Chimie Théorique » QMX

""" Maximally localized Wannier Functions

    Find the set of maximally localized Wannier functions

    using the spread functional of Marzari and Vanderbilt

    (PRB 56, 1997 page 12847). 

"""

import numpy as np

from time import time

from math import sqrt, pi

from pickle import dump, load

from ase.parallel import paropen

from ase.calculators.dacapo import Dacapo

from ase.dft.kpoints import get_monkhorst_shape

from ase.transport.tools import dagger, normalize

dag = dagger

def gram_schmidt(U):

    """Orthonormalize columns of U according to the Gram-Schmidt procedure."""

    for i, col in enumerate(U.T):

        for col2 in U.T[:i]:

            col -= col2 * np.dot(col2.conj(), col)

        col /= np.linalg.norm(col)

def lowdin(U, S=None):

    """Orthonormalize columns of U according to the Lowdin procedure.

    If the overlap matrix is know, it can be specified in S.

    """

    if S is None:

        S = np.dot(dag(U), U)

    eig, rot = np.linalg.eigh(S)

    rot = np.dot(rot / np.sqrt(eig), dag(rot))

    U[:] = np.dot(U, rot)

def neighbor_k_search(k_c, G_c, kpt_kc, tol=1e-4):

    # search for k1 (in kpt_kc) and k0 (in alldir), such that

    # k1 - k - G + k0 = 0

    alldir_dc = np.array([[0,0,0],[1,0,0],[0,1,0],[0,0,1],

                           [1,1,0],[1,0,1],[0,1,1]], int)

    for k0_c in alldir_dc:

        for k1, k1_c in enumerate(kpt_kc):

            if np.linalg.norm(k1_c - k_c - G_c + k0_c) < tol:

                return k1, k0_c

    print 'Wannier: Did not find matching kpoint for kpt=', k_c

    print 'Probably non-uniform k-point grid'

    raise NotImplementedError

def calculate_weights(cell_cc):

    """ Weights are used for non-cubic cells, see PRB **61**, 10040"""

    alldirs_dc = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1],

                           [1, 1, 0], [1, 0, 1], [0, 1, 1]], dtype=int)

    g = np.dot(cell_cc, cell_cc.T)

    # NOTE: Only first 3 of following 6 weights are presently used:

    w = np.zeros(6)              

    w[0] = g[0, 0] - g[0, 1] - g[0, 2]

    w[1] = g[1, 1] - g[0, 1] - g[1, 2]

    w[2] = g[2, 2] - g[0, 2] - g[1, 2]

    w[3] = g[0, 1]

    w[4] = g[0, 2]

    w[5] = g[1, 2]

    # Make sure that first 3 Gdir vectors are included - 

    # these are used to calculate Wanniercenters.

    Gdir_dc = alldirs_dc[:3]

    weight_d = w[:3]

    for d in range(3, 6):

        if abs(w[d]) > 1e-5:

            Gdir_dc = np.concatenate(Gdir_dc, alldirs_dc[d])

            weight_d = np.concatenate(weight_d, w[d])

    weight_d /= max(abs(weight_d))

    return weight_d, Gdir_dc

def random_orthogonal_matrix(dim, seed=None, real=False):

    """Generate a random orthogonal matrix"""

    if seed is not None:

        np.random.seed(seed)

    H = np.random.rand(dim, dim)

    np.add(dag(H), H, H)

    np.multiply(.5, H, H)

    if real:

        gram_schmidt(H)

        return H

    else: 

        val, vec = np.linalg.eig(H)

        return np.dot(vec * np.exp(1.j * val), dag(vec))

def steepest_descent(func, step=.005, tolerance=1e-6, **kwargs):

    fvalueold = 0.

    fvalue = fvalueold + 10

    count=0

    while abs((fvalue - fvalueold) / fvalue) > tolerance:

        fvalueold = fvalue

        dF = func.get_gradients()

        func.step(dF * step, **kwargs)

        fvalue = func.get_functional_value()

        count += 1

        print 'SteepestDescent: iter=%s, value=%s' % (count, fvalue)

def md_min(func, step=.25, tolerance=1e-6, verbose=False, **kwargs):

    if verbose:

        print 'Localize with step =', step, 'and tolerance =', tolerance

        t = -time()

    fvalueold = 0.

