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Comparison of high-order many-body perturbation theory and configuration interaction for H2O
V0luIl1c
50, number
2
CHEMICAL
PHYSICS
LETTERS
1 September
1977
COMPARISON OF HIGH-ORDER MANY-BODY PERTURBATION THEORY AND CONFIGURATION INTERACTION FOR Hz0 Kodncy J. BARTLETT Battrlfe
Cchmbr p Laboratories,
Cdumbus.
Ohio 43201.
USA
and Isaiah SMAVITT flatteffe Cofumbur Laboratories, Ciilumbus. Ohm 43201. USA uf!d ~epartl?wIt of Cflemiwy, Tile Ofuo State Utliventty, Cohnbus, Rcccivcd
01~0 43.201, USA
26 hlay 1977
IJi~~gr~mrn~trc m.my-body perturbatron theory, coupled with n rccursivc computabonal procedure. IS cnqlloycd to obt.Iin correlation cncrsy of Hz0 witbm a 39ST3 IXISIS set by evaluating all double-cxcttatlon diagldrns through twelfth order wallout any ~ppro~irn~ttons. Thr provides, for the first time, the complete double-eucitatton diagrams contrrbutron to the corrcklilon energy, wlucb IS -0.28826 hnrtrec, compared Hrlth ,I correlation cncrgy of -0.27402 hartrce obtamed from :I ~WI~I~UI~~IOII IrItcr.tctron calcuiatlon WIIILII ~ncludcs ,111tioublc excitattons. Ihe dlifercnce of 0.0142 hartrcc includes the corrcctIon to the ~ll_cioublc-cxclt,ltions Cl cnrr!Iy, rlu<~ to the “p3thdq:Ic:ll” unhnkcd+J1:l~rarn terms rcYIfC Con~1Stt’IlLy” mammg111 that result, but also involves certain lourtb- nnd highcr-ordcr rcnrmngcrncnt diLtgrams. Contrary to conventtondl bAci, the bnslIIfrcd, or hlglllcr- I’Iessct partmorung ot the IiamltonIm provide\ a much more rapld convergcncc of the pcrenhance the conturbatlon terlcs than does the shifted, or tpstcm-Ncsbct partitionm,. (1 In both cases, PadE approuirnantr vzrgcncc of the scws consldcr.lbly. A \mlplc vart,ition-pcrturbatlon scheme bawd on the first-order hlBPT wavcfunction I\ sufflctcnt to provide 97.5% of the all
1. lutroduction Mary-body methods, and e~pt~~ally the theory and application of diagrammatic many-body’perturbation theory (MBPT) [l-3], arc becoming increasingly cvidcn t in t!lc quantum chemistry literature These tncthods have several advantages over more cnnventional conf~gurntion interaction proccdurc. For example, MBFT permits the
[4- lo].
the (CI)
direct determination of excitation and ionization energes [I OJ, instead of requiring separate calculations for the in&d and final states. Furthermore, the absence from many-body theories of configuration functions, and thus of the complications resulting from their enormous nunl’,er, is 3 very useful fczturc. Another advantage is the suggestion of certain physical concepts, such as pair correlations and cluster expansions. But most important is the fact that the total energy 190
of a system can be defined by the linked cluster expansion [2]. This ensures the proper dependence of the energy on the size of the system [ I,2 1 (“size consistency” [9]), unlike the cabe for conventional, less ihan fit/Z Cl calculations. This last point is demonstrated most strikingly by the example of a system composed of N non-interacting identical subsystems. The correlation energy predicted by conventional Hartrcc-Fock plus all-doubleexcitations CI (denoted by the symbol U(D)) for such a system is proportional to A+‘* instead of IV as N-t m. In comparison with the linked cluster expansion of the exact energy [2,4,11], the CI(D) energy expression retains “pathological” unlinked diagram terms which increase quadratically or faster with N and thus destroy its siLe consistency. These terms would eventually be cancelled by higher excitation terms in furlCI, but unlike the case with MBPT, the pathological
Volume
50, number 2
CHCMICAL 1’FIYSICS LI-ITI-RS
terms must still be computed in the fulI CI, just to mutually in the final result. In the linked cluster expansion, on the other hand, the cancellation of the unlinked terms has already been incorporated within each order, due to contributions from different excitation levels in the CI sense 141. From another point of view, comparing the CI wzuefunction expansion with the exponential linked cluster form of the exact wavefunction in the coupled cluster approach 112,131, WC are led to the further conclusion that disconnected cluster terms, corresponding principally to independent simultaneous pair correlations (and giving rise to Zirzkcd cncrgy diagrams) are also missing in the Cl(D) expansion. These terms cm only be included in CI by adding higher quadruple and other even-excitation configurations, and the cluster analysis indicates that the most important disconnected contributions to the coefficients of theqc higher excitations can be expressed in terms of the coefficients oft he double excitation terms in the ficlf Ct solution 1133. Even in such a mundane application as the predlction of a potential energy surfiicc for d small ~nolcculc like I-120, a complctc single and double excitations Cl (CI(SD)) calculation retains significant errors [ 14,151. A simple correction for the effect of quadruple cxcitations [ 161, the leading term in the unlinked diagram error in small-molcculc CI(SD) [ 1 11, is enough to bring the resulting potential surface parameters to within cxpcrimental error limits in this case [ 151. This fairly crude cstimatc can harrlly be cxpcctcd to apply in general [! 11, while MUPT provides sbeconsistent results ngorously. (In this connection it shouid bc noted that conventional MBPT, based on a restricted single-configuration zero-or&r function, has difficulty 111describing the dissociative regions of the potential energy surface. This difficulty can usually be avoided by employing an unrestricted SCF reference function. The alternative approach of using a multiconfiguration rcfercncc function appears to be more difficult to implcmerlt in MBPT than in CL) With modern day quantum chemistry beginning to attain high levels of accuracy, at least for small molecules, it is now meaningful to proceed to a correct and systematic treatment of the energy that properly cxcludcs these unlinked energy diagram effects. Such a treatment can be accompli&cd by many-body pcrturbation theory, but in order to do this meaningfully car~cel
1 Scptcmbcr
1977
it is important that no appreciable errors be introduced by arbitrary truncations of the perturbation series. Unjustified neglect of higher-order terms rvay in fact introduce errors which arc not insignificant in comparison with the unlinked diagram errors which we are trying to eluninate. Our intent, rather, is to present :I well-defined Ml3PT model, analogous to CI(D) in its simplicity, but possessing the desired sizcconsistency property_ In order to accon~pl~sl~ thcsc objectives, this pepcr employcs a previously denvcd (17 ] MBPT recursive procedure which permits the complctc evaluation of the contributions of all doublcexcitatlon diagrams, to all orders, and applies it without further approxunimon to a fiurtc basis tlcatnicnt of t!lc H,O molccult at cquilibiium (an application to the H’;O potential surface is under way). The energy obtamed in this treatment is rigorously si/e-consistent, and is a siguificant lmprovcment over CI(SD). Assuming that the dominant cant rlbutions of quad1 uple and higher even excitations in an extended Cl expansion arc to cancel the unlinked diagram errors of tbc double-excitations Cl, the result of this MI3PT ploccdurc should be cornparable to that of a CI expansion which includes such higher even excitation configurations. This treatment does not include the cffcct of disconncctcd clusters in the wave function [12,13 ] (principally ?i, which leac to linked quaclruple-excitation energy diagrams, of wluch sonic il~c CPV rcarrangcmcnt diagrams that arc accounted for in CI(D)), but these do not affect the SILC consistency of the present approach. The correctlon obtained here (the diffcrencc between the MBI’T double-excitation diagrams energy and the Cl(D) energy is in remarkab!e agreement with ~III estirnatc of the quadruple excitations m~provement of CI(D) proposed by DavIdson [ 1G] , which we have dcrlved from unlinked diagram considerations elscwhcrc [ 11 J. This paper also examines the relative I@-order MBPT convergence properties of the “unshifted”, or M&lcr -1’lessct [IS] partitioning of the hamlltonlan and the shifted Fpstcin-Nesbct [ 191 partltionnlg. Contrary to some previous expectations [3,5,6,20], it is found that the M$llcr-Plcssct paltltiorring is far superior, but cnhanccd convc~gcr~cc can bc :Ichievcd in both cases by the USCof Padd approximants [21,X?]. It is also demonstrated that a previously proposed [4,2324] variation-perturbation schernc based only on a first-order MBPT wavefunction with a sin& 191
Volume
50. number
CHEMICAL
2
variational parameter provides a quite accurate upper bound to the energy compared to the CI(D) result.
