Adiabatic potential, calculation

Here g-j)presents the /ill -component in the /th well and A, is the temperature dependent Boltzmann population of the /th well. Analysis of the spectra shows that the tetragonal deformation appears abruptly at Tc= 191 K corresponding to the separation energy 8 li2(3) 130 cm-1. 8 1 2(3) gradually increases to 150 cm-1 as the temperature decreases to T 60 K. The corresponding adiabatic potential calculated for the vibronic parameters of cubic symmetry to which the Ae cos cp component (Ae = 8 1 2(3)) is added, is shown in Ref. [20]. The energy difference between the deepest and the two higher wells is equal to 140 cm-1 and the potential barrier between the wells is about 400 cm-1. 

Fig. 10. Short range adiabatic potentials calculated for ArH/ArH+ and NaO/NaO" " using density functional theory at the B3LYP/6-3lG(3d) level.

This potential Un, also known as adiabatic potential, comes from the resolution of the electronic Schrodinger equation (2.12) and, as above mentioned, corresponds to the sum of the electronic energy and the coulombic repulsion between the nuclei, for a fixed nuclear arrangement. This adiabatic potential calculated for a large number of nuclear geometries is known as potential energy surface (PES). 

The two adiabatic potential energy surfaces that we will use in the present calculations, are called a reactive double-slit model (RDSM) [59] where the first surface is the lower and the second is the upper surface, respectively,... 

The approach developed by Jungen and Merer (JM) [24] is of a similar level of sophistication. The main difference is that IM prefer to remove the coupling between the electronic states by a transformation of the Hamiltonian matrix (i.e., vibronic energy matrix), rather that of the Hamiltonian itself. They first calculate the large amplitude bending functions for one of the adiabatic potentials, as if it belonged to a E electronic state. These functions are used as... 

GP effect, with the shifts being equal to +0.0807, —0.0709, and —0.0273 eV for the Ai, A2, and E symmetries, respectively. Clearly, such shifts are larger than those obtained in the calculations for the lower adiabatic potential energy surface, namely, +0.0104, —0.0194, and —0.0043 eV (in the above order). 

The obtained PES forms the basis for the subsequent dynamical calculation, which starts with determining the MEP. The next step is to use the vibrationally adiabatic approximation for those PES degrees of freedom whose typical frequencies a>j are greater than a>o and a>. Namely, for the high-frequency modes the vibrationally adiabatic potential [Miller 1983] is introduced,... 

The functional B[(2(r)] actually depends only on the velocity dQ/dr at the moment when the non-adiabaticity region is crossed. If we take the path integral by the method of steepest descents, considering that the prefactor B[(2(r)] is much more weakly dependent on the realization of the path than Sad[Q(A]> we shall obtain the instanton trajectory for the adiabatic potential V a, then B[(2(t)] will have to be calculated for that trajectory. Since the instanton trajectory crosses the dividing surface twice, we finally have... 

On the other hand, one can use Eq. (1) to calculate the Hessiam matrix of the difference between the two adiabatic potential energy surfaces, AV R). Up to an irrelevant scalar factor, the result reads... 

The factor k takes into acount the effects of nonadiabatic transition and tunneling properly. Also note that the electronic coupling //ad is assumed to be constant in the Marcus formula, but this is not necessary in the present formulation. The coupling Had cancels out in k of Eq. (126) and the ZN probability can be calculated from the information of adiabatic potentials. 

The angular-dependent adiabatic potential energy curves of these complexes obtained by averaging over the intermolecular distance coordinate at each orientation and the corresponding probability distributions for the bound intermolecular vibrational levels calculated by McCoy and co-workers provide valuable insights into the geometries of the complexes associated with the observed transitions. The He - - IC1(X, v" = 0) and He + 1C1(B, v = 3) adiabatic potentials are shown in Fig. 3 [39]. The abscissa represents the angle, 9,... 

From the given Hamiltonian, adiabatic potential energy surfaces for the reaction can be calculated numerically [Santos and Schmickler 2007a, b, c Santos and Schmickler 2006] they depend on the solvent coordinate q and the bond distance r, measured with respect to its equilibrium value. A typical example is shown in Fig. 2.16a (Plate 2.4) it refers to a reduction reaction at the equilibrium potential in the absence of a J-band (A = 0). The stable molecule correspond to the valley centered at g = 0, r = 0, and the two separated ions correspond to the trough seen for larger r and centered at q = 2. The two regions are separated by an activation barrier, which the system has to overcome. 

Figure 5-3. Active site and calculated PES properties for the reactions studied, with the transferring hydrogen labelled as Hp (a) hydride transfer in LADH, (b) proton transfer in MADH and (c) hydrogen atom transfer in SLO-1. (i) potential energy, (ii) vibrationally adiabatic potential energy, (iii) RTE at 300K and (iv) total reaction path curvature. Reproduced with permission from reference [81]. Copyright Elsevier 2002...

Fig. 5. Contour plot of the adiabatic potential-energy surface of an H atom in the (110) plane for the neutral H—B pair from a local-density pseudopotential calculation. The boron atom is at the center. For every hydrogen position, the B and Si atoms are allowed to relax, but only unrelaxed positions are indicated in the figure (Reprinted with permission from the American Physical Society, Denteneer, P.J.H., Van de Walle, C.G., and Pantelides, S.T. (1989). Phys. Rev. B 39, 10809.)...

The quasi-classical SH model employs the simple and physically appealing picture in which a molecular system always evolves on a single adiabatic potential-energy surface (PES). When the trajectory reaches an intersection of the electronic PESs, the transition probability pk t to the other PES is calculated... 

Due to the large-level density of the lower-lying adiabatic electronic state, the chances of a back transfer of the adiabatic population are quite small for a multidimensional molecular system. To a good approximation, one may therefore assume that subsequent to an electronic transition a random walker will stay on the lower adiabatic potential-energy surface [175]. This observation suggests a physically appealing computational scheme to calculate the time evolution of the system for longer times. First, the initial decay of the adiabatic population is calculated within the QCL approach up to a time to, when the... 

Furthermore, all four periods found in the quantum calculation can be readily explained in terms of the single-orbit contributions to the electronic population probability. In accordance with the discussion above, the 36-fs period of a and the 46-fs period of reflect quasi-harmonic motion on the upper and lower adiabatic potential, respectively. The 12-fs period of <5 c can be attributed to Rabi-type oscillations between the two diabatic states. The 8-fs... 

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