Relativistic, numerically parameterized, optimized, effective potentials for the ground state of the atoms He through Ra

Abstract

Parameterized potentials obtained within the relativistic, optimized, effective potential framework are reported for the ground state of the atoms He through Ra. The potentials are expressed in terms of Yukawian functions times a power of r. The total, kinetic, exchange, and single-particle energies and the expectation value of the spin–spin interaction are given for each atom.

Introduction

In this work we report a relativistic single-particle potential for the atoms with $2⩽Z⩽88$ obtained within the relativistic, optimized, effective potential (ROEP) approximation. This is a variational approximation to the study of the electronic structure of atoms including relativistic effects. The atomic relativistic single-particle potentials reported here can be useful for the calculation of accurate total cross sections for ionization of atoms by impact of charged particles within both the plane-wave Born approximation [1], [2] and the distorted-wave Born approximation [3], [4], starting from the relativistic independent-electron approximation for the target atom. It is also worth mentioning here another application of local potentials in the calculation of photoelectric cross sections [5] where a distorted plane wave of the atomic effective potential is used to describe the final state of the active target electron. Knowledge of accurate cross sections is important for the analysis of several spectroscopic techniques such as electron-probe microanalysis and in industrial or medical diagnosis as radiation dosimetry because this process is the main source of characteristic X rays from materials irradiated by energetic electron beams.

The ROEP approximation was originally proposed in a nonrelativistic and single-configuration framework by Sharp and Horton [6], and by Talman and Shadwick [7], [8]. The trial wavefunction is of the same form as in the Hartree–Fock method but an extra constraint is imposed: the orbitals are the eigenfunctions of a single-particle Hamiltonian with a local potential, which is the same for all of the electrons. Thus the expectation value of the atomic Hamiltonian is a functional of this effective potential. The minimum condition on the energy gives rise to a linear integral equation on the effective potential [6], [7] whose solution provides the optimized effective potential. The energy so obtained is an upper bound to the exact one and is above the Hartree–Fock value. It is worth pointing out that the role of the effective potential here is just that of an auxiliary function used to calculate the orbitals in the wavefunction. An alternative strategy to solve the variational problem is to employ analytical expansions for both the potential and the single-particle orbitals, with parameters determined by minimizing the total energy. This method has been applied to open shell atoms within the LS coupling scheme [9], [10], leading to relative errors in the total energy below 0.002%. The local differences between the numerical and parameterized effective potential are roughly below 0.5% in the most significant range of the potential while these differences increase in the asymptotic region due to different implementation of the long range boundary conditions in the electronic part of the potential.

The relativistic generalization was carried out for closed shell atoms [11], [12], [13] and for spin-polarized atoms [14]. Open shell atoms have been studied within the jj coupling scheme by using a parameterization of the effective potential and numerically solving the Dirac equations [15], the relativistic, numerically parameterized, optimized, effective potential (RNPOEP) method. This implementation of the method does not present some of the problems posed by the purely numerical solution [7], [11], [14] providing accurate results. It has been shown to be important to include all of the ground state jj configurations that contribute to the total angular momentum of the state within an LS electronic configuration in order to compare with the nonrelativistic approach and to reproduce, even qualitatively, the experimental results. An exhaustive application of this method to the calculation of atomic ionization energies [16] showed the good performance of the RNPOEP approach as compared to other single-particle methods, such as the Dirac–Hartree–Fock method, and to the experimental results.

In this work, the relativistic effective potentials for the atoms helium to radium obtained within the RNPOEP approximation are reported. All of the jj configurations included in the LS ground state configuration that contribute to the J value of the state have been included. Atomic units (a.u.) are used throughout this work.

The relativistic effective potentials tabulated in this work have been obtained by using the relativistic, numerically parameterized, effective potential method [15]. Below we give a brief overview of the effective potential method and the parameterization scheme followed.

