Nuclear motion in molecules

Soon after the Schrodinger equation was introduced in 1926, several works appeared dealing with the fundamental problem of the nuclear motion in molecules. Very soon after, the relativistic equations were introduced for one-and two-electron systems. The experiments on the Lamb shift stimulated... 

The special case of equation (7.45) with P x) = Q x) — 0 and 5(x) equal to a positive constant, n2 (the choice of n2 as the constant ensures that it is positive quantity for any real value for n), gives rise to equation (7.46) for simple harmonic motion, the solution of which can be used to model nuclear motion in molecules ... 

The fundamental approximation used for describing the electron and nuclear motion in molecules and in condensed media is the well-known adiabatic approximation. Let us recall its essence. It is based upon the large difference in the masses of electrons and nuclei. Due to this difference the electron motion is fast in comparison with the nuclear motion, and thus electrons have time to adjust themselves to the nuclear motion and at every moment they can be in a state very close to the one they would be in if nuclei were immobile. Within this picture, as the first step in the construction of the complete wave function of the system, it proves useful to find wave functions describing electron motion with fixed positions of the nuclei, i.e. to resolve the Schrodinger equation... 

The Born-Oppenheimer separation19-22 of the electronic and nuclear motions in molecules is probably the most important approximation ever introduced in molecular quantum mechanics, and will implicitly or explicitly be used in all subsequent sections of this chapter. The Born-Oppenheimer approximation is crucial for modern chemistry. It allows to define in a rigorous way, within the quantum mechanics, such useful chemical concepts like the structure and geometry of molecules, the molecular dipole moment, or the interaction potential. In this approximation one assumes that the electronic motions are much faster than the nuclear... 

It is not our purpose to give a detailed description of literate programming and its application in quantum chemistry. We restrict our attention here to those details which are essential to our present purpose - the development of a literate quantum chemistry program for the simultaneous description of electronic and nuclear motion in molecules. For further details of literate programming methods we refer the reader to the original publication of Knuth [40]. For further details of applications of literate programming in quantum chemistry we refer the reader to our previous work [41 14] as well as our paper in the present volume [34],... 

In Sections 2.2 and 2.9 we have discussed the dynamics of the two-level system and of the harmonic oscillator, respectively. These exactly soluble models are often used as prototypes of important classes of physical system. The harmonic oscillator is an exact model for a mode of the radiation field (Chapter 3) and provides good starting points for describing nuclear motions in molecules and in solid environments (Chapter 4). It can also describe the short-time dynamics of liquid environments via the instantaneous normal mode approach (see Section 6.5.4). In fact, many linear response treatments in both classical and quantum dynamics lead to harmonic oscillator models Linear response implies that forces responsible for the return of a system to equilibrium depend linearly on the deviation from equilibrium—a harmonic oscillator property We will see a specific example of this phenomenology in our discussion of dielectric response in Section 16.9. 

The substance of this section is based on the classic article by Herzberg and Teller (6) on vibronic transitions in polyatomic molecules. Some more specific aspects of the general theory are excellently described in recent articles (7, 8, 9). In the present section we wish to examine how symmetry arguments enter into the description and prediction of vibronic states, yet to attain this end it will be necessary to give a brief account of the quantum theory of the interaction between electronic and nuclear motion in molecules. 

The traditional place to begin a quantum-mechanical study of molecules is with the hydrogen molecule ion H2+. Apart from being a prototype molecule, it reminds us that molecules consist of nuclei and electrons. We often have to be aware of the nuclear motion in order to understand the electronic ones. The two are linked. 

Applied electric fields, whether static or oscillating, distort (polarize) the electron distribution and nuclear positions in molecules. Much of this volume describes effects that arise from the electronic polarization. Nuclear contributions to the overall polarization can be quite large, but occur on a slower time-scale than the electronic polarization. Electronic motion can be sufficiently rapid to follow the typical electric fields associated with incident UV to near IR radiation. This is the case if the field is sufficiently off resonance relative to electronic transitions and the nuclei are fixed (see ref 5 for contributions arising from nuclear motion). Relaxation between states need not be rapid, so... 

An electron transfer occurs momentarily between neighboring molecules. No nuclear motions in the molecules occur during this transfer. [The Frank-Condon principle Transfer of an electron takes place much faster than nuclear movements]. Therefore, such a rapid electron transfer may take place only when the geometry of the ion radical and the parent molecule are similar (Warman 1982). Solvent relaxation is assumed to be much faster than... 

Nesbet, R.K. (1981). The concept of a local complex potential for nuclear motion in electron-molecule collisions, Comments At. Mol. Phys. 11. 25-35. 

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