Many-electron Hamiltonian operator

We ve derived a complete many-electron Hamiltonian operator. Of course, the Schrodinger equation involving it is intractable, so let s consider a simpler problem, involving the one-electron hamiltonian... 

H is the many-electron Hamiltonian operator, just as V is the many-electron wave function. The angle-brackets mean integration over the electronic coordinates. 

This is simply a standard linear variation problem to which we have the solutions from Chapter 1 a single matrix equation involving the matrix of H the full many-electron Hamiltonian operator and the overlap matrix ... 

To consider the question in more detail, you need to consider spin eigenfunctions. If you have a Hamiltonian X and a many-electron spin operator A, then the wave function T for the system is ideally an eigenfunction of both operators ... 

In the uncorrelated limit, where the many-electron Fock operator replaces the full electronic Hamiltonian, familiar objects of HF theory are recovered as special cases. N) becomes a HF, determinantal wavefunction for N electrons and N 1) states become the frozen-orbital wavefunctions that are invoked in Koopmans s theorem. Poles equal canonical orbital energies and DOs are identical to canonical orbitals. 

Applying the permutation operator P12 is therefore equivalent to interchanging rows of the determinant in Eq. (2.15). Having devised a method for constructing many-electron wavefunctions as a product of MOs, the final problem concerns the form of the many-electron Hamiltonian which contains terms describing the interaction of a given electron with (a) the fixed atomic nuclei and (b) the remaining (N— 1) electrons. The first step is therefore to decompose H(l, 2, 3,..., N) into a sum of operators Hj and H2, where ... 

The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics [1]. The simplest form is the Dirac-Coulomb (DC) Hamiltonian, where the nonrelativistic one-electron terms in the Schrodinger equation are replaced by the one-electron Dirac operator hj). 

The orbitals of an atom are labeled by 1 and m quantum numbers the orbitals belonging to a given energy and 1 value are 21+1- fold degenerate. The many-electron Hamiltonian, H, of an atom and the antisymmetrizer operator A = (V l/N )Ep sp P commute with total Lz =Ej Lz (i), as in the linear-molecule case. The additional symmetry present in the spherical atom reflects itself in the fact that Lx, and Ly now also commute with H and A. However, since Lz does not commute with Lx or Ly, new quantum... 

The third term in Eq. (8) is the sum over all electron-electron repulsion operators abbreviated by g(i,j), which is in the case of the relativistic many-electron Hamiltonian equal to... 

Since the many-electron Hamiltonian, the effective one-electron Hamiltonian obtained in Hartree-Fock theory (the Fock operator ), and the one- and two-electron operators that comprise the Hamiltonian are all totally symmetric, this selection rule is extremely powerful and useful. It can be generalized by noting that any operator can be written in terms of symmetry-adapted operators ... 

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... 

The puzzle depended on the simple fact that most physicists using the method of complex scaling had not realized that the associated operator u - the so-called dilatation operator - was an unbounded operator, and that the change of spectra -e.g. the occurrence of complex eigenvalues - was due to a change of the boundary conditions. Some of these features have been clarified in reference A, and in this paper we will discuss how these properties will influence the Hartree-Fock scheme. The existence of the numerical examples finally convinced us that the Hartree-Fock scheme in the complex symmetric case would not automatically reduce to the ordinary Hartree-Fock scheme in the case when the many-electron Hamiltonian became real and self-adjoint. Some aspects of this problem have been briefly discussed at the 1987 Sanibel Symposium, and a preliminary report has been given in a paper4 which will be referred to as reference D. 

Since the exact relativistic many-electron Hamiltonian is not known, the electron-electron interaction operators g(i, j) are taken to be of Coulomb type, i.e. 1/r,- . As a first relativistic correction to these nonrelativistic electron-electron interaction operators, the Breit correction, Equations (2.2) or (2.3), is used. For historical reasons, the first term in Equation (2.2) is called the Gaunt or magnetic part of the full Breit interaction. Since it is not more complicated than l/ri2, it is from an algorithmic point of view equivalent to the Coulomb interaction, therefore it has frequently been included in the calculations. The second term, the so-called retardation term, appears to be rather complicated and it has been considered less frequently. In the case of few-electron systems further quantum electrodynamical corrections, like self-energy and vacuum polarization, have also been considered and are reviewed in another part of this book (see Chapter 1). 

Since the Dirac equation is valid only for the one-electron system, the one-electron Dirac Hamiltonian has to be extended to the many-electron Hamiltonian in order to treat the chemically interesting many-electron systems. The straightforward way to construct the relativistic many-electron Hamiltonian is to augment the one-electron Dirac operator, Eq. (70) with the Coulomb or Breit (or its approximate Gaunt) operator as a two-electron term. This procedure yields the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonian derived from quantum electrodynamics (QED)... 

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]. 

Formal expansions are considered in terms of energy eigenstates n) of the many-electron hamiltonian (Eq. (4.3)) and the number operator (Eq. (3.55)) ... 

The many-electron hamiltonian can be partitioned into an unperturbed part consisting of the Fock operator in diagonal form and a perturbation consisting of modified interaction terms. One can write... 

By substituting in Eq.4.18 the Coulomb interaction operator by its relativistic extension (4.19) one obtains the so-called Dirac-Breit many electron Hamiltonian... 

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),... 

We do not yet know if the spin orbitals that are the solution to Equation 2.30 are also the looked-for spin orbitals, minimizing the energy expectation value in Equation 2.27. The proof will be carried out in steps. First, we have to construct a many-electron operator that corresponds to the Fock one-electron operator. This is easy to do if we subtract a constant term to compensate for the fact that electron repulsion is counted twice. We obtain the many-electron Hamiltonian as... 

The Hamiltonian of the two-electron atom already features all pair-interaction operators that are required to describe a system with an arbitrary number of electrons and nuclei. Hence, the step from one to two electrons is much larger than from two to an arbitrary number of electrons. For the latter we are well advised to benefit from the development of nonrelativistic quantum chemistry, where the many-electron Hamiltonian is exactly known, i.e., where it is simply the sum of all kinetic energy operators according to Eq. (4.48) plus all electrostatic Coulombic pair interaction operators. 

The two-electron interaction operators g i,j) in the many-electron Hamiltonian are the reason why a product ansatz for the electronic wave function Yg/( r, ) that separates the coordinates of the electrons is not the proper choice if the exact function is to be obtained in a single product of one-electron functions. Instead, the electronic coordinates are coupled and the motion of electrons is correlated. It is therefore natural to assume that a suitable ansatz for the many-electron wave function requires functions that depend on two coordinates. Work along these lines within the clamped-nuclei approximation has a long history [320-328]. The problem, however, is then what functional form to choose for these functions. First attempts in molecular quantum mechanics used simply terms that are linear in the interelectronic distance Tij = ki — Tjl [329-333], while it could be shown that an exponential ansatz is more efficient [334-337]. 

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