Relativistic Hartree-Fock wavefunctions

The chemical isomer shifts decrease considerably as the formal oxidation state of the neptunium increases from +3 to +6. Comparison of the shifts with electron charge densities at the nucleus derived from non-relativistic Hartree-Fock wavefunctions for 5/ to 5/ configurations showed a linear interrelationship. An approximate value for d<7 >/ of —3-5 X lO" was derived. 

The spectra of these Ions have not been observed so far but the simple system of one electron (n =3,4,5) outside closed shells has been studied by various theoretical methods two of them consider Ions with Z>54. In [9], wavelengths and transition probabilities for 3s—3p and 3s—4p transitions are computed from relativistic Hartree-Fock wavefunctions In W and Au, from which relevant data can be interpolated for Os, Ir, and Pt. A model potential method was used to predict low energy levels and ionization potentials [10] and selected results (energy in cm" ) are given below. 

Blume and Watson,97- 98 using spherical tensor methods, were able to reduce the second part of this expression to a form suitable for calculation. They derived expectation values of this operator from non-relativistic atomic Hartree-Fock wavefunctions and hence the spin-orbit coupling constant, , for many atoms and ions. Consistent results for atoms were also obtained by Hinkley" using wavefunctions both from exponential basis sets100 and from gaussian basis sets.101 Agreement with experiment is good. 

In relativistic Hartree-Fock calculations a wavefunction correct to 0(c ) yields a total energy correct to 0 c ), but orbital energies (which anyway have no rigorous physical meaning) only correct to 0 c ) [17, 18]. 

Kim has formulated a relativistic Hartree-Fock-Roothaan equation for the ground states of closed-shell atoms using Slater-type orbitals. Relativistic effects in atoms have been reviewed by Grant. Malli and coworkers have formulated a relativistic SCF method for molecules. In this method, four-component spinor wavefunctions are obtained variationally in a self-consistent scheme using Gaussian basis sets. 

As discussed below, in the Hartree-Fock model each electron sees only the average field of the other electrons. In reality, the electrons must explicitly avoid each other because of their mutual coulombic repulsion hence their motions are correlated. The difference between the Hartree-Fock energy and the exact, non-relativistic energy is termed electron correlation energy. The Hartree-Fock wavefunction can be improved by taking a linear combination of Slater determinants, yielding a configuration interaction (Cl) wavefunction ... 

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

The so-called Hartree-Fock (HF) limit is important both conceptually and quantitatively in the quantum mechanical theory of many-body interactions. It is based upon the approximation in which one considers each particle as moving in an effective potential obtained by averaging over the positions of all other particles. The best energy calculated from a wavefunction having this physical significance is called the Hartree-Fock energy and the difference between this and the exact solution of the non-relativistic wave equation is called the correlation energy. 

If W were known exactly, the value of a first-order property calculated from equation (12) would be exact. In practice, only an approximation to W is known, and it is important to know how the expectation value differs from the exact value. Since errors in calculated dipole moments due to the breakdown of the Bom-Oppenheimer approximation are likely to be small8 (typically 0.002 a.u.), and for most molecules relativistic effects can be ignored,6 there are two separate remaining problems in practice. The first concerns the likely accuracy when the wavefunction is at the Hartree-Fock limit, the second the effect of using a truncated basis set to obtain a wavefunction away from the Hartree-Fock limit. 

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... 

Martin " was the first to estimate the effects of relativity on the spectroscopic constants of Cu2. The scalar relativistic (mass-velocity and Darwin) terms were evaluated perturbatively using Hartree-Fock or GVB (Two configuration SCF (ffg -mTu)) wavefunctions. At these levels the relativistic corrections for r, cu and D, were found to be — 0.05 A, 15 cm and -h0.06eV for SCF, and —0.05 A, + 14 cm and -l-0.07eV for GVB. The shrinking of the bond length is less than half of the estimate based on the contraction of the 4s atomic orbital. 

