- Email: [email protected]
Exploiting the link between CI and the coupled cluster model. estimates for cluster energies and wavefunctions and a means for the rapid determination of CCD wavefunctions
Volume 88, numbcr 2
CtIEhllCAL
30 Apnl
PHYSICS LET’KRS
1982
EXPLOITING THE LINK BETWEEN Cl AND THE COUPLED CLUSTER MODEL. ESTIMATES FOR CLUSTER ENERGIES AND WAVEFUNCTIONS AND A MEANS FOR THE RAPID DETERMINATION
OF CCD WAVEFUNCTIONS
Chfford E. DYKSTRA Dcparrmctrr
of
Clrcmrsr~
“nrr cnrr~ oflll~norr.
~lrbano. llhnois
Rcccivcd 1 I‘ebruary 1982, m final form 26 rcbruary
~ISOI,
USA
1983
Simple means for choosing m~lnl-guess double substltutron
coupled cluster (CCD) wavcfunctlons and
energies on
the
basis of double substwllon configurrtlon mtcnction (CID) results can lead to npld rtcntive solution of the CCD wavcfuncIIOII. This IS bcausc the importance ofdoubly substltutcd configuntlons m n CID wavcfunction seems to cvhiblt a srmplc, primary rchtion to thcu lmportancc m s CCD wavcfunchon.
1. Introduction
Higher-order electron correlation effects are those that arise from the mixing of configuratlons that do not interact chrectly with an uncorrelated SCF wavefunction. While their Importance can eventually overtake the importance of the chrectly interacting configurations as molecule size Increases [ I(?], the higherorder effects in small molecules that amount to just fractions
of the total
have a sizable
correlation
influence
energies
on molecular
can already
properties.
The shapes of potenrtal energy surfaces of molecules WIII tend to be refined by the mclusron of higher-order correlation effects and this refinement has been found to bc important for ma!ung the most accurate predictions of eqtuhbrmm structures [3,4], and for accurate evaluations of harmonic and anharmonic vibrational constants [S-S]. In addition, higher-order correlation effects can play a role m crrtlcal determmatlons of isomer stablhtles and reaction energetics [9,10], especially for reactions
leading
to separated
species.
In these cases,
ignormg higher-order correlation effects mtroduccs a size-consistency error [I, II- 141. An emplrical
formula
well used for estlmatmg relative
to a configuration
[2] of Davidson higher-order mteracticn
has become
correlation (Cl)
effects
treatment
that Includes only the directly interactmg doubly substitutcd contiguratrons. This formula corrects the correlation energy, Ec, through Increasing it by a small 202
amount,(l -Ci)E, , where CO IS the expansion coefficient of the SCF reference wavefunction, ‘P, m the normahzcd ((9 I$) = I) correlated wavefunction.Thus, E;=E,+(l-C;)E,
(1)
A nonemplrical
basis for the formula
and has also been part of several tions
[ l2,36-183
has existed
subsequent
that have produced
[ 151
eaamina-
a moddied
cor-
rection: E;=Ec+[(I-C@$]Ec.
(2)
This can be put m simpler form by using the normallzation condltlon <@I tj) = 1. Then, eq. (2) becomes E; = (rLIJI)E,. In other
(3)
words
the correlation
by the normahzation
energy
several analyses
of this correction
in a many-body
perturbation
correctlon
provides
tion at fourth mixing
when (I),
the order
for quadruples
many
to their
they have applied or the modified
either
Davidson
energy
this
correc-
the indirectly
contribute
authors
that with-
treatment
at whch
substitutions
gy. Consequently, “+Q”
have shown
correlation
a renormahzation
order,
quadruple
is being resealed
factor of the wavefucntion. The
to the ener-
have appended
a
labels of Cl energies the Davidson correction
correctlon
(2) and (3).
More rigorous treatments of higher-order effects have been oblamed in three ways: explicit incorporation of higher-order
contigurations
in Cl, low-order
per-
Volume 88. number 2
CHEhllCAL
PHYSICS
turbatlve treatment of higher-order substitutions, and cluster models of the correlated wavefunction [ 1,14, 19-27]_ A feature of the cluster model IS that it is a physically based picture which can be related to manybody perturbation theory and which bmlds up higherorder effects from interactions or correlations of several electrons (linked terms) plus simultaneous, independent fewer-body interactions (unlmked terms). Double
substitutions
are of clear trnportancc
of the two-body electron-electron
30 Aprd 1982
of this guess wavefunction
is very simildr to the
Davidson formula and is given by E; = exP((‘!‘DI $B)) E,,
0)
where $D is the correlation part of a CID wavcfunction, J, = (P + tiD_ In the calculations reported here, this formula gives cncrgles in very good agrccmcnt with converged CCD energlcs.
because
repulsion operator
in the hamiltonian,
and cluster expansions with double substitutions (CCD), are effective because the unlinked or simultaneous two-body Interactions have a more important role in higher-order effects than the more-thantwo-body, hnked interactions. The form of a CCD wavefunction is GCCD = exp(??) ‘b.
