22: Copper — Symmetry-adapted Wannier function¶

Outline: Obtain symmetry-adapted Wannier functions for Cu. By symmetry-adapted mode, for example, we can make atomic centered s-like Wannier function, which is not possible in the usual procedure to create maximally localized Wannier functions. For the theoretical background of the symmetry-adapted Wannier functions, see R. Sakuma, Phys. Rev. B 87, 235109 (2013).

Image title — Unit cell of Copper crystal.

Each directory creates \(s\)-like symmetry-adapted Wannier function centered at different position on top of atomic centered \(d\)-like Wannier functions.

Below it is reported the README file from the example directory

README file

# Symmetry-adapted mode

Additional input in Cu.win file
  site_symmetry = .true.   (default value is .false.)
  symmetrize_eps = 1d-9    (default value is 1d-3   )

Additional input in Cu.pw2wan file
  write_dmn = .true.

Working directories
  s_at_0.00 : s-like Wannier function centered at (0,0,0)       + atomic-centered d-like WFs
  s_at_0.25 : s-like Wannier function centered at (1/4,1/4,1/4) + atomic-centered d-like WFs
  s_at_0.50 : s-like Wannier function centered at (1/2,1/2,1/2) + atomic-centered d-like WFs

In s_at_0.25, we use an additional flag "read_sym = .true." to customize the symmetry operations
to be used.
We exclude the inversion symmetry to create s-like Wannier function centered at (1/4,1/4,1/4).
Information on symmetry operations without inversion symmetry is taken from GaAs calculation.
See more detailed discussion in R. Sakuma, Phys. Rev. B 87, 235109 (2013).

The space group of Cu is \(Fm{-}3m\) (sequential number 225 in the International Tables for Crystallography, Vol. A).

\(s\)-like Wannier function centred at the origin¶

1-5: Compute the symmetry-adapted MLWFs.

In this example both the \(s\)-orbital and \(d\)-orbitals is placed at the origin, whose Wyckoff letter is \(a\) and its multiplicity is \(4\). The site symmetry group of \(a\) is \(m{-}3m\), which is isomorphous to the full point group of the crystal, known as \(O_h\) and it contains 48 symmetry operations. The six MLWFs obtained by placing both the initial \(s\)-orbital and the \(d\)-orbitals on the Cu atom are shown in Figure 2.

s-like WF — Six symmetry-adapted MLWFs in Cu. The initial \(s\)-orbital is placed at the origin: a. \(s\)-like WF, b. \(e_{g}\)-like WF, c. \(e_{g}\)-like WF, d. \(t_{2g}\)-like WF, e. \(t_{2g}\)-like WF, f. \(t_{2g}\)-like WF.

eg-like WF — Six symmetry-adapted MLWFs in Cu. The initial \(s\)-orbital is placed at the origin: a. \(s\)-like WF, b. \(e_{g}\)-like WF, c. \(e_{g}\)-like WF, d. \(t_{2g}\)-like WF, e. \(t_{2g}\)-like WF, f. \(t_{2g}\)-like WF.

\(s\)-like Wannier function centred at (0.25,0.25,0.25)¶

1-5: Compute the symmetry-adapted MLWFs.

In this example the \(s\)-orbital is placed at (0.25,0.25,0.25), whose Wyckoff letter is \(c\) and its multiplicity is \(8\). The site symmetry group of \(c\) is \({-}43m\), which is not isomorphous to the full point group of the crystal (\(O_h\)). This site symmetry group contains only 24 symmetry operations, i.e. it comes from \(O_h\) when inversion is removed. This is the reason why the flag read_sym =.true. in the .pw2wan file and an additional input is required, namely Cu.sym. In fact, this file is not automatically generated by pw2wannier90.x but is given as input to be read. The six MLWFs obtained by placing the initial \(s\)-orbital at (0.25,0.25,0.25) and the \(d\)-orbitals on the Cu atom are shown in Figure 3.

\(s\)-like Wannier function centred at (0.5,0.5,0.5)¶

1-5: Compute the symmetry-adapted MLWFs.

In this example the \(s\) orbital is placed at (0.5,0.5,0.5), whose Wyckoff letter is \(b\) and its multiplicity is \(4\). The site symmetry group of \(c\) is \(m{-}3m\), which is isomorphous to the full point group of the crystal (\(O_h\)). Hence, no additional input file is needed in this case. The six MLWFs obtained by placing the initial \(s\)-orbital at (0.5,0.5,0.5) and the \(d\)-orbitals on the Cu atom are shown in Figure 4.