Wave function solute

Equation (15) is solved self-consistently employing the FLAPW method. Using the solutions, wave functions and energies, momentum densities in Equation (8) are calculated. In this step, one more drastic approximation we are going to make is that the occupation number in Equation (10) is replaced by the step function... 

The outline of this review is as follows. In Sec.2, we highlight the fundamental equations and structure of the theory Sec.2.1 motivates the choice or the functional form of the solute wave function Sec.2.2 explains the equation for the free energy of the solute plus solvent system in the nonequilibrium solvation regime Sec.2.3 discusses the corresponding Schrodinger... 

Continuum solvation models consider the solvent as a homogeneous, isotropic, linear dielectric medium [104], The solute is considered to occupy a cavity in this medium. The ability of a bulk dielectric medium to be polarized and hence to exert an electric field back on the solute (this field is called the reaction field) is determined by the dielectric constant. The dielectric constant depends on the frequency of the applied field, and for equilibrium solvation we use the static dielectric constant that corresponds to a slowly changing field. In order to obtain accurate results, the solute charge distribution should be optimized in the presence of the field (the reaction field) exerted back on the solute by the dielectric medium. This is usually done by a quantum mechanical molecular orbital calculation called a self-consistent reaction field (SCRF) calculation, which is iterative since the reaction field depends on the distortion of the solute wave function and vice versa. While the assumption of linear homogeneous response is adequate for the solvent molecules at distant positions, it is a poor representation for the solute-solvent interaction in the first solvation shell. In this case, the solute sees the atomic-scale charge distribution of the solvent molecules and polarizes nonlinearly and system specifically on an atomic scale (see Figure 3.9). More generally, one could say that the breakdown of the linear response approximation is connected with the fact that the liquid medium is structured [105],... 

As for the QM/MM description also for PCM, non-electrostatic (or van der Walls) terms can be added to the Vent operator in this case, besides the dispersion and repulsion terms, a new term has to be considered, namely the energy required to build a cavity of the proper shape and dimension in the continuum dielectric. This further continuum-specific term is generally indicated as cavitation. Generally all the non-electrostatic terms are expressed using empirical expressions and thus their effect is only on the energy and not on the solute wave function. As a matter of fact, dispersion and repulsion effects can be (and have been) described at a PCM-QM level and included in the solute-effective Hamiltonian 7/eff as two new operators modifying the SCRF scheme. Their definition can be found in Ref. [17] while a recent systematic comparison of these contributions determined either using the QM or the classical methods is reported in Ref. [18]... 

To proceed further we recall that AG can be decomposed into an electrostatic contribution (AG ei) corresponding to solute-solvent interactions with the wave function already polarized by the solvent and into the polarization work (AG poi) needed to polarize the solute wave function from its optimum value in vacuo. Of course only the first term can be dissected into contributions originating from different spheres of the cavity. Table 8 shows this dissection for both the conformers obtained at HF/6-3l4G(d) level the largest contributions to the solvation energies are due to the ionized moieties of the zwitterion, which are more exposed to the solvent in the anti conformer than in its gauche counterpart. 

The hcc s obtained at the B3LYP/EPR-II level are shown in Table 12. The calculated hcc s can be dissected into three terms a contribution due to the electronic and structural configurations assumed by the radicals in the gas phase (first column in Table 12) a contribution due to the solvent-induced polarization on the solute wave function without allowing any relaxation of the gas-phase geometry (direct solvent effect, second coliunn in Table 12), and a last contribution due to the solvent-induced geometry relaxation (indirect solvent effect, third column in Table 12). 

The Fock eq.(lO) is solved with the same iterative procedure of the problem in vacuo the only difference introduced by the presence of the continuum dielectric is that, at each SCF cycle, one has to simultaneously solve the st2ui-daxd quantum mechanical problem and the additional electrostatic problem of the evaluation of the interaction matrices, and hence of the apparent charges. The latter are obtained from eq.(23) through a self-consistent technique which has to be nested to that determining the solute wave function, in fact has to be recomputed at each SCF cycle as a consequence, in each cycle, and a fortiori at the convergency, solute and solvent distribution charges are mutually equilibrated. 

It is clear from Eq. (7.21) that the solute wave function Em depends on the reaction field, which in turn depends on Em through the solute charge distribution Hence, Eq. (7.21) has to be solved iteratively. 

The PCM method has been reformulated to eliminate the iterative calculation of the solute s wave function in solution in this reformulation, the mutually consistent solute wave function in solution and the interaction operator are found directly in a single SCF cycle, thereby speeding up the calculations [M. Cossi et al., Chem. Phys. Lett., 2SS, 327(1996)]. 

In the standard original model the perturbation is limited to the electrostatic effects (i.e., the electrostatic interaction between the apparent point charges and the solute charge distribution) however, extensions to include dispersion and repulsion effects have been formulated. In this more general context the operator can be thus partitioned in three terms (electrostatic, repulsive and dispersive), which all together contribute to modify the solute wave function. 

In general, it is assumed that has only a small effect on the solute wave-function and therefore it is usual to represent it through a classical potential that depends only on the nuclear coordinates but not on the electron ones. If this is the case, and for a given configuration of the classical subsystem, the term can simply be added to the final value of the energy as a constant. 

As mentioned in the introduction, elimination of the explicit (atomistic) part of the solvent leads to the conventional polarisable continuum model, in which the solute creates a cavity in the bulk (continuous) solvent, whose shape follows the movement of solute atoms and whose reaction field responds to the electrostatic potential created by the solute wave-function. Note that under such circumstances the solute electron density penetrates the solvent boundary originating the so-caUed escaped charge effects, whose treatment requires more sophisticate descriptions of electrostatic contributions (e.g. the so-called integral equations formalism, lEF [12, 13, 15]). 

This is the key point in the whole SCRF procedure. It gives as output the electrostatic free energy Gei, and the solute wave function , from which other properties of M can be derived. The computational problem can be solved with an iterative procedure where both the wave function I (or equivalently the electronic charge distribution and the interaction... 

The minimization of the free energy, in the framework of the expansion of the solute wave function over a finite basis set (x), allows one to derive a set of self-consistent field equations in which the (pv) element of the Hartree-Fock operator, is deduced from the correspondent element in the case of an isolated molecule by the equation ... 

Ail the above choices present some problems. The definition of the diabatic states is complicated, and the associated solvent coordinate is only valid if the solute wave function may be written as a linear combination of the two diabatic states. If more complex wave functions are used (Cl, for instance), a larger set of solvent coordinates must be introduced. In this case it is necessary to consider as many solvent coordinates as electronic configurations. Anyway, we cannot forget to recall that just this diabatic states description has more recently permitted a very interesting development of the continuum solvent methods with the introduction of a full quantization of the solvent electronic polarization in the work of Kim and Hynes, ... 

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