Feynman Theorem Research Articles

Stark Effect in Hydrogen Atoms for Nonuniform Fields

The correction for the energy eigenvalues of the Schrödinger equation for a hydrogenic atom in a non-uniform field resulting from the inhomogeneity of the field is expressed in terms of expectation values involving the eigenfunctions of the system for a uniform field. Only the first-order terms in the inhomogeneity of the field are retained. An examination of the symmetry of the eigenfunctions for the uniform field, followed by an application of Gauss' law, shows that the correction depends only on one component of the field gradient tensor, regardless of the symmetry of the field, except for states with magnetic quantum number m = ±1. For the latter states we find the degeneracy is removed provided that the field is not cylindrically symmetric. We evaluate the correction by applying Feynman's theorem to a pair of 1-dimensional eigenvalue equations similar to those obtained in the separation of the uniform field problem in parabolic coordinates. All the necessary eigenvalues are calculated by the WKB method that has been previously employed in obtaining the eigenvalues for the uniform field problem. As the final result we present an expression for the zz component of the quadrupole tensor of the electron labeled according to parabolic quantum numbers. Finally, we discuss the use of this expression in the study of line broadening caused by interatomic interactions (pressure broadening).

Application of the Hellmann—Feynman and Virial Theorems to the Theoretical Calculation of Molecular Potential Constants

The Hellmann—Feynman-theorem expression for the z component of the force on a single nucleus is differentiated successively to obtain expressions which are related to the harmonic and anharmonic potential constants in a molecule. The method used to simplify the differentiations is also used to derive two expressions for the field-gradient part of the nuclear-quadrupole coupling constant; the two expressions involve only convergent integrals. The virial-theorem expression for diatomic molecules is differentiated to obtain equations for the cubic and quartic potential constants and for the Dunham constants a1 and a2. Application of the equations to quantum-chemical calculation of the potential constants is discussed.

Axially Symmetric Model for Lattice Dynamics of Metals with Application to Cu, Al, and ZrH2

A lattice dynamics model (A-S model) is proposed for metals which assumes that the restoring forces between the ions have the character of bond stretching and axially symmetric bond bending. It is shown that this type of restoring force is derivable from a sum of spherically symmetrical two-body potentials, $v({R}_{\mathrm{ij}})$, taken over all ($i, j$) pairs of lattice sites separated by a distance ${R}_{\mathrm{ij}}$. The dynamical matrix for a face-centered cubic lattice is worked out for the first three neighbor shells of atoms. The dispersion curves for aluminum and copper are derived from the elastic constant and thermal diffuse x-ray data. The A-S force constants or Cu and Al fall off rapidly with distance and give dispersion curves which are at least as good as those derived from the third-neighbor general tensor force model. Particularly noteworthy is the fact that Walker's tensor force constants for Al are nearly axially symmetric. However, Jacobsen's tensor force constants for Cu are not axially symmetric. Contrary to this observation, it is found that White's force constants derived by using Feynman's theorem are nearly axially symmetric.The A-S model is also applied to Zr${\mathrm{H}}_{2}$ which has a tetragonally deformed fluorite structure. It is shown that the six optical branches are flat to within 3% over the entire Brillouin zone and that the hydrogen-hydrogen interactions are very weak compared to the hydrogen-zirconium interactions. The dynamical matrix for the acoustic spectrum is derived.

Atomic Force Constants of Copper from Feynman's Theorem

Atomic Force Constants of Copper from Feynman's Theorem