Hamiltonian modified Dirac

A different approach is chosen when the screening of nuclear potential due to the electrons is incorporated in /z . Transformation to the eigenspinor basis is then only possible after the DHF equation is solved which makes it more difficult to isolate the spin-orbit coupling parts of the Hamiltonian. Still, it is also in this case possible to define a scalar relativistic formalism if the so-called restricted kinetic balance scheme is used to relate the upper and lower component expansion sets. The modified Dirac formalism of Dyall [24] formalizes this procedure and makes it possible to identify and eliminate the spin-orbit coupling terms in the selfconsistent field calculations. The resulting 4-spinors remain complex functions, but the matrix elements of the DCB Hamiltonian exhibit the non-relativistic symmetry and algebra. 

All exact-decoupling approaches can be related to the modified Dirac equation and we closely follow here the work presented in Refs. [16,647]. Two-component electrons-only Hamiltonians can be obtained from block-diagonalizing the four-component (one-electron) modified Dirac equation in matrix representation. As we have discussed in chapters 8 and 10 for four-component Dirac-Hartree-Fock-Roothaan calculations, basis functions for the small component must fulfill certain constraints as otherwise variational instability and a wrong nonrelativistic limit [547] would result. The correct nonrelativistic limit will be obtained if the kinetic-balance condition,

[Pg.533]

In chapter 10, we have already discussed how the size of the small-component basis set can be made equal to that of the large-component basis set by absorbing the kinetic-balance operator into the one-electron Hamiltonian. In this chapter, we have elaborated on this by introducing a pseudo-large component that has led to the modified Dirac equation. 

From (15.8) we can extract a modified Dirac Hamiltonian that consists of a spin-free and a spin-dependent term. 

The spin-free modified Dirac Hamiltonian is obtained by simply omitting the second term on the right-hand side of (15.11) ... 

The gauge term, which is the difference between the Gaunt interaction and the Breit interaction, produces a spin-free operator that can be interpreted as an orbit-orbit interaction. Thus, both the Gaunt interaction and the gauge term of the Breit interaction give rise to spin-free contributions to the modified Dirac operator. We will use the developments of this section in chapter 17 to derive the Breit-Pauli Hamiltonian. 

We will evaluate the numbers of integrals required for a calculation with the unmodified Dirac Hamiltonian, and compare them with the number of integrals required for a calculation with the spin-free modified Dirac Hamiltonian, and with the number required for a nonrelativistic calculation. The spin-free Hamiltonian is formed by summing all the spin-free terms defined above, but we will consider the Coulomb term and the Gaunt and Breit terms separately. For the purpose of this evaluation, we make the following assumptions and definitions ... 

This chapter is devoted to the development of perturbation expansions in powers of 1 /c from the Dirac equation. In the previous chapter, the Pauli Hamiltonian was developed using the Foldy-Wouthuysen transformation. While this is an elegant method, it is probably simpler to make the derivation from the elimination of the small component with expansion of the denominator, and it is this approach that we use here. Another convenient approach is to make use of the modified Dirac equation in the limit of equality of the large and pseudo-large components. This approach enables us to draw on results from the modified Dirac approach in developing the two-electron terms of the Breit-Pauli Hamiltonian. We then demonstrate how the use of perturbation theory for relativistic corrections requires that multiple perturbation theory be employed for correlation effects and for properties. The last sections of this chapter are... 

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. 

This equation is exact just as the modified Dirac equation of chapter 15 is exact. It can also be separated into spin-free and spin-dependent terms, but now the separation must be done in both the Hamiltonian and the metric. Visscher and van Lenthe (1999) have shown that the spin separation gives different results for the two modified equations, and therefore the spin separation is not unique. This regular modified Dirac equation can be used in renormalization perturbation theory, with ZORA as the zeroth-order Hamiltonian. 

We now abandon conventional quantum chemical wisdom, and embrace the relativistic theory of electromagetic interactions wholeheartedly. In a singleparticle theory, the interaction between an electron and a vector potential, A(r), is included in the Dirac hamiltonian by modifying the canonical momentum, so that... 

This transformation has to be applied to the left and the right of the Dirac Hamiltonian to obtain the modified Hamiltonian. The same applies to operators for various molecular properties, which must also be modified the unmodified form is simply multiplied on the left and the right by the transformation operator T to obtain the modified form. We consider here both an operator defined by a scalar potential and an operator defined by a vector potential. 

The modified two-electron terms contain all the relativistic integrals, which means that the integral work is no different from that in the full solution of the Dirac-Hartree-Fock equations. It would save a lot of work if we could approximate the integrals, in the same way as we did for the Douglas-Kroll-Hess approximation. To do so, we must use the normalized Foldy-Wouthuysen transformation. The DKH approximation neglects the commutator of the transformation with the two-electron Coulomb operator, and in so doing removes all the spin-dependent terms. We must therefore also use a spin-free one-electron Hamiltonian. The approximate Hamiltonian (in terms of operators rather than matrices) is... 

To pass from (A4.12), or (11.2.10), to a Dirac-Pauh equation containing the Hamiltonian (11.2.17), it is customary to expand the operator k in powers of 1/c and it is also necessary to replace (which is only a truncated part of the 4-component quantity (A4.3)) by a modified 2-component function, corrected to admit the small components in (A4.8). It is this step that leads to the mass-variation term and that has certain dubious features (Farazdel and Smith, 1986) there is, however, general agreement that the final Hamiltonian (11.2.17) is satisfactory apart from the n term and that its 2-component eigenfunctions— the usual spin-orbitals—will correctly allow for relativistic effects up to order 1/c. ... 

Reference values for the various 2-component relativistic Hamiltonians are provided by the 4-component Dirac-Coulomb Hamiltonian, but we have also included orbital energies obtained with the Dirac-Coulomb-Gaunt (DCG) Hamiltonian. As already mentioned, the Gaunt term brings in spin-other-orbit (SOO) interaction. Since spin-orbit interaction induced by other electrons will oppose the one induced by nuclei we see from Table 3.3 that the spin-orbit splitting of orbital levels is overall reduced. However, one should note that the Gaunt term also modifies /2 levels. 

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