-.

Recekd 17 October 1983 . .-

Molecular gradients and hessiansfor multicohfigura~onal selfconsistent-field wavefunctions are derived in terms of the

generators of the unitary group using exponent@ un$aj.operat&rs to describe the “ponse of the enev to a geometrical

deformation. Final expressions are cast in forms .which contain reference only to the primitive non-orthogonal atom$ @$s_ s$

and to the final orthonoknal mole&lar orbitals; dl reference to interinediate ortho&onalii& orbitals is &moved. All of‘&

deformation-dependent t&&s in ihe w&king eqvations reside in the one- and two-electron inte&al dehbatives involving the atomic basis orbitals. The deiormation-independent teks. &hose contributions can be &tially~summed. inv&lve symmetrized

density matrix elements which have the same eight-fold index permutational symmetry as the one- and two-electron integral

derivatives they. multiply. This separation of deformation-dependent and -independent factors allows for single-pass integral-

deri~ative-driven implementation of the gradient and hessian expressions.

1. Introduction

In recent years quantum chemists have devoted much effort toward obtaining cotiputationally tractable analytical formulas for the gradiepts (forcesj, hessians (force constar@, and higher cnqgy derivatives appropriate to single Born_OppenheimeT electronic energy surfa&s. &h-knowledge is of -use in-locating

minima-and saddle points on-such-surfaces and characterizing, by lo&l normal-mode tibrational frequen- cies, these stationary-point g&metries.~ Integration of &ssical equations of motion also requires knowledge of the local force (energy gradient) and can be made evexi more efficient g&en higher energyideriv&ive information. .’

-Much of @e. history of_.$e dev@opment_ of -this reseeyh topig is_ re_FTw&$ by Pulay cl]__ The _ve@ sub+ntial tily contribktl$W&f ‘aetiait .and Mills- [i]; : Br_iroz [3]? +yer a.$ %lq [4], _Thq+Penl and Swansir0k [?I, and-Pk&ii 161 .hi@if -tie spelled outl+e@y in ref. [l].,_~~me~hatmo;‘;e_r&ze~t ~~~l~~~&s hav&b&n made-by Po~lk~~‘~;~[7],_ K&g Itid’ Dupi;iS’@];&ddtid:kt al. [$I;’ ~~p~~~~ii[lO];.Schlegel_& A_ : [ll], and Kom&ni&.i et _&[12].~?G ~most. re&nt developments have ~tres$ed the- +rt+&eiit of-_ypFn-shell c&es [4,7,9] . +d &&i;cblifigurati&I ~ti~<kf~&ti&s & t&&d -by- confi&&oti irit&&ioti _[‘7,9] (CL),

multicopfig~~a~&al~elf-~ns~~t~x$-f~eld [9,13-151 (MC SCG), tid ,MOllkrTPl&skt p:&ttibatioti &eOry [T] (MP~--te;cI&iq+.+ 1:-d::_: .:_i -_:--_ :_ -_:_:I --_.-_ 7. _: -_-._ -_ --. :; ~..I’.- -.T; -1 _‘_ :-__- ._- ;-_;;:. ..::_.-.,_z>_ ;_ _-_-,-:

Reknily. &g&s+ asid ‘S+dti [16,17] haye_gitie<_, ari$yti+-:exp&Go& t fck: t+ .e_rikgy g+Ii++t_ and _, he&an of- SCF, ~ MC SC& C&l MPPT; ti_d cou~le+Akste~‘@)- k@efiuiclio&~~; well- + _@+h+d and fc$rth- deri+ati&$Gf’ f’ihe;I$ SCF &Ft&.;I$Iay $~f_@o’ <~i$ly_d@i~_+~[l?] :fhe-v_$; ~CF_:tlJ&d:$kr&. __ de~v~ti~~,--~~-:~~Ij;.:~t--~~itl. _ -.. [ls] $$-!,IJelgakkx_aqd. -AI@_jf -:[1;8]_ _also~~~ade~.-u~~:,~~f :~exphcltly u’fifaj .- .: : .,, -*_. _. .j

0~01_0@4~&/~03,oP 0. I$skie~ S&&e -I&b&hers &ii. _ .- :: -i.

