pyscf.gw.utils package

Submodules

pyscf.gw.utils.ac_grid module

Grids and analytical continuation functions for GW and RPA.

References:

T. Zhu and G.K.-L. Chan, J. Chem. Theory. Comput. 17, 727-741 (2021) New J. Phys. 14 053020 (2012)

class pyscf.gw.utils.ac_grid.AC_Method(*args, **options)

Bases: ABC

Base class for AC methods

Attributes

shapetuple

Shape of the data array that was passed to ac_fit, excluding the frequency axis.

abstractmethod ac_eval(freqs: ndarray, axis: int = -1)

After you call ac_fit, you can call this function to evaluate the AC at

arbitrary complex frequencies.

Parameters

freqsnp.ndarray[np.complex128] | complex

1D array of complex frequencies, or a single complex frequency

axisint, optional

Indicates which axis of the output array should correspond to the frequency axis, by default -1. Example: if you want (nmo, nmo, nfreq), call ac_eval(freqs, axis=-1)

if you want (nfreq, nmo, nmo), call ac_eval(data, freqs, axis=0)

If freqs is a scalar, the output shape will be the same as the input data shape.

Returns

np.ndarray

Interpolated/approximated values at the new complex frequencies

abstractmethod ac_fit(data: ndarray, omega: ndarray, axis: int = -1)

The kernel of the AC method. This function should be implemented in the derived class.

Parameters

datanp.ndarray

Data to be fit, e.g. self energy

omeganp.ndarray[np.double]

1D imaginary frequency grid

axisint, optional

Indicates which axis of the data array corresponds to the frequency axis, by default -1. Example: data.shape is (nmo, nmo, nw), call ac_fit(data, omega, axis=-1)

data.shape is (nw, nmo, nmo), call ac_fit(data, omega, axis=0)

abstractmethod diagonal(axis1=0, axis2=1)

Create a new instance of the AC method with only the diagonal elements

of the self.coeff tensor. Convenient for getting diagonal of self-energy after calculating the full self-energy matrix.

This behaves more or less the same as np.diagonal.

Parameters

axis1int, optional

First axis, by default 0

axis2int, optional

Second axis, by default 1

Returns

object

New instance of the AC method with only the diagonal elements.

save(chkfilename: str, dataname: str = 'ac')

Save the AC object and coefficients to an HDF5 file.

Parameters

pyscf.gw.utils.ac_grid.AC_pade_thiele_diag(sigma, omega, npts=18, step_ratio=0.6666666666666666)

Pade fitting for diagonal elements for a matrix.

Parameters

sigmacomplex ndarray

matrix to fit, (norbs, nomega)

omegacomplex array

frequency of the matrix sigma (nomega)

nptsint, optional

number of selected points, by default 18

step_ratio_type_, optional

step ratio to select points, by default 2.0/3.0

Returns

acobj.coeffcomplex ndarray

fitting coefficient

acobj.omega[acobj.idx]complex ndarray

selected frequency points for fitting

pyscf.gw.utils.ac_grid.AC_pade_thiele_full(sigma, omega, npts=18, step_ratio=0.6666666666666666)

Pade fitting for full matrix.

Parameters

sigmacomplex ndarray

matrix to fit, (norbs, nomega)

omegacomplex array

frequency of the matrix sigma (nomega)

nptsint, optional

number of selected points, by default 18

step_ratio_type_, optional

step ratio to select points, by default 2.0/3.0

Returns

acobj.coeffcomplex ndarray

fitting coefficient

acobj.omega[acobj.idx]complex ndarray

selected frequency points for fitting

class pyscf.gw.utils.ac_grid.PadeAC(*args, npts=18, step_ratio=0.6666666666666666, **options)

Bases: AC_Method

Analytic continuation to real axis using a Pade approximation from Thiele’s reciprocal difference method Reference: J. Low Temp. Phys. 29, 179 (1977)

ac_eval(freqs, axis=-1)

Evaluate Pade AC at arbitrary complex frequencies.

Parameters

freqsnp.ndarray[np.complex128]

1D array of complex frequencies

axisint, optional

Indicates which axis of the output array should correspond to the frequency axis, by default -1. Example: if you want (nmo, nmo, nfreq), call ac_eval(freqs, axis=-1)

if you want (nfreq, nmo, nmo), call ac_eval(data, freqs, axis=0)

Returns

np.ndarray

Pade-Thiele AC evaluated at freqs

ac_fit(data, omega, axis=-1)

Compute Pade-Thiele AC coefficients for the given data and omega.

Parameters

datanp.ndarray

Data to be fit, e.g. self energy

omeganp.ndarray[np.double]

1D imaginary frequency grid

axisint, optional

Indicates which axis of the data array corresponds to the frequency axis, by default -1. Example: data.shape is (nmo, nmo, nw), call ac_fit(data, omega, axis=-1)

data.shape is (nw, nmo, nmo), call ac_fit(data, omega, axis=0)

diagonal(axis1=0, axis2=1)

Create a new instance of the AC method with only the diagonal elements

of the self.coeff tensor. Convenient for getting diagonal of self-energy after calculating the full self-energy matrix.

This behaves more or less the same as np.diagonal.

