Time-dependent Version Research Articles

Time-dependent multiconfigurational Hartree–Fock theory

A time dependent version of multiconfigurational Hartree–Fock theory is formulated as an extension of the time-dependent Hartree–Fock approximation. The time variation of the reference state is calculated using a form of Frenkel’s variation principle, which was suggested by Langhoff, Epstein, and Karplus. A linearized form of the equations provides linear response functions, which transform correctly under contact transformations generated by one particle operators. A consequence of this feature is that the various formally equivalent expressions for the oscillator strengths of electronic transitions will give the same result, apart from truncation errors when a finite orbital basis is used. Example calculations are carried out for some π electron systems within the Pariser–Parr–Pople model using a correlated reference state suggested by Clementi and Veillard.

Numerical experiments in climate stability

Two semiempirical climate models are formulated as time-dependent problems in order to study the stability of their asymptotic steady state equilibrium solutions to perturbations in internal (in this case, initial) conditions. Whereas these models have shown dramatic sensitivity to slight changes in external parameters (e.g., solar constant), here they are found to be relatively stable to perturbations in initial conditions. For fixed external conditions our time-dependent versions of these models exhibit fully transitive behavior to positive perturbations in initial conditions, slight intransivity to negative perturbations up to −18°K, and an ice-covered earth regime is obtained for extremely large negative perturbations in initial temperatures (below −18°K). The parameterization found to be most critical in these models is the albedo-temperature coupling, especially in tropical regions. The temperature albedo parameterization from an intransitive climate model used in a time-dependent form of one of the preceding models is found to be highly sensitive to negative perturbations in both internal and external conditions. Numerical experimentation with these semiempirical models shows the important role of the tropics in maintaining the equilibrium climate and suggests that the radiation balance in equatorial latitudes might have a controlling influence on the equatorward extent of the polar ice cap. The major assumptions and approximations of semiempirical climate models are discussed critically.

Transport in nonhomogeneous superconductors

Most theoretical discussions of transport in superconductors in which the gap parameter Δ( r, t) varies in space and time have made use of local models or of time-dependent versions of the Ginzburg-Landau theory. The general non-local theory is difficult to treat mathematically, but progress is being made with use of Green's function methods and with use of the WKB method for solution of the Bogoliubov quasiparticle equations. Applications of the latter method to vortex lines in type-II superconductors and to flow normal to the N-S in the intermediate state of type-I superconductors will be discussed. Although complete solutions have not been obtained, it is possible to see qualitatively how the transition from super to normal flow occurs in the boundaries and how electric fields are set up to drive current through the normal regions with consequent dissipation.

THE STATIONARY METHOD IN THE ABSTRACT THEORY OF SCATTERING

The wave operators and scattering matrix for pairs of self-adjoint operators are constructed in an explicit and invariant form. It is assumed that the perturbation is nuclear or relatively nuclear. The equivalence of the stationary and time-dependent versions of the theory is established. The results of the paper may be interpreted as justification for the stationary approach to scattering theory within the framework of the abstract theory of operators.

On a Variational Principle for a Classical Plasma

A time-dependent version of the Hartree-Fock method is set up for a classical system, by making use of the formal similarity between Liouville's equation for such a system and the Schrödinger equation. The use of a product trial function in the resultant variational principle produces, for a plasma, the collisionless Boltzmann equation. A second application is made to a system for which short-range correlations are small but not negligible. It is found that regardless of the range of the interparticle forces and the magnitude of the density, equilibrium is described only by a Maxwellian velocity distribution.