-!_: 1 --- -- I. .:

-(No-~~Holland-Physi~‘Publishing_r>~~iionj _:-_~{’ -.:- :_-‘.- _-. ~~-_--_~ ; _: :,:_ __:_ I -_. -___ -__.:.;-

_-

._ .-

204

transformations to describe how the molecular orbital { CPi} and configur&on ( Cg} m&g coeffi&& of the MC SCF wavefunction p) = C,CgQs vary hs the molecule’s nuclei are displaced from-F _t? -R-+ p. By_ using such a unitary formulation, these authors avoided the imposition, via Lagrange multipliers, of

constraints on variations of the { C,i} and { CX} which appear [l-14] in most earlier developtients. In our opinion, this makes the deriuarion of such energy derivatives more straightforward although, of course, the final results should be independent of the method of derivation.

It is a purpose of the present work to address the computational implementation of working MC &ZF and Cl gradient and hessian formulas. We also wish to make clear connection between the derivation of

Jorgensen and Simons (16,171 (JS) and that presented here. In particular we show how the symmetrically

orthogonalized atomic-orbital basis used in ref. [16] can be used as a convenient derivational tool but

removed from final working formulas in favor of the more convenient primitive atomic-orbital basis. Consideration of efficient computer implementation of the analytical expressions for energy derivatives

is very important because of the extremely large number of derivatives of atomic-orbital based integrals which arise in these expressions. Given n primitive (gaussian or Slater) atomic basis functions and a

molecule with N nuclei which are permitted to move. one has of the order of 12 x n4 first derivatives of the

atomic-orbital (AO) two-electron integrals and 78 x n4 second derivatives_ Each two-electron integral over atomic orbitals localized on atomic centers involves at most four centers. Thus, the only derivatives which give non-zero results when operating on such an integral are the X, y or z coordinates of the four atoms. There are twelve such coordinates. Such integral derivative arrays are, except for quite small basis sets, too large to be conveniently computed and stored even on modem high-speed high-volume disks. As a result,

strategies for impIementing energy gradient, hessian, and higher energy derivatives must, where possible, make use of these integral derivatives as they are generated_ That is, in our opinion, successful computa- tiona! schemes must be single-pass integral-derivative driven.

2. Derivation of working equations for the MC SCF case

2.1. The effect of nuclear displacement on the rvaoefunction

We begin by supposing that the MC SCF wavefunction IO) at geometry R has had its energy made stable with respect to variations in the expansion coefficients { CPi) of its molecular orbitals (MOs) { &} and in its configuration mixing amplitudes {C, }. The wavefunction IO) at the displaced geometry R + p is described

in terms of two unitary operators [exp(iK) and exp(iS)] operating on lo)

IO> = exp(iK) exp(iS) 10) _ (1)

The first operator exp(iS) brings about changes [16,19] in the configuration mixing coefficients of IO). The

exp(ih-) operator, when acting on each configuration 0, in (0) = C,,, g .4, ” C @ achieves a unitary transforma- tion among the MOs appearing in @, and all other MOs in the basis being employed_ These two operators

K=iCK,(e,--ee,,), S = i C X(lnXOl- lo>(nl). (2) rss PI+0

contain parameters (K_, S,) whose values are chosen to make the energy of the displaced wavefunction jo) stable. In eqs. (2), the ers are unitary generators given in terms of the MO-based spin-orbital creation ar’, and annihilation CI,, operators as

e b-s = ar+oasa + a&asp, (3)

2.2. Response of the energy to changes in geometi, -_

The second quantized expression for the electron hamiltonian &L) at R 4 p can be used together with

eq. (1) to express the energy at R + p:

Using the well known Baker-Campbell-Hausdorf expansion of the two exponential operators, one obtains an expansion for E in powers of the K, and S,, parameters (which, for notational ease, we collect together as

a single set {K,, S, > = ( Ai 1):

E(X,~)=(O~H(~)(O)+~;~(JL)++~;G~~(~)~~+ ---+&$&K,(p)+ ..- - (6)

Here, the Einstein summation convention over i, j, k is assumed. The Brillouin 4, hessian Gii and superhessian Kjik matrices * are defined in terms of commutators of the operators (T} = ( ers - esr,l~z)(Ol

- p)(nl} with the hamiltonian n(p)

Eq. (7) displays the p and Xi dependence of the MC SCF energy. Choosing the Xi parameters to make E stable @E/i%,= 0) gives rise to a set of equations which can be used to solve for the optimal Xi as functions of p:

