Orthogonal transformation and two-state problem

The two-state problem assumes that the two given adiabatic electronic eigenstates d and P2 of //c are coupled to each other but not to the rest of the electronic eigenstates (a so-called bath )- In other words, the quantum treatment is confined to the Hilbert subspace (I , (l 2 spanned on t and P2 and acts thereupon as P 2HelP12 where Pn = (l )((l I + Id XdTl is the projector onto d , l 2. Rotating d j and P2 via a two-dimensional orthogonal transformation, 

Such an angle a is called the mixing angle of the adiabatic-to-diabatic transformation (ADT) for the two-state problem. 

Let us now consider the generalized two-state problem which assumes that two given adiabatic electronic eigenstates and (J 2 are coupled to the bath d i, 1 defined as the complementary subspace to (P]. (P2 in L)ei (for the inclusion of the third state see, e.g. Refs. [59-66]), i.e., 

Their rotation upon the angle a, similar to that defined by equation (15), gives the 

Equations (16) and (19) are formally identical. It is suggested that a satisfies equation (17) for the two-state problem and a the similar one for the generalized two-state problem. Applying now the curl operator to both sides of the first equations in equations (16) and (19) and substituting a = a in the last one, one readily obtains that, in the former case, this results in the identity 0 = 0 because the right hand side of equation (11) vanishes for the two-state problem. In the latter case, one finds that the substitution a = a and curl operation, applied to the first equation in equation (19), do not commute, viz., replacing a by a eliminates the right hand side of the first equation in equation (19) and a further application of the curl to the resultant equation yields 

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