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< h1 > Source code for grogupy.utilities< / h1 > < div class = "highlight" > < pre >
< span > < / span > < span class = "c1" > # Copyright (c) [2024] []< / span >
< span class = "c1" > #< / span >
< span class = "c1" > # Permission is hereby granted, free of charge, to any person obtaining a copy< / span >
< span class = "c1" > # of this software and associated documentation files (the " Software" ), to deal< / span >
< span class = "c1" > # in the Software without restriction, including without limitation the rights< / span >
< span class = "c1" > # to use, copy, modify, merge, publish, distribute, sublicense, and/or sell< / span >
< span class = "c1" > # copies of the Software, and to permit persons to whom the Software is< / span >
< span class = "c1" > # furnished to do so, subject to the following conditions:< / span >
< span class = "c1" > #< / span >
< span class = "c1" > # The above copyright notice and this permission notice shall be included in all< / span >
< span class = "c1" > # copies or substantial portions of the Software.< / span >
< span class = "c1" > #< / span >
< span class = "c1" > # THE SOFTWARE IS PROVIDED " AS IS" , WITHOUT WARRANTY OF ANY KIND, EXPRESS OR< / span >
< span class = "c1" > # IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,< / span >
< span class = "c1" > # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE< / span >
< span class = "c1" > # AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER< / span >
< span class = "c1" > # LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,< / span >
< span class = "c1" > # OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE< / span >
< span class = "c1" > # SOFTWARE.< / span >
< span class = "sd" > " " " Docstring in utilities.< / span >
< span class = "sd" > " " " < / span >
< span class = "kn" > import< / span > < span class = "nn" > numpy< / span > < span class = "k" > as< / span > < span class = "nn" > np< / span >
< span class = "kn" > from< / span > < span class = "nn" > scipy.special< / span > < span class = "kn" > import< / span > < span class = "n" > roots_legendre< / span >
< span class = "kn" > from< / span > < span class = "nn" > sisl.io.siesta< / span > < span class = "kn" > import< / span > < span class = "n" > eigSileSiesta< / span >
< span class = "c1" > # Pauli matrices< / span >
< span class = "n" > tau_x< / span > < span class = "o" > =< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > array< / span > < span class = "p" > ([[< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "mi" > 0< / span > < span class = "p" > ]])< / span >
< span class = "n" > tau_y< / span > < span class = "o" > =< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > array< / span > < span class = "p" > ([[< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "n" > j< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "n" > j< / span > < span class = "p" > ,< / span > < span class = "mi" > 0< / span > < span class = "p" > ]])< / span >
< span class = "n" > tau_z< / span > < span class = "o" > =< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > array< / span > < span class = "p" > ([[< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "mi" > 0< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ]])< / span >
< span class = "n" > tau_0< / span > < span class = "o" > =< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > array< / span > < span class = "p" > ([[< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "mi" > 0< / span > < span class = "p" > ],< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "mi" > 1< / span > < span class = "p" > ]])< / span >
< div class = "viewcode-block" id = "commutator" >
< a class = "viewcode-back" href = "../../implementation/grogupy.html#grogupy.utilities.commutator" > [docs]< / a >
< span class = "k" > def< / span > < span class = "nf" > commutator< / span > < span class = "p" > (< / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "n" > b< / span > < span class = "p" > ):< / span >
< span class = "w" > < / span > < span class = "sd" > " " " Shorthand for commutator.< / span >
< span class = "sd" > Commutator of two matrices in the mathematical sense.< / span >
< span class = "sd" > Args:< / span >
< span class = "sd" > a : np.array_like< / span >
< span class = "sd" > The first matrix< / span >
< span class = "sd" > b : np.array_like< / span >