LEITCRS
(4)
where ?‘? ISan operator with embedded expansion coefficients scaling individual double substitution opcrators, i.,,_+_ A-power senes expansion gives exp(fz) = (I + T2 + $ 5 + _..). The first two terms (with chfferent embedded coeffiaents) generate a CID wavefunction. G produces quadruple substitutions, and so forth. In this way higher-order correlation effects are accounted for but with the computational feature that there are the same number of embedded expansion coefficients as m CID. Even so, CCD requires much more computational effort than CID. Exacerbatmg the computatlonal cost of CCD IS its apparently slow convergence relative to comparable iterative CID treatments. We have devclopcd an electron pair operator approach to CCD [27] that has the same relationslup to CCD as the self-consistent electron pairs [28,29] (SCEP) method has to CID. That IS, the results are equivalent but the wavefucntion is represented and determmed differently, though conventlonal approaches are now takmg advantage of electron pair operators, too [30]. For reasons discussed below, with our approach to CCD we have not experienced the extremely slow convergence, reported by others, but recently improved [3 I ] _At best, however, convergence to a CCD wavefunction requires more iterations than convergence to a CID wavefunction, with a typical number of iterations required being S-12. We find that this convergence can be unproved by about a factor of two by using a guess wavefunction that IS trinally constructed from a CID wavefunction. The energy
2. Theory and results of calculations The CID and CCD wavefunctlons are detemlmablc from sunrlar self-consistency conditions. Lettmg JI and $’ be total CID and CCD wavcfunctions. respcctively, and letting J/D refer to the double substltullon part of the correlated wavefunctlon and X to the hlghcrorder part, WCmay write CID:
$=++ljf,.
(6)
CCD
I)‘=
(7)
9 + +‘D +X.
Then, takmg QI~,~to bc any of the doubly subslltutcd configuratlons, these wavcfunclions satafy the conditions: CID
(9;b[H-(EotEc)[+)=0,
(8)
CCD-
($$I-(E,tE;)I&‘,=O.
(9)
where E, = GbIH 14). The expectation value cxpresslon for the energy of I) can be rearranged usmg eq. (8) to give Ec = ($DIHl’b).
(10)
In the coupled clusters model, the correlation energy expression is the same
Ef = (1&IHl44.
(IO
Ei ISnot, however, an expectation value since eq (9) IS not equivalent to a variattonal determination of the in T7. But the correlation energy expresslon can be morhficd to be closer to an expectation value expression [7_7].
coefficients
E; = (~~lHl(lN
($;,lH-ElII’).
(12)
In fact, this expression differs from the expectation value of the energy only by CYIH - El IL’) which is likely to be small. The extra term in eq. (I 2) becomes 203
-76 139 316 -76.138 150 -76.136 734 -76.135 088 0.9831 3 444 -24.90 152.0
078 117 821 213
078 144 876 298
-76.139 434 -76.138 299 -76 136 920 -76.135 313 0.9839 3.405 -24.54 172.8
-76.141 -76.141 -76.140 -76 140 958 018 745 162
860 520 736 539
-76 139 292 -76.139 152 -76 136 767 -76 135 153 0 9837 3 410 -24 56 172 I
-76.140 -76 141 -76.140 -76.140
-76.136 -76.138 -76.139 -76 140
Davldson
cq (3) modticd
-_----
-__
730 781 499 906 -76 139 026 -76 137 875 -76.136 479 -76.134 853 0.9835 3 422 -24 60 1706
-76.140 -74 I40 -76.140 -76.139
-76.136 665 -76.138 317 -76.139525 -76 140 320
Davrdson
_-
cq (1)
-_
,-,)
0 046 0.046 0.047 0 048
0 043 0.043 0.044 0.045
0 040 0041 0.041 0 042
188 942 133 542
256 965 690 430
567 218 883 562
SCEP(CID)
($Dit ~-_-
-.___-
3) The HOH tnglc was 104 5’ III011caIculafrons. b, The force constents 31c dcfincdj,, = aJrw)larfll,,,mln A satlr-ordcr polynomial last-squares fit was used to fmd V(r).
-76.133 257 -76.131 976 -76.130 444 -7G.128 679 0.9800 3 536 -25. I5 155.18
-76 141 -76 141 -76.140 -76.140
-76.002594 -76.000408 -76.997 073 -76.995 305
463 393 987 268
1 017 1.027 1.037 1.047
-76.135 -76.134 -76.134 -76.134
-76.008 430 -76.007450 -76.006136 -76.004509
0.917 0 907 0.997 1.007
--
136 963 138 627 139847 140 655
-76 -76 -76 -76
-76.137 -76.138 -56 139 -76.140
-76 131 856 -76. I33 397 -76 134 492 -76.135 171
-76008487 -76.009 iO8 -76.009 286 -76.009 051
0.937 0 947 0 957 0.967
052 695 895 680
cq (5)
CCD
--
_--_-_--_-_.