    fvalue = fvalueold + 10

    count = 0

    V = np.zeros(func.get_gradients().shape, dtype=complex)

    while abs((fvalue - fvalueold) / fvalue) > tolerance:

        fvalueold = fvalue

        dF = func.get_gradients()

        V *= (dF * V.conj()).real > 0

        V += step * dF

        func.step(V, **kwargs)

        fvalue = func.get_functional_value()

        if fvalue < fvalueold:

            step *= 0.5

        count += 1

        if verbose:

            print 'MDmin: iter=%s, step=%s, value=%s' % (count, step, fvalue)

    if verbose:

        t += time()

        print '%d iterations in %0.2f seconds (%0.2f ms/iter), endstep = %s' %(

            count, t, t * 1000. / count, step)

def rotation_from_projection(proj_nw, fixed, ortho=True):

    """Determine rotation and coefficient matrices from projections

    proj_nw = <psi_n|p_w>

    psi_n: eigenstates

    p_w: localized function

    Nb (n) = Number of bands

    Nw (w) = Number of wannier functions

    M  (f) = Number of fixed states

    L  (l) = Number of extra degrees of freedom

    U  (u) = Number of non-fixed states

    """

    Nb, Nw = proj_nw.shape

    M = fixed

    L = Nw - M

    U_ww = np.empty((Nw, Nw), dtype=proj_nw.dtype)

    U_ww[:M] = proj_nw[:M]

    if L > 0:

        proj_uw = proj_nw[M:]

        eig_w, C_ww = np.linalg.eigh(np.dot(dag(proj_uw), proj_uw))

        C_ul = np.dot(proj_uw, C_ww[:, np.argsort(-eig_w.real)[:L]])

        #eig_u, C_uu = np.linalg.eigh(np.dot(proj_uw, dag(proj_uw)))

        #C_ul = C_uu[:, np.argsort(-eig_u.real)[:L]]

        U_ww[M:] = np.dot(dag(C_ul), proj_uw)

    else:

        C_ul = np.empty((Nb - M, 0))

    normalize(C_ul)

    if ortho:

        lowdin(U_ww)

    else:

        normalize(U_ww)

    return U_ww, C_ul

class Wannier:

    """Maximally localized Wannier Functions

    Find the set of maximally localized Wannier functions using the

    spread functional of Marzari and Vanderbilt (PRB 56, 1997 page

    12847).

    """

    def __init__(self, nwannier, calc,

                 file=None,

                 nbands=None,

                 fixedenergy=None,

                 fixedstates=None,

                 spin=0,

                 initialwannier='random',

                 seed=None,

                 verbose=False):

        """

        Required arguments:

          ``nwannier``: The number of Wannier functions you wish to construct.

            This must be at least half the number of electrons in the system

            and at most equal to the number of bands in the calculation.

          ``calc``: A converged DFT calculator class.

            If ``file`` arg. is not provided, the calculator *must* provide the

            method ``get_wannier_localization_matrix``, and contain the

            wavefunctions (save files with only the density is not enough).

            If the localization matrix is read from file, this is not needed,

            unless ``get_function`` or ``write_cube`` is called.

        Optional arguments:

          ``nbands``: Bands to include in localization.

            The number of bands considered by Wannier can be smaller than the

            number of bands in the calculator. This is useful if the highest

            bands of the DFT calculation are not well converged.

          ``spin``: The spin channel to be considered.

            The Wannier code treats each spin channel independently.

          ``fixedenergy`` / ``fixedstates``: Fixed part of Heilbert space.

            Determine the fixed part of Hilbert space by either a maximal

            energy *or* a number of bands (possibly a list for multiple

            k-points).

            Default is None meaning that the number of fixed states is equated

            to ``nwannier``.

          ``file``: Read localization and rotation matrices from this file.

          ``initialwannier``: Initial guess for Wannier rotation matrix.

            Can be 'bloch' to start from the Bloch states, 'random' to be

            randomized, or a list passed to calc.get_initial_wannier.

          ``seed``: Seed for random ``initialwannier``.

          ``verbose``: True / False level of verbosity.

