Hylleraas-Undheim theorem

The first several variational eigenvalues are then upper bounds to the true eigenvalues, provided only that the correct number of variational eigenvalues lies below (Hylleraas-Undheim-MacDonald Theorem [24]), and the eigenvector coefficients are the optimum values of the a%jk coefficients in Eq. (3). For fixed a and / , all the eigenvalues move inexorably downward toward the exact energies as J is progressively increased. 

This relation is also called the Hylleraas Undheim-MacDonald theorem . ... 

In the local-scaling procedure embodied in Eq. (60) because the optimization is performed with respect to the particular density pn corresponding to the excited state j ]p( ri, s, ), one is searching for an energy [pn r) Wj which is an upper bound to the exact energy En>exact. This upper-bound character of the calculated energy is guaranteed by the Hylleraas-Undheim-MacDonald theorem. 

Note that in the present case the matrix elements depend on the final density p . Moreover, because this density is obtained from the transformed wavefunction, they also depend on the expansion coefficients. For this reason, Eq. (177) must be solved iteratively. Such a procedure has been applied - in a sample calculation - to the 2 S excited state of the helium atom. The upper-bound character of the energy corresponding to the energy functional for the transformed wavefunction B,p( r,- ) with respect to the exact energy is guaranteed by the Hylleraas-Undheim-MacDonald theorem. 

See, e.g., G. H. Golub, and G. E. van Loan, Matrix Computations, North Oxford Academic, Oxford, 1983 in quantum chemical literature this theorem is known as the Hylleraas-Undheim-McDonald s theorem, but it goes back to Cauchy (1826). 

The corresponding multistate variational theorem must be suitably generalized [Hylleraas-Undheim-MacDonald (HUM) interleaving theorem see J. K. L. MacDonald, Phys. Rev. 43, 830,1933]. In this case, we consider an orthonormal set of n trial functions , (r = 0, 1, 2,. .., n 1) satisfying... 

This characteristic is commonly referred to as the bracketing theorem (E. A. Hylleraas and B. Undheim, Z. Phys. 759 (1930) J. K. E. MacDonald, Phys. Rev. 43, 830 (1933)). These are strong attributes of the variational methods, as is the long and rich history of developments of analytical and computational tools for efficiently implementing such methods (see the discussions of the CI and MCSCF methods in MTC and ACP). 

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