Potential energy function determination calculations

The explicit expressions used for each of the terms in (16.94) define what is called a molecular-mechanics force field, since the derivatives of the potential-energy function determine the forces on the atoms. A force field contains analytical formulas for the terms in (16.94) and values for all the parameters that occur in these formulas. The MM method is sometimes called the empirical-force-field method. Empirical force fields are used not only for single-molecule molecular-mechanics calculations of energy differences, geometries, and vibrational frequencies, but also for molecular-dynamics simulations of liquids and solutions, where Newton s second law is integrated to follow the motions of atoms with time in systems containing hundreds of molecules. 

One of the important electrochemical interfaces is that between water and liquid mercury. The potential energy functions for modeling liquid metals are, in general, more complex than those suitable for modeling sohds or simple molecular liquids, because the electronic structure of the metal plays an important role in the determination of its structure." However, based on the X-ray structure of liquid mercury, which shows a similarity with the solid a-mercury structure, Heinzinger and co-workers presented a water/Hg potential that is similar in form to the water/Pt potential described earlier. This potential was based on quantum mechanical calculations of the adsorption of a water molecule on a cluster of mercury atoms. ... 

The parameterized, analytical representations of fi, ., fiy, fifi determined in the fitting are in a form suitable for the calculation of the vibronic transition moments V fi V") (a—O, +1), that enter into the expression for the line strength in equation (21). These matrix elements are computed in a manner analogous to that employed for the matrix elements of the potential energy function in Ref. [1]. 

A common and important problem in theoretical chemistry and in condensed matter physics is the calculation of the rate of transitions, for example chemical reactions or diffusion events. In either case, the configuration of atoms is changed in some way during the transition. The interaction between the atoms can be obtained from an (approximate) solution of the Schrodinger equation describing the electrons, or from an otherwise determined potential energy function. Most often, it is sufficient to treat the motion of the atoms using classical mechanics,... 

In practice, the value of the reaction coordinate r is determined from the gas-phase potential energy surface of the complex. Then we use the pair-distribution function for the system (for example, determined by a Monte Carlo simulation) and the intramolecular potential energy Vjatra to calculate the relation between the two rate constants. Alternatively, one may determine the potential of mean force directly in a Monte Carlo simulation. With the example in Fig. 10.2.6 and a reaction coordinate at rj, we see that the potential of mean force is negative, which implies that the rate constant in solution is larger than in the gas phase. Physically, this means that the transition state is more stabilized (has a lower energy) than in the gas phase. If the reaction coordinate is at r, then the potential of mean force is positive and the rate constant in solution is smaller than in the gas phase. 

A method for calculating the barriers to internal rotation has recently been proposed (Scott and Scheraga, 1965). It is based on the concept that the barrier arises from two effects, exchange interactions of electrons in bonds adjacent to the bond about which internal rotation occurs, and nonbonded or van der Waals interactions. The exchange interactions are represented by a periodic function, and the nonbonded interactions either by a Buckingham 6-exp or Lennard-Jones 6-12 potential function, the parameters of which are determined semi-empirically. (The parameters of the nonbonded potential energy functions are discussed in Section VB.)... 

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