Relativistic mass , computation

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. 

Figure 1. The relation between central density and the mass of various degenerate star models. Chandrasekhar s curve is for white dwarfs with a mean molecular weight 2 of atomic mass units. Rudkjobing s curve is the same except for inclusion of the relativistic spin-orbit effects Rudkjobing (1952). The curve labeled Oppenheimer and Volkoff is for a set of neutron star models. The solid line marked Wheeler is a set of models computed with a generalized equation of state, from Cameron (1959).

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. 

In M0ller-Plesset theory, first-order perturbation theory does not improve on the HF energy because the zeroth-order Hamiltonian is not itself the HF Hamiltonian. However, first-order perturbation theory can be useful for estimating energetic effects associated with operators that extend the HF Hamiltonian. Typical examples of such terms include the mass-velocity and one-electron Darwin corrections that arise in relativistic quantum mechanics. It is fairly difficult to self-consistently optimize wavefunctions for systems where these tenns are explicitly included in the Hamiltonian, but an estimate of their energetic contributions may be had from simple first-order perturbation theory, since that energy is computed simply by taking the expectation values of the operators over the much more easily obtained HF wave functions. 

The first-order perturbation theory estimate of relativistic effects (inclusion of the mass-velocity and one-electron Darwin terms as suggested by Cowan and Griffin) is cheap and easy to compute as a property value at the end of a calculation. It is therefore very valuable as a check on the importance of relativistic effects, and should certainly be included in accurate calculations on, for example, transition-metal compounds. For even heavier elements relativistic effective core potentials should be used. 

The relativistic energy of a mass point is computed by differentiation of the relativistic momentum, to yield the relativistic force... 

The method works as follows. The mass velocity, Darwin and spin-orbit coupling operators are applied as a perturbation on the non-relativistic molecular wave-functions. The redistribution of charge is then used to compute revised Coulomb and exchange potentials. The corrections to the non-relativistic potentials are then included as part of the relativistic perturbation. This correction is split into a core correction, and a valence electron correction. The former is taken from atomic calculations, and a frozen core approximation is applied, while the latter is determined self-consistently. In this way the valence electrons are subject to the direct influence of the relativistic Hamiltonian and the indirect effects arising from the potential correction terms, which of course mainly arise from the core contraction. 

The main difference from a computational point of view between the second and third transition rows is that the relativistic effects are much larger for the third row. Therefore, an investigation of methods for the third transition row must start with a discussion of these effects. The first question in this context is how the spin-free effects, like those from the mass-velocity and Darwin terms, have to be treated. The second, much more difficult question is how the spin-orbit effects should be treated. For the first two transition rows spin-orbit effects can normally be neglected, but this is not generally true for the third transition row. 

Table 1 Computed energies of Li ground state (non-relativistic, Coulomb interaction only, infinite-mass nucleus) for various wavefunctions, in Hartree atomic units. This research is for a correlated exponential premultiplied by the electron-nuclear distance for the 2s electron, with the parameters given in Table 33...

In systems with heavier elements, relativistic effects must be included. In the medium range of atomic numbers (up to about 54) the so called scalar relativistic scheme is often used [21], It describes the main contraction or expansion of various orbitals (due to the Darwin s-shift or the mass-velocity term), but omits spin-orbit interaction. The latter becomes important for the heavy elements or when orbital magnetism plays a significant role. In the present version of WIEN2k the core states always are treated fully relativistically by numerically solving the radial Dirac equation. For all other states, the scalar relativistic approximation is used by default, but spin-orbit interaction (computed in a second-variational treatment [22]) can be included if needed [23]. 

We have already discussed in chapters 12 and 13 that low-order scalar-relativistic operators such as DKH2 or ZORA provide very efficient variational schemes, which comprise all effects for which the (non-variational) Pauli Hamiltonian could account for (as is clear from the derivations in chapters 11 and 13). It is for this reason that historically important scalar relativistic corrections which can only be considered perturbatively (such as the mass-velocity and Darwin terms in the Pauli approximation in section 13.1), are no longer needed and their significance fades away. There is also no further need to develop new pseudo-relativistic one- and two-electron operators. This is very beneficial in view of the desired comparability of computational studies. In other words, if there were very many pseudo-relativistic Hamiltonians available, computational studies with different operators of this sort on similar molecular systems would hardly be comparable. 

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. 

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