Parameters

axis1int, optional

First axis, by default 0

axis2int, optional

Second axis, by default 1

Returns

object

New instance of the AC method with only the diagonal elements.

property omega_fit

Return the frequency grid used for fitting.

class pyscf.gw.utils.ac_grid.TwoPoleAC(orbs, nocc, **options)

Bases: AC_Method

Two-pole analytic continuation method.

ac_eval(freqs, axis=-1)

Evaluate two-pole AC at arbitrary complex frequencies.

Parameters

freqsnp.ndarray[np.complex128]

1D array of complex frequencies

axisint, optional

Indicates which axis of the output array should correspond to the frequency axis, by default -1. Example: if you want (nmo, nmo, nfreq), call ac_eval(freqs, axis=-1)

if you want (nfreq, nmo, nmo), call ac_eval(data, freqs, axis=0)

Returns

np.ndarray

Pade-Thiele AC evaluated at freqs

ac_fit(data, omega, axis=-1)

Compute two-pole AC coefficients for the given data and omega.

Parameters

datanp.ndarray

Data to be fit, e.g. self energy

omeganp.ndarray[np.double]

1D imaginary frequency grid

axisint, optional

Indicates which axis of the data array corresponds to the frequency axis, by default -1. Example: data.shape is (nmo, nmo, nw), call ac_fit(data, omega, axis=-1)

data.shape is (nw, nmo, nmo), call ac_fit(data, omega, axis=0)

diagonal(axis1=0, axis2=1)

Create a new instance of the AC method with only the diagonal elements

of the self.coeff tensor. Convenient for getting diagonal of self-energy after calculating the full self-energy matrix.

This behaves more or less the same as np.diagonal.

Parameters

axis1int, optional

First axis, by default 0

axis2int, optional

Second axis, by default 1

Returns

object

New instance of the AC method with only the diagonal elements.

pyscf.gw.utils.ac_grid.load_ac(chkfilename: str, dataname: str = 'ac') → AC_Method

Load an AC object from an HDF5 file.

Parameters

chkfilenamestr

Path to the HDF5 file

datanamestr, optional

Name of the dataset in the HDF5 file, by default ‘ac’

Returns

AC_Method

The loaded AC object

pyscf.gw.utils.ac_grid.pade_thiele_ndarray(freqs, zn, coeff)

NDarray-friendly analytic continuation using Pade-Thiele method.

Parameters

freqsnp.ndarray, shape (nfreqs,), complex

Points in the complex plane at which to evaluate the analytic continuation.

znnp.ndarray, shape (ncoeff,), complex

interpolation points

coeffnp.ndarray, shape (ncoeff, M1, M2, …,), complex

Pade-Thiele coefficients

Returns

np.ndarray, shape (nfreqs, M1, M2, …,), complex

Pade-Thiele analytic continuation evaluated at freqs

pyscf.gw.utils.ac_grid.thiele_ndarray(fn, zn)

Iterative Thiele algorithm to compute coefficients of Pade approximant

Parameters

fnnp.ndarray, shape (nw, N1, N2, …,), complex

Function values at the points zn

znnp.ndarray, shape(nw,), complex

Points in the complex plane used to compute fn

Returns

np.ndarray, shape(nw, N1, N2, …), complex

Coefficients of Pade approximant

pyscf.gw.utils.ac_grid.two_pole(freqs, coeff)

pyscf.gw.utils.ac_grid.two_pole_fit(coeff, omega, sigma)

pyscf.gw.utils.gw_np_helper module

Math functions for GW and RPA.

References:

T. Zhu and G.K.-L. Chan, J. Chem. Theory. Comput. 17, 727-741 (2021) New J. Phys. 14 053020 (2012)

pyscf.gw.utils.gw_np_helper.addto_diagonal(arr, x)

Add a scalar or vector to the diagonal of a matrix.

Parameters

arr(M, M) array_like

Square matrix (will be modified)

x(M,) array_like | scalar

Vector or scalar.

Returns

arr(M, M) ndarray

arr[i, i] <- arr[i, i] + x[i].

pyscf.gw.utils.gw_np_helper.array_scale(arr, alpha)

Scale an array in place by a scalar with BLAS if possible. arr <- alpha * arr

Parameters

arrarray_like

Array to be scaled.

alphascalar

Scale factor.

pyscf.gw.utils.gw_np_helper.get_id_minus_pi(Pi)

Calculate I - Pi in place.

Parameters

Pi(M, M) array_like

Input matrix.

Returns

id_minus_pi(M, M) ndarray

(I - Pi)

pyscf.gw.utils.gw_np_helper.get_id_minus_pi_inv(Pi, overwrite_input=False)

Calculate (I - Pi)^-1, given Pi

Parameters

Pi(M, M) array_like

Input matrix. Must be C-contiguous.

Returns

Pi_inv(M, M) ndarray

(I - Pi)^-1

pyscf.gw.utils.gw_np_helper.get_id_minus_pi_inv_minus_id(Pi, overwrite_input=False)

Calculate (I - Pi)^-1 - I.

Parameters

Pi(M, M) array_like

Input matrix. Must be C-contiguous.

Returns

Pi_inv(M, M) ndarray

(I - Pi)^-1 - I.

pyscf.gw.utils.gw_np_helper.mkslice(l)

Try to make a slice from a list of integers. If this is not possible, return the input unchanged.

Parameters

lslice | list | range | ndarray

Various ways of representing a list of integer indices

Returns

slice | list | ndarray

slice if possible, otherwise returns the input unchanged

Module contents