Potential energy, equation

There are forces other than bond stretching forces acting within a typical polyatomic molecule. They include bending forces and interatomic repulsions. Each force adds a dimension to the space. Although the concept of a surface in a many-dimensional space is rather abstract, its application is simple. Each dimension has a potential energy equation that can be solved easily and rapidly by computer. The sum of potential energies from all sources within the molecule is the potential energy of the molecule relative to some arbitrary reference point. A... 

Molecular dynamics simulations calculate future positions and velocities of atoms, based on their current positions and velocities. A simulation first determines the force on each atom (Fj) as a function of time, equal to the negative gradient of the potential energy (equation 21). 

The total energy of the system, called the Hamiltonian, is the sum of the kinetic and potential energies (equation 24). 

The TDE moisture module (of the model) is formulated from three equations (1) the water mass balance equation, (2) the water momentum, (3) the Darcy equation, and (4) other equations such as the surface tension of potential energy equation. The resulting differential equation system describes moisture movement in the soil and is written in a one dimensional, vertical, unsteady, isotropic formulation as ... 

Molecular mechanics force fields rest on four fundamental principles. The first principle is derived from the Bom-Oppenheimer approximation. Electrons have much lower mass than nuclei and move at much greater velocity. The velocity is sufficiently different that the nuclei can be considered stationary on a relative scale. In effect, the electronic and nuclear motions are uncoupled, and they can be treated separately. Unlike quantum mechanics, which is involved in determining the probability of electron distribution, molecular mechanics focuses instead on the location of the nuclei. Based on both theory and experiment, a set of equations are used to account for the electronic-nuclear attraction, nuclear-nuclear repulsion, and covalent bonding. Electrons are not directly taken into account, but they are considered indirectly or implicitly through the use of potential energy equations. This approach creates a mathematical model of molecular structures which is intuitively clear and readily available for fast computations. The set of equations and constants is defined as the force... 

The third principle relates to the set of equations which describe the potential energy surface of the molecule. These potential energy equations, derived primarily from classical physics, contain parameters optimized to obtain the best match between experimental data and/or theoretical results for a training set of compounds. Once the parameters are evaluated for a set of structures (as diverse as possible), they are fixed and then used unmodified for other similar (and usually larger) compounds. As a first approximation, these parameters must be transferable from one structure to another for this method to work and be generally applicable. 

Hamiltonian Equation 4 for potentials characteristic of each of the surface regions. For simplicity we will assume the parameter, y, of the potential energy Equation 8 is the same in each surface region but the parameter, D, giving the depth of the potential well varies. A minor modification of the theory of localized unimolecular adsorption by Hill 14) can then be used to calculate the distribution of ortho-para or isotopic species on a surface in equilibrium with a gaseous mixture of the same species. 

The success of various models rests on the correct choice of the pairwise potential energy equation. In this section we will address the potential equations commonly employed for adsorbates used in pore characterization. 

For systems that do not obey the assumption of pairwise additivity for the potential energy, equation (3.67) becomes invalid. In a formal way, one can derive an analogous relation involving higher order molecular distribution functions. This does not seem to be useful at present. However, in many applications for mixtures, one can retain the general expression (3.55) even... 

The reliability of a molecular mechanics calculation is dependent on the potential energy equations and the numerical values of the parameters that are incorporated into those equations. In general, parameters are not transferable from one force field to another because of the different forms of equations that have been used and because of parameter correlation within a force field. That is, when one is carrying out the parameterization, if one makes some kind of error, or arbitrary decision, regarding one parameter, other parameters in the... 

Calculation is one of the simplest ways to determine the bulk modulus, as experiments are difficult to perform. At the heart of the calculation is an analytical expression for the potential energy of a crystal in terms of interatomic distances and bond angles. The Born-Meyer function. Equation (2.9), is a simple example of such a function. The potential energy is calculated as a function of interatomic distance, bond angles and other parameters included in the potential energy equation. The interatomic... 

Calculations of coordinate bond energies can be made using classical potential energy equations that take into account the attractive and repulsive interactions between charged particles (10) ... 

The 12-6 potential energy equation exhibits a weak attraction at large separation (proportional to r ) and strong repulsion at small separation (proportional to r ), that is the potential energy has a very sharp decrease when the intermolecular distance r increases from 0 to 2 G j. dX which the potential energy is minimum. A... 

Eq. (6.10-la) is the potential energy for the two atoms or molecules of the same type. For two atoms or molecules of different type (say type 1 and type 2), the relevant 12-6 potential energy equation is ... 

The 10-4 potential energy equation has a minimum and its depth is given by ... 

The potential energy equation, when written in terms of the minimum potential energy (eq. 6.10-4), is ... 

A plot of this 10-4 potential energy equation is shown in Figure 6.10-4a. 

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