Born-Oppenheimer approximation molecular spectroscopy

One branch of chemistry where the use of quantum mechanics is an absolute necessity is molecular spectroscopy. The topic is interaction between electromagnetic waves and molecular matter. The major assumption is that nuclear and electronic motion can effectively be separated according to the Born-Oppenheimer approximation, to be studied in more detail later on. The type of interaction depends on the wavelength, or frequency of the radiation which is commonly used to identify characteristic regions in the total spectrum, ranging from radio waves to 7-rays. 

Obviously, there is an isotope effect on the vibrational frequency v . For het-eroatomic molecules (e.g. HC1 and DC1), infrared spectroscopy permits the experimental observation of the molecular frequencies for two isotopomers. What does one learn from the experimental observation of the diatomic molecule frequencies of HC1 and DC1 To the extent that the theoretical consequences of the Born-Oppenheimer Approximation have been correctly developed here, one can deduce the diatomic molecule force constant f from either observation and the force constant will be independent of whether HC1 or DC1 was employed and, for that matter, which isotope of chlorine corresponded to the measurement as long as the masses of the relevant isotopes are known. Thus, from the point of view of isotope effects, the study of vibrational frequencies of isotopic isomers of diatomic molecules is a study involving the confirmation of the Born-Oppenheimer Approximation. 

Bunker PR, Jensen P (2000) The Born-Oppenheimer approximation. In Jensen P, Bunker PR (eds) Computational molecular spectroscopy, Wiley, New York, pp 3-13... 

Table 9. Fits to parameters from molecular beam spectroscopy using a Born-Oppenheimer approximation. Dj represents the centrifugal distortion of the quadrupole coupling interaction. (Reproduced from Ref.6>t)...

The field gradient is measured at a fixed point within the molecule, the translational part of the wave-function is thus of no consequence for (qap)-The effect of molecular rotation does, however, modify (qap) but the relationship between the rotating and stationary (qap) s has already been treated in the chapter dealing with microwave spectroscopy. In the present context, we are interested in the field gradients in a vibrating molecule in a fixed coordinate system. The Born-Oppenheimer approximation for molecular wave-functions enables us to separate the nuclear and electronic motions, the electronic wave functions being calculated for the nuclei in various fixed positions. The observed (qap) s will then be average values over the vibrational motion. 

From the spectrum of a molecule we can obtain experimental information about the geometry of the molecule (bond lengths), and the energy states from which bond strengths are ultimately obtained. The molecular spectrum depends on the characteristics of the nuclear motions as well as on the electronic motions. In Section 23.1, by invoking the Born-Oppenheimer approximation, we discussed the electronic motion that produces the bonding between the atoms as a problem separate from that of the nuclear motions. We begin the discussion of molecular spectroscopy with a brief recapitulation of the description of the nuclear motions. 

Theories of chemical reaction dynamics and molecular spectroscopy require a knowledge of the molecular potential energy surface (PES) [1,2]. This surface describes the variation in the total electronic energy of the molecule as a function of the nuclear coordinates, within the Born-Oppenheimer approximation, and hence it determines the forces on the atomic nuclei. 

Bunker, P.R., Jensen, P. The Born-Oppenheimer approximation, in Jensen, P, Bunker, P.R., editors. Computational Molecular Spectroscopy. New York Wiley 2000, p. 1-12. 

The Born-Oppenheimer adiabatic approximation represents one of the cornerstones of molecular physics and chemistry. The concept of adiabatic potential-energy surfaces, defined by the Born-Oppenheimer approximation, is fundamental to our thinking about molecular spectroscopy and chemical reaction djmamics. Many chemical processes can be rationalized in terms of the dynamics of the atomic nuclei on a single Born Oppenheimer potential-energy smface. Nonadiabatic processes, that is, chemical processes which involve nuclear djmamics on at least two coupled potential-energy surfaces and thus cannot be rationalized within the Born-Oppenheimer approximation, are nevertheless ubiquitous in chemistry, most notably in photochemistry and photobiology. Typical phenomena associated with a violation of the Born-Oppenheimer approximation are the radiationless relaxation of excited electronic states, photoinduced uni-molecular decay and isomerization processes of polyatomic molecules. 

To see how we should be able to study the evolution of a collision let us consider first how intermolecular potentials between atoms bound together are studied. This is done, of course, via spectroscopy. One starts with the Born-Oppenheimer approximation for the total molecular wave function this enables one to describe the motion of the nuclei in a potential that depends on the separation between them. This result, the existence of a specific adiabatic potential, rests on there being no appreciable mixing between electronic states. One of its corollaries, the Franck-Condon Principle, enables one to interpret and invert (e.g. using the R.K.R. method) the vibrational spectra in terms of the interatomic potentials in different electronic states. To what extent can we extend such a technique to free-free spectra, in other words, to absorption in the middle of a transient molecule — a collision complex — and deduce information about the potentials between atoms as they collide ... 

A solution of either (1.1) or (2.1) is impossible without in-, troducing some approximations even for the simplest chemical reactions. A great simplification of the problem is achieved on the basis of the EHRENFEST adiabatic principle, which states that a system remains in the same quantum state if a change in its surroundings occurs sufficiently slowly. Consequently, the electronic state will be not affected if the motions of nuclei are very slow compared with the motions of the electrons. This is usually the real situation, as first recognized in the molecular spectroscopy by BORN and OPPENHEIMER /22/ and, subsequently, in chemical dynamics by LONDON /23/ Therefore, in (la.I) T. ... 

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