Wave-function functionals

• Published in: Physical review. A, Atomic, molecular, and optical physics Atomic, molecular, and optical physics, 2010-04, Vol.81 (4), Article 042524
• Main Authors: Pan, Xiao-Yin, Slamet, Marlina, Sahni, Viraht
• Format: Article
• Language: English
• Subjects: ANIONS; ATOMIC AND MOLECULAR PHYSICS; ATOMS; BERYLLIUM IONS; CALCULATION METHODS; CATIONS; CHARGED PARTICLES; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DIFFERENTIAL EQUATIONS; ELEMENTS; EQUATIONS; FLUIDS; FUNCTIONALS; FUNCTIONS; GASES; HELIUM; HERMITIAN OPERATORS; HYDROGEN; HYDROGEN IONS; HYDROGEN IONS 1 MINUS; IONS; LAGRANGIAN FUNCTION; LITHIUM IONS; MATHEMATICAL OPERATORS; NONMETALS; PARTIAL DIFFERENTIAL EQUATIONS; RARE GASES; SCHROEDINGER EQUATION; VARIATIONAL METHODS; WAVE EQUATIONS; WAVE FUNCTIONS
• ISSN: 1050-2947; 1094-1622
• DOI: 10.1103/PhysRevA.81.042524

We extend our prior work on the construction of variational wave functions {psi} that are functionals of functions {chi}:{psi}={psi}[{chi}] rather than simply being functions. In this manner, the space of variations is expanded over those of traditional variational wave functions. In this article we perform the constrained search over the functions {chi} chosen such that the functional {psi}[{chi}] satisfies simultaneously the constraints of normalization and the exact expectation value of an arbitrary single- or two-particle Hermitian operator, while also leading to a rigorous upper bound to the energy. As such the wave function functional is accurate not only in the region of space in which the principal contributions to the energy arise but also in the other region of the space represented by the Hermitian operator. To demonstrate the efficacy of these ideas, we apply such a constrained search to the ground state of the negative ion of atomic hydrogen H{sup -}, the helium atom He, and its positive ions Li{sup +} and Be{sup 2+}. The operators W whose expectations are obtained exactly are the sum of the single-particle operators W={Sigma}{sub i}r{sub i}{sup n},n=-2,-1,1,2, W={Sigma}{sub i{delta}}(r{sub i}), W=-(1/2){Sigma}{sub i{nabla}i}{sup 2}, and the two-particle operators W={Sigma}{sub n}u{sup n},n=-2,-1,1,2, where u=|r{sub i}-r{sub j}|. Comparisons with the method of Lagrangian multipliers and of other constructions of wave-function functionals are made. Finally, we present further insights into the construction of wave-function functionals by studying a previously proposed construction of functionals {psi}[{chi}] that lead to the exact expectation of arbitrary Hermitian operators. We discover that analogous to the solutions of the Schroedinger equation, there exist {psi}[{chi}] that are unphysical in that they lead to singular values for the expectations. We also explain the origin of the singularity.