Nonorthogonal basis

Some details of END using a multiconfigurational electronic wave function with a complete active space (CASMC) have been introduced in terms of an orthonormal basis and for a fixed nuclear framework [25], and were recently [26] discussed in some detail for a nonorthogonal basis with electron translation factors. 

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... 

Table 1.1. Weights for nonorthogonal basis functions by various methods.

Nonorthogonality complicates the matrix diagonalization process see Of-fenhartz, pp. 338-341 for the procedure used. (Note that we can, if we like, use the Schmidt or some other orthogonalization procedure to form orthogonal linear combinations of the nonorthogonal basis functions and then use these orthogonalized basis functions to form the secular equation.)... 

The Fock matrix H in the atom-centered nonorthogonal basis ([) is transformed into the matrix H in the delocalized, linear combination orthogonal basis r// by pre- and postmultiplying H by the orthogonalizing matrix S 1/2 (Eq. 4.102). 

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

Except for the operator S, Eq. (256) is a secular equation in standard form with unperturbed part K and perturbation W. Because of S, however, Eq. (256) looks like a generalized secular equation typical of perturbation theory using a nonorthogonal basis, where S would be interpreted as the overlap matrix... 

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 ... 

The nonorthogonal basis functions Xx( )are referred to as atomic orbitals (AOs) and are often taken to be Cartesian Gaussian-type orbitals (GTOs) of the (unnormalized) form ... 

The problem of negative populations is attributable to working with a nonorthogonal basis. A symmetric transformation of all the atomic orbitals to an orthogonal basis restricts the values of the atomic populations to between zero and two. While many transformations are possible, the most commonly used is the symmetrical transformation of Lowdin, leading to the Lowdin... 

We now have the energy levels (-25.5 and -5.95 eV), but the eigenvectors of C must be transformed to give us the coefficients of the original, nonorthogonal basis functions. 

Step 7 Transforming C to C, the coefficient matrix of the original, nonorthogonal basis functions. 

It is also known that the use of a minimal basis set is not adequate for describing the electronic conduction band and the optical properties of tetrahedral silicon. A larger basis set, including the effects of higher-energy orbitals, may be needed to improve the accuracy of the model further. It is still not clear whether a nonorthogonal basis is better than an orthogonal basis. 

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... 

In Seet. 4.2, we need veetor spaee with abasis whieh is formed by A linear independent vectors gp p =, ..., k) which are not generally perpendicular or of unit length [12, 18, 19]. Sueh nonorthogonal basis, we eall a contravariant one. Covariant components of the so called metric tensor are defined by... 

---