Eigenfunction expansion application

The solution of these equations by means of standard eigenfunction expansions can be carried out for any curvilinear, orthogonal coordinate system for which the Laplacian operator V2 is separable. Of course, the most appropriate coordinate system for a particular application will depend on the boundary geometry. In this section we briefly consider the most common cases for 2D flows of Cartesian and circular cylindrical coordinates. 

In the previous section, the method of separation of variables was applied to some problems with special nonhomogeneous terms that could be recasted as homogeneous by using a suitable substitution. In this section, a method will be outlined that is applicable to those nonhomogeneous problems for which no simple substitution can be made to remove the nonhomogeneity. This method is called eigenfunction expansion [3,4,6]. 

E. A. Hylleraas, Z Phys. 65 (1930), 209 note 6, p. 279. Note that if(2) can alternatively be expressed as an infinite expansion in the unperturbed eigenfunctions but the Hylleraas variation-perturbation expression (1.5d) is generally more useful for practical numerical applications. 

The ground-state Hartree-Fock wave function o is the Slater determinant mi 2 m of spin-orbitals. This Slater determinant is an antisymmetrized product of the spin-orbitals [for example, see Eq. (10.36)] and, when expanded, is the sum of n terms, where each term involves a different permutation of the electrqns among the spin-orbitals. Each term in the expansion of s an eigenfunction of the MP H for example, for a four-electron system, application of H to a typical term in the I>o expansion gives... 

If we now form the matrix a from the coeflScients ay , and the matrix b from the coeflicients bmjy we see that the product of these two matrices is equal to the matrix c formed from the coeflScient CmV, moreover, all the matrices are unitary. In other words, the matrices obtained from the coeflScients in the expansion of R Pn, etc., form a representation of the group of operations which leave the Hamiltonian xmchanged. The set of eigenfunctions rpik is said to form a basis for a representation of the group, since the representation is generated by the application of operations R, S, etc. The dimension... 

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