0 = c. + GiiAi + $KiikAiXk + . . _ _ (8)

Since the Xi are now functions of p, @) and H are functions of the only remaining independent variable, IL-

The hamiltonian H can be expanded in powers of p,

H(j,Q=H(O)+pH, +&H2p+-... . (9)

Notice that the displacement p is a vector whose dimension equals the number of geometrical degrees of freedom which are allowed to vary. Hence Hr is a vector of this same dimension, and H, is a second-rank

tensor in this’satne dimension. In what follows, we often delete the vector and tensor notation but keep it in

mind. This expansion of H(c() allows each of the 4, Gii, Kiik to also be expanded in powers of i. For example,

q= c ,,n,n= c (10) ,/I=0 n=O

? A; treated in eq. (33) bf ref. [16], these commut+rs shouid be appropriately symmetrized if one wishes ;o generate G, K, . ._ matrices which have proper index symmetry_ Such- symmetrization _murs ~nattially tihen t&e Baker-Campbell-Hausdorf expansion is carried out. We view thk as a technical detail which we shall not dwell fkther on here. _.

206 A. Banerj2e et al. / MC SCF mokcuhr gradients and hessinns

Since each of I;;:. Gjj and Kiik contain all powers of p, the Ai solutions to eq. (8) contain all powers-of p:

x = A(O) + \(!‘p + +g’)p + J J ‘J - J

-_., (11)

so A:) is a vector, X(il’ a second-rank tensor. etc. Substituting this order-by-order expansion of Ai into eq. (8) and collecting powers of p allows one to solve for the Ai . (“) Low-order solutions of importance here are

h(O) = @ J ’ (124

since F-“) = 0 according to the generalized Brillouin condition, J

A!,!) = _ ( G’“‘)J;‘l;j:“‘,

and

(i2b)

A:“= -(G’“‘),;i(E,!“+ 2x;‘@;‘+ &!;;h:f)h(ll))_

These results, when substituted back into eq. (7) give an expression for the p dependence of E:

E=E”‘+E”‘lr+~lrE”‘~-t ___,

where

(124.

034

E(O)-= (OIH(O)~),

the gradient is

E”’ = (O]H,]O) + A’,‘)@?

the hessian is

Ecz’ = (O]HZ]O) + x’?)&(O) + 2~‘,“k;“’ + “;G,!;O’A(,!).

etc.

(13b)

03c)

(I3d)

Because the MC SCF wavefunction at R was presumed to have been optimized, ClE/Xi = 0 at p = 0. Hence the generalized Brillouin condition

<@)= (O][;r;,H(O)]]O) =o. 04)

holds and the above eqs. (13) simplify to our final working equations for the MC SCF gradient:

E”‘= (O]HJO). Wa)

and hessian

E”’ = (O]HJO) + ,(“A(il)_ Wb)

Clearly one must be able to evaluate the &r’) and G$) matrices as well as (O]H,p) and (O]H,~) in order to compute the MC SCF gradient and hessian. Gjy is nothing but the wavefunction-optimization hessian which can safely be presumed to be available because it is necessary for the MC SCF calculation at R. It is written in terms of one- and two-electron integrals and density matrices over the MC SCF orbitals in eqs. (2.38). (2.42) and (2.44) of ref. [19]. The elements e”)= (O][l;, HJO) are generalized Brillouin matrix elements but with W, replacing N(O), respectively_ Their computational evaluation is addressed in more detail in section 3, but first we must obtain more concrete expressions for the operators H, r&d H,;

__.- -: ---_ .._ -.--_ ,.- ..y_ ___ I- _. -: -. ;--

Our strategy is to show that the derivatives of the SO orbitals appearing in ref:-[i6] ckbe conveniently re-expressed as derivatives -of the -MC SCF. orbitals :themselves.: Once we liave established this relations between SO and MO derivatives it will be straightforward to expr&s U, and Hz -in- terms.of MC SCF

orbital derivatives, and it will become clear that the SOS can be viewed as a convenient derivationk tool which need not acpear in final working formulas.

