Spin-dependent operators

Spin-dependent operators are now introduced. The external potential can be an operator Vext acting on the two-component spinors. The exchange-correlation potential is defined as in Eq. [27], although Exc is now a functional Exc = Exc[pap] of the spin-density matrix. The exchange-correlation potential is then... 

Among several other additional operators, which occur in the complete exact treatment of the parity-nonconserving weak interaction within a four-component relativistic approach, the following nuclear spin-dependent operator [123] is the most important one ... 

Unitary group approach to the many-electron correlation problem spin-dependent operators... 

The adjoint tensor operators are particularly useful in handling of two-electron MEs and, as will be seen below, also when dealing with spin-dependent operators. The action of adjoint tensors Aij on a two-column U(n) irrep a, b) produces modules that are associated with irreps given by the Littlewood-Richardson rule as a C-G series... 

Spin-dependent operators are required when we wish to account for relativistic effects in atoms and molecules [118, 119]. These effects can roughly be classified as strong and weak ones. The relativistic corrections are especially important in heavy atoms where they play a particularly significant role when describing the inner shells. In those cases, they have to be accounted for from the start, usually relying on Dirac-Hartree-Fock method. Fortunately, in most chemical phenomena, only valence electrons play a decisive role and are satisfactorily... 

The eigenstates of Hq are pure spin states with a good total spin quantum number S that are often generated via Cl approaches based on UG A or GUGA. It is thus convenient to extend efficient spin-independent UGA codes to those that can also account for spin-dependent operators. 

The coefficients being independent of the spin component Ms are completely determined by the spin-independent UGA so that the key to the evaluation of relativistic effects is the evaluation of MEs of spin-dependent operators in the basis (37). 

As we have seen above, the well-known Wigner-Eckart theorem represents a powerful tool for the evaluation of MEs. Thus, the MEs of any U(2 ) tensor that may be decomposed into the irreducible tensors of U( ) and SU(2) can be expressed as a product of three factors (1) the RME that depends on the relevant tensors and irreps of U(2n), U( ), and SU(2), (2) the U(n) C-G coefficient, and (3) the SU(2) C-G coefficient. In a multiplicity-free case for U(n) irreps, the U(n) C-G coefficients can be further factorized into simple products of isoscalar factors, yielding the ME segmentation formalism for spin-dependent operators. We shall see that this is exactly the case for one-body operators (36). 

Finally, let us emphasize that in contrast to a spin-independent case, the U(n) irreps in the U(2n) D U(n) SU(2) bases are generally changed or shifted by U(2 ) operators. Thus, one-body spin-dependent operators can change the total spin [or U(n) irrep label] by A5 = 0, 1 (or Ah = 0, 2) and two-body ones by A5 = 0, 1, 2 (or Ah = 0, 2, 4) and thus take us out of the U(n) framework. Nonetheless, as we have indicated above, the U( )-adapted creation (C ) and annihilation (C) type operators— that represent very useful tensors serving as fundamental building blocks for various U(n) tensors—are also useful in the spin-dependent U(2n) case. Indeed, since xj (X, ) are vector (contragredient vector) operators when acting on the irrep modules of U(n), their MEs in the U(n) basis are clearly related to those of (Cf) operators. In view of this fact, the MEs of one-body operators must be related to those of cfCj. The latter were carefully examined in [36] and briefly reviewed above. 

We can thus conclude that in the cases of spin-positive and spin-negative shifts, the MEs of one-body spin-dependent operators are given by the MEs of adjoint tensors Ny, k— + or —, and in the case of a zero-spin shift by the sum of MEs of the generator Ey and the adjoint tensor, as implied by Eqs. 

While in Sects. 4 and 5, the MEs of spin-dependent operators are derived strictly within the framework of the U(n) group, as enabled by the formalism of our spin-... 

Exercise 3.39 Consider the following spin-dependent operator which is a sum of one-electron operators. 

But the most important feature for practical purposes comes with a certain approximation, which is the scalar-relativistic variant of DKH. This one-component DKH approximation, in which all spin-dependent operators are separated by Dirac s relation and then simply omitted, is particularly easy to implement in widely available standard nonrelativistic quantum chemistry program packages, as Figure 12.4 demonstrates. Only the one-electron operators in matrix representation are modified to account for the kinematic or (synonymously) scalar-relativistic effects. The inclusion of the spin-orbit terms requires a two-component infrastructure of the computer program. The consequences of the neglect of spin-orbit effects have been investigate in pilot studies such as those reported in Refs. [626,655] and, naturally, the accuracy depends on the systems under consideration (see also section 14.1.3.2 and chapter 16 for further discussion). 

The development of techniques that incorporate time-reversal symmetry presented here are primarily aimed at four-component calculations, but they are equally applicable to two-component calculations in which the spin-dependent operators are included at the self-consistent field (SCF) stage of a calculation. 

In most cases, spin-free relativistic effects dominate the relativistic corrections to electronic structure. We will show later that in a perturbation expansion based on the nonrelativistic wave function, the spin-free effects for a closed-shell system enter in first order, whereas the spin-dependent effects make their first contribution in second order. Thus a reasonable approach to the treatment of relativistic effects is to include the spin-free effects fully and treat the spin-dependent effects as a perturbation. We discuss the latter task in chapter 21. In this chapter, we will examine a modification to the Dirac equation that permits the spin-fi ee and spin-dependent terms to be separated (Kutzelnigg 1984, Dyall 1994). This separation is exact, in that no approximations have been made to obtain the separation, and therefore results obtained with the modified Dirac equation are identical to those obtained with the unmodified Dirac equation. The advantage of the separation is the identification of the genuine spin-dependent terms and the possibility of their omission in approximate calculations. This development also provides a basis for discussion and analysis of spin-free and spin-dependent operators in other approximations. 

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... 

This operator is very similar to the 2 operator of the free-particle Foldy-Wouthuysen transformation it has a regularizing factor multiplied by (spin-free and spin-dependent operators from the Coulomb, Gaunt, and Breit interactions in an entirely analogous fashion to the Foldy-Wouthuysen transformation. As an example, the two-electron spin-orbit interaction in the regular approximation is... 

As for the first-order term, the second-order term may be split into spin-free and spin-dependent operators. The spin-free version of Wf is... 

There are some situations where, in the interests of clarity, we have allowed some inconsistency or sacrificed some rigor of expression. We do not usually multiply scalars by the unit matrix in expressions where the context would demand it—such as where an operator is a combination of scalar and spin-dependent operators or in a matrix expression—and we do not always indicate the rank of the unit matrix or the zero matrix by a subscript. In some places the notation would be overloaded if I2 were used instead of 1, but the matrix notation is to be inferred from the context. 

More general rules, applicable to all the more common spin-dependent operators that occur as small terms in the Hamiltonian (see Chapter 11) are also available (Cooper and McWeeny, 1966a,b). The complete results, including those already presented, are conveniently expressed in terms of the transition densities P Kk, LX rur ) etc. connecting the various CFs, and from these it is easy to write down the matrix elements of any desired operator. The connection between the matrix elements and the density functions has been dealt with in Section 5.4. We shall not list all the results but two are of special importance— the charge- and q)in-density matrices. Thus, by inspection of the 1-electron terms in (7.5.6), it is evident (cf. (5.4.11)) that... 

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