Dimensional Scaling and Spectral Properties

Lotten Hagg and Osvaldo Goscinski Department of Quantum Chemistry Uppsala University Box 518, S-751 20 Uppsala, Sweden 

Inequalities for oscillator strengths, previously used for estimating dipole polarizabilities in three dimensions, are generalized to D dimensions, and expressions for the dipole polarizability in the large D limit are obtained. The exact results, the dimensional scaling calculations, and the expressions obtained from inequalities are compared and evaluated. It is shown that the exact first order correction to the unperturbed wave function reduces to one term in the sum over states expression. The asymptotic result for the dipole polarizabilities is, in atomic units, 2 = (64Z ) D . 

In the past few years the method of dimensional scaling [12,22,23] has become increasingly more importemt in quantum theory. Using this technique one can solve the many-particle Schrodinger equation in a space of arbitrary dimension D [17]. By taking the limit of infinite 

To calculate polarizabilities, which describe how a system responds, one can use perturbation theory [7,20]. The system one wishes to study is assumed to be characterized by a Hamiltonian with eigenfunction and energy before the perturbation is applied. 

We shall restrict our attention to a bound ground state. If the perturbation can be described by the Hamiltonian the new, pertturbed system has the Hamiltoniem  

---

L. Hagg and O. Goscinski in Dimensional Scaling in Chemical Physics Dimensional Scaling and Spectral Properties D. R. Herschbach, J. Avery and O. Goscinski, Eds. Kluwer Academic Pub. p. 313 (1993). 

---