EVALUATION OF 2ND-ORDER MATRIX-ELEMENTS OF PERTURBATION-THEORY BY VARIATIONAL METHODS

It is shown that the functional {Mathematical expression} when the C k ′ 's, C i 's are defined through {Mathematical expression} O is a Hermitian positive linear operator and all the functions are square summable, has the lower and upper bounds {Mathematical expression} for any real χi's. Then a variational method for the evaluation of - <ψ/Oφ{symbol}> results by adding the maximum and minimum values of J (n). A method is proposed to calculate the approximate functions {Mathematical expression}, from approximate estremal points of J (1) and the error on -<φ{symbol}/Oφ{symbol}> through this method is discussed in comparison with other methods used in the literature. If the functions χ i 's are constrained, e.g., χ i =P i χ i, where the P i 's are projectors, then obviously {Mathematical expression} In the special case n=2 it is shown further that i) if D=-C′2 C 1+C′1 C 2≠0, then the couples of stationary points of J (2)[P i χ i ], J (2)[O -1 P i χ' i ] satisfy {Mathematical expression} and the equal sign is excluded from the above inequalities; ii) it D=0, then {Mathematical expression} In general, only stationary points such that either k 1 or k 2=0, k 1 ′ or k 2 ′ =0 can exist. Rather strict necessary conditions for J (2) to show (local) extrema of this type are given. More conclusive results are derived for functionals related to exchange perturbation theory. © 1979 Società Italiana di Fisica.