An efficient method for large scale configuration interaction calculations

CHEMICAL PHYSiCS LETTERS

3 .Volumc 32, number 3

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.‘. AN EFkENT

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A. SEGAL, and Ross W. WETMORE

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.!Jgdltem &lifoM$.

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: Received 26 Nokmber

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METHoD FOR LARdE SCtiE CONFIG~~O~

iIi~fi&f of Chethrv, Lhiuws22y. of Las Angeles, California 90007, USA

1 May1975.

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An efficierit method of hkndlioglargc

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scale confguation

calculations

is developed and-applied

to the

able [I] for diago&Iii3ng~matrides of order several

?.. Introduction

-. In dealing Mth theoretjcal calculations on any .. physical problem, it is’frequently worth asking what accuracy one really need5 or wants to attain. This de- .’ termined, c+e can often. design a method which takes .’

’

interaction

fi2O.moiewIe as a test casti: The method, which is based upon matrix partitioning, is shown to be &pable of calculating “2 ‘Bt s~ctrum of H20 to ti accuracy level of 0.1 eV for each state with ve& mode& comp&atio+ effort.

‘advdntageqf the mathematical peculiarities of the :

tiethod whichallows the solution of the CI problem

problem a; harid in order to e?o‘nomicaIly ;olve it to the desired raCher than to arb&ary accuracy. In this

cf arbitrary amension to aboutthe 10e3, au level of &curacy without the necessity of eqkci~ hawIling of any large matrices. Thi! capabili!y then allows one to include entir: classes of deierminentai functions at

*ofinaction, it is worth noting that 1 l&/mole = : eV= 0.0016 au sd that an accuracy level of. abo+10L3 au’is aA extremely useful one for quarittun.che.tical cakulations of many typ,es. Th&, for instance, is certainly beyond the accuracy necessary ‘for the theoretkl assignment oflopticals$ctra and .-approaches thk level of accuracy, with resp&t to th& corrktion problem, &.ich is useful in de#ng with. problems on ri chen-+ca.,energy &aje. _ 0.043

The. ixost

StraightfGcrward,

,but brute

force,

reasonable cost with&t the necessity of excluding the large nukber of functions whose individual importance is smaU, but whosk cumulative effect on the energy is often significarit. The method is based on an extention of the matrix partitioning technique first int‘roduced by’ tiwd&‘[2], with suitabkapproximations suggzsted by Brillouin-Wigner perturbation ,.

approach’

theory

either the blectroniccorrelationenergy or the c& :. ‘ctition of the energies,gndyavefunctions of elect& follo&ed.by extensive configuraiion inter- ‘. ,’ .:&ti&.b=etween dettr&&ntal functions built from I ‘: ”

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,. .,

: with y&

br&

matrices &hen the cbnvergen’ti

of the-

yfie CI Gaveftiction .:

:4Jnfortun&ely;%his lea@ to the FFces$iv of d&&ii ;

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to +e problem

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2. Theory

nic.extited st&es,is,the calculation,of,ti SkF wave- , MO’s;

as ap$.ied

at hand.

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-tu

fie,re&tig

thousand to. arbitraiy a&racy, their utilization im.plizs the,use of a very large region of computer core storage and/or consideiable input-output cost in dealing with data stored on discs; In this note we suggest and report the tes$ng of a

‘. :

,’ .,

.. ’ ’ -, .f&la mCllecular state, wl-k.

based upon a precedingSCF &lcuJation, can b.echaracfetized as:$atig.a re@ely small number of.

.’

“V&&e

32,nuqber’3

1’

