Single-surface

The single-surface assumption, known also as the Bom-Oppenlieimer approximation, implies that the nuclei are described by a single wavefiinction where a is a multi-dimensional vector describing the nuclear... 

Baer R, Charutz D M, Kosloff R and Baer M 1996 A study of conical intersection effects on scattering processes—the validity of adiabatic single-surface approximations within a quasi-Jahn-Teller model J. Chem. Phys. 105 9141... 

Particularly in polar solvents, electrostatic charges usually have an important contribution to tire particle interactions. We will first discuss tire ion distribution near a single surface, and tlien tire effect on interactions between two colloidal particles. 

Some final comments on the relevance of non-adiabatic coupling matrix elements to the nature of the vector potential a are in order. The above analysis of the implications of the Aharonov coupling scheme for the single-surface nuclear dynamics shows that the off-diagonal operator A provides nonzero contiibutions only via the term (n A n). There are therefore no necessary contributions to a from the non-adiabatic coupling. However, as discussed earlier, in Section IV [see Eqs. (34)-(36)] in the context of the x e Jahn-Teller model, the phase choice t / = —4>/2 coupled with the identity... 

The transition probabilities obtained due to the above two modified beat-ments of single-surface calculations need to be compared with those riansition probabilities obtained by two surface calculations that confirms the validity of these former heatments. 

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,... 

Single-surface calculation with vector potential,... 

Single-surface calculation (o) — 1). Single-surface calculation (o) — 0). 

As mentioned above, the correct description of the nuclei in a molecular system is a delocalized quantum wavepacket that evolves according to the Schrbdinger equation. In the classical limit of the single surface (adiabatic) case, when effectively 0, the evolution of the wavepacket density... 

Choosing a basis set for the nuclear functions 4>a allows us to write Eq. (52) in a matrix form, similar to Eq. (10) for the single-surface case, now as... 

Similar to the case without consideration of the GP effect, the nuclear probability densities of Ai and A2 symmetries have threefold symmetry, while each component of E symmetry has twofold symmetry with respect to the line defined by (3 = 0. However, the nuclear probability density for the lowest E state has a higher symmetry, being cylindrical with an empty core. This is easyly understand since there is no potential barrier for pseudorotation in the upper sheet. Thus, the nuclear wave function can move freely all the way around the conical intersection. Note that the nuclear probability density vanishes at the conical intersection in the single-surface calculations as first noted by Mead [76] and generally proved by Varandas and Xu [77]. The nuclear probability density of the lowest state of Aj (A2) locates at regions where the lower sheet of the potential energy surface has A2 (Ai) symmetry in 5s. Note also that the Ai levels are raised up, and the A2 levels lowered down, while the order of the E levels has been altered by consideration of the GP effect. Such behavior is similar to that encountered for the trough states [11]. 

After some algebraic manipulation, Eqs. (A, I) and (A.2) lead to the following single-surface equations [4]... 

For this preparation, which must be performed in the fume-cupboard, assemble the apparatus shown in Fig. 67(A). C is a 150 ml. distilling-flask, to the neck of which is fitted a reflux single-surface water-condenser D, closed at the top E by a calcium chloride tube. The side-arm of C carries a cork F which fits the end E of the condenser for subsequent distillation. The side-arm of C is meanwhile plugged by a small rubber cork, or by a short length of glass rod. (Alternatively, use the ground-glass flask and condenser (Fig. 22 (a) and (c), p. 43), and... 

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