Curl equations

In Section IV.A, the adiabatic-to-diabatic transformation matrix as well as the diabatic potentials were derived for the relevant sub-space without running into theoretical conflicts. In other words, the conditions in Eqs. (10) led to a.finite sub-Hilbert space which, for all practical purposes, behaves like a full (infinite) Hilbert space. However it is inconceivable that such strict conditions as presented in Eq. (10) are fulfilled for real molecular systems. Thus the question is to what extent the results of the present approach, namely, the adiabatic-to-diabatic transformation matrix, the curl equation, and first and foremost, the diabatic potentials, are affected if the conditions in Eq. (10) are replaced by more realistic ones This subject will be treated next. 

The Solution for a Single Conical Intersection The curl equation for a two-state system is given in Eq. (26) ... 

Equation (26) is a set of partial first-order differential equations. Each component of the Curl forms an equation and this equation may or may not be coupled to the other equations. In general, the number of equations is equal to the number of components of the Curl equations. At this stage, to solve this set of equation in its most general case seems to be a fomiidable task. 

We start heating the curl equation expressed in terms of polar coordinates ... 

To study the two isolated conical intersections, we have to treat two-state curl equations that are given in Eq. (26). Here, the first 2 x 2 x mahix contains the (vectorial) element, that is, X012 and the second 2 x 2 x mahix contains X023- As before each of the non-adiabatic coupling terms, X012 and X023 has the following components ... 

Equation (171) is the an explicit curl equation for a coupling that does not has a source of its own but is formed due to the interaction between two real conical intersection. 

Another subject with important potential application is discussed in Section XIV. There we suggested employing the curl equations (which any Bohr-Oppenheimer-Huang system has to obey for the for the relevant sub-Hilbert space), instead of ab initio calculations, to derive the non-adiabatic coupling terms [113,114]. Whereas these equations yield an analytic solution for any two-state system (the abelian case) they become much more elaborate due to the nonlinear terms that are unavoidable for any realistic system that contains more than two states (the non-abelian case). The solution of these equations is subject to boundary conditions that can be supplied either by ab initio calculations or perturbation theory. 

We start by writing the curl equation in Eq. 057) for a vector t(x,y) in Cartesian coordinates. 

Geometric phase effect (GPE) conical intersections, 4-8 adiabatic eigenstates, 8-11 topographical energy, 568-569 curl equations, 11-17 degenerate states chemistry, x-xiii electronic states ... 

In this section, we discuss the possibility that the electronic non-adiabatic coupling terms will be derived, not by ab initio treatments but, by solving the curl equations for a given set of boundary conditions obtained from ab initio calculations [114,115]. In other words, instead of performing an ab initio calculation at any point in configuration space we suggest solving the relevant differential equations for boundary conditions obtained from a (limited) ab initio calculation [64-74] or perturbation theory [66,67]. 

The FDTD approach is based on direct numerical solution of the time dependent Maxwell s curl equations. In the 2D TM case the nonzero field components are E Hy and E, the propagation is along the z direction and the transverse field variations are along x. In lossless media. Maxwell s equations... 

Similar to ED- in FEM-schemes the cross-section under consideration is divided in a set of little areas, rectangles or triangles, preferentially. Now, within the finite elements, the field is defined not only by nodal amplitudes, but expressed within the whole element in terms of low-order polynomial functions. If e.g. the full magnetic field is considered. Maxwells curl equations give... 

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)). 

The corresponding four component non-Abelian Curl equation which guarantees the existence of a solution for equation (51 )) takes the form ... 

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. 

The solutions to the Maxwell field equations that are most often used in the applications discussed in this book are referred to as plane waves of monochromatic light. These are derived from the Maxwell curl equations. In a system free of charges and currents, these are... 

Examination of either of the curl equations leads to a useful relationship between E and H. Inserting the results of (1.20) into the curl equations leads to... 

In this case the material is characterized by e - = e8, py = jj.8-, - = tM jiyS. and Q, j = 0. The parameter y is the rotary power. This material couples the electric and magnetic fields, and the Maxwell curl equations are... 

We seek here general solutions to Maxwell s equations and begin with equation (1.8) which resulted by combining the Maxwell curl equations,... 

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