‘_.. whd;re.titi

C~~~CA~~~S~~S

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class&i

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and b GCXQXii~

,Su~sti~~g

~~~.~~~~~~~

and

imtrix

‘HQbD

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-%i~o CQ= WC,,.

-1

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(@-ib~&~-k.~=~Ca.

ofform,

(7)

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(88)

.

third or+3t corrections, etc. : in this note we &iJlconsider OR&Jthe fits: two terms.

under the assumption that the inverse exists. Stibstitutian of (3) i&o (2a) g&es -. Haa G - t-b, W,

(GI)OD

I’

mtiber

and the se&n& of the& solved for Cb’

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dominant portion of the eigerivecror C, of %e Iarge of small te&s in the b block. Another useful interpreta tiori‘arising from e&rrGn&on of successi;le terms in the expansion of V(a) is given by ~rillouinWgner (BW) ~~~rb~tio~.tlleo~. The F&t term is exactly the second order ~~correc~an~ to the ablock from’ali,b deterninental functions, the secbnd terns

,, @a) CZb)

c,=--pbb --i~t]-~HITCH,

L MS!iY L97.5

: wliich can, ti $rinciple at least, bt~iterstively soived . . for w. V(a) represents the eff@t an o,tid 0~ the

‘.

.. @$,a,C,-+ &b cb = 0, ch,

‘-1

[~~+v(w~~.c-_wc,

0f.M

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iii {4),‘one ha8

Thisis anequation

Eq. (1) can be rewritten as two matrix eqimions:’ ~,,C&-i,,C*=uC,,

‘.

lioEl co Cti& ,U-‘(c%?~.H&~iqa .’

.pre$enting the functions in the tail, may be of erder tiw~sands. The reprexda$xx (1) asdumes i&t &c dominant terms have‘ been dktermined f&the problem at’hand. and have been giouped in the upper left COP in tie

,,

‘_

mi+ terms respectively. ?Yhe~~~to~i~ subrn~t~, l-4, will &n be small, where& the matrix H&ice.

ner

LET’I’ERS

gives

AU matrices in (7) are ,of .the order of an a X a matrix,

.‘(4)

and, while &themost cctmpieti term still involves the very large i5 X b matrix 0, it need mwr he store& smaller,a block. ifit’is’contracted before and after by H,&D-‘(u) as it is formed’to LieId a matrix of order a. IF this could since the’inverse matrix [Hb6 - wt]-l WiIlin general be at least as difficult to obtain B ~ag~nal~~ti~n of .. be dotie on&for all w, then tie E31problem would : ‘rhe origid problem and its enerm dependence pres‘redu&.t,o forming l&e contracted .correcf.ianmatrix enisanad~tion~ complic~~onV(w); andthen sol+ir& the ~~~e~Ian~ problem for ,Tlse conjuration interaction piobIem,,wben qy state of the system it contains: l?iaving dete~n~d u, C, can then be’found from (3). ‘Qe dependence .. written in this form, has,certain properties that can.be’ of the coirection matrix on 0, lsowever, ret$reS that exploited to yield an efficient approach to the solu-. .‘. tion of (1). Let us assume that the ~1and b biocks.are the contraction be made on each iteration for each : so chosen.‘tltat the robots.of hterest, w, alllie fai f&m rfjot.actu&y solved:’ .’ ‘. : ‘. the.Ggenvaiues of Q.. [Mb&- lw] cannot, then, be ,‘, ,, This cm be avoide‘d. The energy u app&rs or@ in’ ‘. the dIa~0na.l matrix D-‘(W). SiiCe, by a%lmption, .sirigular a+d the,inverse must exist. F~~~a~ore, the :. .‘diagon$ terms h&b being far from w, tie rnatr+c -. the (I block,contains the dominant co~~~ra~o~~ for fH&b I Iu] .%%Itend.t? ‘be &gonally iiomiqktedi “. the states of Meres&a ieea+lf$e guess at ihe f?rd Nowimecantite .-, ene@es can be obtdned by diagon&ation of Ha03 IIiatrFx &pation af the.order of the 19t this stage, nckhin~ has been gained

whert?eq. (4) is a

.?-[I+)&

y- yli

= El(w)

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pr@&m.

nLc$ i,7ditidu;il

ctenents

if

f+(,j

ex$mded about’,a convenient .zer& 0~ “. ., ~:,deyro’ot, c&, in ternis of Aw, 7 (wo.-- w) [email protected] can theii.d&

: : where [-D(c.i)] “is‘%diag&kl rhttic&&i,@the 1.: odependenttemlsvf~e’~~matr4x:andQisasym-: “.; ‘, : metric, ,w ~.d~~~d~~t ,~t’& tvith a &I diag&& : ,,” “~-.Then,byexgansion[Z :,_.... j..,‘.:..‘.:,:: ,,, .. . ._ ., I,~... _, ‘_ ; .- ., .’ , -.I’:,.’ :-‘,I ,. ‘. :...,.: . _’ . ,. ,:. ‘... ‘. 1.’ .; ., ” :. ,_~ ._,,,’,. ). ‘; _‘_ ._ : ; :.., ‘. .: . _- ‘. ; :” .” ., ;_ ,,,,‘. .‘ : . . .:,. : ‘, : ” ;.:,,. :_,,, ‘. _. _. .

