Hamiltonian many-electron, transformed

The similarity-transformed Hamiltonian method has so far been applied only to two-electron systems. Using closure (i.e., RI) approximations, this technique will be generalized to many-electron systems (IS). 

The unitary group invariance of the hamiltonian assures that its exact eigenvalue spectrum is invariant to a unitary transformation of the basis. It means that a full Cl calculation must provide the same eigenvalues (many-electron energy states) as a full VB calculation. However, at intermediate levels of approximation, this equivalence is not straightforward. In this section we will sketch out the main points of contact between MO and VB related wave functions. 

In this paper, the general theory developed in Part I is applied to the Hartree-Fock Scheme for a transformed many-electron Hamiltonian. It is shown that, if the transformation is a product of one-electron transformations, then the Fock-Dirac operator as well as the effective Hamiltonian undergo similarity transformations of the one-electron type. The special properties of the Hartree-Fock scheme for a real self-adjoint Hamiltonian based on the bi-variational principle are discussed in greater detail. 

This study was started in order to find out whether one could find meaningful complex eigenvalues in the Hartree-Fock scheme for a transformed Hamiltonian in the method of complex scaling. This problem was intensely discussed at the 1981 Tarfala Workshop in the Kebnekaise area of the Swedish mountains. It was found that, if the many-electron Hamiltonian undergoes a similarity transformation U which is a product of one-electron transformations u - as in the method of complex scaling - then the Fock-Dirac operator p as well as the effective Hamiltonian Heff undergo one-electron similarity... 

In this notation the presence of two upper and two lower components of the four-component Dirac spinor fa is emphasized. For solutions with positive energy and weak potentials, the latter is suppressed by a factor 1 /c2 with respect to the former, and therefore commonly dubbed the small component fa, as opposed to the large component fa. While a Hamiltonian for a many-electron system like an atom or a molecule requires an electron interaction term (in the simplest form we add the Coulomb interaction and obtain the Dirac-Coulomb-Breit Hamiltonian see Chapter 2), we focus here on the one-electron operator and discuss how it may be transformed to two components in order to integrate out the degrees of freedom of the charge-conjugated particle, which we do not want to consider explicitly. 

Equations (34) and (35) tell us how many-electron states, constructed by letting electron creation operators act on the vacuum state, transform under the elements of the symmetry group of the core Hamiltonian. The vacuum state is assumed to be invariant under the action of these symmetry operations, i.e., R10) = 10). 

Extended Douglas—Kroll transformations applied to the relativistic many-electron Hamiltonian... 

By an apphcation of the DK transformation to the relativistic many-electron Hamiltonian, recently, we have shown that the many-electron DK Hamiltonian also gives satisfactory results for a wide variety of atoms and molecules compared with... 

Although the proper point of departure for relativistic atomic structure calculations is quantum electrodynamics (QED), very few atomic structure calculations have been carried out entirely within the QED framework. Indeed, almost all relativistic calculations of the structure of many-electron atoms are based on some variant of the Hamiltonian introduced a half century ago by Brown and Ravenhall [1] to understand the helium fine structure. By decoupling the electron and radiation fields in QED to order a (the fine-structure constant) using a contact transformation. Brown and Ravenhall obtained a relativistic momentum-space Hamiltonian in which the electron-electron Coulomb interaction was surrounded by positive-energy projection operators. Owing to the fact that contributions from virtual electron-positron pairs are automatically projected out of... 

The reduced-mass and mass-polarization terms arise on transforming the many-electron plus nucleus Hamiltonian to center of mass coordinates. 

Bohm and Pines transformed the electron-repulsion terms in the classical many-electron Hamiltonian into its Fourier components. This Fourier transform has the effect of interpreting the familiar Coulomb repulsion potential term as a series of momentum-transfers between the states of the electrons. They showed how an important series of results could be obtained by the assumption that the terms in this Fourier transform which depended on a non-zero phase difference in the k-vector could be neglected. The idea behind this approximation is that these terms, having random phases, have a zero mean value and contribute only to random fluctuations in the electron plasma which are negligible under the circumstances of their study, or, what amounts to the same thing, the momentum transfers could be replaced by their ensemble average. 

The reduction of the relativistic many-electron hamiltonian by expansion in powers of the external field is the second-order Douglas-Kroll transformation [29], and has been used with success by Hess and co-workers [30]. The operators which result from this transformation are non-singular, but the integrals over the resulting operators are complicated and have to be approximated, even for finite basis set expansions. The reduction of the Dirac-Coulomb-Breit equation to two-component form using direct perturbation theory has been described by Kutzelnigg and coworkers [26, 27, 31], Rutkowski [32], and van Lenthe et al. [33]. 

As we discovered in the last section the spin and. space parts of the eigenfunctions are separately invariant under transformations of the sphere group. The electrostatic forces separate states having different L, and the possible values of S are determined by the Pauli exclusion principle. Although the many-electron Hamiltonian in Eq. 7.18.8 does not contain the spin coordinates—it is therefore invariant under all transformations involving one or more of the electron spin coordinates— the eigenfunctions of 3C do contain spin coordinates. The spin coordinate electron function /(x, y, 2, [Pg.114]

In contrast to the Hamiltonian H, the similarly transformed Hamiltonian does not lejxesent a Hermitian operator. Moreover, it contains not only the one- and two-electrrm terms, as it does in H, but also all other many-electron operators up to the total number of electrons in the system. 

The assumption here is that if the total many-electron Hamiltonian commutes with various symmetry operations of the group then the Fock operator that leads to F can also be chosen in such a fashion. In such a case each MO will transform as one of the irreducible representations of the group. The ideas presented in the Roothaan paper are very important, and allow factorization of the Roothaan equations into simpler blocks, one for each irreducible representation, but experience has indicated that on occasion there are Hartree-Fock instabilities that destroy the symmetry of the resultant MOs. However, the Fock operator can always be constrained to have the symmetry of the nuclei. 

Although in the Dirac-Coulomb Hamiltonian the one-particle operator is correct to all orders in a, the two-particle interaction is only correct to a°. The Dirac-Coulomb Hamiltonian is not invariant under Lorentz transformations, however it can be considered as the leading term of a yet unknown relativistic many-electron Hamiltonian which fulfills this requirement. An operator which also takes into account the leading relativistic corrections for the two-electron terms is the Coulomb-Breit term (Breit 1929,1930,1932, 1938),... 

Transformation of the Many-Electron Hamiltonian to Polar Coordinates 335... 

After having transformed all operators of the many-electron atomic Hamiltonian in Eq. (9.1) to polar coordinates, we may write the Dirac-Coulomb Hamiltonian for an atom explicitly as... 

To consider the effect of the DKH transformation on the many-electron Hamiltonian of Eq. (12.63), the Hilbert space formulation of many-particle quantum mechanics presented in section 8.4 needs to be recalled. 

---