Vector-potential theory derivation

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. 

Kirtman, Gu, and Bishop [155] have derived a fuUy self-consistent procedure by using a noncanonical form of perturbation theory. Following Genkin and Mednis [120], they employed the vector potential and replaced the quasimomentum k by... 

There is a parallel between the wave packet-based semi-classical treatment of nuclear motion developed here and the Thomas-Fermi theory of an electronic system in a slowly varying vector potential. In the semi-classical electronic theory as well as here, one naturally arrives at a locally linear approximation to the scalar-potential-derived forces and a locally uniform approximation to the magnetic force derived from the vector potential. See, R. A. Harris and J. A. Cina, J. Chem. Phys. 79, 1381 (1983) C. J. Grayce and R. A. Harris, Molec. Phys. 71, 1 (1990). 

In this paper, we present a novel derivation of the London equation based on DP theory The application of a time-dependent vector potential, A(t) = A(0)oos(2nvt) along a fiber generates a time-dq)endent ground state vector which for small time and weak field has the form... 

Two problems need to be addressed next. First, what to do with the two different time derivatives and, second, how are the scalar and vector potentials of the electrons to be chosen. With respect to the former issue, we adopt the single absolute time frame, ti,t2 t of nonrelativistic theory. This reduces the two time derivatives to one with respect to the new absolute time t. 

The space-charge current density in vacuo expressed by Eqs. (3) and (4) constitutes the essential part of the present extended theory. To specify the thus far undetermined velocity C, we follow the classical method of recasting Maxwell s equations into a four-dimensional representation. The divergence of Eq. (1) can, in combination with Eq. (4), be expressed in terms of a fourdimensional operator, where (j, 7 p) thus becomes a 4-vector. The potentials A and are derived from the sources j and p, which yield... 

The first step to making the theory more closely mimic the experiment is to consider not just one structure for a given chemical formula, but all possible structures. That is, we fully characterize the potential energy surface (PES) for a given chemical formula (this requires invocation of the Born-Oppenheimer approximation, as discussed in more detail in Chapters 4 and 15). The PES is a hypersurface defined by the potential energy of a collection of atoms over all possible atomic arrangements the PES has 3N — 6 coordinate dimensions, where N is the number of atoms >3. This dimensionality derives from the three-dimensional nature of Cartesian space. Thus each structure, which is a point on the PES, can be defined by a vector X where... 

The diabatization within the time-dependent framework produced the expected potential matrix W presented in equation (56) but enforced the four vector curl equation which is given in equation (54). This set of equations contains not only derivatives with respect to the spatial coordinates but also with respect to time. In fact this non-Abelian curl equation is completely identical to YM curl equation which has its origin in field theory. 

If we use the potentials derived above in our molecular Hamiltonian, they are open to the further serious objection that they refer only to an electron moving with uniform velocity, a situation which is not very realistic in the context of the molecular problem. However, the theory of special relativity does not provide a means of describing the motion of a rapidly moving and accelerating particle exactly. An approximate treatment is possible, but since the effects of the non-uniform motion of an electron on its vector and scalar potentials give terms with higher powers of 1 /c than we require in the final expansion of our Hamiltonian, we can ignore them. 

In the dielectric screening method the electron density response due to the motion of the ions around their equilibrium positions is calculated in first order perturbation theory. The potential energy of the crystal for an arbitrary configuration of the ions is expanded to second order in the ionic displacements from equilibrium. The expansion coefficients of the second order term form a matrix. The Fourier transform of this matrix is the dynamical matrix whose eigenvalues yield the phonon frequencies. The dynamical matrix has an ionic and electronic part. The electronic part can be expressed in terms of the electron density response matrix and of the ionic potential. This method has the advantage over the total energy difference m ethod that the phonon frequencies for any arbitrary wave vector can be calculated without additional difficulties. Furthermore in this method the acoustic sum rule is automatically satisfied as a consequence of the way the dynamical matrix is derived. However the dielectric screening method is limited to harmonic phonons. 

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