pyscf.gw package

Submodules

pyscf.gw.gw_ac module

Spin-restricted G0W0 approximation with analytic continuation This implementation has N^4 scaling, and is faster than GW-CD (N^4) and analytic GW (N^6) methods. GW-AC is recommended for valence states only, and is inaccuarate for core states.

Method:

See T. Zhu and G.K.-L. Chan, arxiv:2007.03148 (2020) for details Compute Sigma on imaginary frequency with density fitting, then analytically continued to real frequency

Useful References:

J. Chem. Theory Comput. 12, 3623-3635 (2016) New J. Phys. 14, 053020 (2012)

pyscf.gw.gw_ac.AC_pade_thiele_diag(sigma, omega)

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

coeff: 2D array (ncoeff, norbs) omega: 2D array (norbs, npade)

pyscf.gw.gw_ac.AC_twopole_diag(sigma, omega, orbs, nocc)

Analytic continuation to real axis using a two-pole model Returns:

coeff: 2D array (ncoeff, norbs)

class pyscf.gw.gw_ac.GWAC(mf, frozen=None)

Bases: StreamObject

ac = 'pade'

ao2mo(mo_coeff=None)

dump_flags()

get_frozen_mask()

Get boolean mask for the restricted reference orbitals.

In the returned boolean (mask) array of frozen orbital indices, the element is False if it corresonds to the frozen orbital.

get_nmo()

get_nocc()

kernel(mo_energy=None, mo_coeff=None, Lpq=None, orbs=None, nw=100, vhf_df=False)

Input:

orbs: self-energy orbs nw: grid number vhf_df: whether using density fitting for HF exchange

Output:

mo_energy: GW quasiparticle energy

linearized = False

property nmo

property nocc

pyscf.gw.gw_ac.get_rho_response(omega, mo_energy, Lpq)

Compute density response function in auxiliary basis at freq iw

pyscf.gw.gw_ac.get_sigma_diag(gw, orbs, Lpq, freqs, wts, iw_cutoff=None)

Compute GW correlation self-energy (diagonal elements) in MO basis on imaginary axis

pyscf.gw.gw_ac.kernel(gw, mo_energy, mo_coeff, Lpq=None, orbs=None, nw=None, vhf_df=False, verbose=3)

GW-corrected quasiparticle orbital energies Returns:

A list : converged, mo_energy, mo_coeff

pyscf.gw.gw_ac.pade_thiele(freqs, zn, coeff)

pyscf.gw.gw_ac.thiele(fn, zn)

pyscf.gw.gw_ac.two_pole(freqs, coeff)

pyscf.gw.gw_ac.two_pole_fit(coeff, omega, sigma)

pyscf.gw.gw_cd module

Spin-restricted G0W0 approximation with contour deformation

This implementation has the same scaling (N^4) as GW-AC, more robust but slower. GW-CD is particularly recommended for accurate core and high-energy states.

Method:

See T. Zhu and G.K.-L. Chan, arxiv:2007.03148 (2020) for details Compute Sigma directly on real axis with density fitting through a contour deformation method

Useful References:

10. Chem. Theory Comput. 14, 4856-4869 (2018)

class pyscf.gw.gw_cd.GWCD(mf, frozen=None)

Bases: StreamObject

ao2mo(mo_coeff=None)

dump_flags()

eta = 0.001

get_frozen_mask()

Get boolean mask for the restricted reference orbitals.

In the returned boolean (mask) array of frozen orbital indices, the element is False if it corresonds to the frozen orbital.

get_nmo()

get_nocc()

kernel(mo_energy=None, mo_coeff=None, Lpq=None, orbs=None, nw=100, vhf_df=False)

Input:

orbs: self-energy orbs nw: grid number

Output:

mo_energy: GW quasiparticle energy

linearized = False

property nmo

property nocc

pyscf.gw.gw_cd.get_WmnI_diag(gw, orbs, Lpq, freqs)

Compute W_mn(iw) on imarginary axis grids Return:

Wmn: (Nmo, Norbs, Nw)

pyscf.gw.gw_cd.get_rho_response(omega, mo_energy, Lpq)

Compute density response function in auxiliary basis at freq iw

pyscf.gw.gw_cd.get_rho_response_R(gw, omega, Lpq)

Compute density response function in auxiliary basis at poles

pyscf.gw.gw_cd.get_sigmaI_diag(gw, omega, Wmn, sign, freqs, wts)