< span class = "sd" > The second matrix< / span >
< span class = "sd" > Returns:< / span >
< span class = "sd" > np.array_like< / span >
< span class = "sd" > The commutator of a and b< / span >
< span class = "sd" > " " " < / span >
< span class = "k" > return< / span > < span class = "n" > a< / span > < span class = "o" > @< / span > < span class = "n" > b< / span > < span class = "o" > -< / span > < span class = "n" > b< / span > < span class = "o" > @< / span > < span class = "n" > a< / span > < / div >
< span class = "c1" > # define some useful functions< / span >
< div class = "viewcode-block" id = "hsk" >
< a class = "viewcode-back" href = "../../implementation/grogupy.html#grogupy.utilities.hsk" > [docs]< / a >
< span class = "k" > def< / span > < span class = "nf" > hsk< / span > < span class = "p" > (< / span > < span class = "n" > H< / span > < span class = "p" > ,< / span > < span class = "n" > ss< / span > < span class = "p" > ,< / span > < span class = "n" > sc_off< / span > < span class = "p" > ,< / span > < span class = "n" > k< / span > < span class = "o" > =< / span > < span class = "p" > (< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "mi" > 0< / span > < span class = "p" > )):< / span >
< span class = "w" > < / span > < span class = "sd" > " " " Speed up Hk and Sk generation.< / span >
< span class = "sd" > Calculates the Hamiltonian and the Overlap matrix at a given k point. It is faster that the sisl version.< / span >
< span class = "sd" > Args:< / span >
< span class = "sd" > H : np.array_like< / span >
< span class = "sd" > Hamiltonian in spin box form< / span >
< span class = "sd" > ss : np.array_like< / span >
< span class = "sd" > Overlap matrix in spin box form< / span >
< span class = "sd" > sc_off : list< / span >
< span class = "sd" > supercell indexes of the Hamiltonian< / span >
< span class = "sd" > k : tuple, optional< / span >
< span class = "sd" > The k point where the matrices are set up. Defaults to (0, 0, 0)< / span >
< span class = "sd" > Returns:< / span >
< span class = "sd" > np.array_like< / span >
< span class = "sd" > Hamiltonian at the given k point< / span >
< span class = "sd" > np.array_like< / span >
< span class = "sd" > Overlap matrix at the given k point< / span >
< span class = "sd" > " " " < / span >
< span class = "c1" > # this two conversion lines are from the sisl source< / span >
< span class = "n" > k< / span > < span class = "o" > =< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > asarray< / span > < span class = "p" > (< / span > < span class = "n" > k< / span > < span class = "p" > ,< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > float64< / span > < span class = "p" > )< / span >
< span class = "n" > k< / span > < span class = "o" > .< / span > < span class = "n" > shape< / span > < span class = "o" > =< / span > < span class = "p" > (< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > ,)< / span >
< span class = "c1" > # this generates the list of phases< / span >
< span class = "n" > phases< / span > < span class = "o" > =< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > exp< / span > < span class = "p" > (< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "n" > j< / span > < span class = "o" > *< / span > < span class = "mi" > 2< / span > < span class = "o" > *< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > pi< / span > < span class = "o" > *< / span > < span class = "n" > k< / span > < span class = "o" > @< / span > < span class = "n" > sc_off< / span > < span class = "o" > .< / span > < span class = "n" > T< / span > < span class = "p" > )< / span >
< span class = "c1" > # phases applied to the hamiltonian< / span >
< span class = "n" > HK< / span > < span class = "o" > =< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > einsum< / span > < span class = "p" > (< / span > < span class = "s2" > " abc,a-> bc" < / span > < span class = "p" > ,< / span > < span class = "n" > H< / span > < span class = "p" > ,< / span > < span class = "n" > phases< / span > < span class = "p" > )< / span >
< span class = "n" > SK< / span > < span class = "o" > =< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > einsum< / span > < span class = "p" > (< / span > < span class = "s2" > " abc,a-> bc" < / span > < span class = "p" > ,< / span > < span class = "n" > ss< / span > < span class = "p" > ,< / span > < span class = "n" > phases< / span > < span class = "p" > )< / span >
