Separation of Electronic and Nuclear Motions

Basic principies of eiectronic, vibrationai and rotational spectroscopy (A) p. 235 

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. 

The separation of the electronic and nuclear motions will be demonstrated in detail by taking the example of a diatomic molecule. 

The separation of the electronic and nuclear motions represents a fundamental approximation of quantum chemistry. Without this, chemists would lose their basic model of the molecule the 3D structure with the nuclei occup3dng some positions in 3D space, with chemical bonds etc. This is why the present chapter occupies the central position on the TREE. 

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Although a separation of electronic and nuclear motion provides an important simplification and appealing qualitative model for chemistry, the electronic Sclirodinger equation is still fomiidable. Efforts to solve it approximately and apply these solutions to the study of spectroscopy, stmcture and chemical reactions fonn the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for most calculations and the foundation of molecular orbital theory is the independent-particle approximation. 

The close-coupling equations are also applicable to electron-molecule collision but severe computational difficulties arise due to the large number of rotational and vibrational channels that must be retained in the expansion for the system wavefiinction. In the fixed nuclei approximation, the Bom-Oppenlieimer separation of electronic and nuclear motion pennits electronic motion and scattering amplitudes f, (R) to be detemiined at fixed intemuclear separations R. Then in the adiabatic nuclear approximation the scattering amplitude for ... 

A second simplihcation results from introducing the Born-Oppenheimer separation of electronic and nuclear motions for convenience, the latter is most often considered to be classical. Each excited electronic state of the molecule can then be considered as a distinct molecular species, and the laser-excited system can be viewed as a mixture of them. The local structure of such a system is generally described in terms of atom-atom distribution functions t) [22, 24, 25]. These functions are proportional to the probability of Ending the nuclei p and v at the distance r at time t. Building this information into Eq. (4) and considering the isotropy of a liquid system simplifies the theory considerably. 

There are several possible ways of introducing the Born-Oppenheimer model " and here the most descriptive way has been chosen. It is worth mentioning, however, that the justification for the validity of the Bom-Oppenheimer approximation, based on the smallness of the ratio of the electronic and nuclear masses used in its original formulation, has been found irrelevant. Actually, Essen started his analysis of the approximate separation of electronic and nuclear motions with the virial theorem for the Coulombic forces among all particles of molecules (nuclei and electrons) treated in the same quantum mechanical way. In general, quantum chemistry is dominated by the Bom-Oppenheimer model of the theoretical description of molecules. However, there is a vivid discussion in the literature which is devoted to problems characterized by, for example, Monkhorst s article of 1987, Chemical Physics without the Bom-Oppenheimer Approximation... ... 

Whereas for diatomic molecules the vibration-rotation interaction added only a small correction to the energy, for a number of polyatomic molecules the vibration-rotation interaction leads to relatively large corrections. Similarly, although the Born-Oppenheimer separation of electronic and nuclear motions holds extremely well for diatomic molecules, it occasionally breaks down for polyatomic molecules, leading to substantial interactions between electronic and nuclear motions. 

In one quantum mechanical approach based on the diabatic approximation , the electron is assumed to be confined initially at one of the reactant sites and electron transfer is treated as a transition between the vibrational levels of the reactants to those of the products. The quantum mechanical treatment begins with the time dependent Schrodinger equation, Hip = -ihSiplSt, where the wavefunction tj/ is written as a sum of the initial (reactant) and final (product) states. In the limit that the Bom-Oppenheimer approximation for the separation of electronic and nuclear motion is valid, the time dependent Schrodinger equation eventually leads to the Golden Rule result in equation (25). 

Separation of Electronic and Nuclear Motions. The polarizabilities of the ground state and the excited state can follow an electronic transition, and the same is true of the induced dipole moments in the solvent since these involve the motions of electrons only. However, the solvent dipoles cannot reorganize during such a transition and the electric field which acts on the solute remains unchanged. It is therefore necessary to separate the solvent polarity functions into an orientation polarization and an induction polarization. The total polarization depends on the static dielectric constant Z), the induction polarization depends on the square of the refractive index n2, and the orientation polarization depends on the difference between the relevant functions of D and of n2 this separation between electronic and nuclear motions will appear in the equations of solvation energies and solvatochromic shifts. 

Witkowsh, A. Separation of electronic and nuclear motions and the dynamical Scbrodinger... 

Goscinski, 0. and Palma, A. Electron and nuclear density matrices and the separation of electronic and nuclear motion, Int.J.Quantum Chem., 15 (1979) 197-205. 

