Solvent orientational polarization

Finally, the last two terms in G(2.12) account for the effects of the solvent orientational polarization in the nonequilibrium solvation. The matrix K0 is the inverse of the solvent orientational polarization interaction energy matrix I0 whose elements are defined, analogously to Ienm, by... 

In words, s describes the interaction of the solute charge distribution component p, with the arbitrary solvent orientational polarization mediated by the cavity surface. The arbitrary weights p,, previously defined by (2.11), enter accordingly the definition of the solvent coordinates, and reduce, in the equilibrium solvation regime, to the weights tv,, such that the solvent coordinates are no longer arbitrary, but instead depend on the solute nuclear geometry and assume the form se<> = lor. weq. In equilibrium, the solvent coordinates are correlated to the actual electronic structure of the solute, while out of equilibrium they are not. 

Turning to pathway c), the concerted-reaction mechanism, we have formulated two approaches to predicting the rate constant, a double-adiabatic and a two-dimensional approach (4,5). In the double-adiabatic theory, the electron is considered to be coupled to two nuclear modes, a solvent (orientational polarization) mode that is treated classically in view of its low... 

A seheme for the treatment of the solvent effeets on the eleetronie absorption speetra in solution had been proposed in the framework of the eleetrostatie SCRF model and quantum ehemieal eonfiguration interaetion (Cl) method. Within this approaeh, the absorption of the light by ehromophorie moleeules was eonsidered as an instantaneous proeess. Therefore, during the photon absorption no ehange in the solvent orientational polarization was expeeted. Only the eleetronie polarization of solvent would respond to the ehanged eleetron density of the solute moleeule in its exeited (Franek-Condon) state. Consequently, the solvent orientation for the exeited state remains the same as it was for the ground state, the solvent eleetronie polarization, however, must refleet the exeited state dipole and other eleetrie moments of the moleeule. Considering the SCRF Flamiltonian... 

Figure 20.15 Stokes shift as a function of solvent orientational polarization df tor TDA. The structure of the molecule is shown on the right-hand side. Solvents hexane (1), toluene (2), CCI4 (3), dioxane (4), dibutylether (5), chlorobenzene (6), diethylether (7), ethylacetate (8), 1-chlorobutane (9), tetrahydro-furan (10), CH2CI2 (11), dimethylsulfoxide (12), dimethylformamide (13), acetonitrile (14).

A useful molecular model treatment of solvent orientation polarization at an electrode interface was given by Watts-Tobln and Mott (1961). Two orientation states of the solvent dipoles, up t and down 4, aligned with the electrode field E arising from net surface charge density q, were envisaged. The polarization in the interphase at the electrode surface was calculated in terms of the relative population Nt/(Nt + N4.) and N4/(Nt + Nl) of the two states of orientation. Nt and N4. are determined (a) by the field, (b) by temperature, and (c) by any lateral interaction forces between the oriented and unoriented dipoles. Interaction effects were not, however, taken into account in the original treatment. 

The nonlocal diffuse-layer theory near Eam0 has been developed283 with a somewhat complicated function oLyjind of solvent structural parameters. At low concentrations,/ ) approaches unity, reaching the Gouy-Chapman Qatc- 0. At moderate concentrations, deviations from this law are described by the effective spatial correlation range A of the orientational polarization fluctuations of the solvent. 

To obtain an estimate for the energy of reorganization of the outer sphere, we start from the Born model, in which the solvation of an ion is viewed as resulting from the Coulomb interaction of the ionic charge with the polarization of the solvent. This polarization contains two contributions one is from the electronic polarizability of the solvent molecules the other is caused by the orientation and distortion of the... 

The axial equatorial equilibrium of 5-OH-l, 3-dioxane was strongly solvent dependent alcohols favor the axial position, other solvents, more polar than CCI4, the equatorial orientation of the 5-hydroxy substituent, both effects being corroborated by PM3 calculations [92MI3 93JST(287)185],... 