Let us begin by considering the p dependence of the symmetrically orthogonalized (SO) orbitals. [.$;} which consist of combinations of the atomic basis_orbitals (x, >

5, = c s”~“2x” 9 06) P

where the atomic-orbital (AO) overlap matrix&, has elements (x,]x,). The second quantized expression for any operator (the hamiltonian, in particular) involves sums of orthonormal spin orbitals multiplied by their associated creation or kmihilation operators_ It is well known and easily demonstrated that such sums are-independent of & unitary transformation among the ortbonormal orbitals; that is, the sum is the s&me for all orthonormal bases. In particular, the sum of .SO orbitals and operators u, in terms of which H(p) is analyzed in ref. [16] can be replaced by an MO-level sum

C&a, =C UZ&a, =C+&zi, (17) P ia i

by making use of the expansion coefficients ( Upi} relating the SOS { ,$,} to the MC SCF MOs { cp,. }. It is this kind of relationship which we shall use to move from an SO-level expression for H(p) to the MO-level result, but first we need to further analyze the p dependence of- the { &} since this gives rise -to p dependence in H(p)_

The SOS { 5, } can be expanded in powers of p around F = 0 by writing them in the following manner:

5,= c (1 -W,-,“2(S;“2)p,-~~, -(18) O(.v’

where the {x,} are the primitive atomic orbitals (AOs) (gaussian- or Slater-type), S,, is the overlap matrix over these AOs at p = 0, and

_- (1 +A).,~~~(S~-‘~2)~,~~s(S~‘~2)Br._ . . (19) 4

_2.(18) simply provides a convenient parameterization_of the p dependence of p O.-Notice that as p L.0, r,r ---, 0, and that denvatrves of (1-T A) involve~simply-derivatives of the oirerlap S-itself: _-

_- __ --_: .._- : -:- _(20)

:.: .._ _:

Using eq. .(18), we can n&G -expr&k v& in a, convenient- foki_ To- do- so-we first- evaluate’ the fist.. .-

.- -_ :. . -.

. . _ : :-

derivatives * of (1 + A)- */’ at p = 0:

~(1 +A),,!‘*= v’~“.~-~A,~~+~CA,.=A,,- ._. = -+(A;.,). i a >

The desired derivative of 5, can now be written (at p = 0)

:--I _.. ‘-_-.(2f)

v.$,=~(s;1’2)“,vx, -f c (S~“*)*pVS8r(S~“2)rr.(S~1’2)ir.aXP- cm a Q.Y’

BY

In ret [16] the derivatives H, and H2 of the hamiltonian H(p) with respect to P were given in terms of sums C,v&a, and C,v v&, u,. analogous to those discussed in eq. (17). From eq. (17) one-sees that the SO-level annihilation operators (I, can be related to their MO-level counterparts ai:

a, = c u,ja; _ (23) i

This then allows the above sums involving VE, and V Vi, to be rewritten as follows:

C VS,a.=CffiC V,,V5,- CVVS,Qy=CaiC~,VV5,- (24) Y i v I* i Y

Eqs. (24) allow us to move from the SO-level operators {a,.} to the MO-level operators { ai}- The

combinations C,U”iVS, and x&,V Vt,, which can be viewed as MO-level derivatives v+; and V V#+ in which the expansion coefficients Uvj are held fixed, will from now on be denoted V& and VVCP, for notational ease.

When eq. (22) is premultipiied by the MO-to-SO expansion coefficient U,i and summed over Y, we obtain a useful expression for the derivative of the MO +, in which its MO-to-SO expansion coefficients are

held fixed:

(25)

Cni=~Uur~S-I’~)“~ (26) Y

is the expansion coefficient of MO +,. in the primitive A0 basis {x,} at +z = 0. Using this relation between the MOs and AOs (Qi = X,C,jx,) it is straightforward to show that

c s?zX1xoI = C cyi+z - (27) cx I

This allows eq. (25) for v+, to be rewritten in our final working form as

* When vlvz operates on a product of factors (such as (1-k A);f/‘x,) one obtains four terms [e.g.. (l+ A)-‘flVIV2x. XV,V# + it)_“’ and both V,(L + A)-“’ Vzx and Vz(l + A)- ‘EVIx]_ The subscripts 1 and 2 on VI and Vz remind us that- these differentiaI operators have components. For example, when (d/dx,)(d/dy,) operates one obtains four terms including both “cross terms’* (d/dx,)(l + A)-‘r-(d/dyb)x and (d/dyb)(l + A)- ‘r-(d/dx,,)x. In this manuscript we will simply write V(l+ A)-‘/2 vx but keep in mind that both cross terms are present.

