Vector potential term

Although this retardation operator includes products of the Dirac matrices for the two interacting particles, ai 2, magnetic interactions mediated via the vector potential seem to be included. However, already the derivation of the classical retarded interaction in section 3.5 clearly shows that all vector potential terms inherently feature a 1/c prefactor, as in Eq. (3.214), so that the vector potential contribution and, hence, the magnetic contribution of the interaction enters these equations unretarded [178]. This is also clear from the rigorous collection of terms after a Taylor expansion around time fi in Eq. (3.233). 

The term a, therefore plays the role of a vector potential in electromagnetic theory, with a particularly close connection with the Aharonov-Bohra effect, associated with adiabatic motion of a charged quantal system around a magnetic... 

Some final comments on the relevance of non-adiabatic coupling matrix elements to the nature of the vector potential a are in order. The above analysis of the implications of the Aharonov coupling scheme for the single-surface nuclear dynamics shows that the off-diagonal operator A provides nonzero contiibutions only via the term (n A n). There are therefore no necessary contributions to a from the non-adiabatic coupling. However, as discussed earlier, in Section IV [see Eqs. (34)-(36)] in the context of the x e Jahn-Teller model, the phase choice t / = —4>/2 coupled with the identity... 

As demonstrated in [53] it is convenient to incorporate the geometrical phase effect by adding the vector potential in hyperspherical coordinates. Thus we found that the vector potential gave three terms, the first of which was zero, the second is just a potential term... 

Although Eq. (139) looks like a Schrodinger equation that contains a vector potential x, it cannot be interpreted as such because t is an antisymmetric matrix (thus, having diagonal terms that are equal to zero). This inconvenience can be repaired by employing the following unitary bansformation ... 

The important outcome from this transformation is that now the non-adiabatic coupling term t is incorporated in the Schrodinger equation in the same way as a vector potential due to an external magnetic field. In other words, X behaves like a vector potential and therefore is expected to fulfill an equation of the kind [111a]... 

This section is devoted to the idea that the electronic non-adiabatic coupling terms can be simulated as vector potentials. For this purpose, we considered... 

In Section XIII, we made a connection between the curl condition that was found to exist for Bom-Oppenheimer-Huang systems and the Yang-Mills field. Through this connection we found that the non-adiabatic coupling terms can be considered as vector potentials that have their source in pseudomagnetic... 

The first term is identical to the usual kinetic energy operator. Inserting the expression for the vector potential (10.61) yields... 

The first term is referred to as the diamagnetic contribution, while the latter is the paramagnetic part of the magnetizability. Each of the two components depend on the selected gauge origin however, for exact wave functions these cancel exactly. For approximate wave functions this is not guaranteed, and as a result the total property may depend on where the origin for the vector potential (eq. (10.61)) has been chosen. 

There are three ways of implementing the GP boundary condition. These are (1) to expand the wave function in terms of basis functions that themselves satisfy the GP boundary condition [16] (2) to use the vector-potential approach of Mead and Truhlar [6,64] and (3) to convert to an approximately diabatic representation [3, 52, 65, 66], where the effect of the GP is included exactly through the adiabatic-diabatic mixing angle. Of these, (1) is probably the most... 

An advantage of Eq. (90) for computational purposes is that the solutions are subject to single-valued boundary conditions. It is also readily verified that inclusion of an additional factor eiA KC ) on the right-hand side of Eq. (89) adds a term Aa, = —Wg, A / to the vector potential, which leads in turn to a compensating factor g- A,K6) in the nuclear wave function. The total wave function is therefore invariant to changes in such phase factors. 

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