Proper functions orthogonal

SCVB wave functions to include electron correlation is due to the fact that the VB orbitals are strongly localized, and since they are occupied by only one electron, they have the built-in feature of electrons avoiding each other. In a sense, an SCVB wave function is tte best wave function that can be constructed in terms of products of spatial orbitals. By allowing the orbitals to become non-orthogonal, the large majority (80-90%) of what is called electron correlation in an MO approach can be included in a single determinant wave function composed of spatial orbitals, multiplied by proper spin cou ing functions. 

Although the functions Rniir) according to equation (6.20) form an orthogonal set with w r) = r, the orthogonal relationships do not apply to the set of functions Sxiip) with w p) = p. Since the variable p introduced in equation (6.22) depends not only on r, but also on the eigenvalue E, or equivalently on X, the situation is more complex. To determine the proper orthogonal relationships for Sxiip), we express equation (6.24) in the form... 

Note that the choice of non-orthogonal versus orthogonal basis functions has no consequence for the numerical variational solutions (cf. Coulson s treatment of He2, note 76), but it undermines the possibility of physical interpretation in perturbative terms. While a proper Rayleigh-Schrodinger perturbative treatment of the He- He interaction can be envisioned, it would not simply truncate at second order as assumed in the PMO analysis of Fig. 3.58. Note also that alternative perturbation-theory formulations that make no reference to an... 

The above function is a one-center correlated Gaussian with exponential coefficients forming the symmetric matrix A]. <1) are rotationally invariant functions as required by the symmetry of the problem—that is, invariant with respect to any orthogonal transformation. To show the invariance, let U be any 3x3 orthogonal matrix (any proper or improper rotation in 3-space) that is applied to rotate the r vector in the 3-D space. Prove the invariance ... 

Though the core expansion leads to the appropriate fit, it may not be the proper explanation for the scale factor discrepancy. Hansen et al. (1987) note that the expansion of the core would lead to a decrease of 7.5 eV in the kinetic energy of the core electrons, at variance with the HF band structure calculations of Dovesi et al. (1982), which show the decrease to be only about 1.5 eV. An alternative interpretation by von Barth and Pedroza (1985) is based on the condition of orthogonality of the core and valence wave functions. The orthogonality requirement introduces a core-like cusp in the s-like valence states, but not in the p-states. Because of the promotion of electrons from s - p in Be metal, the high-order form factor for the crystal must be lower than that for the free atom. It is this effect that can be mimicked by the apparent core expansion. 

When we make these same comparisons for an internuclear separation of 20 bohr, we obtain the coefficients shown in Table 2.5 and the weights shown in Table 2.6. Now the orthogonalized AOs give the asymptotic ffinction with one configuration, while it requires three for the raw AOs. The energies are the same, of course. The EGSO weights imply the same situation. A little reflection will show that the three terms in the raw VB function are just those required to reconstruct the proper HI5 orbital. 

To find the state IJ-1,J-1> that has the same M-value as the one found above but one lower J-value, we must construct another combination of the two product states with M=J-1 (i.e., Ij,j-1> lj, j > and lj,j> lj, j -l>) that is orthogonal to the combination representing IJ,J-1> after doing so, we must scale the resulting function so it is properly normalized. In this case, the desired function is ... 

In the octahedral CF the ground term 6Aig is not split by the spin-orbit interaction by means of the bilinear spin-spin interaction. Consequently all the MPs vanish gz - ge = gx - ge = D = E = /tip = 0. This is caused by the fact that the angular momentum components of the type (6A[g Lfj 4Ty) and (6Aig ifl 2Ty) vanish exactly due to the orthogonality of the spin functions of different spin multiplicities. Therefore, the simple SH formalism does not work properly, and we are left with the problem of a complete spin-orbit interaction matrix between the CFTs of different spin multiplicities. 

We are not going to review here the transformation properties of spatial wave functions under the symmetry operations of molecular point groups. To prepare the discussion of the transformation properties of spinors, we shall put some effort, however, in discussing the symmetry operations of 0(3)+, the group of proper rotations in 3D coordinate space (i.e., orthogonal transformations with determinant + 1). Reflections and improper rotations (orthogonal transformations with determinant -1) will be dealt with later. 

Therefore, the hybrid density moment is simply a properly weighted average of each individual orbital moment. The orthogonality of the spherical harmonics has insured that different radial functions do not mix together. 

---