Coupling terms

Thus the inclusion of the geometric phase in this case adds two terms to the Hamiltonian. The first is an additional potential term and the second has the effect that 2R is added to Py in the Coriolis coupling term [see Eq. (35)]. 

Hence, in order to contract extended BO approximated equations for an N-state coupled BO system that takes into account the non-adiabatic coupling terms, we have to solve N uncoupled differential equations, all related to the electronic ground state but with different eigenvalues of the non-adiabatic coupling matrix. These uncoupled equations can yield meaningful physical... 

It is important to note that the non-adiabatic coupling terms have a direct effect on the momentum of the nuclei, which is the reason it is called a dynamic coupling. By substituting Eq. (B.13) in Eq. (B.9), we get... 

When the non-adiabatic coupling terms x and x are considered negligibly small and dropped from Eq. (B.15), we get the uncoupled approximate Schrbdinger equation... 

In a different field, location, and characteristics of ci s on diabatic potential surfaces have been recognized as essential for the evaluation of dynamic parameters, like non-adiabatic coupling terms, needed for the dynamic and... 

To obtain nonlinear coupling terms, we consider two linearly independent, not identical E modes, namely. 

We now return to the nuclear BO Eq. (91) in the molecular context. Consider the derivative coupling term in it, having the form... 

Projecting the nuclear solutions Xt( ) oti the Hilbert space of the electronic states (r, R) and working in the projected Hilbert space of the nuclear coordinates R. The equation of motion (the nuclear Schrddinger equation) is shown in Eq. (91) and the Lagrangean in Eq. (96). In either expression, the terms with represent couplings between the nuclear wave functions X (K) and X (R). that is, (virtual) transitions (or admixtures) between the nuclear states. (These may represent transitions also for the electronic states, which would get expressed in finite electionic lifetimes.) The expression for the transition matrix is not elementaiy, since the coupling terms are of a derivative type. 

Nevertheless, the residual first-derivative coupling term Vr does not... 

Fig re 5. Second-derivative coupling term defined at the end of Section TTI.D for... 

In this minimal END approximation, the electronic basis functions are centered on the average nuclear positions, which are dynamical variables. In the limit of classical nuclei, these are conventional basis functions used in moleculai electronic structure theoiy, and they follow the dynamically changing nuclear positions. As can be seen from the equations of motion discussed above the evolution of the nuclear positions and momenta is governed by Newton-like equations with Hellman-Feynman forces, while the electronic dynamical variables are complex molecular orbital coefficients that follow equations that look like those of the time-dependent Hartree-Fock (TDHF) approximation [24]. The coupling terms in the dynamical metric are the well-known nonadiabatic terms due to the fact that the basis moves with the dynamically changing nuclear positions. 

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. 

THE ELECTRONIC NON-ADIABATIC COUPLING TERM IN MOLECULAR SYSTEMS ... 

XIV. A Theoretic-Numeric Approach to Calculate the Electronic Non-Adiabatic Coupling Terms... 

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