Mean force field approximation

At least two obvious shortcomings of (truncated) anharmonic force fields must also be emphasized. Anharmonic force field approximations to PESs do not have accurate asymptotic (dissociation) behavior, even if improvements are made on the choice of the expansion coordinates (e.g., if Morse variables are employed instead of simple bond stretchings). Furthermore, as all specific boundary conditions cannot be imposed on force field expansions, they do not correctly reflect the symmetry properties of the system under study. This means, among other things, that anharmonic force fields should not be employed for studies on very highly excited vibrational states. 

With a finite value of A(i 0, the interface starts to move. In the mean-field approximation of a similar model, one can obtain the growth rate u as a function of the driving force Afi [49]. For Afi smaller than the critical value Afi the growth rate remains zero the system is metastable. Only above the critical threshold, the velocity increases a.s v and finally... 

In order to perform the calculation., of the conductivity shown here we first performed a calculation of the electronic structure of the material using first-principles techniques. The problem of many electrons interacting with each other was treated in a mean field approximation using the Local Spin Density Approximation (LSDA) which has been shown to be quite accurate for determining electronic densities and interatomic distances and forces. It is also known to reliably describe the magnetic structure of transition metal systems. 

For frequency calculations one usually starts out with a set of approximate existent force constants (e.g. taken over from similar, already treated molecules under the preliminary tentative assumption of transferability), and subsequently varies the force constants in a systematic way by means of a least-squares procedure until the calculated frequencies (square roots of the eigenvalues of Eq. (10)) agree satisfactorily with the experimental values. Clearly, if necessary, the analytical form of the force field is also to be modified in the course of this fitting process. 

Thus, the CMD method is isomorphic to classical time evolution of the phase space centroids on the quantum centroid potential of mean force, Vomd. It should be noted that in the harmonic, classical, and free particle limits, the CMD representation for the QDO [Eq. (50)] is also exact. Furthermore, it should also be noted that the approximation in Eq. (50) does not rely on any kind of mean field approximation. 

Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

This perturbative expression for the attractive force shift is derived from a van der Waals mean field approximation (23). Although the predictions of this model have been found to agree with numerous high pressure vibrational frequency shift measurements (23,25,28), a non-linear attractive force model has recently been suggested to be appropriate for some systems (26,27). 

Binding energies or docking energies are often calculated from minimized strain energies in order to estimate binding strength in macromolecule-substrate interactions. Where such interactions directly involve a metal ion, the approximations likely to have been made in the development of the force field mean that such estimations have questionable validity. 

For this extended IBM, exact solutions are not possible, and we are forced to use a mean field approximation. The system is cranked to generate the high spin states. This HB approximation is simpler than the corresponding Hartree Fock lowest state all the bosons (of each type) are in a single boson condensate rather than in a set of occupied states as in the HF case. Finding the self consistent solution is a simple numerical problem and requires little computer time. The ground state is of the form ... 

Mean field theories of mixed quantum-classical systems are based on approximations that neglect correlations in Ehrenfest s equations of motion for the evolution of the position and momentum operators of the heavy-mass nuclear degrees of freedom. The approximate evolution equations take the form of Newton s equations of motion where the forces that the nuclear degrees of freedom experience involve mean forces determined from the time-evolving wave function of the system. 

The quantum-mechanical force fields of the two different molecules are obtained by means of the same method (the same level of theoretical approximation and the same basis set of one-electron wavefunctions). 

PB equation is based on the mean-field approximation, where ions are treated as continuous fluid-like particles moving independendy in a mean electric potential. The theory ignores the discrete ion properties such as ion size, ion-ion correlation and ion fluctuations. Fail to consider these properties can cause inaccurate predictions for RNA folding, especially in the presence of multivalent ions which are prone to ion correlation due to the strong, long-range Coulomb interactions. For example, PB cannot predict the experimentally observed attractive force between DNA helices in multivalent ion solutions. 

The traditional theory of the double layer is based on a combination of the Poisson equation and Boltzmann distribution. While this involves the approximation that the potential of mean force used in the Boltzmann expression equals the mean value of the electrical potential [9], the results thus obtained are satisfactory at least for 1 1 electrolytes. The equations proposed in the present paper use the approximations inherent in the Poisson—Boltzmann equation, but also include the effect of the polarization field of the solvent which is caused by a polarization source assumed uniformly distributed on the surface and by the double layer itself. 

The traditional double-layer theory combines the Poisson equation with the assumption that the polarization is proportional to the macroscopic electric field, and uses Boltzmann distributions for the concentrations of the ions. The potential of mean force, which should be used in the Boltzmann distribution, is approximated by the mean value of the electrical potential. The macroscopic field E and the polarization P are related via the Poisson equation... 

In the QM/MM method the system is usually a priori divided into QM (the solute) and classical (MM, the solvent) parts, and an effective operator describes the interaction between the two subsystems. The solvent molecules are treated with a classical force field ( classical meaning that there are no elementary particles or quantum effects ) that opens the possibility to take a much larger number of solvent molecules into account. Optionally, the whole system can be embedded in a continuum, e.g., for taking large-range interactions into account. Similar to the continuum approach, the solute is separated from the solvent and its molecular properties are therefore well defined. The remaining problem is to find an accurate approximate representation of... 

In recent years, a number of investigators have studied the phase equilibria of simple fluids in pores by the application of density functional theory. Semina] studies were carried out by Evans and his co-workers (1985,1986). Their approach was considered to be the simplest realistic model for an inhomogeneous three-dimensional fluid . The starting point was a model intrinsic Helmholtz free energy functional F(p), with a mean-field approximation for the attractive forces and hard-sphere repulsion. As explained in Section 7.6, the equilibrium density profile of the fluid in a pore was obtained by minimizing the grand potential functional. 

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