Diabatic representation defined

Adiabatic surfaces are defined as the eigenvalues of the electronic Hamiltonian (see Equation 1.5). In the diabatic representation defined above, the potential energy... 

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. 

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]. 

Sometimes it is useful to employ a diabatic representations for the fast variable quantum states, rather than the adiabatic representation. In this work we define a diabatic representation as one for which < /j V /i > = 0, where the superscript d indicates the fast variable states in the diabatic representation, There are off-diagonal matrix elements of the fast variable Hamiltonian, Vy(r) = < rf /j >, in this representation. In contrast, the off-diagonal elements of If are all zero in the adiabatic representation, since the /j are eigenfimction of in this case. 

M. Lombardi What is not needed is the validity of the adiabatic approximation, that is, that there is no transition between adiabatic states. But the geometric phase is defined by following states along a path in parameter space (here nuclear coordinates) with some continuity condition. In the diabatic representation, there is no change of basis at all and thus the geometric phase is identically zero. Do not confuse adiabatic basis (which is required) and adiabatic approximation (which may not be valid). 

The Hamiltonian in Eq. [26] is usually referred to as the diabatic representation, employing the diabatic basis set <1), Hamiltonian matrix is not diagonal. There is, of course, no unique diabatic basis as any pair obtained from (]), by a unitary transformation can define a new basis. A unitary transformation defines a linear combination of cj) and < >b which, for a two-state system, can be represented as a rotation of the (]), basis on the angle /... 

Diabatic electronic states (previously termed crude adiabatic states ) are defined as slowly varying functions of the nuclear geometry in the vicinity of the reference geometry [9-11]. The final vibronic-coupling Hamiltonian is obtained by adding the nuclear kinetic-energy operator which is assumed to be diagonal in the diabatic representation. 

As in section 3, the diabatic representations are obtained finding the orthogonal matrix T such that TT = P. Again, the diabatic representation is not unique because T is defined within an overall p-independent orthogonal transformation. In actual calculations, one has to manipulate the potential energy matrix V = Te(p)T, whose large dimensions are often the bottleneck in practice. Proper choice of T is therefore crucial. The practical implementation (see the final section) of hyperspherical harmonics as the proper diabatic set is of great perspective power. 

In order to define the method and the notation of this paper, we shall present in Sec. 2 some theoretical aspects of the rotationless nonadiabatic formalism, of the conical intersections, and of the diabatic representation. The theory is then applied to the X2Ai/a2B2 conical intersection of NO2 in Sec. 3... 

The computation of the inter-state coupling constants being defined as first derivatives of off-diagonal elements of the electronic Hamiltonian in the diabatic representation (see Eq. (11)), appears at first sight to be more difficult. It can be shown, however, by analj ng the adiabatic PE functions associated with the vibronic-coupling Hamiltonian (13) that the A -"" can be determined from second derivatives of the adiabatic energies with respect to the nontotally symmetric coordinate Qj. For an electronic two-state system, the following simple formula results ... 

The initial wave function defined in Eq. (34) is transformed to the dia-batic representation by using the S matrix [Eq. (7)] prior to its propagation. In the diabatic representation the initial wave function can be written in... 

It is assumed that the diabatic representation has been constructed such that, at this point, the adiabatic and diabatic representations are identical. This is always possible since Eq.(2.24) defines the transformation S R) up to a constant unitary transformation. In other words, if the matrix S(R) satisfies Eq.(2.24), the matrix TS(R), where T does not depends on R, also does. Therefore, by choosing T = S Ro), the adiabatic and diabatic representations are identical at Rq. This point can be the ground state equilibrium geometry, a point of electronic degeneracy, or any other point of interest. The zeroth order diabatic potential matrix is simply the diagonal matrix of the adiabatic energies... 

The diabatic electronic states are defined by a unitary transformation of the adiabatic electronic states within a suitable subspace. This transformation is chosen to render the electronic states in the relevant subspace smoothly varying as a function of the nuclear coordinates, such that the derivative couplings defined in equation (4) are sufficiently small to be neglected. In contrast to the adiabatic representation defined in equation (1), the electronic Hamiltonian is nondiagonal in the diabatic representation. 

Thus, the angle a, which relates a diabatic representation to the adiabatic representation, is related to the angle 0 defined from the intersection-adapted coordinates. 

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