< span class = "k" > return< / span > < span class = "n" > HK< / span > < span class = "p" > ,< / span > < span class = "n" > SK< / span > < / div >
< div class = "viewcode-block" id = "make_kset" >
< a class = "viewcode-back" href = "../../implementation/grogupy.html#grogupy.utilities.make_kset" > [docs]< / a >
< span class = "k" > def< / span > < span class = "nf" > make_kset< / span > < span class = "p" > (< / span > < span class = "n" > dirs< / span > < span class = "o" > =< / span > < span class = "s2" > " xyz" < / span > < span class = "p" > ,< / span > < span class = "n" > NUMK< / span > < span class = "o" > =< / span > < span class = "mi" > 20< / span > < span class = "p" > ):< / span >
< span class = "w" > < / span > < span class = "sd" > " " " Simple k-grid generator to sample the Brillouin zone.< / span >
< span class = "sd" > Depending on the value of the dirs< / span >
< span class = "sd" > argument k sampling in 1,2 or 3 dimensions is generated.< / span >
< span class = "sd" > If dirs argument does not contain either of x, y or z< / span >
< span class = "sd" > a kset of a single k-pont at the origin is returned. The total number of k points is the NUMK**(dimensions)< / span >
< span class = "sd" > Args:< / span >
< span class = "sd" > dirs : str, optional< / span >
< span class = "sd" > Directions of the k points in the Brillouin zone. They are the three lattice vectors. Defaults to " xyz" < / span >
< span class = "sd" > NUMK : int, optional< / span >
< span class = "sd" > The number of k points in a direction. Defaults to 20< / span >
< span class = "sd" > Returns:< / span >
< span class = "sd" > np.array_like< / span >
< span class = "sd" > An array of k points that uniformly sample the Brillouin zone in the given directions< / span >
< span class = "sd" > " " " < / span >
< span class = "c1" > # if there is no xyz in dirs return the Gamma point< / span >
< span class = "k" > if< / span > < span class = "ow" > not< / span > < span class = "p" > (< / span > < span class = "nb" > sum< / span > < span class = "p" > ([< / span > < span class = "n" > d< / span > < span class = "ow" > in< / span > < span class = "n" > dirs< / span > < span class = "k" > for< / span > < span class = "n" > d< / span > < span class = "ow" > in< / span > < span class = "s2" > " xyz" < / span > < span class = "p" > ])):< / span >
< span class = "k" > return< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > array< / span > < span class = "p" > ([[< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "mi" > 0< / span > < span class = "p" > ]])< / span >
< span class = "n" > kran< / span > < span class = "o" > =< / span > < span class = "nb" > len< / span > < span class = "p" > (< / span > < span class = "n" > dirs< / span > < span class = "p" > )< / span > < span class = "o" > *< / span > < span class = "p" > [< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > linspace< / span > < span class = "p" > (< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "mi" > 1< / span > < span class = "p" > ,< / span > < span class = "n" > NUMK< / span > < span class = "p" > ,< / span > < span class = "n" > endpoint< / span > < span class = "o" > =< / span > < span class = "kc" > False< / span > < span class = "p" > )]< / span >
< span class = "n" > mg< / span > < span class = "o" > =< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > meshgrid< / span > < span class = "p" > (< / span > < span class = "o" > *< / span > < span class = "n" > kran< / span > < span class = "p" > )< / span >
< span class = "n" > dirsdict< / span > < span class = "o" > =< / span > < span class = "nb" > dict< / span > < span class = "p" > ()< / span >
< span class = "k" > for< / span > < span class = "n" > d< / span > < span class = "ow" > in< / span > < span class = "nb" > enumerate< / span > < span class = "p" > (< / span > < span class = "n" > dirs< / span > < span class = "p" > ):< / span >
< span class = "n" > dirsdict< / span > < span class = "p" > [< / span > < span class = "n" > d< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ]]< / span > < span class = "o" > =< / span > < span class = "n" > mg< / span > < span class = "p" > [< / span > < span class = "n" > d< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ]]< / span > < span class = "o" > .< / span > < span class = "n" > flatten< / span > < span class = "p" > ()< / span >