SCLPICID)
(A) a) Encrgics (au)
SCT:
Roll
Table 1 Comprativc cslcullltlons onw~tcr -_----
_
044 044 045 046
0.050 0.051 0.052 0053
0.047 0.048 0 048 0.049
0 0 0 0
CCD
-___
640 550 484 441
214 040 886 752
097 849 620 408
165 006 868 752
995 760 543 344
0.050 658 0.051 563 0052515 0 053 492
0.047 0 048 0.048 0 049
0.043 0044 0045 0.046
cq (13)
5 g v1
7 E iii
2 g z g 3
CIIEAIICAL
Volume 88. number 2
30 April 1982
PHYSICS LCITERS
zero at convergence but durmg intermediate ltcratlons it helps to smooth out oscillations in the calculated energy just as usmg an expectation value for the energy instead of eq. (IO) does m SCEP (CID). Perhaps for this reason and the flexibility In electron par operator approaches [30], we have not found CCD to be mherently poorly convergent, though slower than SCEP (CID). Slegbahn has shown that the modified DavIdson forn~la [eq. (3)) is correct to fifth order in the renor-
(I) seems to consislcntly undcrcstimatc the correlatron energy. R, values and force constants arc simdar for all. The ability of any of the correction fomlulas to come so close to the CCD encrglcs. to which there has been but little prcvlous comparison, IS an mdlcatlon that to a very hlgh degree $D and I& are simply related by a proportlonahty factor. Comparmg eqs. (IO) Jnd (I I) and usmg the cslmiate of cq. (5) gives
mahzation energy terms for a perturbatlve expansron
Thus guess for the wavefunction can bc checked by evaluating ($bI I$;) usmg eq. (13) and usq a CCD
of the wavefunction
and has provided an expression for these terms summed to all orders [ 161. The term in this expansion which he finds to be the leadmg correctlon to DavIdson’s formula is Ez Zi c2/(Eo - E,) where c, IS the expansion of the ith contiguratlon in $ and E, IS Its unperturbed energy elgcnvaluc, whrle the remaining terms mvolve higher powers of the ci coefficients The terms in a power series expansion of both the exponential estimating formula, cq. (S), and the exponential generator of the CCD wavefunction naturally show a diminishing importance because they involve higher-power products of the configuration expansion coefficients. The first two terms of the expansion of the estimate [eq. (S)] are the modified Davidson formula. Use of eq. (5) has been tested m calculations on water at a number of different geometries around the equdibnum. Calculations were perfonned with a doublc-zeta (DZ) basis [32] using the SCEPSO [33] computer program, the results are m table I. Around the equihbrium O-H length of zO.98 A, eq. (5) gives a very good estimate for the CCD energies -Just about exact at 0.977 A. Eq. (2) gives similar results, whde eq
$b = cxp(($~ls~))
$D.
(13)
calculation, and as shown in table
I, there IS cvtrcmely
good agreement in these values for all points. It is clear that to a large extent, the Fz gcncrator of a CID wavcfunction doffers from the r, generator of a CCD wavefunction prrmarrly by a constant factor which may bc regarded as a renormalGng factor. To the extent that this holds, one may argue that though the unlinked cluster effects are important, It is the simple two-body mteractlons themselves that detcrminc the rclativc mlportancc of the doubly substituted configuratIons Further evidence of this IS in the practical apphcatron tion of usmg eq. (13) and eq. (5) to guess mrtial CCD wavefunctions and energies front CID results. As shown in table 3, these initial guesses lcad lo rapid convergence to the CCD wavcfunctions for the DZ water calculatlons * A DZ+P basis calculation on water and DZ basis calculatron on formaldehyde show the same bchav~or as mdrcated by the results In table 3 In these * For Ilcratlon 1, the corrchtion energy c\prcssion was cq (5). while Tar subsequentIlcratrons, the c\pcclatlon-like eripresSIOIIor cq (12) was used.
Table 2 CCD convergencefor water a)
Iteration
&,,., = 0.947 bE
E (au)
: 3 4 5
-76.138 -76.138 -76.138 -76.138
Roll = 0.977
640 678 41 691 9 695 7 695 4
-0 000 -0 000 -0 000 0.000
037 013 003 000
7 8 8 3
Roil = 1.017
E (au)
bE
E klu)
-76 -76 L4lO96 1410514 0
0 000 038 6 -0 000 017 I -0.000 004 2 00000003
-76 139 -76 139 457 286 -76.139 31 I -76 139 315 -76.139 315
-76.141 074 5 -76.141078 7 -76.141078 4
6E 83 2 9 7
0 000 -0.000 -0.000 0 000
169 022 004 000
a) Pomts correspondto those in table 1.
205
5 8 7 2
Furthermore,
Table 3 Addltmnal CCD calculationsnnd CCD cstbnatcs a) -
-_.
.
.__
30 Aprrl 1982
CHCMICAL. PHYSICS LETKRS
Volume 88. number 2
result of usmg an estimate and achieving rapid con-
vergence helps establish that there is a simple, domt-
Formaldehyde b, DZ basis
Water c, DZ+P basis
-114031710
-76.240
658
guesstng the CCD wavefunctron
-0.193
920
evaluating properties such as dipole moments with the mclusion of higher-order effects.
___~.
CID energy obtamcd from SCCP
the practical
for a guess CCD wavefunctton
CID corrclalron rncrgg
-0 20’
171
0081
962
nant proportionahty
relationship
CCD wavefunctions.
