Operator Darwin

The first two terms, the mass-velocity and the Darwin operators, are called scalar relativistic terms since they do not involve the electron spin. They are given by... 

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... 

The Darwin operator for hydrogen-like atoms is often written in a different form. The Coulombic potential of a point nucleus, (pnuc = Vnuc /ife, can be... 

The scalar operator is called the Darwin operator and the spin-dependent operator the spin-orbit operator. We will meet these again in the chapter on perturbation theory (chapter 17). The Hamiltonian correct to order 2 in 1/c can then be written as... 

The mass-velocity term is therefore the lowest-order term from the relativistic Hamiltonian that comes from the variation of the mass with the velocity. The second relativistic term in the Pauli Hamiltonian is called the Darwin operator, and has no classical analogue. Due to the presence of the Dirac delta function, the only contributions for an atom come from s functions. The third term is the spin-orbit term, resulting from the interaction of the spin of the electron with its orbital angular momentum around the nucleus. This operator is identical to the spin-orbit operator of the modified Dirac equation. 

The multiplying factor is clearly related to the expectation of the Dirac delta function or of the spin-orbit operator, both of which scale as Z jr , and hence we can derive a correction factor for the Darwin operator of... 

The lowest-order effect of relativity on energetics of atoms and molecules—and hence usually the largest—is the spin-free relativistic effect (also called scalar relativity), which is dominated by the one-electron relativistic effect. For light atoms, this effect is relatively easily evaluated with the mass-velocity and Darwin operators of the Pauli Hamiltonian, or by direct perturbation theory. For heavier atoms, the Douglas-Kroll-Hess method or the NESC le method provide descriptions of the spin-independent relativistic effect that are satisfactory for all but the highest accuracy. 

From the form of the mass-velocity and Darwin operators, it is clear that the largest contributions to the direct scalar relativistic effect come from s electrons. Consequently, sa bonds should be strengthened by relativity. The coinage metals have valence configurations and we would expect compounds of these elements to exhibit such effects in their bonding properties. Table 22.3 shows the bond lengths, harmonic... 

NR - nonrelativistic, PT-MVD - pCTturbative treatment of mass-velocity and Darwin operators (only SCF), DKH - Douglas-Kroll-Hess, RECP — relativistic effective core potential, DC - four-component Dirac-Coulomb, Exp - experiment. 

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). 

Consider, for example, the dissociation of the fiourine molecule at the all-electron CCSD(T)/cc-pCVQZ level. For this molecule, the correction from the mass-velocity and Darwin operators is... 

There are other first-order relativistic corrections to the Hamiltonian operator. From Exercise 2.2, we recall the two-electron Darwin operator and the spin-spin contact operator ... 

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 ... 

Show that the second-quantization representation of the two-electron Darwin operator becomes... 

Writing the spin-free two-electron Darwin operator in the more symmetric form... 

The contributions from the (2S.2.10) integrals to (2S.2.8) vanish for the same reason as for the two-electron Darwin operator. The second-quantization representation of the two-electron spin-spin contact term then becomes... 

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