[Wien] Density matrix and QTL

[David Tompsett]

Dear All,

I am trying to understand how QTL works in performing its spectral
decomposition of the density matrix. I could not find QTL - Technical report
on the web anywhere so I hope you can help me.

If we have a d-orbital density matrix, then the matrix is a 5x5
corresponding to matrix elements:
n_mm' = SUM_E<Ef   < KS_k | phi_m> < phi_m' | KS_k>.

Here, KS_k is a Kohn-Sham orbital and phi_m is a predefined orbital. Here
with angular momentum m. I presume the quantization axis for m is set by the
local rotation matrix? The summation is over energies less than the Fermi
energy.

Then if we do a spectral decomposition for the QTL partial density of states
we rewrite this matrix as:
n_mm' = ( V * D * V^-1 )_mm'
Here, D is a diagonal matrix and V is another square matrix whose columns
should be the eigenvectors. We are now in a Hilbert space in which the
occupations are diagonal.
What I don't understand then is how these new eigenvectors are assigned to
be dxy, dxz etc by QTL? Is it via a following projection? With respect to
what spatial axis is the z-axis defined (still by the local rotation matrix
in the struct file)?
Also, if we set our own z-axis (and maybe x) for QTL does this simply alter
the following projection?

Finally, if it is easy to get the occupation matrix in a diagonal basis, why
do codes use the rotationally invariant form?

Many thanks,
David.
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