          """

        # Bloch phase sign convention

        sign = -1

        classname = calc.__class__.__name__

        if classname in ['Dacapo', 'Jacapo']:

            print 'Using ' + classname

            sign = +1

        self.nwannier = nwannier

        self.calc = calc

        self.spin = spin

        self.verbose = verbose

        self.kpt_kc = sign * calc.get_ibz_k_points()

        assert len(calc.get_bz_k_points()) == len(self.kpt_kc)

        self.kptgrid = get_monkhorst_shape(self.kpt_kc)

        self.Nk = len(self.kpt_kc)

        self.unitcell_cc = calc.get_atoms().get_cell()

        self.largeunitcell_cc = (self.unitcell_cc.T * self.kptgrid).T

        self.weight_d, self.Gdir_dc = calculate_weights(self.largeunitcell_cc)

        self.Ndir = len(self.weight_d) # Number of directions

        if nbands is not None:

            self.nbands = nbands

        else:

            self.nbands = calc.get_number_of_bands()

        if fixedenergy is None:

            if fixedstates is None:

                self.fixedstates_k = np.array([nwannier] * self.Nk, int)

            else:

                if type(fixedstates) is int:

                    fixedstates = [fixedstates] * self.Nk

                self.fixedstates_k = np.array(fixedstates, int)

        else:

            # Setting number of fixed states and EDF from specified energy.

            # All states below this energy (relative to Fermi level) are fixed.

            fixedenergy += calc.get_fermi_level()

            print fixedenergy

            self.fixedstates_k = np.array(

                [calc.get_eigenvalues(k, spin).searchsorted(fixedenergy)

                 for k in range(self.Nk)], int)

        self.edf_k = self.nwannier - self.fixedstates_k

        if verbose:

            print 'Wannier: Fixed states            : %s' % self.fixedstates_k

            print 'Wannier: Extra degrees of freedom: %s' % self.edf_k

        # Set the list of neighboring k-points k1, and the "wrapping" k0,

        # such that k1 - k - G + k0 = 0

        #

        # Example: kpoints = (-0.375,-0.125,0.125,0.375), dir=0

        # G = [0.25,0,0]

        # k=0.375, k1= -0.375 : -0.375-0.375-0.25 => k0=[1,0,0]

        #

        # For a gamma point calculation k1 = k = 0,  k0 = [1,0,0] for dir=0

        if self.Nk == 1:

            self.kklst_dk = np.zeros((self.Ndir, 1), int)

            k0_dkc = self.Gdir_dc.reshape(-1, 1, 3)

        else:

            self.kklst_dk = np.empty((self.Ndir, self.Nk), int)

            k0_dkc = np.empty((self.Ndir, self.Nk, 3), int)

            # Distance between kpoints

            kdist_c = np.empty(3)

            for c in range(3):

                # make a sorted list of the kpoint values in this direction

                slist = np.argsort(self.kpt_kc[:, c], kind='mergesort')

                skpoints_kc = np.take(self.kpt_kc, slist, axis=0)

                kdist_c[c] = max([skpoints_kc[n + 1, c] - skpoints_kc[n, c]

                                  for n in range(self.Nk - 1)])               

            for d, Gdir_c in enumerate(self.Gdir_dc):

                for k, k_c in enumerate(self.kpt_kc):

                    # setup dist vector to next kpoint

                    G_c = np.where(Gdir_c > 0, kdist_c, 0)

                    if max(G_c) < 1e-4:

                        self.kklst_dk[d, k] = k

                        k0_dkc[d, k] = Gdir_c

                    else:

                        self.kklst_dk[d, k], k0_dkc[d, k] = \

                                       neighbor_k_search(k_c, G_c, self.kpt_kc)

        # Set the inverse list of neighboring k-points

        self.invkklst_dk = np.empty((self.Ndir, self.Nk), int)

        for d in range(self.Ndir):

            for k1 in range(self.Nk):

                self.invkklst_dk[d, k1] = self.kklst_dk[d].tolist().index(k1)

        Nw = self.nwannier

        Nb = self.nbands

        self.Z_dkww = np.empty((self.Ndir, self.Nk, Nw, Nw), complex)

        self.V_knw = np.zeros((self.Nk, Nb, Nw), complex)

        if file is None:

            self.Z_dknn = np.empty((self.Ndir, self.Nk, Nb, Nb), complex)

            for d, dirG in enumerate(self.Gdir_dc):

                for k in range(self.Nk):

                    k1 = self.kklst_dk[d, k]

                    k0_c = k0_dkc[d, k]

                    self.Z_dknn[d, k] = calc.get_wannier_localization_matrix(

                        nbands=Nb, dirG=dirG, kpoint=k, nextkpoint=k1,

                        G_I=k0_c, spin=self.spin)

        self.initialize(file=file, initialwannier=initialwannier, seed=seed)

    def initialize(self, file=None, initialwannier='random', seed=None):

        """Re-initialize current rotation matrix.

