J. Phys. Chem. 1982, 86,1096-1098
1096
of most systems indicates that the CS approximation would give excellent results for spin-spin and spin-rotation NMR relaxation times. The fact that cross sections of type (a) give consistently good results for the elastic cross sections while the inelastic cross sections are much smaller but not so accurate as the elastic ones indicates that the CS approximation should give good results for diffusion and viscosity related phenomena in the absence of external fields. More sensitive phenomena such as the deviations of thermal conductivity or shear viscosity in the presence of a magnetic field (SBE) depend quite sensitively upon rotationally and energy averaged cross sections of type (d). Because the CS approximation often does not agree well with the CC results for cross sections of this type, the success of the CS (or 10s for that matter) is questionable in determining SBE effects. On the other hand, the CS method does properly predict the shape of the opacity which determines these cross sections. Because the elastic and inelastic cross of this type are small, often opposite in sign, and similar in magnitude, it is also clear that averages over them are going to be extremely sensitive to
the accuracy of even the exact CC values of the S matrices. Thus, the challenge presented in calculating cross sections which are extremely sensitive to interference effects extends beyond the approximate methods to include accurate CC approaches as well. Much work remains to be done before these cross sections can be fully utilized in obtaining detailed information about molecular potential anisotropies. Acknowledgment. D. K. Hoffman and D. J. Kouri both benefited from the wonderful experience of being graduate students at Professor J. 0. Hirschfelder’s Theoretical Chemistry Institute. In addition to very best wishes on the occasion of his 70th birthday, they wish to express their gratitude to Professor Hirschfelder for his great kindness both during and following their graduate studies and for providing a stimulating and friendly environment in which to learn. The authors also gratefully thank Sheldon Green for providing us his CC S matrices for the He CO and He + HC1 systems. This work was supported in part by the National Science Foundation Grant CHE 77-22911.
+
A New Algorithm for the Evaluation of Vector-Coupling Coefficients in an MCSCFXI Study 0. Das Chemistry Division, Argonne National Labomby, Argonne, Iiiinols 60439 (Received June 22, 1981; I n Final Form: August 17, 1981)
A new method is presented for an efficient handling of the vector-coupling coefficients in an MCSCFICI calculation. The method is suited to those problems (such as transition metal clusters) where only specialized sets of angular momentum couplings are important.
Introduction Owing to the increasing use of ever larger sets of configurations in theoretical studies based on configuration interaction (CI) or many-body-perturbation theory (MBPT), the problem of efficient handling of the vector-coupling efficients has attracted the attention of both chemists and physicists. The methods1$that have emerged are ideally suited to those cases where all spin couplings are treated on the same footing. For many potential surface calculations this generality is desirable. There are, however, two classes of problems for which such generality is very seldom required: (i) studies aimed at computing spectroscopic properties of systems in static geometries and (ii) those that deal with systems such as transition metal clusters, many of the open shells being relatively compact. As an example of the latter case consider a system l i e Crz. A recent calculation3 on this system has revealed that bonding in the ground state of this system corresponds largely to an antiferromagnetic pairing of the spins on each Cr. This implies that the molecular wave function is very nearly valence-bond-like, consisting of ’S atomic ground states of Cr. For chromium multimers, therefore, it is (1) J. Paldus in ‘Theoretical Chemistry: Advancea and Perspectives”, Vol. 2, H. Eyring and D. J. Henderson, Ed.,Academic Press, New York, 1975. (2) I. Shavitt, Znt. J. Quantum Chem. Symp., 11, 131 (1977); 12, 5 (2978); Chem. Phys. Lett., 63, 421 (1979). (3) M. M. Goodgame and W. A. Goddard, III, J.Chem. Phys., in press. 0022-3654/82/2086-1096$01.25/0
conceivable that the molecular states are best expressed in terms of groups of atomic orbitals in a limited number of atomic spin states leading to a drastic simplification of the “vector-coupling” problem. The unitary group approach,l obviously, cannot reproduce such simplicity. Moreover, even for potential surface calculations based on GVB/CI or MCSCF/CI it is generally true that not all types of configurations are important such that what is considered the strength of the unitary group approach, viz., the full generality of the configuration space, may in fact be its weakness. For example consider a recent calculation on N02.4 Over the interactive region on the N + O2 side of the potential surface, the following four configurations turn out to be the most important: = { core)YO12XolX~Z~Yb2Zb2x02 91 = 9 2
(Yb2
= (zb2
- z;)@O Y;)@O
a3 = (YbZb y,za)@O where the orbitals (X,, Y,, Z t ) , E = N, 0 , O l are approximately the 2px, 2py, and 2pz orbitals centered on the centers (5) and Yb, Zb, Ya, and Z, are bonding and antibonding orbitals, Y’s for the u bond in NO and Z’s for the u bond in 02.Moreover, all important single excitations +
(4) G. Das and P. A. Benioff, Chem. Phys. Lett., 7 5 , 519 (1980).