2. High-order diagrammatic
perturbation
I
LETTERS
September 1977
rules for the interpretation of the diagrams [4J (coupled with a specific ordering of the summation indices) give the algebrdc formulas:
theory
Assun$ng a Hartree-Fock (HF) reference ‘PO, the defining equations of MBPT arc: N=if()
PHYSICS
function (abIltj-X~llkfXklllob)
t V,
D(Qab)D(klab)
with Ui the IIF cffcctive one-electron potential and ci the orbital energy. The total energy is then given by the linked-cluster expression [1,2]
x
(cjllbkXkillac) -C D(k&zc)
-
--z&UC)
k*illbkXkjllacl
’
+ killakXkjllbc) D(kjbc)
1 ’
(6)
where D(iiab) = cl + ‘I - ca - ch,
where the subscript L indicates that only linked diagrams are to be included. Assuming IIF spin-orbitals as the single-particle basis, all the antisymmetric diagrams with nonvanishing contributions to the second and third order of MBPT are shown in fig. 1. They arc all of the double excitation type, because Brillouin’s theorem climinatcs the single excitation contributions in thesc’orders. The
(7)
‘and where i, j, k. I represent spin-orbit& occupied in +0, while a, b, c, d arc excited, or “virtual”, HF orbitals. The double bar indicates an antisymmetrized two-electron integral. Contributions to the fourth-order energy arise from single, double, triple, and quadruple excitation diagrams. Of these, the twelve possible antisymmetrized double excitation diagrams are shown in fig. 2. If the treatment is restricted to just double excitation diagrams in all orders (which implicitly includes the im-
.I
*I
diapmc which contribute to of MBPT when the singie-particle
rig. 1. AI1 the antisymmctrizcd
Fe- 2. All the antisymmetnzed double-excitation diagrams
the second and third order
which contribute
baas consists Gf Hartrcc--Fock
192
orbitalc.
to tbc fourth order of hIBPT wher. the singlo
particle basis cans&s of Hartrec-Fock olbitals.
CHEMICAL PHYSICS L13-I-TLKS
Volume 50, number 2
portant effect of quadruple and higher even excitations in Cl terminology), it becomes possible to evaluate all the resulting contributions readily. In fact, using a recursive procedure which employs the same computer programs as for second and third order calculations, the contributions of all double excitations to all orders can be evaluated to within any desired small tolerance. ’ A generalization of cqs. (4)-(6) to any order can be obtained in terms of {ab IIQ),I = (ab llQ),,/D(abij) where the pseudo-mtcgrals given by
(n 2 l),
(8)
(ub II+,, (Q > b, i >i) arc
(Ohllij)~ =
(9) (n 2 2),
tab ll~>,, = X,,(abij) + Y,,(abij) + Z,,(abij) with (for )I > 1, a > h, i >j) X,+,(obij)
= CFd (ah llcd){cdllzj~,,
(10)
Y,+,(ub@
= kil
(11)
(ijlltW{ub
Ilk&,
Z,,+l(aW) = c
[(kb Ilic~{ca IlikI,,
- (ka Ilic) {cb
IIik),,
k.c
+ (ka Ilic){cb Ilik),r].
The pseudo-integral notation is appropriate, may easily be seen from eqs. (9)-( 12) that {ab Ilv>,, = -(ab Ilji),, = -(buIlij),,
=
(ha
llji),,
(12)
bc used recursively to evaluate their contributions in higher orders. The three third-order and twelve fourth-order
double-excitation
diagrams are thus evaluated
sin&
of eqs. (9)-(12)
apphcation
(ab II ii>,, Iab Iliil,+. 1,
by a
(with tz = I),
followed by eqs. (14) and (15). A second evaluation of eqs. (9)-( 12), now with tz = 2, conveniently provides the contribution of the sixty distinct tifthorder double-excitation diagrams as well as those which arise in sixth order. Additional iterations can be pcrformcd until convergence within a desirable tolerance is reached. Alternatively, this procedure may be vlcwed as a generalization
of Brucckncr’s
k-matrix theory [X]
to include all double excitation diagrams, mstead of just the two-body or pair interaction terms. This can bc done in terms of an Opriltor K, dcfincd so that its antisymmetriled integral is the ultimate iteration of eqs. (9)-( 12):
+ k2
‘hb I[Cd){Cd(K Izj-&,
(i/iikl){kllK
kZb)A
since it + Aq((kbIljC)i
(13)
for any n, although not all the symmetry properties of the initial integrals (rz = 1) arc necessarily retained for II b 2. In terms of these quantities, we can write the double excitation contributions to a‘ny order (n > 2) of perturbation theory as
I>/ a>b
1977
Scptcmbcr
This may be demonstrated directly in terms of the algebraic equivalents of the fourth, fift!l, etc. order diagrams, although, as shown previously [ 171, this type of high-order recurrence is a necessary consequence of perturbation theory. In fact, there riced be no limitation to double-excitation diagrams, except that additional code would be required to evaluate the linked single, triple, and quadruple excitation diagrams in fourth alid fifth order, and this could then
‘, = ‘qd .
- (kb Ilic){calljk),~
E&-I=Q
1
, .
{CQIK
+(kaIlic)[{cbl~lik)~
lik)A - {cb
IK
lik ‘}A1
- kvzl~lik~~l) (i >i,
a > 6).
Cl@
The braces indicate the incorporation of the dcnominator, as in eq. (8). As a result the sum total of all double-excitation diagram,
diagrams can be written a< a_singlc
(14)
(15)
where the antisymmetrized
K-vcrtcx is designated by 193
Volume 50. number 2
CHEllIlCAL PHYSICS LETTERS
the boldface line. However, since the evaluation of the (ab SQ>,,quantities is the time-consuming step in the procedure, an actual determination of the K-integrals acquires twice as many iterations for the same accuracy in the energy as required by eqs. (14) and (15), which exploit an analogue of the 2r1+ 1 rule of perturbation
theory and are thus decidedly preferable for computational purposes. A simpler type of summation conveniently used in MBPI’ involves some rearrangement of the terms in the perturbation series by using the technique of denominator shifts [3] _ This approach rccognizcs that certain terms, normally “diagonally scattered” terms such as occur in eq. (4) when c = a and d = b, form a geometric series, and incorporates their sums into lower-order terms. The most frequently used denominator shift has the form
[I%$M--
1 September 1977
I] =afA-'a,
(19)
[M, Ml =J$ +btB-l,,
(20)
.t = (E$?,
. . . EM,,),
(21)
b* = (Es&
. . . &+&
(22)
APq
= EP+q
- EP+q+l
Bpq = Epeq+ 1 - L:p+q+2
(~99 - 1,2, --., M-J
(23)
@,9 = 1,2, -.., Ml.
(24)
As seen from eqs. (23) and (24), the calculation of the approximants [& M - 1] and [M. M] rccluircs that the MBPT series be evaluated to orders 2M + 1 and 2hf + 2, respectively. The first approximant of each of the two types, [ 1,0] and [I, 1 ] , is equivalent to a geometric series sum based on the ratios E3/E2 and E4/E3, respectively.