The N-electron relativistic Hamiltonian considered in this work for an atom of nuclear charge Z is

$$H_r=\sum _{i=1}^N{h}_D(i)+\sum _{i<j}^N\frac{1}{r_{\mathit{ij}}}\text{,}$$

where $h_D$ is the Dirac Hamiltonian for a central potential

$$h_D(i)=c{\overrightarrow{\alpha }}_i·{\overrightarrow{p}}_i+c^2{\beta }_i+V_n(r_i)\text{.}$$

Here, c is the speed of light, ${\overrightarrow{p}}_i$ the linear momentum operator of the electron i, $({\beta }_i\text{,}{\overrightarrow{\alpha }}_i)$ the Dirac matrices in the Pauli–Dirac representation acting on the coordinates of the ith electron, and $V_n(r)$ the nuclear potential, which is taken as the electrostatic potential of a uniform spherical charge distribution of radius R [17], [18]

$$V_n(r)=\left\{\begin{array}{cc}-\frac{Z}{2R}\left(3-\frac{r^2}{R^2}\right)\hfill & r<R\hfill \\ -\frac{Z}{r}\hfill & r⩾R\text{.}\hfill \end{array}\right.$$

We have used $R=2.2677\times {10}^{-5}{A}^{1/3}$ where A is the nuclear mass number.

As is well known (see, e.g., Ref. [19]), the next relativistic correction to the electron–electron interaction in the Coulomb gauge is the Breit interaction, which includes the Gaunt magnetic interaction and the leading retardation contribution of order $1/c^2$. In this work the Breit interaction is considered perturbatively by including only the magnetic part of the interaction (see Table 1)

$$-\frac{1}{2}\frac{{\overrightarrow{\alpha }}_i·\overrightarrow{{\alpha }_j}}{r_{\mathit{ij}}}$$

that provides the most important contribution to the energy [18], [20].

The optimized effective potential method is a variational approach to treat the ground state of the N-electron Hamiltonian (Eq. (1)). The trial function is, in general, a linear combination of state vectors arising from different jj configurations coupled to the total J value of the state under study,

$$|\Psi 〉=\sum _{k=1}^{N_c}{d}_k|C_k\text{;}J〉\text{,}$$

where $C_k$ and $d_k$ represent a jj configuration and its weight, respectively. The coefficients $d_k$ are obtained by diagonalization of the Hamiltonian matrix coming from the minimization of the energy with respect to the configuration weights.

The $|C_k\text{;}J〉$ vectors are linear combinations of Slater determinants coupled to the J of the state under study and built from the occupied single-particle spinors, ${\psi }_{\mathit{nljm}}$, of the corresponding jj configurations. The spinors are taken to be the single-particle eigenfunctions of the Dirac equation for the effective potential $V_e(r)$

$$\left\{c\overrightarrow{\alpha }·\overrightarrow{p}+c^2\beta +V_e(r)\right\}{\psi }_{\mathit{nljm}}={\epsilon }_{\mathit{nlj}}{\psi }_{\mathit{nljm}}\text{.}$$

The approximation outlined above is implemented within the RNPOEP scheme by using the following parameterization of the effective potential

$$V_e(r)=\left\{\begin{array}{cc}-\frac{Z}{2R}\left(3-\frac{r^2}{R^2}\right)\hfill & r<R\hfill \\ -\frac{1}{r}\left\{Z-N+1+(N-1)f(r)\right\}\hfill & r⩾R\text{,}\hfill \end{array}\right.$$

where

$$f(r)=\sum _{k=1}^{n_c}{c}_k{r}^{n_k}{e}^{-{\beta }_{k}r}$$

with the constraint $f(R)=1$ imposed in order to get continuity of the potential at $r=R$. By using this parameterization, the correct asymptotic behavior of the potential

$$V_e(r)\to -\frac{Z-(N-1)}{r}\text{as}r\to \infty$$

is fulfilled, which is important in collision problems [3].

