Diabatic electronic representation, adiabatic

In this section, we introduce the model Hamiltonian pertaining to the molecular systems under consideration. As is well known, a curve-crossing problem can be formulated in the adiabatic as well as in a diabatic electronic representation. Depending on the system under consideration and on the specific method used, both representations have been employed in mixed quantum-classical approaches. While the diabatic representation is advantageous to model potential-energy surfaces in the vicinity of an intersection and has been used in mean-field type approaches, other mixed quantum-classical approaches such as the surfacehopping method usually employ the adiabatic representation. 

To apply the mapping formalism to vibronically coupled systems, we identify the / ) with electronic states and the h m with operators of the nuclear dynamics. Hereby, the adiabatic as well as a diabatic electronic representation may be employed. In a diabatic representation, we have [cf. Eq. (1)]... 

The scheme has been successfully applied to the photodissociation on coupled surfaces of O3 and H2S. In both cases theoretical reference treatments are available in a diabatic electronic representation. The resulting adiabatic surfaces have been re-diabatized using Eqs. (37) and (38), and the resulting photodissociation spectra been compared to the earlier data (taken as the exact reference). For all technical details we refer to the original work. ° In Fig. 5, we show a contour line drawing of the diabatic potential matrix elements of O3 as a function of the bond lengths r and V2... 

It should be stressed that for multidimensional curve crossing problems the low-order Taylor expansions (8), (9) and (19) are justified only in the diabatic electronic representation. In the adiabatic representation, curve crossings generally lead to rapid variations of potential-energy functions and transition dipole moments, rendering a low-order Taylor expansion of these functions in terms of nuclear coordinates meaningless. 

In Eq. (46) the diabatic electronic representation has been employed. Of course, adiabatic electronic populations Pf(t) may be defined in a completely analogous manner. These electronic population probabilities and their behavior for typical conical intersection models are further discussed in Chapter 9. 

The time dependence of the adiabatic electronic populations can be calculated either by using the S matrix [Eq. (7)] or by defining suitable adiabatic projection operators. The S matrix is a double-valued function of the coordinates and possesses a branch point at the conical intersection. This problem is circumvented by using adiabatic projectors in the diabatic electronic representation... 

As we saw in Sec. 2.1, CW spectra can be calculated by time-independent (Tl) or time-dependent (TD) methods. In an adiabatic electronic representation, the vibronic couplings diverge along the Cl locus. Therefore, a diabatic electronic representation strongly simplifies the calculations, coupling the electronic species with well-behaved potential terms. 

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... 

Single-valued potential, adiabatic-to-diabatic transformation matrix, non-adiabatic coupling, 49-50 topological matrix, 50-53 Skew symmetric matrix, electronic states adiabatic representation, 290-291 adiabatic-to-diabatic transformation, two-state system, 302-309 Slater determinants ... 

In coordinate representation, there exists alternative base representations, adiabatic and diabatic. Both representations are equivalent if the basis are complete. For a thorough discussion on adiabatic-diabatic electronic state transformations the reader is referred to the work by Baer [49, 50], see also the work by Chapuisat et al. [51] In this... 

In what follows we introduce the model Hamiltonian using both diabatic and adiabatic representations. Adopting diabatic electronic basis states /j ), the molecular model Hamiltonian can be written as [162, 163]... 

To describe the electronic relaxation dynamics of a photoexcited molecular system, it is instructive to consider the time-dependent population of an electronic state, which can be defined in a diabatic or the adiabatic representation [163]. The population probability of the diabatic electronic state /jt) is defined as the expectation value of the diabatic projector... 

X + P )/4, which by construction varies between 0 (system is in /i)) and 1 (system is in /2)). Describing, as usual, the nuclear motion through the position X, the vibronic PO can then be drawn in the (A dia,v) plane. Here, the subscript dia emphasizes that we refer to the population of the diabatic states which are used to define the molecular Hamiltonian H. For inteipretational purposes, on the other hand, it is often advantageous to change to the adiabatic electronic representation. Introducing the adiabatic population A ad. where Nad = 0 corresponds to the lower and A ad = 1 to the upper adiabatic electronic state, the vibronic PO can be viewed in the (Nad,x) plane. Alternatively, one may represent the vibronic PO as a curve N d i + (1 — A ad)IFi between the... 

Similar results have been found[58] in recent semiclassical simulations even when some variation was obtained depending on the particular method and on the electronic representation, either diabatic or adiabatic. The electronic representa-... 

Fig. 7.1. Schematic illustration of indirect photodissociation for a one-dimensional system. The two dashed potential curves represent so-called diabatic potentials which are allowed to cross. The solid line represents the lower member of a pair of adiabatic potential curves which on the contrary are prohibited to cross. The other adiabatic potential, which would be purely binding, is not shown here. More will be said about the diabatic and the adiabatic representations of electronic states in Chapter 15. The right-hand side shows the corresponding absorption spectrum with the shaded bars indicating the resonance states embedded in the continuum. The lighter the shading the broader the resonance and the shorter its lifetime.

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. 

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... 

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