D’(ijab) = D(ijab) - bb llab) - (qllij)
3. Caiculatiens + (aillai>+ lbil(bi) + (ajllaj) + lbillbl’),
(W
and the resulting formalism is equivalent to the Epstein-Ncsbct partitioning of the hamiltonian. The shifted denominator can be used nearly as easily in place of D(ijab) in cqs. (Q-(8), cxccpt that the surnrnation must then be rcstrictcd to exclude the terms with cxi = ub, kl = ij, and kc = ia, ja, ib, or jb, rcspcctivcly. Since in the present recursive approach these disgonal contributions (as well as the non-diagonal parts of the diagrams) ‘arc computed, in effect, to all orders, such geometric series sums are included in the final result without need for denominator modification, but theconvergence rate of the recurslon will depend on the form used. The Epstein-Nesbet partitiorGig has been shown to be superior t-or some problerns 16,201, and is traditionally thought to be an improvement over the simple denominator. It is therefore intcrcsting also to report here a comparison of the relative convergence rate of the shifted and unshifted series through fligh order. Another approach to improving the convergence of the MBPT calculations is the USCof some nonlinear series summation method. In the present cassz we use matrix Pad6 approximants [Icr.M - I] and IM, M] constructed from the terms of the (shifted or unshifted) perturbation series [2 1,22 1. These are given by
194
and results
Extensive I-IF and CI studies of the II,0 molecule have been reported in previous work [14], providing the opportunity for pertinent and overdue comparisons with an MBPT’ calculation_ The same integrals over the EIF orbit& as used in the CI study (with the 39-ST0 basis set) have been employed here to evaluate the MBPT diagrams from eqs. (9)-( 12) and (14) and (15). For comparison purposes, since the single excitation diagrams are excluded in the present MBW treatment, a corresponding Cl(D) energy has been obtained from the hamiltonian matrix of the original CI(SD) calculation. Since all the HF orbitals have been included in both the CI and MBPT treatments, the converged results are invariant to any transformation among the excited orbitals (such as would occur with a VNT1 potential [5,6] ), and therefore all calculations wcrc carried out in terms of Ihe canonical HF orbitals. A summary of the principal results and comparison with CI are given in table I_ The converged limit ofall double-excitation diagrams is 0.0127 hartree below the variational CI(SD) result and 0.0142 hartree below the CI@) v&e. This latter difference accounts for the unlinked diagrams in the CI(D) energy (which would be cancelled by quadruple and higher excitations), but also involves certain fourth- and higherorder: rearrangement diagrams, neglected here but in-
cluded in U(D), which would have reduced the mag nitudc of the correction. While not very farge in itself,
Table 1 Comparison of correlation -----
1 Scptcmbcr
CHEMlCAL PIIYSICS LCTTIXS
Volume 50, number 2
cncrg~~s (in hartrw) for Hz0 obtained __--_----__-_-_
by MBPT and CI methods ‘1 _-___---.------_M$m2r -Pll!w3 (unGiftcd)
_____
__-____-----------_-_------
-
second-order MBPT (E2) thrd-order MBPT (Ez + E3) fourth~rdcr MBPT (E2 + E3 + ~54) MWT (all orders in double-excitation third-order
variational,
sum of independent Cl(D) Cl CI(D) + cstimatcd
digrams)
-
Cpctcin-Nesbct (5illfkd) __ __--____-_--_____-
-__---
-O-34361(1 19.2) -O-26399( 91.6) -0.29832(103.5)
-O-28826(
-0.28826(100.0)
100.0) 1(
02.7)
--0 33944(117
pairs to iourth orde: b,
8) -0.27402(
quadrupics
correction
CI(SIz) c) CI(SD) + estimated quadruples correction estinlatcd total correlation energy ‘) -~-_-----_----_____-___________-_____-
----_-
-O-28178( 97.8) -0.28265( 98.1) -0.287 lO( 99.6) -0.2672
eq. (26)
1977
--
-0.26243( 9 1.0) - 0.34366(119.2) 95.1)
-0.28853(100.1) --0.27558 -0.29009 -0.370 + 0.003 __-__-________-___-_--_------
d, d)
‘1 Bdscd
on the 39-ST0 calculations of Rosenberg and Shdvltt [ 141 at the cquillbrmm gcomctry (R = I.81 1 hollr, 0 7 104.45O). The computed Ill- energy IS -76.06423, compared to an cstimatcd HI. hit ot -76.0675 + 0.00 10. lkrccnt;~gc~ rchtwc to the converged MUPT result are shown in parenthcrec b)Computcd from cqs. (9)-(12) and (14) and (15) with the pair rcstrlctwnnk/ =I] m (11) and k =I or k ‘1 in (12). c, Rcmlculated from Cl(SD) hamiltonian matrix of ref. [ 141. @ 1;rem Davidson’s quadruples wrrcction formula [ 161, based on Cl(D) [ 111. e) Ref. 1141.
thls correction
has a significant effect on the potential even for a molecule as small as Hz0 [ 151, and could become drastic for larger systems. The estimated surface,
quadruples correction by Davidson’s formula [16] based on the CI(D) wavefunction is -0.0 145 1 hartree [ 111, compared to our value of -0.0 1424 hartrce. Such excellent agreement is probably fortuitous, and cannot be cxpccted to hold for larger molecules, for which unlinked diagrams of higher-order excitations should become important. Although little is normally said about upper bounds in connection with MBYT, for the very good reason that an upper bound requires the introduction of unlinked cluster terms 141, an MBPT wavefunction exists, and a corresponding upper bound may bc obtained by the usual R&e&h-Ritz procedure. This requires an energy calculation to some odd order 2n + 1. The simplest possible upper bound is obtained from the third-order calculation, using the trial wavefunction
[423,241 %=ao+h\kl,
(25)
i?=Eo +E, +-iiE2,
(26)
x=(25)-9(y-l)+[(,y-l)2+4]“2}, y = &&.
(27)
s=
<*pkl>,
(28)
with Xtlzc oytimum value of the coefficient. With M&lcr-Plesset partitioning one obtains x= 0.348295, implying that the true perturbation wavefunction (X = 1) is also a good approximation. With the optrmum X, this sirnplc one-paramctcr wavefunction accounts for Q7.5% of the Cl(D) correlation energy (table 1). In the shifted cast xs = 0.763740, giving 95.8% of the Cl(D) result. The two-body contribution throu& fourth order is also given in table I_ This result is somowht greater in magnitude than the syrmnetry-aci~ptcd indepcndent electron pair approximation (IEPA) vaiue rcported for the same basis by Rosenberg and Shavitt [ 131, since the latter accounts for some of the positive three-body interactions as well as some rearrangcment diagram contributions. The convcrgcnce of-the MUPT scrrcs with the two different partitionings of the hamiltonian is compared in table 2. It is quite apparent that the M$llcrPlesset (MP) formulation has decidedly better convergcnce properties than the shifted (EN) vcrsiou. As has been the case in other calculations which took proper account of three-body and other many-body effects [9,11,26], tl le unshifted 6, is quite small, 105
Volume
50, number
2
CHEMICAL
1 Scptcrnber 1977
PIiYSICS LETTERS
T&k 2 Contnbutions to the correlation energy of 1120 from all double-cxcltation diagmns through twelfth order, and the corresponamg Pali? dpprowmants a) (in hartrec) -- - ___--_ _-_ _-_-.. --..-----_-___ ___---_.-- . - ---Order II
hlfillcr-Pleswt
-_
___---.
(unsh~ftcd) ._-
--.___
Epstein-Nesbet
--_--
(\hdtcd) _---.---
b,
Pad6 approxlmant Pad& approximant FS 2 I;-ic Sk.2 &;I -,I - _____. ______ k=l _ __ ____ ___--__---- - --.-_-_---- k=l --- --2 -0.281780 - 0.28178 -0.343608 -0.34361 3 -0.000867 -0.28265 f 1,Oj = -0.28265 +0.079622 -0.26399 [ I.01 = -0.27897 4 -0.004455 -0.28710 [l,l] = -0.28157 -0.034335 -0.29832 [ l,l] = -0.28798 5 --0.000650 -0.28775 [2,1] _=-0.2&796 +0.014854 -0.28347 [2.1] = -0.28795 6 -0.000311 -0.28806 [2,2] = -0.28817 -0.007006 -0.29047 12.21 = -0.28797 7 -0.000111 -0.28818 13.21 = -0.28825 +0.003276 -0.28720 13.11 = - 0.28825 8 -0.000048 -0.28822 [ 3.3) = -0.28826 -0.001567 -0.28876 [ 3,3] = -0.28826 9 -0.00002 1 -0.28824 [4,3] = --0.28826 +0.000745 -0.28802 [4,3] = -0.28826 I0 --(I 030009 -0.28825 [4,4] = -0.28826 -0.ono357 -0.28838 [4/l] 7 -0.28826 11 -0.000004 -0.28826 [5,4] = -0.28826 NJ.000 17 1 -0.28821 [5,4] = -0.28826 12 -0.000002 -0.28826 [5,5] = -0.28826 -0.000082 -0.28829 [5,5] = -0.28826 -._- -_ ___----___. ---._-__ -___----. ---_---a) ?‘hc cvnluntion of the Paci~ :Ipprowm.mts [hr. M-l] and [M,M] requires cncrgy contributions of order II = 2M+ 1 and n = 2M+ 2, rcspwtivcly, and they arc li\tetl 111the t.tblc in the corresponding rows. b, The cncraies obtained with the shifted dcnommator, ccl. (18). arc indicated lay superscript S.
