Darwin terms

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. 

Relativistic correction to the first term (Darwin term). 

Up to now we have omitted the contribution of the Darwin terms which needs more discussion which will be given below. 

Semiclassical methods from quantum mechanics with first-order relativistic corrections obtained from the Foldy-Wouthuysen transformation match with the weak relativistic limit of functionals obtained from quantum electrodynamics, neglecting the (spurious) Darwin terms. 

The only term depending upon % is the last on the left hand side and this is just the spin-orbit couphng. If this term is dropped x-independent solutions, corresponding to zero spin-orbit couphng, are obtained. However, the large relativistic shifts, the mass velocity and Darwin terms are retained. In Fig. 3, for example, the relativistic levels remain in place but each of the spin-orbit split pairs is replaced by the average energy level... 

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). 

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... 

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. 

There are three terms which appears in the first order relativistic expression the mass-velocity tehn, the Darwin term and the spin-orbit term[12]. Out of these terms the first two are comparatively easy to calculate, while the spin-orbit interaction term is more complicated. Fortunately, the spin-orbit interaction is often not too important for chemical properties, at least for the second row transition elements. It is therefore usual to neglect it in quantum chemical calculations. 

This term is very similar to the Darwin term. It is evaluated using equation (3.134) ... 

Semiempirical spin-orbit operators play an important role in all-electron and in REP calculations based on Co wen- Griffin pseudoorbitals. These operators are based on rather severe approximations, but have been shown to give good results in many cases. An alternative is to employ the complete microscopic Breit-Pauli spin-orbit operator, which adds considerably to the complexity of the problem because of the necessity to include two-electron terms. However, it is also inappropriate in heavy-element molecules unless used in the presence of mass-velocity and Darwin terms. 

The secpnd relativistic corection, V comes from the Darwin term Hcarw -It is positive definite since ( )... 

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. 

Terms to be calculated are relativistic mass correction terms for the electron and Darwin terms for the electron-antiproton and electron-helium interactions. They can be expressed in terms of the electron momentum pe and 6—functions of the corresponding distances ... 

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)... 

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