Approximate Foldy-Wouthuysen Transformations

If it is not possible to obtain the Foldy-Wouthuysen transformation in closed form, the best that we might be able to do is to find a transformation that decouples the large and small components, or the positive- and negative-energy states, to some order in a suitable coupling parameter. This means that our approximations will all be based on perturbation theory in a general sense. Before examining some specific transformations, we need to develop the formal theory of transformations so that we can have some information about the order to which the components are decoupled. 

We now make a unitary transformation of the wave function using an exponential ansatz. 

The obvious expansion parameter is 1 /c, which should generate the nonrelativistic Hamiltonian and operators for relativistic corrections to various orders. In terms of this parameter, the three operators in the Dirac Hamiltonian, Pmc, O = ca p, and S =V, are of orders -2, -1, and 0, respectively. It is for this reason that the rest mass term was not included in the even operator. If we choose 

In fact, we also need to consider the next commutator to obtain the hill term of order 1, which is the highest-order odd operator in the transformed Hamiltonian—and if we wish to go further and obtain all the terms of order 2 to determine the lowest-order relativistic correction to the Hamiltonian we need to consider the next two commutators. Iterating the previous results, we find that 

With these expressions, the transformed Hamiltonian correct to order 2 in 1/c is 

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The Breit-Pauli Hamiltonian is an approximation up to 1/c2 to the Dirac-Coulomb-Breit Hamiltonian obtained from a free-particle Foldy-Wouthuysen transformation. Because of the convergence issues mentioned in the preceding section, the Breit-Pauli Hamiltonian may only be employed in perturbation theory and not in a variational procedure. The derivation of the Breit-Pauli Hamiltonian is tedious (21). 

We have therefore achieved our objective in that equation (3.83), which is correct to order 1 /c2, contains even operators only. It would, of course, be possible to proceed further with the Foldy Wouthuysen transformation but there is little point in doing so, since the theory is inaccurate in other respects. For example, we have treated the electromagnetic field classically, instead of using quantum field theory. Furthermore, we shall ultimately be interested in many-electron diatomic molecules, for which it will be necessary to make a number of assumptions and approximations. 

In the nonrelativistic context current-density functional theory is based on the nonrelativistic limits of the paramagnetic current (87) and/or the magnetization density (89) [128,129]. In the relativistic situation, however, a density functional approach relying on jp or m can only be considered an approximation, as long as the external magnetic field does not vanish. In order to clarify the relation between these two points of view the weakly relativistic limit of RDFT has to be analyzed. The weakly relativistic limit of the Hamiltonian (23) can be derived either by a direct expansion in 1/c or by a low order Foldy-Wouthuysen transformation,... 

Another major two-component approximation is the Foldy-Wouthuysen transformation (Foldy and Wouthuysen 1950), which makes the large-component and small-component submatrices of the Dirac Hamiltonian matrix, Hd, linear independent by a unitary transformation such as... 

Several authors have considered a number of approximate solutions to the Dirac equation. One such method is the use of the Foldy-Wouthuysen transformation (see, for example, Morrison and Moss ). Upon application of a unitary transformation of the form... 

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 " ... 

The relativistic correction to the potential is no more singular than the potential itself in this limit and therefore will support bound states. In the small momentum limit, when the electron is far from the nucleus, the potential goes as 1 /r and is therefore a short-range potential. It can be seen that the kinematic factors provide a cutoff to the potential that is absent in the Pauli approximation and that permits variational calculations with the free-particle Foldy-Wouthuysen transformed Hamiltonian. 

The complicated nature of the operator VVi raises the question of whether a simpler alternative can be found. Barysz, Sadlej, and Snijders (1997) devised an approach which starts from the free-particle Foldy-Wouthuysen transformation, just as the Douglas-Kroll transformation does. Whereas the Douglas-Kroll approach seeks to eliminate the lowest-order odd term from the transformed Hamiltonian, their approach seeks to be correct to a particular order in 1 /c, and it provides a ready means for defining a sequence of approximations of increasing order in 1/c. It is important to note that, while the expansion in powers of 1 /c that resulted from the Foldy-Wouthuysen transformation in section 16.2 generates highly singular operators, this is not true per se of expansions in 1/c. What multiplies 1/c is all-important. In the case of the Foldy-Wouthuysen transformation it is and due to the fact that p can become... 

