Hamiltonian equation electrical potential

This Hamiltonian is used usually only for the short-range part of Coulomb interaction. The long-range interactions can be better introduced through the self-consistent electrical potential Poisson equation with the average electron density. 

Yaspatial positions rj of the N molecules yields a set of energy eigenvalues ( rj ), which can be interpreted as the effective Al-particle potential in the single-channel many-body Hamiltonian (Equation 12.1). The dependence of Vgl ( r ) on the electric fields E provides the basis for the engineering of the many body interactions in (Equation 12.2). The validity of this adiabatic approximation and of the associated decoupling of the Born-Oppenheimer channels will be discussed below. 

The most unsatisfactory features of our derivation of the molecular Hamiltonian from the Dirac equation stem from the fact that the Dirac equation is, of course, a single particle equation. Hence all of the inter-electron terms have been introduced by including the effects of other electrons in the magnetic vector and electric scalar potentials. A particularly objectionable aspect is the inclusion of electron spin terms in the magnetic vector potential A, with the use of classical field theory to derive the results. It is therefore of interest to examine an alternative development and in this section we introduce the Breit Hamiltonian [16] as the starting point. We eventually arrive at the same molecular Hamiltonian as before, but the derivation is more satisfactory, although fundamental difficulties are still present. 

In the absence of interactions, electrons are described by the Dirac equation (1928), which rules out the quantum relativistic motion of an electron in static electric and magnetic fields E= yU and B = curl A (where U and A are the scalar and vectorial potentials, respectively) [43-45]. As the electrons involved in a solid structure are characterized by a small velocity with respect to the light celerity c (v/c 10 ) a 1/c-expansion of the Dirac equation may be achieved. More details are given in a paper published by one of us [46]. At the zeroth order, the Pauli equation (1927), in which the electronic spin contribution appears, is retrieved then conferring to this last one a relativistic origin. At first order the spin-orbit interaction arises and is described by the following Hamiltonian... 

The equation is written in velocity gauge. Atomic units are used. The particle has charge unity and mass m in units of the free electron mass. V is the constant potential energy appropriate for the interval under consideration. The vector potential is supposed to be spatially constant at the length scale of the structure. With such a vector potential, the A2 term contributes an irrelevant phase factor which can be omitted. For a one-mode field A(t) is written as Ao cos(ut). The associated electric field is 0 sin(ut), with 0 = uAq. px is the linear momentum i ld/dx. For such a time-periodic Hamiltonian, a scattering approach can be developped, with a well-defined initial energy, and time-independent transition probabilities for reflection and transmission. 

In a different gauge, it is possible to construct the multipolar Hamiltonian which is obtained by applying a unitary transformation to the minimal coupling Hamiltonian [75-77,106]. In the multipolar Hamiltonian, it is the transverse electric field, and the magnetic field, B(r) (satisfying Maxwell s equation, V x Et = - f), that appear, rather than the vector potential. Now, the interaction is written as... 

The second, third and foiuth terms inside the first summation in equation (3) are the perturbations introduced into the hamiltonian by the effects of the external fields. The fourth term, describing the electric field perturbation, is linear in the external potential or electric field. The second and third terms give rise to linear and quadratic responses ro a constant, uniform magnetic field. Smaller terms, arising from the Dirac equation, which represent spin-orbit coupling etc. have been omitted. 

Consistent with time-independent Hartree-Fock theory the main approximation in time-dependent Hartree-Fock theory is, that the system is represented by a single Slater determinant, which now is composed of time-dependent single-particle wavefunctions. The time-dependent Schrodinger equation that has to be solved is given in eqn (1). The time-dependent Hamiltonian consists of a static Hamiltonian and an additional time-dependent operator describing the time-dependent perturbation, e.g. an electric field, which is a sum of time-dependent single-particle potentials ... 

A second and more serious consequence of the appearance of the potential in the denominator is that the ZORA Hamiltonian is not invariant to the choice of electric gauge. Adding a constant to the potential should result in the addition of a constant to the energy, which is indeed the case for the Dirac equation. For ZORA, the relation between the ZORA and the Dirac eigenvalue for a one-electron system is given by (18.9). If we add a constant. A, to the Dirac energy in this equation, we get... 

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