Adiabatic vector potential, electronic

The development of electronic geometric phase factors is governed by an adiabatic vector potential induced in the nuclear kinetic energy when we extend the Born-Oppenheimer separation to the degenerate pair of states [1, 23-25]. To see how the induced vector potential appears, we consider the family of transformations which diagonalize the excited state electronic coupling in the form... 

The transverse part of the induced adiabatic vector potential, (2.16), which appears in the kinetic energy of the adiabatic nuclear Hamiltonian, (2.18), governs electronic geometric phase development [25,28]. The two dimensional curl of Aj yields a solenoidal magnetic field of the form 7rftS(, )S( 2) general case of zt O, it yields a monopolar field). 

The semi-classical equations of motion obtained above involve only the transverse adiabatic vector potential which is, by definition, independent of the choice of gauge functions/(q) and g(q). The (Aj -f A2)/2M term in the potential is also independent of those two arbitrary functions. The locally quadratic approach to Gaussian dynamics therefore gives physically equivalent results for any choice of /(q) and g(q). The finding that the locally quadratic Hamiltonian approach developed here is strictly invariant with respect to choice of phases of the adiabatic electronic eigenstates supersedes the approximate discussion of gauge invariance given earlier by Romero-Rochin and Cina [25] (see also [40]). 

Finally, in brief, we demonstrate the influence of the upper adiabatic electronic state(s) on the ground state due to the presence of a Cl between two or more than two adiabatic potential energy surfaces. Considering the HLH phase, we present the extended BO equations for a quasi-JT model and for an A -1- B2 type reactive system, that is, the geometric phase (GP) effect has been inhoduced either by including a vector potential in the system Hamiltonian or... 

At this stage, we would like to mention that the model, without the vector potential, is constructed in such a way that it obeys certain selection rules, namely, only the even —> even and the odd —> odd transitions are allowed. Thus any deviation in the results from these selection rules will be interpreted as a symmetry change due to non-adiabatic effects from upper electronic states. 

In this chapter, we discussed the significance of the GP effect in chemical reactions, that is, the influence of the upper electronic state(s) on the reactive and nonreactive transition probabilities of the ground adiabatic state. In order to include this effect, the ordinary BO equations are extended either by using a HLH phase or by deriving them from first principles. Considering the HLH phase due to the presence of a conical intersection between the ground and the first excited state, the general fomi of the vector potential, hence the effective... 

Now, we recall the remarkable result of [72] that if the adiabatic electronic set in Eq. (90) is complete (N = oo), then the curl condition is satisfied and the YM field is zero, except at points of singularity of the vector potential. (An algebraic proof can be found in Appendix 1 in [72]. An alternative derivation, as well as an extension, is given below.) Suppose now that we have a (pure) gauge g(R), that satisfies the following two conditions ... 

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... 

The time-dependent density functional theory [38] for electronic systems is usually implemented at adiabatic local density approximation (ALDA) when density and single-particle potential are supposed to vary slowly both in time and space. Last years, the current-dependent Kohn-Sham functionals with a current density as a basic variable were introduced to treat the collective motion beyond ALDA (see e.g. [13]). These functionals are robust for a time-dependent linear response problem where the ordinary density functionals become strongly nonlocal. The theory is reformulated in terms of a vector potential for exchange and correlations, depending on the induced current density. So, T-odd variables appear in electronic functionals as well. 

In this chapter we discuss the close relationship between the Born-Oppenheimer treatment of molecular systems and field theory as applied to elementary particles. The theory is based on the Born-Oppenheimer non-adiabatic coupling terms which are known to behave as vector potentials in electromagnetic dynamics. Treating the time-dependent Schrodinger equation for the electrons and the nuclei we show that enforcing diabatization produces for non-Abelian time-dependent systems the four-component Curl equation as obtained by Yang and Mills (Phys. Rev. 95, 631 (1954)). 

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