Potential energy of a molecule

A set of equations defining how the potential energy of a molecule varies with the locations of its component atoms. 

FIGURE 4.29 Variation in the potential energy of a molecule as it approaches another molecule. The potential energy rises sharply once the two molecules come into direct contact. 

Here > = kBT, AUa is the binding potential energy of a molecule of type a to... 

The potential energy of a molecule can then be described in terms of the internal coordinates, i.e.,... 

The total electronic potential energy of a molecule depends on the averaged electronic charge density and the nonlocal charge-density susceptibility. The molecule is assumed to be in equilibrium with a radiation bath at temperature T, so that the probability distribution over electronic states is determined by the partition function at T. The electronic potential energy is given exactly by... 

The concepts. All interatomic interactions are modeled with a set of mathematical functions which, when summed over all interactions, gives the potential energy of a molecule. The potential energy functions, the PEFs, contain adjustable parameters which, for a start, are taken from similar work or are merely guessed. 

The adsorption sites on the surface are assumed to be uniform and to bind the adsorbate with an energy e per molecule or E per mole that is, the potential energy of a molecule in the gaseous state is zero, and in the adsorbed state it is — e. Note that this adsorption energy is a characteristic of the interaction between the adsorbed molecules and the adsorbent. As such, it is the same not only for all parts of the surface but also for all degrees of surface coverage. This is equivalent to saying that the adsorbed molecules do not interact with each other. 

The adsorption and diffusion properties of benzene are of immense interest in zeolite research aromatics play important roles in a number of zeolite-catalyzed processes. Theoretical simulations of benzene diffusion first began to be published in the late 1980s. The first studies evaluated and minimized the potential energy of a molecule such as benzene within the channels, a method less computationally demanding than the MD simulations that followed. Most recent studies have used the TST formalism. 

Theoretical prediction of the diffusivity thus depends only on estimating the ratio of partition functions///// and the potential energy of a molecule at the center of the window (Ur). 

The established method for calculating the vibrational frequencies of molecules is the Wilson GF method.27 In this method, the potential energy of a molecule is defined in terms of the force constants by a matrix F, and the kinetic energy, which depends on the geometry of the molecule, is defined by a matrix G. Using the methods of classical mechanics, the following equation may be derived. 

Intermolecular forces are the attractions and repulsions between molecules. All molecules attract one another when they are a few molecular diameters apart. However, they repel one another as soon as their electron clouds come into contact. Figure 4.29 shows how the potential energy of a molecule varies with its distance from a second molecule. At moderate separations, its potential energy is lower than when it is infinitely far away attractions always lower the potential energy of an object. As the molecules come into contact, the potential energy starts to rise, because repulsions always increase the potential energy of an object. 

The derivation which we have given for the barometer formula in Eq. (4.3) can be easily extended to a general potential energy. Let the potential energy of a molecule be . Then the force acting on it is d/ds, where ds is a displacement opposite to the direction in which the force acts. Take a unit cross section of height ds in this direction. Then, as before, we have... 

In a similar way we can consider the face perpendicular to the px axis in the phase space. The component of velocity of a representative point along the px axis in this space is by definition simply the time rate of change of px that is, by Newton s second law, it is the x component of force acting on a molecule. If the potential energy of a molecule is , this component of force is —d/dx. Thus, the number of molecules entering over the face perpendicular to the px axis is... 

The vibrational potential energy of a molecule can be expanded as a function of internal displacement coordinates, qjt in the following way ... 

Though we cannot hope to give a precise picture of the mode of decomposition of a polyatomic molecule, we can represent schematically, as in Fig. X.4, the general behavior of the potential energy of a molecule as a function of configuration. In Fig. X.4 is shown a schematic hyper-... 

The potential energy of a molecule which obeys Flooke s law is obtained by integrating Eq. 2,1-1 ... 

Let us express the potential energy of a molecule as a Taylor expansion in its normal co-ordinates about the origin ... 

The first solution to this problem as given by Debye [1] for the case of dipolar molecules in the gas phase. The potential energy of a molecule with dipole moment pim. field E is equal to... 

Bader RFW (1980) J. Chem. Phys. 73 2781. G(r) is always positive while V(r) is always negative. When integrated over total molecular space, they yield the kinetic and potential energy of a molecule E = jH(r) dr = jG(r) dr + JV(r) dr... 

The conformational potential energy of a molecule is, in general, given by ... 

In other words, the potential energy of a molecule depends on the time-averaged relative locations of its constituent electrons and nuclei. This dependence is expressed by the familiar potential energy curve which serves as an important description of the chemical bond between two atoms. 

As a result of this analysis, the interaction potential energy of a molecule and a macroscopic surface decreases proportional to IT3, instead of D 6 for a molecule-molecule interaction so the decrease is less steep for the former case than the latter. In order to perform this integration and simplification, Hamaker used various assumptions ... 

Now, the potential energy of the liquid may be divided into two parts the potential energy when the molecules are at the cell centers and the mean potential energy for the molecular motion around the cell centers. If the motion of a molecule is simple harmonic, the average potential energy of a molecule corresponding to the motion in any direction is given by (lj2)kT irrespective of the frequencies therefore it has the same value whether the... 

From Eqs. (I.T) and (1.3 ) the potential energy of a molecule at rest within the exterior magnetic field may be written as ... 

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