On the Appearance of Unlinked Clusters in SCF–CI Calculations

(a)

O.

 

Sinanoglu

,  , ().

(b)

O.

 

Sinanoglu

,  , ().

(c)

O.

 

Sinanoglu

,  , (). ,

(d)

O.

 

Sinanoglu

,  , (), and others.

(e)

O.

Sinanoglu

and

D. F.

Tuan

, , ().

H. P.

Kelly

, , ().

It seems quite reasonable to expect perturbation theory to converge in atomic and molecular problems if one starts out with as SCF zeroth‐order state due to the smallness of the resulting interactions [see discussion of this point in Ref. 1(b)] especially in the light of Kelly’s perturbational results on Be (Ref. 2). For a more general discussion of the convergence of perturbational expansions see: J. O. Hirschfelder, W. B. Brown, and S. T. Epstein, “Recent Developments in Perturbation Theory,” to be published in Advances in Quantum Chemistry (Academic Press Inc., New York, 1964), Vol. 1, and references herein,

and also

S.

Weinberg

, , , Appendix A ().

The terms “linked” and “unlinked” clusters are defined by many authors; see, for example, Refs. 1, 2, and 5.

The linked‐cluster theorem in perturbation theory has been proved (in several different ways) by: (a)

K. A.

 

Brueckner

,  , ().

(b)

J.

 

Goldstone

,  , ().

(c)

H.

 

Primas

,  , ();

H.

 

Primas

,  , ().

(d)

C.

Bloch

, , (), and others.

The physical interpretation of, for example, the four‐body unlinked cluster [see,

R.

 

Brout

,  , ();

O. Sinanoğlu, Ref. 1(b),

and , ()] is of two, two‐particle interactions occurring at the same “time.”

By order we mean the lowest order perturbational contribution included in a term.

In order to keep the following notation as simple as possible, we shall deal explicitly only with the nonsymmetrized case (Hartree SCF zeroth‐order Hamiltonian and product wavefunction). The inclusion of antisymmetrization can be accomplished in a straightforward manner by using the Hartree‐Fock $H0$ and including an antisymmetrizing operator in each of the resulting matrix elements. For example, the matrix element $〈ij|mij|uij〉$ which occurs below would be replaced by $〈ij|mij(1−Pij)|uij〉,$ etc.

Specifically we require, $1 = 〈φ|〉<〈χ|χ〉,$ a condition somewhat weaker than that necessary for perturbational convergence.

It can also be shown by methods similar to that of Sinanoglu [Ref. 1(b), Eq. (53), ff.] that in the lin it $N→∞,$ in which the expansion of the normalization denominator 𝔑 becomes impossible due to the presence of the unlinked clusters, these unlinked clusters still cancel.

The three‐body terms include the so‐called “exclusion principle violating” terms discussed at length by H. P. Kelly (to be published) and earlier authors referred to therein. N. R. Kestner has informed us that he is examining the contributions of these three‐body terms to intramolecular interaction energies. One of us (J.I.M.) is very grateful to Kestner for several valuable discussions on this point. It is also at Kestner’s suggestion that we point out below that our improved incomplete CI energy value is not an upper bound to the true energy.