There exist several SCF codes for the solution of radial equations the Hartree-Fock [16] equations are only one example, and the case described above is that of the single configuration approximation, in which each electron has well-defined values of n and l. There exist several other possibilities as stressed above, in Hartree s original method, the exchange term was left out in the Hartree-Slater method [17], an approximate expression is used for the form of the exchange term. The Cowan code [20] is a pseudorelativistic SCF method, which avoids the complete four-component wavefunctions by simulating relativistic effects. 

Eqs. (l)-(3), (13), and (19) define the spin-free CGWB-AIMP relativistic Hamiltonian of a molecule. It can be utilised in any standard wavefunction based or Density Functional Theory based method of nonrelativistic Quantum Chemistry. It would work with all-electron basis sets, but it is expected to be used with valence-only basis sets, which are the last ingredient of practical CGWB-AIMP calculations. The valence basis sets are obtained in atomic CGWB-AIMP calculations, via variational principle, by minimisation of the total valence energy, usually in open-shell restricted Hartree-Fock calculations. In this way, optimisation of valence basis sets is the same problem as optimisation of all-electron basis sets, it faces the same difficulties and all the experience already gathered in the latter is applicable to the former. 

To go beyond the Hartree-Fock limit and obtain the full solution to the Schrodinger equation (in the non-relativistic and Bom-Oppenheimer limit), one would have to combine various solutions of the product type. In any calculation one obtains more molecular orbitals than needed to accommodate all the electrons in the system. In a system with 2n electrons, the n molecular orbitals with the lowest molecular orbital energies are used in the Hartree-Fock solution for the ground state (this assumes a closed shell system, where two electrons are paired up in each molecular orbital). The rest of the molecular orbitals obtained will be excited molecular orbitals. Of course, other possible wavefunctions of the product type can be formed by using excited molecular orbitals in the product. The set of all such possible products can be used as a basis set to solve the full Schrodinger equation. The solution now looks like ... 

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A relativistic Dirac-Hartree-Fock calculation is somewhat more complicated than the corresponding nonrelativistic calculation due to the fact that each wavefunction has a large and a small component. Thus for the n electron problem there are 2n coupled equations in the relativistic calculation rather than n as in the nonrelativistic calculation. There is an even more severe complication however, produced by the fact that each nonrelativistic (nl) orbital corresponds to two relativistic orbitals (n,l,j = 1+ J)and (n,l,j =1 -J) (except of course, if 1= 0). Consequently, what is a one configuration Hartree-Fock (HF) calculation non-relativistically usually corresponds to a multi-configuration Hartree-Fock (MCHF) relativistically. What this implies is that a single configuration Hartree-Fock calculation is usually less likely to give accurate results in the relativistic case than in the nonrelativistic case. 

The problem of the exchange term in Hartree-Fock equations has been treated in different ways. The HF-Slater (HFS) method was used in [15]. Numerical SCF calculations of ground-state total energies in relativistic and nonrelativistic approximations are compared in [16, 17]. HFS wavefunctions served as zeroth-order eigenfunctions to compute the relativistic Hamiltonian. In [18], seven contributions to the total energy (including magnetic interaction, retardation, and vacuum polarization terms) are detailed. 

Electron correlation effects can be defined as the difference between results obtained from the exact solution of a Schrodinger equation with a specific Hamiltonian, and the results obtained at the imcorrelated level, e.g., at the Hartree-Fock or Dirac-Hartree-Fock level. Since for all but the simplest problems the exact solution of the Schrodinger equation is not accessible and usually approximate correlated wavefunctions are used instead. Sometimes experimental values are used rather than the results for the exact solution, which is reasonable as long as the Hamiltonian used for the uncorrelated solution includes all important terms, e.g., with regard to relativistic contributions, influence of the environment of the studied system, etc. As for relativistic effects, the magnitude of electron correlation effects depends to some extent on the details of their evaluation [23]. 

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