When compared to eq_ (28) one sees that I

~~~X~Vc~i~ -+C(Cfl,sS,,C,,)&,* ’ k--

_ _- cl

_ -- _____ (30)

BY i _

from which it is straightforward to show that

VCai = -$C(C’VSC)i~C~k* (31) k

Eq. (31) provides us with a very important tool for expressing the change in the MO-to-AC? expansion coefficients which arises from the MO orthonormality constraint CTSC = 1.

If JS had, in ref. [16], realized &rr eq. (28) they could have avoided using the SO basis as an intermediate device in their derivation. We chose to show the relation of the Sa>‘fl array which arises in the SO basis to the direct MO-to-A0 result given in eq. (28) so as to clarify the derivation of ref. [16]. However, it should be clear that because eq. (28) contains no reference to the intermediate Sd orbitals, it is a general result. Because eq. (23) refers only to the AO-level orbital derivatives and to the AO-to-MO expansion

-coefficients, its validity is independent of the SO basis which was used in ref. [16]. That is,~the SO b&is can be viewed as a perhaps convenient intermediate; eqs. (25) and (31) are, however, more useful because they.. allow all orbital derivatives to involve only the primitive AOs.

To obtain an expression for V V& analogous to eq. (28) for V& we simply differentiate eq. (28) once more and make use of eqs. (28) and (31) to obtain

VV~i==CC,iVVX,-~C(CTVVSC)iX_~k-~ c (C’VSC)ikCakVXn Q k k,a

(32)

We are now ready to use our expressions [eqs. (23) and (32)] for ZZ:,UpiV5, and CpU,iV V& to write down III and H,. Within the SO basis H(p) is given as

H(P) = c ($,I - f v2-- GYIr - k - PIIS,)&% + $ c &LI~oir)&o+~v~l,

‘&

QYOT

,,aTa, + 4 C (uYIu~)u~+~a,u,, (33) *.a a!ymT.

where the shorthand one- and Mull&en two-electron integral notations, _h,, and (a+-) have been introduced. The first and second p-derivatives of ‘H(p)_ (evaluated at -c = 0 j-regive H, :and -Hi__ Thesf: derivatives clearly-involve C,V&a, arid. &,VV&,a, which, by eqs. (24); can be rewritten as CiUiV+i and &aiV v+;:; The results of taking the required derivatives bf; the H-shown in e& (33): isin& eqs.-(24), and. substituiing eqs. (28)‘and (32) for .V& and V V&are given as follows- . -. . ,

-. . I&.& c C Cprtitii( &&, +svS~:) i + c c _ ~prCIsC&&d,s,& @+I& 1. : -. .!I -_I -‘- (34) .~ 1 cs -p_c -- I qy a+ __ . .~ _- -.

.-

210 A. Banerjke et aL / MC SCF molecular gradients and hessians

where the Er, are elements of a Fock-like operator

Er, = Ch,,d,, + c bkb?ddrk,, k PQk

(which is symmetric if the

and drkp4 are symmetrized

d, = f(e, + esrIr

MC SCF wavefunction obeys the generalized Brillouin condition) and- the-d,, density operators

_. (3i

(36)

which have the same index permutational symmetries as the integrals which they multiply in-H: The ei j = a,‘_hcli, t a,‘a,& and eijk, = ei,ekI - 6jl,eil are the well-known unitary generators. Note that in eq. (34) all integral derivatives involve atomic-orbital integrals (labeled with Greek indices LY, /!I y, 6) whereas all creation and annihilation operators (in dij) involve molecular orbitals (labeled p. q, r. s). These facts are of crucial importance to the efficient computational implementation of eqs. (1Sa) and (1Sb).

The corresponding expression for H, is

with

vE,, = c C,&vh,,ds, + c 1 C,sc,,Cppc,qv( Orpi~v) Gpq - kWJ kPq QBP

Eqs. (34) and (37) represent our final expressions for H, and Hz.

2.4. MC SCF gradient and hessian

When used in eqs_ (15a) and (15b) these results provide working equations hessian El”:

EC’) = (OlH,W = c #w)v h,, - c (E,w)vSw + t c W,.,AJV(PI~~), PV

py jwho

(381

for the gradient E”’ and

(3%

where the following quantities involve density matrices which are first computed in the MO basis and then transformed to the A0 basis:

l-s 'SPQ

&J = ~c,rc,s(&s)~ tErs> = Chrk(dsk) 4 k

The MC SCF hessian is given in like fashion by

+ c (SklPd&,q)- ?‘Qk

(40

E”‘= (OIHJO) -(KY), ~~“)(Gco))-’ (42)

where Pm,, represents permuting indices m and n, and

