#### Document Type

Article

#### Abstract

Variational principles have been used extensively for estimating some given functional *F*(*φ*,* **φ*^{†}) where the functions *φ *and *φ*^{†} are well defined by a set of differential equations and boundary conditions but cannot be determined exactly. The variational principle for the estimation of a matrix element of an arbitrary Hermitian operator *W *involves not only the trial wave functions *φ** _{t} *but also trial auxiliary Lagrange functions

*L*

_{t}; the

*L*

*depend on the*

_{t}*φ*

*and on*

_{t}*W*. To determine the parameters in the

*L*

*efficiently, a functional*

_{t}*M*(

*L*

*) is constructed which is an extremum for*

_{tt}*L*

*=*

_{tt}*L*

*The technique was recently used successfully in the variational estimation of two diagonal matrix elements. We here use this technique for the variational estimation of an*

_{t}.*off-diagonal*matrix element, the generalized oscillator strengths of helium for the transition between the ground state and the excited 2

^{1}

*P*state. Two

*L*

*'s must be determined. Our results on helium indicate that variational estimates are a significant improvement over the first-order estimates. The results are also compared with those obtained nonvariationally using more elaborate ground-and excited-state wave functions; the comparison represents a check on the method. It is not yet clear which of the two approaches is more efficient.*

_{t}#### Disciplines

Physics

#### Recommended Citation

Wadehra JM, Spruch L, & Shakeshaft R. Application of an extremum principle to the variational determination of the generalized oscillator strengths of helium. Phys. Rev. A. 1978;18(2):344-49. doi: 10.1103/PhysRevA.18.344