Crystal energy differences

Molecular conformer and crystal lattice energies were computed using Gaussian 03 and Cerius software (Table 3-4). The conformer as well as the lattice energy of the sublimed polymorph is stable relative to the melt form. Whereas there is no rotamer and lattice energy compensation here, crystal energy differences are very... 

It is possible to calculate derivatives of the free energy directly in a simulation, and thereby detennine free energy differences by thenuodynamic integration over a range of state points between die state of interest and one for which we know A exactly (the ideal gas, or hanuonic crystal for example) ... 

We have previously calculated conformational free energy differences for a well-suited model system, the catalytic subunit of cAMP-dependent protein kinase (cAPK), which is the best characterized member of the protein kinase family. It has been crystallized in three different conformations and our main focus was on how ligand binding shifts the equilibrium among these ([Helms and McCammon 1997]). As an example using state-of-the-art computational techniques, we summarize the main conclusions of this study and discuss a variety of methods that may be used to extend this study into the dynamic regime of protein domain motion. 

The parameters in the original parameterization are adjusted in order to reproduce the correct results. These results are generally molecular geometries and energy differences. They may be obtained from various types of experimental results or ah initio calculations. The sources of these correct results can also be a source of error. Ah initio results are only correct to some degree of accuracy. Likewise, crystal structures are influenced by crystal-packing forces. 

Figure 5.2 The modification of the electron energy distribution curve by the presence of diffraction limits in a crystal. The lower filled band is separated from upper unoccupied states in a semiconductor by a small energy difference, so that some electrons can be promoted to conduction by an increase in temperature...

This is achieved by coupling the system to a suitably defined order parameter that is sensitive to the crystal order (the stacking sequence of 111 planes in this case), and doing umbrella sampling with this quantity. The result of the simulation is the free energy difference between both candidate structures—and the winner is fed... 

The different phase behaviors are evidenced in the corresponding free energy diagrams, which have been estimated for both polymers [15]. These diagrams are shown in Fig. 10 (due to the different approximations used in the calculation of the free energy differences, these diagrams are only semiquantitative [15]). It can be seen that the monotropic transition of the crystal in... 

In a previous work we showed that we could reproduce qualitativlely the LMTO-CPA results for the Fe-Co system within a simple spin polarized canonical band model. The structural properties of the Fe-Co alloy can thus be explained from the filling of the d-band. In that work we presented the results in canonical units and we could of course not do any quantitative comparisons. To proceed that work we have here done calculations based on the virtual crystal approximation (VGA). In this approximation each atom in the alloy has the same surrounding neighbours, it is thus not possible to distinguish between random and ordered alloys, but one may analyze the energy difference between different crystal structures. 

Figure 2.Virtual crystal approximation calculations (solid line) compared with coherent potential approximation calculations for Fe-Co (longdashed line), Fe-Ni (dot-dashed line) and Fe-Cu (dashed line). The fcc-bcc energy difference is shown as a function of the atomic number.

Free energy difference between crystal and liquid per unit volume... 

This is an expression for the overall enthalpy and entropy divided by the volume of the complete lamella and is strictly correct. However, because the total free energy difference is calculated, the effects of the unfolded chain ends lying at the surface are implicitly included in AH[Tm(0, p)] and AS[Tra(0, p)] and it is therefore misleading to consider these as bulk per volume quantities. The proportion due to the contribution from the surfaces will be greatest for thinner crystals, that is for lower molecular weight. 

The oxygen atoms of the homo cyclic oxides occupy axial positions in solid S7O and SsO. In fact, recent ab initio MO calculations at the G3X(MP2) level of theory have demonstrated that these isomers are, by 7 and 9 kj mol respectively, more stable than the alternative structures with the oxygen atoms in equatorial positions [89]. In the case of SsO, however, the energy difference is only 1 kJ mol and the conformation with the oxygen atom in the equatorial position is more stable. This result agrees with the observation that SsO exists as two isomeric forms in the solid state depending on the crystallization conditions (see above). 

Color is a spectacular property of coordination complexes. For example, the hexaaqua cations of 3 transition metals display colors ranging from orange through violet (see photo at right). The origin of these colors lies in the d orbital energy differences and can be understood using crystal field theory. 

A three-dimensional (3D) piece of metal can be considered as a crystal of infinite extension in the directions x, y and z with standing waves with the wave numbers k, ky and k, each being occupied with two electrons as a maximum. In a piece of bulk metal the energy differences Sk y are so small that A->0, identical with quasi-free continuously distributed electrons. Since the energy of free electrons varies with the square of the wave numbers, its dependence on k describes a parabola. Figure 4a shows these relations. 

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