The Pauli Hamiltonian

To develop the Pauli Hamiltonian via perturbation theory we start from the Dirac equation in two-component form, 

a factor of 2mc has been taken out of the denominator term of (17.2). As an equation in its own right, this equation has two major problems. First, it is nonlinear in the energy, which appears in the denominator. Second, it has the potential in the denominator also, making it hard to evaluate integrals. Since we are interested in a perturbation theory that starts from a nonrelativistic Hamiltonian, the obvious thing to do is to expand the expression in square brackets in powers of 1 /2mc. The vahdity of this expansion depends on y - 1 being less than 2mc —a point to which we will return later. 

The exception is the Lamb shift, which we treat briefly and is 0 c ). 

From the Dirac relation (4.14), (or p)(or p) = p, and the first term in the series gives p /2w = f, the nonrelativistic kinetic energy operator. Making use of the Dirac relation in the second term gives 

There is a clear connection of these operators with those of the modified Dirac equation, except that they are operating on the large component rather than on the pseudo-large component 

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A number of types of calculations can be performed. These include optimization of geometry, transition structure optimization, frequency calculation, and IRC calculation. It is also possible to compute electronic excited states using the TDDFT method. Solvation effects can be included using the COSMO method. Electric fields and point charges may be included in the calculation. Relativistic density functional calculations can be run using the ZORA method or the Pauli Hamiltonian. The program authors recommend using the ZORA method. 

Similarly, the first-order energy obtained from the Pauli Hamiltonian given in equation (53) can be written as... 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

Combining these results, the Pauli Hamiltonian /iPauli (i) with an external field can be written as... 

The interaction Hamiltonian can be extracted from the Pauli Hamiltonian plus a dipole interaction Hamiltonian... 

A complete derivation involves a complete expansion of the Pauli Hamiltonian and the recognition that for the two complexified vector potentials A11 and A 2 that one has the term... 

Since there is a non-Abelian nature to this theory, we return to the nonrelativistic equation that describes the interaction of a fermion with the electromagnetic field. The Pauli Hamiltonian is modified with the addition of a interaction term [9]... 

Thus, the main relativistic effects are (1) the radical contraction and energetic stabilization of the s and p orbitals which in turn induce the radial expansion and energetic destabilization of the outer d and f orbitals, and (2) the well-known spin-orbit splitting. These effects will be pronounced upon going from As to Sb to Bi. Associated with effect (1), it is interesting to note that the Bi atom has a tendency to form compounds in which Bi is trivalent with the 6s 6p valence configuration. For this tendency of the 6s electron pair to remain formally unoxidized in bismuth compounds (i.e. core-like nature of the 6s electrons), the term inert pair effect or nonhybridization effect has been often used for a reasonable explanation. In this context, the relatively inert 4s pair of the As atom (compared with the 5s pair of Sb) may be ascribed to the stabilization due to the d-block contraction , rather than effect (1) . On the other hand, effect (2) plays an important role in the electronic and spectroscopic properties of atoms and molecules especially in the open-shell states. It not only splits the electronic states but also mixes the states which would not mix in the absence of spin-orbit interaction. As an example, it was calculated that even the ground state ( 2 " ) of Bij is 25% contaminated by Hg. In the Pauli Hamiltonian approximation there is one more relativistic effect called the Dawin term. This will tend to counteract partially the mass-velocity effect. 

Scalar relativistic (mass-velocity and Darwin) effects for the valence electrons were incorporated by using the quasi-relativistic method (55), where the first-order scalar relativistic Pauli Hamiltonian was diagonalized in the space of the nonrelativistic basis sets. The Pauli Hamiltonian used was of the form... 

The problem is that the turn-over rule is valid only if the integrand vanishes at the boundaries. This is the case, e.g. for a homogeneous magnetic field, but not for the magnetic field created by a (point) nucleus. In the former case we get the same result as from the Pauli Hamiltonian (128)... 

The Pauli Hamiltonian has already been discussed in volume 1 of this series (see e.g. Ref. [35]) and reads... 

The (iterative) use of the Pauli Hamiltonian, the so-called quasi-rela-tivistic (QR) method, must be regarded as obsolete, as the Pauli... 

This procedure is usually known as the elimination of the small component (ESC), and Eq. (34) is still equivalent to the original Dirac equation. Although the equation has been reduced to a two-component form, nothing is gained since we now have an energy-dependent Hamiltonian, and one must introduce further approximations to transform Eq. (34) into a form useful for actual calculations. The principal difference between the Pauli and the ZORA Hamiltonian is that to obtain the Pauli Hamiltonian, one uses an expansion in c ... 

While it is true that for singular external Potentials hke the Coulomb potential of a point charge, an expansion as in Eq. (35) is not valid in the vicinity of the nuclei, it is time to remind that the singularity of the potential is only one of the sources of the problem. Even for non-singluar potentials the Pauli Hamiltonian is unbound through the... 

The Pauli Hamiltonian is ideally suited for carrying out relativistic corrections as a first-order perturbation to a non-relativistic Hamiltonian. In recent years, several authors have considered inclusion of the Pauli terms in variational self-consistent field (SCF) calculations. Wadt, Hay and... 

For a spherically-symmetric Coulomb potential V r) = —Ze /r, the second-order even term in the Foldy-Wouthuysen expansion, 2 Eq. (11.84), can be cast into the much more familiar form to yield the (relativistic corrections of the) Pauli Hamiltonian,... 

Combining all individual results yields the Pauli Hamiltonian already encountered in section 11.5.2,... 

This is the four-component form of the Pauli Hamiltonian. 

It is instructive to compare the 2-component Hamiltonian T-Cl with the Pauli Hamiltonian, which was derived in the previous section. The term Ep contains the relativistic free-particle energy, which is well-behaved for all values of the momentum. Ep, which is the kinetic energy operator for the positive-energy states, is a positive definite operator. In the Pauli Hamiltonian we see that this operator is expanded in powers of p/mc, which does not converge if p/mc > 1—a situation that will occur in any potential if the electron is sufficiently close to the nucleus. As mentioned above, the mass-velocity term is not bounded from below and so cannot be used variationally. 

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