Potential adiabatic

To see physically the problem of motion of wavepackets in a non-diagonal diabatic potential, we plot in figure B3.4.17 a set of two adiabatic potentials and their diabatic counterparts for a ID problem, for example, vibrations in a diatom (as in metal-metal complexes). As figure B3.4.17 shows, if a wavepacket is started away from the crossing point, it would slide towards this crossing point (where where it would... 

Figure 1. Adiabatic potential surfaces (a) for the linear E x e case and (b) for a state with linear Jahn-Teller coupling and spin-orbit coupling to a state,...

Finally, in brief, we demonstrate the influence of the upper adiabatic electronic state(s) on the ground state due to the presence of a Cl between two or more than two adiabatic potential energy surfaces. Considering the HLH phase, we present the extended BO equations for a quasi-JT model and for an A -1- B2 type reactive system, that is, the geometric phase (GP) effect has been inhoduced either by including a vector potential in the system Hamiltonian or... 

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,... 

In this section, we prove that the non-adiabatic matiices have to be quantized ( similar to Bohr-Sommerfeld quantization of the angulai momentum) in order to yield a continous, uniquely defined, diabatic potential matrix W(i). In another way, the extended BO approximation will be applied only to those cases that fulfill these quantization rules. The ADT matrix A(s,so) transforms a given adiabatic potential matiix u(i) to a diabatic matiix W(s, so)... 

Figure 6. Diabatic and corresponding adiabatic potential energy along a relevant reaction coordinate for normal electron transfer...

In Table I, 3D stands for three dimensional. The symbol symbol in connection with the bending potentials means that the bending potentials are considered in the lowest order approximation as already realized by Renner [7], the splitting of the adiabatic potentials has a p dependence at small distortions of linearity. With exact fomi of the spin-orbit part of the Hamiltonian we mean the microscopic (i.e., nonphenomenological) many-elecbon counterpart of, for example, The Breit-Pauli two-electron operator [22] (see also [23]). 

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... 

First, let us note that the adiabatic potentials and V [Eq. (67)], even in the lowest order (harmonic) approximation, depend on the difference of the angles 4>j- and t >c this is an essential difference with respect to triatomics where the adiabatic potentials depend only on the radial bending coordinate p. The foims of the functions V, Vt, and Vc are determined by the adiabatic potentials via the following relations... 

Thus the function Vt represents one-half of the splitting of the adiabatic potentials computed for pure trans bending, and Vc one-half of the splitting of the cis-bending curves. 

The expressions (75) and (77) can he used to extract the parameters ki, k2, ki2, s l, s 2, and (2 from the mean adiabatic potential and the difference of the adiabatic potentials for two components of the electronic state spatially degenerate at linear molecular geomehy. 

The force constants kj,kc and the dimensionless Renner parameters r, c ate defined by the adiabatic potentials for the components of the II state at pure trans (Vj, Vj) and pure cis (V, V ) bending vibrations,... 

The dimensionless parameters Ot, , Ctc appearing in the last expression are connected with the sums and differences of the adiabatic potentials as shown elsewhere [149,150]. This effective Hamiltonian acts onto the basis functions (A.l) with A = 2. 

Thus, the neglect of the off-diagonal matrix elements allows the change from mixed states of the nuclear subsystem to pure ones. The motion of the nuclei leads only to the deformation of the electronic distribution and not to transitions between different electronic states. In other words, a stationary distribution of electrons is obtained for each instantaneous position of the nuclei, that is, the elechons follow the motion of the nuclei adiabatically. The distribution of the nuclei is described by the wave function x (R i) in the potential V + Cn , known as the proper adiabatic approximation [41]. The off-diagonal operators C n in the matrix C, which lead to transitions between the states v / and t / are called operators of nonadiabaticity and the potential V = (R) due to the mean field of all the electrons of the system is called the adiabatic potential. 

H3 (and its isotopomers) and the alkali metal triiners (denoted generally for the homonuclears by X3, where X is an atom) are typical Jahn-Teller systems where the two lowest adiabatic potential energy surfaces conically intersect. Since such manifolds of electronic states have recently been discussed [60] in some detail, we review in this section only the diabatic representation of such surfaces and their major topographical details. The relevant 2x2 diabatic potential matrix W assumes the fomi... 

Now, we examine the effect of vibronic interactions on the two adiabatic potential energy surfaces of nonlinear molecules that belong to a degenerate electronic state, so-called static Jahn-Teller effect. 

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). 

Figure 11. Perspective view [60] of a relaxed triangular plot [68] for the two DMBE adiabatic potential energy surfaces of H3 using hyperspherical coordinates.

Thus B is a diagonal mati ix that contains in its diagonal (complex) numbers whose norm is 1 (this derivation holds as long as the adiabatic potentials are nondegenerate along the path T). From Eq. (31), we obtain that the B-matrix hansfomis the A-matrix from its initial value to its final value while tracing a closed contour ... 

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