Electron free energy function

Ben]amin I, Barbara P F, Gertner B J and Hynes J T 1995 Nonequilibrium free energy functions, recombination dynamics, and vibrational relaxation of tjin acetonitrile molecular dynamics of charge flow in the electronically adiabatic limit J. Phys. Chem. 99 7557-67... 

G King, A Warshel. Investigation of the free energy functions for electron transfer reactions. J Chem Phys 93 8682-8692, 1990. 

FIGURE 35.3 Free-energy functions for reactant (AE) and product Ag (AE) of an electron transfer reaction as calculated using umbrella sampling within a simple dipolar diatomic solvent. AG° is the reaction free energy. Solid lines are polynomial fittings to the simulated points. Dashed lines are parabolic extrapolations from the minimum of the curves. (From King and Warshel, 1990, with permission from the American Institute of Physics.)... 

Rose and Benjamin (see also Halley and Hautman ) utilized molecular dynamic simulations to compute the free energy function for an electron transfer reaction, Fe (aq) + e Fe (aq) at an electrodesolution interface. In this treatment, Fe (aq) in water is considered to be fixed next to a metal electrode. In this tight-binding approximation, the electron transfer is viewed as a transition between two states, Y yand Pf. In Pj, the electron is at the Fermi level of the metal and the water is in equilibrium with the Fe ion. In Pf, the electron is localized on the ion, and the water is in equilibrium with the Fe" ions. The initial state Hamiltonian H, is expressed as... 

The theory of electronic polarization in dielectric media [25] provides the framework for the derivation of a free energy functional that meets the requirements set forth in the Introduction. In particular, the additional free energy of the system due to a polarization P(r) can be expressed as [26] ... 

In the past few years, a great effort has been devoted to the extensions of solvation models to QM techniques of increasing accuracy. All these computational extensions have been based on a reformulation of the various QM theories describing electron correlation so as to include in a proper way the effects of the nonlinearity of the solvation model by assuming the free-energy functional as the basic energetic quantity. 

Most of these extensions have involved electron correlation methods based on variational approaches (DFT, MCSCF, CI,VB). These methods can be easily formulated by optimizing the free energy functional (1.117), expressed as a function of the appropriate variational parameters, as in the case of the HF approximation. In contrast, for nonva-riational methods such as the Moller-Plesset theory or Coupled-Cluster, the parallel extension to solvation model is less straightforward. 

A further issue arises in the Cl solvation models, because Cl wavefunction is not completely variational (the orbital variational parameter have a fixed value during the Cl coefficient optimization). In contrast with completely variational methods (HF/MFSCF), the Cl approach presents two nonequivalent ways of evaluating the value of a first-order observable, such as the electronic density of the nonlinear term of the effective Hamiltonian (Equation 1.107). The first approach (the so called unrelaxed density method) evaluates the electronic density as an expectation value using the Cl wavefunction coefficients. In contrast, the second approach, the so-called relaxed density method, evaluates the electronic density as a derivative of the free-energy functional [18], As a consequence, there should be two nonequivalent approaches to the calculation of the solvent reaction field induced by the molecular solute. The unrelaxed density approach is by far the simplest to implement and all the Cl solvation models described above have been based on this method. 

Although the correlative methods based on the coupled-cluster (CC) ansatz are among the most accurate approaches for molecules in vacuum, their extension to introduce the interactions between a molecule and a surrounding solvent have not yet reached a satisfactory stage. The main complexity in coupling CC to solvation methods comes from the evaluation of the electronic density, or of the related observables, needed for the calculation of the reaction field. Within the CC scheme the electronic density can only be evaluated by a relaxed approach, which implies the evaluation of the first derivative of the free energy functional. As discussed previously for the cases of the Cl and MPn approaches, this leads to a more involved formalism. 

Following this method, the electronic wavefunction is obtained by minimizing self-consistently the free energy functional... 

We will in this section introduce the representation of the electronic wave function of the molecular system and describe how we determine the wave function of the quantum mechanical subsystem including interactions with the structured environment. We consider the situation where the MCSCF electronic wave function of the quantum mechanical subsystem is optimized while interacting with a classical system represented by charges, polarization sites and van der Waals sites. We start out by expressing the total electronic free energy for the QM/MM-system as... 

The equilibrium constant as a function of temperature is determined from the free energy function (fef) values tabulated by Woolley (W4, W5) and from a calculated free energy function for the electron. 

Figure 12. Sketch of the probabiiities Wo (E) and fkRMi( ) to find an empty or filled electron level corresponding to an oxidized and reduced ion, respectively, as a function of the electron free energy (vertical axis). The standard electrochemical potential /i"(Ox/Red) with respect to the vacuum level acts as a reference point.

Figure 1. Schematic illustration of the diabatic (solid) and adiabatic (long-dashed) electronic free energy curves as functions of the solvent coordinate z<. for a single electron transfer reaction. The Marcus theory quantities AG° and A are indicated.

The approach used to obtain the EVB free-energy functionals (the Ag of Equation (7)) has been originally developed in Ref. 25 in order to provide the microscopic equivalent of the Marcus theory for electron transfer (ET) reactions.38 This approach allows one to explore the validity of the Marcus formula and the underlying linear response approximation (LRA) on a microscopic molecular level.39 While this point is now widely accepted by the ET community,40 the validity of the EVB as perhaps the most general tool in microscopic LFER studies is less appreciated. This issue will be addressed below. 

King, G., Waeshel, A., Investigation of the Free Energy Functions for Electron Transfer Reactions, J. Chem. Phys. 1990, 93, 8682-8692. 

Benjamin [217] calculated the solvent reorganization free energy functions for the electron transfer reaction of a monovalent ion near the interface between immiscible... 

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