Average potential energy

A molecular dynamics simulation used for a conformational search can provide a quick assessment of low energy conformers suitable for further analysis. Plot the average potential energy of the molecule at each geometry. This plot may also suggest conformational changes in a molecule. 

The induction (polarization) forces arise from the effeet of a moment in a polar molecule inducing a charge separation in an adjaeent molecule. The average potential energy funetions are... 

Eq. (16.5) shows that U is simply a sum of energies weighted by the probabifity of being in that state, i.e. U is the average (potential) energy of the system. 

Expression (6.71) for the expectation value of may be used to calculate the average potential energy of the electron in the state nlm). The potential energy V r) is given by equation (6.13). Its expectation value is... 

The result depends only on the principal quantum number n, so we may drop the subscript 1. A comparison with equation (6.57) shows that the total energy is equal to one-half of the average potential energy... 

Proceeding conceptually for a moment without these logistical difficulties, once we have determined the density of states we can calculate thermodynamic properties at any temperature of interest. The average potential energy is... 

In the following, we give a brief description on how to evaluate the various terms in Equation 1. A more extensive summary of the computational details typically used in our studies as well as the CPU requirements of the MM/PBSA approach can be found elsewhere.11 simply evaluates the average potential energy of the system using the same force field (e.g. Equation 3 of Cornell et al.)12 as used to propagate the molecular dynamics trajectory. 

Hamilton s principle is equivalent to the statement that for the actual motion the average kinetic energy approaches the average potential energy as closely as possible. If the potential energy is a function of position only the equivalent mathematical statement is... 

The averaged potential energy (V) includes contributions from fluctuations in the charge density at all real frequencies. The fluctuation-dissipation theorem restricts the contributing frequencies to co = -o), but allows for all real co. The effects on the energy are contained in the term defined by... 

If and denote the average potential energy, the average kinetic energy and the average... 

That is average potential energy (which is negative) is numerically equal to twice the average kinetic... 

Note In the above discussion for the hydrogen atom, the average potential energy is negative so... 

In these equations TC is the average kinetic energy (always positive), V is the average potential energy, and W is the total energy, which is a constant. 

Exercise. The physicist s approach would be to expand y(x) in normal modes and apply (5.6), knowing that the average potential energy of each harmonic oscillator is kT. Derive in this way (5.5). 

These expressions depend on approximating the effects of sorbate-sorbate interaction in a multiply occupied cavity as a reduction in free volume because of the finite size of the molecules (factor (1 — sfi/v)), together with a decrease in the average potential energy from the intermolecular attraction (exponential term). 

In practice a molecule with a dipole moment is often mobile. If the dipole is free to rotate and close to a positive charge it tends to rotate until its negative pole points towards the positive charge. On the other hand, thermal fluctuations drive it away from a perfect orientation. On average, a net preferential orientation remains and the dipole is attracted by the monopole. The average potential energy is... 

The average potential energy U = —e2/AtzSqt would double. Since the potential energy is negative this would tend to decrease the total energy. 

Fig. XXIX-5.—Potential energy of an electron in a metallic crystal, for two different distances of separation, illustrating the decrease of average potential energy with decreasing lattice spacing.

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