momentGW.pbc.rpa

Construct RPA moments with periodic boundary conditions.

Module Contents

class momentGW.pbc.rpa.dRPA(gw, nmom_max, integrals, mo_energy=None, mo_occ=None)

Bases: momentGW.pbc.tda.dTDA, momentGW.rpa.dRPA

Compute the self-energy moments using dRPA and numerical integration with periodic boundary conditions.

Parameters:

  • gw (BaseKGW) – GW object.

  • nmom_max (int) – Maximum moment number to calculate.

  • integrals (KIntegrals) – Density-fitted integrals at each k-point.

  • mo_energy (dict, optional) – Molecular orbital energies. Keys are “g” and “w” for the Green’s function and screened Coulomb interaction, respectively. If None, use gw.mo_energy for both. Default value is None.

  • mo_occ (dict, optional) – Molecular orbital occupancies. Keys are “g” and “w” for the Green’s function and screened Coulomb interaction, respectively. If None, use gw.mo_occ for both. Default value is None.

Notes

See momentGW.tda.dTDA.__init__ for initialisation details and momentGW.tda.dTDA.kernel for calculation run details.

property nov

Get the number of ov states in W.

property kpts

Get the k-points.

property nkpts

Get the number of k-points.

property nmo

Get the number of MOs.

property naux

Get the number of auxiliaries.

integrate()

Optimise the quadrature and perform the integration for a given set of k-points for the zeroth moment.

Returns:

integral – Integral array, including the offset part.

Return type:

numpy.ndarray

build_dd_moments(integral=None)

Build the moments of the density-density response.

Parameters:

integral (numpy.ndarray, optional) – Integral array, including the offset part. If None, calculate from scratch. Default is None.

Returns:

moments – Moments of the density-density response.

Return type:

numpy.ndarray

optimise_offset_quad(d, diag_eri, name='main')

Optimise the grid spacing of Gauss-Laguerre quadrature for the offset integral.

Parameters:

  • d (numpy.ndarray) – Orbital energy differences at each k-point.

  • diag_eri (numpy.ndarray) – Diagonal of the ERIs at each k-point.

  • name (str, optional) – Name of the integral. Default value is “main”.

Returns:

  • points (numpy.ndarray) – The quadrature points.

  • weights (numpy.ndarray) – The quadrature weights.

eval_diag_offset_integral(quad, d, diag_eri)

Evaluate the diagonal of the offset integral.

Parameters:

  • quad (tuple) – The quadrature points and weights.

  • d (numpy.ndarray) – Orbital energy differences at each k-point.

  • diag_eri (numpy.ndarray) – Diagonal of the ERIs at each k-point.

Returns:

integral – Offset integral.

Return type:

numpy.ndarray

eval_offset_integral(quad, d, Lia=None)

Evaluate the offset integral.

Parameters:

  • quad (tuple) – The quadrature points and weights.

  • d (numpy.ndarray) – Orbital energy differences at each k-point.

  • Lia (dict of numpy.ndarray) – Dict. with keys that are pairs of k-point indices (Nkpt, Nkpt) with an array of form (aux, W occ, W vir) at this k-point pair. The 1st Nkpt is defined by the difference between k-points and the second index’s kpoint. If None, use self.integrals.Lia.

  • Liad (dict of numpy.ndarray) – Product of Lia and the orbital energy differences at each k-point.

Returns:

integral – Offset integral.

Return type:

numpy.ndarray

optimise_main_quad(d, diag_eri, name='main')

Optimise the grid spacing of Clenshaw-Curtis quadrature for the main integral.

Parameters:

  • d (numpy.ndarray) – Orbital energy differences at each k-point.

  • diag_eri (numpy.ndarray) – Diagonal of the ERIs at each k-point.

Returns:

  • points (numpy.ndarray) – The quadrature points.

  • weights (numpy.ndarray) – The quadrature weights.

eval_diag_main_integral(quad, d, d_sq, d_eri, d_sq_eri)

Evaluate the diagonal of the main integral.

Parameters:

  • quad (tuple) – The quadrature points and weights.

  • d (numpy.ndarray) – Orbital energy differences at each k-point.

  • d_sq (numpy.ndarray) – Orbital energy differences squared at each k-point. See “optimise_main_quad” for more details.

  • d_eri (numpy.ndarray) – Orbital energy differences times the diagonal of the ERIs at each k-point. See “optimise_main_quad” for more details.

  • d_sq_eri (numpy.ndarray) – Orbital energy differences times the diagonal of the ERIs plus the orbital energy differences at each k-point. See “optimise_main_quad” for more details.

Returns:

integral – Main integral.

Return type:

numpy.ndarray

eval_main_integral(quad, d, Lia=None)

Evaluate the main integral.

Parameters:

  • quad (tuple) – The quadrature points and weights.

  • Variables –

  • ---------- –

  • d (numpy.ndarray) – Orbital energy differences at each k-point.