Another
between
0052416 -0.203
-0 217 486
DavIdson cstlmntc of concbtlon energy. cq. (I)
578
sults in table
and actual CCD re-
I may be a bit misleadmg. For instance,
were one to apply any of the formulas
to a twoelec-
tron system where there are no higher-order
modficd Davidson cstimatc of correla-
-0.218
-0.204
741
084
IS that
may be suitable for
The closeness of the estimated
(Q Dig D’
CID and
posstble dividend
tion effects, the result would
correla-
not be a zero correctton.
This IS related to the fact that none of the formulas go
cq (5) estimate of corrclatlon energy
-0.219 439
-0 204 355
to precisely the CCD result at the limit of separated, identical and non-interacting pans. Grven a CID wave-
CCD correlatlon c”WZy
-0 217 195
-0.201935
to parts, rJrI, for the correlatton
tion energy. eq (3)
function
the exact CCD wavefunction’s
cq. (I 3) eStlnlatC of ($‘Dl(/*D)
I 096 562
1 058 209
cJl$J’D,
I 096 670
1.057 484
5
4
number of CCD ltcrntlons rcquucd using this method d)
b)The geometry was/Q0 LHCH = 121.1”. =) The gcomctry wilsR0~
= 1 3-l A, RCH = I 102 A and = 0.947 A and LHOH
= 104 5’
d)Thc conucrgcnce hmlt was 10m6au.
two calculattons,
formulas (I), (3) and (5) are slightly the CCD energy with eq. (5) givmg the
greatest overestimate.
However,
the exponenttal
m just about all of the calculations.
estimates ofequthbrium
bond lengths
relative to CCD. In considenng the IV-pair dependence, Davidson and Sliver [I71 presented another alternate form for the energy correction: - I),
or for wavefunctions
(19 where
(+I JI)=
I,
JY;=LJ(2-(@I$)).
(16)
esti-
matmg factor as used in cq. (13) grves the best estimate of (&fr&)
Thus,
it was chosen to make mttral guesses of the CCD wave-
wdl
energy curves and may be prone to give slight-
E; = E,&(2c; overestimatmg
pairs,
double substitutions
(5) would give 1 t N ($tf 4,) instead, which is an overestrmate [6,17]. Probably for this reason, the estimating formulas tend to slightly favor dissociative sides of ly overlongated
arc u1 3~.
such pairs broken inof each of theN
be related by the factor I + (N- I) (Jlil$,). The modified Davidson correction or the first two terms of eq.
potential
3,Lnergcs
for a super-system ofN
This formula
overestrmates
gy in all the calculations
the CCD correlation
ener-
given here and may be less
surtable at least for comparison
with CCD results.
functions, though it may be that the Davtdson and modified Davtdson formulas would do about as well. In place of eq. (13). the modtfied
Davidson correction
Acknowledgement
leads to. This work was supported, Chemrstry The conclusions of this work are that simple formulas such as eq. (5) * or the modified
Davidson for-
nuda can g~vc very good cstuuatcs of CCD energies. * Par P normabzcd CID wavcfunctron, (9 comcsE;:=E,eup[(l-C~)/C’l.
206
10)= I.
cq. (5) be-
in part, by the Quantum
Program of the National
(Grant CHE-7815444). by the Universtty
Computer
of Illinois
Science
Foundatton
time was provided
Research Board.
Volume 88. number 2
CHEhllCAL
PHYSICS LITI-ERS
References
Ill R J. Bartlett. Ann. Rev 121C.R Dawdson, in The
Phys. Chcm. 32 (1981)
1161 P IX. Slcgbahn, Chcm Phys Lcttcrs 55 (1978) 386 1171 C R. Davidson and D W. Sdvcr, Chcm Phys Letters 52 (1977) 403
359.
world of quantum chcmatry, cds.
R Daudcl and B. Pullman (Reldel. Dordrccht. 1974). S R Langholland E.R. Davidson, lntcrn J. Quantum Chcm. 8 (1974) 6. 131 J.A Montgomery and C E. Dykstra. J. Chcm Phys. 71 (1979) 1380; P.S Dardi and C.L. Dykstra, Astrophys J. 240 (1980) L171. 141 151 161 [71 lBl
D J. Dcfrccs, C.H Locw and A D. hIcLean, Astrophys. J to be published R J. Bartlett, I. Shawtt and C.D. Purvis.J. Chcm. Phys. 71 (1979) 281 W. hleycr. J. Chem Phys. 58 (1973) 1017; Intern J Quantum Chem. 5 (1971) 341. P.R. Taylor. C.B. Bacskay. N.S. Hush and A C. Hurlcy, J. Chem. Phys. 69 (1978) 197 1.4669. C.E. Dykstra and D. Secrcst, J. Chcm. Phys 75 (198 I)
3967. I91 R Krishnan, MJ. rrach. J A. Poplc and P. von R. Schlcycr, Chem. Phys Letters 79 (1981) 408. IlO1 S R Candht. MA. Bcnzcl. C lI. Dyksln and T rukunaga. J. Am. Chem. Sot.. submitted for publication. 1111 J A. Poplc. J.S. Binkley and R. Seeger, Intern J. Quartturn Chem. IOS (1976) 1. 1121 R J. Bartlett and I Shnvitt. Intern. J Quantum Chcm.
