Relativistic one-electron Hamiltonian terms

For stationary states the time-dependent operator reduces to the energy E 

For further partitioning and classification of the Hamiltonian terms some assumptions concerning the electromagnetic field will be accepted  

The operator k can be expanded according to the identity fulfilled for any operators 

Inside this expression the commutation relations of individual terms hold true. The Dirac equation for the upper component spinor can be written in the form 

using the commutator relations, all factors (e Pnt + W) can be moved to the right, which allows us to write 

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The last term can be combined with the first term of to yield a term containing the total electric field strength E = jnt + ext- The commutator corrections from terms in are much smaller (of the order 1/c4) and are thus neglected, so that kQ is simply shifted either to the far left or the far right. The final relativistic one-electron Hamiltonian terms are collected in Table 4.2. 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics [1]. The simplest form is the Dirac-Coulomb (DC) Hamiltonian, where the nonrelativistic one-electron terms in the Schrodinger equation are replaced by the one-electron Dirac operator hj). 

The second term in Eq. (8) denotes the sum of the relativistic one-electron Dirac Hamiltonians of moving electrons in some additional external electromagnetic field ( Aext),... 

Since the Dirac equation is valid only for the one-electron system, the one-electron Dirac Hamiltonian has to be extended to the many-electron Hamiltonian in order to treat the chemically interesting many-electron systems. The straightforward way to construct the relativistic many-electron Hamiltonian is to augment the one-electron Dirac operator, Eq. (70) with the Coulomb or Breit (or its approximate Gaunt) operator as a two-electron term. This procedure yields the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonian derived from quantum electrodynamics (QED)... 

The last three terms in the Hamiltonian in Eq. (5.28) define an effective one-body potential, Veff(r), which transforms the density of the non-interacting system into the real density. Then, by choosing Vj(r) = Veffir) in Eq. (5.25) the effective potential is found. As was the case with HE, the one-electron Hamiltonian in the KS equations is solved iteratively and self-consistently. If the exact expression for Exc [p(r)] was known, the KS equations would provide the exact non-relativistic ground state solution within the space spanned by a given basis set, including all electron correlation effects. The latter are missing in HE. This is an important difference between HE and KS. 

One of the major fundamental difference between nonrelativistic and relativistic many-electron problems is that while in the former case the Hamiltonian is explicitly known from the very beginning, the many-electron relativistic Hamiltonian has only an implicit form given by electrodynamics [13,37]. The simplest relativistic model Hamiltonian is considered to be given by a sum of relativistic (Dirac) one-electron Hamiltonians ho and the usual Coulomb interaction term ... 

Only recently has the work of Bauschlicher, Walch and Siegbahn showed the need to include d correlation, whereas the work of Werner and Martin and of Scharf, Brode and Ahlrichs stressed the importance of cluster corrections and relativistic corrections. The results of Werner et al. were obtained using CEPA-1 to account for cluster contributions, while Scharf et al. used the CPF approach. Both groups accounted for relativistic corrections by employing first-order perturbation theory, i.e. by evaluating the Cowan-Griffin operator which consists of the mass-velocity and the one-electron Darwin term of the Breit-Pauli Hamiltonian. 

Although in the Dirac-Coulomb Hamiltonian the one-particle operator is correct to all orders in a, the two-particle interaction is only correct to a°. The Dirac-Coulomb Hamiltonian is not invariant under Lorentz transformations, however it can be considered as the leading term of a yet unknown relativistic many-electron Hamiltonian which fulfills this requirement. An operator which also takes into account the leading relativistic corrections for the two-electron terms is the Coulomb-Breit term (Breit 1929,1930,1932, 1938),... 

In most cases, however, the relativistic effects are rather weak and may be separated into spin-orbit coupling effects and scalar effects. The latter lead to compression and/or expansion of electron shells and can rather accurately be treated by modifying the one-electron part of the non-relativistic many-electron Hamiltonian. With this scalar-relativistic Hamiltonian the (modified) energies and wave functions are computed and subsequently an effective spin-orbit part is added to the Hamiltonian. The effects of the spin-orbit term on the energies and wave functions are commonly estimated using second-order perturbation theory. More information for the interested reader can be found in excellent textbooks on relativistic quantum chemistry [2, 3]. 

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... 

The second relativistic contribution of scalar nature is the one-electron Darwin term, Wd. This term derives from the non-relativistic expansion of Dirac s equation, in powers of (v/c), and results in a non-local interaction between the electron and the nucleus. The interaction extends over a region in space of size roughly that of the Compton wavelength of the electron. The order of magnimde of this term in the Hamiltonian is also (Za) making it non-negligible for heavy elements. [12] These scalar relativistic terms have significant effects on the radial extent of the inner core orbitals. 

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

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