The Chemical Grip

The sum of the n, energies of the lower set can tlien be evaluated with the formula 

One way to derive this is to eliminate the coefficients for the upper set between Eqs. (19-19). This leads to an eigenvalue equation for eigenvalues of the matrix + 2,. W.i,. The sum of eigenvalues (sum of E ) over this lower. set is 

We first show that the second-order term in Eq. (19-20) does not lead to angular forces. We focus on a particular titanium ion. For each of the d states, y, we must sum over the ncigliboring oxygen ions, but these are independent sums. We may have initially constructed our d states as listed in Table 19-2, with respect to some laboratory coordinate system, but we can reexpress them in terms oft/ states defined with respect to a coordinate system with z-axis in the direction of the oxygen ion being considered, The details are not crucial here but will be when we go to fourth order, so the point should be staled carefully. (More detail is given by Rose, 1957, or Weissbluth, 1978.) 

The set of coefficients is a unitary matrix (since within each set the spherical harmonics are orthogonal to each other). Notice, for example, that / = 2 slates are expanded only in / = 2 stales. 

Let us now focus on the sum over y in the second-order term of Eq. (19-20). We take a coordinate system with coordinates O2 and cp2, in which the stales y are cxprcs.sed these slates can be numbered m, with in taking 21 4- 1 values. Let the 

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The same effect can be seen in the zig-zag chain of Fig.. 3-11. It is remarkable that we can compute the angular force constant in that model exactly, as well as in the Bond Orbital Approximation (see Problems 8-1 and 8-2). The results turn out to be identical for the homopolar semiconductors, but for polar semiconductors, the exact solution has a, replaced by . Sokel has shown that the result is not so simple for the tetrahedral solid, but turns out quantitatively to be very close to an dependence. We will also find an ot dependence when we treat tetrahedral solids in terms of the chemical grip in Section I9-F. This suggests the approximation to the full calculation,... 

Let us then isolate the angular terms in the energy from Eq. (19-20). We call all of these the chemical grip, but here we shall include only the c-oriented contributions from among them. At each ion (here, a titanium) we construct vectors to each neighboring ion and write the angle between a pair as 0, (which we called b before). Each pair enters twice and there are two spins so we may write the result in the form... 

Before making application of this formula to tlie perovskites, let us make a brief application to ionic crystals- we summarized the results of this application in Chapter 13 -and to simple tetrahedral solids. In the alkali halides, we focus upon the occupied p states in the halogen ion and calculate the chemical grip associated with interaction of the halogen ion with the alkali, v stales. These arc the same couplings that were included in the calculation of ion softening in Section 14-C. The coupling W2 of Eq. (19-29) becomes the matrix element = 1.84 h /(md ), and 2W i, is to be identified with the = 9.1 h l(nul ) used in Table 14-2. Then (Eq. 19-29) becomes... 

Let us also make an application of the chemical grip to polar covalent solids. This corresponds to beginning in the limit of unit polarity, V [V + = I,... 

By comparison of Eqs. (19-34) and (19-35) we. sec that the calculation based upon the chemical grip corresponds to the form we obtained from consideration of bond orbitals for tetrahedral solids with A equal to 7/12. In particular, we obtain the V2 0c dependence suggested in Eq. (8-15). 

We return finally to the chemical grip based upon the d state in the perovskite structure. We combine Eqs. (19-27) and (19-29) and evaluate the expression for small deviations of from 90° ... 

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