.175976 2.642
0.550
DZP
y25.127350 5.048
0.406
DZP
y100.160395 5.099
0.887
DZP
y100.081107 10.178
0.255
DZ
y76.014521 5.590
1.465
DZ
y75.905246 9.365 y7.675
DZP
y76.071405 10.159
1.999
DZP
y75.952269 21.410 y4.630
Mean abs. deviation
Mean deviation
Standard derivation
a
10.245
2.524
10.245 y1.288
5.584
8.205
QCISD QCISDT
y3.497 y9.264 17.970
0.026
1.448 2.343
y0.091
2.203 3.856
0.645 y0.407 4.136
1.125 y1.132 8.392
1.473
1.910 4.740
y2.211
3.970 9.589
1.784
1.816 9.366
y2.472
4.057 21.386
1.480
y0.358
7.314
2.912
0.511
9.302
QCISDTQ QCISDTQŽ6.
y14.351 y10.146
y0.305 y0.226
y1.355 y1.165
y0.452
0.174
y0.962
0.370
y0.109
0.865
y7.991 y4.461
0.021
1.097
y6.153 y3.224
9.086
9.086
5.460
3.522
y3.517
11.198
y4.423
0.124
1.174
0.307
2.966
0.331
11.776
0.220
8.670
2.414
y1.857
9.599
3.332
2.349
8.506
a
Absolute energies in hartree, deviations in mhartree. FCI energies are taken from Refs. w21–25x where details on basis sets and geometries
are published.
energies. However, the connected Q contributions, in
particular the relatively large negative TQT contribution is responsible for a closer approach to the FCI
values. The additional improvement obtained at the
QCISDTQŽ6. level is a result of a complicated interplay of positive and negative contributions, in which
a relatively large DTS q DTQ contribution plays an
important role.
We conclude that QCISDTQŽ6. leads to correlation energies close to CCSDTQ and FCI values,
that the method is much better than either MP6,
QCISDŽT., CCSDŽT., QCISDT or CCSDT and that
it is the cheapest of all Q methods covering infinite
order correlation effects introduced by the Tˆ4 operator. Apart from this, QCISDTQ results can be used
as a reasonable guess for the more accurate CCSDTQ results. Finally, QCISDTQ calculations confirm the importance of connected four-electron correlation effects, which are needed to approach FCI
correlation energies.
authors thank the NSC for a generous allotment of
computer time.
Appendix A.
The intermediate arrays used in the QCISDTQ
program are defined in Eq. ŽA.1. to Eq. ŽA.13..
u ia s y Ý ² la < < id : a ld y 12 Ý ² la < < de : a idle
ld
y
1
2
lde
Ý
² lm < < id : a lamd
,
Ž A.1 .
lmd
u iajb s Ý Ž ² ab < < dj : a id y ² ab < < di : a dj .
d
q Ý Ž ² la < < ij : a lb y ² lb < < ij : a la .
l
q
1
2
Ý ² ab < < de : a idje q 12 Ý ² lm < < ij : a lamb
de
yÝ
Ý Ž y1.
ld
Acknowledgements
lm
p
P Ž ijrab . ² lb < < jd : a iadl ,
p
Ž A.2 .
This work was supported by the Swedish Natural
Science Research Council ŽNFR.. All calculations
were done on the CRAY C90 of the Nationellt
Superdatorcentrum ŽNSC., Linkoping,
Sweden. The
¨
Õia s 12 Ý
Ý ² lm < < de: Ž a ida lema q a laa iemd q 2 a lda iame . ,
lm de
Ž A.3 .
Y. He et al.r Chemical Physics Letters 317 (2000) 535–544
544
Õiajb s 14 Ý
a idjea lamb y 2 Ž a iajda lbme q a ibjea lamd
Ý ² lm < < de:
lm de
qa ialb a djme q a idlea ajmb . q 4 Ž a ialda bjme q a iblea ajmd . ,
Ž A.4 .
1
2
X 1 Ž i , d, b, c . s ² id < < bc : q
Ý
² mn < < ed : a iebm cn
,
mne
Ž A.5 .
X 2 Ž j, k , l, a . s ² jk < < la: q
1
2
Ý ² ml < < ef
: a ma ejkf
,
mef
Ž A.6 .
X 3 Ž b, c, e, f . s ² bc < < ef : q
1
2
Ý ² mn < < ef
: a mb cn
,
mn
Ž A.7 .
1
2
X 4 Ž m, n, j, k . s ² mn < < jk : q
Ý ² mn < < ef
: a ejkf
,
ef
Ž A.8 .
X5 Ž m, a, i , e . s ² ma < < ie : y Ý ² mn < < ef : a ianf ,
nf
Ž A.9 .
Y1 Ž f , a . s 12
Ý ² mn < < ef : a ma en ,
Ž A.10 .
mne
Y2 Ž n, i . s 12
Ý ² mn < < ef : a iemf ,
Ž A.11 .
mef
Z1 Ž k , l, b, c, d, e . s 12
Ý
² mn < < ef : a mf bnckdl ,
m, n , f
Ž A.12 .
Z2 Ž j, k , l, c, d, n . s y
1
2
Ý
² mn < < ef : a me fjkc dl .
m, e, f
Ž A.13 .
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