Electron relativistic mass

Unlike semiempirical methods that are formulated to completely neglect the core electrons, ah initio methods must represent all the electrons in some manner. However, for heavy atoms it is desirable to reduce the amount of computation necessary. This is done by replacing the core electrons and their basis functions in the wave function by a potential term in the Hamiltonian. These are called core potentials, elfective core potentials (ECP), or relativistic effective core potentials (RECP). Core potentials must be used along with a valence basis set that was created to accompany them. As well as reducing the computation time, core potentials can include the effects of the relativistic mass defect and spin coupling terms that are significant near the nuclei of heavy atoms. This is often the method of choice for heavy atoms, Rb and up. 

The most common description of relativistic quantum mechanics for Fermion systems, such as molecules, is the Dirac equation. The Dirac equation is a one-electron equation. In formulating this equation, the terms that arise are intrinsic electron spin, mass defect, spin couplings, and the Darwin term. The Darwin term can be viewed as the effect of an electron making a high-frequency oscillation around its mean position. 

The non-relativistic Hamiltonian for an idealized system made up of a point nucleus of infinite mass and charge Ze, surrounded by N electrons of mass m and charge - e is ... 

For a polemic against the common but convenient practice of referring to relativistic mass versus rest mass see Okun L (1989) Phys Today June 1989 30. Relativistic effects like mass increase and time decrease (time dilation) at a velocity v are given by what we may call the Einstein factor , V(l-v2/c2), where c is the velocity of light. The inner electrons of a heavy atom can move at about 0.3c, so here the mass increase is l/V(l-v2/c2) = l/V(l-0.32) = 1/ 0.95 = 1.05 or 5 percent. Small but significant... 

Another method, devised by Cohen et al. to determine oxygen-rate gas collision parameters is to define an effective spin-orbit operator that includes r dependence, Zeff/r3, where the value of Zeff is adjusted to match experimental data (76). Langhoff has compared this technique with all-electron calculations using the full microscopic spin-orbit Hamiltonian for the rare-gas-oxide potential curves and found very good agreement (77). This operator has also been employed in REP calculations on Si (73), UF6 (78), U02+ and Th02 (79), and UF5 (80). The REPs employed in these calculations are based on Cowen-Griffin atomic orbitals, which include the relativistic mass-velocity and Darwin effects but do not include spin-orbit effects. Wadt (73), has made comparisons with calculations on Si by Stevens and Krauss (81), who employed the ab initio REP-based spin-orbit operator of Ermler et al. (35). 

For heavy elements, all of the above non-relativistic methods become increasingly in error with increasing nuclear charge. Dirac 47) developed a relativistic Hamiltonian that is exact for a one-electron atom. It includes relativistic mass-velocity effects, an effect named after Darwin, and the very important interaction that arises between the magnetic moments of spin and orbital motion of the electron (called spin-orbit interaction). A completely correct form of the relativistic Hamiltonian for a many-electron atom has not yet been found. However, excellent results can be obtained by simply adding an electrostatic interaction potential of the form used in the non-relativistic method. This relativistic Hamiltonian has the form... 

Near the nucleus, the electrons - especially the s electrons which have a maximum probability density for points near the nucleus - accelerate and their relativistic mass increases. The result is a decrease of the average distance to the nucleus (see the expressions (3.29) and (3.30) and page 53 for... 

Scalar relativistic (mass-velocity and Darwin) effects for the valence electrons were incorporated by using the quasi-relativistic method (55), where the first-order scalar relativistic Pauli Hamiltonian was diagonalized in the space of the nonrelativistic basis sets. The Pauli Hamiltonian used was of the form... 

Terms to be calculated are relativistic mass correction terms for the electron and Darwin terms for the electron-antiproton and electron-helium interactions. They can be expressed in terms of the electron momentum pe and 6—functions of the corresponding distances ... 

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