Separation of Electronic and Nuclear Motion. Because, in general, electrons move with much greater velocities than nuclei, to a first approximation electron and nuclear motions can be separated (Born-Oppenheimer theorem [3]). The validity of this separation of electronic and nuclear motions provides the only real justification for the idea of a potential-energy curve of a molecule. The eigenfunction Y for the entire system of nuclei and electrons can be expressed as a product of two functions F< and T , where is an eigenfunction of the electronic coordinates found by solving Schrodinger s equation with the assumption that the nuclei are held fixed in space and Yn involves only the coordinates of the nuclei [4]. 

Combining all of these ideas of this section gives rise to Bom-Oppenheimer MD, sometimes also referred to as semiclassical dynamics. Separation of electronic and nuclear motion is assumed, namely, the Bom-Oppenheimer approximation. Atoms move on the electronic PES, computed using some QM method, following the classical equations of motion. In summary, the steps for computing trajectories using direct dynamics are the following. 

The time-dependent perturbation theory of the rates of radiative ET is based on the Born-Oppenheimer approximation [59] and the Franck Condon principle (i.e. on the separation of electronic and nuclear motions). The theory predicts that the ET rate constant, k i, is given by a golden rule -type equation, i.e., it is proportional to the product of the square of the donor-acceptor electronic coupling (V) and a Franck Condon weighted density of states FC) ... 

The transition from the initial state to the final one is related to electron-vibrational relaxation processes in the crossing region this depends on the separation of electron and nuclear motions according to the Born-Oppenheimer principal. 

The idea of an effective potential function between the nuclei in a molecule can be formulated empirically (as we have done in Section 3.5 and Fig. 3.9), but it can be defined precisely and related to molecular parameters R, D and Dq only through quantum mechanics. The fundamental significance of the Born-Oppenheimer approximation is its separation of electronic and nuclear motions, which leads directly to the effective potential function. 

The structure of approximate reasoning is not simple. Consider the Born-Oppenheim approximation (separability of electronic and nuclear motions due to extreme mass difference), which in application produces "fixed nuclei" Hamiltonians for individual molecules. In assuming a nuclear skeleton, the idealization neatly corresponds to classical conceptions of a molecule containing localized bonds and definite structure. All early quantum calculations, and the vast majority to date, invoke the approximation. In 1978, following decades of quiet assumption, Cambridge chemist R. G. Woolley asserted ... 

Several approximations need to be introduced for solving Eq. (1). The first one is the well-known Bom-Oppenheimer (1927) approximation which allows the separation of electronic and nuclear motions. This means that the total wave function may be separated into two parts such as ... 

Considering an elementary process involving only one or two molecules and assuming it may be described by a series of stationary solutions of the time-independent Schrodinger equation, we are led to search the wave function T of the corresponding supermolecule by solving Eq. (1). When the separation of electronic and nuclear motions is allowed, the wave functions of the two types of particles can be obtained independently from Eqs. (4) and (5). 

Thus within any rotational manifold it is the eigensolutions of the effective Hamilton given by (23) which are invariant to orthogonal transformations and it these functions that will be used to consider the separation of electronic and nuclear motion. 

The adiabatic approximation for reaction dynamics assumes that motion along the reaction coordinate is slow compared to the other modes of the system and the latter adjust rapidly to changes in the potential from motion along the reaction coordinate. This approximation is the same as the Born-Oppenheimer electronically adiabatic separation of electronic and nuclear motion, except that here we... 

Apart from use of experimental values of atomic - rather than nuclear - and electronic masses and of electric charges, the basis of this calculation has an empirical component. The calculation is certainly not made genuinely from first principles or ab initio, firstly because the composition of the basis set is predetermined, by those who have published this basis set [12] and by the authors of Dalton software [11] who have incorporated it, according to its success in reproducing experimentally observable quantities and other calculated properties. Secondly, the solution of Schrodinger s equation is based on a separation of electronic and nuclear motions, essentially with atomic nuclei fixed at relative positions, which is a further empirical imposition on the calculation efforts elsewhere to avoid such an arbitrarily distinct treatment of subatomic particles, even on much simpler molecular systems, have... 

Indeed, Dunham s energy-level formula [Eq. (2.1.1)] is based both on the concept of a potential energy curve, which rests on the separability of electronic and nuclear motions, and on the neglect of certain couplings between the angular momenta associated with nuclear rotation, electron spin, and electron orbital motion. The utility of the potential curve concept is related to the validity of the Born-Oppenheimer approximation, which is discussed in Section 3.1. 

Thus, the formation of the intermediate complex N20 ( S+) takes place through the non-adiabatic transitions between the vibronic terms (see Fig. 6-7). The probability of transition in an intersection can be calculated assuming separation of electronic and nuclear motion in the molecular system. Perturbation matrix elements can be factorized into electronic and vibrational parts the transition probability between electronic (n n ) and vibrational (pi V2) states can be presented as... 

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