Nonadiabatic coupling term

The full dynamical treatment of electrons and nuclei together in a laboratory system of coordinates is computationally intensive and difficult. However, the availability of multiprocessor computers and detailed attention to the development of efficient software, such as ENDyne, which can be maintained and debugged continually when new features are added, make END a viable alternative among methods for the study of molecular processes. Eurthemiore, when the application of END is compared to the total effort of accurate determination of relevant potential energy surfaces and nonadiabatic coupling terms, faithful analytical fitting and interpolation of the common pointwise representation of surfaces and coupling terms, and the solution of the coupled dynamical equations in a suitable internal coordinates, the computational effort of END is competitive. 

By following Section II.B, we shall be more specific about what is meant by strong and weak interactions. It turns out that such a criterion can be assumed, based on whether two consecutive states do, or do not, form a conical intersection or a parabolical intersection (it is important to mention that only consecutive states can form these intersections). The two types of intersections are characterized by the fact that the nonadiabatic coupling terms, at the points of the intersection, become infinite (these points can be considered as the black holes in molecular systems and it is mainly through these black holes that electronic states interact with each other.). Based on what was said so far we suggest breaking up complete Hilbert space of size A into L sub-Hilbert spaces of varying sizes Np,P = 1,..., L where... 

The extended Bom-Oppenheimer approximation based on the nonadiabatic coupling terms was discussed on several occasions [23,25,26,55,56,133,134] and is also presented here by Adhikari and Billing (see Chapter 3). 

In Chapter VI, Ohm and Deumens present their electron nuclear dynamics (END) time-dependent, nonadiabatic, theoretical, and computational approach to the study of molecular processes. This approach stresses the analysis of such processes in terms of dynamical, time-evolving states rather than stationary molecular states. Thus, rovibrational and scattering states are reduced to less prominent roles as is the case in most modem wavepacket treatments of molecular reaction dynamics. Unlike most theoretical methods, END also relegates electronic stationary states, potential energy surfaces, adiabatic and diabatic descriptions, and nonadiabatic coupling terms to the background in favor of a dynamic, time-evolving description of all electrons. 

Lischka H, Dallos M, Szalay PG, Yarkony DR, Shepard R (2004) Analytic evaluation of nonadiabatic coupling terms at the MR-CI level. I. Formalism. J Chem Phys 120 7322... 

In the previous section, we discussed the calculation of the PESs needed in Eq. (2.16a) as well as the nonadiabatic coupling terms of Eqs. (2.16b) and (2.16c). We have noted that in the diabatic representation the off-diagonal elements of Eq. (2.16a) are responsible for the coupling between electronic states while Dp and Gp vanish. In the adiabatic representation the opposite is true The off-diagonal elements of Eq. (2.16a) vanish while Du and Gp do not. In this representation, our calculation of the nonadiabatic coupling is approximate because we assume that Gp is negligible and we make an approximation in the calculation of Dp. (See end of Section n.A for more details.)... 

The nonadiabatic coupling terms can quickly become large or even infinite (or singular) when two successive adiabatic states become degenarate. Such singular nonadiabatic coupling may not only lead to the breakdown of the... 

M. Baer, Beyong Bom-Oppenheiner Electronic Nonadiabatic Coupling Terms and Conical Intersection, Wiley, New York, 2006. 

Analytic Evaluation of Nonadiabatic Coupling Terms and Efficient Searching Algorithm of Conical Intersections within the COLUMBUS Program System Szalay V. 

This formulation of the problem treats simStaneously the dynamics of electrons and nuclei with SI the nonadiabatic coupling terms and without the use of the stationary electronic states and the associated potentiS energy curves. It has been applied with considerable success to ion-atom and ion-molecule collisions involving light elements (23, 26, 28, 31). 

The nonadiabatic coupling terms can also be evaluated analytically to yield... 

Obviously, the electronic energies E (R) for n 0 corresponds in a similar manner to potential surfaces for electronically excited states. Each PES usually exhibits considerable structure for a polyatomic system and will provide useful pictures with reactant and product valleys, local minima corresponding to stable species, and transition states serving as gateways for the system to travel from one valley to another. However, for the number of nuclear degrees of freedom beyond six, i.e. for more than four-atom systems it becomes extremely cumbersome to produce the PES s and quite complicated to visuahze the topology. Furthermore, when more than one PES is needed, which is not unusual, there is a need for nonadiabatic coupling terms, which also may need interpolation in order to provide useful information. 

When the forces between reactants are derived from a precalculated PES it is possible to produce informative pictures with reactant valleys and product valleys perhaps connected by saddles indicating transition states. Time-laps photography or movies of dynamical events may show probabilities in terms of nuclear wave functions evolving on one surface and then transfer to another surface if nonadiabatic coupling terms are present. 

The Born-Oppenheimer approximation is valid when the nonadiabatic coupling terms are very small, as is generally the case for most organic molecules near equilibriiun geometries in their ground electronic state. However, it fails when the energy gap between two electronic PESs vanishes, as can be shown by the Hellmann-Feynman formula for nonadiabatic coupling ... 

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