< span class = "k" > for< / span > < span class = "n" > d< / span > < span class = "ow" > in< / span > < span class = "s2" > " xyz" < / span > < span class = "p" > :< / span >
< span class = "k" > if< / span > < span class = "ow" > not< / span > < span class = "p" > (< / span > < span class = "n" > d< / span > < span class = "ow" > in< / span > < span class = "n" > dirs< / span > < span class = "p" > ):< / span >
< span class = "n" > dirsdict< / span > < span class = "p" > [< / span > < span class = "n" > d< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "mi" > 0< / span > < span class = "o" > *< / span > < span class = "n" > dirsdict< / span > < span class = "p" > [< / span > < span class = "n" > dirs< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ]]< / span >
< span class = "n" > kset< / span > < span class = "o" > =< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > array< / span > < span class = "p" > ([< / span > < span class = "n" > dirsdict< / span > < span class = "p" > [< / span > < span class = "n" > d< / span > < span class = "p" > ]< / span > < span class = "k" > for< / span > < span class = "n" > d< / span > < span class = "ow" > in< / span > < span class = "s2" > " xyz" < / span > < span class = "p" > ])< / span > < span class = "o" > .< / span > < span class = "n" > T< / span >
< span class = "k" > return< / span > < span class = "n" > kset< / span > < / div >
< div class = "viewcode-block" id = "make_contour" >
< a class = "viewcode-back" href = "../../implementation/grogupy.html#grogupy.utilities.make_contour" > [docs]< / a >
< span class = "k" > def< / span > < span class = "nf" > make_contour< / span > < span class = "p" > (< / span > < span class = "n" > emin< / span > < span class = "o" > =-< / span > < span class = "mi" > 20< / span > < span class = "p" > ,< / span > < span class = "n" > emax< / span > < span class = "o" > =< / span > < span class = "mf" > 0.0< / span > < span class = "p" > ,< / span > < span class = "n" > enum< / span > < span class = "o" > =< / span > < span class = "mi" > 42< / span > < span class = "p" > ,< / span > < span class = "n" > p< / span > < span class = "o" > =< / span > < span class = "mi" > 150< / span > < span class = "p" > ):< / span >
< span class = "w" > < / span > < span class = "sd" > " " " A more sophisticated contour generator.< / span >
< span class = "sd" > Calculates the parameters for the complex contour integral. It uses the< / span >
< span class = "sd" > Legendre-Gauss quadrature method. It returns a class that contains< / span >
< span class = "sd" > the information for the contour integral.< / span >
< span class = "sd" > Args:< / span >
< span class = "sd" > emin : int, optional< / span >
< span class = "sd" > Energy minimum of the contour. Defaults to -20< / span >
< span class = "sd" > emax : float, optional< / span >
< span class = "sd" > Energy maximum of the contour. Defaults to 0.0, so the Fermi level< / span >
< span class = "sd" > enum : int, optional< / span >
< span class = "sd" > Number of sample points along the contour. Defaults to 42< / span >
< span class = "sd" > p : int, optional< / span >
< span class = "sd" > Shape parameter that describes the distribution of the sample points. Defaults to 150< / span >
< span class = "sd" > Returns:< / span >
< span class = "sd" > ccont< / span >
< span class = "sd" > Contains all the information for the contour integral< / span >
< span class = "sd" > " " " < / span >
< span class = "n" > x< / span > < span class = "p" > ,< / span > < span class = "n" > wl< / span > < span class = "o" > =< / span > < span class = "n" > roots_legendre< / span > < span class = "p" > (< / span > < span class = "n" > enum< / span > < span class = "p" > )< / span >
< span class = "n" > R< / span > < span class = "o" > =< / span > < span class = "p" > (< / span > < span class = "n" > emax< / span > < span class = "o" > -< / span > < span class = "n" > emin< / span > < span class = "p" > )< / span > < span class = "o" > /< / span > < span class = "mi" > 2< / span > < span class = "c1" > # radius< / span >
< span class = "n" > z0< / span > < span class = "o" > =< / span > < span class = "p" > (< / span > < span class = "n" > emax< / span > < span class = "o" > +< / span > < span class = "n" > emin< / span > < span class = "p" > )< / span > < span class = "o" > /< / span > < span class = "mi" > 2< / span > < span class = "c1" > # center point< / span >
