Representation electronically adiabatic

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

Of course, the degree of non-adiabatic character in the electron-exchange process is an intrinsic property, independent of the wavefunction representation used to describe it. For example, in terms of t f+ and t f, an electronically adiabatic process is one which can be analyzed in terms of a single adiabatic... 

In an electronic adiabatic representation, however, the electronic Hamiltonian becomes diagonal,i.e. ( a 77e C/3) = da,0Va, where the adiabatic Va potentials for initial (A,B,B ) and final (X) electronic states were described in Ref.[31]. The couplings between different electronic states arises from the matrix elements of the nuclear kinetic operator Tn, giving rise to the so-called non-adiabatic coupling matrix elements (NACME) and are due to the dependence of the electronic functions on the nuclear coordinates. The actual form of these matrix elements depends on the choice of the coordinates. 

Under the conditions of validity of the two-electronically-adiabatic-state approximation it is possible to change from the i]/al,ad(r q) (n = i, j) electronically adiabatic representation to a diabatic one 1,ad(r q) (n = i, j) for which the VR Xn(R) terms in the corresponding diabatic nuclear motion equations are significantly smaller than in the adiabatic equation or, for favorable conditions, vanish [24-26]. Such an electronically diabatic representation is usually more convenient for scattering calculations involving two electronically adiabatic PESs, but not for those involving a single adiabatic PES. This matter will be further discussed in Sec. III.B.3 for the case in which a conical intersection between the E ad(q) and Ejad(q) PESs occurs. 

In addition to the electronically adiabatic representation described by (4) and (5) or, equivalently (57) and (58), other representations can be defined in which the adiabatic electronic wave function basis set used in expansions (4) or (58) is replaced by some other set of functions of the electronic coordinates rel or r. Let us in what follows assume that we have separated the motion of the center of mass G of the system and adopted the Jacobi mass-scaled vectors R and r defined after (52), and in terms of which the adiabatic electronic wave functions are i] l,ad(r q) and the corresponding nuclear wave function coefficients are Xnd (R). The symbol q(R) refers to the set of scalar nuclear position coordinates defined after (56). Let iKil d(r q) label that alternate electronic basis set, which is allowed to be parametrically dependent on q, and for which we will use the designation diabatic. We now proceed to define such a set. LetXn(R) be the nuclear wave function coefficients associated with those diabatic electronic wave functions. As a result, we may rewrite (58) as... 

Let us now compare the characteristics of the electronically adiabatic and diabatic representations for systems displaying a conical intersection, in the one- and two-state approximations. 

In the force-state representation, we usually do not directly refer to the value of adiabatic potential energy surfaces. This is a practice common to any other electron wavepacket dynamics that does not rest on the electronic adiabatic representation. Therefore the potential energy that each path bears is to be estimated with integration of the force over a shifted distance. With the above Hamiltonian, Eq. (6.52) for path 0, let us write down the time-derivative of the momenta, that is, the forces as... 

Fig. 18. Schematic representation of adiabatic full arrows) and non-adiabatic dotted arrows) mechanisms of the vibrational transition n = 1—>n = 0ina molecule BC colliding with atom A. The terms represented by full and dash-dotted lines correspond to various electronic states arising upon lifting of the electronic state degeneracy of interacting partners...

Along each trajectory R(t) which is propagated in the framework of MD on the fly , we calculate the density matrix elements p,j by numerical integration. If the initial electronic state is a pure state as it is in our case, the set of equations 17.14-17.15 is equivalent to the time-dependent Schrodinger equation in the representation of adiabatic electronic states ... 

Thus is coupled to the state . The potentials associated with these two states are exhibited in Fig. lb. In this product representation, in which electronic and radiative degrees of freedom are treated on equal footing, we see that the presence of the radiation is manifested in a curve crossing of the potentials. (We note that although the (j) are electronically adiabatic, the system states are overall diabatic since the electric-dipole coupling is off-diagonal. Adiabatic states for the system could be obtained by diagonalizing -f in which case we would have an avoided... 

The Electronic Adiabatic-to-Diabatic Transformation Matrix and the Irreducible Representation of the Rotation Group... 

So far, we have treated the case n = /lo, which was termed the adiabatic representation. We will now consider the diabatic case where n is still a variable but o is constant as defined in Eq. (B.3). By multiplying Eq. (B.7) by j e I o) I arid integrating over the electronic coordinates, we get... 

II. n-ELECTRONIC STATE ADIABATIC REPRESENTATION A. Born-Huang Expansion... 

U(qJ is referred to as an adiabatic-to-diabatic transformation (ADT) matrix. Its mathematical sbucture is discussed in detail in Section in.C. If the electronic wave functions in the adiabatic and diabatic representations are chosen to be real, as is normally the case, U(q ) is orthogonal and therefore has n n — l)/2 independent elements (or degrees of freedom). This transformation mabix U(qO can be chosen so as to yield a diabatic electronic basis set with desired properties, which can then be used to derive the diabatic nuclear motion Schrodinger equation. By using Eqs. (27) and (28) and the orthonormality of the diabatic and adiabatic electronic basis sets, we can relate the adiabatic and diabatic nuclear wave functions through the same n-dimensional unitary transformation matrix U(qx) according to... 

In the -electronic-state adiabatic representation involving real electronic wave functions, the skew-symmetiic first-derivative coupling vector mahix... 

In the two-adiabatic-electronic-state Bom-Huang description of the total orbital wave function, we wish to solve the corresponding nuclear motion Schrodinger equation in the diabatic representation... 

The adiabatic picture is the standard one in quantum chemistry for the reason that, not only is it mathematically well defined, but it is also that used in ab initio calculations, which solve the electronic Hamiltonian at a particular nuclear geometry. To see the effects of vibronic coupling on the potential energy surfaces one must move to what is called a diabatic representation [1,65,180, 181]. 

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