Derivation of the many-electron Hamiltonian

From classical magnetostatic theory, the magnetic vector potential at electron i is given by 

The first term arises from the spin magnetic moments of the other electrons, and the second term comes from the orbital motions. Rji is equal to Rj — Rt, the vector which gives the position of electron j relative to electron i. Strictly speaking the second term in (3.133) is incorrect for several reasons, but we will return to this point later. 

In section 3.4 we carried out a Foldy-Wouthuysen transformation on the Dirac Hamiltonian and obtained the result (3.84), correct to order c 2, 

We now expand the different terms in (3.135), retaining only terms which are linear in A, and obtain the results, 

The procedure is now to expand the terms in (3.140), using (3.133) and (3.134) to evaluate A, and pi and replacing tr,- by Pi + eA. Clearly, many terms are obtained but we can simplify matters by neglecting all terms which are quadratic or higher order in A] and by neglecting terms in c-3 or higher. We examine each of the six terms in (3.140) in turn. It is, of course, obvious that e, = e, = e and m, = m, = m in what follows but we retain the subscripts to keep track of the two identical particles. 

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In the derivation of the many-electron Hamiltonian (section 3.6) the interactions between electrons were introduced in the expressions for the magnetic and electric... 

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... 

In this set the functions can be classified into two types in the right column the spatial multiplier is symmetric with respect to transpositions of the spatial coordinates and the spin multiplier is antisymmetric with respect to transpositions of the spin coordinates in the left column the spatial multiplier is antisymmetric with respect to transpositions of the spatial coordinates and the spin multipliers are symmetric with respect to transpositions of the spin coordinates. Because in the second case the spatial (antisymmetric) multiplier is the same for all three spin-functions, the energy of these three states will be the same i.e. triply degenerate - a triplet. The state with the antisymmetric spin multiplier is compatible with several different spatial wave functions, which probably produces a different value of energy when averaging the Hamiltonian, thus producing several spin-singlet states. From this example one may derive two conclusions (i) the spin of the many electronic wave function is important not by itself (the Hamiltonian is spin-independent), but as an indicator of the symmetry properties of the wave function (ii) the symmetry properties of the spatial and spin multipliers are complementary - if the spatial part is symmetric with respect to permutations the spin multiplier is antisymmetric and vice versa. 

The derivation of the Breit-Pauli Hamiltonian is tedious. It is nowadays customary to follow the Foldy-Wouthuysen approach first given by Chraplyvy [679-681], which has, for instance, been sketched by Harriman [59]. Still, many presentations of this derivation lack significant details. In the spirit of this book, we shall give an explicit derivation which is as detailed and compact as possible. A review of the same expression derived differently was provided by Bethe in 1933 [72]. Compared to the DKH treatment of the two-electron term to lowest order as described in section 12.4.2 we now only consider the lowest-order terms in 1/c. We transform the Breit operator in Eq. (8.19) by the free-particle Foldy-Wouthuysen transformation. 

Density Functional Models. Methods in which the energy is evaluated as a function of the Electron Density. Electron Correlation is taken into account explicitly by incorporating into the Hamiltonian terms which derive from exact solutions of idealized many-electron systems. 

In this review we shall first establish the theoretical foundations of the semi-classical theory that eventually lead to the formulation of the Breit-Pauli Hamiltonian. The latter is an approximation suited to make the connection to phenomenological model Hamiltonians like the Heisenberg Hamiltonian for the description of electronic spin-spin interactions. The complete derivations have been given in detail in Ref. (21), but turn out to be very involved and are thus scattered over many pages in Ref. (21). For this reason, we aim here at a summary that is as brief and concise as possible so that all relevant connections between different levels of approximation are evident. This allows us to connect present-day quantum chemical methods to phenomenological Hamiltonians and hence to establish and review the current status of these first-principles methods applied to transition-metal clusters. 

Prior to choosing the wave-function approximation it is, however, necessary to set up the electronic Hamiltonian H that describes all interactions of elementary particles. Therefore, we start with the derivation of the full semi-classical many-electron Hamiltonian describing all interactions relevant for chemical problems and subsequently discuss approximations to this full-fledged quantum chemical Hamiltonian. 

This Fock operator has been derived starting from the assumption of a Hartree-Fock valence function valence electrons has little influence on the core electrons, so that the many-electron valence hamiltonian may be similarly approximated as... 

Semirigorous LCAO-MO-SCF methods start with the complete many-electron Hamiltonian and make certain approximations for the integrals and for the form of the matrices to be solved. Several years ago, such a method was derived starting with the correct many electron Hamiltonian (in which interelectronic interactions are included explicitly) and the LCAO-MO-SCF equations of Roothaan and then making a consistent series of systematic... 

In the DFT, as in the Hartree-Fock approach, an effective independent-particle Hamiltonian is arrived at, and the electronic states are solved for self consistency. The many-electron wave function is still written as a Slater determinant. However, the wave functions used to construct the Slater determinant are not the one-electron wave functions of the Hartree-Fock approximation. In the DFT, these wave functions have no individual meaning. They are merely used to construct the total electron-charge density. The difference between the Hartree-Fock and DFT approaches lies in the dependence of the Hamiltonian in DFT on the exchange correlation potential, Vxc[ ](t), a functional derivative of the exchange correlation energy, Exc, that, in turn, is a functional (a function of a function) of the electron density. In DFT, the Schrodinger equation is expressed as ... 

Second, the Hamiltonian operator for a relativistic many-body system does not have the simple, well-known form of that for the non-relativistic formulation, i.e. a sum of a sum of one-electron operators, describing the electronic kinetic energy and the electron-nucleus interactions, and a sum of two-electron terms associated with the Coulomb repulsion between the electrons. The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics.46... 

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