# (p.240) Appendix 3

# (p.240) Appendix 3

Solution of the network equations

The network equations constitute a set of *N* _{a} − 1 valence sum rule equations (eqn (3.3)) and *N* _{b} − *N* _{a} + 1 loop equations (eqn (3.4)) where the network contains *N* _{a} atoms and *N* _{b} bonds. Alternatively one can use the equivalent Kirchhoff Equations (2.7) and (2.11). One can readily write down *N* _{a} equations of type 3.3 but one of these is redundant since the sum of all atomic valences in the crystal must be zero. There are many more than *N* _{b} − *N* _{a} + 1 possible loops in most bond graphs, but only *N* _{b} − *N* _{a} + 1 are independent. Equations (3.3) and (3.4) thus constitute a set of *N* _{b} equations which is exactly the number needed to solve for the *N* _{b} unknown bond valences, *s.*

These equations can be written using the connectivity matrix *M*:

where *s* is the vector containing the *N* _{b} bond valences and *V* is the vector containing *N* _{a} − 1 atomic valences and *N* _{b} − *N* _{a} + 1 zeros representing the sum of the bond valences around the loops. The solution to this equation is

which can be evaluated using standard matrix inversion methods, providing *M* is known.

The *N* _{a} − 1 equations of the type 3.3 can be easily written down, but selecting the correct set of loop equations is more difficult since the inclusion of even one loop that can be derived from other loops in the matrix makes the matrix singular. The best procedure for selecting the correct loops is to construct a spanning tree, i.e. a graph which contains no loops but in which all the atoms are connected to the tree (Figs. 2.4, 2.5). This leaves *N* _{b} − *N* _{a} + 1 bonds from the bond graph unselected. Each time one of these unselected bonds is added to the tree, a loop is closed. An independent set of *N* _{b} − *N* _{a} + 1 loops can be chosen by ensuring that each loop contains a different loop-closing bond.

A program to solve these equations has been described by Orlov *et al.* (1998). O’Keeffe (1989) has described an alternative method that is suitable for performing the calculation by hand. Rutherford (1990) has presented a way of inverting the matrix that retains the symmetry of the equations by including all *N* _{a} of the equations of type 3.3. Brown (1977) has described a robust iterative technique for solving the equations based on recognizing that eqn (3.4) is an expression of the principle of maximum symmetry (Rule 3.1). In this procedure
(p.241)
an initial set of bond valences is chosen by taking the average of *V\v* for the two terminal atoms, where *V* is the atomic valence and *v* the coordination number. These initial values do not obey the valence sum rule, so the valences of the bonds around each atom are, in turn, either increased or decreased by equal amounts until the valence sum rule around that atom is satisfied. After carrying out this procedure around each atom in the bond graph, each bond has been adjusted twice, once to give the correct valence sum around the cation and once to give the correct valence sum around the anion. The procedure is iterated to convergence which usually requires a number of cycles equal to the number of bond valences to be determined.

Alternatively the network equations can be solved by the method of simultaneous equations which is illustrated here for the case of CaCrF_{5} whose bond graph is shown in Fig. A3.1.

The symmetry of the bond graph shown in Fig. A3.1 can be used to simplify the calculation by recognizing that two bonds joining the same pair of atoms, and symmetry equivalent bonds, must have the same valences. This reduces the number of independent bonds to six, whose valences are given symbolic values *a, b, c, d, e*, and *f.* The valence sum equations (eqn (3.3)) around Ca, Cr, F1, and F2 are then respectively:

The sum equation around F3 is redundant since it is determined by the condition of charge neutrality.

(p.242) Two loop equations (eqn (3.4)) are needed to solve for six unknowns. One possible choice is

One then proceeds to eliminate the variables one at a time. From (A3.5) and (A3.6) one gets

Substituting these into the remaining four equations gives

From (A3.7′) one gets

Substituting this into the remaining three equations gives

From (A3.3″) and (A3.4″) one gets

Substituting this into (A3.8″) gives

hence substituting back one finds

(p.243) These values can be compared with the observed bond fluxes and valences given in Table 3.1.

One problem with the network equations is that they can, on occasion, give rise to negative bond valences which have no physical significance (expect to indicate that, from a chemical point of view, the bond should not exist). Rutherford (1998) has explored the resonance bond model as an alternative to the use of the loop equation (Section 14.4) while Rao and Brown (1998) have suggested using the method of maximum entropy (Section 11.2.2.1).