This is the equivalent of the second-order Douglas-KroU operator, but it only involves operators that have been defined in the free-particle Foldy-Wouthuysen transformation. As for the Douglas-Kroll transformed Hamiltonian, spin separation may be achieved with the use of the Dirac relation to define a spin-fi ee relativistic Hamiltonian, and an approximation in which the transformation of the two-electron integrals is neglected, as in the Douglas-Kroll-Hess method, may also be defined. Implementation of this approximation can be carried out in the same way as for the Douglas-Kroll approximation both approximations involve the evaluation of kinematic factors, which may be done by matrix methods. 

While one-electron systems provide a good formal testing ground for an approximate theory, for quantum chemistry we need a theory that encompasses many-electron systems. Formally, we can treat the regular approximations as a Foldy-Wouthuysen transformation with a particular choice of X, and then we can write the transformed two-electron operator 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... 

Since we can regard the regular approximation as a Foldy-Wouthuysen transformation, we should be able to use a similar line of development for the property operators as in chapter 16. There we found that it was relatively easy to deal with the electric perturbations because they appeared as powers of the perturbation operator, with a linear operator at lowest order. The magnetic perturbations, on the other hand, were much more complicated because the transformation involved complicated functions of the momentum, which must be replaced with the expression that includes the vector potential. 

The process of making approximations starts with either a partitioning of the Hamiltonian and the metric, as in direct perturbation theory, or, for variational approximations, the elimination of the small component. This can be done either directly or via a Foldy-Wouthuysen transformation. Here, we are interested first and foremost in variational approximations, so we will focus on the elimination of the small component and the Foldy-Wouthuysen transformation. Before considering the approximations, we first outline some theory for the exact solutions. 

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 formalize these concepts, we first develop the frozen-core approximation, and then examine pseudoorbitals or pseudospinors and pseudopotentials. From this foundation we develop the theory for effective core potentials and ab initio model potentials. The fundamental theory is the same whether the Hamiltonian is relativistic or nonrela-tivistic the form of the one- and two-electron operators is not critical. Likewise with the orbitals the development here will be given in terms of spinors, which may be either nonrelativistic spin-orbitals or relativistic 2-spinors derived from a Foldy-Wouthuysen transformation. We will use the terminology pseudospinors because we wish to be as general as possible, but wherever this term is used, the term pseudoorbitals can be substituted. As in previous chapters, the indices p, q, r, and s will be used for any spinor, t, u, v, and w will be used for valence spinors, and a, b, c, and d will be used for virtual spinors. For core spinors we will use k, I, m, and n, and reserve i and j for electron indices and other summation indices. 

A regular alternative to the Foldy-Wouthuysen transformation was given by Douglas and Kroll and later developed for its use in electronic structure calculations by Hess et al. The Douglas-Kroll (DK) transformation defines a transformation of the external-field Dirac Hamiltonian Hq of equation (11) to two-component form which leads, in contrast to the Foldy-Wouthuysen transformation, to operators which are bounded from below and can be used variationally, similarly to those of the regular approximations discussed above. As in the FW transformation, it is not possible in the DK formalism to give the transformation in closed form. Rather, it is... 

Relativity becomes important for elements heavier than the first row transition elements. Most methods applicable on molecules are derived from the Dirac equation. The Dirac equation itself is difficult to use, since it involves a description of the wave function as a four component spinor. The Dirac equation can be approximately brought to a two-component form using e.g. the Foldy-Wouthuysen (FW) transformational,12]. Unfortunately the FW transformation, as originally proposed, is both quite complicated and also divergent in the expansion in the momentum (for large momenta), and it can thus only be carried out approximately (usually to low orders). The resulting equations are not variationally stable, and they are used only in first order perturbation theory. 

Since working with the full four-component wave function is so demanding, different approximate methods have been developed where the small component of the wave function is eliminated to a certain order in 1/c or approximated (like the Foldy-Wouthuysen or Douglas-Kroll transformations thereby reducing the four-component wave function to only two components. A description of such methods is outside the scope of this book. 

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 Dirac wave equation is somewhat cumbersome to solve due to the presence of four,in general complex, components. Foldy and Wouthuysen (5.) have fortunately proposed a transformation which allows one to approximate... 

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