Ellipticity of the quantum mechanical Hamiltonians: corner singularity of the helium atom

We extend our approach of asymptotic parametrix construction for Hamiltonian operators from conical to edge-type singularities which is applicable to coalescence points of two particles of the helium atom and related two electron systems including the hydrogen molecule. Up to second-order, we have calculated the symbols of an asymptotic parametrix of the nonrelativistic Hamiltonian of the helium atom within the Born–Oppenheimer approximation and provide explicit formulas for the corresponding Green operators which encode the asymptotic behavior of the eigenfunctions near an edge.

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Stationary properties of approximate wave functions

Canadian Journal of Physics ◽

10.1139/p69-287 ◽

1969 ◽

Vol 47 (21) ◽

pp. 2355-2361 ◽

Cited By ~ 1

Author(s):

A. R. Ruffa

Keyword(s):

Wave Function ◽

Total Energy ◽

Helium Atom ◽

Wave Functions ◽

Quantum Mechanical ◽

Mechanical Wave ◽

Expectation Value ◽

Charge Densities ◽

Hartree Fock ◽

Approximate Wave

The accuracy of quantum mechanical wave functions is examined in terms of certain stationary properties. The most elementary of these, namely that displayed by the class of wave functions which yields a stationary value for the total energy of the system, is demonstrated to necessarily require few other stationary properties, and none of these appear to be particularly useful. However, the class of wave functions which yields both stationary energies and charge densities has very important stationary properties. A theorem is proven which states that any wave function in this class yields a stationary expectation value for any operator which can be expressed as a sum of one-particle operators. Since the Hartree–Fock wave function is known to possess these same stationary properties, this theorem demonstrates that the Hartree–Fock wave function is one of the infinitely many wave functions of the class. Methods for generating other wave functions in this class by modifying the Hartree–Fock wave function without changing its stationary properties are applied to the calculation of wave functions for the helium atom.

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The Quantum Mechanical Second Virial Coefficient for Anisotropic Interactions: Hydrogen Molecule–Helium Atom

Canadian Journal of Physics ◽

10.1139/p74-252 ◽

1974 ◽

Vol 52 (19) ◽

pp. 1914-1925 ◽

Cited By ~ 13

Author(s):

Sigurd Yves Larsen ◽

J. D. Poll

Keyword(s):

Helium Atom ◽

Scattering Theory ◽

Hydrogen Molecule ◽

Virial Coefficient ◽

Second Virial Coefficient ◽

Bound State ◽

Quantum Mechanical ◽

Closed Channel ◽

Channel Calculation ◽

Anisotropic Interactions

A derivation of a generalized Uhlenbeck and Beth formula is given which relies neither on the notion of a box nor on formal scattering theory. The expression for the 2nd virial coefficient then involves bound state energies as well as eigenphase shifts associated with the asymptotic description of a coupled (open and closed) channel calculation. The proof and specific details of such a calculation are worked out for the case of a hydrogen molecule interacting with a helium atom.

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