kirk0830
commented
10 months ago The bad condition number of QO in present implementation may result from wrong selection of subspace to project AO, I mean:
$\langle Q\mu(\mathbf{k})|\hat{S}|Q\nu(\mathbf{k})\rangle$ , is the overlap matrix ill-conditioned, or say QO overlap in subspace representation. To see this, in QO paper the QO is defined as:
$|Q{\nu}(\mathbf{k})\rangle =\sum{\beta}{|\varphi{\beta}(\mathbf{k})\rangle\langle\varphi{\beta}(\mathbf{k})|A_{\nu}(\mathbf{k})\rangle}$
Thus the overlap is:
$\langle Q{\mu}(\mathbf{k})|\hat{S}|Q{\nu}(\mathbf{k})\rangle =\sum{\alpha\beta}{\langle A{\mu}(\mathbf{k})|\varphi{\alpha}(\mathbf{k})\rangle\langle \varphi{\alpha}(\mathbf{k})|\hat{S}|\varphi{\beta}(\mathbf{k})\rangle\langle \varphi{\beta}(\mathbf{k})|A_{\nu}(\mathbf{k})\rangle}$
With orthonormality $\langle \varphi{\alpha}(\mathbf{k})|\hat{S}|\varphi{\beta}(\mathbf{k})\rangle =\delta_{\alpha\beta}$, we have:
$\langle Q{\mu}(\mathbf{k})|\hat{S}|Q{\nu}(\mathbf{k})\rangle =\sum{\alpha}{\langle A{\mu}(\mathbf{k})|\varphi{\alpha}(\mathbf{k})\rangle\langle\varphi{\alpha}(\mathbf{k})|A_{\nu}(\mathbf{k})\rangle}$
One can find projection operator $\sum\alpha|\varphi{\alpha}(\mathbf{k})\rangle\langle \varphi_{\alpha}(\mathbf{k})|$, therefore QO's overlap depends on selection of subspace to project on.
Details
Just a note leave here, for some k-point, QO has very bad condition number, these will bring about problem when doing kpoint extrapolation. For many of kpoints, QO can reproduce band structures well (by selecting 4-th to 8-th bands to reproduce):
But for other kpoints, condition numbers are very bad:
Detailed information
qo.log