Most solvents consist of molecules that are intrinsic dipoles and have permanent dipole moments (pi). If such molecules are placed between the two plates of a capacitor as a vapor (or as a dilute solution in a nonpolar liquid), they are oriented by the electric field. Then, the orientational polarization and the induced polarization occur simultaneously, as described above. If er is the relative permittivity of the vapor, there is a relationship ... 

Here Fe(t) and Fg(t) are the time-dependent nonequilibrium Helmholtz free energies of the e and g states, respectively. The energy difference A U(t) can be replaced by a free energy difference due to the fact that the entropy is unchanged in a Franck-Condon transition [51]. Free energies in Eq. (3) can be represented [54] by a sum of an equilibrium value Fcq and an additional contribution related to nonequilibrium orientational polarization in the solvent. Thus for the free energy in the excited state Fe(t) we have... 

Separation of Electronic and Nuclear Motions. The polarizabilities of the ground state and the excited state can follow an electronic transition, and the same is true of the induced dipole moments in the solvent since these involve the motions of electrons only. However, the solvent dipoles cannot reorganize during such a transition and the electric field which acts on the solute remains unchanged. It is therefore necessary to separate the solvent polarity functions into an orientation polarization and an induction polarization. The total polarization depends on the static dielectric constant Z), the induction polarization depends on the square of the refractive index n2, and the orientation polarization depends on the difference between the relevant functions of D and of n2 this separation between electronic and nuclear motions will appear in the equations of solvation energies and solvatochromic shifts. 

Dipole-Dipole Interaction. The first of the four terms in the total electrostatic energy depends on the permanent dipole moment of the solute molecule of radius a (assuming a spherical shape) immersed in a liquid solvent of static dielectric constant D. The function f(D) = 2(D - l)/(2D + 1) is known as the Onsager polarity function. The function used here is [f(D) — f(n2)] so that it is restricted to the orientational polarity of the solvent molecules to the exclusion of the induction polarity which depends on the polarizability as of the solvent molecules, related to the slightly different Debye polarity function q>(n2) according to... 

This external or outer-sphere parameter A0 can be expressed according to a modified form of the Born equation to take into account only the solvent s orientation polarization. (As in the theory of solvatochromic shifts, it is assumed that the induction polarization can follow electronic motion.)... 

Osipov proposed a formula to calculate the dipole moment of a polar substance in a polar solvent and proved its applicability (22,23). In this case the molar orientation polarization of the solution ( P°rdu ) can be written as ... 

Here PB,Aor eq(x r) is the equilibrium orientational polarization field in the solvent, at bond length r for the solute in either the B or A state, and s,yu is the solvent infinite frequency dielectric constant. 

The PMF as a function of Xs is determined by a coupled free energy perturbation and umbrella sampling technique.5,14,16,41 The computational procedure follows two steps, although they are performed in the same simulation. The first is to use a reference potential rp to enforce the orientation polarization of the solvent system along the reaction path. A convenient choice of the reference potential, which is called mapping potential in Warshel s work,13,14,16,42 is a linear combination of the reactant and product diabatic potential energy ... 

Experimental tests of the dielectric continuum model based on variations of the solvent medium are, unfortunately, quite limited these are summarized in Sect. 4.5. It is nonetheless useful to examine the physical basis of eqn. (19) in order to ascertain the likelihood of its applicability. An enlightening derivation of eqn. (19) involves treating the formation of the transition state in terms of a hypothetical two-step charging process [31b, 40]. First, the charge of the reactant Ox is slowly adjusted to an appropriate value so that the solvent molecules are polarized to an extent identical to that for the transition state (step 1). Second, the charge is readjusted to that of the reactant sufficiently rapidly so that the solvent orientation remains unaltered (step 2), thereby yielding the non-equilibrium solvent polarization appropriate to the transition state. 

In Eq. (28) e is the charge of the electron, r the radius of the reactant, Avo-gadro s number, and d the reactant-electrode distance. The solvent was considered here as a continuous medium with a fast electronic polarization characterized by the optical dielectric permittivity and a slower oscillatory plus orientational polarization characterized by the static dielectric permittivity... 

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