while the shifted case has to have a large and positive Ez in order to begin to correct for the large ncgatlve overestimate in E;_ Not surprisingly, since Es is so small, E4 is an order of magnitude larger, though still only -0.0045 hartree (E4 is understood to refer only to the double-excitation diagrams in this paper)_ Fourth-order EPV rearrangement diagrams arc not included, since in MBPT these arc formally of “quadruple” excitation type. These do occur in U(D), and would contribute a positive value to the fourth-order energy. After &E4the terms in the MP series decrease monotonically. Convergence of the EN series is OScillatoiy and much less satisfactory. In both cases convergence is cortiidcrably improved by using the I’;&5 approximants. Five-figure accuracy is acli~cvcd with the [3, 2 1 approximant, obtained from a seventh-order calculation, with or without denominator shift, while the linear sum requires tenth order for the same accuracy in the MP case and more than twelfth order in the EN series. At the same time, the fourth-order approximant [ 1, 11 provides threefigure accuracy in the EN case, but only two figures for Ml’. This is because I E41 > lE31 in the Ml? series, and in fact the linear sum is considerably more reliable for MI’ at this point. As seen from eqs. (9)-(1 S), the computational effort depends almost entirely on the highest odd 296
--
order included, with practically no additional work being needed for the next higher even order. Noting that E3 + E4 account for 2% of the correlation energy in the MP series, while succeeding terms contribute only 0.4%, it appears that a fourth-order calculation, which requires no iteration, offers the best compromise between accuracy and computational requirements. Contributions of individual diagrams through fourth order for both the MP and EN series arc shown in table 3. The most interesting aspect of these numbers is that, even though the values of the individual diagrams are smaller in the EN than in the Ml? case, there is far more cancellation in the latter, resulting in a much smaller total contribution of each order than the value of any of the diagran’s. It is unlikely that this high degree of cancellation is accidental, so that it presumably must reflect a further, hitherto unsuspected summation property of the MP MBPT expansion in terms of HF c‘anonical orbitals. Furthermore, the contributions of the three-body and higher terms to each order of the MP series are invariably smaller in magnitude, and mostly opposite in sign, compared to the two-body part. This is in contrast to the EN case, for which the third-order three-body (ring) HP diagrams arc much larger than the two-body HP terms which survive the geometric series summation. This effect persists in fourth and higher orders, and it
total, all dragrams
total
HL-IIf’ twobody terms many-bodymms
tOtill
PL-HP two-bodyterms many-bodyterms
total
PL-HL two-bodyterms many-body terms
LOtal
ftofcpartrcfe (HP) two-bodyterms many-bodyterms
total
hole ladder (HL) twobody terms many-body terms
particfcbdder (PL)
--
Dngrums
-0.28178 --
,0.28178 l(A)
-0.00087
-0.09759 l(D)
-0.14003 to.04244
to.04444 l(C)
to.04094 to.00350
to.05228 l(B)
.lidlfcr-Plcssct (unshrftcd) --.---___ 2nd order 3rd order -_--
-----
-0.00446
to.0397 1 ?(G,fl)
-10.05107 -0.01136
to.04441 2(E,P)
-0.01636
to.06078
-0.01996 2(C,D)
-0.01812 -0.00184
-0.04672 2(1-L)
-0.08427 UI.03756
-0.00965 2(B)
-0.00806 -0.00147
-0.01225 2(A)
4th order
-0.34361
-0.34361
to.07962
to.03312
-0.03846 to.07 158
to.00533
0. to.00533
to.041 16
Kpstcm-Ncsbcl(shifted) ---_-_--____ 2nd order 3rd order ----__
to.00228
to.00228
0.
to.01 190 -0.01488
-0.00207
-0.00207
0.
-0.02324
-0.01630
-0.00694
-0.0006 1
-0.00061
0.
-0.00771
4th order
Table 3 Contributionsof indrvidualdragramsthrougfrfourth order to the correlationcncgy of Hz0 (in frartrec).Specifrcdragr~msarc rcfcrcnccd by ffgurcnumberand lettcr --____-_.._
Volume
50, number
2
CHEMICAL
PIIYSICS
is these diagrams which are largely responsible for the total contributicn of each order in the EN series. Since the sta,idard denominator shift sums only twobody diagonally scnttcrcc! &Grams, incorporating them into lower orders, no component of the thrccbody part of the HP diagrams is summed by the shift, leaving these terms disproportionately large. Actually, using doubly-occupied spatial orbit& and intcglating out the spin variables to give diagrams in terms of the spatial orbitals, an additional denominator shift can bc employed which would incorporate some of these three-body interactions, namely those components wkch involve the same spatial orbital with different spins (see ref. [41 for a related example). While this would improve the convergence of the shifted series, there appears to be little reason to use any denominator modification, since the unshifted MP scheme is apparently the best way to crnploy MBPT.
In f-act, a trivial second-order unshifted calculation provides a good estimate of the basis-set limit for the correlation energy, and this estimate is aimost certainly an upper bound. It also provides a two-body or pair correlation approximation, with its convenience of interpretation. For good molecular geometries and binding energies it is necessary to go beyond second order [9], but as shown here, the convergence of the MP series is sufficiently rapid that a fourth-order calculation is probably sufficient for most problems.
Acknowledgement WC arc grateful to Dr. B.J. Rosenberg for providmg the MO integrals and hamiltonian matrix of the 39ST0 $1 calculation for H,C). Useful discussions with Drs. CiYck, J. Paldus and W. Meyer are gratefully acknowlcdgcd. This research was supported l,y Battclle
Memorial
Institute.
References [I ] K.A. Brucckncr. [2] ?boIdstone.
Phys. Rev. 97 (1955)
1353: 100 (1955)
Proc. Roy. Sec. A239 (1957)
[ 31 H.P. KcIly. Advan. Chcm. Phys. I4 (1969) Rev. 131 (1963) [4)
198
684;
13GB (1964)
R.J. 13xtlctt and D.M. SJvcr, 9s (1975) 183.
Intern.
896;
267.
129; I’hys. 144 (1966) 39.
J. Quantum
Chcm.
LETTERS
1 Scptembcr
1977
[ 5 ] R.J. Bartlett and D.hl. Silver, J. Chem. Phys. 62 (1975) 3258;64 (1976) 1260.4578,Phys. Rev. A10 (1374) 1927;AI3 (1976) 912, Chcm. Phys. Letters 29 (1974) 199; 37 (1976) 198; lntcrn. J. Quantum Chem. 8s (1974) 271; IO (1976) 185. [6] D.hf. Sdvcr and R.J. Bartlett, Phys. Rev. AI3 (1976) 1. 171 U. Kaldor, Phys. Rev. A7 (1973) 427; I’hys. Rev. Letters 31 (1973) 1338;J.Chcm. Phys. 63 (1975) 2199; P.S. Stern and II. Kaldor. J. Chcm. Phys. 64 (1976) 2002. 181 D-1,. Frucman and bf. Karplus, J. Chem. Phys. 64 (1976) 2641. [9] J.A. Poplc. J.S. Bmklcy and R. Scegcr, Intern. J. Quanturn Chcm. 10s (1976) 1. 1101 J. PaIdus and J. &!ck, Advan. Quantum Chcm. 9 (1975) 105. IllI R.J. Bartlett and I. Shavitt. Intern. J. Quantum Chcm. 11 S, to bc published. 1121J. ci%k, J. Chcm. Phys. 45 (1966) 4256; Adva1. Chem. 1’11~s. 14 (1969) 35; J. C&k and J. PaIdus, Intern. J. Quantum Chcm. 5 (1971) 359. J. P.rIdus, J. &.ck and I. Shavitt, Phys. Rev. A5 (1972) 50. B.J. Rosenberg and I. Shavitt, J. CI1cm. Phys. 63 (1975) 2162. B.J. Rosenberg, W.C. EnnIcr and 1. Shavitt, J. Chem. l’hy\. 65 (1976) 4072. E.R. Davidson, in: The world of qudntuni chemistry, eds. R. Daudcl and B. Pullman (Reid& Dordrecht, 1974) p. 17; S.R. Langhoff and E.R. Davidson, Intern. J. Quantum Chem. 8 (1974) 61. [ 171 R.J. Bartlett and D.M. Sdvcr, In- Quantum science, cds. J.-L. Calaip, 0. Coscinski. J. Linclcrbery and Y. 6hm (I’lcnum Press, New York, 1976) p_ 393. C. hfgller and M.S. Plcrtet. Phys. Rev. 46 (1934) 618. P.S. Epstein. Phyx Rev. 28 (1926) 695; R-K. Ncsbct, Proc. Roy. Sot. A230 (1955) 312. P. Cinvurrc. S. Diner and J.P. hfdr~eu, Intern. J. Quantum Chcm. I (1967) 751. C.-A. Baker Jr., Advan. Thcor. Phys. 1 (196.5) 1; III: The Pad6 upproximant m thcoret1cal physics, eds. G.A. 13.lkcr Jr. and J.L. C;arnmcl (Acalernic Press. New York, 1970). 0. C;oscmskl, Intern. J. Quantum Chcm. 1 (1967) 769; I:. Rrandas .rnd 0. Coscinskr, Phys. Rev. Al (1970) 52; R.J. Bartlett and E.J. Brandas. J. Chcm. Phys. 56 (1972) 5467; 59 (1973) 2032. 0. C;o\cmski and I-.. LZr3ndas. Chem. Phy5. Letters 2 (1968) 299. L.S. Cedcrbaum, K. Schonhammer and W. von Niesscn. Chcm. Phys. Letters 34 (1975) 392. R I). Mattuck. A k.gtde to I’cyrunan dragrams ir. the manyhody problem (McGraw-tIiI1. New York, 1967). S. WiIson. D-hi. Silver and RJ. Bartlett, Mol. Phys., to hc published; R.J. Bartlett. S. Wdson and D.M. Sllvcr. Intern. J. Quanium Chem., to be published.
