Separation of Electronic and Nuclear Coordinates

The full Hamiltonian of a system in atomic units can be written as 

Equation 4.1 contains, in order, the kinetic energy of the K nuclei, the kinetic energy of the N electrons, the Coulomb attraction between nuclei and electrons, the Coulomb repulsion between the nuclei, and the Coulomb repulsion between the electrons. Since the mass of nucleus I (M,) is at least 1836 times the mass of the electron (m = 1), and the momenta are on an average the same, the kinetic energy of the nuclei can be neglected in the first approximation. If this is done, an equation for the electrons remains, valid for fixed positions of the nuclei. The task to obtain an equation for the nuclei, quantum or classical, was undertaken by the German physicist Max Born and his American assistant Robert Oppenheimer in 1927. 

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The previous treatment relied on the assumption that the transition occurs on a single potential energy surface V(x) characterized by a barrier separating two wells. This potential is actually created from the terms of the initial and final electronic states. The separation of electron and nuclear coordinates in each of these states gives rise to the diabatic basis with nondiagonal Hamiltonian matrix... 

This assumption depends upon the accuracy of the Born-Oppenheimer separation of electronic and nuclear coordinates which is described in most standard textbooks on quantum mechanics. The force constants are theoretically determined by the equation for electronic motion, which involves the charges and configuration of the nuclei, but not their masses. 

This complicated paper describes the separation of electronic and nuclear coordinates in a molecular quantum-mechanical problem. The internal coordinates of the molecule are indicated by the translation and rotation coordinates by The equation for the minimum of the electronic energy is derived [Eq. (40)], and the problem of molecular vibration [Eq. (46)] as well as molecular rotation [Eq. (69)] is discussed. 

In order to separate the electronic and nuclear coordinates in an eigenvalue problem for the Hamiltonian defined by Equation 1, the adiabatic approximation in the version of a Bom-Oppenheimer model is used. In general, the eigenfunction defined within the adiabatic approximation is defined as a linear combination. 

In contrast to the above situation, based on an average charge density (pa), one may identify another dynamical regime where the solvent electronic timescale is fast [50-52] relative to that of the solute electrons (especially, those participating in the ET process). In this case, H F remains as in Equation (3.106), treated at the Born-Oppenheimer (BO) level (i.e., separation of electronic and nuclear timescales), but HFF is replaced by an optical RF operator involving instantaneous electron coordinates [52] ... 

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]. 

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... 

The great majority of studies of electron transfer problems continue to be structured around the separate treatment of electronic and nuclear coordinates, as exemplified in Eq. (1), where the K/ (i = el, nu) are electronic and nuclear transmission... 

The Born-Oppenheimer approximation separates the electronic and nuclear coordinates. For Hj, we figured out there are nine coordinates overall. The three coordinates of the electron are separated into the effective Hamiltonian. Once we have obtained the electronic energy as a function of R, we can add and solve the Schrodinger equation along the remaining six coordinates for motion of the nuclei ... 

The Born-Oppenheimer approximation separates the electronic and nuclear coordinates in the Schrodinger equation, allowing us to split the one big problem into two smaller problems. But the motions of the nuclei separate still further. 

It would appear that identical particle pemuitation groups are not of help in providing distinguishing syimnetry labels on molecular energy levels as are the other groups we have considered. However, they do provide very usefiil restrictions on the way we can build up the complete molecular wavefiinction from basis fiinctions. Molecular wavefiinctions are usually built up from basis fiinctions that are products of electronic and nuclear parts. Each of these parts is fiirther built up from products of separate uncoupled coordinate (or orbital) and spin basis fiinctions. Wlien we combine these separate fiinctions, the final overall product states must confonn to the pemuitation syimnetry mles that we stated above. This leads to restrictions in the way that we can combine the uncoupled basis fiinctions. 

Use of the Born-Oppenheimer approximation is implicit for any many-body problem involving electrons and nuclei as it allows us to separate electronic and nuclear coordinates in many-body wave function. Because of the large difference between electronic and ionic masses, the nuclei can be treated as an adiabatic background for instantaneous motion of electrons. So with this adiabatic approximation the many-body problem is reduced to the solution of the dynamics of the electrons in some frozen-in configuration of the nuclei. However, the total energy calculations are still impossible without making further simplifications and approximations. 

This is referred as BO ansatz. This ansatz is taken as a variational trial function. Terms beyond the leading order in m/M are neglected m is the electronic and M is nuclear mass, respectively). The problem with expansion (4) is that functions /(r, R) contain except bound states also continuum function since it includes the centre of mass (COM) motion. Variation principle does not apply to continuum states. To avoid this problem we can separate COM motion. The remaining Hamiltonian for the relative motion of nuclei and electrons has then bound state solution. But there is a problem, because this separation mixes electronic with nuclear coordinates and also there is a question how to define molecule-fixed coordinate system. This is in detail discussed by Sutcliffe [5]. In the recent paper by Kutzelnigg [8] this problem is also discussed and it is shown how to derive adiabatic corrections using, as he called it, the Bom-Handy ansatz. There are few important steps to arrive at formula for a diabatic corrections. Firstly, one separates off COM motion. Secondly, (very important step) one does not specify the relative coordinates (which are to some extent arbitrary). In this way one arrives at relative Hamiltonian Hrd [8] with trial wavefunction If we make BO ansatz... 

As a consequence of the transformation, the equation of motion depends on three extra coordinates which describe the orientation in space of the rotating local system. Furthermore, there are additional terms in the Hamiltonian which represent uncoupled momenta of the nuclear and electronic motion and moment of inertia of the molecule. In general, the Hamiltonian has a structure which allows for separation of electronic and vibrational motions. The separation of rotations however is not obvious. Following the standard scheme of the various contributions to the energy, one may assume that the momentum and angular momentum of internal motions vanish. Thus, the Hamiltonian is simplified to the following form. 

It is well known from the Bom-Oppenheimer separation [1] that the pattern of energy levels for a typical diatomic molecule consists first of widely separated electronic states (A eiec 20000 cm-1). Each of these states then supports a set of more closely spaced vibrational levels (AEvib 1000 cm-1). Each of these vibrational levels in turn is spanned by closely spaced rotational levels ( A Emt 1 cm-1) and, in the case of open shell molecules, by fine and hyperfine states (A Efs 100 cm-1 and AEhts 0.01 cm-1). The objective is to construct an effective Hamiltonian which is capable of describing the detailed energy levels of the molecule in a single vibrational level of a particular electronic state. It is usual to derive this Hamiltonian in two stages because of the different nature of the electronic and nuclear coordinates. In the first step, which we describe in the present section, we derive a Hamiltonian which acts on all the vibrational states of a single electronic state. The operators thus remain explicitly dependent on the vibrational coordinate R (the intemuclear separation). In the second step, described in section 7.55, we remove the effects of terms in this intermediate Hamiltonian which couple different vibrational levels. The result is an effective Hamiltonian for each vibronic state. 

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