Adiabatic electron wave function

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. 

As discussed in Section II.A, the adiabatic electronic wave functions and depend on the nuclear coordinates Rx only through the subset... 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

Multiplying on the left by R,e(t)), integrating over the electronic coordinates and using the orthonormality of the adiabatic electronic wave functions of Eq. (25) we obtain... 

As discussed in Section II. A, the adiabatic electronic wave functions, a and / 1,ad depend on the nuclear coordinates R> only through the subset (which in the triatomic case consists of a nuclear coordinate hyperradius p and a set of two internal hyperangles this permits one to relate the 6D vector W(1)ad(Rx) to another one w(1 ad(q J that is 3D. For a triatomic system, let aIX = (a1 -. blk, crx) be the Euler angles that rotate the space-fixed Cartesian frame into the body-fixed principal axis of inertia frame IX, and let be the 6D gradient vector in this rotated frame. The relation between the space-fixed VRi and is given by... 

The total wave function (r,q), which describes the motions of electrons and nuclei, can be expanded into a series by the adiabatic electron wave functions, i.e. represented in the form of the sum... 

The above analysis is based on the Born-Openheimer approximation in which the adiabatic electron wave functions are frozen at the bottom of the corresponding minimum. An important advantage of this approach is that we work with the limited size for the electron basis functions and not with the infinite basis of the vibrational states. This makes the problem solvable in simple terms. 

Equation (36) agrees with previous results, which have been derived in a more heuristic manner [30-32], The adiabatic electronic potential-energy surfaces (that is, the eigenvalues of H — TVU) are doubly degenerate (Kramers degeneracy). The adiabatic electronic wave functions carry nontrivial geometric phases which depend on the radius of the loop of integration [29-32]. 

The two cases where the 4>n are the diabatic or the adiabatic electronic wave-functions will be discussed here. In the case where the > are the diabatic basis functions, the coupled equations are written as... 

When a pair of adjacent PESs intersects, it is useful to discuss their shape in the vicinity of their intersection. In some instances, more than two PESs may intersect at certain q0, but these less frequent cases will not be examined here. Let i /f,ad and i /j ,ad be an orthonormal pair of adiabatic electronic wave functions corresponding to the consecutive electronic eigenvalues Ead and Ejad respectively. We will now determine a set of conditions for the associated PESs to intersect. To that effect, let us consider an orthogonal transformation relating that pair of adiabatic electronic wave functions to another orthonormal basis pair of electronic functions, f and j by... 

C. Conical Intersections and the Non-Single-Valuedness of Real Adiabatic Electronic Wave Functions... 

We now examine the changes which the adiabatic electronic wave functions v /n(rcl q0) (n = i, j) undergo when the system traverses a closed loop in the 2D Qx, Q2 space around the origin Qx = Q2 = 0 corresponding to the intersection configuration q0. To that effect, let us define the polar coordinates R,

[Pg.421]

The complete set of adiabatic electronic wave functions are orthonormalized according to... 

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... 

There is, however, a serious shortcoming associated with such a q-independent electronic basis set l d(r). If we consider, for example, a two-state adiabatic expansion involving only iK ad(r q) and ijjjl ad(r q), the corresponding diabatic expansion in thei]i, -d(r q) must contain a sufficiently large number of terms to represent those two adiabatic electronic wave functions well for all values of the q sampled by xad(R)- This is, in general, an unacceptably large number. In addition, it can be shown [25] that in general no other choice of 1,d(r q) makes W(1)d(R) vanish for all R. 

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