50, number
2
CHEMICAL
PHYSICS
LETTERS
1 September
1977
COMPARISON OF HIGH-ORDER MANY-BODY PERTURBATION THEORY AND CONFIGURATION INTERACTION FOR Hz0 Kodncy J. BARTLETT Battrlfe
Cchmbr p Laboratories,
Cdumbus.
Ohio 43201.
USA
and Isaiah SMAVITT flatteffe Cofumbur Laboratories, Ciilumbus. Ohm 43201. USA uf!d ~epartl?wIt of Cflemiwy, Tile Ofuo State Utliventty, Cohnbus, Rcccivcd
01~0 43.201, USA
26 hlay 1977
IJi~~gr~mrn~trc m.my-body perturbatron theory, coupled with n rccursivc computabonal procedure. IS cnqlloycd to obt.Iin correlation cncrsy of Hz0 witbm a 39ST3 IXISIS set by evaluating all double-cxcttatlon diagldrns through twelfth order wallout any ~ppro~irn~ttons. Thr provides, for the first time, the complete double-eucitatton diagrams contrrbutron to the corrcklilon energy, wlucb IS -0.28826 hnrtrec, compared Hrlth ,I correlation cncrgy of -0.27402 hartrce obtamed from :I ~WI~I~UI~~IOII IrItcr.tctron calcuiatlon WIIILII ~ncludcs ,111tioublc excitattons. Ihe dlifercnce of 0.0142 hartrcc includes the corrcctIon to the ~ll_cioublc-cxclt,ltions Cl cnrr!Iy, rlu<~ to the “p3thdq:Ic:ll” unhnkcd+J1:l~rarn terms rcYIfC Con~1Stt’IlLy” mammg111 that result, but also involves certain lourtb- nnd highcr-ordcr rcnrmngcrncnt diLtgrams. Contrary to conventtondl bAci, the bnslIIfrcd, or hlglllcr- I’Iessct partmorung ot the IiamltonIm provide\ a much more rapld convergcncc of the pcrenhance the conturbatlon terlcs than does the shifted, or tpstcm-Ncsbct partitionm,. (1 In both cases, PadE approuirnantr vzrgcncc of the scws consldcr.lbly. A \mlplc vart,ition-pcrturbatlon scheme bawd on the first-order hlBPT wavcfunction I\ sufflctcnt to provide 97.5% of the all
1. lutroduction Mary-body methods, and e~pt~~ally the theory and application of diagrammatic many-body’perturbation theory (MBPT) [l-3], arc becoming increasingly cvidcn t in t!lc quantum chemistry literature These tncthods have several advantages over more cnnventional conf~gurntion interaction proccdurc. For example, MBFT permits the
[4- lo].
the (CI)
direct determination of excitation and ionization energes [I OJ, instead of requiring separate calculations for the in&d and final states. Furthermore, the absence from many-body theories of configuration functions, and thus of the complications resulting from their enormous nunl’,er, is 3 very useful fczturc. Another advantage is the suggestion of certain physical concepts, such as pair correlations and cluster expansions. But most important is the fact that the total energy 190
of a system can be defined by the linked cluster expansion [2]. This ensures the proper dependence of the energy on the size of the system [ I,2 1 (“size consistency” [9]), unlike the cabe for conventional, less ihan fit/Z Cl calculations. This last point is demonstrated most strikingly by the example of a system composed of N non-interacting identical subsystems. The correlation energy predicted by conventional Hartrcc-Fock plus all-doubleexcitations CI (denoted by the symbol U(D)) for such a system is proportional to A+‘* instead of IV as N-t m. In comparison with the linked cluster expansion of the exact energy [2,4,11], the CI(D) energy expression retains “pathological” unlinked diagram terms which increase quadratically or faster with N and thus destroy its siLe consistency. These terms would eventually be cancelled by higher excitation terms in furlCI, but unlike the case with MBPT, the pathological
Volume
50, number 2
CHCMICAL 1’FIYSICS LI-ITI-RS
terms must still be computed in the fulI CI, just to mutually in the final result. In the linked cluster expansion, on the other hand, the cancellation of the unlinked terms has already been incorporated within each order, due to contributions from different excitation levels in the CI sense 141. From another point of view, comparing the CI wzuefunction expansion with the exponential linked cluster form of the exact wavefunction in the coupled cluster approach 112,131, WC are led to the further conclusion that disconnected cluster terms, corresponding principally to independent simultaneous pair correlations (and giving rise to Zirzkcd cncrgy diagrams) are also missing in the Cl(D) expansion. These terms cm only be included in CI by adding higher quadruple and other even-excitation configurations, and the cluster analysis indicates that the most important disconnected contributions to the coefficients of theqc higher excitations can be expressed in terms of the coefficients oft he double excitation terms in the ficlf Ct solution 1133. Even in such a mundane application as the predlction of a potential energy surfiicc for d small ~nolcculc like I-120, a complctc single and double excitations Cl (CI(SD)) calculation retains significant errors [ 14,151. A simple correction for the effect of quadruple cxcitations [ 161, the leading term in the unlinked diagram error in small-molcculc CI(SD) [ 1 11, is enough to bring the resulting potential surface parameters to within cxpcrimental error limits in this case [ 151. This fairly crude cstimatc can harrlly be cxpcctcd to apply in general [! 11, while MUPT provides sbeconsistent results ngorously. (In this connection it shouid bc noted that conventional MBPT, based on a restricted single-configuration zero-or&r function, has difficulty 111describing the dissociative regions of the potential energy surface. This difficulty can usually be avoided by employing an unrestricted SCF reference function. The alternative approach of using a multiconfiguration rcfercncc function appears to be more difficult to implcmerlt in MBPT than in CL) With modern day quantum chemistry beginning to attain high levels of accuracy, at least for small molecules, it is now meaningful to proceed to a correct and systematic treatment of the energy that properly cxcludcs these unlinked energy diagram effects. Such a treatment can be accompli&cd by many-body pcrturbation theory, but in order to do this meaningfully car~cel
1 Scptcmbcr
1977
it is important that no appreciable errors be introduced by arbitrary truncations of the perturbation series. Unjustified neglect of higher-order terms rvay in fact introduce errors which arc not insignificant in comparison with the unlinked diagram errors which we are trying to eluninate. Our intent, rather, is to present :I well-defined Ml3PT model, analogous to CI(D) in its simplicity, but possessing the desired sizcconsistency property_ In order to accon~pl~sl~ thcsc objectives, this pepcr employcs a previously denvcd (17 ] MBPT recursive procedure which permits the complctc evaluation of the contributions of all doublcexcitatlon diagrams, to all orders, and applies it without further approxunimon to a fiurtc basis tlcatnicnt of t!lc H,O molccult at cquilibiium (an application to the H’;O potential surface is under way). The energy obtamed in this treatment is rigorously si/e-consistent, and is a siguificant lmprovcment over CI(SD). Assuming that the dominant cant rlbutions of quad1 uple and higher even excitations in an extended Cl expansion arc to cancel the unlinked diagram errors of tbc double-excitations Cl, the result of this MI3PT ploccdurc should be cornparable to that of a CI expansion which includes such higher even excitation configurations. This treatment does not include the cffcct of disconncctcd clusters in the wave function [12,13 ] (principally ?i, which leac to linked quaclruple-excitation energy diagrams, of wluch sonic il~c CPV rcarrangcmcnt diagrams that arc accounted for in CI(D)), but these do not affect the SILC consistency of the present approach. The correctlon obtained here (the diffcrencc between the MBI’T double-excitation diagrams energy and the Cl(D) energy is in remarkab!e agreement with ~III estirnatc of the quadruple excitations m~provement of CI(D) proposed by DavIdson [ 1G] , which we have dcrlved from unlinked diagram considerations elscwhcrc [ 11 J. This paper also examines the relative I@-order MBPT convergence properties of the “unshifted”, or M&lcr -1’lessct [IS] partitioning of the hamlltonlan and the shifted Fpstcin-Nesbct [ 191 partltionnlg. Contrary to some previous expectations [3,5,6,20], it is found that the M$llcr-Plcssct paltltiorring is far superior, but cnhanccd convc~gcr~cc can bc :Ichievcd in both cases by the USCof Padd approximants [21,X?]. It is also demonstrated that a previously proposed [4,2324] variation-perturbation schernc based only on a first-order MBPT wavefunction with a sin& 191
Volume
50. number
CHEMICAL
2
variational parameter provides a quite accurate upper bound to the energy compared to the CI(D) result.
