The Use of the Virial Theorem and Sum-rules in Atomic Structure Calculations

Abstract

Dilatation transformations have been used sucessfully to obtain eigenvalues and eigenfunctions of metastable states such as resonances in electron-atom scattering.The applica- tion of dilatation theory requires in general the expansion of the eigenfunctions and energies of dilated Hamiltonians in powers of the dilatation parameter. These expansions yield a set of sum rules which supply stationary and stability conditions which exact solutions will satisfy auto- matically. For approximate wave functions these expansions provide tools for optimization in a given parameter space. The flrst member in the set of sum rules is the quantum virial theorem which is particularly valuable to obtain the correct balance of potential and kinetic energy. Ap- plied to resonances these expectation values are both complex numbers. Accurate calculations of properties of few electron systems are of interest for astrophysical plasma diagnostics. So far, most calculations have been performed on isolated atoms and molecules. In Debye-Huckel theory (4), the dilatation technique can be conveniently extended to allow for the inclusion of a model plasma environment. Correlated wave functions for two-electron systems have widely been applied to atomic structure calculations in screening environments modeled by the Debye-Huckel theory, e.g., (3,6). The cus- tomary expansion of the eigensolutions as linear combinations of a flnite number of correlated basis functions depends crucially on the size and choice of the basis set. In particular, moderately sized basis sets of Slater type orbitals require a careful, non-linear optimization of the exponents. The quantum virial theorem and a set of related sum rules provide tools not only to assess the quality of approximate calculations when the exact result is not known a priori but also to optimize the electronic wave function in a large space of nonlinear parameters. The present work studies the extension of the dilatation technique to ionic systems which are embedded in a Debye Plasma. The satisfaction of the virial theorem is warranted for physical reasons so that the potential and kinetic energy contributions are well balanced. In its complex version, the quantum virial theorem and the corresponding complex sum rules provide in addition a means to optimize resonance energies and widths without having to strictly orthogonalize the eigenvectors to all lower solutions of the same symmetry. In complex, non-Hermitian quantum mechanics this is not only not possible but also not wanted. Debye screening, originally developed for strong electrolytes is now used as a convenient model for plasmas at or near thermal equilibrium. In Debye plasmas, the interaction between the electrons and the atomic nucleus are screened as well as the electron-electron interaction. Two-electron systems ofier both relatively simple spectra and the potential for highly accurate calculations which makes them valuable tools for plasma diagnostics. Of course, electronic structure calculations of metastable states have been performed using a variety of difierent techniques: An extension into the complex plane is the hallmark of the dilatation transformation approach (1,5,7,10{13) | also called rotated coordinate methods | as well as the Siegert approach (14), while other approaches solve the same problem entirely with real numerics (6). In the present work we focus on the bound and resonance state calculations by augmenting the dilatation transformation approach with the implementation of the quantum virial theorem and the second order sum rule. The dilatation transformation consists mainly in the replacement of all radial variables ~r by iµ~r in the Hamiltonian operator of the system. By this procedure the computational method is extended onto relevant parts of the complex energy plane which allows for the calculation of decaying states (i.e., resonances) as eigenstates of the transformed Hamiltonian which is no longer Hermitean (8). The corresponding eigenvalues are complex-valued with its imaginary part related to the lifetime of the state. The departure from Hermiticity has consequences for the computational strategy. We lose, in particular, the well established rule of variational calculations: The lower the calculated energy the better is the wave function. One aims instead at a high degree of stability of the resonance eigenvalue against changes of the rotation angle µ. In order to calculate a resonance