Darwin terms effects

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. 

As expected, Ap vanishes if the strength of the spin-orbit coupling is reduced to 0 by reducing (co/c) or respectively. Both sets "f model calculations give nearly the same results indicating that the so-called scalar relativistic effects due to the mass-velocity and Darwin-term, are of minor importance for the absolute value of Ap. 

For an elementary proton r )p = 0, g = 2, and only the first term in the square brackets survives. This term leads to the well known local Darwin term in the electron-nuclear effective potential (see, e.g., [1]) and generates the contribution proportional to the factor Sio in (3.4). As was pointed out in [2], in addition to this correction, there exists an additional contribution of the same order produced by the term proportional to the anomalous magnetic moment in (6.6). 

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. 

Martin " was the first to estimate the effects of relativity on the spectroscopic constants of Cu2. The scalar relativistic (mass-velocity and Darwin) terms were evaluated perturbatively using Hartree-Fock or GVB (Two configuration SCF (ffg -mTu)) wavefunctions. At these levels the relativistic corrections for r, cu and D, were found to be — 0.05 A, 15 cm and -h0.06eV for SCF, and —0.05 A, + 14 cm and -l-0.07eV for GVB. The shrinking of the bond length is less than half of the estimate based on the contraction of the 4s atomic orbital. 

The expression proportional to (Aei) in (99) is called the Darwin term. It is sometimes heuristically explained as an effect related to Zitterbewegung, but this is rather doubtful, because electronic bound states do not exhibit any Zitterbewegung according to the Dirac equation. 

In this last section we mention a few cases, where properties other than the energy of a system are considered, which are influenced in particular by the change from the point-like nucleus case (PNC) to the finite nucleus case (FNC) for the nuclear model. Firstly, we consider the electron-nuclear contact term (Darwin term), and turn then to higher quantum electrodynamic effects. In both cases the nuclear charge density distribution p r) is involved. The next item, parity non-conservation due to neutral weak interaction between electrons and nuclei, involves the nuclear proton and neutron density distributions, i.e., the particle density ditributions n r) and n (r). Finally, higher nuclear electric multipole moments, which involve the charge density distribution p r) again, are mentioned briefly. 

The attempt to correct the non-relativistic Schrodinger equation in an approximate way for relativistic effects leads to the appearance of an one-electron operator, known as electron-nucleus Darwin term [109],... 

Finally, in Sect. 6, we have briefly given some examples for physical properties or effects, which involve the nuclear charge density distribution or the nucleon distribution in a more direct way, such that the change from a point-like to an extended nucleus is not unimportant. These include the electron-nucleus Darwin term, QED effects like vacuum polarization, and parity non-conservation due to neutral weak interaction. Hyperfine interaction, i.e., the interaction between higher nuclear electric (and magnetic)... 

Scalar relativistic effects (e.g. mass-velocity and Darwin-type effects) can be incorporated into a calculation in two ways. One of these is simply to employ effective core potentials (ECPs), since the core potentials are obtained from calculations that include scalar relativistic terms [50]. This may not be adequate for the heavier elements. Scalar relativity can be variationally treated by the Douglas-Kroll (DK) [51] method, in which the full four-component relativistic ansatz is reduced to a single component equation. In gamess, the DK method is available through third order and may be used with any available type of wavefunction. 

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. 

Neglect of relativistic effects, by using the Schrodinger instead of the Dirac equation. This is reasonably justified in the upper part of the periodic table but not in the lower half. For some phenomena, such as spin-orbit coupling, there is no classical counterpart and only a relativistic treatment can provide an understanding. The relativistic effects may be incorporated by a one-component (mass-velocity and Darwin terms), two-component (spin-orbit) or fuU four-component methods (Figure 8.2). 

After the abovementioned cancellation (of and Darwin terms), the retardation becomes one of the most important relativistic effects. As seen from Table 3.1, die effect is about 100 times larger (both for the ionization energy and the polarizability) for the electron-electron retardation than for that of the nucleus-electron. This is quite understandable because the nucleus represents a massive rock" (it is about 7000 times heavier in comparison to an electron). it moves slowly, and in the nucleus-electron interaction, only the electron contributes to the retardation effect. Two electrons make the retardation much more serious. 

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. 

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 ... 

In preceding sections we have discussed several different relativistic methods four-component Dirac—Fock with and without correlation energy, the second-order Douglas—Kroll method, and perturbation methods including the mass—velocity and Darwin terms. The relativistic effective core potential (RECP) method is another well-established means of accounting for certain relativistic effects in quantum chemical calculations. This method is thoroughly described elsewhere - anJ is basically not different in the relativistic and... 

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