Spin-coupling coefficients

The spin-coupling coefficients Cqa , together with the coefficients determining the orbitals cp and within the selected basis, are treated as variational parameters. 

The c a are the spin-coupling coefficients of the a-th configuration. One should mention that in the earlier MCSC version of the method [1] all orbital configurations shared the same linear combination of the f(Ne,S) Yamanouchi-Kotani spin functions there was a single set of f(Ne,S) spin-coupling coefficients, denoted simply csk, just as in single-configuration spin-coupled theory. 

Similarly, invariance under reflection in one of the symmetry planes requires the vector of spin-coupling coefficients to be an eigenvector, with eigenvalue +1, of the matrix representing the self-inverse permutation... 

The SC solution can be referred to, more precisely, as an OBS-SC solution, inasmuch as the basis set exponential parameters have been optimized simultaneously with the coefficients with which they enter the eight orbitals (and with the spin-coupling coefficients). Values are given in Table 3. 

We first perform a standard SC calculation to obtain orbitals and the spin-coupling coefficients c°sk (cf. Eqs. (f)—(3)). 

The corresponding spin-coupling coefficients Cgk for the excited configurations are taken to be the same as c°sk. 

The energy is optimized by means of the stabilized Newton-Raphson procedure . Let 5c denote the vector of corrections to all the orbital and all the spin-coupling coefficients. It is determined by solving the equations... 

In Fig. 8a the spin-coupled and final spin-coupled VB potentials are compared for the ground state. Fig. 8b shows the associated spin-coupling coefficients. It can be seen that the spin-coupled calculation by itself yields a potential energy curve with all the essential features of the final result. The spin-coupling coefficients display much of the essential chemistry of bond formation in BH. At large distances, coefficients are C2 I and 0, which shows that the two 2s-like orbitals stemming from B are coupled to a... 

Fig. 8. (a) Spin-coupled and final spin-coupled VB potential energy curves for BH (X X + H. (b) Spin-coupling coefficients for the potential curves of (a). 

Spin-coupled wavefunctions have also recently been used in a critical appraisal of local density approximations in small molecules and for a comparison of the effects of bond formation on electron densities in momentum space and in position space. An example of a momentum space-orbital is shown in Fig. 12 for at 3.75 bohr in the range of R where the spin-coupling coefficients are undergoing large changes. The obvious diffraction effects are greatest in this region a detailed interpretation of momentum-space orbitals is presented in Ref. 81. 

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