Diabatic electronically

Mead C A and Truhlar D G 1982 Conditions for the definition of a strictly diabatic electronic basis for molecular systems J. Chem. Rhys. 77 6090... 

The simplest way to write down the 2 x 2 Hamiltonian for two states such that its eigenvalues coincide at trigonally symmetric points in (x,y) or (q, ( )), plane is to consider the matrices of vibrational-electronic coupling of the e Jahn-Teller problem in a diabatic electronic state representation. These have been constructed by Haiperin, and listed in Appendix TV of [157], up to the third... 

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... 

W (Rj.) is an n X n diabatic first-derivative coupling matrix with elements defined using the diabatic electronic basis set as... 

The diabatic electronic functions are related to the adiabatic functions by unitary transformations at each point in coordinate space... 

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... 

Chapuisat, X., Nauts, A. and Dehareng-Dao, D. Adiabatic-to-diabatic electronic state transformation and curvilinear nuclear coordinates for molecular systems, Chem.Phys.Lett., 95 (1983), 139-143... 

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. 

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... 

In a mixed quantum-classical simulation such as a mean-field-trajectory or a surface-hopping calculation, the population probability of the diabatic state v[/ t) is given as the quasiclassical average over the squared modulus of the diabatic electronic coefficients dk t) defined in Eq. (27). This yields... 

Figure 1. Quantum-mechanical (thick lines) and mean-field-trajectory (thin lines) calculations obtained for Model 1 describing the S2 — Si internal-conversion process in pyrazine. Shown are the time-dependent population probabilities Pf t) and Pf (t) of the initially prepared adiabatic and diabatic electronic state, respectively, as well as the mean momenta pi (t) and P2 t) of the two totally symmetric modes Vi and V( of the model.

Figure 3. Time-dependent simulations of the nonadiabatic photoisomerization dynamics exhibited by Model III, comparing results of the mean-field-trajectory method (dashed lines), the surface-hopping approach (thin lines), and exact quanmm calculations (full lines). Shown are the population probabilities of the initially prepared (a) adiabatic and (b) diabatic electronic state, respectively, as well as (c) the probability Pdsit) that the sytem remains in the initially prepared cis conformation.

Figure 7. Comparison of SH (thin solid line), MFT (dashed line), and quantum path-integral (solid line with dots) calculations (Ref. 198) obtained for Model Va describing electron transfer in solution. Shown is the time-dependent population probability Pf t) of the initially prepared diabatic electronic state.

To illustrate the quality of the MFT method, let us first consider Model 1 describing the S2 — conical intersection in pyrazine. Figure 1 compares quantum-mechanical and MFT results obtained for the adiabatic and diabatic electronic population probabilities P t) and as well as for the mean... 

Finally, we consider the performance of the MFT method for nonadiabatic dynamics induced by avoided crossings of the respective potential energy surfaces. We start with the discussion of the one-mode model. Model IVa, describing ultrafast intramolecular electron transfer. The comparison of the MFT method (dashed line) with the quantum-mechanical results (full line) shown in Fig. 5 demonstrates that the MFT method gives a rather good description of the short-time dynamics (up to 50 fs) for this model. For longer times, however, the dynamics is reproduced only qualitatively. Also shown is the time evolution of the diabatic electronic coherence which, too, is... 

Figure 11. Time-dependent population probability of the upper (a) adiabatic and (b) diabatic electronic state of Model 1. The quantum-mechanical results (thick lines) are compared to SH results obtained directly from the electronic coefficients (dashed lines) and to SH results obtained from binned coefficients (thin solid lines), reflecting the percentage N2(t) of trajectories propagating on the upper adiabatic surface. Panel (c) shows the absolute number of successful (thick hue) and rejected (thin line) surface hops occurring in the SH calculation.

Although the systems discussed so far exhibit fairly complex vibrational and diabatic electronic relaxation dynamics, their adiabatic population dynamics is relatively simple and thus is well-suited for a SH description. To provide a challenge for the SH method, we next consider various spin-boson-type models, which may give rise to a quite comphcated adiabatic population dynamics. 

Finally, we consider Model V by describing two examples of outer-sphere electron-transfer in solution. Figures 7 and 8 display results for the diabatic electronic population for Models Va and Vb, respectively. Similar to the mean-field trajectory calculations, for Model Va the SH results are in excellent agreement with the quantum calculations, while for Model Vb the SH method only is able to describe the short-time dynamics. As for the three-mode Model IVb discussed above, the SH calculations in particular predict an incorrect long-time limit for the diabatic population. The origin of this problem will be discussed in more detail in Section VI in the context of the mapping formulation. 

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