The lattice dynamic terms

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

Coming back to equations (104) and (105) we rewrite the terms quadratic in the phonon displacement operators

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

$\textstyle =$

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

(115)

	$\displaystyle \hat \mathbf P_{ij}^T \hat \mathbf P_{ij}$	$\textstyle =$	$\displaystyle \sum_{\alpha=1,2,3} (\hat P_{j}^{\alpha}-\hat P_{i}^{\alpha})^2$
		$\textstyle =$	$\displaystyle \sum_{\alpha=1,2,3} (\hat P_{j}^{\alpha})^2 +(\hat P_{i}^{\alpha})^2- 2 \hat P_{i}^{\alpha} \hat P_{j}^{\alpha}$	(116)

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

	$\displaystyle \hat H_{\rm E}$	$\textstyle =$	$\displaystyle \sum_{i} \hat H_{\rm E}(i)$
	$\displaystyle \hat H_{\rm E}(i)$	$\textstyle \equiv$	$\displaystyle \frac{a_0^2 \hat \mathbf w_i^2}{2 m_i} -\frac{1}{2} \sum_{\alpha\beta} K_{\alpha\beta}(ii) \hat P_{i}^{\alpha}\hat P_{i}^{\beta}a_0^{-2}$	(117)

	$\displaystyle K_{\alpha\beta}(ii)$	$\textstyle \equiv$	$\displaystyle -a_0^2\sum_{j} \frac{R_{ij}^{\alpha}R_{ij}^{\beta}}{\vert\mathbf R_{ij}\vert^2} [c_L(ij)-c_T(ij)] +$
			$\displaystyle + \delta_{\alpha\beta} c_T(ij)$	(118)

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

$\displaystyle \hat H_{\rm int}$

$\textstyle =$

$\displaystyle -\frac{1}{2}\sum_{i\ne j,\alpha\beta} K_{\alpha\beta}(ij) \hat P_{i}^{\alpha}\hat P_{j}^{\beta}a_0^{-2}$

(119)

$\displaystyle K_{\alpha\beta}(ij)$

$\textstyle \equiv$

$\displaystyle a_0^2\frac{R_{ij}^{\alpha}R_{ij}^{\beta}}{\vert\mathbf R_{ij}\vert^2}[c_L(ij)-c_T(ij)] +a_0^2\delta_{\alpha\beta}c_T(ij)$