Here the subscript n means that-all such density matrices are evaluated as transition density matrices:For example,

As mentioned earlier, G,, is the~usual hessian matrix of MC- SCF theory [19].

2.5. CI gradient

Before closing this derivation section; we would-point out how theleresults of this work allow one to also compute mol&cular gradients for‘configuration interaction (CI) wavefunctions ICI). Eq- (13~) stili,holds for this case, but the‘ generalized Brillouin conditioti [eq_ (14)] only holds in the varifibIe.spacc {S, } for which the CI energy has been v&aiiona.lIy optimized. That is F,‘O)= 0 but 1;15”!.# 0..As.a result,. the CI molecular gradient has a non-va&hing second term in eq; (13~) -_

P’= (CI(HJCI) + K;i)c:O)~ .- __ (47) TZS __ : IZU - - _.

._ The -computation of- .(CIIsr]CI) .istreatedlexa&as~ outhnedahove relative- to. kql (39)exceI$ &at-, aIi density mat& elements @Its>_ and, (d,,,,) are evaluated i relative : to _theY_CI w~vefunctionl :Becat&e @e combination :_~E:r,u(G(o))rT;.ljU&~) g&s_.the response, K, (I) of the ~orthonormal orbitals. to nuclear displace: ment [see.eq. (I2b)], the calculations of this ~quant$y~~depends upou the riatureLo_f ~e_orbr:fa~s~l&ng I$*;+

t&e CI calculation; riot the configuratious i.ncluded in ICI). That .is,lI;, (‘) described the changes-in the CI energy due to variation -of the K, orbital parameters;’ -ii:) gives the ..ehange~ in the :orbita&‘due to. dis&cement. ~Cle&ly,- the orbit&habg&‘oirly~ de$ends du- tihat theorbit& Bire, riot o&the CI co&&- . ..- .-.. ._: _ -_ __ _‘_- _; 2.. :

: . -, -_

.__ .- -. I-

217 A. Banerjee et aL / MC SCF molecular gra&-ems and hessians

tions. The most commonly used choices are I-Iartree-Fock (HF) molecular orbitals and orbitals resulting from an MC SCF calculation of considerably smaller dimension (i.e. number of configurations) than the CI

calculation under study. For either of these. two cases, the evaluation of F,‘,” and (Gco))& proceeds as follows: F,z) is computed as discussed above in eq. (47) except that the density matrices are taken relative either to the HF or MC SCF function which was used to compute the orbitals. Likewise. G,!z’U is evaluated as in eq. (2.42) of ref. [193 but with HF or MC SCF density matrices. To evaluate Trio) one used the well-known expressions (e.g.. eq. (2.41) of ref. [19]) for these generalized Brillouin matrix elements except that one computes the density matrices with respect to the CI wavefunction whose molecular gradient is being sought.

3. Computationai implementation of working equations

In writing eqs. (39) and (42) for the molecular gradient and hessian. care has been taken to group

together those factors (c-g.. vh,,.. v(p~Xu). v~(p~Xcr)_ OS,,) which depend upon the components of the displacement derivatives and those which are independent of displacement_ This facilitates our computational strategy: (1) to compute. sort. and store on disk all quantities (performing as many

intermediate sums as possible) which are deformation independent, (2) to carry out a single shell-by-shell pass through the AO-based integral derivatives during which these derivatives are generated and the corresponding displacement-independent quantities are read into the computer’s core from disk and multiplied by the integral derivatives. This strategy allows us to generate and use the AO-level integral derivatives without storing them.

Let us consider a few examples of how this procedure works. In the last term of eq. (39) we see that the

AO-based two-electron integral derivatives v(pz$b) are multiplied by (ci,,.xm) which is displacement-in- dependent- The MO-level density matrix (d,,, ). which has full eight-fold index permutational symmetry. is evaluated within our unitary-group MC SCF computer program: it is a necessary ingredient of the MC SCF process itself. This array (ci,,_,,) can then be transformed. using the conventional n5 procedure employed to transform two-electron integrals. to give (dPyehO) as shown in eq. (40). This latter array also has full eight-fold index permutation symmetry_ It can be sorted and stored on disk in the same order in which the integral derivatives X(p~lha) are evaluated. Notice that it is our desire to perform only a single

