Upper and lower bounds were obtained previously on matrix elements of the form ${W}_{\mathrm{nm}}=({\ensuremath{\psi}}_{n},W{\ensuremath{\psi}}_{m})$, where
W is a Hermitian operator and ${\ensuremath{\psi}}_{n}$ and ${\ensuremath{\psi}}_{m}$ are the wave functions of the
nth and
mth states of the system. The bounds are variational but nonstationary; they are expressed in terms of trial wave functions ${\ensuremath{\Psi}}_{\mathrm{nt}}$ and ${\ensuremath{\Psi}}_{\mathrm{mt}}$ containing variational parameters, but the error in the bound is of first order in the errors in the ${\ensuremath{\Psi}}_{\mathrm{nt}}$ and ${\ensuremath{\Psi}}_{\mathrm{mt}}$. The results have been either subject to rather restrictive conditions (for example, only for certain specific choices for
W and only for real wave functions) or have been very conservative. We remove most of these restrictions (
W need not be positive or negative definite, the wave functions may be complex, the system may not even be invariant under time reversal) but maintain rigorous bounds of good quality. The method of using Gram-determinant inequalities, which has been employed previously, especially by Weinhold, and which we adopt, leads to variational but nonstationary bounds on
Wnm in terms of "simple" upper bounds (which may be poor) on
W2nn. Here again, only for a very few particular choices of
W have such simple bounds on
W2nn been given previously (for example,
n restricted to be the ground state, and
W the operator
zi, the coordinate of the
ith electron). The main result of this paper is to show that such simple upper bounds can be obtained for a very wide class of operators
W in terms of the energy eigenvalues of the Hamiltonian. (They can be improved if given additional experimental information on oscillator strengths, for example). These simple bounds on
W2nn do not involve any trial wave functions. The method of variational but nonstationary bounds is illustrated for diagonal matrix elements of
r1 and
r12---we, therefore, require simple bounds on
r12 and
r14---for the states $1{s}^{2} ^{1}S$ and
1s2s3S of the helium atom, with rather good results.
Recent work [Phys. Rev. A 9, 108 (1974)] on variational principles for diagonal bound state matrix elements of arbitrary Hermitian operators is extended. In particular, it is shown that the previously derived minimum principle for the trial auxiliary function appearing in such variational principles can be constructed using a modified Hamiltonian possessing not heretofore recognized positive definite properties. Thus there is at least one alternative to the particular modified Hamiltonian on which the results of Phys. Rev. A 9, 108 (1974) originally were based.
We describe a systematic procedure for the construction of variational principles for the variational estimation of an extremely wide class of functionals $F({\ensuremath{\varphi}}^{\ifmmode\dagger\else\textdagger\fi{}},\ensuremath{\varphi})$. The quantities $\ensuremath{\varphi}$ are well defined but not known exactly because the equations defining the set $\ensuremath{\varphi}$ are not solvable exactly. The procedure is illustrated for
F, the off-diagonal matrix element ${\ensuremath{\varphi}}_{1}^{\ifmmode\dagger\else\textdagger\fi{}}\mathrm{W}{\ensuremath{\varphi}}_{2}$ of an arbitrary Hermitian linear operator
W, for ${\ensuremath{\varphi}}_{1}$ and ${\ensuremath{\varphi}}_{2}$ nondegenerate normalized bound-state eigenfunctions of the same Hamiltonian with a specified relative phase.