The lattice dynamic terms

The lattice dynamic terms $H_{E}+H_{int}$

Coming back to equations (80) and (81) we rewrite the terms quadratic in the phonon displacement operators

$\displaystyle (\vec R_{ij}^T \vec P_{ij})^2$

$\textstyle =$

$\displaystyle \sum_{\alpha,\beta=1,2,3} R_{ij}^{\alpha}R_{ij}^{\beta}(P_{j}^{\alpha}-P_{i}^{\alpha})(P_{j}^{\beta}-P_{i}^{\beta})$

(92)

$\displaystyle \vec P_{ij}^T \vec P_{ij}$	$\textstyle =$	$\displaystyle \sum_{\alpha=1,2,3} (P_{j}^{\alpha}-P_{i}^{\alpha})^2$
	$\textstyle =$	$\displaystyle \sum_{\alpha=1,2,3} (P_{j}^{\alpha})^2 +(P_{i}^{\alpha})^2- 2 P_{i}^{\alpha} P_{j}^{\alpha}$	(93)

The single ion Hamiltonian $H_{\rm E}$ is similar to the linear chain, in general

$\displaystyle H_{\rm E}$	$\textstyle =$	$\displaystyle \sum_{i} H_{\rm E}(i)$
$\displaystyle H_{\rm E}(i)$	$\textstyle \equiv$	$\displaystyle \frac{a_0^2 \vec p_i^2}{2 m_i} -\frac{1}{2} \sum_{\alpha} K_{\alpha\beta}(ii) P_{i}^{\alpha}P_{i}^{\beta}a_0^{-2}$	(94)
$\displaystyle K_{\alpha\beta}(ii)$	$\textstyle \equiv$	$\displaystyle -a_0^2\sum_{j} \frac{R_{ij}^{\alpha}R_{ij}^{\beta}}{\vert\vec R_{ij}\vert^2} [c_L(ij)-c_T(ij)] +$
		$\displaystyle + \delta_{\alpha\beta} c_T(ij)$	(95)

and the interaction Hamiltonian $H_{\rm int}$

$\displaystyle H_{\rm int}$	$\textstyle =$	$\displaystyle -\frac{1}{2}\sum_{i\ne j,\alpha\beta} K_{\alpha\beta}(ij) P_{i}^{\alpha}P_{j}^{\beta}a_0^{-2}$	(96)
$\displaystyle K_{\alpha\beta}(ij)$	$\textstyle \equiv$	$\displaystyle a_0^2\frac{R_{ij}^{\alpha}R_{ij}^{\beta}}{\vert\vec R_{ij}\vert^2}[c_L(ij)-c_T(ij)] +a_0^2\delta_{\alpha\beta}c_T(ij)$