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Volume%. nun&r

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Substitution in (7) produces the final opertitipnal equation:

‘,

wherrs

(lla)

Vji’ = H,b’[D-‘(&,) f D-“(q)

0 D-l+,)]

..

H,,

: pertti&Giori. Those wilO& inter&ion rKasbeatei th& .a preset tluesho1.d (i 0W4 au)‘wer,e then added to,the ,.cbrebiock rdtaining only the largest.when the nunber,exceeded an arbitrary maximuin size, (40). Since. the exptijiori (6) is tincated after at most two terms, a conjuration which hasno matrix elements with this fmal block car-have no ~?~raction. M&e string6ntIy,

configurations

~4th &o second order in-

teracticJn larger than (10m6 au) were drop,ped from furthe;: consideration with no significant change in accuracy and consideiable savings in labor. It should

(Jlb)

perhaps be pointed qut ai this point that while theie

tid so on, form a convenient way of r&ainin,o~ti t.he .’ are no theoretical &fficulties in h&Uing degenerate necessary information while storing only n-i&trices.of functions, care should be exercised ihat all members the dimension a X a. Eq. (10) may now. bk solved are treated e&lly and not split between the II and b Iteratively with little additional effort.,pe use of (9) blocks.. While most’oi the calculations described is clearly a highly efficient modus o~randi prodded below were carried out at both the D-l and D-l0 the expansion can be held to Iow powers of Am. Q-l levels, it tias.quicMy found &at the additional ~oreovec, not only a‘single root, but any nuniber of ’ ‘.. effort expended to retainthe second term was generi ” ally not justified. roots may be obtained simult~eau~~y provided they lie witbin the radius of convergence of the expansion. The example chosen for ntim&ical testing &as the As may be seen,from the examples, both provisions water molecule. The basis set of contracted gautussians ‘ae adequately satisfied. was the (3s2p/Zs) ‘set of Dunning [S] aupented by &?modifi&d forms, eq. (10) has previously been 3s and 3p ST0 3G basis functions on oxygen (Slater used by severalworkers. Gershgon and Shavitt (31’ exponent 0.58) and by d single diffuse s gaussianon’ limited their treatment to the correction Hab D-‘(wo) each hydrogen (exponent’ 0.038). The SCF equations tfba, ~~~~~r a~p~Ga~i~n by,S~buya et al, [47, in their consideration of double excitation c~rrec~ons

weresolvedusingthe programpackageG-AU870[P]I au

to’the WA, ag& limits the correttion expansion to D-‘(o,,), with, however, the further neglect of all two electron integrals appeaiing in the diagonal nia-, Y : tr& elements. J-Iowever, from the results below, the key to accuracy in the method lies ti the iteration, ti ., pr&ess tide efficient by expansion (9). This is expected of course from the mtich greater ‘efficiency 96 .’ BWas compared to .XS perturqation theory. ‘.

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3. A practical approach t6 implemgntation :

The first,p;of$em that piesents itself is iiie prdper ’ ‘choice of t.5~ a block congelations. This’was done : : a~ltornatic~y for the lowest n roots by two successive tesb. (i),The diagonal dlements of the entire matrix. ‘were geu+ed &d t&e lowest n f 1 selected. (ii)Using Was atestcore, HJ, e!ements were getier@d andused toestimatd theintemctioll of.eachsuccessjve’determ$-,i / &&I fun@& by:secdtid order Ra;rieigh-S&.+linger _-

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The ground state energy ob~ned was -76.0108 with r& = 0.?572 a and LHOH = 104.85’. The

lowest and highest MO’s were then dropped from con; sideration and all.17 other &IO’s-were included in the. CIdalcul~tion. 35 procedure followed was.(l) automatic selec: ti6n of the a block, (2) diagonalization df this block to ,fid WO’,the lowest root of’the Ha, block, (3). formation of the contracted matrices of eq. (10) up to terms in A& (as &xplained below do3 is, in fact, sufficient for most c&es), and (4) it&ion of (IO), by succes$ve d.iago+izati& df the conecteda’blbck until a self-consistence ieye of 10m6 au was achieved. . C, w& dete&e$ by eq. (3). me first goal of the_ ,’ .. study t;vas,a determination.of the prospective numeri- .’ ti.ai;&raj of the method relative .to the true eigen-’ values of arbitrary C1 mat&es. To this end, a 429~di- : mensi&l matrix comprised of.& ‘AI- single and

