Variational Perturbation Theory of the Confined Hydrogen Atom

Variational perturbation theory was used to solve the Schrödinger equation for a hydrogen atom confined at the center of an impenetrable cavity. Ground state and excited state energies and expectation values calculated from the perturbation wavefunction are comparable in accuracy to results from direct numerical solution.


Introduction
Confined quantum mechanical systems are a useful model for simulating the effect of external conditions on an enclosed atom.Over sixty years ago, Michels et al [1] studied a hydrogen atom confined at the center of an impenetrable cavity and calculated the effects of pressure on kinetic energy and polarizability.This model has subsequently been applied to a wide range of physical problems.The interested reader is referred to Varshni [2] and references therein.
Application of Rayleigh-Schrödinger perturbation theory to confined systems is complicated by the lack of closed form zero-order wavefunctions.However, when a zero-order wavefunction can be obtained, variational perturbation theory provides a method to carry the calculation to high order.In this work we partition the Hamiltonian using a method developed by Sternheimer [3] and calculate energies and expectation values over a range of confinement radii.By comparison with results from direct numerical calculations and with exact results at selected confinement radii, the variational perturbation wavefunctions are shown to be highly accurate.We restricted our attention to the 1s, 2p and 3d states which, as the lowest states of a given angular momentum, are readily calculated by variational procedures.

Computational Procedures
For a zero-order wavefunction 0 ψ that satisfies the symmetry and boundary conditions of the system of interest, Sternheimer [3] defined the zero-order potential ( ) where T is the kinetic energy operator for the system and 0 ε is an arbitrary constant chosen to simplify the potential.The zero-order Hamiltonian 0 H is then given by 0 0 For a Hamiltonian H , the perturbation potential 1 where λ is an ordering parameter which will be set equal to 1 at the end of the calculation.Hylleraas-Scherr-Knight variational perturbation theory [4,5] can be used to calculate corrections to the energy and wavefunction and to evaluate expectation values.
For zero-order wavefunctions, we use where N is a radial normalization factor, A typical calculation used fifteen-term trial functions and was carried through ninth-order in the energy.For small values of 0 r the higher powers of r contribute little to the energy and the number of terms in the trial function was reduced.All calculations were performed using quadruple precision arithmetic (~30 decimal digits).

Results and Discussion
Table 1 gives the energy corrections for the 1s state over a range of confinement radii.Although the first-order correction is large for small 0 r , the magnitude of the energy corrections for second-order and higher steadily decreases and the energy expansion is in exact agreement with the energies calculated by Goldman and Joslin [7] using direct numerical solution.Similar agreement is observed for the 2p and 3d states., have a single node at ) 2 ( , the free atom wavefunction has two nodes, the innermost of which is at a.u. at 90192379 . 1 0 = r a.u.. Varshni [8] defined the critical cage radius c r as the radius of the confining sphere at which the total energy of the atom becomes zero.Sommerfeld and Welker [9] showed that c r could be obtained from the zeros of ( ) z J p , the Bessel function of the first kind of order p .If i p j , denotes the ith zero of ( ) ) , ( For each state, exact energies can be found for three values of 0 r either from simple algebra or from tables of Bessel function zeros.In order to assess the accuracy of the variational perturbation technique, we calculated the variational perturbation energies at c r , ) 2 ( + l node r and ) 3 ( + l node r for the 1s, 2p and 3d states.Rather than tabulate nearly identical numbers, we note that with the input radius specified to 1x10 -10 a.u., the exact energy and the ninth-order variational perturbation energy agree to within 1x10 -10 a.u.
Radial operators such as r and 2 r depend on the wavefunction in regions of configuration space other than those which determine the energy.By comparing expectation values calculated using an approximation method with those calculated by direct solution, we get additional information on the accuracy of the approximate wavefunction.Table 2 gives variational perturbation expectation values for 1 − r , r and 2 r for the 1s, 2p and 3d states over a range of 0 r from 1 to 8 a.u.
de Groot and ten Seldam[6], ensures that ( )0 0 = r ψ .The variational perturbation wavefunctions were constructed from trial functions of the form for a confined atom corresponds to a node in the radial wavefunction of the free atom with the same value of l , the confined atom and the free atom have the same energy.Thus, the 1s energy of the confined

Table 1 .
p ε s in a.u.for the 1s state.

Table 2 .
Variational perturbation expectation values for the confined hydrogen atom.