< span class = "n" > y1< / span > < span class = "o" > =< / span > < span class = "o" > -< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > log< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "o" > +< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > pi< / span > < span class = "o" > *< / span > < span class = "n" > p< / span > < span class = "p" > )< / span > < span class = "c1" > # lower bound< / span >
< span class = "n" > y2< / span > < span class = "o" > =< / span > < span class = "mi" > 0< / span > < span class = "c1" > # upper bound< / span >
< span class = "n" > y< / span > < span class = "o" > =< / span > < span class = "p" > (< / span > < span class = "n" > y2< / span > < span class = "o" > -< / span > < span class = "n" > y1< / span > < span class = "p" > )< / span > < span class = "o" > /< / span > < span class = "mi" > 2< / span > < span class = "o" > *< / span > < span class = "n" > x< / span > < span class = "o" > +< / span > < span class = "p" > (< / span > < span class = "n" > y2< / span > < span class = "o" > +< / span > < span class = "n" > y1< / span > < span class = "p" > )< / span > < span class = "o" > /< / span > < span class = "mi" > 2< / span >
< span class = "n" > phi< / span > < span class = "o" > =< / span > < span class = "p" > (< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > exp< / span > < span class = "p" > (< / span > < span class = "o" > -< / span > < span class = "n" > y< / span > < span class = "p" > )< / span > < span class = "o" > -< / span > < span class = "mi" > 1< / span > < span class = "p" > )< / span > < span class = "o" > /< / span > < span class = "n" > p< / span > < span class = "c1" > # angle parameter< / span >
< span class = "n" > ze< / span > < span class = "o" > =< / span > < span class = "n" > z0< / span > < span class = "o" > +< / span > < span class = "n" > R< / span > < span class = "o" > *< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > exp< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "n" > j< / span > < span class = "o" > *< / span > < span class = "n" > phi< / span > < span class = "p" > )< / span > < span class = "c1" > # complex points for path< / span >
< span class = "n" > we< / span > < span class = "o" > =< / span > < span class = "o" > -< / span > < span class = "p" > (< / span > < span class = "n" > y2< / span > < span class = "o" > -< / span > < span class = "n" > y1< / span > < span class = "p" > )< / span > < span class = "o" > /< / span > < span class = "mi" > 2< / span > < span class = "o" > *< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > exp< / span > < span class = "p" > (< / span > < span class = "o" > -< / span > < span class = "n" > y< / span > < span class = "p" > )< / span > < span class = "o" > /< / span > < span class = "n" > p< / span > < span class = "o" > *< / span > < span class = "mi" > 1< / span > < span class = "n" > j< / span > < span class = "o" > *< / span > < span class = "p" > (< / span > < span class = "n" > ze< / span > < span class = "o" > -< / span > < span class = "n" > z0< / span > < span class = "p" > )< / span > < span class = "o" > *< / span > < span class = "n" > wl< / span >
< span class = "c1" > # just an empty container class< / span >
< span class = "k" > class< / span > < span class = "nc" > ccont< / span > < span class = "p" > :< / span >
< span class = "k" > pass< / span >
< span class = "n" > cont< / span > < span class = "o" > =< / span > < span class = "n" > ccont< / span > < span class = "p" > ()< / span >
< span class = "n" > cont< / span > < span class = "o" > .< / span > < span class = "n" > R< / span > < span class = "o" > =< / span > < span class = "n" > R< / span >
< span class = "n" > cont< / span > < span class = "o" > .< / span > < span class = "n" > z0< / span > < span class = "o" > =< / span > < span class = "n" > z0< / span >
< span class = "n" > cont< / span > < span class = "o" > .< / span > < span class = "n" > ze< / span > < span class = "o" > =< / span > < span class = "n" > ze< / span >
< span class = "n" > cont< / span > < span class = "o" > .< / span > < span class = "n" > we< / span > < span class = "o" > =< / span > < span class = "n" > we< / span >
< span class = "n" > cont< / span > < span class = "o" > .< / span > < span class = "n" > enum< / span > < span class = "o" > =< / span > < span class = "n" > enum< / span >
< span class = "k" > return< / span > < span class = "n" > cont< / span > < / div >
< div class = "viewcode-block" id = "tau_u" >