1131 1141 1151
11s (1977) 165. H. Prtmas, u-t. Modern quantum chemtstry, cd. 0 Stnano~lu (Aeademtc Press. New York. 1965). R J Bartlett and G D Putvis, Phystca Scrlpta 21 (1980) 2555. K A. Brucckncr, Phys Rev. 100 (1955) 36. :
30 Aprd 1981
[ 181 [ 191
W.L. Lukcn. Chcm Phys Lctlcrs 58 (1978) 421. J. C&k. J Chem Phys 45 (1966) 4256,Adwn Chcm Phys. I4 (1969) 35.
120) A C. tlurlcy. Clectron correlation tn smaff molcculcs (Academic Press. Ncrv York. 1976) 121 I J. CL*ck. J. Paldus and L. Sroubkova. Intcrn.‘J. Quantum Chcm 3 (1969) 149. 1221 J. titck
,
(1971)
and 1. Paldus, lntcrn
J. QuantumChem
5
359.
[231 P.R Taylor. C.B. Bacskay. N S. llush and A C flu&y. Chcm. Phys Lcttcrs41 (1976) 444. (241 J A. Poplc. R. Erlshnan, II B Schlcgcl and J S flmklcy. lnlcrn J. QuantumChcnt. 14 (1978) 545. (25 I RJ. Bartlett and C.D Purws. lntcrn I Quantum Chcm. 14 (1978)561 1261 W. Kutzclnigg, in Modern thcorctical chemistry. Vol 3. cd HI. Schaefer 111(Plenum Press. New York, 1977) 127 I R A Chdcs and C C. Dyhstra. J Chcm. Phys 74 (198 I) 4544 (28 1 W.hleyer, J. Chcm. Phys 64 (1976) 2901 1291 C.C. Dykstra. H I’ Schaefer and W. Meyer. 1. Chcnt Phys 65 (1976) 2740. 1301 P.G. Jasicn and CC Dykstra. J.Chem. Phys 76 (1982). to be pubhshcd [31] G D. Purvis and R J. BarIlctt, J Chcm Phys 75 (1981) I284 1321 S fluzwgs. J.Chcm Phys 42 (1965) 1293, T H Dunnmg. J. Chem. Phys 53 (1970) 2823. 1331 C.E Dykstra, R A Chtlessnd hf D. Garrett, J.Comput Chcm 2 (1981) 266.
207
CtIEhllCAL
30 Apnl
PHYSICS LET’KRS
1982
EXPLOITING THE LINK BETWEEN Cl AND THE COUPLED CLUSTER MODEL. ESTIMATES FOR CLUSTER ENERGIES AND WAVEFUNCTIONS AND A MEANS FOR THE RAPID DETERMINATION
OF CCD WAVEFUNCTIONS
Chfford E. DYKSTRA Dcparrmctrr
of
Clrcmrsr~
“nrr cnrr~ oflll~norr.
~lrbano. llhnois
Rcccivcd 1 I‘ebruary 1982, m final form 26 rcbruary
~ISOI,
USA
1983
Simple means for choosing m~lnl-guess double substltutron
coupled cluster (CCD) wavcfunctlons and
energies on
the
basis of double substwllon configurrtlon mtcnction (CID) results can lead to npld rtcntive solution of the CCD wavcfuncIIOII. This IS bcausc the importance ofdoubly substltutcd configuntlons m n CID wavcfunction seems to cvhiblt a srmplc, primary rchtion to thcu lmportancc m s CCD wavcfunchon.
1. Introduction
Higher-order electron correlation effects are those that arise from the mixing of configuratlons that do not interact chrectly with an uncorrelated SCF wavefunction. While their Importance can eventually overtake the importance of the chrectly interacting configurations as molecule size Increases [ I(?], the higherorder effects in small molecules that amount to just fractions
of the total
have a sizable
correlation
influence
energies
on molecular
can already
properties.
The shapes of potenrtal energy surfaces of molecules WIII tend to be refined by the mclusron of higher-order correlation effects and this refinement has been found to bc important for ma!ung the most accurate predictions of eqtuhbrmm structures [3,4], and for accurate evaluations of harmonic and anharmonic vibrational constants [S-S]. In addition, higher-order correlation effects can play a role m crrtlcal determmatlons of isomer stablhtles and reaction energetics [9,10], especially for reactions
leading
to separated
species.
In these cases,
ignormg higher-order correlation effects mtroduccs a size-consistency error [I, II- 141. An emplrical
formula
well used for estlmatmg relative
to a configuration
[2] of Davidson higher-order mteracticn
has become
correlation (Cl)
effects
treatment
that Includes only the directly interactmg doubly substitutcd contiguratrons. This formula corrects the correlation energy, Ec, through Increasing it by a small 202
amount,(l -Ci)E, , where CO IS the expansion coefficient of the SCF reference wavefunction, ‘P, m the normahzcd ((9 I$) = I) correlated wavefunction.Thus, E;=E,+(l-C;)E,
(1)
A nonemplrical
basis for the formula
and has also been part of several tions
[ l2,36-183
has existed
subsequent
that have produced
[ 151
eaamina-
a moddied
cor-
rection: E;=Ec+[(I-C@$]Ec.