        Keywords are identical to those of the constructor.

        """

        Nw = self.nwannier

        Nb = self.nbands

        if file is not None:

            self.Z_dknn, self.U_kww, self.C_kul = load(paropen(file))

        elif initialwannier == 'bloch':

            # Set U and C to pick the lowest Bloch states

            self.U_kww = np.zeros((self.Nk, Nw, Nw), complex)

            self.C_kul = []

            for U, M, L in zip(self.U_kww, self.fixedstates_k, self.edf_k):

                U[:] = np.identity(Nw, complex)

                if L > 0:

                    self.C_kul.append(

                        np.identity(Nb - M, complex)[:, :L])

                else:

                    self.C_kul.append([])

        elif initialwannier == 'random':

            # Set U and C to random (orthogonal) matrices

            self.U_kww = np.zeros((self.Nk, Nw, Nw), complex)

            self.C_kul = []

            for U, M, L in zip(self.U_kww, self.fixedstates_k, self.edf_k):

                U[:] = random_orthogonal_matrix(Nw, seed, real=False)

                if L > 0:

                    self.C_kul.append(random_orthogonal_matrix(

                        Nb - M, seed=seed, real=False)[:, :L])

                else:

                    self.C_kul.append(np.array([]))        

        else:

            # Use initial guess to determine U and C

            self.C_kul, self.U_kww = self.calc.initial_wannier(

                initialwannier, self.kptgrid, self.fixedstates_k,

                self.edf_k, self.spin)

        self.update()

    def save(self, file):

        """Save information on localization and rotation matrices to file."""

        dump((self.Z_dknn, self.U_kww, self.C_kul), paropen(file, 'w'))

    def update(self):

        # Update large rotation matrix V (from rotation U and coeff C)

        for k, M in enumerate(self.fixedstates_k):

            self.V_knw[k, :M] = self.U_kww[k, :M]

            if M < self.nwannier:

                self.V_knw[k, M:] = np.dot(self.C_kul[k], self.U_kww[k, M:])

            # else: self.V_knw[k, M:] = 0.0

        # Calculate the Zk matrix from the large rotation matrix:

        # Zk = V^d[k] Zbloch V[k1]

        for d in range(self.Ndir):

            for k in range(self.Nk):

                k1 = self.kklst_dk[d, k]

                self.Z_dkww[d, k] = np.dot(dag(self.V_knw[k]), np.dot(

                    self.Z_dknn[d, k], self.V_knw[k1]))

        # Update the new Z matrix

        self.Z_dww = self.Z_dkww.sum(axis=1) / self.Nk

    def get_centers(self, scaled=False):

        """Calculate the Wannier centers

        ::

          pos =  L / 2pi * phase(diag(Z))

        """

        coord_wc = np.angle(self.Z_dww[:3].diagonal(0, 1, 2)).T / (2 * pi) % 1

        if not scaled:

            coord_wc = np.dot(coord_wc, self.largeunitcell_cc)

        return coord_wc

    def get_radii(self):

        """Calculate the spread of the Wannier functions.

        ::

                        --  /  L  \ 2       2

          radius**2 = - >   | --- |   ln |Z| 

                        --d \ 2pi /

        """

        r2 = -np.dot(self.largeunitcell_cc.diagonal()**2 / (2 * pi)**2,

                     np.log(abs(self.Z_dww[:3].diagonal(0, 1, 2))**2))

        return np.sqrt(r2)

    def get_spectral_weight(self, w):

        return abs(self.V_knw[:, :, w])**2 / self.Nk

    def get_pdos(self, w, energies, width):

        """Projected density of states (PDOS).

        Returns the (PDOS) for Wannier function ``w``. The calculation

        is performed over the energy grid specified in energies. The

        PDOS is produced as a sum of Gaussians centered at the points

        of the energy grid and with the specified width.

