Breit-Pauli spin-orbit operators

We assume that we somehow know how to handle the spin-free terms of this equation, and that they form a suitable Ho. The spin-orbit part may be divided into a one-electron and a two-electron operator. 

Obviously, for molecules there must be a sum over nuclei in the Hamiltonian expression. 

We may use these operators in various ways, to which we will return later. Regardless of which computational model we choose to follow, we end up evaluating basis function integrals over these operators. It is easy to see that this will require a somewhat different treatment from the ordinary nonrelativistic integral handling. Ignoring the constants, we can write the matrix element of the one-electron spin-orbit operator as 

The antisymmetry can easily be established by integration by parts. This antisymmetry may be a little easier to see if we write the spin-orbit integral in terms of second derivatives of the regular nuclear attraction integrals  

Since the basis functions depend on relative coordinates r, - r, we can transfer the derivatives from the electron coordinates r, to the nuclear coordinates r, and demonstrate the antisymmetry  

---

The Breit-Pauli spin-orbit operator has one major drawback. It implicitly contains terms coupling electronic states (with positive energy) and posi-tronic states (in the negative energy continuum) and is thus unbounded from below. It can be employed safely only in first-order perturbation theory. 

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

For many types of electron spectroscopies there are still comparatively few studies of SOC effects in molecules in contrast to atoms, see, e.g., [1, 2, 3, 4, 5, 6, 7] and references therein. This can probably be referred to complexities in the molecular analysis due to the extra vibrational and rotational degrees of freedom, increased role of many-body interaction, interference and break-down effects in the spectra, but can also be referred to the more difficult nature of the spin-orbit coupling itself in polyatomic species. Modern ab initio formulations, as, e.g., spin-orbit response theory [8] reviewed here, have made such investigations possible using the full Breit-Pauli spin-orbit operator. 

Semiempirical spin-orbit operators play an important role in all-electron and in REP calculations based on Co wen- Griffin pseudoorbitals. These operators are based on rather severe approximations, but have been shown to give good results in many cases. An alternative is to employ the complete microscopic Breit-Pauli spin-orbit operator, which adds considerably to the complexity of the problem because of the necessity to include two-electron terms. However, it is also inappropriate in heavy-element molecules unless used in the presence of mass-velocity and Darwin terms. 

The array of methods in gamess for treating spin-orbit coupling effects has recently been the subject of two reviews [46,47]. These methods include the full Breit-Pauli spin-orbit operator and approximations to it, primarily developed by Koseki and Fedorov. All of the methods require a multi-reference wavefunction as a starting point. This can be MCSCF, first or second order Cl, or MRPT2. The simplest method is a... 

As may be seen by comparing Eqs. [103] and [105], the no-pair spin-orbit Hamiltonian has exactly the same structure as the Breit-Pauli spin-orbit Hamiltonian. It differs from the Breit-Pauli operator only by kinematical factors that damp the 1/rfj and l/r singularities. 

The electronic spin-orbit interaction operator, referred to as the Breit-Pauli spin-orbit Hamiltonian, is given by... 

Since the operators f and f2 occur only at the level of the calculation of the spatial spin-orbit integrals over atomic orbitals, Breit-Pauli spin-orbit coupling operators and DKH spin-orbit coupling operators can be discussed on the same footing as far as their matrix elements between multi-electron wave functions are concerned. These terms constitute, by definition, the spin-orbit interaction part of the operator H+ (Hess etal. 1995). The spin-independent terms characteristic of relativistic kinematics define the scalar relativistic part of the operator, and terms with more than one cr matrix (not considered here) contribute to spin-spin coupling phenomena. 

Breit-Pauli spin-orbit coupling operator [101-104] reads in SI units as... 

The matrix elements of the spin-orbit coupling operator have been included in these works using empirically obtained or computed spin-orbit coupling constants for an effective one electron operator. The Breit-Pauli spin-orbit coupling operator (115) with all multi-center terms was employed for the first time by Kiyonaga, Morihashi and Kikuchi [125]. 

First attempts to calculate molecular parity violating potentials within a two-component framework have been undertaken by Kikuchi and coworkers [168,169]. They have added the Breit-Pauli spin-orbit coupling operator Hso to the usual non-relativistic Hamiltonian Hq... 

The disadvantage of this particular two-component realisation is that the full Breit-Pauli spin-orbit coupling operator is not bound from below and therefore critical in a variational procedure. Hence, only relative modest basis sets have been used in the calculations of parity violating effects within this scheme [168,169]. 

In Table I, 3D stands for three dimensional. The symbol symbol in connection with the bending potentials means that the bending potentials are considered in the lowest order approximation as already realized by Renner [7], the splitting of the adiabatic potentials has a p dependence at small distortions of linearity. With exact fomi of the spin-orbit part of the Hamiltonian we mean the microscopic (i.e., nonphenomenological) many-elecbon counterpart of, for example, The Breit-Pauli two-electron operator [22] (see also [23]). 

The spin-orbit operator in the Breit-Pauli approximation is given by (49,71)... 

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. 

The use of Breit-Pauli or no-pair spin-orbit operators to compute the spin-orbit integrals implies to deal with the full nodal structure of the orbitals [2]. When AREPs are used at the SCF step the pseudoorbitals, as eigenfunctions of the valence Fock-operators, have lost their nodal structure in the core region, exactly where spin-orbit operators essentially act, making it impossible to apply such operators in pseudopotential schemes. Three solutions can be employed to evaluate the spin-orbit integrals on nodeless pseudoorbitals ... 

Here is given in parentheses for the nonrelativistic states and the iE carry the E = E ov E" representations of the Cg double group. The relativistic wave functions, kE, A = 1—3 and E = E or E", are the eigenfunctions of in this basis. The spin-orbit operator is described within the Breit-Pauli approximation. ... 

Models related to spin-forbidden reactions are discussed in this chapter. Coupling between two surfaces of different spin and symmetry is given by various levels of approximation for spin-orbit operators from the reduction of relativistic quantum mechanics. Well-established methods such as the Breit-Pauli Hamiltonian exist, but new relativistic methods such as the Douglas-Kroll Hamiltonian and other new transformation schemes are also being investigated and implemented today. 

What we must not do is to add an all-electron spin-orbit operator, such as the Breit-Pauli or Douglas-Kroll operator, because the effect of the core is not included in these operators, nor is the removal of the core tail. The all-electron spin-orbit operators behave as 1/r, and since the pseudospinors have minimal core amplitude, the spin-orbit effect will be grossly underestimated. 

The matrix elements v,y and w,y may be derived from Breit-Pauli or Douglas-Kroll spin-orbit operators, or other operators that conveniently lend themselves to this type of expression. 

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 SOC Hamiltonian contains a one-electron and two-electron parts. The one-electron part describes an interaction of an electron spin with a potential produced by nuclei. The two-electron part has the SSO contribution and the SOO contribution. The SSO contribution describes an interaction of an electron spin with an orbital momentum of the same electron. The SOO contribution describes an interaction of an electron spin with the orbital momentum of other electrons. However, due to a complicated two-electron part, the evaluation of the Breit-Pauli SOC operator takes considerable time. A mean field approximation was suggested by Hess et al. [102] This approximation allows converting the complicated two-electron Breit-Pauli Hamiltonian to an effective one-electron spin-orbit mean-field form... 

---