Mass-velocity correction

This result is remarkably simple as compared to the usual methods. For a spin-polarised potential V, Kraft, Oppeneer, Antonov and Eschrig (1995) used the elimination method and found the corrections as a sum of 9 terms, which is equivalent to our Eq.(ll). They notice that three terms of their sum have a known physical meaning (spin-orbit, Darwin and mass-velocity corrections), but the other terms have no special name . 

Liquid entrainment mass velocity corrected for liquid properties and plate spacing, lb entrainment/(hr) (ft ), based on net tray area as for Wg... 

The scalar relativistic contribution is computed as the first-order Darwin and mass-velocity corrections from the ACPF/MTsmall wave function, including inner-shell correlation. 

Here we propose a new reduced-cost variant of W1 theory which we shall denote Wlc (for cheap ), with Wlch theory being derived analogously from Wlh theory. Specifically, the core correlation and scalar relativistic steps are replaced by the approximations outlined in the previous two sections, i.e. the MSFT bond additivity model for inner-shell correlation and scaled B3LYP/cc-pVTZuc+l Darwin and mass-velocity corrections. Representative results (for the W2-1 set) can be seen in Table 2.1 complete data for the molecules in the G2-1 and G2-2 sets are available through the World Wide Web as supplementary material [63] to the present paper. 

According to this equation, the interaction between two charged particles depends only mass-velocity correction, is also present in eq. (8.23). is a spin-orbit type... 

Molecules that contain heavy elements (in particular 5d transition metals) play an important role in the photochemistry and photophysics of coordination compounds for their luminescent properties as well as for their implication in catalysis and energy/electron transfer processes. Whereas molecular properties and electronic spectroscopy of light molecules can be studied in a non-relativistic quantum chemical model, one has to consider the theory of relativity when dealing with elements that belong to the lower region of the periodic table. As far as transition metal complexes are concerned one has to distinguish between different manifestations of relativity. Important but not directly observable manifestations of relativity are the mass velocity correction and the Darwin correction. These terms lead to the so-called relativistic contraction of the s- and p- shells and to the relativistic expansion of the d- and f- shells. A chemical consequence of this is for instance a destabilisation of the 5d shells with respect to the 3d shells in transition metals. 

For heavy atoms, however, it is generally accepted that relativistic corrections are essential. The most important relativistic Hamiltonian terms appear to be the mass-velocity correction, the spin-orbit coupling and the Darwin... 

Relativistic corrections make significant impact on the electronic properties of heavy atoms and molecules containing heavy atoms. The inner s orbitals are the closest to the nucleus and thus experience the high nuclear charge of the heavy atoms. Thus, the inner s orbitals shrink as a result of mass-velocity correction. This, in turn, shrinks the outer s orbitals as a result of orthogonality. Consequently, the ionization potential is also raised. The p orbitals are iilso shrunk by mass-velocity correction but to a lesser extent since the angular momentum keeps the electrons away from the nucleus. However, the spin-orbit interaction splits the p shells into pi/2 and pj/2 subshells and expands the P3/2 subshells. The net result is that the mass-velocity and spin-orbit interactions tend to cancel for the P3/2 shell but reinforce for the Pi/2. 

T. Baba, H. Fukui. Calculation of nuclear magnetic shieldings XIV. Relativistic mass-velocity corrected perturbation Hatrultotvians. Mol. Phys., 100 (2002) 623-633. 

Fully relativistic calculations even for atoms are quite complicated. The relativistic ECP parameters are, therefore, usually derived from atomic calculations that include only the most important relativistic terms of the Dirac-Fock Hamiltonian, namely, the mass-velocity correction, the spin-orbit coupling, and the so-called Darwin term.6 This is why the reference atomic calculations and the derived ECP parameters are sometimes termed quasi-relativistic. The basic assumption of relativistic ECPs is that the relativistic effects can be incorporated into the atom via the derived ECP parameters as a constant, which does not change during formation of the molecule. Experience shows that this assumption is justified for calculating geometries and bond energies of molecules. 

For the hydrogen atom with Z = = 1, more than 60% of the mass-velocity correction is thus canceled by the Darwin term, and it is obvious that an unbalanced treatment of these effects could easily give worse results than one that ignores relativistic effects entirely—not only for hydrogen, but for any other system as well ... 

The mass-velocity corrections to the orbital energy and the orbital are calculated from... 

A number of static perturbations arise from internal interactions or fields, which are neglected in the nonrelativistic Born-Oppenheimer electronic Hamiltonian. The relativistic correction terms of the Breit-Pauli Hamiltonian are considered as perturbations in nonrelativistic quantum chemistry, including Darwin corrections, the mass-velocity correction, and spin-orbit and spin-spin interactions. Some properties, such as nuclear magnetic resonance shielding tensors and shielding polarizabilities, are computed from perturbation operators that involve both internal and external fields. 

Mass-polarization, 54 Mass-velocity correction, in relativistic methods, 209, 211 Matrix element, 55, 103 McLean-Chandler basis sets, 160 Mean field approximation, 64 Metal coordination compounds, force field, 36 Metropolis sampling, in Monte Carlo techniques, 376... 

The electronic structure of the molecule, formed by two very heavy atoms , will be considerably influenced by relativistic effects, as are the mass-velocity correction, the Darwin correction, and also the spin-orbit coupling see a recent review on relativistic quantum chemistry [1] see also two earlier reviews (primarily concerned with Au) [2, 3] and three related reviews of Pyykkb [4 to 6]. In the Pt2 case, two main routes were used 1) Relativistic effective potentials (REP) or pseudopotentials were introduced into SCF theory [7 to 10] for details, see below and a review on effective potentials in molecular quantum chemistry [11]. 2) The SCF-Xa-Dirac Scattered Wave (DSW) method was used, which treats relativistic... 

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