Mass relativistic

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. 

In general, theory is a word with which most scientists are entirely comfortable. A theory is one or more rules that are postulated to govern the behavior of physical systems. Often, in science at least, such rules are quantitative in nature and expressed in the form of a mathematical equation. Thus, for example, one has the theory of Einstein that the energy of a particle, E, is equal to its relativistic mass, m, times the speed of light in a vacuum, c, squared,... 

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... 

AgRMC and AgGC are first-order contributions, which take into account relativistic mass (RMC) and gauge (GC) corrections, respectively. The first term can be expressed as ... 

When we substitute equations (3.255) and (3.256) in the remaining terms of (3.251) we find that we retain a number of awkward cross terms involving P0 whereas symmetry considerations suggest that translational terms should be completely separable for the field free case. The explanation seems to be that we have made a coordinate transformation to the centre of rest mass of all particles rather than to the centre of relativistic mass. Since the translational velocities of molecules are very much less than the speed of light, the contributions of these cross terms in P0 are expected to be very small and we ignore them in further discussion. 

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). 

The basis of these effects is found in the relativistic mass increase for a moving particle. Its mass, m, increases with its velocity, v, according to the relation... 

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... 

M—H bonds in the heavier hydrides are made of mostly np (M) and Is (H) atomic orbitals. The 6s population on the Bi atom in BiH3 is 0.14 larger than the 5s population on SbHj, while the 5s and 4s populations on SbHj and AsHj differ by only 0.07. This is mainly due to the relativistic mass-velocity stabilization of the 6s shell of Bi, also known as the inert-pair effect. As a result, the lightest NH3 and the heaviest BiHj exhibit considerable deviation from the trends given by other members in this group. 

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