2. High-order diagrammatic
perturbation
I
LETTERS
September 1977
rules for the interpretation of the diagrams [4J (coupled with a specific ordering of the summation indices) give the algebrdc formulas:
theory
Assun$ng a Hartree-Fock (HF) reference ‘PO, the defining equations of MBPT arc: N=if()
PHYSICS
function (abIltj-X~llkfXklllob)
t V,
D(Qab)D(klab)
with Ui the IIF cffcctive one-electron potential and ci the orbital energy. The total energy is then given by the linked-cluster expression [1,2]
x
(cjllbkXkillac) -C D(k&zc)
-
--z&UC)
k*illbkXkjllacl
’
+ killakXkjllbc) D(kjbc)
1 ’
(6)
where D(iiab) = cl + ‘I - ca - ch,
where the subscript L indicates that only linked diagrams are to be included. Assuming IIF spin-orbitals as the single-particle basis, all the antisymmetric diagrams with nonvanishing contributions to the second and third order of MBPT are shown in fig. 1. They arc all of the double excitation type, because Brillouin’s theorem climinatcs the single excitation contributions in thesc’orders. The
(7)
‘and where i, j, k. I represent spin-orbit& occupied in +0, while a, b, c, d arc excited, or “virtual”, HF orbitals. The double bar indicates an antisymmetrized two-electron integral. Contributions to the fourth-order energy arise from single, double, triple, and quadruple excitation diagrams. Of these, the twelve possible antisymmetrized double excitation diagrams are shown in fig. 2. If the treatment is restricted to just double excitation diagrams in all orders (which implicitly includes the im-
.I
*I
diapmc which contribute to of MBPT when the singie-particle
rig. 1. AI1 the antisymmctrizcd
Fe- 2. All the antisymmetnzed double-excitation diagrams
the second and third order
which contribute
baas consists Gf Hartrcc--Fock
192
orbitalc.
to tbc fourth order of hIBPT wher. the singlo
particle basis cans&s of Hartrec-Fock olbitals.
CHEMICAL PHYSICS L13-I-TLKS
Volume 50, number 2
portant effect of quadruple and higher even excitations in Cl terminology), it becomes possible to evaluate all the resulting contributions readily. In fact, using a recursive procedure which employs the same computer programs as for second and third order calculations, the contributions of all double excitations to all orders can be evaluated to within any desired small tolerance. ’ A generalization of cqs. (4)-(6) to any order can be obtained in terms of {ab IIQ),I = (ab llQ),,/D(abij) where the pseudo-mtcgrals given by
(n 2 l),
(8)
(ub II+,, (Q > b, i >i) arc
(Ohllij)~ =
(9) (n 2 2),
tab ll~>,, = X,,(abij) + Y,,(abij) + Z,,(abij) with (for )I > 1, a > h, i >j) X,+,(obij)
= CFd (ah llcd){cdllzj~,,
(10)
Y,+,(ub@
= kil
(11)
(ijlltW{ub
Ilk&,
Z,,+l(aW) = c
[(kb Ilic~{ca IlikI,,
- (ka Ilic) {cb
IIik),,
k.c
+ (ka Ilic){cb Ilik),r].
The pseudo-integral notation is appropriate, may easily be seen from eqs. (9)-( 12) that {ab Ilv>,, = -(ab Ilji),, = -(buIlij),,
=
(ha
llji),,
(12)
bc used recursively to evaluate their contributions in higher orders. The three third-order and twelve fourth-order
double-excitation
diagrams are thus evaluated
sin&
of eqs. (9)-(12)
apphcation
(ab II ii>,, Iab Iliil,+. 1,
by a
(with tz = I),
followed by eqs. (14) and (15). A second evaluation of eqs. (9)-( 12), now with tz = 2, conveniently provides the contribution of the sixty distinct tifthorder double-excitation diagrams as well as those which arise in sixth order. Additional iterations can be pcrformcd until convergence within a desirable tolerance is reached. Alternatively, this procedure may be vlcwed as a generalization
of Brucckncr’s
k-matrix theory [X]
to include all double excitation diagrams, mstead of just the two-body or pair interaction terms. This can bc done in terms of an Opriltor K, dcfincd so that its antisymmetriled integral is the ultimate iteration of eqs. (9)-( 12):
+ k2
‘hb I[Cd){Cd(K Izj-&,
(i/iikl){kllK
kZb)A
since it + Aq((kbIljC)i
(13)
for any n, although not all the symmetry properties of the initial integrals (rz = 1) arc necessarily retained for II b 2. In terms of these quantities, we can write the double excitation contributions to a‘ny order (n > 2) of perturbation theory as
I>/ a>b
1977
Scptcmbcr
This may be demonstrated directly in terms of the algebraic equivalents of the fourth, fift!l, etc. order diagrams, although, as shown previously [ 171, this type of high-order recurrence is a necessary consequence of perturbation theory. In fact, there riced be no limitation to double-excitation diagrams, except that additional code would be required to evaluate the linked single, triple, and quadruple excitation diagrams in fourth alid fifth order, and this could then
- (kb Ilic){calljk),~
E&-I=Q
1
, .
{CQIK
+(kaIlic)[{cbl~lik)~
lik)A - {cb
IK
lik ‘}A1
- kvzl~lik~~l) (i >i,
a > 6).
Cl@
The braces indicate the incorporation of the dcnominator, as in eq. (8). As a result the sum total of all double-excitation diagram,
diagrams can be written a< a_singlc
(14)
(15)
where the antisymmetrized
K-vcrtcx is designated by 193
Volume 50. number 2
CHEllIlCAL PHYSICS LETTERS
the boldface line. However, since the evaluation of the (ab SQ>,,quantities is the time-consuming step in the procedure, an actual determination of the K-integrals acquires twice as many iterations for the same accuracy in the energy as required by eqs. (14) and (15), which exploit an analogue of the 2r1+ 1 rule of perturbation
theory and are thus decidedly preferable for computational purposes. A simpler type of summation conveniently used in MBPI’ involves some rearrangement of the terms in the perturbation series by using the technique of denominator shifts [3] _ This approach rccognizcs that certain terms, normally “diagonally scattered” terms such as occur in eq. (4) when c = a and d = b, form a geometric series, and incorporates their sums into lower-order terms. The most frequently used denominator shift has the form
[I%$M--
1 September 1977
I] =afA-'a,
(19)
[M, Ml =J$ +btB-l,,
(20)
.t = (E$?,
. . . EM,,),
(21)
b* = (Es&
. . . &+&
(22)
APq
= EP+q
- EP+q+l
Bpq = Epeq+ 1 - L:p+q+2
(~99 - 1,2, --., M-J
(23)
@,9 = 1,2, -.., Ml.
(24)
As seen from eqs. (23) and (24), the calculation of the approximants [& M - 1] and [M. M] rccluircs that the MBPT series be evaluated to orders 2M + 1 and 2hf + 2, respectively. The first approximant of each of the two types, [ 1,0] and [I, 1 ] , is equivalent to a geometric series sum based on the ratios E3/E2 and E4/E3, respectively.
