Upper bounds and the Hylleraas-Undheim theorem

Consider two orthonormal sets of A -electron basis functions S = / ) and S = / ) where S is a subset of S . The eigenvalue equations in the two basis sets are given by 

Therefore, in the linear variation method, the lowest eigenvalue will always drop as the variational space is extended, providing us with a highly systematic way of approaching the exact ground state. 

The result (4.2.38) is just a special case of a more general result embodied in Cauchy s interlace theorem [6]. According to this theorem, which is proved in Exercise 4.2, the eigenvalues of two linear variational spaces S C S are related to one another in the following manner 

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