Solute-solvent interaction Hamiltonian

In a hybrid DFT/MM approach, interactions with the solvent molecules are introduced into the Kohn-Sham equations through the external potential v(r), which is formally defined in the same way as the solute—solvent interaction Hamiltonian Hxs in Eq. [4] v(r) = J x -... 

The implementation of the QM/MM solute-solvent interaction Hamiltonian (Eq. [4]) within an HF MO scheme involves introduction of the one-elearon integral over the operator 1/r, into the Fbck matrix (Eq. [9]). Note that three terms in Eq. [4], whidi do not involve positions of die solute electrons, are treated classically. The one-electron integrals have the general form ... 

Considering, for simplicity, only electrostatic interactions, one may write the solute-solvent interaction temi of the Hamiltonian for a solute molecule surrounded by S solvent molecules as... 

Now knowing how to evaluate solvation-free energies, we are ready to explore the effect of the solvent on the potential surface of the reacting solute atoms. Adapting the EVB approach we can describe the reaction by including the solute-solvent interaction in the diagonal elements of the solute Hamiltonian, using... 

An alternative approach to calculating the free energy of solvation is to carry out simulations corresponding to the two vertical arrows in the thermodynamic cycle in Fig. 2.6. The transformation to nothing should not be taken literally -this means that the perturbed Hamiltonian contains not only terms responsible for solute-solvent interactions - viz. for the right vertical arrow - but also all the terms that involve intramolecular interactions in the solute. If they vanish, the solvent is reduced to a collection of noninteracting atoms. In this sense, it disappears or is annihilated from both the solution and the gas phase. For this reason, the corresponding computational scheme is called double annihilation. Calculations of... 

Here Ha and Hb are the Hamiltonians of the isolated reactant molecules, Hso is the Hamiltonian of the pure solvent, and Vmt is the interaction energy between reactants and between reactant and solvent molecules, i.e., it contains the solute-solute as well as the solute-solvent interactions, qa and reactant molecules A and B, respectively, and pa and pb are the conjugated momenta. If there are na atoms in molecule A and tib atoms in molecule B, then there will be, respectively, 3ua coordinates c/a and 3rt j coordinates c/b Similarly, R are the coordinates for the solvent molecules and P are the conjugated momenta. In the second line of the equation, we have partitioned the Hamiltonians Hi into a kinetic energy part T) and a potential energy part V). 

Hdl is composed by two terms, the Hamiltonian of the solute //(j, (i.e., the molecular part M of the continuum model) and the solute-solvent interaction term Vm[ ... 

Summing up, the structure of the effective Hamiltonian of Equation (1.107) makes explicit the nonlinear nature of the QM problem, due to the solute-solvent interaction operator depending on the wavefunction, via the expectation value of the electronic density operator. The consequences of the nonlinearity of the QM problem may be essentially reduced to two aspects (i) the necessity of an iterative solution of the Schrodinger Equation (1.107) and (ii) the necessity to introduce a new fundamental energetic quantity, not described by the effective molecular Hamiltonian. The contrast with the corresponding QM problem for an isolated molecule is evident. 

Molecular dynamics (MD) and Monte Carlo (MC) are in most cases associated with a discrete description of the solvent and with classical representations of the solute and/or solvent Hamiltonians. However, the same type of sampling engines can be coupled to continuum methods, which implies a loss of detail in the representation of individual solute-solvent interactions, but present two main advantages (i) calculation can be faster since no explicit sampling of solvent is needed, (ii) sampling efficiency of solute movements can be very high because of the neglect of solvent friction. 

As an example of application of the method we have considered the case of the acrolein molecule in aqueous solution. We have shown how ASEP/MD permits a unified treatment of the absorption, fluorescence, phosphorescence, internal conversion and intersystem crossing processes. Although, in principle, electrostatic, polarization, dispersion and exchange components of the solute-solvent interaction energy are taken into account, only the firsts two terms are included into the molecular Hamiltonian and, hence, affect the solute wavefunction. Dispersion and exchange components are represented through a Lennard-Jones potential that depends only on the nuclear coordinates. The inclusion of the effect of these components on the solute wavefunction is important in order to understand the solvent effect on the red shift of the bands of absorption spectra of non-polar molecules or the disappearance of... 

When the donor-acceptor complex is placed in a solvent, its Hamiltonian changes due to the solute-solvent interaction... 

The statistical average over the electronic degrees of freedom in Eq. [15] is equivalent, in the Drude model, to integration over the induced dipole moments pg and py. The Hamiltonian H, is quadratic in the induced dipoles, and the trace can be calculated exactly as a functional integral over the fluctuating fields pg and The resulting solute-solvent interaction energy... 

In the continuum model, the solvent effect is accountable for in two ways. Either one evaluates the solvation energy by means of explicit formulas derived in the classical theories noted above, or, preferably, one may introduce the term for the solute-solvent interaction directly into the Hamiltonian "" , The latter approach provides not only the solvation energy but also the wave function of the... 

The surrogate Hamiltonian is expressed in terms of renormalized solute-solvent interactions, a feature that leads to a simple and natural linear response description of the solvent dynamics in the vicinity of the solute. In addition to the measurable solvation time correlation function (tcf), we can also calculate observables needed to elucidate the detailed mechanism of solvation response, such as the evolution of the solvent polarization charge density around the solute. 

The solute-solvent interactions are accounted for by a perturbation added to the solute Hamiltonian operator ... 

Self-consistent reaction-field (SCRF) theories are obtained by introducing solute-solvent interactions into the Hamiltonian. (Cf. Tapia, 1982.) Based on Cl wave functions and on reasonable approximation for the cavity radius... 

As we have seen, both models considered in the previous pages lead to the definition of a microscopic portion of the whole liquid system, the larger portion of the liquid being treated differently. We may rationalize this point by introducing, in the quantum mechanical language, an effective Hamiltonian of the subsystem (A B-Sn), where the Hamiltonian of the isolated system M = A-B Ch°M) is supplemented by an effective solute-solvent interaction potential (Vint)-... 

The potential energy surface used in solution, G (R), is related to an effective Hamiltonian containing a solute-solvent interaction term, Vint- In the implementation of the EH-CSD model, that will be examined in Section 6, use is made of the equilibrium solute-solvent potential. There are good reasons to do so however, when the attention is shifted to a dynamical problem, we have to be careful in the definition of Vint - This operator may be formally related to a response function TZ which depends on time. For simplicity s sake, we may replace here TZ with the polarization vector P, which actually is the most important component of TZ (another important contribution is related to Gdis) For the calculation of Gei (see eq.7), we resort to a static value, while for dynamic calculations we have to use a P(t) function quantum electrodynamics offers the theoretical framework for the calculation of P as well as of TZ. The strict quantum electrodynamical approach is not practical, hence one usually resorts to simple naive models. 

The methods today more in use in computational chemistry belong to the ASC, MPE, GB and FD families. For every type of method there are now QM versions, many thus far limited to infinite isotropic distributions. Several of those methods may introduce, via the effective Hamiltonian or with less formal procedures, solute-solvent interaction effects of non electrostatic origin. 

The formulation of the QM continuum models reduces to the definition of an Effective Hamiltonian, i.e. an Hamiltonian to which solute-solvent interactions are added in terms of a solvent reaction potential. This effective Hamiltonian may be obtained from the basic energetic quantity which has the thermodynamic status of free energy for the whole solute-solvent system and for this reason is called free energy functional, This energy... 

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