Orthogonality of wave functions

PROOF OF ORTHOGONALITY OF WAVE FUNCTIONS CORRESPONDING TO DIFFERENT ENERGY LEVELS... 

In the exact theory of P decay the selection rules are derived from the transformation properties of operators and orthogonality of wave functions of different angular momentum states. For the actual vector and axial vector interactions the following selection rules have been obtained ... 

A variation of wave function coefficients is subject to constraints like maintaining orthogonality of the MOs, and normalization of the MOs and the total wave function. 

Based on the same two step proeedure as presented above for C2H4 (MCSCF ealeulations followed by Schmidt orthogonalization of Rydberg functions), a systematic search was conducted by progressively incorporating groups of orbitals in the active space. Two types of wave functions proved well adapted to the problem, one for in-plane excitations, the other for out-of-plane excitations from the carbene orbital. The case of the Ai states will serve as an illustration of the general approach done for all symmetries and wave functions. 

All terms on the left vanish except that for n = due to orthogonality property of wave functions and from the conditions of normalization,... 

Two full orthonormal bases of wave functions xpj) and q>j), describing the system under consideration, are related by a unitary transformation (moreover, if the corresponding transformation matrices are chosen to be real, then the transformation itself will be orthogonal)... 

To sum up the potentialities of the isospin method are not exhausted by the results stated above. There is a deep connection between orthogonal transformations of radial orbitals and rotations in isospin space (see (18.40) and (18.41)). This shows that the tensorial properties of wave functions and operators in isospin space must be dominant in the Hartree-Fock method. This issue is in need of further consideration. 

The simplest version of the self-consistent field approach is the Hartree method, in which the variational principle is applied to a non-symmetrized product of wave functions, and the orthogonality conditions for functions with different n are neglected. This leads to neglecting the exchange part of the potential, which causes errors in the results. 

In Section 1.4, we discussed the history and foundations of MO theory by comparison with VB theory. One of the important principles mentioned was the orthogonality of molecular wave functions. For a given system, we can write down the Hamiltonian H as the sum of several terms, one for each of the interactions which will determine the energy E of the system the kinetic energies of the electrons, the electron-nucleus attraction, the electron-electron and nucleus-nucleus repulsion, plus sundry terms like spin-orbit coupling and, where appropriate, other perturbations such as an applied external magnetic or electric field. We now seek a set of wave functions P, W2,... which satisfy the Schrodinger equation ... 

One not so obvious problem with the shape-consistent REP formalism (or any nodeless pseudoorbital approach) is that some molecular properties are determined primarily by the electron density in the core region (some molecular moments, Breit corrections, etc.) and cannot be computed directly from the valence-only wave function. For Phillips-Kleinman (21) types of wave functions, Daasch et al. (52) have shown that the core electron density can be approximated quite accurately by adding in the atomic core orbitals and then Schmidt orthogonalizing the valence orbitals to the core. This new set of orbitals (core plus orthogonalized valence) is a reasonable approximation to the all-electron set and can be used to compute the desired properties. This will not work for the shape-consistent case because / from Eq. (18) cannot be accurately described in terms of the core orbitals alone. On the other hand, it is clear from that equation that the corelike portion of the valence orbitals could be reintroduced by adding in fy (53),... 

Separability theorem, 309 SHAKE algorithm, 385 SHAPES force field, 40 Simulated Annealing (SA), global optimization, 342 Simulation methods, 373 Supidfiidiil, iulcs, 3j6 Susceptibility, 237 Symbolic variables, for optimizations, 416 Symmetrical orthogonalization of basis sets, 314 Symmetry adapted functions, 75 Symmetry breaking, of wave functions, 76 ... 

Now let us consider a complete set of orthogonal normalized wave functions xo. xi> , x -, , each function x being a solution of the Schrodinger time equation for the system under discussion. These wave functions are linear combinations of the stationary-state wave functions 4, , being obtained from them by the linear transformation... 

Finally, Malmqvist et al recently proposed to extend the Cl /SO method to the use of a non orthogonal set of wave functions for the main model space... 

The separated electron pair concept, which was first proposed by Hurley et al. [14] and which was later referred to as antisymmetrized product of strongly orthogonal geminals (APSG) [15], is also a special case of the group function concept. This kind of wave function is qualitatively correct at all internuclear distances and it can be improved either perturbationally [16, 17] or variationally [18]. 

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