The band-resolved local density matrix

For every eigenstate, k point and selected atom, FLEUR can store the full local density matrix of the state in the muffin-tin sphere. Quantities that otherwise need a dedicated DOS mode can then be computed afterwards from these matrices: weights for real (cubic) harmonics, t2g/eg or j-resolved weights, weights in a rotated local frame, or spin moments per state. This page describes what exactly is stored. How to run the calculation is described in Band-resolved local density matrices.

Representation in FLEUR

In the muffin-tin sphere of atom $\alpha$, the spin component $s$ of the eigenstate $\nu$ at k point $\vec{k}$ is expanded as

$$\begin{equation} \renewcommand{\vec}[1]{\mathbf{#1}} \psi_{\nu\vec{k}}^{s}(\vec{r}) = \sum_{lm}\sum_{i} c^{s}_{\nu,lmi}\, u^{ls}_{i}(r)\, Y_{lm}(\hat{\vec{r}}), \end{equation}$$

where $Y_{lm}$ are the complex spherical harmonics used throughout FLEUR and $i$ runs over all radial functions of the channel $l$: the solution $u_l$, its energy derivative $\dot{u}_l$ and all local orbitals of this $l$. For a pair of angular momenta $(l,l')$ and a pair of spins $(s,s')$ the stored matrix is

$$\begin{equation} n^{\nu\vec{k}}_{mm'}(ll',ss') = \sum_{ij} c^{s}_{\nu,l'm'i}\,\big(c^{s'}_{\nu,lmj}\big)^{*}\, \langle u^{l's}_{i} | u^{ls'}_{j}\rangle = \langle l'm's\,|\,\hat\rho_{\nu\vec{k}}\,|\,lms'\rangle . \end{equation}$$

The bra carries $(l',m',s)$, the ket $(l,m,s')$. This is the same index convention as the density matrix of DFT+U. Summed over occupied states with their k-point weights $w_{\nu\vec{k}}$, the diagonal blocks give the occupation matrix of the $l$ shell in the sphere, $\sum_{\nu\vec{k}} w_{\nu\vec{k}}\, n^{\nu\vec{k}}(ll,ss)$, including the contributions of the local orbitals.

• Radial overlaps. Blocks with $l=l'$ use the radial overlap integrals of the density calculation. Blocks with $l\neq l'$ use the overlaps of the radial functions of both channels. All overlaps include the small component of the scalar-relativistic radial functions.
• Blocks between different $l$. Only $l<l'$ is stored. The partner block follows from $n(l'l,s's)=n(ll',ss')^{\dagger}$.
• Spin blocks. Non-spin-polarized calculations store the block $11$, collinear magnetic calculations $11$ and $22$, and non-collinear calculations $11$, $22$, $21$ and $12$. In each state, block $12$ is the conjugate transpose of block $21$.
• Charge and moments. The trace of a diagonal block gives the $l$-resolved charge of the state in the sphere, the same quantity as the $l$-resolved weights of the standard DOS. In non-collinear calculations the moments follow as $m_z=\mathrm{Tr}(n_{11}-n_{22})$, $m_x=2\,\mathrm{Re\,Tr}\,n_{21}$ and $m_y=2\,\mathrm{Im\,Tr}\,n_{21}$.

Frames

The matrices are given in a local orbital frame and, in non-collinear calculations, in a spin frame.

• Orbital frame. Per atom, the Euler angles alpha, beta, gamma of the atom position (in radians) rotate the matrices with the Wigner matrices $D_l$: $n(ll')\rightarrow D_l\, n(ll')\, D_{l'}^{\dagger}$. With alignToSpin="T", the orbital $z$ axis is aligned to the local spin direction of the atom. The same angles are used by the orbital-decomposed DOS and the jDOS. The angles that were applied are stored with each matrix.
• Spin frame. In non-collinear calculations, the spin blocks refer to the local spin frame of the atom type. With output/bandDOS/@globalspin="T" they are rotated into the global spin frame, as for the $l$-resolved DOS.

Symmetrization and degenerate states

Without further treatment, the matrix of a single state depends on the choice of the states within a degenerate subspace and on the k point in the irreducible wedge. FLEUR therefore averages the matrices over degenerate states (eigenvalues closer than $10^{-5}$ Htr) in all cases. The site symmetry is handled by output/bandDOS/densityMatrix/@symmetrize:

• T: each matrix is averaged over the operations of the site symmetry of the atom and over the equivalent atoms of the type. One matrix per atom type is stored. With the k-point weights of the irreducible wedge, sums over k then represent the full Brillouin zone. Components that are mixed by the site symmetry are averaged state by state. For example, at a cubic site the $d_{xy}$, $d_{yz}$ and $d_{zx}$ weights of every state are equal, while the t2g and eg sums are not changed.
• F: every selected atom is stored separately, without averaging. The matrix is given in the frame of the muffin-tin functions of the atom, which for an equivalent atom is rotated by the symmetry operation that maps it onto the representative atom of its type. The per-state weights of single real orbitals are then the same as in the orbital-decomposed DOS.
• auto (default): symmetrize unless the k points form a path, i.e. symmetrize for a DOS on a k-point mesh and keep the full information for a band structure.

Limitations

• Only $l\le 3$ is supported, and only channels up to the $l_\text{max}$ of the species are computed.
• Only the muffin-tin part of the states is described. Contributions of the interstitial and vacuum regions are not included.
• Symmetrization in non-collinear calculations uses the spin phases of the DFT+U implementation. These are only meaningful if the magnetic structure is compatible with the symmetry operations. For non-collinear calculations with symmetry, symmetrize="F" is the safe choice.

Further reading

• Band-resolved local density matrices: running the calculation and reading the data
• The different DOS and bandstructure modes of FLEUR: input and output reference
• Construction of the charge density and Treatment of noncollinear magnetism