Born-Oppenheimer generalized equation

The Born-Oppenheimer approximation is the first of several approximations used to simplify the solution of the Schradinger equation. It simplifies the general molecular problem by separating nuclear and electronic motions. This approximation is reasonable since the mass of a typical nucleus is thousands of times greater than that of an electron. The nuclei move very slowly with respect to the electrons, and the electrons react essentially instantaneously to changes in nuclear position. Thus, the electron distribution within a molecular system depends on the positions of the nuclei, and not on their velocities. Put another way, the nuclei look fixed to the electrons, and electronic motion can be described as occurring in a field of fixed nuclei. 

One may consider the above equation as a generalization of Born-Oppenheimer dynamics in which electrons always stay on the Born-Oppenheimer surface. For a given conformation of nuclei, the numerical value of the fictitious mass associated with electronic degrees of freedom determines how far the electron density is allowed to deviate from the Born-Oppenheimer one. Each consecutive step along the trajectory, which involves electronic and nuclear degrees of freedom, can be obtained without determining the exact Born-Oppenheimer electron density. 

The purpose of most quantum chemical methods is to solve the time-independent Schrodinger equation. Given that the nuclei are much more heavier than the electrons, the nuclear and electronic motions can generally be treated separately (Born-Oppenheimer approximation). Within this approximation, one has to solve the electronic Schrodinger equation. Because of the presence of electron repulsion terms, this equation cannot be solved exactly for molecules with more than one electron. 

Based on first principles. Used for rigorous quantum chemistry, i. e., for MO calculations based on Slater determinants. Generally, the Schrodinger equation (Hy/ = Ey/) is solved in the BO approximation (see Born-Oppenheimer approximation) with a large but finite basis set of atomic orbitals (for example, STO-3G, Hartree-Fock with configuration interaction). 

This section briefly introduces the generalized coupled master equation within the Born-Oppenheimer adiabatic (BOA) approximation. In this case, the non-adiabatic processes are treated as the vibronic transitions between the vibronic manifolds. Three types of the rate constant are then introduced to specify the nature of the transitions depending on whether the electronically excited molecular system achieves its vibrational thermal equilibrium or not. The radiationless transitions can occur between two... 

In the related work of Kim and Hynes [50], Equations (3.107) and (3.112) have been designated, respectively, by the labels SC (self-consistent or mean field) and BO (where Born-Oppenheimer here refers to timescale separation of solvent and solute electrons). More general timescale analysis has also been reported [50,51], Equation (3.112) is similar in spirit to the so-called direct RF method (DRF) [54-56], The difference between the BO and SC results has been related to electronic fluctuations associated with dispersion interactions [55], Approximate means of separating the full solute electronic densities into an ET-active subspace and the remainder, treated, respectively, at the BO and SC levels, have also been explored [52],... 

An important modification of the general Schrodinger equation (Eq. 2.10) is that based on the Born-Oppenheimer approximation[l ], which assumes stationary nuclei. Further approximations include the neglect of relativistic effects, where they are less important, and the reduction of the many-electron problem to an effective one-electron problem, i. e., the determination of the energy and movement... 

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]. 

Atomic nuclei are much heavier than electrons and can, in general, be treated accurately using a classical approach. Electrons, of course, must be treated quantum mechanically, and they are considered to move via the equations of quantum mechanics within the fixed external potential of the positively charged nuclei. Because of the relative speed of the motion of the electrons compared to that of the nuclei, their motion is, to an excellent approximation, separate from that of the nuclei in what is called the Born-Oppenheimer approximation. Moreover, excited electronic states are usually irrelevant at temperatures of interest to chemical engineers (<10,000 K), so only their ground state (minimum energy state) needs to be considered. (1 do not consider here the interaction of radiation with matter, the treatment of which is not readily possible at this time using Car-Parrinello methods.)... 

The force-field method involves the other part of the Born-Oppenheimer approximation, that is the positioning of the nuclei. The electronic system is not considered explicitly, but its effects are of course taken into account indirectly. This method is often referred to as a classical approach, not because the equations and parameters are derived from classical mechanics, but rather because it is assumed that a set of equations exist which are of the form of the classical equations of motion. The problem from this point of view is one of establishing just which equations are necessary, and determining the numerical values for the constants which appear in the equations. In general there is no limit as to what functions may be chosen or what parameters arc to be used, except that the force-field must duplicate the experimental data. 

An important modification of the general time-independent Schrodinger equation [Eq. (2.10)1 is that based on the Born-Oppenheimer approximation [23], which... 

The transition electric dipole moment in eqn [57] can be developed by invoking the Born-Oppenheimer approximation to express the total molecular wave function as a product of electronic and vibrational parts. (Rotational wave functions do not have to be included here since eqn [57] refers to an isotropic system. That is, the equation is a result of a rotational average which is equivalent to a summation over all the rotational states involved in the transition.) A general molecular state can now be expressed as the product of vibrational and electronic parts. Assuming that the initial and final electronic states are the ground state jcg). 

In Section 2.1, the electronic problem is formulated, i.e., the problem of describing the motion of electrons in the field of fixed nuclear point charges. This is one of the central problems of quantum chemistry and our sole concern in this book. We begin with the full nonrelativistic time-independent Schrodinger equation and introduce the Born-Oppenheimer approximation. We then discuss a general statement of the Pauli exclusion principle called the antisymmetry principle, which requires that many-electron wave functions must be antisymmetric with respect to the interchange of any two electrons. 

Several applications, such as predictions of narrow scattering resonances for heavy alkali-metal atomic species or spin relaxation in collisions of atoms in excited Zee-man states, may require the inclusion of higher-order corrections (e.g.. Refs. [29-32]) in Equation 11.5. Whereas the singlet and triplet potentials depend only on the inter-atomic distance, r = r, these additional interactions are generally directional, that is, they couple different partial waves in the relative motion of the atoms. All examples of bound-state and scattering phenomena throughout this chapter might be described, however, at least qualitatively, on the basis of the isotropic Born-Oppenheimer potentials only. 

Using Eq. (12) and repeating the same steps which lead to Eq. (9), we obtain the generalized Born-Oppenheimer equation for the nuclear motion -12-15-1 -23... 

Although the generalized Born-Oppenheimer, Eq. (13), was first derived in 1979, nmnerical techniques for solving this equation were developed only recently. Part of this delay is due to the singular nature of the vector... 

The methodology for solving the generalized Born-Oppenheimer equation described above was first applied to the inelastic scattering of H - - 02 v,j) H + 02 v, j ) at low collision energies and zero total angular momentum (i.e. J = The ground state electronic poten-... 

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