Iterative mutual interactions

As noted in Chapter 2, computation of charge-charge (or dipole-dipole) terms is a particularly efficient means to evaluate electrostatic interactions because it is pairwise additive. However, a more realistic picture of an actual physical system is one that takes into account the polarization of the system. Thus, different regions in a simulation (e.g., different functional groups, or different atoms) will be characterized by different local polarizabilities, and the local charge moments, by adjusting in an iterative fashion to their mutual interactions, introduce many-body effects into a simulation. 

An important aspect, until now not introduced, is that the solvent apparent charges, and consequently the reaction operator depend on the solute charge, i.e., in the present QM framework, on the wave function they contribute to define. This mutual interactions between and induces a complexity in the problem which can be solved through the standard iterative procedures characterizing the self-consistent (SC) methods. Only for an aspect the calculation in solution has to be distinguished from that in vacuo way the energy functional to be minimized in a variational solution of [8.120] is not the standard functional E but the new free energy functional G... 

Thus we ought also to study the iterated Prisoner s Dilemma when the players have to take turns. The slight modification in such an alternating Prisoner s Dilemma game can affect the interaction to a considerable extent. For instance, if two Tit For Tat players engage in a Prisoner s Dilemma of the usual simultaneous kind, and if one of them defects by mistake, both players will subsequently cooperate and defect in turns. On the other hand, if two Tit For Tat players engage in an alternating Prisoner s Dilemma, and a unilateral defection occurs inadvertently, then the outcome will be an unbroken sequence of mutual defections. 

Again we feel that a rationale for correct results from incorrect input information may be found in the phenomenon of compensating errors such as have already been shown to minimize the effects of differences between arbitrary and ruby N, M, and Q in Eq. (1). It is no less likely that, even where the computer grossly misjudges the AVs, the buffered equilibria may introduce mutual cancellations which also tend to lessen the effects on ruby s pressure and velocity of errors in the various quantities which interact to produce these predictions in the computer s multi-iterative machinations. Thus relatively large differences between ruby values and actual values of pj, Pt (or Vg), T, the Atf/s, CvdT., N, < , Ej—Eo, y,... 

Further, all coils on the core material serve somewhat as both output and input coils, and also have mutual iterative interactions with each other around the loop, coupled by the field-free external A potential and the -field and magnetic flux in the nanocrystalline material flux path acting as the cores of the coils. These interactions also provide gain in the kinetic energy produced in the Drude electron gas, due to the iterative summation work performed on the electrons to increase their energy. When more coils are utilized, the gain is affected correspondingly. 

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. 

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)]. 

Potentials 5 and 6 are the first examples in our lists of anisotropic potentials. The interaction here depends on the mutual orientations of the two dipoles. Other versions of the di-pole-into-a-sphere potentials (not reported in Table 8.5) include induced dipoles and actually belong to a higher level of complexity in the models, because for flieir use there is the need of an iterative loop to fix the local value of F. There are also other similar models simulating liquids in which the location of the dipoles is held fixed at nodes of a regular 3D grid, the hard sphere potential is discarded, and flie optimization only regard orientation and strengfli of the local dipoles. This last type of model is used in combination wifli solutes M described in another, more detailed way. 

The Fock operators of the molecules were modified by the Coulomb potential and, in the case of two H2O molecules, also by the exchange potential of the partner molecule, and iterated until mutually consistent solutions were obtained. The interaction energy can then be expressed in the form... 

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