Nuclear and Electronic Degrees of Freedom

It ensues from the property (11) that it is sufficient to define (r R) and n(r) only within the domain of internal nuclear coordinates R. The replacement of R by R = Rj , where Rj = Xj,Yj,Zj , which results in the removal of three degrees of freedom (two for linear molecules), corresponds to adopting a rotating ( body-fixed ) coordinate system in place of the fixed ( space-fixed ) one. Various definitions of the former coordinate system are possible, the most natural involving the requirement that the 

The ground eigenstate V q(fIr) can be employed as a continuous basis set for the trial wavefunction (r,R), 

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One common approximation is to separate the nuclear and electronic degrees of freedom. Since the nuclei are considerably more massive than the electrons, it can be assumed that the electrons will respond mstantaneously to the nuclear coordinates. This approximation is called the Bom-Oppenlieimer or adiabatic approximation. It allows one to treat the nuclear coordinates as classical parameters. For most condensed matter systems, this assumption is highly accurate [11, 12]. 

In the strictest meaning, the total wave function cannot be separated since there are many kinds of interactions between the nuclear and electronic degrees of freedom (see later). However, for practical purposes, one can separate the total wave function partially or completely, depending on considerations relative to the magnitude of the various interactions. Owing to the uniformity and isotropy of space, the translational and rotational degrees of freedom of an isolated molecule can be described by cyclic coordinates, and can in principle be separated. Note that the separation of the rotational degrees of freedom is not trivial [37]. 

In this chapter, we discussed the permutational symmetry properties of the total molecular wave function and its various components under the exchange of identical particles. We started by noting that most nuclear dynamics treatments carried out so far neglect the interactions between the nuclear spin and the other nuclear and electronic degrees of freedom in the system Hamiltonian. Due to... 

In this expression p is a mass parameter associated to the electronic fields, i.e. it is a parameter that fixes the time scale of the response of the classical electronic fields to a perturbation. The factor 2 in front of the classical kinetic energy term is for spin degeneracy. The functional f [ i , ] plays the role of potential energy in the extended parameter space of nuclear and electronic degrees of freedom. It is given by. 

Fe ". In the two-state model, the electron transfer is viewed as a quantum transition between two localized states V, - and Pf. In IF,-, the ion with charge <7/ is at equilibrium with the interfacial water molecules, and the electron is in the metal. In the metal has lost one electron, and the ion with charge q/ is at equilibrium with the interfacial water. The total Hamiltonian of the system H, including all nuclear and electronic degrees of freedom, is not diagonal in the basis ( , , Pf), and so if the system is prepared in the state P, it will evolve in time according to ... 

In order to illustrate electronic transitions we discuss the simple two-dimensional model of a linear triatomic molecule ABC as depicted in Figure 2.1. R and r are the appropriate Jacobi coordinates to describe the nuclear motion and the vector q comprises all electronic coordinates. The total molecular Hamiltonian Hmoi, including all nuclear and electronic degrees of freedom, is given by Equation (2.28) with Hei and Tnu being the electronic Hamiltonian and the kinetic energy of the nuclei, respectively. 

Here HM is the nuclear part, written as the sum of a kinetic term (K) and a nuclear-nuclear interaction Fnn, and Hel is the electronic part. Since Hc] involves interactions between the electrons and nuclei, HMT is not separable in the nuclear and electronic, degrees of freedom. However, adopting the Bom-Oppenheimer approximation -substantially simplifies matters. In this approximation we (a) define the Born- ppenheimer potential seen by the nuclei in electronic state e) as We = (el m-h/ye) where... 

The short scattering time of the NCS process implies that, in this physical context, there is no well-defined time scale separation between the characteristic times of "slow" nuclear and "fast" electronic dynamics. Therefore, the well known Bom-Oppenheimer approximation does not apply here (see also below). As a consequence, QE between nuclear and electronic degrees of freedom has been expected to strongly affect the dynamics of II atoms (or protons) [Chatzidimitriou-Dreismann 2003 (a) Chatzidimitriou-Dreismann 2001 Chatzidimitriou-Dreismann 2000 (b) Chatzidimitriou-Dreismann 2002 (a)]. 

The Hartree-Fock (HF) approximation. The HF method is based on the Born-Oppenheimer and orbital approximations. Under the Born-Oppenheimer approximation the nuclear and electronic degrees of freedom of a molecule are decoupled, and the nuclei are held fixed while the electronic contribution to the energy is calculated. In the orbital approximation... 

To make a comprehensive comparison between the two representations of a total wavefunction, namely, Eqs. (6.5) and (6.147), we need a high dimensional space composed of both nuclear and electronic degrees of freedom. However, it is practically impossible, and therefore we alternatively compare only the nuclear part of Eq. (6.3) associated with the electronic eigenfunction K r R), that is,... 

The mass of a proton is 1836 times that of an electron, and so the electron has a much smaller mass than even the H atom nucleus. This difference allows the functional form of the total wavefunction to be simplified by treating the electronic and vibrational states separately. The separation of the nuclear and electronic degrees of freedom in this way is... 

The equations of motion for the nuclear and electronic degrees of freedom are ... 

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