< a class = "viewcode-back" href = "../../implementation/grogupy.html#grogupy.utilities.tau_u" > [docs]< / a >
< span class = "k" > def< / span > < span class = "nf" > tau_u< / span > < span class = "p" > (< / span > < span class = "n" > u< / span > < span class = "p" > ):< / span >
< span class = "w" > < / span > < span class = "sd" > " " " Pauli matrix in direction u.< / span >
< span class = "sd" > Returns the vector u in the basis of the Pauli matrices.< / span >
< span class = "sd" > Args:< / span >
< span class = "sd" > u : list or np.array_like< / span >
< span class = "sd" > The direction< / span >
< span class = "sd" > Returns:< / span >
< span class = "sd" > np.array_like< / span >
< span class = "sd" > Arbitrary direction in the base of the Pauli matrices< / span >
< span class = "sd" > " " " < / span >
< span class = "c1" > # u is force to be of unit length< / span >
< span class = "n" > u< / span > < span class = "o" > =< / span > < span class = "n" > u< / span > < span class = "o" > /< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > linalg< / span > < span class = "o" > .< / span > < span class = "n" > norm< / span > < span class = "p" > (< / span > < span class = "n" > u< / span > < span class = "p" > )< / span >
< span class = "k" > return< / span > < span class = "n" > u< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ]< / span > < span class = "o" > *< / span > < span class = "n" > tau_x< / span > < span class = "o" > +< / span > < span class = "n" > u< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ]< / span > < span class = "o" > *< / span > < span class = "n" > tau_y< / span > < span class = "o" > +< / span > < span class = "n" > u< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ]< / span > < span class = "o" > *< / span > < span class = "n" > tau_z< / span > < / div >
< span class = "c1" > #< / span >
< div class = "viewcode-block" id = "crossM" >
< a class = "viewcode-back" href = "../../implementation/grogupy.html#grogupy.utilities.crossM" > [docs]< / a >
< span class = "k" > def< / span > < span class = "nf" > crossM< / span > < span class = "p" > (< / span > < span class = "n" > u< / span > < span class = "p" > ):< / span >
< span class = "w" > < / span > < span class = "sd" > " " " Definition for the cross-product matrix.< / span >
< span class = "sd" > It acts as a cross product with vector u.< / span >
< span class = "sd" > Args:< / span >
< span class = "sd" > u : list or np.array_like< / span >
< span class = "sd" > The second vector in the cross product< / span >
< span class = "sd" > Returns:< / span >
< span class = "sd" > np.array_like< / span >
< span class = "sd" > The matrix that represents teh cross product with a vector< / span >
< span class = "sd" > " " " < / span >
< span class = "k" > return< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > array< / span > < span class = "p" > ([[< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "o" > -< / span > < span class = "n" > u< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ],< / span > < span class = "n" > u< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ]],< / span > < span class = "p" > [< / span > < span class = "n" > u< / span > < span class = "p" > [< / span > < span class = "mi" > 2< / span > < span class = "p" > ],< / span > < span class = "mi" > 0< / span > < span class = "p" > ,< / span > < span class = "o" > -< / span > < span class = "n" > u< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ]],< / span > < span class = "p" > [< / span > < span class = "o" > -< / span > < span class = "n" > u< / span > < span class = "p" > [< / span > < span class = "mi" > 1< / span > < span class = "p" > ],< / span > < span class = "n" > u< / span > < span class = "p" > [< / span > < span class = "mi" > 0< / span > < span class = "p" > ],< / span > < span class = "mi" > 0< / span > < span class = "p" > ]])< / span > < / div >
< div class = "viewcode-block" id = "RotM" >
< a class = "viewcode-back" href = "../../implementation/grogupy.html#grogupy.utilities.RotM" > [docs]< / a >
< span class = "k" > def< / span > < span class = "nf" > RotM< / span > < span class = "p" > (< / span > < span class = "n" > theta< / span > < span class = "p" > ,< / span > < span class = "n" > u< / span > < span class = "p" > ,< / span > < span class = "n" > eps< / span > < span class = "o" > =< / span > < span class = "mf" > 1e-10< / span > < span class = "p" > ):< / span >
