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The orbital magnetization in periodic solids has been nicely described by the so-called modern theory of magnetization [1,2,3,4]. $$\tag{1} \mathcal{M}_{orb} = -\frac{1}{2} \Im \sum_{n,\mathbf{k}} f_{n\mathbf{k}} w_{\mathbf{k}} \langle \partial_\mathbf{k} \psi_{n\mathbf{k}} | \times (\mathcal{H}_k+\epsilon_k-2\mu)|\partial_\mathbf{k} \psi_{n\mathbf{k}} \rangle. $$ I'm wondering if we could calculate $\mathcal{M}_{orb}$ in the following way (in atomic units): \begin{align} \mathcal{M}_{orb} &= - \frac{1}{2} \sum_{n,\mathbf{k}} f_{n\mathbf{k}} w_{\mathbf{k}} \langle \psi_{n\mathbf{k}} | \mathbf{r} \times \mathbf{v} | \psi_{n\mathbf{k}} \rangle \tag{2}\\ &= - \frac{1}{2} \sum_{n,\mathbf{k}} f_{n\mathbf{k}} w_{\mathbf{k}} \sum_m \langle \psi_{n\mathbf{k}} | \mathbf{r} |\psi_{m\mathbf{k}}\rangle \times \langle \psi_{m\mathbf{k}} |\mathbf{v} | \psi_{n\mathbf{k}} \rangle \tag{3}\\ &= - \frac{i}{2} \sum_{n,\mathbf{k}} f_{n\mathbf{k}} w_{\mathbf{k}} \sum_m \langle \psi_{n\mathbf{k}} | \mathbf{r} |\psi_{m\mathbf{k}}\rangle \times \langle \psi_{m\mathbf{k}} |\mathbf{r} | \psi_{n\mathbf{k}} \rangle \left( \epsilon_n- \epsilon_m \right ) \tag{4} \end{align} where I have used in Eq. (3) the completeness relation $\sum_m |\psi_{mk}\rangle \langle \psi_{mk}|=1$ for $m$ runs over all the states (should be subjected to convergence study for real calculations) and in Eq. (4) the relation $i\hbar \langle \psi_{m\mathbf{k}} |\mathbf{v} | \psi_{n\mathbf{k}} \rangle =\langle \psi_{m\mathbf{k}} |\mathbf{r} | \psi_{n\mathbf{k}} \rangle (\epsilon_{n\mathbf{k}}-\epsilon_{m\mathbf{k}})$. So the calculation of the orbital magnetization is reduced to evaluate the matrix elements of the dipole operator $\langle \psi_{m\mathbf{k}} |\mathbf{r} | \psi_{n\mathbf{k}} \rangle$, which is well-known and implemented in many periodic DFT codes.

I think the difficulty of Eqs. (2-4) lies in the fact that when $m=n$, the position operator $\mathbf{r}$ is ill-defined, the solution of which should refer to the Berry phase method.

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    $\begingroup$ +1. I just had to change your "split" environment to an "aligned" environment, because you can't label equations in the split environment. It's important to label equations even if you're not referring to them in the question, because someone might want to refer to them in the answer. Please see what I did! $\endgroup$ – Nike Dattani Oct 29 '20 at 22:12
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    $\begingroup$ @NikeDattani I did realize the numbering issue you mentioned, thanks for your help. $\endgroup$ – Xiaoming Wang Oct 29 '20 at 22:35

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