Equilibrium position, energy

The calculation is made by determining the primary contribution to the surface energy, that of the two separate parts, holding all the atoms in fixed positions. The total energy is reduced by the rearrangement of the surface layer to its equilibrium position as... 

If the total energy associated with the state is equal to the potential energy at the equilibrium position, it follows that... 

Although all of the nuclear coordinates participate in this kinetic energy operator, and in our previous discussions, all of the nuclear coordinates are expanded, with respect to an equivalent position, in power series of the parameter K, here in the specific case of a diatomic molecule, we found that only the R coordinate seems to have an equilibrium position in the molecular fixed coordinates. This means that actually we only have to, or we can only, expand the R coordinate, but not the other coordinates, in the way that... 

In Chapter IX, Liang et al. present an approach, termed as the crude Bom-Oppenheimer approximation, which is based on the Born-Oppen-heimer approximation but employs the straightforward perturbation method. Within their chapter they develop this approximation to become a practical method for computing potential energy surfaces. They show that to carry out different orders of perturbation, the ability to calculate the matrix elements of the derivatives of the Coulomb interaction with respect to nuclear coordinates is essential. For this purpose, they study a diatomic molecule, and by doing that demonstrate the basic skill to compute the relevant matrix elements for the Gaussian basis sets. Finally, they apply this approach to the H2 molecule and show that the calculated equilibrium position and foree constant fit reasonable well those obtained by other approaches. 

As a simple example of a normal mode calculation consider the linear triatomic system ir Figure 5.16. We shall just consider motion along the long axis of the molecule. The displace ments of the atoms from their equilibrium positions along this axis are denoted by It i assumed that the displacements are small compared with the equilibrium values Iq and th( system obeys Hooke s law with bond force constants k. The potential energy is given by ... 

We envision a potential energy surface with minima near the equilibrium positions of the atoms comprising the molecule. The MM model is intended to mimic the many-dimensional potential energy surface of real polyatomic molecules. (MM is little used for very small molecules like diatomies.) Once the potential energy surface iias been established for an MM model by specifying the force constants for all forces operative within the molecule, the calculation can proceed. 

Suppose, for simplicity, that the masses in Fig. 5-lb are the same, tti = m2 = m, and all three springs are the same, but veloeities and displaeements of the masses may not be the same. Let one mass be displaeed by a distance x from its equilibrium position while the other is displaeed by a distanee X2- The only plaee the potential energy V... 

The potential energy 0(z) depends not only on the distance z hut also on the position of the gas molecule in the xy plane parallel to the surface of the solid and distant z from it. For any given position, the adsorption energy will be equal to the value of 0 = 0o minimum of the potential curve (cf. Fig. 1.2), which of course represents the equilibrium position. 

As a system moves from a nonequilibrium to an equilibrium position, AG must change from its initial value to zero. At the same time, the species involved in the reaction undergo a change in their concentrations. The Gibb s free energy, therefore, must be a function of the concentrations of reactants and products. 

Figure 2.7 shows a representation of this situation. The ordinate is an energy axis and the abscissa is called the reaction coordinate and represents the progress of the elementary step. In moving from P to H, the particle simply moves from one equilibrium position to another. In the absence of any external forces, the energy of both the initial and final locations should be the same as shown by the solid line in Fig. 2.7. Between the two minima corresponding to the initial and final positions is the energy barrier arising from the dislodging of the particles neighboring the reaction path from their positions of minimum energy.

Thus far we have discussed the direct mechanism of dissipation, when the reaction coordinate is coupled directly to the continuous spectrum of the bath degrees of freedom. For chemical reactions this situation is rather rare, since low-frequency acoustic phonon modes have much larger wavelengths than the size of the reaction complex, and so they cannot cause a considerable relative displacement of the reactants. The direct mechanism may play an essential role in long-distance electron transfer in dielectric media, when the reorganization energy is created by displacement of equilibrium positions of low-frequency polarization phonons. Another cause of friction may be anharmonicity of solids which leads to multiphonon processes. In particular, the Raman processes may provide small energy losses. 

Estimation of the free-energy change associated with a reaction permits the calcula-aon of the equilibrium position for a reaction and indicates the feasibility of a given chemical process. A positive AG° imposes a limit on the extent to which a reaction can x cur. For example, as can be calculated using Eq. (4.2), a AG° of 1.0 kcal/mol limits conversion to product at equilibrium to 15%. An appreciably negative AG° indicates that e reaction is thermodynamically favorable. 

Figure 4-6. Representation of the magnetization components A/, A/., and A/,. (A) In presence of field without field H. (B) Immediately after absorption of energy from field Hi. (C) After partial relaxation back to the equilibrium position shown in A.

Enantiotopic (NMR), 455 Endergonic. 153 Endergonic reaction, Hammond postulate and, 197-198 Endo stereochemistry, Diels-Alder reaction and, 495 Endothermic, 154 -ene, alkene name ending, 176 Energy difference, equilibrium position and, 122... 

Here, u is the displacement of the /ith molecule from its equilibrium position and M the reduced mass of each molecular site. Second, the electron is described within the frame of the tight-binding approximation, where it is assumed that the effect of the potential at a given site of the one-dimensional chain is limited to its nearest neighbors. In that case, the energy dispersion of the electron is given by... 

In the derivation above, we have included the kinetic energy of the nuclei in the Hamiltonian and considered a stationary state. In Eq. II.3, this term has been neglected, and we have instead assumed that the nuclei have given fixed positions. It has been pointed out by Slater34 that, if the nuclei are not situated in the proper equilibrium positions, the virial theorem will appear in a slightly different form. (A variational derivation has been given by Hirschfelder and Kincaid.11)... 

But, according to Einstein s equation, the kinetic energy of an atom in its equilibrium position at the commencement of fusion is, with assumed linear vibrations ... 

In 1936, de Boer formulated his theory of a stressed bond which, despite its simplicity, still constitutes the basis for most models of chemical reactivity under stress [92], In order to fracture an unstressed bond which, in the absence of any vibration, is approximated by the Morse potential of Fig. 18, an energy D must be supplied. If, however, the bond is under tension due to a constant force feitt pulling on either end, the bond rupture activation energy will be decreased by an amount equivalent to the work performed by the mechanical force over the stretching distance from the equilibrium position. The bond potential energy in the presence of stress is given by ... 

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