Hamiltonians four-component wave function

The most straightforward method for electronic structure calculation of heavy-atom molecules is solution of the eigenvalue problem using the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonians [4f, 42, 43] when some approximation for the four-component wave function is chosen. 

Although the full four-component treatment with the Dirac Hamiltonian is ideal, the computation of four-component wave functions is expensive. Thus, since small components have little importance in most chemically interesting problems, various two- or one-component approximations to the Dirac Hamiltonian have been proposed. From Eq. 10.32, the Schrodinger-Pauli equation composed of only the large component is obtained as... 

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

When the site spinor vj/V - p is a symmetric fourth-rank spinor QA u/p (corresponding to the two-dimensional AKLT model[13]), only the quintet component out of the six multiplets on each spin quartet is present in the wave function (60). The sixth-rank spinors (62) are symmetric with respect to two triplets of indices and, hence, contain four multiplets with S = 0, 1, 2, 3 formed from two quintets. Consequently, the cell Hamiltonian (Hi and H coincide in this case) has the form... 

For most chemical applications, one is not interested in negative energy solutions of a four-component Dirac-type Hamiltonian. In addition, the computational expense of treating four-component complex-valued wave functions often limited such calculations to benchmark studies of atoms and small molecules. Therefore, much effort was put into developing and implementing approximate quantum chemistry methods which explicitly treat only the electron degrees of freedom, namely two- and one-component relativistic formulations [2]. This analysis also holds for a relativistic DFT approach and the solutions of the corresponding DKS equation. 

The preceding three chapters have already introduced Hamiltonians of reduced dimension. Particularly successful in variational calculations are the DKH and ZORA approaches. In their scalar-relativistic variant, they can easily be implemented in a computer program for nonrelativistic quantum chemistry so that spin remains a good quantum number leading to great computational advantages (if this approximation is justifiable). Already for these methods we have seen that numerous approximations can be made in order to increase their computational efficiency with little or even no loss of accuracy compared with a four-component reference calculation with the same type of total wave function. 

The unitary transformation of the Dirac Hamiltonian to two-component form is accompanied by a corresponding reduction of the wave function. As discussed in detail in chapters 11 and 12, the four-component Dirac spinor ip will... 

The relativistic Hamilton operator for an electron can be derived, using the correspondence principle, from its relativistic classical Hamiltonian and this leads to the one-electron Dirac equation, which does contain spin operators. From the one-electron Dirac equation it seems trivial to define a many-electron relativistic equation, but the generalization to more electrons is less straightforward than in the non-relativistic case, because the electron-electron interaction is not unambiguously defined. The non-relativistic Coulomb interaction is often used as a reasonable first approximation. The relativistic treatment of atoms and molecules based on the many-electron Dirac equation leads to so-called four-component methods. The name stems from the fact that the electronic wave functions consist of four instead of two components. When the couplings between spin and orbital angular moment are comparable to the electron-electron interactions this is the preferred way to explain the electronic structure of the lowest states. 

Symmetric anisotropy The basis of this Hamiltonian can no longer be restricted to determinants with the same Ms value as was done for the isotropic interactions. The inclusion of magnetic anisotropy in the model causes the removal of the degeneracy of the different Ms levels and eventually mixing of the wave functions with different spin moment. Here, we have to consider four CSFs the three components of the triplet plus the singlet. To facilitate the determination of the matrix elements of the model... 

The operators of the Dirac Eq. (5) are 4 x 4 matrix operators, and the corresponding wave function is therefore a four-component (4c) vector (spinor). The V" includes the effect of the finite nuclear size, while some finer effect, like QED, can be added to the hocB perturbatively, although the self-energy QED term is more difficult to treat [36,47,48]. The DCB Hamiltonian in this form contains all effects through the second order in a, the fine-structure constant. 

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