pass through the integral derivatives and not to store these integral derivatives which require us to have access to the AO-level density matrices (L$,..\~ )_ It is true that the density matrices are likely to be much sparser. and thus more easily handled. in the MO basis. however our computing strategy requires us to evaluate and store the (dpv_Ao ) in the same shell-by-shell blocks as we used to generate the integral derivatives_ One could alternatively compute (dijk,) in the MO basis and multiply these density matrix elements by the integral derivatives which have been transformed to the MO basis ~(ijlkl) = C,,,,C,,C,,C,,C,,v(~cYIXa). However. this transformation results in an integral derivative list of dimension 3iV x 0'. where 0 is the number of MOs which are occupied in the MC SCF wavefunction and N is the number of atoms which are allowed to move. Notice that the sparseness of the AO-level integral derivative list (whose size is = 12 X nJ) is lost once the AO-to-MO transformation is effected_ For molecules with more than a few geometrical degrees of freedom. the storage difference between 12 x n4 and 3N X 0' can be so large as to offset the density matrix transformation and storage difference (n4 for (dpyx,) versus O4

for (d,k,>> Most of the terms in E, and Ez can be straightfonvardly evaluated following the above outlined strategy_

However. the computation of Fiz involves special difficulties which are worth focusing attention on_ The last term in eq. (44) requires an ,* three-index transformation

A:.~anerje et bL /.MC SCF moIectdakgradi& and he&C --

to form;- Sort -and store the -XmV_i_ array. As the ‘integral derivatives :e(@;]he j 1 tie: generated;.: the_

appropriate groups .of X,,,_k_ and C,, are brought into the computer’s -core and’contributions to the

quadruple (i.e. n5)-Sum 2~~ = Cu.xo X,,rr.Xb ~v(~-rv]Xa) &e generated.--It is,possible-to reduce:the‘~mount~of

disk-to-core input/output required to evaluate the Z_ array;‘ By allowing b&self: to compute integral

derivatives more than -once, one _caq min_imize IO of the-density mat$x~:elements !X,,zl._,,i ‘Alte_rnatively;~by . _~ insisting that integr$ derivatives-iV(pY]Xe) be computed only once; pne must be willing to-do more IO bf

the x,,,,, array. Once Znlp -is in hand it can be_ multiplied by Cp,, and Summed over‘ p to give F,‘,‘: The next

to the last terms in eq. (44) can be handled in like fashion.

The last terms in eq. (45) present- the most challenges. -As written, its- evaluation. would require the

evaluation and storage of transition density matrices over the A0 basis. Since there are one fewer transition density matrices than there are configurations (iv,) in the MC SCF wavefunction IO), there may be -very many such arrays. Therefore, for any but quite small (s 10) -configuration expansion. lengths, it is not

practical to evaluate this term as it is written. Instead, it may be preferable to transform the integral

derivatives to the MO basis and to evaluate the transition density matrices -in this- same MO basis.

Evaluation of the MO-level transition density matrices is relatively straightforward. Transformation of the AO-level integral derivatives v(~v~XU) to the MO basis is tedious and storage intensive (since there are 3N X O4 such MO integral derivatives, where N is the number of atoms and 0 is the number of occupied-

MOs). Nevertheless, the requirements for transforming this array of length 3Nn’ may, for moderate to large configuration expansion lengths (N,). be less than is needed to evaluate the (d~y_xO),, array whose dimension is (ZV,, - 1)n’. Thus, it may be wiser to evaluate the last term in eq. (45) via transformation of the integral derivatives to the MO basis.

