Often the Born-Lande equation is quoted (alongwith the calculation of the Madelung constant and Born exponent) with reference to rock-salt structure. But what if we take some other crystal, like rutile $\textrm{TiO}_2$. The Coulomb interaction energy for a reference $\text{Ti(IV)}$ ion is now: $$ U_{E} = \frac {e^2}{4π\epsilon_0} \sum_{\text{all surrounding ions}} \left( \frac {16}{r_{\text{Ti}^{4+}}} - \frac {8}{r_{\text{O}^{2-}}} \right) $$ I'm unable to figure out how does one get the Born-Lande equation using this expression for the electrostatic energy, that for the Pauli repulsion remaining as it is (that is, $U_P = \sum A/r_k^n$). If it were a rock-salt structure, we would say, lattice energy is, $$ U_{\text{lattice}} = \sum_{\text{entire lattice}} \left( U_E + U_P \right) $$ And after a little mathematical manipulation we would get, $$ U_{\text{lattice}} = -\frac{N_A Mz^2e^2}{4πR_0\epsilon_0} \left(1 - \frac1n\right) $$ However this doesn't seem proper for the case of rutile (and other cases too, like perovskite). Any suggestions are welcome.



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