Equilibrium, position

The relative magnitude of the coherent and incoherent structure factors mainly depends on the nature of the nuclei. For almost all the atoms bcoh inc of the principal exceptions is the H atom for which bcoh = -3.74 fm, bine = 25.22 fm. Because of that the incoherent signal of protons is 50 times larger than the coherent one. For structural analysis, i.e. for the determination of the equilibrium positions of the sample atoms, it is preferable to work with deuterated samples. In contrast dynamical measurements of individual guests require a highly incoherent scatterer and hydrogen is the best candidate. 

Neutron diffraction is used for the determination of the equilibrium positions and space ordering of the atoms of the sample, i.e. the host and the guest. The best results will be obtained for coherent scatterers. Therefore deuterated species are preferable instead of protonated guest molecules. 

The diffraction pattern of the loaded zeolite corresponds to the positions of the N ions of the unit cell of the host plus the M atoms of the guest molecule. The intensity of a Bragg peak is  

C is the normalisation factor, 0 the scattering angle, W the temperature (or Debye-Waller) factor, hkl the structure factor of the hkl reflection. 

The diffraction patterns are analysed by standard refinement methods developed for X-ray scattering one determines the atom position inside the unit-cell which gives the best fit between the experimental and the calculated spectrum. In the Rietveld method [11], the fit is done on the complete line profiles if the resolution of the spectrometer is not well known, or if the Bragg peaks are broadened for physical reasons, then only the integrated intensities of each peak can be used. 

---

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... 

In this section we concentrate on the electronic and vibrational parts of the wavefimctions. It is convenient to treat the nuclear configuration in temis of nomial coordinates describing the displacements from the equilibrium position. We call these nuclear nomial coordinates Q- and use the symbol Q without a subscript to designate the whole set. Similarly, the symbol v. designates the coordinates of the th electron and v the whole set of electronic coordinates. We also use subscripts 1 and ii to designate the lower and upper electronic states of a transition, and subscripts a and b to number the vibrational states in the respective electronic states. The total wavefiinction f can be written... 

Here P j (0) is the value at the equilibrium position of the initial electronic state. 

Figure Bl.1.1. (a) Potential curves for two states with little or no difference in the equilibrium position of tire upper and lower states. A ttansition of O2, witli displacement only 0.02 A, is shown as an example. Data taken from [11]. Most of the mtensity is in the 0-0 vibrational band with a small intensity in the 1-0 band, (b) Potential curves for two states with a large difference in the equilibrium position of the two states. A ttansition in I2, with a displacement of 0.36 A, is shown as an example. Many vibrational peaks are observed.

The molecular dipole moment (not the transition dipole moment) is given as a Taylor series expansion about the equilibrium position... 

While the Lorentz model only allows for a restoring force that is linear in the displacement of an electron from its equilibrium position, the anliannonic oscillator model includes the more general case of a force that varies in a nonlinear fashion with displacement. This is relevant when tire displacement of the electron becomes significant under strong drivmg fields, the regime of nonlinear optics. Treating this problem in one dimension, we may write an appropriate classical equation of motion for the displacement, v, of the electron from equilibrium as... 

Equation (Bl.8.6) assumes that all unit cells really are identical and that the atoms are fixed hi their equilibrium positions. In real crystals at finite temperatures, however, atoms oscillate about their mean positions and also may be displaced from their average positions because of, for example, chemical inlioniogeneity. The effect of this is, to a first approximation, to modify the atomic scattering factor by a convolution of p(r) with a trivariate Gaussian density function, resulting in the multiplication ofy ([Pg.1366]

The lattice atoms in the simulation are assumed to vibrate independently of one another. The displacements from the equilibrium positions of the lattice atoms are taken as a Gaussian distribution, such as... 

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. 

For the equilibrium M(s) M (aq) + 2e, it might then be (correctly) assumed that the equilibrium for copper is further to the left than for zinc, i.e. copper has less tendency to form ions in solution than has zinc. The position of equilibrium (which depends also on temperature and concentration) is related to the relative reducing powers of the metals when two different metals in solutions of their ions are connected (as shown in Figure 4.1 for the copper-zinc cell) a potential difference is noted because of the differing equilibrium positions. 

We have seen that the energetic feasibility of a reaction can be deduced from redox potential data. It is also possible to deduce the theoretical equilibrium position for a reaction. In Chapter 3 we saw that when AG = 0 the system is at equilibrium. Since AG = — nFE. this means that the potential of the cell must be zero. Consider once again the reaction... 

These are similar to those of the alkali metals but are rather less soluble in water. However, calcium sulphide, for example, is not precipitated by addition of sulphide ions to a solution of a calcium salt, since in acid solution the equilibrium position... 

In an early study of lysozyme ([McCammon et al. 1976]), the two domains of this protein were assumed to be rigid, and the hinge-bending motion in the presence of solvent was described by the Langevin equation for a damped harmonic oscillator. The angular displacement 0 from the equilibrium position is thus governed by... 

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 ... 

It is a property of this family of differential equations that the sum or difference of two solutions is a solution and that a constant (including the constant i = / ) times a solution is also a solution. This accounts for the acceptability of forms like A (t) = Acoscot, where the constant A is an amplitude factor governing the maximum excursion of the mass away from its equilibrium position. The exponential form comes from Euler s equation... 

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. 

Figure 4-5 The Geometry of Ethylene with all Atoms at their Equilibrium Positions in the MM3 Eorce Eield.

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... 

HyperChem models the vibrations of a molecule as a set of N point masses (the nuclei of the atoms) with each vibrating about its equilibrium (optimized) position. The equilibrium positions are determined by solving the electronic Schrodinger equation. 

The vibrational potential may be expanded in a Taylor series about the equilibrium positions of the atoms. 

Dyne, unit of force dyn Equilibrium position vector Ro... 

---