Polyatomic systems electronically adiabatic

In this chapter, we look at the techniques known as direct, or on-the-fly, molecular dynamics and their application to non-adiabatic processes in photochemistry. In contrast to standard techniques that require a predefined potential energy surface (PES) over which the nuclei move, the PES is provided here by explicit evaluation of the electronic wave function for the states of interest. This makes the method very general and powerful, particularly for the study of polyatomic systems where the calculation of a multidimensional potential function is an impossible task. For a recent review of standard non-adiabatic dynamics methods using analytical PES functions see [1]. 

We have considered the case of vibrational motion of the photofragments accompanied by slow relative motion. We have developed the adiabatic approach to evaluate the nuclear wave-function (Jp and obtained eqs. 74 and 96. Note, that instead of a system of electrons and nuclei (Born-Oppenheimer approximation), we considered here only nuclear motion of a polyatomic system with several degrees of freedom, one of which is "fast" relative to the others. 

If simultaneously W k = 0 and <

= 0 in a region of M, then the nuclei move in the field of force the potential of which is given by the energy Wm of a single state of the electronic subsystem and there is no difference between the description of the electronic subsystem in the adiabatic or diabatic basis sets, i.e., Wam = W, . The corresponding behavior of the polyatomic system is referred to as electronically adiabatic ... 

A certain electronic transition in a polyatomic system can be equally described either within the adiabatic or within the diabatic basis set as the transitions can be induced either by the dynamic or by the static coupling,... 

If the regions of nonadiabatic behavior are well localized in the configuration space M, an (F — l)-dimensional hypersurface can be defined at which the nonadiabatic transitions may take place this hypersurface is referred to as the crossing seam. The coupled relations, Eq. (14), describing the corresponding nonseparable electronic and nuclear motion, are to be solved at the seam. Elsewhere, the evolution of the polyatomic system can be then treated adiabatically (49,50). 

Conical intersections between electronically adiabatic potential energy surfaces are not only possible but actually quite frequent, if not prevalent, in polyatomic systems. Some examples are triatomic systems whose isolated atoms have 2S ground states [1,2] such as H3 and its isotopomers (DH2, HD2, HDT, etc.), LiH2 and its isotopomers, and tri-alkali systems such as Na3 and LiNaK. Many other kinds of polyatomic molecules also display such intersections. The reason is that they have three or more internal nuclear motion degrees of freedom, and only two independent relations between electronic Ham-... 

In polyatomic systems, there is a serious problem with the adiabatic basis (this is why the diabatie functions are preferred). As we will see later, the adiabatic electronic wave function is multivalued, and the corresponding rovibrational wave function, having to compensate for this (because the total wave function must be single-valued), also has to be multivalued. 

Nuclei are thousands times heavier than the electrons. As an example let us take the hydrogen atom. From the conservation of momentum law, it follows that the proton moves 1840 times slower than the electron. In a polyatomic system, while a nucleus moves a little, an electron travels many times through the molecule. It seems that a lot can be simplified when assuming electronic motion in a field created by immobile nuclei. This concept is behind what is called adiabatic approximation, in which the motions of the electrons and the nuclei are separated. Only after this approximation is introduced, can we obtain the fundamental concept of chemistry the molecular structure in 3D space. 

We shall concentrate in this contribution on energy transfer in electronically adiabatic phenomena involving collisions of atoms with diatomics and with polyatomics. We shall not deal with collisions involving electronic excitations. The formalism can be written down for these cases but not much has yet been done to develop the computational methods required in applications. This is in great part due to the lack of information on interaction-potential energies of electronically excited states and on their couplings due to nuclear motions, for polyatomic systems. Similarly, the formalism can be extended to include rearrangement collisions. Little is known however about interaction potentials for reactions... 

Potential energies for the nuclear motions in a polyatomic system can be obtained from the Born-Oppenheimer separation of electronic and nuclear motions, for each adiabatic electronic state. Their values E can be separated into asymptotic contributions giving internal potential energies and Vg, and a remainder term V describing the interaction potential. 

This example illustrates the general need for calculations, or even rough estimation, of potential surfaces of polyatomic systems correlating with different atomic states. Once these are known, the possible mechanisms of electronic energy conversion can be elucidated. Determination of the actual path of non-adiabatic processes must be based on molecular dynamics, i.e. on the study of motions of atom over different potential surfaces and transitions between these surfaces. 

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