Transformation of Electric Property Operators

The transformed Hamiltonians that we have derived allow us to calculate intrinsic molecular properties, such as geometries and harmonic frequencies. We would like to be able to calculate response properties as well, with wave functions derived from the transformed Hamiltonian. If we used a method such as the Douglas-Kroll-Hess method, it would be tempting to simply evaluate the property using the nonrelativistic property operators and the transformed wave function. As we saw in section 15.3, the property operators can have a relativistic correction, and for properties sensitive to the environment close to the nuclei where the relativistic effects are strong, these corrections are likely to be significant. To ensure that we do not omit important effects, we must derive a transformed property operator, starting from the Dirac form of the property operator. 

For electric perturbations, deriving the appropriate perturbation operator turns out to be relatively straightforward because the four-component operator is even. The rest of this section is devoted to derivation and discussion of transformed electric perturbation operators. Magnetic perturbations are a different story, because the four-component operator is odd and therefore enters into the lowest order of the transformation. We will deal with magnetic perturbations in the next section. 

As well as deriving the appropriate forms of the electrie perturbation operators, we would like to be able to express them in terms of the nonrelativistic operators plus a relativistie eorreetion. This we ean do by writing, the transformed Hamiltonian including the perturbation, as a sum of the untransformed operator T-L plus a correction. 

The commutator will (in general) change the order of the perturbation with respect to some expansion parameter and will introduce cross-terms between the unperturbed Hamiltonian and the perturbation. 

In the Pauli approximation, the Hamiltonian eorreet to second order is linear in the potential, and the separation of the zeroth-order Hamiltonian from the perturbation is 

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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],... 

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