Nuclear fluctuations

The quantum/classical procedures recover the nuclear fluctuation properties of the surrounding medium via the Monte Carlo statistical approach or by using molecular dynamics simulations. In the following section we examine the problem of energy exchange between solute and solvent from a quantum dynamical viewpoint. 

The kets Rik> and vRjn> are two eigenstates, one associated to the Hamiltonians Hc(i) and the other related with Hc(j)- They describe different regimes of electro-nuclear fluctuations. The labels i andj are there to indicate that by spontaneous or/and induced emission processes, there is a subset of corresponding excited states that would relax... 

To make the ideas sharper consider the case of two quasi degenerate quantum states of the active precursor and successor complexes. The discussion made around equation (57) holds true here too. The activated complex will be the place of a coherent electro-nuclear fluctuation that will go on forever, unless there are quantum states belonging to the relaxation channels of Hc(i) and Hc(j). Note that the mechanisms of excitation to get into the quantum activated complex and those required to relax therefrom are related to the actual rate, while the mechanism of interconversion is closely connected with an... 

Nonequilibrium effects. In applying the various formalisms, a Boltzmann distribution over the vibrational energy levels of the initial state is assumed. The rate constant calculated on the basis of the equilibrium distribution, keq, is the maximum possible value of k. If the electron transfer is very rapid then the assumption of an equilibrium distribution over the energy levels is not valid, and it is more appropriate to treat the nuclear fluctuations in terms of a steady-state rather than an equilibrium formalism. Although a rigorous treatment of this problem has not yet appeared, intuitively it seems that since the slowest nuclear fluctuation will generally be a solvent orientational motion, ke will equal keq when vout keq and k will tend to vout when vout keq (a simple treatment gives l/kg - 1/ vout + 1/keq). These considerations are... 

The surrounding medium and its relaxation time determine the time for the nuclear fluctuations that are to couple with the electron and to capture it, anulling the charged state of H30+. 

If quantum nuclear fluctuations are sought, from eq.(8) one only needs the first term, H c = Kn + He(L ). These fluctuations include rotations as a whole of a frame rigidly bound to the stationary external Coulomb sources. The coupling terms in W, involving the momenta operators of electrons and nuclei, have no diagonal components they will contribute to changes in the populations of stationary levels related to the hamiltonian Ham. Integrating over electronic coordinates with a particular Hn(p cc°n). the collective nuclear motion can be obtained as a solution to... 

The index k(n) recalls that the nuclear fluctuation quantum states in eq.(l 1) are determined by the electronic quantum state via potential energy Een(7 )- Once the electronic problem is fully solved, via a complete set ofeq.(5), it is not difficult to see that pTif nk) multiplied by the box-normalized wave solutions (see p. 428, ref. [17] 2nd ed.) are eigenfunctions ofthehamiltonian H0and, for stationary global momentum solutions, the molecular hamiltonian is also diagonalized thereby solving eq. (2). 

The neglect of the electronic coupling in the calculation of the ECWD (assumption 1) was adopted in the original Marcus and Elush formulation. " Within this framework, the ET matrix element does not strongly affect the nuclear fluctuations, although a nonzero value of Hab is required for electronic transitions to occur. In other words, the transferred electron is assumed to be fully localized in the calculation of the ECWD. To classify electronic delocalization, Robin and Day distinguished between three classes of symmetrical (APo = 0) systems. 

The only way the electron can be locahsed on this model acceptor site is if some perturbation makes it lower in energy than the resonantly coupled atomic sites of the lattice. This perturbation is the nuclear fluctuation of the dielectric medium that displace the nuclear configurations towards position C in Fig. 2.24 where the charge-... 

Fig. 16.6 A schematic representation of the potential surface for the electron between two centers (D—donor, A—acceptor). Shown also are the relevant diabatic electronic wavefunctions, localized on each center. The lower diagram corresponds to a stable nuclear configuration and the upper one— to a nuclear fluctuation that brings the system into the transition state where the diabatic electronic energies are equal. The electronic transition probability depends on the overlap between the electronic wavefunctions i/zp and tn this transition state.

Fig. 9. Molecular adsorbate level between two continuous metallic-level distributions. Tunneling is from the negatively (right) to the positive (left) biased electrode and feasible when nuclear fluctuations take the adsorbate level from the initial location above both Fermi levels, Srf and Elf, respectvely, to a location between the Fermi levels, q with different subscripts indicates nuclear configurations at which tunneling in the coherent two-step ET mode can occur [53, 55].

Any nuclear fluctuation away the stationary value will set up electronic restitution forces if there are bound electro-nuclear states. The electronic stationary state acts as a glue for the nuclei. Any picture of electrons following the nuclear motion is totally foreign to the present approach. 

The solvent Hamiltonian HB includes two components. The first one is an intrinsically quantum part that describes polarization of the electronic clouds of the solvent molecules. This polarization is given by the electronic solvent polarization, Pe. The second part is due to thermal nuclear motions that can be classical or quantum in character. Here, to simplify the discussion, we consider only the classical spectrum of nuclear fluctuations resulting in the classical field of nuclear polarization, Pn. Fluctuations of the solvent polarization field are usually well described within the Gaussian approximation,35 leading to the quadratic solvent Hamiltonian... 

Expression 11.3 provides the whole simulated spectrum, while a detailed vibrational analysis requires the unambiguous assignment of each mode contribution. Recently, a number of methods appeared in the literature aimed at the extraction of normal-mode-like analysis from ab initio dynamics [58-63]. Some of these [58-60] refer to the quasi-harmonic model introduced by Karplus [64,65] in the framework of classical molecular dynamics and individuate normal-mode directions as main components of the nuclear fluctuations in the NVE or NVT ensemble. The quasinormal model relies on the equipartition of the kinetic energy among normal modes thus problems arise when the simulation time required to obtain such a distribution is computationally too expensive, as is often the case for ab initio dynamics. Other approaches [61-63] carry out the time evolution analysis in the momenta subspace instead of the configurational space. In these approaches the basic consideration is that, at any temperature, generalized normal modes g, correspond to uncorrelated momenta such that [61]... 

---