Pairwise-additivity interactions

Figure 6.2. Binding isotherms and the average correlation, g(C) - 1 for the tetrahedral (T), square (S), and linear (L) models. The sites are identical and all correlations are due to direct ligand-ligand pairwise additive interactions, (a) Curves for positive cooperativity, S(2) = 10 (b) curves for negative coopera-tivity, S(2) = 0.1. Note that in these systems the cooperativity increases in absolute magnitude from L to S to T.

In Sections 17.2 and 17.3, we have reviewed the QM/MM approach based on the real-space grids [40,41,58,59,60,61,62] and the novel theory of solutions [14,15,16], respectively. As has been suggested, the theory of energy representation is readily applicable to a solute that is quantum chemically described. The present section is devoted to the details of the methodology, referred to as QM/MM-ER, developed by combining the QM/MM approach with the theory of energy representation [19]. The point of the method is to divide the total solvation free energy into the contributions due to the pairwise additive interaction between the solute and the solvent and the residual contribution due to the electron density fluctuation. A focus will be placed on the treatment of the many-body interaction inherent in the quantum chemical object. 

Most of the potential energy surfaces reviewed so far have been based on effective pair potentials. It is assumed that the parameterization is such as to account for nonadditive interactions, but in a nonexplicit way. A simple example is the use of a charge distribution with a dipole moment of 2.ID in the ST2 model. However, it is well known that there are significant non-pairwise additive interactions in liquid water and several attempts have been made to include them explicitly in simulations. Nonadditivity can arise in several ways. We have already discussed induced dipole interactions, which are a consequence of the permanent diple moment and polarizability of the molecules. A second type of nonadditive interaction arises from the deformation of the molecules in a condensed phase. Some contributions from such terms are implicitly included in calculations based on flexible molecule potentials. Other contributions arises from electron correlation, exchange, and similar effects. A good example is the Axilrod-Teller three-body dispersion interaction ... 

The interaction potential in excited states of the HX-Rg clusters has been divided into a non-pairwisc X- -Rgn component and the remaining pairwise additive interactions... 

The concept of an atom-atom potential (Kitaigorodsky, 1973) is based on the idea that the interaction potential between two molecules P and P can be approximated by pairwise additive interactions between the constituent atoms, a P and j8 E P, which, in practice, are nearly always taken to be isotropic, i.e., dependent only on the interatomic distances rap ... 

The response of the interface to an external perturbation is analyzed using the interfacial stress tensor S. For a pairwise-additive interaction, S is given by the Kirkwood formula (Doi and Edwards, 1986 Wijmans and Dickinson, 1999a)... 

B. Y. Ha and A. J. Liu. Effect of non-pairwise-additive interactions on bundles of rodlike polyelectrolytes. Physical Review Letters 81 1011-1014 (1998). 

Fi is the force on particle i caused by the other particles, the dots indicate the second time derivative and m is the molecular mass. The forces on particle i in a conservative system can be written as the gradient of the potential energy, V, C/, with respect to the coordinates of particle /. In most simulation studies, U is written as a sum of pairwise additive interactions, occasionally also three-particle and four-particle interactions are employed. The integration of Eq. (1) has to be done numerically. The simulation proceeds by repeated numerical integration for tens or hundreds of thousands of small time steps. The sequence of these time steps is a set of configurations, all of which have equal probability. The completely deterministic MD simulation scheme is usually performed for a fixed number of particles, iV in a fixed volume V. As the total energy of a conservative system is a constant of motion, the set of configurations are representative points in the microcanonical ensemble. Many variants of these two basic schemes, particularly of the Monte Carlo approach exist (see, e.g.. Ref. 19-23). 

As usual, the internal potential energy U(rN) of the system is calculated from a specified intersite potential-energy function, generally by assuming pairwise additive interactions. The kinetic energy of the system is given by... 

Strictly taken, a prerequisite for the discussion of cooperativity or nonadditivity requires the definition of the additive or noncooperative case [50]. Generally, in the field of intermolecular interaction, the additive model is a model based on the concept of pairwise additive interactions. For atomic clusters per definition, but also for molecular clusters, the use of pairwise additive interactions is almost always used in combination with the assumption of structurally frozen interaction partners. Even in cases of much stronger intermolecular interactions the concept of pair potentials modified to that of effective pair potentials is often used. Most of the molecular dynamics calculations of liquids and molecular solids take advantage of this concept. 

The third virial coefficient B T) is more difficult to determine experimentally and far more difficult to compute. For a system with pairwise additive interactions, the third virial coefficient is expressed in terms of the pair potential as ... 

Equation (3.24) is the reduced Liouville equation for pairwise additive interaction forces. Note that this is an integro-differential equation, where the evolution of the fs distribution depends on the next higher-order fs+i distribution. This is known as the BBGKY hierarchy (named after its originators Bogoliubov, Bom, Green, Kirkwood, and Yvon see the Further Reading section at the end of this chapter). 

The classical fluid systems are characterized with simplified Hamiltonian in which the semiempirical pairwise-additive interaction between two particles are included. Those semiempirical interactions substantially arise from the Pauh exclusion of two electrons at the same quantum state and from the electrostatic interactions among electrons and nuclei. As a result, both repulsion and attraction appear in the two-body interaction. Specifically, when all involved particles are spherical ones like atoms, ions, or coarsegrained beads, the systems are called simple fluids. Obviously, the pair interactions in simple fluid systems are simply distance-dependent. Toward the investigation of simple fluid systems, atomic DFT is developed. A notable merit of atomic DFT is that the contributions to the free energy functional from different interaction parts can be treated separately. To demonstrate, below we present the DFT investigations for the simple systems of HS fluids, LJ fluids, and charged systems. 

Fully atomistic molecular dynamics simulations quickly require excessive computer time as systems become large. This is despite the fact that the computational effort for large n, governed by the calculation of the nonbonded forces or interactions, scales as 0(n) for short-range interactions and as 0(n in n) for long-range interactions. This improvement over the naively expected 0(n ) behavior (assuming simple pairwise additive interactions) is achieved by the use of various cell techniques, which are discussed in this book. Actually, a more severe restriction than the limited... 

For molecules having internal rotational degrees of freedom (say, polymers), the expression for the chemical potential should be modified (even with pairwise additive interactions) to take into account all possible conformations of the molecules. In particular, the rotational partition function of the molecules (included in q) might be different for different conformations and therefore should be properly averaged. We shall discuss a simple case of such molecules in section 6.15. More complex molecules are treated in Chapter 8. 

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