  • Lia (dict of numpy.ndarray) – Dict. with keys that are pairs of k-point indices (Nkpt, Nkpt) with an array of form (aux, W occ, W vir) at this k-point pair. The 1st Nkpt is defined by the difference between k-points and the second index’s kpoint. If None, use self.integrals.Lia.

  • Liad (dict of numpy.ndarray) – Product of Lia and the orbital energy differences at each k-point.

Returns:

integral – Offset integral.

Return type:

numpy.ndarray

kernel(exact=False)

Run the polarizability calculation to compute moments of the self-energy.

Parameters:

exact (bool, optional) – Has no effect and is only present for compatibility with dRPA. Default value is False.

Returns:

  • moments_occ (numpy.ndarray) – Moments of the occupied self-energy at each k-point.

  • moments_vir (numpy.ndarray) – Moments of the virtual self-energy at each k-point.

convolve(eta, mo_energy_g=None, mo_occ_g=None)

Handle the convolution of the moments of the Green’s function and screened Coulomb interaction.

Parameters:

  • eta (numpy.ndarray) – Moments of the density-density response partly transformed into moments of the screened Coulomb interaction at each k-point.

  • mo_energy_g (numpy.ndarray, optional) – Energies of the Green’s function at each k-point. If None, use self.mo_energy_g. Default value is None.

  • mo_occ_g (numpy.ndarray, optional) – Occupancies of the Green’s function at each k-point. If None, use self.mo_occ_g. Default value is None.

Returns:

  • moments_occ (numpy.ndarray) – Moments of the occupied self-energy at each k-point.

  • moments_vir (numpy.ndarray) – Moments of the virtual self-energy at each k-point.

build_se_moments(moments_dd)

Build the moments of the self-energy via convolution.

Parameters:

moments_dd (numpy.ndarray) – Moments of the density-density response at each k-point.

Returns:

  • moments_occ (numpy.ndarray) – Moments of the occupied self-energy at each k-point.

  • moments_vir (numpy.ndarray) – Moments of the virtual self-energy at each k-point.

build_dp_moments()

Build the moments of the dynamic polarizability for optical spectra calculations.

Returns:

moments – Moments of the dynamic polarizability.

Return type:

numpy.ndarray

build_dd_moment_inv()

Build the first inverse (n=-1) moment of the density-density response.

Returns:

moment – First inverse (n=-1) moment of the density-density response.

Return type:

numpy.ndarray

Notes

This is not the full n=-1 moment, which is

\[\begin{split}D^{-1} - D^{-1} V^\dagger (I + V D^{-1} V^\dagger)^{-1} \\ V D^{-1}\end{split}\]

but rather

\[(I + V D^{-1} V^\dagger)^{-1} V D^{-1}\]

which ensures that the function scales properly. The final contractions are done when constructing the matrix-vector product.

mpi_slice(n)

Return the start and end index for the current process for total size n.

Parameters:

n (int) – Total size.

Returns:

  • p0 (int) – Start index for current process.

  • p1 (int) – End index for current process.

mpi_size(n)

Return the number of states in the current process for total size n.

Parameters:

n (int) – Total size.

Returns:

size – Number of states in current process.

Return type:

int

build_dd_moments_exact()

Build the exact moments of the density-density response.

Returns:

moments – Moments of the density-density response.

Return type:

numpy.ndarray

static rescale_quad(bare_quad, a)

Rescale quadrature for grid space a.

Parameters:

  • bare_quad (tuple) – The quadrature points and weights.

  • a (float) – Grid spacing.

Returns:

  • points (numpy.ndarray) – The quadrature points.

  • weights (numpy.ndarray) – The quadrature weights.

get_optimal_quad(bare_quad, integrand, exact, name=None)

Get the optimal quadrature.

Parameters:

  • bare_quad (tuple) – The quadrature points and weights.

  • integrand (function) – The integrand function.

  • exact (float) – The exact value of the integral.

  • name (str, optional) – Name of the integral. Default value is None.

Returns:

  • points (numpy.ndarray) – The quadrature points.

  • weights (numpy.ndarray) – The quadrature weights.

gen_clencur_quad_inf(even=False)

Generate quadrature points and weights for Clenshaw-Curtis quadrature over an (-inf, +inf).

Parameters:

even (bool, optional) – Whether to assume an even grid. Default is False.

Returns:

  • points (numpy.ndarray) – Quadrature points.

  • weights (numpy.ndarray) – Quadrature weights.

gen_gausslag_quad_semiinf()

Generate quadrature points and weights for Gauss-Laguerre quadrature over an (0, +inf).

Returns:

  • points (numpy.ndarray) – Quadrature points.

  • weights (numpy.ndarray) – Quadrature weights.

estimate_error_clencur(i4, i2, imag_tol=1e-10)

Estimate the quadrature error for Clenshaw-Curtis quadrature.

Parameters:

  • i4 (numpy.ndarray) – Integral at one-quarter the number of points.

  • i2 (numpy.ndarray) – Integral at one-half the number of points.

  • imag_tol (float, optional) – Threshold to consider the imaginary part of a root to be zero. Default value is 1e-10.

Returns:

error – Estimated error.

Return type:

numpy.ndarray