Relativistic corrections mass-velocity operator

It comprises the non-relativistic Hamiltonian of the form pf/2me + V and the relativistic correction terms, such as the mass-velocity operator —pf/8m c2, the Darwin term proportional to Pi E and the spin-orbit coupling term proportional... 

Here we see a relativistic correction to the kinetic energy, which is commonly called the mass-velocity operator, and a relativistic correction to the potential. The latter term reduces using the equivalent of (4.14) to... 

It is clear that this approach of successive transformations in the expansion parameter 1/c yields operators that involve higher and higher powers of p the commutator series is in fact a series in p/mc. Powers of the momentum operator higher than 2 are not bounded, and when operating on the potential V produce highly singular operators. This makes it problematic to use any but the lowest-order terms in a calculation, and since the mass-velocity operator is unbounded from below, this form of the relativistic corrections must be used in perturbation theory only. 

The first of the relativistic correction terms is called the mass-velocity operator. If we expand the square root operator in the classical relativistic Hamiltonian for a free particle, we find... 

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 relativistic correction of the mass variation with velocity depends essentially on the fourth power of the nabla operator [68b]. In fact one can write the involved integral as ... 

The so-called mass-velocity term Hmv /which represents the first order (in a ) relativistic correction to the non-relativistic kinetic energy operator... 

Spin-orbit (SO) coupling corrections were calculated for the Pt atom since the relativistic effects are essential for species containing heavy elements. Other scalar relativistic corrections like the Darwin and mass-velocity terms are supposed to be implicitly included in (quasi)relativistic pseudopotentials because they mostly affect the core region of the considered heavy element. Their secondary influence can be seen in the contraction of the outer s-orbitals and the expansion of the d-orbitals. This is considered in the construction of the pseudoorbitals. The effective SO operator can be written within pseudopotential (PS) treatment in the form71 75... 

The non-relativistic PolMe (9) and quasirelativistic NpPolMe (10) basis sets were used in calculations reported in this paper. The size of the [uncontractd/contracted] sets for B, Cu, Ag, and Au is [10.6.4./5.3.2], [16.12.6.4/9.7.3.2], [19.15.9.4/11.9.5.2], and [21.17.11.9/13.11.7.4], respectively. The PolMe basis sets were systematically generated for use in non-relativistic SCF and correlated calculations of electric properties (10, 21). They also proved to be successful in calculations of IP s and EA s (8, 22). Nonrelativistic PolMe basis sets can be used in quasirelativistic calculations in which the Mass-Velocity and Darwin (MVD) terms are considered (23). This follows from the fact that in the MVD approximation one uses the approximate relativistic hamiltonian as an external perturbation with the nonrelativistic wave function as a reference. At the SCF and CASSCF levels one can obtain the MVD quasi-relativistic correction as an expectation value of the MVD operator. In perturbative CASPT2 and CC methods one needs to use the MVD operator as an external perturbation either within the finite field approach or by the analytical derivative schems. The first approach leads to certain numerical accuracy problems. 

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. 

Other relativistic corrections, such as the mass-velocity and Darwin terms, affect the wave function but do not lead to operators associated with molecular... 

Only recently has the work of Bauschlicher, Walch and Siegbahn showed the need to include d correlation, whereas the work of Werner and Martin and of Scharf, Brode and Ahlrichs stressed the importance of cluster corrections and relativistic corrections. The results of Werner et al. were obtained using CEPA-1 to account for cluster contributions, while Scharf et al. used the CPF approach. Both groups accounted for relativistic corrections by employing first-order perturbation theory, i.e. by evaluating the Cowan-Griffin operator which consists of the mass-velocity and the one-electron Darwin term of the Breit-Pauli Hamiltonian. 

Historically, the first derivations of approximate relativistic operators of value in molecular science have become known as the Pauli approximation. Still, the best-known operators to capture relativistic corrections originate from those developments which provided well-known operators such as the spin-orbit or the mass-velocity or the Darwin operators. Not all of these operators are variationally stable, and therefore they can only be employed within the framework of perturbation theory. Nowadays, these difficulties have been overcome by, for instance, the Douglas-Kroll-Hess hierarchy of approximate Hamiltonians and the regular approximations to be introduced in a later section, so that operators such as the mass-velocity and Darwin terms are no... 

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. 

Including also the next term of the expansion, Eq. (2.88), gives rise to additional operators including the mass-velocity, Darwin and one-electron spin-orbit operators, which can be used in perturbation theory calculations of relativistic corrections to the non-relativistic results of the Schrodinger equation and molecular properties. However, the expansion is based on the assumption that the scalar potential r) is small, which is not fulfilled for the inner electrons of heavy atoms, because close to the nucleus they are exposed to the strong Coulomb potential of the nucleus. For this situation the expansion is then no longer valid. Alternative expansions exist, which circumvent this... 

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. 

Numerous molecular properties which describe nonlinear effects, such as the Kerr effect (O section Second Dipole HyperpolarizabUity ) or magnetic circular dichroism (O section Magnetic Circular Dichroism ), arising in the presence of radiation and additional electric or magnetic fields, are interpreted as derivatives of the dipole polarizability (Michl and Thul-strup 1995). They can be calculated as higher-order response functions. Similarly, relativistic corrections to the polarizabilities for heavy atoms can be estimated from higher-order response functions including the mass-velocity and Darwin operators, O Eqs. 11.9 and O 11.20, as additional perturbations (Kirpekar et al. 1995). 

Mass-velocity-Darwin term operator Scalar relativistic correction of first order in c. Negative-energy states... 

In addition, there exists a two-electron operator that couples the spins of the electrons in a dipolar fashion as well as an operator that couples their oibital angular momenta. In general, the two-electron relativistic operators are less important than the one-electron mass-velocity and Darwin operators. For the neon atom, for example, we obtain the following first-order one- and two-electron corrections in the cc-pVDZ basis using a valence-electron FCI wave function ... 

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