Degrees of freedom electronic

The electronic partition function involves a sum over electronic quantum states. These are the solutions to the electronic Schrodinger equation, i.e. the lowest (ground) state and all possible excited states. In almost all molecules, the energy difference between the ground and excited states is large compared with kT, which means that only the first term (the ground state energy) in the partition function summation (eq. (13.11)) is important. 

Defining the zero point for the energy as the electronic energy of the reactant, the electronic partition functions for the reactant and TS is given in eq. (13.35). 

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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]. 

These results do not agree with experimental results. At room temperature, while the translational motion of diatomic molecules may be treated classically, the rotation and vibration have quantum attributes. In addition, quantum mechanically one should also consider the electronic degrees of freedom. However, typical electronic excitation energies are very large compared to k T (they are of the order of a few electronvolts, and 1 eV corresponds to 10 000 K). Such internal degrees of freedom are considered frozen, and an electronic cloud in a diatomic molecule is assumed to be in its ground state f with degeneracy g. The two nuclei A and... 

Meyer H D and Miller W H 1979 A classical analog for electronic degrees of freedom in nonadiabatic... 

Election nuclear dynamics theory is a direct nonadiababc dynamics approach to molecular processes and uses an electi onic basis of atomic orbitals attached to dynamical centers, whose positions and momenta are dynamical variables. Although computationally intensive, this approach is general and has a systematic hierarchy of approximations when applied in an ab initio fashion. It can also be applied with semiempirical treatment of electronic degrees of freedom [4]. It is important to recognize that the reactants in this approach are not forced to follow a certain reaction path but for a given set of initial conditions the entire system evolves in time in a completely dynamical manner dictated by the inteiparbcle interactions. 

Solving the Eqs. (C.6-C.8,C.12,C.13) comprise what is known as the Ehrenfest dynamics method. This method has appealed under a number of names and derivations in the literatnre such as the classical path method, eilconal approximation, and hemiquantal dynamics. It has also been put to a number of different applications, often using an analytic PES for the electronic degrees of freedom, but splitting the nuclear degrees of freedom into quantum and classical parts. 

Let US consider the simplified Hamiltonian in which the nuclear kinetic energy term is neglected. This also implies that the nuclei are fixed at a certain configuration, and the Hamiltonian describes only the electronic degrees of freedom. This electronic Hamiltonian is... 

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... 

We follow Thompson and Mead [13] to discuss the behavior of the electronic Hamiltonian, potential energy, and derivative coupling between adiabatic states in the vicinity of the D31, conical intersection. Let A be an operator that transforms only the nuclear coordinates, and A be one that acts on the electronic degrees of freedom alone. Clearly, the electronic Hamiltonian satisfies... 

Electronic spectroscopy is the study of transitions, in absorption or emission, between electronic states of an atom or molecule. Atoms are unique in this respect as they have only electronic degrees of freedom, apart from translation and nuclear spin, whereas molecules have, in addition, vibrational and rotational degrees of freedom. One result is that electronic spectra of atoms are very much simpler in appearance than those of molecules. 

In 1985 Car and Parrinello invented a method [111-113] in which molecular dynamics (MD) methods are combined with first-principles computations such that the interatomic forces due to the electronic degrees of freedom are computed by density functional theory [114-116] and the statistical properties by the MD method. This method and related ab initio simulations have been successfully applied to carbon [117], silicon [118-120], copper [121], surface reconstruction [122-128], atomic clusters [129-133], molecular crystals [134], the epitaxial growth of metals [135-140], and many other systems for a review see Ref. 113. 

The fact that detailed balance provides only half the number of constraints to fix the unknown coefficients in the transition probabilities is not really surprising considering that, if it would fix them all, then the static (lattice gas) Hamiltonian would dictate the kind of kinetics possible in the system. Again, this cannot be so because this Hamiltonian does not include the energy exchange dynamics between adsorbate and substrate. As a result, any functional relation between the A and D coefficients in (44) must be postulated ad hoc (or calculated from a microscopic Hamiltonian that accounts for couphng of the adsorbate to the lattice or electronic degrees of freedom of the substrate). Several scenarios have been discussed in the literature [57]. 

For the calculation of the LDA ground-state one can proceed either via the direct" methods, i.e. via the glocal minimization of the total free energy with respect to the electronic degrees of freedom, or via the the diagonalization (for large PW basis-sets necessarily iterative diagonalization) of the KS Hamiltonian in combination with an iterative update of chai ge-density and potential. 

Therefore, the simplest classical treatment in which the propagator exp(it (T+V) ) is approximated in the product form exp(it (T) ) exp(it (V)/fc) and die nuclear kinetic energy T is conserved during the transition produces a nonsensical approximation to the non BO rate. This should not be surprising because (a) In the photon absorption case, the photon induces a transition in the electronic degrees of freedom which subsequently cause changes in the vibration-rotation energy, while (b) in the non BO case, the electronic and vibration-... 

The review commences, in Section 2, at the level of the electronic degrees of freedom (Muller-Plathe), which takes one from electrons to atoms, as well as an effective potential operating between atoms. 

It is the purpose of this section to review ways in which processes involving electrons are either explicitly accounted for in calculations on polymeric systems or in which a more or less rigorous abstraction from the electronic degrees of freedom into effective models of a coarser-grained nature is performed. The next level up from electrons is obviously atoms. Hence, this section deals mainly with the connection between quantum chemistry and atomistic (force field) simulations. Calculations which exclusively use quantum chemistry are not covered. This excludes, for example, all of the recent work on metallocene catalysis. 

Mahoney MW, Jorgensen WL (2001) Rapid estimation of electronic degrees of freedom in Monte Carlo calculations for polarizable models of liquid water. J Chem Phys 114(21) 9337-9349... 

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. 

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