Pair potentials range dependence

The choice of new complexes was guided by some simple considerations. The overall eel efficiency of any compound is the product of the photoluminescence quantum yield and the efficiency of excited state formation. This latter parameter is difficult to evaluate. It may be very small depending on many factors. An irreversible decomposition of the primary redox pair can compete with back electron transfer. This back electron transfer could favor the formation of ground state products even if excited state formation is energy sufficient (13,14,38,39). Taking into account these possibilities we selected complexes which show an intense photoluminescence (0 > 0.01) in order to increase the probability for detection of eel. In addition, the choice of suitable complexes was also based on the expectation that reduction and oxidation would occur in an appropriate potential range. 

The direct interaction depends on the distance between the ligands and has the same range as the ligand-ligand pair potential. The indirect correlation, in this particular model, is independent of the ligand-ligand distance. It does depend on... 

This equation can be derived from potential theory. The entropy and enthalpy changes as functions of the loading state are the prime differentiators for various sorbent/sorbate pairs. These loading dependencies are indicated by the form of the functions AS(x) and AH(x). Here the dependencies are written strictly as functions of the loading (x) only. There may some modest temperature dependency as well. The heat and entropy changes with temperature tend to be small hence the universal form tends to be linear in reciprocal temperature over a wide range of temperatures. 

The mean square torque is another test of the pair potentials used. The calculated mean square torques are very potential dependent they range from 7 x 10"31 to 36 x 10"28 (dyne-cm)2. The experimental values51 of the mean square torque in solid CO at 68°K and in liquid CO at 77.5°K are 19 x 10 28 and 21 x 10 28 (dyne-cm)2, respectively. Therefore, the Stockmayer potential clearly does not represent the noncentral forces in liquid CO, i.e., this potential is much too weak. On the other hand, the noncentral part of the modified Stockmayer potential is too strong. However, as pointed out previously, this problem can easily be solved by using a smaller quadrupole moment. The mean square torques from the other two potentials agree quite favorably with the experimental values. We conclude from the above that the quadrupole-quadrupole interaction can easily account for observed mean square torques in liquid CO. 

In order to improve the description of the atomic arrangement, according to the authors, an optimization consists in determining an adequate short-range part, msr( ), and long-range part, lr(r), so that u(r) = msr( ) + mlr(V). To achieve this, they proposed a partition of the pair potential so-called optimized division scheme (ODS), that contrary to DHHDS is not density dependent, for which... 

Note that the expressions we have written thus far are entirely generic. All that we have assumed is the existence of a total energy function that is dependent upon the set of ionic positions. Our analysis is thus indifferent to the microscopic energy functional used to compute Etot and hence the results are equally valid for total energy methods ranging from pair potentials to fully quantum mechanical Etot. On... 

As an example of the capabilities of EXAFS spectroscopy, the mean Cd-S distances as a function of the diameters of a series of CdS nanocrystals are depicted in Figure 3.15 [163]. These data are gained from a thorough temperature-depen-dent study of the size dependence of various structural and dynamic properties of CdS nanoparticles ranging in size from 1.3 to 12.0 nm. The properties studied include the static and the dynamic mean-square relative displacement, the asymmetry of the interatomic Cd-S pair potential, with conclusions drawn as to the crystal structure of the nanoparticles, the Debye temperatures, and the Cd-S bond lengths. As seen from Figure 3.15, the thiol-stabilized particles (samples 1-7) show an expansion of the mean Cd-S distance, whereas the phosphate-stabilized particles (samples 8-10) are slightly contracted with respect to the CdS bulk values. 

Similar though possibly less dramatic difficulties will occur for other models. For polar molecules the jump discontinuity in the dipole s potential field will lead to a different angular distribution for particles just within and just outside the cutoff sphere. That is, the angular pair correlation functions will be discontinuous at that distance, and this will lead to a jump in the radial pair function and to an unrealistic energy. For Lennard-Jones molecules there will be a jump in the radial function g2(/ ), but it will be small. The magnitudes of the jumps in the pair functions depend exponentially on the effective jump in the pair potential due to truncation, and so are strongly dependent on the ranges of the forces and on the cutoff diameter. 

This part of the pair potential is added mainly to account for the long-range interactions between the two molecules. It should be noted that the distance dependence of the dipole-dipole interaction is valid for fixed orientations of the dipoles. The average dipole-dipole interaction at large distance has an dependence. 

The most interesting behavior, which is a direct result of the particular pair potential (2.5.1), is for pressures in the range 4 < P < 9 and temperatures in the range 2 < T < 5. In this range we observe a negative temperature dependence of the volume. This behavior is unique to liquid water and it is quite surprising... 

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