Relations for supermatrices in hartree-fock-roothaan equation

,.Relati&s ~upetiat&ces

:’ .:

‘. ,,

‘.

.

”

-.,

:

.

,;‘.

have been prcwed.by which the douttfe s~mma&ons over symmetry subspecies ap$wing of the Hartreq-Fock-Rootha equation are mduced to a single summation.

.’ . ‘.

‘.

:

:

: ,: Ln the ~~~ee-Fock-Roo~a~ ,tih&s appear, The eiements ..-.,

in J- and K-

equations

f

1,2]

based & the symmetry basis functions, the i- Ad K-super-

of .the J-super-matrix are defmed by [2] .

‘..a.+ those’of th&K-super&&ix are similarly’defmed. In eq. (1) d, is the dimension of the symmetry species A, and k&p is the basis function belonging to the symmetry sp:cies h and the subspecies a,p denoting the index ‘distinguishingjMous b,asis functions having the same Xa. The sGnmations over Crtid 0 in eq. (1) are taken over ail the: subspecies of species h and g, respektivefy, . We sha.li here prove a relation which is stated (in the case of the J-supermatrix) as

Fdr every symmetry operation T,

Volume 11, number

3

CHEMICAL PHYSICS LE’ITERS

1.5 October

1971

(6)

Averaging eq.(6) over all the symmetry operations, we obtain

I

(7) where g is the total number of the symmetry operations of the group. U&g the well ‘known orthogonaiity relation for irreducible representations [3], the summation over T can be carried out, and we obtain Formula (2). In eq. (7) we have assumed that the group considered is finite but it is not difficult to prove the reIation in the case of a continuous tie group. We should also notice from the left-hand side of eq. (2) that I in eq. (3) is independent of the subspecies (Y. The relation proved will be useful for the computation of the super-matrix, since the double summations over a and p are reduced to the single summation over a or & The elements of J-supermatrix are simplified to

A similar relation can also be deduced for the K-supermatrix.

References [I] C.C.J.Roothaan, Rev. Mod. Phys. 23 (1951) 69. [2] C.C.J.Roothaan, Rev. Mod. Phyn 32 (1960) 179. [3] E.P.Wigner, Group theory and its application to the quantum p. 83.

mechanics of atomic spectra (Academic Press, New York, 1959)