D’(ijab) = D(ijab) - bb llab) - (qllij)
3. Caiculatiens + (aillai>+ lbil(bi) + (ajllaj) + lbillbl’),
(W
and the resulting formalism is equivalent to the Epstein-Ncsbct partitioning of the hamiltonian. The shifted denominator can be used nearly as easily in place of D(ijab) in cqs. (Q-(8), cxccpt that the surnrnation must then be rcstrictcd to exclude the terms with cxi = ub, kl = ij, and kc = ia, ja, ib, or jb, rcspcctivcly. Since in the present recursive approach these disgonal contributions (as well as the non-diagonal parts of the diagrams) ‘arc computed, in effect, to all orders, such geometric series sums are included in the final result without need for denominator modification, but theconvergence rate of the recurslon will depend on the form used. The Epstein-Nesbet partitiorGig has been shown to be superior t-or some problerns 16,201, and is traditionally thought to be an improvement over the simple denominator. It is therefore intcrcsting also to report here a comparison of the relative convergence rate of the shifted and unshifted series through fligh order. Another approach to improving the convergence of the MBPT calculations is the USCof some nonlinear series summation method. In the present cassz we use matrix Pad6 approximants [Icr.M - I] and IM, M] constructed from the terms of the (shifted or unshifted) perturbation series [2 1,22 1. These are given by
194
and results
Extensive I-IF and CI studies of the II,0 molecule have been reported in previous work [14], providing the opportunity for pertinent and overdue comparisons with an MBPT’ calculation_ The same integrals over the EIF orbit& as used in the CI study (with the 39-ST0 basis set) have been employed here to evaluate the MBPT diagrams from eqs. (9)-( 12) and (14) and (15). For comparison purposes, since the single excitation diagrams are excluded in the present MBW treatment, a corresponding Cl(D) energy has been obtained from the hamiltonian matrix of the original CI(SD) calculation. Since all the HF orbitals have been included in both the CI and MBPT treatments, the converged results are invariant to any transformation among the excited orbitals (such as would occur with a VNT1 potential [5,6] ), and therefore all calculations wcrc carried out in terms of Ihe canonical HF orbitals. A summary of the principal results and comparison with CI are given in table I_ The converged limit ofall double-excitation diagrams is 0.0127 hartree below the variational CI(SD) result and 0.0142 hartree below the CI@) v&e. This latter difference accounts for the unlinked diagrams in the CI(D) energy (which would be cancelled by quadruple and higher excitations), but also involves certain fourth- and higherorder: rearrangement diagrams, neglected here but in-
cluded in U(D), which would have reduced the mag nitudc of the correction. While not very farge in itself,
Table 1 Comparison of correlation -----
1 Scptcmbcr
CHEMlCAL PIIYSICS LCTTIXS
Volume 50, number 2
cncrg~~s (in hartrw) for Hz0 obtained __--_----__-_-_
by MBPT and CI methods ‘1 _-___---.------_M$m2r -Pll!w3 (unGiftcd)
_____
__-____-----------_-_------
-
second-order MBPT (E2) thrd-order MBPT (Ez + E3) fourth~rdcr MBPT (E2 + E3 + ~54) MWT (all orders in double-excitation third-order
variational,
sum of independent Cl(D) Cl CI(D) + cstimatcd
digrams)
-
Cpctcin-Nesbct (5illfkd) __ __--____-_--_____-
-__---
-O-34361(1 19.2) -O-26399( 91.6) -0.29832(103.5)
-O-28826(
-0.28826(100.0)
100.0) 1(
02.7)
--0 33944(117
pairs to iourth orde: b,
8) -0.27402(
quadrupics
correction
CI(SIz) c) CI(SD) + estimated quadruples correction estinlatcd total correlation energy ‘) -~-_-----_----_____-___________-_____-
----_-
-O-28178( 97.8) -0.28265( 98.1) -0.287 lO( 99.6) -0.2672
eq. (26)
1977
--
-0.26243( 9 1.0) - 0.34366(119.2) 95.1)
-0.28853(100.1) --0.27558 -0.29009 -0.370 + 0.003 __-__-________-___-_--_------
d, d)
‘1 Bdscd
on the 39-ST0 calculations of Rosenberg and Shdvltt [ 141 at the cquillbrmm gcomctry (R = I.81 1 hollr, 0 7 104.45O). The computed Ill- energy IS -76.06423, compared to an cstimatcd HI. hit ot -76.0675 + 0.00 10. lkrccnt;~gc~ rchtwc to the converged MUPT result are shown in parenthcrec b)Computcd from cqs. (9)-(12) and (14) and (15) with the pair rcstrlctwnnk/ =I] m (11) and k =I or k ‘1 in (12). c, Rcmlculated from Cl(SD) hamiltonian matrix of ref. [ 141. @ 1;rem Davidson’s quadruples wrrcction formula [ 161, based on Cl(D) [ 111. e) Ref. 1141.
thls correction
has a significant effect on the potential even for a molecule as small as Hz0 [ 151, and could become drastic for larger systems. The estimated surface,
quadruples correction by Davidson’s formula [16] based on the CI(D) wavefunction is -0.0 145 1 hartree [ 111, compared to our value of -0.0 1424 hartrce. Such excellent agreement is probably fortuitous, and cannot be cxpccted to hold for larger molecules, for which unlinked diagrams of higher-order excitations should become important. Although little is normally said about upper bounds in connection with MBYT, for the very good reason that an upper bound requires the introduction of unlinked cluster terms 141, an MBPT wavefunction exists, and a corresponding upper bound may bc obtained by the usual R&e&h-Ritz procedure. This requires an energy calculation to some odd order 2n + 1. The simplest possible upper bound is obtained from the third-order calculation, using the trial wavefunction
[423,241 %=ao+h\kl,
(25)
i?=Eo +E, +-iiE2,
(26)
x=(25)-9(y-l)+[(,y-l)2+4]“2}, y = &&.
(27)
s=
<*pkl>,
(28)
with Xtlzc oytimum value of the coefficient. With M&lcr-Plesset partitioning one obtains x= 0.348295, implying that the true perturbation wavefunction (X = 1) is also a good approximation. With the optrmum X, this sirnplc one-paramctcr wavefunction accounts for Q7.5% of the Cl(D) correlation energy (table 1). In the shifted cast xs = 0.763740, giving 95.8% of the Cl(D) result. The two-body contribution throu& fourth order is also given in table I_ This result is somowht greater in magnitude than the syrmnetry-aci~ptcd indepcndent electron pair approximation (IEPA) vaiue rcported for the same basis by Rosenberg and Shavitt [ 131, since the latter accounts for some of the positive three-body interactions as well as some rearrangcment diagram contributions. The convcrgcnce of-the MUPT scrrcs with the two different partitionings of the hamiltonian is compared in table 2. It is quite apparent that the M$llcrPlesset (MP) formulation has decidedly better convergcnce properties than the shifted (EN) vcrsiou. As has been the case in other calculations which took proper account of three-body and other many-body effects [9,11,26], tl le unshifted 6, is quite small, 105
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PIiYSICS LETTERS
T&k 2 Contnbutions to the correlation energy of 1120 from all double-cxcltation diagmns through twelfth order, and the corresponamg Pali? dpprowmants a) (in hartrec) -- - ___--_ _-_ _-_-.. --..-----_-___ ___---_.-- . - ---Order II
hlfillcr-Pleswt
-_
___---.
(unsh~ftcd) ._-
--.___
Epstein-Nesbet
--_--
(\hdtcd) _---.---
b,
Pad6 approxlmant Pad& approximant FS 2 I;-ic Sk.2 &;I -,I - _____. ______ k=l _ __ ____ ___--__---- - --.-_-_---- k=l --- --2 -0.281780 - 0.28178 -0.343608 -0.34361 3 -0.000867 -0.28265 f 1,Oj = -0.28265 +0.079622 -0.26399 [ I.01 = -0.27897 4 -0.004455 -0.28710 [l,l] = -0.28157 -0.034335 -0.29832 [ l,l] = -0.28798 5 --0.000650 -0.28775 [2,1] _=-0.2&796 +0.014854 -0.28347 [2.1] = -0.28795 6 -0.000311 -0.28806 [2,2] = -0.28817 -0.007006 -0.29047 12.21 = -0.28797 7 -0.000111 -0.28818 13.21 = -0.28825 +0.003276 -0.28720 13.11 = - 0.28825 8 -0.000048 -0.28822 [ 3.3) = -0.28826 -0.001567 -0.28876 [ 3,3] = -0.28826 9 -0.00002 1 -0.28824 [4,3] = --0.28826 +0.000745 -0.28802 [4,3] = -0.28826 I0 --(I 030009 -0.28825 [4,4] = -0.28826 -0.ono357 -0.28838 [4/l] 7 -0.28826 11 -0.000004 -0.28826 [5,4] = -0.28826 NJ.000 17 1 -0.28821 [5,4] = -0.28826 12 -0.000002 -0.28826 [5,5] = -0.28826 -0.000082 -0.28829 [5,5] = -0.28826 -._- -_ ___----___. ---._-__ -___----. ---_---a) ?‘hc cvnluntion of the Paci~ :Ipprowm.mts [hr. M-l] and [M,M] requires cncrgy contributions of order II = 2M+ 1 and n = 2M+ 2, rcspwtivcly, and they arc li\tetl 111the t.tblc in the corresponding rows. b, The cncraies obtained with the shifted dcnommator, ccl. (18). arc indicated lay superscript S.