< span class = "w" > < / span > < span class = "sd" > " " " Definition of rotation matrix with angle theta around direction u.< / span >
< span class = "sd" > Args:< / span >
< span class = "sd" > theta : float< / span >
< span class = "sd" > The angle of rotation< / span >
< span class = "sd" > u : np.array_like< / span >
< span class = "sd" > The rotation axis< / span >
< span class = "sd" > eps : float, optional< / span >
< span class = "sd" > Cutoff for small elements in the resulting matrix. Defaults to 1e-10< / span >
< span class = "sd" > Returns:< / span >
< span class = "sd" > np.array_like< / span >
< span class = "sd" > The rotation matrix< / span >
< span class = "sd" > " " " < / span >
< span class = "n" > u< / span > < span class = "o" > =< / span > < span class = "n" > u< / span > < span class = "o" > /< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > linalg< / span > < span class = "o" > .< / span > < span class = "n" > norm< / span > < span class = "p" > (< / span > < span class = "n" > u< / span > < span class = "p" > )< / span >
< span class = "n" > M< / span > < span class = "o" > =< / span > < span class = "p" > (< / span >
< span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > cos< / span > < span class = "p" > (< / span > < span class = "n" > theta< / span > < span class = "p" > )< / span > < span class = "o" > *< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > eye< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > )< / span >
< span class = "o" > +< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > sin< / span > < span class = "p" > (< / span > < span class = "n" > theta< / span > < span class = "p" > )< / span > < span class = "o" > *< / span > < span class = "n" > crossM< / span > < span class = "p" > (< / span > < span class = "n" > u< / span > < span class = "p" > )< / span >
< span class = "o" > +< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "o" > -< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > cos< / span > < span class = "p" > (< / span > < span class = "n" > theta< / span > < span class = "p" > ))< / span > < span class = "o" > *< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > outer< / span > < span class = "p" > (< / span > < span class = "n" > u< / span > < span class = "p" > ,< / span > < span class = "n" > u< / span > < span class = "p" > )< / span >
< span class = "p" > )< / span >
< span class = "c1" > # kill off small numbers< / span >
< span class = "n" > M< / span > < span class = "p" > [< / span > < span class = "nb" > abs< / span > < span class = "p" > (< / span > < span class = "n" > M< / span > < span class = "p" > )< / span > < span class = "o" > < < / span > < span class = "n" > eps< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "mf" > 0.0< / span >
< span class = "k" > return< / span > < span class = "n" > M< / span > < / div >
< div class = "viewcode-block" id = "RotMa2b" >
< a class = "viewcode-back" href = "../../implementation/grogupy.html#grogupy.utilities.RotMa2b" > [docs]< / a >
< span class = "k" > def< / span > < span class = "nf" > RotMa2b< / span > < span class = "p" > (< / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "n" > b< / span > < span class = "p" > ,< / span > < span class = "n" > eps< / span > < span class = "o" > =< / span > < span class = "mf" > 1e-10< / span > < span class = "p" > ):< / span >
< span class = "w" > < / span > < span class = "sd" > " " " Definition of rotation matrix rotating unit vector a to unit vector b.< / span >
< span class = "sd" > Function returns array R such that R @ a = b holds.< / span >
< span class = "sd" > Args:< / span >
< span class = "sd" > a : np.array_like< / span >
< span class = "sd" > First vector< / span >
< span class = "sd" > b : np.array_like< / span >
< span class = "sd" > Second vector< / span >
< span class = "sd" > eps : float, optional< / span >
< span class = "sd" > Cutoff for small elements in the resulting matrix. Defaults to 1e-10< / span >
< span class = "sd" > Returns:< / span >
< span class = "sd" > np.array_like< / span >
< span class = "sd" > The rotation matrix with the above property< / span >
< span class = "sd" > " " " < / span >
< span class = "n" > v< / span > < span class = "o" > =< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > cross< / span > < span class = "p" > (< / span > < span class = "n" > a< / span > < span class = "p" > ,< / span > < span class = "n" > b< / span > < span class = "p" > )< / span >
