Nonorthonormality

Finally, in certain problems involving mainly excited states, it is preferable to first partition the total function space into two or more (rarely) zero order separately optimized spaces, each representing one or more moieties of physical significance, which are fhen allowed to mix. This implies the application of nonorthonormal configuration interaction (NONCI). 

The principal contribution to (, D xI//) comes from the first term, (xI/ D xI/ ). Furthermore, since the coefficients of and of are, according to the criterion of Eq. (8), completely dominant, the next two terms of Eq. (10) dictate the part of the function spaces of and of which are connected directly with the zero-order Fermi-sea components and therefore contribute to ( Ti D xI//) the most. Note that the recognition of fhese function spaces from direct dipole connection involves the consideration of N-electron integrals and not just one-electron dipole (or quadrupole, etc) integrals. Therefore, nonorthonormality (NON) between the two sets of orbitals also becomes a criterion of selection of the correlation components. 

When computing state mixing of separately optimized function spaces, or electron correlation, nonorthonormal Cl (NONCI) can be applied. Normally, because the number of matrix elements in such SPSA constructions is not very large, the execution of NONCI does not destroy the economy of the overall calculation. Also, in certain types of problems, such as radiative or radiationless transition rates, NON is also taken into account for off-diagonal configurational N-electron matrix elements involving state-specific basis sets that are not orthonormal. 

I.D. Petsalakis, G. Theodorakopoulos, C.A. Nicolaides, R.J. Buenker, S.D. Peyerimhoff, Nonorthonormal Cl for molecular excited states. I. The sudden polarization effect in 90° twisted ethylene, J. Chem. Phys. 81 (1984) 3161. 

In general, we will have to assume that the 5 and 5° spaces are not orthogonal. This means that there does not exist a vector in the 5-space which is orthogonal to all of the vectors of the 5°-space (22). In addition, the states )° constitute a nonorthonormal basis set for the model space, 5°. From a physical point of view, it is important to have a one-to-one correspondence between the exact eigenvectors, 1 ), and the vectors )°. However, another basis set, denoted w)°, n = 1,, biorthogonal to the previous one w)°, n = 1, N has to be defined and used in Bloch s formulation. These vectors satisfy the following equations ... 

From a modest beginning with H-like atoms, to diatomic molecules, the field has now expanded to multidimensional problems. Here density functional theory is most appropriate, and initial studies have emerged (using nonorthonormal wavelets) [18]. Our abilities to calculate the electronic structure of multi-electron substances in cubic lattices [19] and molecular vibrations in four-atoms systems [20,21] have been extended by making full use of powerful parallel computers. The approach of Arias et al. [19] to determine the electronic structure of all the atoms in the periodic table is to expand functions, f, in three dimensions as a sum of scaling functions at the lowest resolution plus wavelet functions of all finer resolutions ... 

The first theme is the significance of nonorthonormality (NON) between separately optimized state-specific wavefunctions of the form of Eq. (37). One... 

The procedure for using matrix algebra to solve the linear variation equations when nonorthonormal basis functions are used is outlined in Problem 8.49. 

Thus, all pairs of creation and/or annihilation operators anticommute except for the conjugate pairs of operators such as ap and ap. From these relationships, all other properties of the creation and annihilation operators - often referred to as the elementary operators of second quantization - follow. We note that equation (103) holds only for orthonormal sets of spin orbitals. For nonorthonormal spin orbitals, the Kronecker delta in equation (103) must be replaced by the overlap integral between the two spin orbitals. 

The MCTDH equations of motion are slightly modified for numerical reasons. Firstly, we note that it is the projector P > which ensures the constraints and preserves the orthonormality of the single-particle functions during the propagation. However, if the single-particle functions become nonorthonormal because of inaccuracies of the integration, as defined by equation (23) ceases to be a projector, and orthonormality is further destroyed by the propagation. A cure to this numerical problem is to define the projector as... 

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