Non-orthogonal integral

Let us now obtain the non-orthogonal integral of two such functions, o r example 23. This is... 

In exactly the same manner all the remaining non-orthogonal integrals equal zero and it is therefore possible to rewrite the secular equation 16.14 in the form ... 

In addition to the integrals im and n the secular equation 17.16 contains the non-orthogonal integral... 

By evaluating the non-orthogonality integral (7.4.9), we have therefore been able to infer the form of the 1-electron density matrices, referred to the orbital basis ( J, and consequently the form of all 1-electron contributions to the matrix elements. 

To indicate the reduction, it will be sufficient to consider the non-orthogonality integral in (14.5.4). Thus... 

The same expression can be used with the appropriate restrictions to obtain matrix elements over Slater determinants made from non-orthogonal one-electron functions. The logical Kronecker delta expression, appearing in equation (15) as defined in (16)] must he substituted by a product of overlap integrals between the involved spinorbitals. 

Because 0 <. S, , < 1, the elements of S are completely analogous to the basis-set overlap integrals familiar in quantum chemistry (50). As the off-diagonal elements of the matrix are non-zero, that is, S , 0 for all i /, the basis is non-orthogonal. In some applications S is equivalent to what is typically called the metric matrix in statistics S is equivalent to the correlation matrix. 

It is easy to show that if any member of the set is a linear combination of the others, it will not be orthogonal to any function which has a non-zero coefficient in the linear combination. If all the functions are mutually orthogonal, they must be linearly independent. Integrals of this type are often called overlhp integrals, because they provide a numerical measure of the extent to which y, and y, overlap with each other in space. Two functions which overlap are non-orthogonal. 

Slater determinants constructed from non-orthogonal spin orbitals. If we now denote the matrix of overlap integrals between spin orbitals... 

Table 1 Hamiltonian matrix elements between Slater determinants constructed from orthogonal and non-orthogonal orbitals, h stands for the one-electron operator from the non-relativistic Hamiltonian H, the two-electron integrals are denoted as ij kl) - (i(l))(2) rr2 k(l)l(2)), i4>T -(Pn] by replacing orbital (j) with orbital (a > AO, finally, D, (i[j) and D (ij kl) are the first and second-order cofactors of the overlap determinant...

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