Wavefunctions for Large Systems

Extending the ideas described in the previous section, linear combinations of molecular wavefunctions can be used to generate approximate wavefunctions for systems containing more than one molecule. For example, a wavefunction representing an excited state of an oligonucleotide can be described as a linear combination of wavefunctions for the excited states of the individual nucleotides. 

Although the individual basis wavefunctions are not eigenfunctions of the full Hamiltonian, it is possible in principle to find linear combinations of these wavefunctions that do give such eigenfunctions, at least to the extent that the basis functions are a complete, orthonormal set. Equation (2.42) then becomes exact. The coefficients and eigenvalues are obtained by solving the simultaneous linear equations 

In general, there will be n eigenvalues of the energy ( t) that satisfy Eq. (2.44), each with its own eigenvector Ck- The eigenvectors can be arranged in a square matrix C, in which each colunui corresponds to a particular eigenvalue  

If we then find C (the inverse of C), the product C H C turns out to be a diagonal matrix with the eigenvalues on the diagonal (see Appendix 2)  

The Hamiltonian matrix is always Hermitian, and for all the cases that will concern us is symmetric (Appendix A.2). Its eigenvectors (C ) are, therefore, always real. In addition, there is always an orthonormal set of eigenvectors CrCj = 0 for i i- j, and CrCi= 1) [32]. 

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