Off-diagonal Hypervirial Relations Research Articles

Tests of accuracy for computed magnetic properties via off-diagonal hypervirial relations.

Most of the methods presently available to investigate the molecular magnetic response work extremely well for the computation of properties, such as magnetizability and nuclear magnetic shielding, but they provide insufficiently accurate current density maps, in that they do not guarantee exact conservation, leading to unphysical features in maps. The present study starts from the results obtained by Epstein and Sambe and moves forward to generalize them. An off-diagonal hypervirial relationship, connecting the matrix elements of a given differentiable function of position f(r) to its derivatives ∇f(r), via the anticommutator ∇αf,p^α + with the canonical momentum operator p^, has first been proven. Afterward, this relationship is applied to show that the equations proposed by Sambe to check the quality and conservation of computed electronic current densities can be obtained as particular cases of this general theorem, with a substantial gain in computational efficiency. Connections with previous work by Arrighini, Maestro, and Moccia are outlined, and the implications that hint at future work are discussed.

The application of hypervirial relations to periodic potentials

An investigation is made of the application of hypervirial relations to problems involving periodic potentials. Necessary and sufficient conditions are established for both diagonal and off-diagonal hypervirial relations to yield solutions of the Schrodinger equation. A practical procedure is described for using these relations which, when used on the Kronig-Penney model, gives rapidly convergent results.

Ab initio calculation of atomic contributions to the magnetic susceptibility by continuous transformation of the origin of the current density in HF, H2O, NH3, and CH4 molecules

The conventional random phase approximation (RPA) of the polarization propagator theory and a computational method based on continuous transformation of origin for the current density (CTOCD) induced within the electron cloud by an external homogeneous, static magnetic field has been employed to calculate atomic contributions to magnetic susceptibilities. The diamagnetic part of the magnetic susceptibility is written in terms of the polarization propagator. Since the paramagnetic term may also be obtained from the propagator it is thus possible to compute both contributions at the same level of approximation. The evaluated average susceptibility is independent of the origin of the vector potential, but depends on the origin of the reference frame. The atomic contributions to the diamagnetic and paramagnetic parts of the magnetic susceptibility are derived by applying off-diagonal hypervirial relations which are exactly fulfilled if the state functions are exact eigenfunctions of a model Hamiltonian. The rationalization of the magnetic susceptibilities into atomic contributions is applied to some small molecules: HF, H2O, NH3 and CH4, and the sum of these contributions is compared to the corresponding calculated total values and the experimental data for the molecular magnetic susceptibility for the same compounds. Computations are performed using basis sets of increasing quality. A series of sum rules for gauge independence of the computed results and charge-current conservation have been tested to document the accuracy of the calculation of magnetic properties.

Hypervirial relations as constraints in calculations of electronic excitation properties: the random phase approximation in configuration interaction language

Off-diagonal hypervirial relations (for example, the equivalence of the dipole length and dipole velocity forms of the electronic transition moment for exact wavefunctions) are imposed upon two approximate configuration interaction (CI) representations of the ground and excited states of interest in an atomic or molecular system. It is shown that, if the ground state is approximated by the exact Hartree-Fock function correlated by double excitations and the excited state is represented by mono-excited CI, the hypervirial constraint on the transition moments leads directly to the random phase approximation (RPA) equations. No second-quantized formulation or assumed excitation operator is invoked. The same constraint, applied to an uncorrelated ground state, leads to non-variational mono-excited CI schemes, which are related to Hartree-Fock instability conditions. The methods are illustrated by 5–31/G computations of the chiroptical properties, in dipole length, velocity and acceleration forms, of two low-l...

Note on some hypervirial calculations

The harmonic oscillator has been used to test the efficiency of some off-diagonal hypervirial relations in providing approximate solutions of the wave equation. Inaccuracies have been deliberately introduced in both initial and final states to reveal the degree of error propagation.