Use Of Approximate Wavefunctions Research Articles

Accurate effective potential for density amplitude and the corresponding Kohn–Sham exchange–correlation potential calculated from approximate wavefunctions

Over the past few years it has been pointed out that direct inversion of accurate but approximate ground state densities leads to Kohn–Sham exchange–correlation (xc) potentials that can differ significantly from the exact xc potential of a given system. On the other hand, the corresponding wavefunction based construction of exchange-correlation potential as done by Baerends et al and Staroverov et al obviates such problems and leads to potentials that are very close to the true xc potential. In this paper, we provide an understanding of why the wavefunction based approach gives the exchange–correlation potential accurately. Our understanding is based on the work of Levy, Perdew and Sahni (LPS) who gave an equation for the square root of density (density amplitude) and the expression for the associated effective potential in the terms of the corresponding wavefunction. We show that even with the use of approximate wavefunctions the LPS expression gives accurate effective and exchange–correlation potentials. Based on this we also identify the source of difference between the potentials obtained from a wavefunction and those given by the inversion of the associated density. Finally, we suggest exploring the possibility of obtaining accurate ground-state density from an approximate wavefunction for a system by making use of the LPS effective potential.

Detailed analysis of polyad-breaking spectroscopic Hamiltonians for multiple minima with above barrier motion: Isomerization in HO2

We present a two-dimensional model for isomerization in the hydroperoxyl radical (HO(2)). We then show that spectroscopic fitting Hamiltonians are capable of reproducing large scale vibrational structure above isomerization barriers. Two resonances, the 2:1 and 3:1, are necessary to describe the pertinent physical features of the system and, hence, a polyad-breaking Hamiltonian is required. We further illustrate, through the use of approximate wave functions, that inclusion of additional coupling terms yields physically unrealistic results despite an improved agreement with the exact energy levels. Instead, the use of a single diagonal term, rather than "extra" couplings, yields good fits with realistic results. Insight into the dynamical nature of isomerization is also gained through classical trajectories. Contrary to physical intuition the bend mode is not the initial "reaction mode," but rather isomerization requires excitation in both the stretch and bend modes. The dynamics reveals a Farey tree formed between the 2:1 and 3:1 resonances with the prominent 5:2 (2:1 + 3:1) feature effectively dividing the tree into portions. The 3:1 portion is associated with isomerization, while the 2:1 portion leads to "localization" and perhaps dissociation at higher energies than those considered in this work. Simple single resonance models analyzed on polyad phase spheres are able to account in a qualitative way for the spectral, periodic orbit, and wave function patterns that we observe.

Remarks on the Use of Approximate Wavefunctions for the Second-Order Perturbation Energy

The use of approximate ground-state wavefunctions in the calculation of interaction energies by second-order perturbation theory (e.g., calculation of spin—spin coupling constants in nuclear magnetic resonance) has been investigated. It has been found that the approximate zero-order ground-state wavefunction must satisfy a stringent condition if reliable results are to be obtained. The condition is that the approximate ground-state energy must lie below the true energy of the first excited state. This condition is rarely met for molecules more complicated than hydrogen, and hence reasonable results can be obtained only by a fortuitous cancellation of errors.

Integral Hellmann—Feynman Theorem

In the Born—Oppenheimer approximation, it is shown that for any molecular isoelectronic process the total energy change, excluding nuclear kinetic energy, is given by the formula ΔW=ΔVnn+ ∫ ρ(1)H′(1)dτ(1).Here ΔVnn is the nuclear—nuclear repulsion-energy change, and H′ is the change in the one-electron nuclear—electron attraction operator. The function ρ(1) is the normalized transition density for the change, the integral over Electrons 2 through N of the product of the initial and final electronic wavefunctions, multiplied by N and divided by the overlap integral between the initial and final wavefunctions. For infinitesimal changes, it is demonstrated that this theorem reduces to the conventional Hellmann—Feynman theorem. A corresponding differential equation and its equivalent variational principle are derived. Use of approximate wavefunctions in the several formulas is discussed, and various possible applications of the theorem are indicated. The theorem is shown to provide a rigorous basis for a pseudoclassical theory of molecular vibrations, and shown to indicate why electron-correlation effects ordinarily do not control molecular deformation.

Ground-State Hyperfine Structure and Nuclear Magnetic Moment of Thulium-169

The hyperfine structure of t e 4f/sup 13/6s/sup 2/ /sup 2/F/sub 7/2/ ground sta te of Tm/sup 169/ has been studied by means of the atomic-beam magnetic-resonance method. By the use of multiple transitions, measurements were made from which the nuclear magnetic moment could be extracted without the use of approximate wave functions. computer analysis of the experimental data gives the following results: A = -374.1374 plus or minus 0.0016 Mc/sec; g/sub j/ =-1.141l9 plus or minus 0.00004; g/sub I'/= -(2.478 plus or minus 0.032) x 10/sup -4/ (with mu /sub I/ expressed in Bohr magnetons); whence mu /sub I/ = -(0.229 plus or minus 0.003) nm (corrected for the diamagnetic shielding effect of the electrons). (auth)

Use of Approximate Wave Functions for Extrapolation to True Energy Values

A WAVE equation Hϕ = Eϕ can be written for any trial function ϕ. Only if ϕ is equal to an eigenfunction ψn, however, does E have a constant value (E n). In general, for any trial function, ϕ, there is a corresponding average energy, E av., defined by: