General

The following are descriptions on how to use OpenMX to generate maximally localized Wannier function (MLWF) [74,75]. Keywords and settings for controlling the calculations are explained. The style of key words are closely following those originally in OpenMX. Throughout the section, a couple of results for silicon in the diamond structure will be shown for convenience. The calculation can be traced by openmx code with an input file 'Si.dat' in 'openmx*.*/work/wf_example'. There is no additional post processing code. After users may get the convergent result for the conventional SCF process for the electronic structure calculation, the following procedure explained below will be repeated by changing a couple of parameters with the restart file until desired MLWFs are obtained.

To acknowledge in any publications by using the functionality, the citation of the reference [46] would be appreciated:

Switching on generating MLWFs

To switch on the calculation, keyword 'Wannier.Func.Calc' should be explicitly set as 'on'. Its default value is 'off'.

    Wannier.Func.Calc        on         #default off

Setting the number of target MLWFs

The number of target MLWFs should be given explicitly by setting a keyword 'Wannier.Func.Num' and no default value for it.

    Wannier.Func.Num       4         #no default

Energy window for selecting Bloch states

The MLWFs will be generated from a set of Bloch states, which are selected by defining an energy window covering the eigen energies of them. Following Ref. [75], two energy windows are introduced. One is so-called outer window, defined by two keywords, 'Wannier.Outer.Window.Bottom' and 'Wannier.Outer.Window.Top', indicating the lower and upper boundaries, respectively. The other one is inner window, which is specified by two similar key words, 'Wannier.Inner.Window.Bottom' and 'Wannier.Inner.Window.Top'. All these four values are given in units eV relative to Fermi level. The inner window should be fully inside of the outer window. If the two boundaries of inner window are equal to each other, it means inner window is not defined and not used in calculation. There is no default values for outer window, while 0.0 is the default value for two boundaries of inner window. One example is as following:

 Wannier.Outer.Window.Bottom   -14.0   #lower boundary of outer window, no default value
 Wannier.Outer.Window.Top        0.0   #upper boundary of outer window, no default value
 Wannier.Inner.Window.Bottom     0.0   #lower boundary of inner window, default value 0.0
 Wannier.Inner.Window.Top        0.0   #upper boundary of outer window, default value 0.0

To set these two windows covering interested bands, it is usually to plot band structure and/or density of states before the calculation of MLWFs. If you want to restart the minimization of MLWFs by reading the overlap matrix elements from files, the outer window should not be larger than that used for calculating the stored overlap matrix. Either equal or smaller is allowed. The inner window can be varied within the outer window as you like when the restart calculation is performed. This would benefit the restarting calculation or checking the dependence of MLWFs on the size of both the windows. For the restarting calculation, please see also the section (7) 'Restart optimization without calculating overlap matrix'.

Initial guess of MLWFs

User can choose whether to use initial guess of target MLWFs or not by setting the keyword 'Wannier.Initial.Guess' as 'on' or 'off'. Default value is 'on', which means we recommend user to use an initial guess to improve the convergence or avoid local minima during the minimization of spread function.

If the initial guess is required, a set of local functions with the same number of target MLWFs should be defined. Bloch wave functions inside the outer window will be projected on to them. Therefore, these local functions are also called as projectors. The following steps are required to specify a projector.

A. Define local functions for projectors

Since the pseudo-atomic orbitals are used for projectors, the specification of them is the same as for the basis functions. An example setting, for silicon in diamond structure, is as following:

   Species.Number          2            

   <Definition.of.Atomic.Species 
     Si       Si5.5-s2p2d1    Si_CA   
     proj1    Si5.5-s1p1d1f1  Si_CA   
   Definition.of.Atomic.Species>

In this example, since we employ PAOs from Si as projectors, an additional specie 'proj1' is defined as shown above. Inside the pair keywords '$<$Definition.of.Atomic.Species' and 'Definition.of.Atomic.Species$>$', in addition to the first line used for Si atoms, one species for the projectors is defined. Its name is 'proj1' defined by 'Si5.5-s1p1d1f1' and the pseudopotential 'Si_CA'. In fact, the pseudopotential defined in this line will not be used. It is given just for keeping the consistence of inputting data structure. One can use any PAO as projector. Also the use of only a single basis set is allowed for each l-component. We strongly recommend user to specify 's1p1d1f1' in all cases to avoid possible error.

B. Specify the orbital, central position and orientation of a projector

Pair keywords '$<$Wannier.Initial.Projectos' and 'Wannier.Initial.Projectos$>$' will be used to specify the projector name, local orbital function, center of local orbital, and the local z-axis and x-axis for orbital orientation.

An example setting is shown here:

<Wannier.Initial.Projectors 
   proj1-sp3   0.250  0.250  0.250   -1.0 0.0 0.0    0.0  0.0 -1.0
   proj1-sp3   0.000  0.000  0.000    0.0 0.0 1.0    1.0  0.0  0.0
Wannier.Initial.Projectors>

Each line contains the following items. For example, in the first line, the species name, 'proj1', is defined in pairing keywords 'Definition.of.Atomic.Species'. '-' is used to connect the projector name and the selected orbitals. 'sp3' means the sp3 hybridized orbitals of this species is used as the initial guess of four target Wannier functions (see also Table 6 for all the possible orbitals and their hybrids). The projectors consisting of hybridized orbitals are centered at the position given by the following 3 numbers, '0.25 0.25 0.25', which are given in unit defined by keyword 'Wannier.Initial.Projectors.Unit' (to be explained below). The next two sets of three numbers define the z-axis and x-axis of the local coordinate system, respectively, where each axis is specified by the vector defined by three components in xyz-coordinate. In this example, in the first line the local z-axis defined by '-1.0 0.0 0.0' points to the opposite direction to the original x-axis, while the local x-axis defined by '0.0 0.0 -1.0' points to the opposite direction to the original z-axis. In the second line the local axes are the same as the original coordinate system.

The orbital used as projector can be the original PAOs or any hybrid of them. One must be aware that the total number of projectors defined by 'sp3' is 4. Similarly, 'sp' and 'sp2' contain 2 and 3 projectors, respectively. A list of supported PAOs and hybridizations among them can be found in Table 6. Any name other than those listed is not allowed.

The projector can be centered anywhere inside the unit cell. To specify its location, we can use the fractional (FRAC) coordinates relative to the unit cell vectors or Cartesian coordinates in atomic unit (AU) or in angstrom (ANG). The corresponding keyword is 'Wannier.Initial.Projectors.Unit'.

   Wannier.Initial.Projectors.Unit     FRAC     #AU, ANG or FRAC

Table: Orbitals and hybrids used as projector. The hybridization is done within the new coordinate system defined by z-axis and x-axis.

Orbital name	Number of included projector	Description
s	1	$s$ orbital from PAOs
p	3	$p_x, p_y, p_z$ from PAOs
px	1	$p_x$ from PAOs
py	1	$p_y$ from PAOs
pz	1	$p_z$ from PAOs
d	5	$d_{z^2}, d_{x^2-y^2}, d_{xy}, d_{xz}, d_{yz}$ from PAOs
dz2	1	$d_{z^2}$ from PAOs
dx2-y2	1	$d_{x^2-y^2}$ from PAOs
dxy	1	$d_{xy}$ from PAOs
dxz	1	$d_{xy}$ from PAOs
dyz	1	$d_{xy}$ from PAOs
f	7	$f_{z^3}, f_{xz^2}, f_{yz^2}, f_{zx^2}, f_{xyz}, f_{x^3-3xy^2}, f_{3yx^2-y^3}$ from PAOs
fz3	1	$f_{z^3}$ from PAOs
fxz2	1	$f_{xz^2}$ from PAOs
fyz2	1	$f_{yz^2}$ from PAOs
fzx2	1	$f_{zx^2}$ from PAOs
fxyz	1	$f_{xyz}$ from PAOs
fx3-3xy2	1	$f_{x^3-3xy^2}$ from PAOs
f3yx2-y3	1	$f_{3yx^2-y^3}$ from PAOs
sp	2	Hybridization between s and px orbitals, including $\frac{1}{\sqrt 2 }(s + p_x)$ and $\frac{1}{\sqrt 2 }(s - p_x)$
sp2	3	Hybridization among s, px, and py orbitals, including $\frac{1}{\sqrt 3 }s - \frac{1}{\sqrt 6 }p_x + \frac{1}{\sqrt 2 }p_y$, $\frac{1}{\sqrt 3 }s - \frac{1}{\sqrt 6 }p_x - \frac{1}{\sqrt 2 }p_y$ and $\frac{1}{\sqrt 3 }s + \frac{2}{\sqrt 6 }p_x$
sp3	4	Hybridization among s, px, py and pz orbitals: $\frac{1}{\sqrt 2 }(s + p_x + p_y + p_z), \frac{1}{\sqrt 2 }(s + p_x - p_y - p_z)$ $\frac{1}{\sqrt 2 }(s - p_x + p_y - p_z), \quad \frac{1}{\sqrt 2 }(s - p_x - p_y + p_z)$
sp3dz2	5	Hybridization among $s, p_x, p_y, p_z$ and $d_{z^2}$ orbitals: $\frac{1}{\sqrt 3 }s - \frac{1}{\sqrt 6 }p_x + \frac{1}{\sqrt 2 }p_y$, $\frac{1}{\sqrt 3 }s - \frac{1}{\sqrt 6 }p_x + \frac{1}{\sqrt 2 }p_y, \frac{1}{\sqrt 3 }s - \frac{2}{\sqrt 6 }p_x$ $\frac{1}{\sqrt 2 }p_z + \frac{1}{\sqrt 2 }d_{z^2}, - \frac{1}{\sqrt 2 }p_z + \frac{1}{\sqrt 2 }d_{z^2}$
sp3deg	6	Hybridization among $s, p_x, p_y, p_z$ and $d_{z^2}, d_{x^2-y^2}$ orbitals: $\frac{1}{\sqrt 6 }s - \frac{1}{\sqrt 2 }p_x - \frac{1}{\sqrt {12} }d_{z^2} + \frac{1}{2}d_{x^2-y^2}$, $\frac{1}{\sqrt 6 }s + \frac{1}{\sqrt 2 }p_x - \frac{1}{\sqrt {12} }d_{z^2} + \frac{1}{2}d_{x^2-y^2},$ $\frac{1}{\sqrt 6 }s - \frac{1}{\sqrt 2 }p_y - \frac{1}{\sqrt {12} }d_{z^2} - \frac{1}{2}d_{x^2-y^2},$ $\frac{1}{\sqrt 6 }s + \frac{1}{\sqrt 2 }p_y - \frac{1}{\sqrt {12} }d_{z^2} - \frac{1}{2}d_{x^2-y^2},$ $\frac{1}{\sqrt 6 }s - \frac{1}{\sqrt 2 }p_z + \frac{1}{\sqrt 3 }d_{z^2}, \frac{1}{\sqrt 6 }s + \frac{1}{\sqrt 2 }p_z + \frac{1}{\sqrt 3 }d_{z^2}$

K grid mesh and b vectors connecting neighboring k-points

The Monkhorst-Pack k grid mesh is defined by keyword 'Wannier.Kgrid'. There is no default setting for it. To use finite difference approach for calculating k-space differentials, b vectors connecting neighboring k points are searched shell by shell according to the distance from a central k point. The maximum number of searched shells is defined by keyword 'Wannier.MaxShells'. Default value is 12 and it should be increased if failure in finding a set of proper b vectors. The problem may happen in case of a system having a large aspect ratio among unit vectors, and in this case you will see an error message, while the value 12 works well in most cases. A proper setting of 'Wannier.Kgrid' will also help to find b vectors, where the grid spacing by the discretization for each reciprocal lattice vector should be nearly equivalent to each other.

   Wannier.MaxShells          12           # default value is 12. 
   Wannier.Kgrid           8  8  8         # no default value

Minimizing spread of WF

For entangled band case [75], two steps are needed to find the MLWFs. The first step is to minimize the gauge invariant part of spread function by disentangling the non-isolated bands. The second step is the same as isolated band case [74]. The gauge dependent part is optimized by unitary transformation of the selected Bloch wave functions according to the gradient of spread function. For the first step, three parameters are used to control the self-consistence loop. They are 'Wannier.Dis.SCF.Max.Steps', 'Wannier.Dis.Conv.Criterion', and 'Wannier.Dis.Mixing.Para'. They are the maximum number of SCF loops, the convergence criterion, and the parameter to control the mixing of input and output subspace projectors, respectively.

    Wannier.Dis.SCF.Max.Steps         2000         # default 200
    Wannier.Dis.Conv.Criterion        1e-12        # default 1e-8
    Wannier.Dis.Mixing.Para           0.5          # default value is 0.5

For the second step, three minimization methods are available. One is a steepest decent (SD) method, and the second one is a conjugate gradient (CG) method. The third one is a hybrid method which uses the SD method firstly and then switches to the CG method. The keyword 'Wannier.Minimizing.Scheme' indicates which method to be used. '0', '1', and '2' mean the simple SD method, the CG method, and hybrid method, respectively. The step length for the SD method is set by the keyword 'Wannier.Minimizing.StepLength'. In the CG method, a secant method is used to determine the optimized step length. The maximum secant steps and initial step length is specified by 'Wannier.Minimizing.Secant.Steps' and 'Wannier.Minimizing.Secant.StepLength', respectively. The maximum number of minimization step and convergence criterion are controlled by 'Wannier.Minimizing.Max.Steps' and 'Wannier.Minimizing.Conv.Criterion', respectively.

   Wannier.Minimizing.Scheme             2     # default 0, 0=SD 1=CG 2=hybrid 
   Wannier.Minimizing.StepLength        2.0    # default 2.0
   Wannier.Minimizing.Secant.Steps       5     # default 5      
   Wannier.Minimizing.Secant.StepLength 2.0    # default 2.0
   Wannier.Minimizing.Conv.Criterion   1e-12   # default 1e-8
   Wannier.Minimizing.Max.Steps         200    # default 200

In the hybrid minimization scheme, SD and CG have the same number of maximum minimization steps as specified by 'Wannier.Minimizing.Max.Steps'.

Restarting optimization without calculating overlap matrix

If the overlap matrix $M_{mn}^{({\rm {\bf k}},{\rm {\bf b}})}$ has been calculated and stored in a disk file, the keyword 'Wannier.Readin.Overlap.Matrix' can be set as 'on' to restart generating MLWF without calculating $M_{mn}^{({\rm {\bf k}},{\rm {\bf b}})}$ again.

    Wannier.Readin.Overlap.Matrix       off      # default is on

This can save the computational time since the calculation of overlap matrix is time consuming. The code will read the overlap matrix as well as the eigenenergies and states from the disk file. One should keep in mind that the outer window and k grid should be the same as those used for calculating the stored overlap matrix and eigenvalues. Consistence will be checked in the code. The inner window, initial guess of MLWF as well as the convergence criteria can be adjusted for restarting optimization. If 'Wannier.Readin.Overlap.Matrix' is set as 'off', the overlap matrix will be calculated and automatically stored into a disk file. The file name is defined by 'System.Name' with extension '.mmn'. The eigenenergies and states are also stored in the disk file with extension '.eigen'.