Transformations Barysz-Sadlej-Snijders

The other transformations that we have considered, the Douglas-Kroll transformation and the Barysz-Sadlej-Snijders transformation, start with a free-particle Foldy-Wouthuysen transformation. This transformation is independent of V and it is a simple matter to separate the perturbation. The transformed perturbation operator can be written down directly from (16.44)... 

Before considering how to separate this operator into a nonrelativistic operator and a relativistic correction, we should examine the effect of the next transformation, U. In both the Douglas-KroU transformation and the Barysz-Sadlej-Snijders transformation, the result of using ZYi is of second order in the potential. When we substitute... 

To illustrate, we use the second-order Barysz-Sadlej-Snijders transformation, which is more transparent than the Douglas-Kroll transformation. Introducing a perturbation parameter X, the Hamiltonian including the electric perturbation is... 

Expressions for the corrections due to the Douglas-Kroll and Barysz-Sadlej-Snijders transformations may be derived in the same way. However, since these operators are only correct to 0(c ) and have their leading term at we only... 

It remains to consider the Douglas-Kroll and Barysz-Sadlej-Snijders transformations. Since these corrections have leading order the magnetic terms of this order arise from the commutator [V, a (p - - eA)] = [V, kinematic factors, and are of leading order c. ... 

In quantum chemistry, regardless of which operators we choose for the Hamiltonian, we almost invariably implement our chosen method in a finite basis set. The Douglas-Kroll and Barysz-Sadlej-Snijders methods in the end required a matrix representation of the momentum-dependent operators in the implementation, and the regular methods usually end up with a basis set, even if the potentials are tabulated on a grid. Why not start with a matrix representation of the Dirac equation and perform transformations on the Dirac matrix rather than doing operator transformations, for which the matrix elements are difficult to evaluate analytically It is almost always much easier to do manipulations with matrices of operators than with the operators themselves. Provided proper account is taken in the basis sets of the correct relationships between the range and the domain of the operators (Dyall et al. 1984), matrix manipulations can be performed with little or no approximation beyond the matrix representation itself. 

It is even possible to define a free-particle transformation from which we could continue to develop a matrix Douglas-Kroll-Hess or Barysz-Sadlej-Snijders approximation. But with the matrix formalism we now have available, there is the opportunity to take a different approach, one that centers around the properties of the matrix X. 

X2C ( eXact 2-Component ) is an umbrella acronym [56] for a variety of methods that arrive at an exactly decoupled two-component Hamiltonian, with X2C referring to one-step approaches [65]. Related methods to arrive at formally exact two-component relativistic operators are, for example, infinite-order methods by Barysz and coworkers (BSS = Barysz Sadlej Snijders, lOTC = infinite-order two-component) [66-69] and normalized elimination of the small component (NESC) methods [70-77]. We discuss here an X2C approach as it has been implemented in a full two-component form with spin-orbit (SO) coupling and transformation of electric property operators to account for picture-change (PC) corrections [14],... 

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... 

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