while the shifted case has to have a large and positive Ez in order to begin to correct for the large ncgatlve overestimate in E;_ Not surprisingly, since Es is so small, E4 is an order of magnitude larger, though still only -0.0045 hartree (E4 is understood to refer only to the double-excitation diagrams in this paper)_ Fourth-order EPV rearrangement diagrams arc not included, since in MBPT these arc formally of “quadruple” excitation type. These do occur in U(D), and would contribute a positive value to the fourth-order energy. After &E4the terms in the MP series decrease monotonically. Convergence of the EN series is OScillatoiy and much less satisfactory. In both cases convergence is cortiidcrably improved by using the I’;&5 approximants. Five-figure accuracy is acli~cvcd with the [3, 2 1 approximant, obtained from a seventh-order calculation, with or without denominator shift, while the linear sum requires tenth order for the same accuracy in the MP case and more than twelfth order in the EN series. At the same time, the fourth-order approximant [ 1, 11 provides threefigure accuracy in the EN case, but only two figures for Ml’. This is because I E41 > lE31 in the Ml? series, and in fact the linear sum is considerably more reliable for MI’ at this point. As seen from eqs. (9)-(1 S), the computational effort depends almost entirely on the highest odd 296
--
order included, with practically no additional work being needed for the next higher even order. Noting that E3 + E4 account for 2% of the correlation energy in the MP series, while succeeding terms contribute only 0.4%, it appears that a fourth-order calculation, which requires no iteration, offers the best compromise between accuracy and computational requirements. Contributions of individual diagrams through fourth order for both the MP and EN series arc shown in table 3. The most interesting aspect of these numbers is that, even though the values of the individual diagrams are smaller in the EN than in the Ml? case, there is far more cancellation in the latter, resulting in a much smaller total contribution of each order than the value of any of the diagran’s. It is unlikely that this high degree of cancellation is accidental, so that it presumably must reflect a further, hitherto unsuspected summation property of the MP MBPT expansion in terms of HF c‘anonical orbitals. Furthermore, the contributions of the three-body and higher terms to each order of the MP series are invariably smaller in magnitude, and mostly opposite in sign, compared to the two-body part. This is in contrast to the EN case, for which the third-order three-body (ring) HP diagrams arc much larger than the two-body HP terms which survive the geometric series summation. This effect persists in fourth and higher orders, and it
total, all dragrams
total
HL-IIf’ twobody terms many-bodymms
tOtill
PL-HP two-bodyterms many-bodyterms
total
PL-HL two-bodyterms many-body terms
LOtal
ftofcpartrcfe (HP) two-bodyterms many-bodyterms
total
hole ladder (HL) twobody terms many-body terms
particfcbdder (PL)
--
Dngrums
-0.28178 --
,0.28178 l(A)
-0.00087
-0.09759 l(D)
-0.14003 to.04244
to.04444 l(C)
to.04094 to.00350
to.05228 l(B)
.lidlfcr-Plcssct (unshrftcd) --.---___ 2nd order 3rd order -_--
-----
-0.00446
to.0397 1 ?(G,fl)
-10.05107 -0.01136
to.04441 2(E,P)
-0.01636
to.06078
-0.01996 2(C,D)
-0.01812 -0.00184
-0.04672 2(1-L)
-0.08427 UI.03756
-0.00965 2(B)
-0.00806 -0.00147
-0.01225 2(A)
4th order
-0.34361
-0.34361
to.07962
to.03312
-0.03846 to.07 158
to.00533
0. to.00533
to.041 16
Kpstcm-Ncsbcl(shifted) ---_-_--____ 2nd order 3rd order ----__
to.00228
to.00228
0.
to.01 190 -0.01488
-0.00207
-0.00207
0.
-0.02324
-0.01630
-0.00694
-0.0006 1
-0.00061
0.
-0.00771
4th order
Table 3 Contributionsof indrvidualdragramsthrougfrfourth order to the correlationcncgy of Hz0 (in frartrec).Specifrcdragr~msarc rcfcrcnccd by ffgurcnumberand lettcr --____-_.._
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PIIYSICS
is these diagrams which are largely responsible for the total contributicn of each order in the EN series. Since the sta,idard denominator shift sums only twobody diagonally scnttcrcc! &Grams, incorporating them into lower orders, no component of the thrccbody part of the HP diagrams is summed by the shift, leaving these terms disproportionately large. Actually, using doubly-occupied spatial orbit& and intcglating out the spin variables to give diagrams in terms of the spatial orbitals, an additional denominator shift can bc employed which would incorporate some of these three-body interactions, namely those components wkch involve the same spatial orbital with different spins (see ref. [41 for a related example). While this would improve the convergence of the shifted series, there appears to be little reason to use any denominator modification, since the unshifted MP scheme is apparently the best way to crnploy MBPT.
In f-act, a trivial second-order unshifted calculation provides a good estimate of the basis-set limit for the correlation energy, and this estimate is aimost certainly an upper bound. It also provides a two-body or pair correlation approximation, with its convenience of interpretation. For good molecular geometries and binding energies it is necessary to go beyond second order [9], but as shown here, the convergence of the MP series is sufficiently rapid that a fourth-order calculation is probably sufficient for most problems.
Acknowledgement WC arc grateful to Dr. B.J. Rosenberg for providmg the MO integrals and hamiltonian matrix of the 39ST0 $1 calculation for H,C). Useful discussions with Drs. CiYck, J. Paldus and W. Meyer are gratefully acknowlcdgcd. This research was supported l,y Battclle
Memorial
Institute.
References [I ] K.A. Brucckncr. [2] ?boIdstone.
Phys. Rev. 97 (1955)
1353: 100 (1955)
Proc. Roy. Sec. A239 (1957)
[ 31 H.P. KcIly. Advan. Chcm. Phys. I4 (1969) Rev. 131 (1963) [4)
198
684;
13GB (1964)
R.J. 13xtlctt and D.M. SJvcr, 9s (1975) 183.
Intern.
896;
267.
129; I’hys. 144 (1966) 39.
J. Quantum
Chcm.
LETTERS
1 Scptembcr
1977
[ 5 ] R.J. Bartlett and D.hl. Silver, J. Chem. Phys. 62 (1975) 3258;64 (1976) 1260.4578,Phys. Rev. A10 (1374) 1927;AI3 (1976) 912, Chcm. Phys. Letters 29 (1974) 199; 37 (1976) 198; lntcrn. J. Quantum Chem. 8s (1974) 271; IO (1976) 185. [6] D.hf. Sdvcr and R.J. Bartlett, Phys. Rev. AI3 (1976) 1. 171 U. Kaldor, Phys. Rev. A7 (1973) 427; I’hys. Rev. Letters 31 (1973) 1338;J.Chcm. Phys. 63 (1975) 2199; P.S. Stern and II. Kaldor. J. Chcm. Phys. 64 (1976) 2002. 181 D-1,. Frucman and bf. Karplus, J. Chem. Phys. 64 (1976) 2641. [9] J.A. Poplc. J.S. Bmklcy and R. Scegcr, Intern. J. Quanturn Chcm. 10s (1976) 1. 1101 J. PaIdus and J. &!ck, Advan. Quantum Chcm. 9 (1975) 105. IllI R.J. Bartlett and I. Shavitt. Intern. J. Quantum Chcm. 11 S, to bc published. 1121J. ci%k, J. Chcm. Phys. 45 (1966) 4256; Adva1. Chem. 1’11~s. 14 (1969) 35; J. C&k and J. PaIdus, Intern. J. Quantum Chcm. 5 (1971) 359. J. P.rIdus, J. &.ck and I. Shavitt, Phys. Rev. A5 (1972) 50. B.J. Rosenberg and I. Shavitt, J. CI1cm. Phys. 63 (1975) 2162. B.J. Rosenberg, W.C. EnnIcr and 1. Shavitt, J. Chem. l’hy\. 65 (1976) 4072. E.R. Davidson, in: The world of qudntuni chemistry, eds. R. Daudcl and B. Pullman (Reid& Dordrecht, 1974) p. 17; S.R. Langhoff and E.R. Davidson, Intern. J. Quantum Chem. 8 (1974) 61. [ 171 R.J. Bartlett and D.M. Sdvcr, In- Quantum science, cds. J.-L. Calaip, 0. Coscinski. J. Linclcrbery and Y. 6hm (I’lcnum Press, New York, 1976) p_ 393. C. hfgller and M.S. Plcrtet. Phys. Rev. 46 (1934) 618. P.S. Epstein. Phyx Rev. 28 (1926) 695; R-K. Ncsbct, Proc. Roy. Sot. A230 (1955) 312. P. Cinvurrc. S. Diner and J.P. hfdr~eu, Intern. J. Quantum Chcm. I (1967) 751. C.-A. Baker Jr., Advan. Thcor. Phys. 1 (196.5) 1; III: The Pad6 upproximant m thcoret1cal physics, eds. G.A. 13.lkcr Jr. and J.L. C;arnmcl (Acalernic Press. New York, 1970). 0. C;oscmskl, Intern. J. Quantum Chcm. 1 (1967) 769; I:. Rrandas .rnd 0. Coscinskr, Phys. Rev. Al (1970) 52; R.J. Bartlett and E.J. Brandas. J. Chcm. Phys. 56 (1972) 5467; 59 (1973) 2032. 0. C;o\cmski and I-.. LZr3ndas. Chem. Phy5. Letters 2 (1968) 299. L.S. Cedcrbaum, K. Schonhammer and W. von Niesscn. Chcm. Phys. Letters 34 (1975) 392. R I). Mattuck. A k.gtde to I’cyrunan dragrams ir. the manyhody problem (McGraw-tIiI1. New York, 1967). S. WiIson. D-hi. Silver and RJ. Bartlett, Mol. Phys., to hc published; R.J. Bartlett. S. Wdson and D.M. Sllvcr. Intern. J. Quanium Chem., to be published.