double excit&qs.from

the SGF ‘@oui;dstate was -,

diagonalized by u+e of.the Shslvitt program [l] ,ti,Wai, a ,‘BI r+ti of dhnenSidn 312 chosen from the same ,, .__

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Volume 32, numb&r3

CHEMICAL PHYSI& LETTERS

L May 1975

Table 1 Test matrix results a) Total number of configuuntions

w

IA1 lB1

429 312

b,

312

-0.08275 0.24137 0.33259

State

(au)

u block

.-

-0.11775

D-‘0 D-‘.

exact

-0.11466

-0.11669

0.20977

0.20906

0.20813

0.29994.

0.30063

0,30030

a) Energies relative to SCF ground state energy (76.01076 au).

set of configurations. The lowest root of the ‘aI

as well as the first two roots of the ‘B1 matrices are displayed i&table I_ The approximation method was applied in a nurnber of stages. First the a block was chosen and diagonal&d This procedure exhausted the 40-dimensional limit in each case and the lowest root of each matriv. listed under a block in table 1, was chosen at, WO. Secondly, only the D-‘(w) correction of eq. (10) was included and the results iterated to self-consistency. This resulted in the bulk of the improvement of the approximate roots of w and the greatest error relative to the exact solution was 1 kcal/mole. The third order D-‘OD-’ correction was then included tid the results again iterated_ The geatest change due fo this term was 0.00309 au or = 2 kczl/mole, leading to a foal error of the aider of 1 kc-al/mole relative to the true roots of the matrices considered. The method was then applied to the calculation of the excitation energies to-the first 3 ‘B, states of H,O. The 429dimensional ‘A, results were used as a correlated ground state (all single and double excitations relative to thk closed shell SCF ground state) and the ‘Bl matrix was &nstructed

Table 2 Excitation energies (in e\‘) a) to ‘B1 states of H20 h) u block

D-1

&ptL [7]

9.67

7.53

7.40

11.20

9.89

10,oo

12.75

11.37

11.46

a) Energies relative to 429 ‘AI gra?lnd state of table 1 treated at the Same IeveL b) 1862 configurations with SO-dimensior.al~bIock_

The results for the first three ‘B, states given in table 2 with the c&responding experimental transitions are of gratifying accuracy and differ from experiment by at most 0.13 eV or 3 kcallmole, the anticipated level of accuracy. Convergence

of the Iterative step wzs essenti&’

immediate (2-4 iterations) in all cases. The norm of C’, was found io be about 0.02 or 98% of the wavefunction was thus found to be incIuded in C,.

4. Conclusions

from a &gk

and double excitations relative to the 3 lowest open shell ‘B, virtual orbital configurations. This then included selected triple excitations relative to the ground state and resulted in the examination of 1862 configurations. From the results of table 1, the third order D-‘OD-1 corrections are of the order of 1 kcal/mole.

Analogous to the usual behavior of perturbation theory, the’results of table 1 indicate the second-order term D-l lowers the energies obtained by diagonalization of the u block, while the third order terms D-lOD-’ tend t o correct these results upward toward the true’roots of the matrix. Large scale calcula-

and fundamentally

&tent behavior.of successive terms allows one to stop at D-l, in both the ground and excited state caIculation with fair cancelktion of errors and no signifi;

cancel errors at the D-l

level,

leading to a similar error above the true TOO? as was

obtaine‘d with the second order terms, but below the, true root. We therefore ljmited the large S&le calculation to the ,D-? level, an approxiqation which should lead to maximum attainable level of accuracywith a considerablk savin@ in labgi. :. ,-

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tions

.cant

show a sikilar’beh2tior.

loss of accwscy

truncation

at

This apparently

in the present

D-’ dbtiates

application.

the necessity

cdn-

The

of evaluating

elements within the 6 block Cth a corresponding savings,in time. The necessity of the

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