Curls

Assume that a differentiable vector field is a function of certain independent variables as V = V (x, X2, X3). The curl (also called the rotation) of V produces a vector. 

---

This wave equation is tire basis of all wave optics and defines tire fimdamental stmcture of electromagnetic tlieory witli tire scalar function U representing any of tire components of tire vector functions E and H. (Note tliat equation (C2.15.5) can be easily derived by taking tire curl of equation (C2.15.1) and equation (C2.15.2) and substituting relations (C2.15.3) and (C2.15.4) into tire results.)... 

A third expression may be obtained by taking the curl inside the bracket in Eq. (39) and using the identities... 

Pure versus Tensorial Gauge Fields The Curl Condition ... 

In a non-Abelian theory (where the Hamiltonian contains noncommuting matrices and the solutions are vector or spinor functions, with N in Eq. (90) >1) we also start with a vector potential Af, [In the manner of Eq. (94), this can be decomposed into components A, in which the superscript labels the matrices in the theory). Next, we define the field intensity tensor through a covaiiant curl by... 

The consistency condition for this set of equations to possess a (unique) solution is that the field intensity tensor defined in Eq. (99) is zero [72], which is also known as the curl condition and is written in an abbreviated form as... 

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 ... 

It will be recognized that this generalizes the result proved by Baer in [72]. Though that work did establish the validity of the curl condition for the derivative operator as long as some 25 years ago and the validity is nearly trivial for the second term taken separately, the same result is not self-evident for the combination of the two teiins, due to the nonlinearity of F X). An important special case is when G(R) = R /2. Then... 

The structure of W (Rl) discussed at the beginning of this section, will reflect itself in some interrelations between the P, y(qx) obtained by solving this equation. More importantly, this equation has a solution if and only if the elements of the mati ix satisfy the following curl-condition... 

The curl condition given by Eq. (43) is in general not satisfied by the n x n matrix W (R ), if n does not span the full infinite basis set of adiabatic elechonic states and is huncated to include only a finite small number of these states. This tmncation is extremely convenient from a physical as well as computational point of view. In this case, since Eq. (42) does not have a solution, let us consider instead the equation obtained from it by replacing WC) (R t) by its longitudinal part... 

XIII. Curl Condition Revisited Introduction of the Yang-Mills Field... 

Like the curl condition is reminiscent of the Yang-Mills field, the quantization just mentioned is reminiscent of a study by Wu and Yang [76] for the quantization of Dirac s magnetic monopole [77-78]. As will be shown, the present quantization conditions just like the Wu and Yang conditions result from a phase factor, namely, the exponential of a phase and not just from a phase. 

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. 

Thus, we still relate to the same sub-space but it is now defined for P-states that are weakly coupled to <2"States. We shall prove the following lemma. If the interaction between any P- and Q-state measures like 0(e), the resultant P-diabatic potentials, the P-adiabatic-to-diabatic bansfomiation maOix elements and the P-curl t equation are all fulfilled up to 0(s ). 

Next, we analyze the P-curl condition with the aim of examining to what extent it is affected when the weak coupling is ignored as described in Section IV.B.l [81]. For this purpose, we consider two components of the (unperturbed) X matrix, namely, the mafiices Xq and Xp, which are written in the following form [see Eq. (43)] ... 

Notice, all terms in the curled parentheses are of O(fi ) curl condition becomes... 

In this section, the curl condition is extended to include the points of singularity as discussed in Appendix C. The study is meant to shed light as to the origin of... 

In Section in.B, and later in Appendix C, it was shown that the sufficient condition for the adiabatic-to-diabatic transfonnation matrix A to be single valued in a given region in configuration space is the fulfillment of the following curl condition [8,34] ... 

This condition is fulfilled as long as the components of t are analytic functions at the point under consideration (in case part of them become singular at this point, curl X is not defined). 

---