We feel that the working equations given in eqs. (39) and (44) together with the above strategy for their implementation provide an efficient and computationaily feasible procedure for generating moIecular

gradients and hessians via ab initio MC SCF-based wavefunctions. We presently have at Utah and at

Argonne * a working program based upon this approach in which the gradient routines are fully operative.

We are currently working on bringing the second integral derivatives [e.g.; vvQ.wlXa)] and the F,,Sfr routines into production.

These remarks conclude our treatment of MC SCF and CI molecular gradients and hessians. We feel that our working expressions are given in computationally tractable form and that our derivation was carried out in a manner which other workers will find enlightening. It is possible to use the connections

which we demonstrated between derivatives of the SOS and the MO-level derivatives V& and v v& given

in eqs. (28) and (32) to express the Moller-Plesset and coupled-cluster gradients and hessians of ref. [16] in terms of AO-level integral derivatives and MO-level density matrices_ In fact, by using eqs. (28) and (32) for V& and v V& it should be possible to efficiently derive any gradient, hessian, or higher energy-deriva- tive formula in computationally tractable form.

Acknowledgement

We wish to acknowledge support from the American Chemical Society, administered by the Qonors of

the Petroleum Research Fund (grant No. 14446-6). We also wish to thank the National Science Foundation (grant No. 8206845).

*‘This MC SCF-based program called MCGRAD will be detailed in Group of the Chemistry Division of Argonne National Laboratory.

the annual report (10/82-9/83) of the Theoretical Chemistry

214

References

[1] P. Pulay. in: Modem theoretical chemistry, ed. H-F. Schaefer III (Plenum Press, New York. 1977) ch. 4_ [2] J. Gerratt and I.M. Milts, J. Chea Phys. 49 (1968) 1719.

[3] S. Bmtoz. Coiloq. Intern. CNRS 82 (1958) 287. [4J W. Meyer and P. Pulay, Proceedings of the 2nd Seminar on Computational Problems in Quantum Chemistry. Strasbou~, France.

September 1972, pp. 44-48; J_ Chem. Phys. 56 (1972) 2109.

[S] K. Thomsen and P_ Swanstrom, MO!. Phys. 26 (1973) 735.

[6] P. Pulay, Mol. Phys. 17 (1969) 197. (71 J_A_ Popie. R. Krishnan, H-B. Schlegei and J.S. Binkley. Intern. J. Quantum Chem. 13 (1979) 225;

R. Krishnan. H.B. Schlegel and J_A_ Poplc, J. Chem. Phys. 72 (1980) 4654. [S] H.F. King and hf. Dupuis. J. Comput. Phys. 21 (1976) 1;

hi. Dupuis, J. Rys and H-F_ King, J. Chem. Phys. 65 (1976) 111: M. Dupuis and H.F. King. J. Chem. Phys. 68 (1978) 3995.

[9] J.D. Goddard, N.C. Handy and H.F. Schaefer III. J. Chem. Phys. 71 (1979) 1525:

Y. Osamura. Y. Yamaguchi and H.F. Schaefer III. J. Chem. Phys. 75 (1981) 2919: 77 (1982) 383:

B.R. Brooks, W.D. Laiding. P. Saxe. J-D. Goddard. Y. Yamaguchi and H.F. Schaefer III. J. Chem. Phys. 72 (1980) 4652.

[lo] D. Poppinger. Chem. Phys. Letters 34 (1975) 332; 35 (1975) 550. [ll] H.B. Schlegel. S. Wolfe and F. Bemardi. J. Chem. Phys. 63 (1975) 3632: 67 (1977) 4181; 67 (1977) 4199.

[12] A. Komomicki. K. Ishida K. Morokuma. R. Ditchfield and M. Conrad, Chem. Phys. Letters 45 (1977) 595; K. Ishida. K. Morokuma and A. Komomicki, J. Chem. Phys. 66 (1977) 2153.

[13] S. Kato and K. Morokuma, Chem. Phys. Letters 66 (1979) 19.

[14] P. Pulay. J. Chem. Phys. 78 (1983) 5043. [IS] R.N. Camp. H.F. King, J.W. McIver Jr. and D. MulIally. J. Chem. Phys. 79 (1983) 1088.

[16] P. Jorgensen and J. Simons. J. Chem. Phys. 79 (1983) 334. [17] P. Jorgensen and J. Simons, Chem. Phys.. to be published. [18] T.V. Helgaker and J. Almlof. private communication_

[19] P. Jorgensen and J. Simons, Quantum chemistry: second quantization based methods (Academic Press, New York. 1981).