(2)
This can be put m simpler form by using the normallzation condltlon <@I tj) = 1. Then, eq. (2) becomes E; = (rLIJI)E,. In other
(3)
words
the correlation
by the normahzation
energy
several analyses
of this correction
in a many-body
perturbation
correctlon
provides
tion at fourth mixing
when (I),
the order
for quadruples
many
to their
they have applied or the modified
either
Davidson
energy
this
correc-
the indirectly
contribute
authors
that with-
treatment
at whch
substitutions
gy. Consequently, “+Q”
have shown
correlation
a renormahzation
order,
quadruple
is being resealed
factor of the wavefucntion. The
to the ener-
have appended
a
labels of Cl energies the Davidson correction
correctlon
(2) and (3).
More rigorous treatments of higher-order effects have been oblamed in three ways: explicit incorporation of higher-order
contigurations
in Cl, low-order
per-
Volume 88. number 2
CHEhllCAL
PHYSICS
turbatlve treatment of higher-order substitutions, and cluster models of the correlated wavefunction [ 1,14, 19-27]_ A feature of the cluster model IS that it is a physically based picture which can be related to manybody perturbation theory and which bmlds up higherorder effects from interactions or correlations of several electrons (linked terms) plus simultaneous, independent fewer-body interactions (unlmked terms). Double
substitutions
are of clear trnportancc
of the two-body electron-electron
30 Aprd 1982
of this guess wavefunction
is very simildr to the
Davidson formula and is given by E; = exP((‘!‘DI $B)) E,,
0)
where $D is the correlation part of a CID wavcfunction, J, = (P + tiD_ In the calculations reported here, this formula gives cncrgles in very good agrccmcnt with converged CCD energlcs.
because
repulsion operator
in the hamiltonian,
and cluster expansions with double substitutions (CCD), are effective because the unlinked or simultaneous two-body Interactions have a more important role in higher-order effects than the more-thantwo-body, hnked interactions. The form of a CCD wavefunction is GCCD = exp(??) ‘b.
LEITCRS
(4)
where ?‘? ISan operator with embedded expansion coefficients scaling individual double substitution opcrators, i.,,_+_ A-power senes expansion gives exp(fz) = (I + T2 + $ 5 + _..). The first two terms (with chfferent embedded coeffiaents) generate a CID wavefunction. G produces quadruple substitutions, and so forth. In this way higher-order correlation effects are accounted for but with the computational feature that there are the same number of embedded expansion coefficients as m CID. Even so, CCD requires much more computational effort than CID. Exacerbatmg the computatlonal cost of CCD IS its apparently slow convergence relative to comparable iterative CID treatments. We have devclopcd an electron pair operator approach to CCD [27] that has the same relationslup to CCD as the self-consistent electron pairs [28,29] (SCEP) method has to CID. That IS, the results are equivalent but the wavefucntion is represented and determmed differently, though conventlonal approaches are now takmg advantage of electron pair operators, too [30]. For reasons discussed below, with our approach to CCD we have not experienced the extremely slow convergence, reported by others, but recently improved [3 I ] _At best, however, convergence to a CCD wavefunction requires more iterations than convergence to a CID wavefunction, with a typical number of iterations required being S-12. We find that this convergence can be unproved by about a factor of two by using a guess wavefunction that IS trinally constructed from a CID wavefunction. The energy
2. Theory and results of calculations The CID and CCD wavefunctlons are detemlmablc from sunrlar self-consistency conditions. Lettmg JI and $’ be total CID and CCD wavcfunctions. respcctively, and letting J/D refer to the double substltullon part of the correlated wavefunctlon and X to the hlghcrorder part, WCmay write CID:
$=++ljf,.
(6)
CCD
I)‘=
(7)
9 + +‘D +X.
Then, takmg QI~,~to bc any of the doubly subslltutcd configuratlons, these wavcfunclions satafy the conditions: CID
(9;b[H-(EotEc)[+)=0,
(8)
CCD-
($$I-(E,tE;)I&‘,=O.
(9)
where E, = GbIH 14). The expectation value cxpresslon for the energy of I) can be rearranged usmg eq. (8) to give Ec = ($DIHl’b).
(10)
In the coupled clusters model, the correlation energy expression is the same
Ef = (1&IHl44.
(IO
Ei ISnot, however, an expectation value since eq (9) IS not equivalent to a variattonal determination of the in T7. But the correlation energy expresslon can be morhficd to be closer to an expectation value expression [7_7].
coefficients
E; = (~~lHl(lN
($;,lH-ElII’).
(12)
In fact, this expression differs from the expectation value of the energy only by CYIH - El IL’) which is likely to be small. The extra term in eq. (I 2) becomes 203
-76 139 316 -76.138 150 -76.136 734 -76.135 088 0.9831 3 444 -24.90 152.0
078 117 821 213
078 144 876 298
-76.139 434 -76.138 299 -76 136 920 -76.135 313 0.9839 3.405 -24.54 172.8
-76.141 -76.141 -76.140 -76 140 958 018 745 162
860 520 736 539
-76 139 292 -76.139 152 -76 136 767 -76 135 153 0 9837 3 410 -24 56 172 I
-76.140 -76 141 -76.140 -76.140
-76.136 -76.138 -76.139 -76 140
Davldson
cq (3) modticd
-_----
-__
730 781 499 906 -76 139 026 -76 137 875 -76.136 479 -76.134 853 0.9835 3 422 -24 60 1706
-76.140 -74 I40 -76.140 -76.139
-76.136 665 -76.138 317 -76.139525 -76 140 320
Davrdson
_-
cq (1)
-_
,-,)
0 046 0.046 0.047 0 048
0 043 0.043 0.044 0.045
0 040 0041 0.041 0 042
188 942 133 542
256 965 690 430
567 218 883 562
SCEP(CID)
($Dit ~-_-
-.___-
3) The HOH tnglc was 104 5’ III011caIculafrons. b, The force constents 31c dcfincdj,, = aJrw)larfll,,,mln A satlr-ordcr polynomial last-squares fit was used to fmd V(r).
