Nuclear motion separation

A rigorous mathematical formalism of chemical bonding is possible only through the quantum mechanical treatment of molecules. However, obtaining analytical solutions for the Schrodinger wave equation is not possible even for the simplest systems with more than one electron and as a result attempts have been made to obtain approximate solutions a series of approximations have been introduced. As a first step, the Bom-Oppenheimer approximation has been invoked, which allows us to treat the electronic and nuclear motions separately. In solving the electronic part, mainly two formalisms, VB and molecular orbital (MO), have been in use and they are described below. Both are wave function-based methods. The wave function T is the fundamental descriptor in quantum mechanics but it is not physically measurable. The squared value of the wave function T 2dT represents probability of finding an electron in the volume element dr. 

This approximation of treating electronic and nuclear motions separately is the Born-Oppenheimer approximation. In the rest of this chapter, we consider the electronic Schrodinger equation. 

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 result is that, to a very good approxunation, as treated elsewhere in this Encyclopedia, the nuclei move in a mechanical potential created by the much more rapid motion of the electrons. The electron cloud itself is described by the quantum mechanical theory of electronic structure. Since the electronic and nuclear motion are approximately separable, the electron cloud can be described mathematically by the quantum mechanical theory of electronic structure, in a framework where the nuclei are fixed. The resulting Bom-Oppenlieimer potential energy surface (PES) created by the electrons is the mechanical potential in which the nuclei move. Wlien we speak of the internal motion of molecules, we therefore mean essentially the motion of the nuclei, which contain most of the mass, on the molecular potential energy surface, with the electron cloud rapidly adjusting to the relatively slow nuclear motion. 

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 ... 

An alternative to the Car-Parrinello method is the following scheme, which separates the electronic and nuclear motions ... 

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. 

In this chapter we present in detail the separation of the nuclear and electronic motions, the nuclear motion within a molecule, and the coupling between nuclear and electronic motion. 

The separation of the electronic and nuclear motions depends on the large difference between the mass of an electron and the mass of a nucleus. As the nuclei are much heavier, by a factor of at least 1800, they move much more slowly. Thus, to a good approximation the movement of the elections in a polyatomic molecule can be assumed to take place in the environment of the nuclei that are fixed in a particular configuration. This argument is the physical basis of the Born-Oppenheimer approximation. 

Marcus uses the Born-Oppenheimer approximation to separate electronic and nuclear motions, the only exception being at S in the case of nonadiabatic reactions. Classical equilibrium statistical mechanics is used to calculate the probability of arriving at the activated complex only vibrational quantum effects are treated approximately. The result is... 

Molecular mechanics force fields rest on four fundamental principles. The first principle is derived from the Bom-Oppenheimer approximation. Electrons have much lower mass than nuclei and move at much greater velocity. The velocity is sufficiently different that the nuclei can be considered stationary on a relative scale. In effect, the electronic and nuclear motions are uncoupled, and they can be treated separately. Unlike quantum mechanics, which is involved in determining the probability of electron distribution, molecular mechanics focuses instead on the location of the nuclei. Based on both theory and experiment, a set of equations are used to account for the electronic-nuclear attraction, nuclear-nuclear repulsion, and covalent bonding. Electrons are not directly taken into account, but they are considered indirectly or implicitly through the use of potential energy equations. This approach creates a mathematical model of molecular structures which is intuitively clear and readily available for fast computations. The set of equations and constants is defined as the force... 

For finite values of /n the system moves within a limited width, given by the fictitious electronic kinetic energy, above the Born-Oppenheimer surface. Adiabacity is ensured if the highest frequency of the nuclear motion separated from the lowest frequency associated with the fictitious motion of the electronic degrees of freedom cofm. It can be shown [30] that eo in is proportional to the gap Eg ... 

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

The motions of a molecular system, for example a solution, occur on many time scales. There are very fast electronic motions, the basic mechanism in chemical reactions then, the nuclear motions, vibrations, librations, rotations, and translations (diffusion). In the Bom-Oppenheimer spirit, one can consider the electronic motion as separated from the nuclear motions, thus one can talk of micro-deformations to be treated quantum mechani-... 

The technique can be used either to perform geometry optimization, by simultaneously annealing the wavefunction and the geometry, or to simulate real dynamics, if the temperature of the fictitious (electronic) parameters is kept close to zero. A drawback of the method is that small masses must be chosen for the electronic parameters in order to achieve an adiabatic separation of the nuclear and the fictitious parameter motions. As a consequence, time steps smaller than MD simulations involving only nuclear motion, are required. 

As pointed out in Chapter 2, nuclear motion takes place on the Born-Oppenheimer potential surface. The motion of the center of mass (corresponding to translation) rigorously separates from the other motions of the atoms. Translational motion may be subject to a potential corresponding to the fact that the molecule... 

A meaningful comparison of kinetic data obtained in different biological systems should be based on the determination of the respective contributions of the nuclear and electronic factors. The most direct method of separating these contributions consists in the measurement of the temperature dependence of the rate over the widest available range. In the following, we distinguish between experiments performed at room temperature, which are usually interpreted by assuming that all the nuclear motions coupled to the transfer may be described classically, and experiments performed at lower temperature, in which the quantified character of particular vibrational modes may appear. 

After the separation of the kinetic energy operator due to the center-of-mass motion from the Hamiltonian, the Hamiltonian describes the internal motions of electrons and nuclei in the system. These in the BO approximation can be separated into the vibrational and rotational motions of the nuclear frame of the molecule and the electronic motion that only parametrically depends on the instantenous positions of the nuclei. When the BO approximation is removed, the electronic and nuclear motions become coupled and the only good quantum numbers, which can be used to quantize the stationary states of the system, are the principle quantum number, the quantum number quantizing the square of the total (nuclear and electronic) squared angular momentum, and the quantum number quantizing the projection of the total angular momentum vector on a selected direction (usually the z axis). The separation of different rotational states is an important feamre that can considerably simplify the calculations. 

---