Nonorthogonal Basis Sets

The treatment of orbital overlap in conjunction with the use of nonorthogonal basis sets deserves particular attention in treatments in terms of electron field operators. The definition of creation and annihilation operators and their anticommutation rules are basic for this development. Let s( ) be a set of atomic spin orbitals used to define the creation operators 

For an orthonormal basis set, all of the matrix elements Sgr equal zero. The basis defines a subspace of the total Hilbert space and a projection operator can be introduced through the kernel 

There will also be corresponding approximate representations of the state vectors of interest. It will be assumed that such states can be generated from a common reference state, analogous to the vacuum state. The reference state 0) should have the property that 

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Stuchebrukhov AA (1996) Tunneling currents in electron transfer reaction in proteins. II. Calculation of electronic superexchange matrix element and tunneling currents using nonorthogonal basis sets. J Chem Phys 105(24) 10819-10829... 

The principal computational step is to diagonalize the H matrix in the nonorthogonal basis set defined by... 

It is possible to achieve an eigenfunction of or some type of configuration interaction with nonorthogonal basis sets by using a combination of Slater determinants. The element between two determinants of the electronic energy matrix is ... 

One way of dealing with the problem of a nonorthogonal basis set would thus be to orthogonalize the functions 0 to obtain the transformed basis... 

AOs constitute the nonorthogonal basis set used for the solution of the HF crystaUine equations. To exploit the translation invariance property of LWFs the matrix =... 

If it is assumed that we work again with a nonorthogonal basis set Xr), then the unperturbed problem assumes the form... 

An important generalization of the second quantized formalism is the extension to the case of a nonorthogonal basis set. Such an extension is inevitable if one is aiming to develop a theory within the original AO basis, or if one is studying the interaction between different molecules whose wave functions (or MOs) mutually overlap. In what follows we shall review this theory in detail, since, up to our knowledge, no such review has yet been published. 

Thus, we see that the anticommuting character of the creation operators as well as that of their adjoints x j remain true in a nonorthogonal basis set, but the mutual commutation rule between x" and is replaced by Eq. (13.9c). 

In numerical implementation, p and h are represented in a finite basis set. If an orthogonal basis set is used, such as in some tight-binding model, we can simply replace the operators in equation (38) with corresponding matrices. In a nonorthogonal basis set, the definitions of Hamitonian and density matrices from equations (19) and (22) are used. The variational expression becomes ... 

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