Non-orthogonal configuration basis

We can also represent the wavefunction in the non-orthogonal configuration basis In this case it reads... 

These relations can be used to evaluate the residual vector in the non-orthogonal configuration basis. For convenience in later expressions we define the contracted functions... 

The desired solution is reached if the matrices T , the vectors t and all elements r, vanish. The quantities and t are most easily calculated first in the non-orthogonal configuration basis, and then transformed to the orthogonal basis ... 

An advantage of using the non-orthogonal configuration basis is that the coupling coefficients are simple linear combinations of transition density matrix elements between internal N-electron states. These can in principle be... 

There have been a few VB calculations on this molecule, particularly since the recent advances in computer capability make the problem of non-orthogonality of the orbitals less serious. Maclagan and Schnuelle384 have reported the results of a minimal-basis STO calculation, including all configurations which do not involve excitation of the Be core electrons. Comparison with a MO calculation was made. A VB equivalent to the CNDO/2 method was proposed. Gallup et a/.385 have also carried out a similar calculation and proposed a population analysis scheme for VB wavefunctions. Other calculations on BeKfe by these workers have appeared.116-118... 

This use of two non-orthogonal jr-orbitals forms the basis of Paoloni s theory" of the quinquevalent nitrogen atom. Paoloni was not concerned with the construction of a wave-function for the quinquevalent nitrogen atom in a molecule, but only with indicating by means of the valence-bond structure that the two ji-electrons of the pyrrolic nitrogen (tr, tr, tr, ) configuration (tr = sp ) are involved in bonding to the adjacent carbon atoms. 

In order to obtain a consistent higher-order operator, the first idea is to refrain from expansion in 1/c. This approach leads in most cases to energy-dependent or non-hermitian operators, which prevent the formulation of a set of mutually orthogonal zero-order solutions. These could in turn be used as the basis for methods making use of configuration interaction or many-body perturbation theory. ... 

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