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| | '''Kugel–Khomskii coupling''' describes a [[coupling]] between the [[rotation|spin]] and [[Atomic orbital|orbital]] [[Degrees of freedom (physics and chemistry)|degrees of freedom]] in a solid; it is named after the Russian physicists [[Kliment I. Kugel]] (Климент Ильич Кугель) and [[Daniel I. Khomskii]] (Daniil I. Khomskii, Даниил И. Хомский). The [[Hamiltonian (quantum mechanics)|Hamiltonian]] used is: |
![H={\frac {t^{2}}{U}}\sum _{{\langle i,j\rangle }}\left[4\left({\overrightarrow {S_{i}}}\cdot {\overrightarrow {S_{j}}}\right)(\tau _{i}^{{\alpha }}-{\frac {1}{2}})(\tau _{j}^{{\alpha }}-{\frac {1}{2}})+(\tau _{i}^{{\alpha }}+{\frac {1}{2}})(\tau _{j}^{{\alpha }}+{\frac {1}{2}})-1\right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/642945e76f089a36ef99f5ce96afd04a8482ad88)