Frank C. Sanders

On the solution of the hylleraas-scheer-knight variational-perturbation equations

Abstract Dalgarno and Drake have proposed a procedure for solving the equations that arise in the HSK variational-perturbation method. It is shown that the original Hylleraas procedure is simpler and possesses all the adavantages of the Dalgarno and Drake procedure.

Multiperturbation approach to potential energy surfaces for polyatomic molecules

In Z-dependent perturbation theory, the lowest-order wave functions for a polyatomic molecule are not only independent of the nuclear charges, but also of the total number of nuclear centers and electrons in the molecule. The complexity of the problem is then determined by the highest order retained in the calculation. Choosing the simplest possible unperturbed Hamiltonian, we describe an n-electron, m-center polyatomic molecule as n ‘‘hydrogenic’’ electrons on a single center perturbed by electron‐electron and electron‐nucleus Coulomb interactions. With this H 0 , the first-order wave function for any polyatomic molecule will be a sum of products of hydrogenic orbitals with either two-electron, one-center or one-electron, two-center first-order wave functions. These first-order wave functions are obtained from calculations on He-like and H2 -like systems. Similarly, the nth-order wave function decouples so that the most complex terms are just the nth-order wave functions of all the p-electron, q-center subsystems ( p1q5n12) contained in the molecule. We illustrate applications of this method with some results, complete through third order in the energy, for H3 -like molecules. These are compared with accurate variational results available in the literature. We conclude that, through this order, this perturbation approach is capable of yielding results comparable in accuracy to variational calculations of moderate complexity. The ease and efficiency with which such results can be obtained suggests that this method would be useful for generating detailed potential energy surfaces for polyatomic molecules.

Recent Developments in High-precision Computational Methods for Simple Atomic and Molecular Systems

Within the last few years, there has been a resurgence of interest in high-accuracy calculations of simple atomic and molecular systems. For helium, such calculations have reached an extraordinary degree of precision. These achievements are only partially based on the availability of increased computational power. We review the present state of developments for such accurate calculations, with an emphasis on variational methods. Because of the central place occupied by the helium atom and its ground state, much of the discussion centers on methods developed for helium. Some of these methods have also been applied to more complex systems, and calculations on such systems now approach or even surpass a level of precision once only associated with calculations on helium. Hence, other atoms and molecules amenable to high-precision methods are also discussed.

Equation of state for hard disks and spheres: The Kirkwood transition

Abstract A nonlinear differential equation for the pressure P of a hard-core system gives rise to a unique solution for P P 0 and a separate unique solution for P > P 0 . The former corresponds to the uniform liquid and the latter to the crystalline solid. The two distinct solutions of the differential equation are separated by the singular solution P = P 0 which represents the liquid-solid coexistence phase. The existence of a singular solution signals the translational symmetry breaking associated with the liquid-solid transition. The hard-disk and hard-sphere freezing and melting transition densities obtained compare well with computer simulation results.

Z-Dependent Perturbation Theory and Complex Rotation Method in Autoionizing States of Atoms

A combined Complex Rotation and Feshbach Projection method is implemented within Z-dependent perturbation theory to obtain the lowest-order contribution to the widths of autoionizing states of two- and three-electron atoms. This approach relies on the fact that in the complex rotation method, it is the open-channel part of the wave function that produces the imaginary component of the energy. Calculation of this part of the wave function involves the solution of a simple one-electron differential equation which, in lowest-order, can be obtained to any desired level of accuracy. These results represent the limiting values of the width for high Z values in the non-relativistic approximation, and are particularly useful for states with extremely narrow widths, as these are difficult to calculate accurately with other methods.