Nuclear electronic motion

In the case of direct vibrational excitation, the vibrational transition probability is given by p, where are the intermediate and ground vibrational states, respectively, and is the vibrational transition moment. The electronic transition probability out of the intermediate state is < n < e ng e > n>, where are the excited and ground electronic states, respectively, and is the electronic dipole moment operator and vibrational state in the upper electronic state. Applying the Born-Oppenheimer approximation, where the nuclear electronic motion are separated, S can be presented as... 

Dependent Electron Localization Functions for Coupled Nuclear-Electronic 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 ... 

With tlie development of femtosecond laser teclmology it has become possible to observe in resonance energy transfer some apparent manifestations of tire coupling between nuclear and electronic motions. For example in photosyntlietic preparations such as light-harvesting antennae and reaction centres [32, 46, 47 and 49] such observations are believed to result eitlier from oscillations between tire coupled excitonic levels of dimers (generally multimers), or tire nuclear motions of tire cliromophores. This is a subject tliat is still very much open to debate, and for extensive discussion we refer tire reader for example to [46, 47, 50, 51 and 55]. A simplified view of tire subject can nonetlieless be obtained from tire following semiclassical picture. 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

The separation of nuclear and electronic motion may be accomplished by expanding the total wave function in functions of the election coordinates, r, parametrically dependent on the nuclear coordinates... 

In currently available software, the Hamiltonian above is nearly never used. The problem can be simplified by separating the nuclear and electron motions. This is called the Born-Oppenheimer approximation. The Hamiltonian for a molecule with stationary nuclei is... 

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. 

Assumption of a rigorous separation of nuclear and electronic motions (Bom-Oppenheimer approximation). In most cases this is a quite good approximation, and there is a good understanding of when it will fail. There are, however, very few general techniques for going beyond the Bom-Oppenheimer approximation. 

For Q = Q , this density function describes electronic motions for given nuclear positions, while for Q = Q it describes the quantal correlation of nuclear positions at time f, which should be small for classical-like variables. The equation of motion for the density function could be derived from the original LvN equation. Instead, it is more convenient to construct it from the wavefunctions. The phase factor and the preexponential factor are trial functions to be determined from the TDVP. The procedure followed here parallels that in ref. (23). 

A molecule is composed of positively charged nuclei surrounded by electrons. The stability of a molecule is due to a balance among the mutual repulsions of nuclear pairs, attractions of nuclear-electron pairs, and repulsions of electron pairs as modified by the interactions of their spins. Both the nuclei and the electrons are in constant motion relative to the center of mass of the molecule. However, the nuclear masses are much greater than the electronic mass and, as a result, the nuclei move much more slowly than the electrons. Thus, the basic molecular structure is a stable framework of nuclei undergoing rotational and vibrational motions surrounded by a cloud of electrons described by the electronic probability density. 

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. 

In the original mathematical treatment of nuclear and electronic motion, M. Bom and J. R. Oppenheimer (1927) applied perturbation theory to equation (10.5) using the kinetic energy operator Tq for the nuclei as the perturbation. The proper choice for the expansion parameter is A = (me/M) /", where M is the mean nuclear mass... 

Perturbation terms in the Hamiltonian operator up to still lead to the uncoupling of the nuclear and electronic motions, but change the form of the electronic potential energy funetion in the equation for the nuclear motion. Rather than present the details of the Bom-Oppenheimer perturbation expansion, we follow instead the equivalent, but more elegant procedure of M. Bom and K. Huang (1954). 

The application of the Bom-Oppenheimer and the adiabatic approximations to separate nuclear and electronic motions is best illustrated by treating the simplest example, a diatomic molecule in its electronic ground state. The diatomic molecule is sufficiently simple that we can also introduce center-of-mass coordinates and show explicitly how the translational motion of the molecule as a whole is separated from the internal motion of the nuclei and electrons. 

Central to the description of this dynamics is the BO approximation. This separates the nuclear and electronic motion, and allows the system evolution to be described by a function of the nuclei, known as a wavepacket, moving over a PES provided by the (adiabatic) motion of the electrons. 

Non-adiabatic coupling is also termed vibronic coupling as the resulting breakdown of the adiabatic picture is due to coupling between the nuclear and electronic motion. A well-known special case of vibronic coupling is the Jahn-Teller effect [14,164—168], in which a symmetrical molecule in a doubly degenerate electronic state will spontaneously distort so as to break the symmetry and remove the degeneracy. 

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. 

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. 

There is no distinction between electrons and nuclei and no mechanism whereby relative atomic positions can be localized. However, since the lighter electrons may reasonably be assumed to move much faster than the nuclei, the nuclear coordinates may to first approximation be assumed to remain fixed at Q on the time scale of electronic motion. The electronic wave equation then reduces to... 

Solutions to the Schrodinger equation Hcj) = E(f> are the molecular wave functions 0, that describe the entangled motion of the three particles such that (j) 4> represents the density of protons and electron as a joint probability without any suggestion of structure. Any other molecular problem, irrespective of complexity can also be developed to this point. No further progress is possible unless electronic and nuclear variables are separated via the adiabatic simplification. In the case of Hj that means clamping the nuclei at a distance R apart to generate a Schrodinger equation for electronic motion only, in atomic units,... 

To more fully appreciate the equilibrium models, like SCRF theories, and their usefulness and limitations for dynamics calculations we must consider three relevant times, the solvent relaxation time, the characteristic time for solute nuclear motion in the absence of coupling to the solvent, and the characteristic time scale of electronic motion. We treat each of these in turn. 

The SCRF models assume that solvent response to the solute is dominated by motions that are slow on the solute electronic motion time scales, i.e., Xp Telec. Thus, as explained in Section 2.1, the solvent sees the solute electrons only in an averaged way. If, in addition to the SCRF approximation, we make the usual Bom-Oppenheimer approximation for the solute, then we have xs Xelect-In this case the solute electronic motion is treated as adjusting adiabatically both to the solvent motion and to the solute nuclear motion. 

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