Upper Bounds to Total Energies

Upper Bounds to Total Energies.—In performing a calculation of the energy to third-order one also implicitly calculates the wave function to first-order. Let the wave function through first-order be written as 

A may be regarded as a variational parameter. Its optimal value may be obtained by invoking the variation principle, giving 

Unfortunately, by forming an upper bound one introduces terms depending on the square of the number of electrons and thus the quality of this upper bound deteriorates as the number of electrons is increased. This is illustrated in Table 1 for an array of well-separated nitrogen molecules. 

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