The adiabatic representation

According to Equation (2.29), in the adiabatic representation (index a) one expands the total molecular wavefunction F(R, r, q) in terms of the Born-Oppenheimer states Ej (q R, r) which solve the electronic Schrodinger equation (2.30) for fixed nuclear configuration (R,r). In this representation, the electronic Hamiltonian is diagonal, 

The nuclear wavefunctions for states 1 and 2 must solve the coupled equations (2.31) with 

The set of coupled equations must be solved subject to boundary conditions similar to (2.59) with unit outgoing flux in only one particular electronic channel, which we designate by e, and one particular vibrational state n of the diatomic fragment, e.g., e = l,n = 5 for A + BC(n = 5) and e = 2, n = 6 for A + BC(n = 6). In the actual calculation one would subsequently expand the nuclear wavefunctions (.R, r E, e, n) in a set of vibrational basis functions (n = 0. nmax) as described in Section 3.1 which leads to a total of 2(nmax + 1) coupled equations. It is not difficult to surmise how complicated the coupled equations will become if the rotational degree of freedom is also included. 

Following the general rules given in Chapter 2, the partial cross sections for absorbing a photon with frequency u and at the same time producing the fragments in electronic state e and vibrational state n are given by 

Equation (2.35)]. Equation (15.6b) is formally equivalent to (2.66) with the exception that in the present case the outgoing channel also includes, in addition to the vibrational state, the particular electronic state. It is important to realize that because of the nonadiabatic coupling both excited electronic states and both electronic product channels are populated, even if one transition dipole moment is exactly zero for all nuclear geometries. Furthermore, the superposition of two complex-valued amplitudes in the case that both transition moments are non-zero can lead to interesting interference patterns. 

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Single surface calculations with a vector potential in the adiabatic representation and two surface calculations in the diabatic representation with or without shifting the conical intersection from the origin are performed using Cartesian coordinates. As in the asymptotic region the two coordinates of the model represent a translational and a vibrational mode, respectively, the initial wave function for the ground state can be represented as. 

Single surface calculations with proper phase treatment in the adiabatic representation with shifted conical intersection has been performed in polai coordinates. For this calculation, the initial adiabatic wave function tad(9, 4 > o) is obtained by mapping t, to) ittlo polai space using the relations,... 

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... 

Ignoring all nonadiabatic couplings to higher electronic states, the nuclear motion in a two-state elechonic manifold is described explicitly in the adiabatic representation by... 

Having learned how states change in the adiabatic representation, we now turn briefly to examine the equation of motion of matrices. Clearly,... 

Yarkony DR (2001) Nuclear dynamics near conical intersections in the adiabatic representation. I. The effects of local topography on interstate transition. J Chem Phys 114 2601... 

In the previous section, we discussed the calculation of the PESs needed in Eq. (2.16a) as well as the nonadiabatic coupling terms of Eqs. (2.16b) and (2.16c). We have noted that in the diabatic representation the off-diagonal elements of Eq. (2.16a) are responsible for the coupling between electronic states while Dp and Gp vanish. In the adiabatic representation the opposite is true The off-diagonal elements of Eq. (2.16a) vanish while Du and Gp do not. In this representation, our calculation of the nonadiabatic coupling is approximate because we assume that Gp is negligible and we make an approximation in the calculation of Dp. (See end of Section n.A for more details.)... 

Single surface calculations with proper phase treatment in the adiabatic representation with shifted conical intersection has been performed in polar coordinates. For this calculation, the initial adiabatic wave function bad(< , to) is obtained by mapping 4 a and R Rq = qcas < x At this point, it is necessary to mention that in all the above cases the initial wave function is localized at the positive end of the R coordinate where the negative and positive ends of the R coordinate are considered as reactive and nonreactive channels. 

We have used the above analysis scheme for all single- and two-surface calculations. Thus, when the wave function is represented in polar coordinates, we have mapped the wave function, 4,ad(, t) to Tatime step to use in Eq. (17) and as the two surface calculations are performed in the diabatic representation, the wave function matrix is back transformed to the adiabatic representation in each time step as... 

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 specify the model Hamiltonian in the adiabatic representation, we introduce adiabatic electronic states... 

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 complete analogy to the diabatic case, the equations of motion in the adiabatic representation are then obtained by inserting the ansatz (29) into the time-dependent Schrodinger equation for the adiabatic Hamiltonian (7)... 

As long as no approximation is introduced, it is clear that the equations of motion are equivalent in the diabatic and adiabatic representations. This is no longer true, however, once the classical-path approximation is employed the resulting classical-path equations of motion in the adiabatic representation are... 

To give an impression of the virtues and shortcomings of the QCL approach and to study the performance of the method when applied to nonadiabatic dynamics, in the following we briefly introduce the QCL working equation in the adiabatic representation, describe a recently proposed stochastic trajectory implementation of the resulting QCL equation [42], and apply this numerical scheme to Model 1 and Model IVa. 

By inserting Eqs. (51)-(53) into the QCL equation (50a), the equations of motion for the QC density matrix in the adiabatic representation can be written in the following suggestive form [34] ... 

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. Richter, P. Marquetand, J. Gonzalez-Vazquez, I. Sola, and L. Gonzalez. SHARC ab initio molecular dynamics with surface hopping in the adiabatic representation including arbitrary couplings, J. Chem. Theory Comput, 7 1253-1258 (2011). 

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