-76.133 257 -76.131 976 -76.130 444 -7G.128 679 0.9800 3 536 -25. I5 155.18
-76 141 -76 141 -76.140 -76.140
-76.002594 -76.000408 -76.997 073 -76.995 305
463 393 987 268
1 017 1.027 1.037 1.047
-76.135 -76.134 -76.134 -76.134
-76.008 430 -76.007450 -76.006136 -76.004509
0.917 0 907 0.997 1.007
--
136 963 138 627 139847 140 655
-76 -76 -76 -76
-76.137 -76.138 -56 139 -76.140
-76 131 856 -76. I33 397 -76 134 492 -76.135 171
-76008487 -76.009 iO8 -76.009 286 -76.009 051
0.937 0 947 0 957 0.967
052 695 895 680
cq (5)
CCD
--
_--_-_--_-_.
SCLPICID)
(A) a) Encrgics (au)
SCT:
Roll
Table 1 Comprativc cslcullltlons onw~tcr -_----
_
044 044 045 046
0.050 0.051 0.052 0053
0.047 0.048 0 048 0.049
0 0 0 0
CCD
-___
640 550 484 441
214 040 886 752
097 849 620 408
165 006 868 752
995 760 543 344
0.050 658 0.051 563 0052515 0 053 492
0.047 0 048 0.048 0 049
0.043 0044 0045 0.046
cq (13)
5 g v1
7 E iii
2 g z g 3
CIIEAIICAL
Volume 88. number 2
30 April 1982
PHYSICS LCITERS
zero at convergence but durmg intermediate ltcratlons it helps to smooth out oscillations in the calculated energy just as usmg an expectation value for the energy instead of eq. (IO) does m SCEP (CID). Perhaps for this reason and the flexibility In electron par operator approaches [30], we have not found CCD to be mherently poorly convergent, though slower than SCEP (CID). Slegbahn has shown that the modified DavIdson forn~la [eq. (3)) is correct to fifth order in the renor-
(I) seems to consislcntly undcrcstimatc the correlatron energy. R, values and force constants arc simdar for all. The ability of any of the correction fomlulas to come so close to the CCD encrglcs. to which there has been but little prcvlous comparison, IS an mdlcatlon that to a very hlgh degree $D and I& are simply related by a proportlonahty factor. Comparmg eqs. (IO) Jnd (I I) and usmg the cslmiate of cq. (5) gives
mahzation energy terms for a perturbatlve expansron
Thus guess for the wavefunction can bc checked by evaluating ($bI I$;) usmg eq. (13) and usq a CCD
of the wavefunction
and has provided an expression for these terms summed to all orders [ 161. The term in this expansion which he finds to be the leadmg correctlon to DavIdson’s formula is Ez Zi c2/(Eo - E,) where c, IS the expansion of the ith contiguratlon in $ and E, IS Its unperturbed energy elgcnvaluc, whrle the remaining terms mvolve higher powers of the ci coefficients The terms in a power series expansion of both the exponential estimating formula, cq. (S), and the exponential generator of the CCD wavefunction naturally show a diminishing importance because they involve higher-power products of the configuration expansion coefficients. The first two terms of the expansion of the estimate [eq. (S)] are the modified Davidson formula. Use of eq. (5) has been tested m calculations on water at a number of different geometries around the equdibnum. Calculations were perfonned with a doublc-zeta (DZ) basis [32] using the SCEPSO [33] computer program, the results are m table I. Around the equihbrium O-H length of zO.98 A, eq. (5) gives a very good estimate for the CCD energies -Just about exact at 0.977 A. Eq. (2) gives similar results, whde eq
$b = cxp(($~ls~))
$D.
(13)
calculation, and as shown in table
I, there IS cvtrcmely
good agreement in these values for all points. It is clear that to a large extent, the Fz gcncrator of a CID wavcfunction doffers from the r, generator of a CCD wavefunction prrmarrly by a constant factor which may bc regarded as a renormalGng factor. To the extent that this holds, one may argue that though the unlinked cluster effects are important, It is the simple two-body mteractlons themselves that detcrminc the rclativc mlportancc of the doubly substituted configuratIons Further evidence of this IS in the practical apphcatron tion of usmg eq. (13) and eq. (5) to guess mrtial CCD wavefunctions and energies front CID results. As shown in table 3, these initial guesses lcad lo rapid convergence to the CCD wavcfunctions for the DZ water calculatlons * A DZ+P basis calculation on water and DZ basis calculatron on formaldehyde show the same bchav~or as mdrcated by the results In table 3 In these * For Ilcratlon 1, the corrchtion energy c\prcssion was cq (5). while Tar subsequentIlcratrons, the c\pcclatlon-like eripresSIOIIor cq (12) was used.
