Two-center expansion

An eigenfunction ijr of the Schrodinger equation satisfies the homogeneous integral equation b = f 3 Govt/r. In cell r this defines a locally regular solution of the Helmholtz equation 

The function x can be represented in x by a sum of regular solid harmonic functions, 

Similarly, because i(r is regular at the local coordinate origin, it can be represented at this origin by 

From the Lippmann-Schwinger equation and the expansion of Go in some specified cell r/x, the coefficients in these expansions must be related by = cl + ft N jyf. Consistency conditions are derived by considering the alternative 

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The first section is devoted to the derivation of the two-center expansion of the Green function in overlapping spheres. This expansion is a notoriously difficult problem (Sack, 1964) that was finally solved by Sun and Su. In the second section, we discuss the use of this overlapping Green function. 

Sack, R.A., 1964, Two-center expansion for the powers of the distance between two points, J. Math. Phys. 5 260. 

From the two-center expansion of the Green function, this equation implies X(r) = -J2 Jl(r) J2 J2... 

When coordinate x = X/x + r lies in cell r/x and x = Xv + r lies in a different cell r , a two-center expansion of the Green function is needed. Since N /fx — X/x) is a regular solution of the Helmholtz equation in a sphere of radius X/( — Xv about the origin of a displaced cell v,... 

The integrals entering two center expansions parameter calculation as given in equation (5.1) will be discussed here. 

Using the general expressions (1) with (4) for an atom-atom model and substituting the two-center expansion (16), we write the intermolecular atom-atom potential as... 

Figure 4.2. The relative error of the expansion of (a) the slope matrix using a one-center Taylor expansion and (b) using a two-center expansion.The error is defined in Eq. (4.35) and the expansion center is (0,0) for the one center expansion, and (0,0) and (-10,0) for the two-center expansion.

These conditions lead to a system of linear equations for an+i,an+2,. Solving these equations, we obtain a (n + m+ l)th order expansion for Sa. In Paper I, this expansion is examined and compared with the old one-center Taylor expansion. The results show a marked improvement in the expansion of the fee slope matrix, see Fig.(4.2), and we also find some cases where it is necessary to use the two-center expansion in order to produce correct results. The error in Fig.(4.2) is defined as... 

The two-center expansion (Carlson and Rushbrooke, 1950) can be substituted into eq. (32) to obtain... 

The solid curve shown in Fig 6 was derived by Lin and coworkerswho used a two center expansion method and a Hartree-Fock-Slater potential The agreement is reasonably good ... 

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