@ 1982 American Chemical Society
The Journal of Physical Chemistry, Vol. 80, No. 7, 1982 1097
Evaluatlon of VectorCoupling Coefficients
-
-
from these configurations into the “virtual” space can be classified 88 Yo1 Yo:, Xol- &’, Yb Ybl, and Z b &‘, etc. Hence we can do a natural grouping of the orbitals as follows: (i) Yol,Yol’ (iv) ZN,ZN’
(ii) &l,Xo1’
(iii) XN,XN’ (vi) yb,ya,yb’,y: (vii) zb,z,,zb’,z,’
(VI Xo,Xo’
where the quantities c and d are Clebsch-Gordon coefficients and the functions f and g are spin eigenstates having spins s - 1 / 2 and s 1 / 2 , respectively. We shall now define quantities relating various ESF’s of the same ES differing at most by the single excitation 4i 4i’
+
-
( f n i ~ l / 2 , m - 1 / 2 ~ n ~ i f ~ - l / 2 , m - 1 /= 2 ) 6ssJ%intl
Orbital sets such as (9-(vii) will be referred to below as defining excitation subspaces.
General Form of Matrix Elements in Terms of Excitation Subspaces Definitions. An excitation subspace (ES)will be defined to consist of a specified number of many-electron symmetry-adapted functions of which both symmetry species and number of electrons are variable. A configuration in the space of these functions is a symmetry-adaptedproduct of ESF’swith the proviso that there is at most one ESF from each ES and that they contain the correct total number of electrons. Matrix Elements. We shall refer to the symmetryadapted functions with specific quantum numbers belonging to an ES as excitation subspace functions (ESF) and will denote by three indices lnisimi)where si and mi are species and subspecies indices and ni a serial index for all the distinct ESF’s in the ES i. A symmetrized confiiation (SC) in terms of ESFs can be written as follows Qrl = )I,n1s1n’$2...)= C CIm,m, ... Inisi9n2sz,...; m1m2,...) ( 1 ) mlm,...
...
where CImlQ, are Clebsch-Gordon coefficients and Inlsl, y2, ...,m l 4,...) is an unsymmetrized function (UF),which IS defined to be simply an antisymmetrized product of the ESF’sInpimi). In the case of Cr2, for example, we can take all the 3d orbitals on each atom to form one ES and the 4s orbitals to form another, leading to a total of three ES’s for Crz. in the following form: We shall reexpress @J
=
ca!ksmSi&mls~
naSanbsb)
(2)
where the ESF’sat a and b are combined to a function of symmetry S and M while the rest of the ESF’s are combined into a&, of symmetry s and m and “parentage” k . The transformation (2) enables one to express the “overall” vector-coupling coefficients in terms of elementary vector-coupling coefficients involving the ESF’s. The matrix element between the configurations I and J is given by (I;n1’s1’n;s;...l%l4 n,sln#2 ) = CsAfjs( n:S:nb’Sbl17fY(2)lnasanbsb )S (3)
...
