Apparent charge

This yields 0.51 x 10 24 cm3 which agrees approximately with the value 0.58 xl0 24 ascribed to the double bond on an empirical basis. Also we may note that this / value corresponds to a transition dipole of 3.16xl0-18, if a one-dimensional charge oscillation is assumed, and that this dipole yields an apparent charge of 0,49 times the electronic charge for a C=C distance of 1.34 A. 

Another solution consists in using surfactants that stick on the particle and substrate surfaces and therefore modify their apparent charges. The results of Fig. 15 were obtained under the same experimental conditions as those reported in Fig. 14. Some anionic surfactants such as TA sulfate injected in 0.1% HF at the critieal miscellanies concentration (CMC) level achieve the same good performances as those obtained in alkaline media. 

For easy comparison we show in Table 13.8 the apparent charge on the H atom in each of our molecules. The trend in these charges is broken between one and... 

Table 13.11. Various multipole moments and the apparent charges on H atoms from 6-3IG calculations.

The apparent charges on the H atoms in this basis are shown in Table 13.11. These may be compared to the similar values in Table 13.8. We see that the larger basis yields smaller values, particularly for methane. Nevertheless, we still predict that the H atoms in these small hydrocarbons are more positive than the C atom. 

The alternative method is to bring the equation (2.43) values into agreement with the estimated values of enthalpies of hydration of the same ions by optimizing their ionic charges. This approach yields the apparent charges given in Table 2.I0 for the ions under consideration. 

First, as discussed earlier in connection with the aluminum halide catalyzed rearrangements of hydrocarbons (Section II. A. 2), intermolecular hydride transfer reactions appear to be fairly unselective processes. Apparently, charge development in the transition states of these reactions is minimized a penta-coordinate carbon intermediate may be involved. As a result, the strong preference for the bridgehead positions exhibited by most ionic substitution reactions is partially overcome. 

As described in the previous contributions by Cances and by Tomasi, in such a family of methods the solvent effects on the molecular solutes are evaluated by introducing a set of apparent charges representing the polarization of the dielectric medium. These charges are obtained by solving integral equations defined on the domain of the boundary of the cavity which hosts the molecular solute. The solution of such equations can be divided in two main steps. 

E. L. Coitino, J. Tomasi and R. Cammi, On the evaluation of the solvent polarization apparent charges in the polarizable continuum model A new formulation, J. Comput. Chem., 16... 

The strategy to obtain a Lagrangian formulation of PCM is to consider the PCM apparent charges as a set of dynamic variables, exactly as the solute nuclear coordinates. The algorithm proposed in the present chapter is applied within the MM framework, since it allows a simplified notation and faster calculations. However, we point out that it can be straightforwardly extended to QM calculations. 

The bottleneck of a calculation in solution is the evaluation of the polarization which, in the case of PCM, corresponds to the evaluation of the apparent surface charges. In particular, the bottleneck is represented by the evaluation of the products between the integral matrices of the electrostatic potential (matrix S in Equation (1.8.6)) or of the normal component of the electric field (matrix D in Equation (1.92)) and the apparent charges vector q. Thus the criterion we use to compare the standard and the simultaneous approach is based on the number of matrix products (Sq or D q) necessary in the whole optimization process. We also remind the reader that the dimension of the matrices is equal to the square of the number of the surface elements. 

The advantage of the new strategy is that, for each step of the optimization, a small and constant number of matrix-vector products are necessary (three for CPCM and nine for DPCM). In contrast, for the standard approach the evaluation of the apparent charges... 

We remark that, in this formulation, we have collected into a single set of one-electron operators all the interaction operators we have defined in the preceding section, and, in parallel, we have put in the qk set both the apparent charges related to the electrons and nuclei of M. This is an apparent simplification as all the operators are indeed present. It is interesting here to note that this nesting of the electrostatic problem in the QM framework is performed in a similar way in all continuum QM solvation codes. 

In this way, we have rewritten all the solvent interaction elements of the Fock matrix in terms of the unknown qe and cf apparent charges (the last, not being modified in the SCF cycle, can be separately computed at the beginning of the calculation). 

Within this framework the input quantities are the dielectric permittivity s, the solute charge p(r) and the excluded volume cavity occupied by a solute. The response field is created by the surface charge density cr(r) (the apparent charge) arising, as a result of medium polarization, on the cavity surface S ... 

A connection to vector fields (1.119) is established by the notion that a is equal to the normal component of the polarization vector P(r) located on the external side of S. Polarization vanishes in the bulk of the medium provided the dielectric constant does not change there. The apparent charge cr(r) found in terms of numerical algorithms [12] is, in turn, a linear functional of p(r). Its computation is equivalent to a solution of the Poisson equation with proper matching conditions for [Pg.98]

For given to value the apparent charge density cr(r, to) is available in terms of the extended PCM procedure with a complex-valued dielectric function s, namely, e(w) = s1((o) + is2((o) where e w) = 1 +4ttXi(oj) and s2(oj) = 4ttx2(m) with complex-valued susceptibilities defined in Equation (1.127). The complication that both a(r, to) and 0(r, to) become complex is inevitable. However, after applying the inverse Fourier transform, they become real in the time domain. This is warranted by the symmetry properties,... 

This approach, based on a complex-valued realization of the PCM algorithm, reduces to a pair of coupled integral equations for real and imaginary parts of apparent charge density for tr(f,to) [13]. An alternative technique avoiding explicit treatment of the complex permittivity has been also derived [14,15]. The kernel K(f,f, t) of operator K does not appear explicitly. However, its matrix elements can be computed for any pair of basis charge densities p1(r) and p2(r) px k p = Jp2(j) (r, f)d3r, where tp(r, t), given by Equation (1.137), corresponds to p(r) = p2(r). 

The nonequilibrium equivalent of Equation (1.161) can be obtained using two alternative but equivalent schemes (often associated to the names of Pekar and Marcus). The two schemes are characterized by a different partition of the low and fast contributions of the apparent charges, namely we have [34] ... 

The requirement needed to incorporate the solvent effects into a state-specific (multireference) method is fulfilled by using the effective Hamiltonian defined in Equation (1.159). The only specificity to take into account is that in order to calculate Va we have to know the density matrix of the electronic state of interest (see the contribution by Cammi for more details). Such nonlinear character of Va is generally solved through an iterative procedure [35] at each iteration the solvent-induced component of the effective Hamiltonian is computed by exploiting Equation (1.157) with the apparent charges determined from the standard ASC equation with the first order density matrix of the preceding step. At each iteration n the free energy of each state K is obtained as... 

In particular, the inclusion of nonequilibrium effects requires a two-step calculation (i) an equilibrium calculation for the initial electronic state (either ground or excited) from which the slow apparent charges, qs, are obtained and stored for the successive calculation on the final state (ii) a nonequilibrium calculation performed with the interaction potential Va composed by two components ... 

The only unknown term of Equations (1.174) and (1.175) remains the relaxation part of the density matrix, PA (or PAeq) (and the corresponding apparent charges qA or qfn). These quantities can be obtained through the extension of LR approaches to analytical energy gradients here in particular it is worth mentioning the recent formulation... 

Once PA is known we can straightforwardly calculate the corresponding apparent charges qA = q(ex, PA) where... 

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