Breit-Pauli equation

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... 

Though the ESR Hamiltonian is typically expressed in terms of effective electronic and nuclear spins, it can, of course, also be derived from the more fundamental Breit-Pauli Hamiltonian, when the magnetic fields produced by the moving nuclei are explicitly taken into account. In order to see this, we shall recall that in classical electrodynamics the magnetic dipole equation can be derived in a multipole expansion of the current density. For the lowest order term the expansion yields (59)... 

The spin-orbit mean field (SOMF) operator (56-58) is used to approximate the Breit—Pauli two-electron SOC operator as an effective one-electron operator. Using second-order perturbation theory (59), one can end up with the working equations ... 

SO coupling is a relativistic effect. The theory of the interaction of the magnetic moments of the electron spin and the orbital motion in one- and two-electron atoms has been formulated independently by Heisenberg and Pauli [12,13], shortly before the advent of the four-component Dirac theory of the electron [14]. Breit later has added the retardation correction [15]. The resulting Breit-Pauli SO operator, which can more elegantly be derived from the Dirac equation via a Foldy-Wouthuysen transformation [16], was thus well known for atoms since the early 1930s [17]. 

The Breit-Pauli (BP) approximation [140] is obtained truncating the Taylor expansion of the Foldy-Wouthuysen (FW) transformed Dirac Hamiltonian [141] up to the (p/mc) term. The BP equation has the well-known mass-velocity, Darwin, and spin-orbit operators. Although the BP equation gives reasonable results in the first-order perturbation calculation, it cannot be used in the variational treatment. 

Table 3.1 Effective Nuclear Charge (Scaling Parameter) Z ff for Approximate Spin-Orbit Interaction Calculations Using the One-Electron Term in the Breit-Pauli Hamiltonian (Equation 3.8, developed by Koseki et...

In molecular property calculations the same mutual interplay of electron correlation, relativity and perturbation operators (e.g. external fields) occurs. For light until medium atoms relativistic contributions were often accounted for by perturbation theory facilitating quasirela-tivistic approximations to the Dirac-Hamiltonian [114-117]. It is well-known that operators like the Breit-Pauli Hamiltonian are plagued by essential singularities and therefore are not to be used in variational procedures. It can therefore be expected that for heavier elements per-turbational inclusion of relativity will eventually become inadequate and that one has to start from a scheme where relativitiy is included from the beginning. Nevertheless very efficient approximations to the Dirac equation in two-component form exist and will be discussed further below in combination with their relevance for EFG calculations. In order to calculate the different contributions to a first-order property as the EFG, Kello and Sadlej devised a multiple perturbation scheme [118] in which a first-order property is expanded as... 

Finally, the Breit equation has been given. The equation goes beyond the Dirac model by taking into account the retardation effects. The Breit-Pauli expression for the Breit Hamiltonian contains several easily interpretable physical effects. 

Then it turned out that a promising approach (the test was for the hydrogen molecule) is to start with an accurate solution to the Schrodinger equation and go directly toward the expectation value of the Breit-Pauli Hamiltonian with this wave function (i.e., to abandon the Dirac equation), and then to the QED corrections. This Breit-Pauli... 

Note that no approximation has been made so far. The Breit-Pauli (BP) approximation [49] is introduced by expanding the inverse operators in the Schrodinger-Pauli equation in powers of (V — E) jlc and ignoring the higher-order terms. Instead, the BP approximation can be obtained truncating the Taylor expansion of the FW transformed Dirac Hamiltonian up to the (ptcf term. The one-electron BP Hamiltonian for the Coulomb potential V = Zr /r is represented by... 

Gaunt and Breit interaction operators represent potential energies due to magnetic interactions, which one would also assume to play a role in the non-relativistic many-particle theory given by the Pauli equation, Eq. (5.140). One... 

Finally, the full Breit-Pauli Hamiltonian contains the one-electron terms discussed in the beginning of this chapter as well as the magnetic interactions of the electrons that can be rewritten as interacting (coupled) angular momenta. Starting from the full Dirac-Breit equation for two electrons, Eq. (8.19), we obtain the purely two-component external-field-free Breit-Pauli Hamiltonian [72, p. 377],... 

Here Hd, is the Dirac Hamiltonian for a single particle, given by Eq. [30]. Recall from above that the Coulomb interaction shown is not strictly Lorentz invariant therefore, Eq. [59] is only approximate. The right-hand side of the equation gives the relativistic interactions between two electrons, and is called the Breit interaction. Here a, and a, denote Dirac matrices (Eq. [31]) for electrons i and /. Equation [59] can be cast into equations similar to Eq. [36] for the Foldy-Wouthuysen transformation. After a sequence of unitary transformations on the Hamiltonian (similar to Eqs. [37]-[58]) is applied to reduce the off-diagonal contributions, one obtains the Hamiltonian in terms of commutators, similar to Eq. [58]. When each term of the commutators are expanded explicitly, one arrives at the Breit-Pauli Hamiltonian, for a many-electron system " ... 

An alternative formulation that one can use to obtain the Breit-Pauli Hamiltonian is to start with the Breit equation (Eq. [59]). The equation is first transformed by Fourier transformation to momentum space. The terms involving the positive and negative eigenstates into PauH functions v /+ and / are then written, and only the electronic part of the equations is kept. Then, expanding the energy (in momentum space) in powers of p/mc, and Fourier transforming back to the coordinate space, one finally arrives at a differential equation of the form containing the Breit-Pauli Hamiltonian. 

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... 

In the previous two sections, we have presented the Breit-Pauli perturbation Hamiltonian for one- and two-electron relativistic corrections of order 1/c to the nonrelativistic Hamiltonian. But there is a problem for many-electron systems. For the perturbation theory to be valid, the reference wave function must be an eigenfunction of the zeroth-order Hamiltonian. If we take this to be the nonrelativistic Hamiltonian and the perturbation parameter to be 1/c, we do not have the exact solutions of the zeroth-order equation. 

The calculation of properties using direct perturbation theory follows exactly the same lines as we used for Breit-Pauli theory. As we noted above, stationary direct perturbation theory leads to precisely the same equations we would have obtained by simply expanding the perturbed wave functions in the set of eigenfunctions of the zeroth-order Hamiltonian, and on this basis we proceed with the development of multiple direct perturbation theory for properties. 

Here, Ne is the number of electrons, N is the number of nuclei, me is the mass of the electron, e is the elementary charge, eg is the vacuum permittivity, r and p are the position and momentum operators of electron s, and R and Zk are the position and atomic number of nucleus K. Starting from the Schrodinger equation (O Eq. 5.1), relativistic effects as described by the Breit-Pauli Hamiltonian can be treated as perturbations on an equal footing with external fields. Effects of nuclear motion (vibrations and rotations) can be estimated once the electronic response functions have been calculated. 

As can be seen, the spin-orbit contribution (term 2) in equation (12.3) is present even at this lowest order of the expansion. Equation (12.3) shows the separation of the ZORA operator into a scalar (spin-free) part and the SO operator (the electron spin-dependent term with a). With )C 1 - - Vnuc/(2c ), the ZORA SO operator becomes equivalent to the Breit-Pauli one-electron counterpart in order However, the scalar part of ZORA misses some contributions in order c . The ZORA equation can also be written as,... 

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