The RNPOEP single-particle energies, Eq. (6), are very close to their Dirac–Hartree–Fock counterparts. It is worth mentioning here that the eigenvalues of the one-electron Dirac equation with the Dirac–Fock–Slater potential are in a better agreement with the experimental ionization energies, in such a way that ionization cross sections obtained within a ROEP scheme will have ionization thresholds with some discrepancies with the experimental ionization energies.

For a given set of values of the parameters of the effective potential, all of the spinors in the trial wavefunction are obtained by solving the Dirac equation (Eq. (6)). Then the trial wavefunction (Eq. (5)) is built and the expectation value of the relativistic N-electron Hamiltonian (Eq. (1)) is calculated. Therefore, for a given choice of $n_c$ and $\{n_k\}$, $k=1\text{,}\dots \text{,}{n}_c$, the total energy of the atom is a function of the parameters $\{c_k\text{,}{\beta }_k\}$ of the effective potential. By using the variational principle, the optimal effective potential is determined, as well as the best wavefunction within this scheme.

The minimization of the expectation value of the relativistic Hamiltonian is performed with two additional constraints. The first one is that the relativistic virial theorem [21], [22] must be fulfilled

$$〈\sum _{i}c{\overrightarrow{\alpha }}_i·{\overrightarrow{p}}_i〉=-〈\sum _i{V}_n(r_i)+\sum _{i<j}\frac{1}{r_{\mathit{ij}}}〉\text{.}$$

This condition prevents the collapse of the results due to the fact that the relativistic Hamiltonian is not bounded from below. The second condition is that for the occupied orbital with highest energy, the eigenvalue of Eq. (6) must be equal to the following expectation value [23];

$${\overline{\epsilon }}_{\lambda }^{\mathit{mv}}=I_{\lambda }+\sum _{\mu }(J_{\lambda \mu }-K_{\lambda \mu })\text{,}$$

where λ stands for the quantum numbers of the occupied orbital with highest energy, μ runs over all of the occupied orbitals, and I, J, and K are the usual single-particle, direct and exchange integrals calculated starting from the single-particle orbitals. These two constraints are in some sense complementary. The main contribution to the quantities involved in the virial condition arise from the internal region of the atom whereas the highest energy eigenvalue is governed by the outermost region.

• (1) The parameterization of $f(r)$ in Eq. (8) and some initial values of the variational parameters $\{c_k\text{,}{\beta }_k\}$ are selected.

• (2) The Dirac equation (Eq. (6)) is solved and all of the occupied single-particle spinors are obtained.

• (3) The $|C_k:J〉$ functions are built and the Hamiltonian matrix elements $〈C_k:J|H_r|C_l:J〉$, with $k\text{,}l=1\text{,}\dots N_c$, are computed.

• (4) The generalized eigenvalue problem is solved and the $d_k$ coefficients in Eq. (5) and the expectation value of the Hamiltonian for those parameters of the potential are obtained.

• (5) The SIMPLEX algorithm is used to predict new values of the variational parameters to minimize the expectation value of the N-electron relativistic Hamiltonian (Eq. (1)) with the above mentioned constraints.

The algorithm is iterated from step 2 until a given tolerance is reached.

Finally, description of some technical aspects of the algorithm of this work is appropriate. The parameterization of the nonrelativistic effective potential, Ref. [10], is a good starting point for its relativistic counterpart. The Dirac equation in step 2 is solved by using the very accurate analytic continuation method [24], [25], [26]. The radial part of the spinors is tabulated by using the logarithmic variable $t=\ln (r)$. This can be done straightforwardly within the analytic continuation method. Because the radial functions change rather rapidly near the origin, any of the standard quadrature methods is, in general, efficient provided the logarithmic variable is used. In this work the integrals have been calculated numerically by using a five-point Newton integration rule with a step size of 0.01 a.u. leading to an accuracy of 12 digits in double precision.