        """

        spec_kn = self.get_spectral_weight(w)

        dos = np.zeros(len(energies))

        for k, spec_n in enumerate(spec_kn):

            eig_n = self.calc.get_eigenvalues(k=kpt, s=self.spin)

            for weight, eig in zip(spec_n, eig):

                # Add gaussian centered at the eigenvalue

                x = ((energies - center) / width)**2

                dos += weight * np.exp(-x.clip(0., 40.)) / (sqrt(pi) * width)

        return dos

    def max_spread(self, directions=[0, 1, 2]):

        """Returns the index of the most delocalized Wannier function

        together with the value of the spread functional"""

        d = np.zeros(self.nwannier)

        for dir in directions:

            d[dir] = np.abs(self.Z_dww[dir].diagonal())**2 *self.weight_d[dir]

        index = np.argsort(d)[0]

        print 'Index:', index

        print 'Spread:', d[index]           

    def translate(self, w, R):

        """Translate the w'th Wannier function

        The distance vector R = [n1, n2, n3], is in units of the basis

        vectors of the small cell.

        """

        for kpt_c, U_ww in zip(self.kpt_kc, self.U_kww):

            U_ww[:, w] *= np.exp(2.j * pi * np.dot(np.array(R), kpt_c))

        self.update()

    def translate_to_cell(self, w, cell):

        """Translate the w'th Wannier function to specified cell"""

        scaled_c = np.angle(self.Z_dww[:3, w, w]) * self.kptgrid / (2 * pi)

        trans = np.array(cell) - np.floor(scaled_c)

        self.translate(w, trans)

    def translate_all_to_cell(self, cell=[0, 0, 0]):

        """Translate all Wannier functions to specified cell.

        Move all Wannier orbitals to a specific unit cell.  There

        exists an arbitrariness in the positions of the Wannier

        orbitals relative to the unit cell. This method can move all

        orbitals to the unit cell specified by ``cell``.  For a

        `\Gamma`-point calculation, this has no effect. For a

        **k**-point calculation the periodicity of the orbitals are

        given by the large unit cell defined by repeating the original

        unitcell by the number of **k**-points in each direction.  In

        this case it is usefull to move the orbitals away from the

        boundaries of the large cell before plotting them. For a bulk

        calculation with, say 10x10x10 **k** points, one could move

        the orbitals to the cell [2,2,2].  In this way the pbc

        boundary conditions will not be noticed.

        """

        scaled_wc = np.angle(self.Z_dww[:3].diagonal(0, 1, 2)).T  * \

                    self.kptgrid / (2 * pi)

        trans_wc =  np.array(cell)[None] - np.floor(scaled_wc)

        for kpt_c, U_ww in zip(self.kpt_kc, self.U_kww):

            U_ww *= np.exp(2.j * pi * np.dot(trans_wc, kpt_c))

        self.update()

    def distances(self, R):

        Nw = self.nwannier

        cen = self.get_centers()

        r1 = cen.repeat(Nw, axis=0).reshape(Nw, Nw, 3)

        r2 = cen.copy()

        for i in range(3):

            r2 += self.unitcell_cc[i] * R[i]

        r2 = np.swapaxes(r2.repeat(Nw, axis=0).reshape(Nw, Nw, 3), 0, 1)

        return np.sqrt(np.sum((r1 - r2)**2, axis=-1))

    def get_hopping(self, R):

        """Returns the matrix H(R)_nm=<0,n|H|R,m>.

        ::

                                1   _   -ik.R 

          H(R) = <0,n|H|R,m> = --- >_  e      H(k)

                                Nk  k         

        where R is the cell-distance (in units of the basis vectors of

        the small cell) and n,m are indices of the Wannier functions.

        """

        H_ww = np.zeros([self.nwannier, self.nwannier], complex)

        for k, kpt_c in enumerate(self.kpt_kc):

            phase = np.exp(-2.j * pi * np.dot(np.array(R), kpt_c))

            H_ww += self.get_hamiltonian(k) * phase

        return H_ww / self.Nk

    def get_hamiltonian(self, k=0):

        """Get Hamiltonian at existing k-vector of index k

        ::

                  dag

          H(k) = V    diag(eps )  V

                  k           k    k

        """

        eps_n = self.calc.get_eigenvalues(kpt=k, spin=self.spin)

        return np.dot(dag(self.V_knw[k]) * eps_n, self.V_knw[k])

    def get_hamiltonian_kpoint(self, kpt_c):

        """Get Hamiltonian at some new arbitrary k-vector

        ::

                  _   ik.R 

          H(k) = >_  e     H(R)

