Variational reformulation of nonvariational energies

Although many computational techniques rely on the use of the variation principle at some early stage to provide a zero-order wave function, they do not always provide us with final energies that are variationally determined (i.e. stationary) and we cannot then invoke the Hellmann-Feynman theorem to simplify the calculation of molecular properties. Indeed, it would appear that, in order to calculate first-order properties for such nonvariational wave functions, we would need to calculate the first-order response of the wave function to the perturbation of interest, making the calculation of properties from nonvariational wave functions rather cumbersome. 

Fortunately, the additional complications associated with nonvariational energies are not as severe as we might first suspect since, in most cases, it is rather easy to modify the energy function of a nonvariational wave function in such a way that the optimized energy becomes stationary with respect to the variables of this new function, hi this variational formulation, then, the conditions of the Hellmann-Feynman theorem are indeed fulfilled and molecular properties may be calculated by a procedure that is essentially the same as for variational wave functions. 

At each value of a, X is obtained as the solution to some set of equations 

For variational wave functions, this condition corresponds to the stationary requirement 

Now consider the evaluation of the derivative of ( ) with respect to a. In terms of partial derivatives, we obtain 

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