Relativistic polarizabilities with the Lagrange-mesh method

Relativistic dipolar to hexadecapolar polarizabilities of the ground state and some excited states of hydrogenic atoms are calculated by using numerically exact energies and wave functions obtained from the Dirac equation with the Lagrange-mesh method. This approach is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. The partial polarizabilities conserving the absolute value of the quantum number $\kappa$ are also numerically exact with small numbers of mesh points. The ones where $\left|\kappa \right|$ changes are very accurate when using three different meshes for the initial and final wave functions and for the calculation of matrix elements. The polarizabilities of the $n=2$ excited states of hydrogenic atoms are also studied with a separate treatment of the final states that are degenerate at the nonrelativistic approximation. The method provides high accuracies for polarizabilities of a particle in a Yukawa potential and is applied to a hydrogen atom embedded in a Debye plasma.

Figure 1

Convergence of the $1s_{1/2}$ partial static dipole polarizabilities of the $Z=100$ hydrogenic ion as a function of the number $N$ of mesh points. Contributions from the $p_{1/2}$ ($\kappa =+1$) states (crosses for one mesh, circles for three meshes) and the $p_{3/2}$ ($\kappa =-2$) states (triangles).

Figure 2

Convergence of the $2p_{3/2}$ static dipole polarizability of the hydrogen atom as a function of the number $N$ of mesh points. Use of one mesh (crosses) and three meshes (circles).

Figure 3

Relative difference between nonrelativistic and relativistic Lagrange-mesh computations of the ground-state polarizability of Yukawa potentials (43) for $V_0=1$ as a function of $\mu$.