Table 2 CCD convergencefor water a)
Iteration
&,,., = 0.947 bE
E (au)
: 3 4 5
-76.138 -76.138 -76.138 -76.138
Roll = 0.977
640 678 41 691 9 695 7 695 4
-0 000 -0 000 -0 000 0.000
037 013 003 000
7 8 8 3
Roil = 1.017
E (au)
bE
E klu)
-76 -76 L4lO96 1410514 0
0 000 038 6 -0 000 017 I -0.000 004 2 00000003
-76 139 -76 139 457 286 -76.139 31 I -76 139 315 -76.139 315
-76.141 074 5 -76.141078 7 -76.141078 4
6E 83 2 9 7
0 000 -0.000 -0.000 0 000
169 022 004 000
a) Pomts correspondto those in table 1.
205
5 8 7 2
Furthermore,
Table 3 Addltmnal CCD calculationsnnd CCD cstbnatcs a) -
-_.
.
.__
30 Aprrl 1982
CHCMICAL. PHYSICS LETKRS
Volume 88. number 2
result of usmg an estimate and achieving rapid con-
vergence helps establish that there is a simple, domt-
Formaldehyde b, DZ basis
Water c, DZ+P basis
-114031710
-76.240
658
guesstng the CCD wavefunctron
-0.193
920
evaluating properties such as dipole moments with the mclusion of higher-order effects.
___~.
CID energy obtamcd from SCCP
the practical
for a guess CCD wavefunctton
CID corrclalron rncrgg
-0 20’
171
0081
962
nant proportionahty
relationship
CCD wavefunctions.
Another
between
0052416 -0.203
-0 217 486
DavIdson cstlmntc of concbtlon energy. cq. (I)
578
sults in table
and actual CCD re-
I may be a bit misleadmg. For instance,
were one to apply any of the formulas
to a twoelec-
tron system where there are no higher-order
modficd Davidson cstimatc of correla-
-0.218
-0.204
741
084
IS that
may be suitable for
The closeness of the estimated
(Q Dig D’
CID and
posstble dividend
tion effects, the result would
correla-
not be a zero correctton.
This IS related to the fact that none of the formulas go
cq (5) estimate of corrclatlon energy
-0.219 439
-0 204 355
to precisely the CCD result at the limit of separated, identical and non-interacting pans. Grven a CID wave-
CCD correlatlon c”WZy
-0 217 195
-0.201935
to parts, rJrI, for the correlatton
tion energy. eq (3)
function
the exact CCD wavefunction’s
cq. (I 3) eStlnlatC of ($‘Dl(/*D)
I 096 562
1 058 209
cJl$J’D,
I 096 670
1.057 484
5
4
number of CCD ltcrntlons rcquucd using this method d)
b)The geometry was/Q0 LHCH = 121.1”. =) The gcomctry wilsR0~
= 1 3-l A, RCH = I 102 A and = 0.947 A and LHOH
= 104 5’
d)Thc conucrgcnce hmlt was 10m6au.
two calculattons,
formulas (I), (3) and (5) are slightly the CCD energy with eq. (5) givmg the
greatest overestimate.
However,
the exponenttal
m just about all of the calculations.
estimates ofequthbrium
bond lengths
relative to CCD. In considenng the IV-pair dependence, Davidson and Sliver [I71 presented another alternate form for the energy correction: - I),
or for wavefunctions
(19 where
(+I JI)=
I,
JY;=LJ(2-(@I$)).
(16)
esti-
matmg factor as used in cq. (13) grves the best estimate of (&fr&)
Thus,
it was chosen to make mttral guesses of the CCD wave-
wdl
energy curves and may be prone to give slight-
E; = E,&(2c; overestimatmg
pairs,
double substitutions
(5) would give 1 t N ($tf 4,) instead, which is an overestrmate [6,17]. Probably for this reason, the estimating formulas tend to slightly favor dissociative sides of ly overlongated
arc u1 3~.
such pairs broken inof each of theN
be related by the factor I + (N- I) (Jlil$,). The modified Davidson correction or the first two terms of eq.
potential
3,Lnergcs
for a super-system ofN
This formula
overestrmates
gy in all the calculations
the CCD correlation
ener-
given here and may be less
surtable at least for comparison
with CCD results.
functions, though it may be that the Davtdson and modified Davtdson formulas would do about as well. In place of eq. (13). the modtfied
Davidson correction
Acknowledgement
leads to. This work was supported, Chemrstry The conclusions of this work are that simple formulas such as eq. (5) * or the modified
Davidson for-
nuda can g~vc very good cstuuatcs of CCD energies. * Par P normabzcd CID wavcfunctron, (9 comcsE;:=E,eup[(l-C~)/C’l.
206
10)= I.
cq. (5) be-
in part, by the Quantum
Program of the National
(Grant CHE-7815444). by the Universtty
Computer
of Illinois
Science
Foundatton
time was provided
Research Board.
Volume 88. number 2
CHEhllCAL
PHYSICS LITI-ERS
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207