< span class = "n" > c< / span > < span class = "o" > =< / span > < span class = "n" > a< / span > < span class = "o" > @< / span > < span class = "n" > b< / span >
< span class = "n" > M< / span > < span class = "o" > =< / span > < span class = "n" > np< / span > < span class = "o" > .< / span > < span class = "n" > eye< / span > < span class = "p" > (< / span > < span class = "mi" > 3< / span > < span class = "p" > )< / span > < span class = "o" > +< / span > < span class = "n" > crossM< / span > < span class = "p" > (< / span > < span class = "n" > v< / span > < span class = "p" > )< / span > < span class = "o" > +< / span > < span class = "n" > crossM< / span > < span class = "p" > (< / span > < span class = "n" > v< / span > < span class = "p" > )< / span > < span class = "o" > @< / span > < span class = "n" > crossM< / span > < span class = "p" > (< / span > < span class = "n" > v< / span > < span class = "p" > )< / span > < span class = "o" > /< / span > < span class = "p" > (< / span > < span class = "mi" > 1< / span > < span class = "o" > +< / span > < span class = "n" > c< / span > < span class = "p" > )< / span >
< span class = "c1" > # kill off small numbers< / span >
< span class = "n" > M< / span > < span class = "p" > [< / span > < span class = "nb" > abs< / span > < span class = "p" > (< / span > < span class = "n" > M< / span > < span class = "p" > )< / span > < span class = "o" > < < / span > < span class = "n" > eps< / span > < span class = "p" > ]< / span > < span class = "o" > =< / span > < span class = "mf" > 0.0< / span >
< span class = "k" > return< / span > < span class = "n" > M< / span > < / div >
< div class = "viewcode-block" id = "read_siesta_emin" >
< a class = "viewcode-back" href = "../../implementation/grogupy.html#grogupy.utilities.read_siesta_emin" > [docs]< / a >
< span class = "k" > def< / span > < span class = "nf" > read_siesta_emin< / span > < span class = "p" > (< / span > < span class = "n" > eigfile< / span > < span class = "p" > ):< / span >
< span class = "w" > < / span > < span class = "sd" > " " " It reads the lowest energy level from the siesta run.< / span >
< span class = "sd" > It uses the .EIG file from siesta that contains the eigenvalues.< / span >
< span class = "sd" > Args:< / span >
< span class = "sd" > eigfile : str< / span >
< span class = "sd" > The path to the .EIG file< / span >
< span class = "sd" > Returns:< / span >
< span class = "sd" > float< / span >
< span class = "sd" > The energy minimum< / span >
< span class = "sd" > " " " < / span >
< span class = "c1" > # read the file< / span >
< span class = "n" > eigs< / span > < span class = "o" > =< / span > < span class = "n" > eigSileSiesta< / span > < span class = "p" > (< / span > < span class = "n" > eigfile< / span > < span class = "p" > )< / span > < span class = "o" > .< / span > < span class = "n" > read_data< / span > < span class = "p" > ()< / span >
< span class = "k" > return< / span > < span class = "n" > eigs< / span > < span class = "o" > .< / span > < span class = "n" > min< / span > < span class = "p" > ()< / span > < / div >
< div class = "viewcode-block" id = "int_de_ke" >
< a class = "viewcode-back" href = "../../implementation/grogupy.html#grogupy.utilities.int_de_ke" > [docs]< / a >
< span class = "k" > def< / span > < span class = "nf" > int_de_ke< / span > < span class = "p" > (< / span > < span class = "n" > traced< / span > < span class = "p" > ,< / span > < span class = "n" > we< / span > < span class = "p" > ):< / span >
< span class = "w" > < / span > < span class = "sd" > " " " It numerically integrates the traced matrix.< / span >
< span class = "sd" > It is a wrapper from numpy.trapz and it contains the< / span >
< span class = "sd" > relevant constants to calculate the energy integral from< / span >
< span class = "sd" > equation 93 or 96.< / span >
< span class = "sd" > Args:< / span >
< span class = "sd" > traced : np.array_like< / span >
< span class = "sd" > The trace of a matrix or a matrix product< / span >
< span class = "sd" > we : float< / span >
< span class = "sd" > The weight of a point on the contour< / span >
< span class = "sd" > Returns:< / span >
< span class = "sd" > float< / span >
< span class = "sd" > The energy calculated from the integral formula< / span >
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