Intersection, conical

Why has this, a slightly weird, coordinate system been chosen We see from the formula (6.43) for E+ and E- that f 1 and correspond to the fastest change of the first term and the second term under the square-root sign, respectively.  

Any change of all other coordinates (being orthogonal to the plane 1 2) does not influence the value of the square root, i.e. does not change the difference between E+ and E- (although it changes the values of E+ andE-). 

Therefore, the hypersurface E+ intersects with the hypersurface E-, and their common part, i.e. the intersection set, are all those vectors of the n-dimensional space that fulfil the condition = 0 and (2 = 0- The intersection represents a (n — 2)-dimensional subspace of the n-dimensional space of the nuclear configurations. When we withdraw from the point (0,0, fa, 4. sN-e) by changing the coordinates f 1 and/or 2 a difference between E+ and E- appears. For small increments df j the changes in the energies E+ and E- are proportional to d i and for E+ and differ in sign. This means that the hypersurfaces E+ and E-as functions of f j (at 2 = 0 and fixed other coordinates) have the shapes shown in Fig. 6.14.a. For 2 the situation is similar, but the cone may differ by its angle. From this it follows that 

This is called the conical interseaion. Fig. 6.14.b. The cone opening angle is in general different for different values of the coordinates 3, 4. 3n-6 see Fig. 6.14.C. 

The conical intersection plays a fundamental role in the theory of chemical reactions (Chapter 14). The lower (ground-state) as well as the higher (exdted-state) adiabatic hypersurfaces are composed of two diabatic parts, which in polyatomics correspond to different patterns of chemical bonds. This means that the stem, (point) when moving on the ground-state adiabatic hypersurface towards the join of the two parts, passes near the conical intersection point and overcomes the energy barrier. This is the essence of a chemical reaction. 

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Figure B3.4.16. A generic example of crossing 2D potential surfaces. Note that, upon rotating around the conic intersection point, the phase of the wavefunction need not return to its original value.

Sadygov R G and Yarkony D R 1998 On the adiabatic to diabatic states transformation in the presence of a conical intersection a most diabatic basis from the solution to a Poisson s equation. I J. Chem. Rhys. 109 20... 

Mead C A and Truhlar D G 1979 On the determination of Born-Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei J. Chem. Phys. 70 2284... 

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... 

The stoi7 begins with studies of the molecular Jahn-Teller effect in the late 1950s [1-3]. The Jahn-Teller theorems themselves [4,5] are 20 years older and static Jahn-Teller distortions of elecbonically degenerate species were well known and understood. Geomebic phase is, however, a dynamic phenomenon, associated with nuclear motions in the vicinity of a so-called conical intersection between potential energy surfaces. 

Molecular aspects of geometric phase are associated with conical intersections between electronic energy surfaces, W(Q), where Q denotes the set of say k vibrational coordinates. In the simplest two-state case, the W Q) are eigen-surfaces of the nuclear coordinate dependent Hermitian electronic Hamiltonian... 

Comparison between the first and last lines of the table shows that the sign of the ground-state wave function has been reversed, which implies the existence of a conical intersection somewhere inside the loop described by the table. 

While the presence of sign changes in the adiabatic eigenstates at a conical intersection was well known in the early Jahn-Teller literature, much of the discussion centered on solutions of the coupled equations arising from non-adiabatic coupling between the two or mom nuclear components of the wave function in a spectroscopic context. Mead and Truhlar [10] were the first to... 

Phase factors of this type are employed, for example, by the Baer group [25,26]. While Eq. (34) is strictly applicable only in the immediate vicinity of the conical intersection, the continuity of the non-adiabatic coupling, discussed in Section HI, suggests that the integrated value of (x Vq x+) is independent of the size or shape of the encircling loop, provided that no other conical intersection is encountered. The mathematical assumption is that there exists some phase function, vl/(2), such that... 

As mentioned in the introduction, the simplest way of approximately accounting for the geomehic or topological effects of a conical intersection incorporates a phase factor in the nuclear wave function. In this section, we shall consider some specific situations where this approach is used and furthermore give the vector potential that can be derived from the phase factor. 

Hence, the expression of Eq. (5) indicates that, in a polar coordinate system, Eq. (4) will remain unchanged even if the position of the conical intersection is shifted from the origin of the coordinate system. 

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,... 

At this point, it is important to note that as the potential energy surfaces are even in the vibrational coordinate (r), the same parity, that is, even even and odd odd transitions should be allowed both for nonreactive and reactive cases but due to the conical intersection, the diabatic calculations indicate that the allowed transition for the reactive case ate odd even and even odd whereas in the case of nomeactive transitions even even and odd odd remain allowed. 

Reactive State-to-State Transition Probabilities when Calcnladons are Performed Keeping the Position of the Conical Intersection at the Origin of the Coordinates... 

Reactive State-to-State Transition ftobabilides when Calculations are Performed by Shifting the Position of Conical Intersection from the Origin of the Coordinate System... 

In Figure 1, we see that there are relative shifts of the peak of the rotational distribution toward the left from f = 12 to / = 8 in the presence of the geometiic phase. Thus, for the D + Ha (v = 1, DH (v, f) - - H reaction with the same total energy 1.8 eV, we find qualitatively the same effect as found quantum mechanically. Kuppermann and Wu [46] showed that the peak of the rotational state distribution moves toward the left in the presence of a geometric phase for the process D + H2 (v = 1, J = 1) DH (v = 1,/)- -H. It is important to note the effect of the position of the conical intersection (0o) on the rotational distribution for the D + H2 reaction. Although the absolute position of the peak (from / = 10 to / = 8) obtained from the quantum mechanical calculation is different from our results, it is worthwhile to see that the peak... 

The relative shift of the peak position of the rotational distiibution in the presence of a vector potential thus confirms the effect of the geometric phase for the D + H2 system displaying conical intersections. The most important aspect of our calculation is that we can also see this effect by using classical mechanics and, with respect to the quantum mechanical calculation, the computer time is almost negligible in our calculation. This observation is important for heavier systems, where the quantum calculations ai e even more troublesome and where the use of classical mechanics is also more justified. 

In this chapter, we discussed the significance of the GP effect in chemical reactions, that is, the influence of the upper electronic state(s) on the reactive and nonreactive transition probabilities of the ground adiabatic state. In order to include this effect, the ordinary BO equations are extended either by using a HLH phase or by deriving them from first principles. Considering the HLH phase due to the presence of a conical intersection between the ground and the first excited state, the general fomi of the vector potential, hence the effective... 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

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