The structural energy difference theorem

The energy difference between two competing structure types is usually very small compared to the total binding energy. Moreover, the simplest assumption that the first nearest-neighbour bond length is structure-independent can lead to incorrect predictions. For example, for the degree 

In general, as we have seen, the binding energy can be written as the sum of a bonding and a repulsive contribution, namely 

The structural energy difference theorem states that the energy difference, At/, between two structures is given to first order by 

Proof Consider two structures I and II with equilibrium first nearest-neighbour bond lengths, R 0 and respectively. Then 

Let Ru be the distance at which structure II displays the same repulsive energy as structure, I, at equilibrium, that is 

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The relative stability of these molecules may be predicted using the structural energy difference theorem which we have proved in the previous section. That is, the differences in the total energies per atom are simply the differences in the bond energies provided that the bond lengths have first been adjusted, so that the molecules have identical repulsive energies. We will assume, as in eqn (4.3), that... 

The heat of formation AH may now be found by comparing the binding energy of the AB alloy at its equilibrium nearest-neighbour separation, RqB, with that of the A and elemental transition metals at their equilibrium nearest neighbour distances, Rq and RB, respectively, as shown in Fig. 7.14. We may use the structural energy difference theorem of 4.3 to write down this small energy difference directly as... 

The energy difference between two structure types is then given by the structural energy difference theorem as... 

The structural energy difference theorem requires that we prepare the lattices so that they all display the same repulsive energy. It follows from eqs (8.45)-(8.47) that the repulsive energy per AB unit may be written... 

The relative prepared volumes, which are required to satisfy the structural energy difference theorem, may be obtained from eqn (8.48) by setting A Vxtv = 0. The resultant first-order change in volume Aft is given by... 

Hence, using the structural energy difference theorem compare the energies of the four-atom linear chain, square, and rhombus with that of the tetrahedron for values of the degree of normalized hardness oth =, and 1, respectively. Comment on how the relative stability of... 

The ° mn coefficients are the mean values of the generalized spherical harmonics calculated over the distribution of orientation and are called order parameters. These are the quantities that are measurable experimentally and their determination allows the evaluation of the degree of molecular orientation. Since the different characterization techniques are sensitive to specific energy transitions and/or involve different physical processes, each technique allows the determination of certain D mn parameters as described in the following sections. These techniques often provide information about the orientation of a certain physical quantity (a vector or a tensor) linked to the molecules and not directly to that of the structural unit itself. To convert the distribution of orientation of the measured physical quantity into that of the structural unit, the Legendre addition theorem should be used [1,2]. An example of its application is given for IR spectroscopy in Section 4. 

Then, following the second step of the theorem, we compare the bond energies of the different structures, y, at these prepared bond lengths with that of the tetrahedron, t, that is... 

The ends of CNTs may be either closed or open (Figure lb). Based on Euler s theorem on polyhedra, along with additional knowledge of the minimum energy structure of fullerenes, Fujita et al.6 have shown that any cap at the end of small-radius CNT must contain 6 pentagons isolated from each other. For CNTs with large radius, there are different possibilities of forming caps that satisfy this requirement. 

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