Relativistic Hartree-Fock radial orbitals

These (see Chapter 2) may be obtained utilizing the relativistic analogue of the Hartree-Fock method, normally called the Dirac-Hartree-Fock method [176-178], The relevant equations may be found in an analogous manner to the non-relativistic case, therefore here we shall present only final results (in a.u. let us recall that X = nlj, X = nl j)  

In a relativistic case we have a system of two equations for each electron nlj due to the two-component character of wave function (2.15). Here k is defined by (19.70) and V(X r), X X r) and % X r), respectively, given by 

Coefficients fk and gk are defined by (19.73) and (20.30), respectively, whereas Nxt denotes the number of electrons in subshell Quantities e, proportional to Lagrange multipliers, are in charge of orthonormality (2.17) of the radial functions. 

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Figure 3.4 Orbital radial expectation values (in ao) of (left) the 4f elements cerium through lutetium and (right) the 5f elements thorium through lawrencium from 4-component relativistic Hartree-Fock calculations averaging over the (n - 2)f (/= 1,14) valence configuration of the +3 cations...

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

Here af and cf for the cases n = l + 1 are found from the variational principle requiring the minimum of the non-relativistic energy, whereas cf (n > l + 1) - form the orthogonality conditions for wave functions. More complex, but more accurate, are the analytical approximations of numerical Hartree-Fock wave functions, presented as the sums of Slater type radial orbitals (28.31), namely... 

As described by eq.(3), APb is connected with relativistic change in atomic radial wavefunctions. It is, however, difficult to examine the atomic-number dependence of the magnitude of the ratio l(( ) -(t)"0/

radial wavefunctions, respectively, for each AO. Then we investigated the atomic-number dependence of A/ (A= - "0 for each atom in the priodic table (H to Pu), where " and denote the expectation values of each atomic orbital for Hartree-Fock and Dirac-Fock... 

Crosswhite (23) has used the correlated multiconfiguration Hartree-Fock scheme of Froese-Fisher and Saxena (24) with the approximate relativistic corrections of Cowan and Griffin (25) to calculate the Slater, spin-orbit, and Marvin radial integrals for all of the actinide ions. A comparison of the calculated and effective parameters is shown in Table II. The relatively large differences between calculation and experiment are due to the fact that configuration interaction effects have not been properly included in the calculation. In spite of this fact, the differences vary smoothly and often monotonically across the series. Because the Marvin radial integral M agrees with the experimental value, the calculated ratios M3(HRF)/M (HRF) =0.56 and M4 (HRF)/M° (HRF) =0.38 for all tripositive actinide ions, are used to fix M and M4 in the experimental scheme. 

The relativistic theory and computation of atomic structures and processes has therefore attained some sort of maturity and the various codes now available are widely used. Those mentioned so far were based on ideas originating from Hartree and his students [28], and have been developed in much the same way as the non-relativistic self-consistent field theory recorded in [28-30]. All these methods rely on the numerical solution, using finite differences, of the coupled differential equations for radial orbital wave-functions of the self-consistent field. This makes them unsuitable for the study of molecules, for which it is preferable to expand the radial amplitudes in a suitably chosen set of analytic functions. This nonrelativistic matrix Hartree-Fock method, as it is often termed, was pioneered by Hall and Lennard-Jones [31], Hall [32,33] and Roothaan [34,35], and it was Roothaan s students, Synek [36] and Kim [37] who were the first to attempt to solve the corresponding matrix Dirac-Hartree-Fock equations. Kim was able to obtain solutions for the ground state of neon in 1967, but at the expense of some numerical instability, and it seemed at the time that the matrix Dirac-Hartree-Fock scheme would not be a serious competitor to the finite difference codes. 

If spin-orbit effects are considered in ECP calculations, additional complications for the choice of the valence basis sets arise, especially when the radial shape of the / -f-1/2- and / — 1/2-spinors differs significantly. A noticeable influence of spin-orbit interaction on the radial shape may even be present in medium-heavy elements as 53I, as it is seen from Fig. 21. In many computational schemes the orbitals used in correlated calculations are generated in scalar-relativistic calculations, spin-orbit terms being included at the Cl step [244] or even after the Cl step [245,246]. It therefore appears reasonable to determine also the basis set contraction coefficients in scalar-relativistic calculations. Table 9 probes the performance of such basis sets for the fine structure splitting of the 531 P ground state in Kramers-restricted Hartree-Fock [247] and subsequent MRCI calculations [248-250], which allow the largest flexibility of... 

The expression for the lowest order contribution to the parity violating potential within the Dirac Hartree-Fock framework is identical to that within the relativistically parameterised extended Hiickel approach in eq. (146). The difference is, however, that in DHF typically atomic basis sets with fixed radial functions are employed (see [161]) and that the molecular orbital coefficients are obtained in a self-consistent Dirac Hartree-Fock procedure. Computations of parity violating potentials along these lines have occasionally been called fully relativistic, although this term is rather unfortunate. In the four-component Dirac Hartree-Fock calculations by Quiney, Skaane and Grant [155] as well as in those by Schwerdtfeger, Laerdahl and coworkers [65,156,162,163] the Dirac-Coulomb operator has been employed, which is for systems with n electrons given by... 

Orbital energies e (a.u.) and radial expectation values (r) (a,u.) for the valence shells of Ce and Lu from multi-conflguration Dirac-Hartree-Fock calculations for the average of the 4f 5d 6s and 4f 5d 6s configurations, respectively. The ratio of relativistic and corresponding nonrelativistic values is given in parentheses, Data taken... 

The proportionality constant k is usually assigned a value of approximately 1.75. The overlap matrix elements Sy are calculated with respect to a set of two component basis functions with lsjm) quantization. The radial parts were chosen to be one or two Slater functions yielding (r ) (k=-l,0,1,2) expectation values as close as possible (Lohr and Jia 1986) to the Dirac-Hartree-Fock or Hartree-Fock results tabulated by Desclaux (1973) for the relativistic and nonrelativistic case, respectively. The diagonal Hamiltonian matrix elements Hu were set equal to the corresponding orbital energies from Desclaux s tables. Due to the use of a two-component lsjm) basis set the matrices H and S are generally complex and of dimension 2nx2n, when is the number of spatial orbitals. 

Experimental data for the electronic spectra of lanthanides and actinides are available and may serve to parametrize semiempirical approaches or to calibrate ab initio calculations. Total energies, orbital energies, radial orbital expectation values, and maxima from relativistic Dirac-Hartree-Fock as well as nonrelativistic Hartree-Fock calculations have been summarized by Desclaux and form a useful starting point for (qualitative) discussions of the electronic structure of lanthanide and actinide compounds. 

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