                  R         

        Warning: This method moves all Wannier functions to cell (0, 0, 0)

        """

        if self.verbose:

            print 'Translating all Wannier functions to cell (0, 0, 0)'

        self.translate_all_to_cell()

        max = (self.kptgrid - 1) / 2

        max += max > 0

        N1, N2, N3 = max

        Hk = np.zeros([self.nwannier, self.nwannier], complex)

        for n1 in xrange(-N1, N1 + 1):

            for n2 in xrange(-N2, N2 + 1):

                for n3 in xrange(-N3, N3 + 1):

                    R = np.array([n1, n2, n3], float)

                    hop_ww = self.get_hopping(R)

                    phase = np.exp(+2.j * pi * np.dot(R, kpt_c))

                    Hk += hop_ww * phase

        return Hk

    def get_function(self, index, repeat=None):

        """Get Wannier function on grid.

        Returns an array with the funcion values of the indicated Wannier

        function on a grid with the size of the *repeated* unit cell.

        For a calculation using **k**-points the relevant unit cell for

        eg. visualization of the Wannier orbitals is not the original unit

        cell, but rather a larger unit cell defined by repeating the

        original unit cell by the number of **k**-points in each direction.

        Note that for a `\Gamma`-point calculation the large unit cell

        coinsides with the original unit cell.

        The large unitcell also defines the periodicity of the Wannier

        orbitals.

        ``index`` can be either a single WF or a coordinate vector in terms

        of the WFs.

        """

        # Default size of plotting cell is the one corresponding to k-points.

        if repeat is None:

            repeat = self.kptgrid

        N1, N2, N3 = repeat

        dim = self.calc.get_number_of_grid_points()

        largedim = dim * [N1, N2, N3]

        wanniergrid = np.zeros(largedim, dtype=complex)

        for k, kpt_c in enumerate(self.kpt_kc):

            # The coordinate vector of wannier functions

            if type(index) == int:

                vec_n = self.V_knw[k, :, index]

            else:   

                vec_n = np.dot(self.V_knw[k], index)

            wan_G = np.zeros(dim, complex)

            for n, coeff in enumerate(vec_n):

                wan_G += coeff * self.calc.get_pseudo_wave_function(

                    n, k, self.spin, pad=True)

            # Distribute the small wavefunction over large cell:

            for n1 in xrange(N1):

                for n2 in xrange(N2):

                    for n3 in xrange(N3): # sign?

                        e = np.exp(-2.j * pi * np.dot([n1, n2, n3], kpt_c))

                        wanniergrid[n1 * dim[0]:(n1 + 1) * dim[0],

                                    n2 * dim[1]:(n2 + 1) * dim[1],

                                    n3 * dim[2]:(n3 + 1) * dim[2]] += e * wan_G

        # Normalization

        wanniergrid /= np.sqrt(self.Nk)

        return wanniergrid

    def write_cube(self, index, fname, repeat=None, real=True):

        """Dump specified Wannier function to a cube file"""

        from ase.io.cube import write_cube

        # Default size of plotting cell is the one corresponding to k-points.

        if repeat is None:

            repeat = self.kptgrid

        atoms = self.calc.get_atoms() * repeat

        func = self.get_function(index, repeat)

        # Handle separation of complex wave into real parts

        if real:

            if self.Nk == 1:

                func *= np.exp(-1.j * np.angle(func.max()))

                if 0: assert max(abs(func.imag).flat) < 1e-4

                func = func.real

            else:

                func = abs(func)

        else:

            phase_fname = fname.split('.')

            phase_fname.insert(1, 'phase')

            phase_fname = '.'.join(phase_fname)

            write_cube(phase_fname, atoms, data=np.angle(func))

            func = abs(func)

        write_cube(fname, atoms, data=func)

    def localize(self, step=0.25, tolerance=1e-08,

                 updaterot=True, updatecoeff=True):

        """Optimize rotation to give maximal localization"""

        md_min(self, step, tolerance, verbose=self.verbose,

               updaterot=updaterot, updatecoeff=updatecoeff)

    def get_functional_value(self): 

        """Calculate the value of the spread functional.

