Chapter 5 FOUR-COMPONENT ELECTRONIC STRUCTURE METHODS FOR ATOMS
Uzi Kaldor, Ephraim Eliav, and Arie Landau School of Chemistry, Tel Aviv University, Tel Aviv, Israel
[email protected]
Abstract
Four-component methods for high-accuracy atomic calculations are reviewed. The projected (or no-virtual-pair) Dirac-Coulomb-Breit Hamiltonian serves as the starting point and defines the physical framework. One-electron four-component Dirac-Fock-Breit functions, similar in spirit to Hartree-Fock orbitals in the nonrelativistic formulation, are calculated first, followed by treatment of electron correlation. Correlation methods include multiconfiguration Dirac-Fock and many-body perturbation theory or its all-order limit, the coupled cluster approach. The Fock-space CC and its extension to the intermediate Hamiltonian approach are described. Applications address mostly transition energies in various atoms. Very large basis sets, going up to I = 8, are used. High I orbitals are particularly important for transitions involving / electrons. The Breit term is required for fine-structure splittings and for / transitions. Representative applications are described, including the gold atom, with relativistic effects of 3-4 eV on transition energies; eka-gold (Elll), where relativity changes the ground state from 6d107s to 6d*7s2; Pr3+, where the many /2 levels are reproduced with great precision; Rf (E104), where opposite effects of relativity and correlation lead finally to a 7s26d2 ground state, ~0.3 eV below the 7a26d7p predicted by MCDF; eka-lead (El 14), a potential member of the "island of stability" forecast by nuclear physics, predicted to have ionization potentials higher than all other group-14 atoms except carbon; and eka-radon (E118), which has a unique property for a rare gas, positive electron affinity. Heavy anions are described, showing instances of multiple stable excited states. Finally, applications to properties other than energy are discussed.
Keywords: Four-component methods, atomic structure, heavy atoms, superheavy atoms, relativity, electron correlation, atomic transition energies. 171 U. Kaldor and S. Wilson (eds.), Theoretical Chemistry and Physics of Heavy and Superheavy Elements, 171-210. €> 2003 Kluwer Academic Publishers. Printed in the Netherlands.
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Introduction The structure, spectroscopy and chemistry of heavy atoms exhibit large relativistic effects. These effects play an important role in lighter elements too, showing up in phenomena such as fine or hyperfine structure of electronic states. Perturbative approaches, starting from a nonrelativistic Hamiltonian, are often adequate for describing the influence of relativity on light atoms; for heavier elements, the Schrodinger equation must be supplanted by an appropriate relativistic wave equation. No closed-form relativistic many-body equation is known, and one must resort to approximate schemes. Most commonly, the Dirac Hamiltonian is used for the one-body terms, with the Coulomb repulsion serving as the two-body interaction, yielding the Dirac-Coulomb equation. The frequency-independent Breit operator is often added to give the Dirac-Coulomb-Breit scheme. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions; the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for lighter elements, and may be included in a similar manner. Four-component methods, complemented by high-level treatment of correlation, provide the most accurate approach to heavy-atom studies; they are also the most expensive in terms of computational resources. Less demanding methods are often used; they are described in other chapters of the present volume. The basic relativistic equations are described in section 1, with section 2 discussing correlation methods. Representative applications are shown in section 3, to illustrate the level of accuracy one can expect and the predictions current methods afford regarding properties of hard to access superheavy atoms. It should be noted that a complete listing of applications is beyond the scope of this chapter; the interested reader is referred to Pyykko's exhaustive database, comprising three volumes and a periodically updated web site [1]. The no-virtual-pair Dirac-Coulomb-Breit Hamiltonian, correct to second order in the fine-structure constant a, provides the framework for the four-component method described below. Many-body methods, and in particular the coupled cluster (CC) approach, give the most successful approach to the treatment of correlation. The method is described in the next section, followed by representative applications. The gold atom provided the first test of the relativistic coupled cluster method and demonstrated its accuracy. Eka-gold (element 111) is an example of a superheavy atom where relativity leads to properties (in this case
Four-component electronic structure methods for atoms
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the ground state electronic configuration) different from those of lighter homologs. The /2 levels of Pr3+ are studied in the following subsection, demonstrating the ability of the method to give accurate results for a manifold of /-shell levels by including dynamic correlation; the importance of the Breit term for getting correct fine structure splittings is also shown. Next we study the nature of ground-state configuration of Rf, where contradictory predictions have been made in the past, since relativity favors p over d orbitals whereas correlation has the opposite effect. It is shown that dynamic correlation is the decisive factor, making the 7s26d2 configuration lower than 7s2Qd7p, contrary to multiconfiguration Dirac-Fock results. The recently developed intermediate Hamiltonian Fock-space coupled cluster approach provides higher accuracy by allowing much larger P spaces; this is demonstrated by application to the electron affinities of the alkali atoms. The method is also applied to ekalead (element 114), which nuclear physics predicts to be in an "island of stability" and have isotopes with much longer lifetime than other superheavy elements. Another interesting superheavy atom is the rare gas eka-radon (El 18), which is predicted to have a positive electron affinity, an uncharacteristic property for a rare gas. Transitions involving / electrons in atoms such as La and E121 are shown to require the Breit term and at least i (I = 6) functions in the basis to attain the desired accuracy. Several studies of electron affinities, a tough property to calculate, are then listed. A brief discussion of properties other than energy concludes the roster of applications.
1.
Basic Equations
A brief review of the basic equations providing the framework to relativistic atomic structure calculations is presented below, followed by reference to early calculations at the SCF level.
1.1
The relativistic Hamiltonian
The relativistic many-electron Hamiltonian cannot be written in closed form; it may be derived perturbatively from quantum electrodynamics [2]. The simplest form is the Dirac-Coulomb (DC) Hamiltonian, where the nonrelativistic one-electron terms in the Schrodinger equation are replaced by the one-electron Dirac operator ho,
(5.1)
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with
Vme.
(5.2)
a and /0 are the four-dimensional Dirac matrices, and V^uc is the nuclear attraction operator, with the nucleus modeled as a point or finite-size charge. Only the one-electron terms in the DC Hamiltonian are relativistic, and the two-electron repulsion remains hi the nonrelativistic form. The lowest-order correction to the two-electron term in Coulomb gauge is the Breit operator [3] , w h (ai • Vi)(Q 2 • V 2 ) -o—
,
. (5-3)
Usually, the frequency-independent Breit operator is taken,
(5.4) yielding the Dirac-Coulomb-Breit (DCB) Hamiltonian #DCB = X) MO + !>/ry + Bi:i) . i
(5.5)
i
