Adiabatic ground state

The MOVB wave function for the adiabatic ground state is written as a linear combination of the diabatic states in Eqs. (4-11) and (4-12) ... 

The CDC-MOVB method is the appropriate computational approach for studying properties associated with the adiabatic ground state such as the reaction barrier for a chemical reaction and the solvent reorganization energy. 

Considering the practical application of the mapping approach, it is most important to note that the quantum correction can also be determined in cases where no reference calculations exist. That is, if we a priori know the long-time limit of an observable, we can use this information to determine the quantum correction. For example, a multidimensional molecular system is for large times expected to completely decay in its adiabatic ground state, that is. 

It is interesting to note that the latter criterion imphes that the ground-state level density completely dominates the total level density— that is, that No E) N E). Hence the assumption (98) of complete decay into the adiabatic ground state is equivalent to the criterion that the classical and quantum total level densities should be equivalent. Furthermore, it is clear that this criterion determines an upper limit of 7. This is because larger values of the quantum correction would result in ground-state population larger than one (or negative excited-state populations). 

Having determined the appropriate value of the quantum correction from the comparison of classical and quantum level densities, it is interesting to study the accuracy of the simple approximation (99). Extracting from Fig. 19 the longtime limits of the adiabatic ground-state populations as Pq j = 0, oo) = 0.75 and Pq j = 1, oo) = 1.25, the difference of the two populations yields Ky2 Ti) = 0.5, just as predicted by Eq. (99). Furthermore, we may employ the approximation to estimate the optimal quantum correction. Assuming that Pq oo) = 1, we obtain y = 0.5, which is in qualitative agreement with the results obtained above. ... 

Fig. 5. The pseudo-Jahn-Teller effect in ammonia (NH3). (a) CCSD(T) ground state potential energy curve breakdown of energy into expectation value of electronic Hamiltonian (He), and nuclear-nuclear repulsion VNN. (b) CASSCF frequency analysis of pseudo-Jahn-Teller effect showing the effect of including CSFs of B2 symmetry is to couple the ground and 1(ncr ) states to give a negative curvature to the adiabatic ground state potential energy surface for the inversion mode.

Fig. 4.2.2 Potential energy diagram for the Landau-Zener model. Adiabatic potentials (solid lines) and diabatic potentials (dashed lines), with /3i < 0 and (32 > 0. The arrow illustrates the dynamics on the lower adiabatic (ground-state) potential.

The integration limits Si and 2 are the reaction coordinate classical turning points /u-eff is the reduced mass, which introduces the reaction path curvature and V (s) is the adiabatic ground-state potential. 

Assuming a Boltzmann distribution over the energies on the adiabatic ground-state surface yields the following expression for the band shape... 

Selloni et al. [48] were the first to simulate adiabatic ground state quantum dynamics of a solvated electron. The system consisted of the electron, 32 K+ ions, and 31 Cl ions, with electron-ion interactions given by a pseudopotential. These simulations were unusual in that what has become the standard simulating annealing molecular dynamics scheme, described in the previous section, was not used. Rather, the wave function of the solvated electron was propagated forward in time with the time-dependent Schrodinger equation,... 

In practice, we approximate the exact transmission coefficient by a mean-field-type of approximation that is we replace the ratio of averages by the ratio for an average or effective potential. For gas-phase reactions with small reaction-path curvature, this effective potential would just be the vibrationally adiabatic ground-state potential. In the liquid phase and enzymes we generalize this with the canonical mean-shape approximation. In any event, though, the transmission coefficient should not be thought of as a perturbation. The method used here may be thought of as an approximate full-dimensional quantum treatment of the reaction rate. 

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