        ::

          Tr[|ZI|^2]=sum(I)sum(n) w_i|Z_(i)_nn|^2,

        where w_i are weights."""

        a_d = np.sum(np.abs(self.Z_dww.diagonal(0, 1, 2))**2, axis=1)

        return np.dot(a_d, self.weight_d).real

    def get_gradients(self):

        # Determine gradient of the spread functional.

        # 

        # The gradient for a rotation A_kij is::

        # 

        #    dU = dRho/dA_{k,i,j} = sum(I) sum(k')

        #            + Z_jj Z_kk',ij^* - Z_ii Z_k'k,ij^*

        #            - Z_ii^* Z_kk',ji + Z_jj^* Z_k'k,ji

        # 

        # The gradient for a change of coefficients is::

        # 

        #   dRho/da^*_{k,i,j} = sum(I) [[(Z_0)_{k} V_{k'} diag(Z^*) +

        #                                (Z_0_{k''})^d V_{k''} diag(Z)] *

        #                                U_k^d]_{N+i,N+j}

        # 

        # where diag(Z) is a square,diagonal matrix with Z_nn in the diagonal, 

        # k' = k + dk and k = k'' + dk.

        # 

        # The extra degrees of freedom chould be kept orthonormal to the fixed

        # space, thus we introduce lagrange multipliers, and minimize instead::

        # 

        #     Rho_L=Rho- sum_{k,n,m} lambda_{k,nm} <c_{kn}|c_{km}>

        # 

        # for this reason the coefficient gradients should be multiplied

        # by (1 - c c^d).

        Nb = self.nbands

        Nw = self.nwannier

        dU = []

        dC = []

        for k in xrange(self.Nk):

            M = self.fixedstates_k[k]

            L = self.edf_k[k]

            U_ww = self.U_kww[k]

            C_ul = self.C_kul[k]

            Utemp_ww = np.zeros((Nw, Nw), complex)

            Ctemp_nw = np.zeros((Nb, Nw), complex)

            for d, weight in enumerate(self.weight_d):

                if abs(weight) < 1.0e-6:

                    continue

                Z_knn = self.Z_dknn[d]

                diagZ_w = self.Z_dww[d].diagonal()

                Zii_ww = np.repeat(diagZ_w, Nw).reshape(Nw, Nw)

                k1 = self.kklst_dk[d, k]

                k2 = self.invkklst_dk[d, k]

                V_knw = self.V_knw

                Z_kww = self.Z_dkww[d]

                if L > 0:

                    Ctemp_nw += weight * np.dot(

                        np.dot(Z_knn[k], V_knw[k1]) * diagZ_w.conj() +

                        np.dot(dag(Z_knn[k2]), V_knw[k2]) * diagZ_w,

                        dag(U_ww))

                temp = Zii_ww.T * Z_kww[k].conj() - Zii_ww * Z_kww[k2].conj()

                Utemp_ww += weight * (temp - dag(temp))

            dU.append(Utemp_ww.ravel())

            if L > 0:

                # Ctemp now has same dimension as V, the gradient is in the

                # lower-right (Nb-M) x L block

                Ctemp_ul = Ctemp_nw[M:, M:]

                G_ul = Ctemp_ul - np.dot(np.dot(C_ul, dag(C_ul)), Ctemp_ul)

                dC.append(G_ul.ravel())

        return np.concatenate(dU + dC)

    def step(self, dX, updaterot=True, updatecoeff=True):

        # dX is (A, dC) where U->Uexp(-A) and C->C+dC

        Nw = self.nwannier

        Nk = self.Nk

        M_k = self.fixedstates_k

        L_k = self.edf_k

        if updaterot:

            A_kww = dX[:Nk * Nw**2].reshape(Nk, Nw, Nw)

            for U, A in zip(self.U_kww, A_kww):

                H = -1.j * A.conj()

                epsilon, Z = np.linalg.eigh(H)

                # Z contains the eigenvectors as COLUMNS.

                # Since H = iA, dU = exp(-A) = exp(iH) = ZDZ^d

                dU = np.dot(Z * np.exp(1.j * epsilon), dag(Z))

                U[:] = np.dot(U, dU)

        if updatecoeff:

            start = 0

            for C, unocc, L in zip(self.C_kul, self.nbands - M_k, L_k):

                if L == 0 or unocc == 0:

                    continue

                Ncoeff = L * unocc

                deltaC = dX[Nk * Nw**2 + start: Nk * Nw**2 + start + Ncoeff]

                C += deltaC.reshape(unocc, L)

                gram_schmidt(C)

                start += Ncoeff

        self.update()