if the ES’s a and b are the ones in which the configurations I and J differ, and where
(4) and %(2) is the truncated Hamiltonian for the electrons in the ES’sa and b. Also for partially filled shells we rewrite an ESF as fOllOWS lnsm) c$p@iim + 4ij3QrLm I dia(C8dni,s-1/2,m-l/2 + dsIdni,8+1/2,m-1/2)+ CbiP(c ’smfni,s-1/2.m+l/~ + d 61dni,s+1/2.m+1/2) (5)
(gnia+1/2,m+1/21gn,i,Z+1/2,m+l/~)
(fnia-l/2,m-1/21gn~i~a/+l/2,m-1/2)
= 8ss’Cn5) = haf+lQnin’it
(6)
It is to be noted that the quantities pf and Q are numerical constants independent of the forms of the orbitals. In simplifying (3) we shall use the following formula (for proof see Appendix): ( n:S,’nblSbl17f‘2))neS,nbSb)S =
D~C:s$,sCb(n,’s,‘m,nblsblS - mlln,s,m,nbsbS - m ) + D~a~s$,sgb(n:Ss,‘m,nb‘SblrS - mllnasam 1,nbsbS - m - 1 ) + D ~ , ~ ~ ~ ~ ~ ~ ~ ( n : s , ’ m-, mlln,s,m n ~ ’ s b l , -S 1,nbsbS - m + 1 ) (7)
where
1098
The Journal of Physical Chemisfry, Vol. 86, No. 7, 1982 Unim
= csmc’sm-1Pni+ dsmdklP!ii
Wnim
= ckcsm-1Pni + dfmdam-lKi
Das
vided i, i f , j , and j’are all at most singly occupied
Using the fact Tinainbi+ Tknainbi= 1 one arrives at the expression
where n: is the occupancy of the ith orbital in the Ith configuration, and klJ,ij
=
CSA8s[T8n,in, + DSs,se,sbTSnain$ + Dis.ag,sbTin,indI (10) for both i and j open, otherwise k,ij =
‘/npi“n?,
Off-Diagonal Matrix Elements We shall consider here three types of off-diagonal elements: (a) same occupancy-these are between configurations formed by coupling differently the same set of ESFs similarly occupied, (b) excitation of a single ES with the nature of occupation of the remaining ES’s unchanged, and (c) double excitation involving two of the ES’s. Excitations which correspond to single and double transfer between ES’s will not be considered here. Same Occupancy Off-DiagonalMatrix Elements. We have = (&nlslnls2...1%1J;nlsln$2...) = -E klJ,ijKij
Two special cases in the present category need to be distinguished: (a) While the symmetry characteristics of each ES change, occupation of orbitals does not, and (b) the orbital occupation (but not symmetry) of one of the ES’s is unchanged while the other undergoes a single excitation. The matrix element for the type (a) is easily written as
This clearly vanishes unless /As,/ = 1 and = 1. The matrix element for the type (b) is given by
HIJ = -,E RIJ$jj(iIK?,!$’)
a>biCa jCb
JCb
(11)
where klJ,ii is given by eq 11. Excitatron of a Single ES. There are two kinds of excitations within a single ES which will involve only the corresponding internal vector coupling coefficients. These are excitations into another state (i.e., n, n’J without any change in the occupation of the elementary orbitals and (ii) double excitations. A genuine “single” excitation, on the other hand, will connect the ES in question with all other ES’s. Such a matrix element is easily written by use of eq 11: Let the single excitation be from the orbital & to the orbital $, in the ES a. Then the corresponding matrix element is
-
where klJ,ik is given by eq 11
This too vanishes unless IAsJ = 1 and IAsbl = 1.
Summary and Conclusions We have addressed ourselves above to a special class of problems where the given molecular systems can be represented as consisting of a set of smaller atomic or molecular fragments which retain well-defined symmetry characteristics. A number of formulae are derived to demonstrate that one can exploit these features to achieve a great deal of compactness as well as efficiency in dealing with the CI or MCSCF problem associated with such problems. Acknowledgment. This work was supported by the Office of Basic Energy Sciences, Division of Chemical Sciences, U.S. Department of Energy, under Contract W-31-109-Eng-38. Appendix Construct the spin-projection operator Os such that OS(n~ss,’manb’sb’mb) = t(ns,’ss,’nb’ss,’)
and nlJ,ij= 0 unless I and J are identical in all spin couplings in which case nlJ,ij= n,,ij is the internal single-excitation one-electron vector-coupling coefficient for the ES a. Excitation of Two ES’s. Consider the matrix element (3) when Ins,’ss,’)and In{sb’) are single excitations i i’ and j j’with respect to Inas,) and In&,), respectively. Then it is easily shown that
-
-
HIj
= dIj( ilJjyli ’) - %lJ,iitjj’(ilKjPli’)
(13)
where dIJ = 1only if the spins and the spin couplings are identical between I and J , otherwise dlJ = 0. Also, pro-
where t is a normalization constant. Then t2 = ( OS(n~ss,‘m,mb’sb’m~~Os(ns,’ss,’manbfss,’mb) ) = ( n~s~m,mbfsb’mb~OS(ns,‘ss,’manbfss,’mb) = t ( n,’s,’m,nbfsb’mbl(ns,’ss,’nb)sb’)S) Hence
t = (SMlSs,‘m,Sb’mb)
M = ma + mb
Thus O s l t operating on Iss,’m,sb’ma) projects the normalized symmetrical function (S&,)S. btaining only those components of (ss,’s{)s that can have nonzero matrix elements with (ss,’mas{mb)one arrives at the expression eq 7 given in the text.
