Relativistic Dirac-Fock Wave Functions Research Articles

Large-scale Dirac–Fock–Breit method using density fitting and 2-spinor basis functions

We present an efficient theory and algorithm for computing four-component relativistic Dirac-Fock wave functions using the Coulomb, Gaunt, and full Breit interactions. Our implementation is based on density fitting, and is routinely applicable to systems with 100 atoms and a few heavy elements. The small components are expanded using 2-spinor basis functions. We show that the factorization of 3-index half-transformed integrals before building Coulomb and exchange matrices is essential for efficient evaluation of the Fock matrix. With the Coulomb interaction, the computational cost for evaluating the Fock operator has been found to be only 70-90 times that in the non-relativistic density-fitted Hartree-Fock method. The prefactors have been 170 and 350-450 for the Gaunt and Breit interactions, respectively. The largest molecule to which we have applied the Dirac-Fock-Coulomb method is an Ac(III) motexafin complex (130 atoms, 556 electrons, 1289 basis functions), for which one self-consistent iteration takes around 1100 s using 1024 CPU cores. In addition, we have found that, while the standard fitting basis sets are accurate for Dirac-Fock-Coulomb calculations, their accuracy is very poor for Dirac-Fock-Gaunt and Breit calculations. We report a prototype of accurate fitting basis sets for these cases.

Ab initio all-electron fully relativistic Dirac–Fock self-consistent field calculations for molecules of superheavy elements: Seaborgium hexabromide

Gargantuan ab initio all-electron fully relativistic Dirac–Fock (DF) and nonrelativistic (NR) Hartree–Fock (HF) limit self-consistent field (SCF) molecular calculations are reported for SgBr6 at various Sg–Br bond distances assuming an octahedral geometry. Our fully relativistic Dirac–Fock and nonrelativistic HF calculations predict for SgBr6 bond distance of 2.52 and 2.59 Å, respectively. Both our DF and NR HF SCF calculations predict the ground state of SgBr6 to be bound, with the predicted atomization energy of 18.75 and 11.53 eV, respectively. A relativistic Dirac–Fock wave function predicts for SgBr6∼63% larger atomization energy than the corresponding NR HF calculation. The vertical ionization potential of SgBr6 calculated with our DF and HF wave functions is almost the same, viz., 10.60 and 10.78 eV, respectively. This is due to the fact that the HOMO consists entirely of the combination of the 4p AOs of the six Br ligands, for which relativistic effects are nominal. However, the vertical electron affinity calculated with our HF and DF wave function for SgBr6 is 5.35 and 3.80 eV, respectively. The calculated HF HOMO–LUMO gap of 7.74 eV is in fairly close agreement with that of 8.91 eV obtained from the corresponding DF relativistic MOs for SgBr6. These results can be understood in terms of the nature of the HOMOs and LUMOs calculated in our HF and DF calculations for SgBr6. Mulliken population analysis of our relativistic DF and HF wave functions yields a charge of 1.26 and 0.70, respectively on Sg in SgBr6; our DF wave function predicts SgBr6 to be more ionic (and less volatile) than that by the corresponding HF wave function. Our prediction of the bond dissociation energy of 44 and 72 kcal mol−1 with our NR HF and relativistic DF wave functions, respectively for SgBr6 is a first for a species of a superheavy transactinide element, as is our prediction of a positive electron affinity for SgBr6 with both our HF and DF wave functions.

The electron momentum spectroscopy of lead

Results of electron momentum spectroscopy measurements on lead vapour at a total energy of 1000 eV are presented in conjunction with a theoretical analysis using relativistic structure calculations. The 6p-1 transition is seen to be split, resulting in transitions to the 6p1 and 6p3/2 ion states. The measured branching ratio for these transitions is reproduced when a multiconfiguration Dirac-Fock wavefunction is used for the target ground state. The transition involving ejection of an electron from the 6s subshell is also seen to undergo significant splitting. The splitting in this case is mainly due to final-state configuration interaction, although a multiconfiguration target state must again be used to obtain detailed agreement with experiment. The measured momentum probability distributions for the 6p and 6s electrons agree with the orbital momentum distributions given by relativistic Dirac-Fock wavefunctions, although distortion effects are evident at high momenta for the 6s-1 transitions. The corresponding non-relativistic Hartree-Fock momentum distributions are in poorer agreement with the data. An inner valence 5d-1 transition is also seen, but the cross section is too small for detailed comparison with theory.

