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The energy of a dipole in an electric field is just the sum of the energies of it's constituents, and is easily seen to be $U = -\mathbf{p} \cdot \mathbf{E}$. From this you can derive the dynamics of the dipole through the generalized forces $-\boldsymbol\nabla_\mathbf{p} U$ and $-\boldsymbol\nabla_\mathbf{x} U$. This assumes that the dipole's constituents ...

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The dipole transition matrix elements $\mathbf{d}_m$ and $\mathbf{d}_n$ are complex-valued vectors which are relatively easy to define. Their "direction" is a mathematical convenience, and it is essentially given by the vector divided by its modulus, for an appropriate interpretation of the latter. Consider first the case of a single molecule, with a ...

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