Energy shifts of M x rays.

The recent experiment by Springer, Bennett, and Baisden on M-x-ray energy shifts is reanalyzed, taking advantage of existing extensive work on K x-rays. The shifts, calculated by use of relativistic Dirac-Fock wave functions, are about a factor of 3 smaller than the experimental values. It is pointed out that x-ray energy shifts can provide information on the healing process in metals.

Magnetic form factor of Eu 2+ in EuO

Polarized neutron measurements of the magnetic form factor of Eu 2+ in EuO were made, to check for possible solid state effects on the radial extent of the 4f wavefunctions in this ferromagnetic insulator. Previous studies show good agreement between the magnetic form factors observed for the heavy rare earth metals and those calculated for the free ions using relativistic Dirac-Fock wavefunctions. However, the same comparison in this case shows an approximate 9% expansion of the 4f wavefunctions in EuO relative to that calculated for the Eu 2+ ion. This apparent solid state effect has the same sign but twice the magnitude of that recently calculated by Byrom, Ellis and Freeman for (EuO 6) −12 clusters.

Polarized-neutron study of the coherent paramagnetic scattering amplitude of metallic erbium

Polarized-neutron techniques have been used to measure the coherent paramagnetic scattering amplitude of high-purity single crystals of metallic erbium at 300 K. The angular dependence of the coherent-scattering amplitude has been determined with the external magnetic field ($H=22$ kOe) applied parallel as well as perpendicular to the $c$ axis of the crystal. It has been found that the angular dependence of the scattering amplitude for $\stackrel{\ensuremath{\rightarrow}}{H}\ensuremath{\parallel}\stackrel{\ensuremath{\rightarrow}}{\mathrm{c}}$ as well as $\stackrel{\ensuremath{\rightarrow}}{H}\ensuremath{\parallel}\stackrel{\ensuremath{\rightarrow}}{\mathrm{c}}$ is in good agreement, except at small scattering angles, with the $4f$-electron magnetic form factor calculated using relativistic Dirac-Fock wave functions. At small scattering angles the experimental points were found to be systematically higher than the theoretical curves obtained by least-square fitting of the calculated $4f$-electron form factor to the large-angle experimental points. This small-angle contribution, attributed to the polarization of the conduction bands of erbium, has an unusual angular dependence; it is different from either a $5d$ or $6s$ atomic form factor and exhibits a non-negative oscillatory character.

Neutron-diffraction study of UO2: Paramagnetic state

The magnetic form factor of the induced moment in U${\mathrm{O}}_{2}$ has been measured in the paramagnetic state at $T=64$ K with polarized neutrons in an applied field of 50 kOe. The theoretical form factor is derived with the tensor-operator method and the $〈{j}_{i}〉$ integrals for ${\mathrm{U}}^{4+}$ derived from relativistic Dirac-Fock wave functions. Three models for the electronic state are considered: (i) the free-ion ${\mathrm{U}}^{4+}$ ground state, (ii) the $^{3}H_{4}$ ${\ensuremath{\Gamma}}_{5}$ crystal-field ground state, and (iii) the Rahman and Runciman mixed-$J$ ${\ensuremath{\Gamma}}_{5}$ ground state. The calculation of the magnetic scattering in the presence of intermediate coupling and $J$ mixing is discussed. Whereas the experimental data are unable to distinguish between the small angular differences predicted for the three models, the radial dependence of the magnetic scattering is found to be in good agreement with theory and confirms the validity of the Dirac-Fock $〈{j}_{i}〉$ integrals.

Neutron magnetic form factors of uranium ions

The radial integrals, which contribute to the neutron magnetic form factors, of several uranium ions (U$sup 3+$, U$sup 4+$, U$sup 5+$) are determined from fully relativistic Dirac-Fock wave functions. The expectation values of the r/subn/ operators, , with n = -3, 2, 4, and 6 are given for the uranium ions. The possiblity of determining the ionization state by magnetic scattering experiments is discussed. The radial integrals are used to analyze the experimental form factor of US determined by Wedgwood. Satisfactory agreement is found by choosing a 5f$sup 2$ GAMMA$sub 1$ crystal-field ground state with a large internal exchange field inducing ferromagnetism. Contributions to the magnetic scattering from 6d electrons are found to be negligible for values of sintheta/lambda > 0.2 A$sup -1$. (AIP)