Compute self-energy by integrating on imaginary axis

pyscf.gw.gw_cd.get_sigmaR_diag(gw, omega, orbp, ef, Lpq)

Compute self-energy for poles inside coutour (more and more expensive away from Fermi surface)

pyscf.gw.gw_cd.get_sigma_diag(gw, ep, p, Lpq, Wmn, freqs, wts)

Compute self-energy on real axis using contour deformation

pyscf.gw.gw_cd.kernel(gw, mo_energy, mo_coeff, Lpq=None, orbs=None, nw=None, vhf_df=False, verbose=3)

GW-corrected quasiparticle orbital energies

Returns:

A list : converged, mo_energy, mo_coeff

pyscf.gw.gw_exact module

Spin-restricted G0W0 approximation with exact frequency integration

class pyscf.gw.gw_exact.GWExact(mf, frozen=None, tdmf=None)

Bases: StreamObject

non-relativistic restricted GW

Saved results

mo_energy :

Orbital energies

mo_coeff

Orbital coefficients

ao2mo(mo_coeff=None)

dump_flags(verbose=None)

eta = 1e-08

get_frozen_mask()

Get boolean mask for the restricted reference orbitals.

In the returned boolean (mask) array of frozen orbital indices, the element is False if it corresonds to the frozen orbital.

get_g(omega, eta=None)

get_g0(omega, eta=None)

get_nmo()

get_nocc()

kernel(mo_energy=None, mo_coeff=None, td_e=None, td_xy=None, eris=None, orbs=None)

Kernel function is the main driver of a method. Every method should define the kernel function as the entry of the calculation. Note the return value of kernel function is not strictly defined. It can be anything related to the method (such as the energy, the wave-function, the DFT mesh grids etc.).

linearized = False

property nmo

property nocc

reset(mol=None)

pyscf.gw.gw_exact.get_g(omega, mo_energy, mo_occ, eta)

pyscf.gw.gw_exact.get_sigma_deriv_element(gw, omega, tdm_p, tdm_q, td_e, eta=None, vir_sgn=1)

pyscf.gw.gw_exact.get_sigma_element(gw, omega, tdm_p, tdm_q, td_e, eta=None, vir_sgn=1)

pyscf.gw.gw_exact.kernel(gw, mo_energy, mo_coeff, td_e, td_xy, eris=None, orbs=None, verbose=3)

GW-corrected quasiparticle orbital energies

Returns:

A list : converged, mo_energy, mo_coeff

pyscf.gw.gw_slow module

This module implements the G0W0 approximation on top of pyscf.tdscf.rhf_slow and pyscf.tdscf.proxy TD implementations. Unlike gw.py, all integrals are stored in memory. Several variants of GW are available:

• (this module) pyscf.gw_slow: the molecular implementation;

• pyscf.pbc.gw.gw_slow: single-kpoint PBC (periodic boundary condition) implementation;

• pyscf.pbc.gw.kgw_slow_supercell: a supercell approach to PBC implementation with multiple k-points. Runs the molecular code for a model with several k-points for the cost of discarding momentum conservation and using dense instead of sparse matrixes;

• pyscf.pbc.gw.kgw_slow: a PBC implementation with multiple k-points;

class pyscf.gw.gw_slow.AbstractIMDS(td, eri=None)

Bases: object

entire_space()

The entire orbital space. Returns:

An iterable of the entire orbital space.

get_rhs(p)

The right-hand side of the quasiparticle equation. Args:

p (int, tuple): the orbital;

Returns:

Right-hand sides of the quasiparticle equation

get_sigma_element(omega, p, **kwargs)

The diagonal matrix element of the self-energy matrix. Args:

omega (float): the energy value; p (int, tuple): the orbital;

Returns:

The diagonal matrix element.

initial_guess(p)

Retrieves the initial guess for the quasiparticle energy for orbital p. Args:

p (int, tuple): the orbital;

Returns:

The value of initial guess (float).

orb_dims = 1

quasiparticle_eq(p, **kwargs)

The quasiparticle equation f(omega) = 0. Args:

p (int, tuple): the orbital; **kwargs: keyword arguments to get_sigma_element;

Returns:

A callable function of one parameter.

class pyscf.gw.gw_slow.GW(td, eri=None)

Bases: object

base_imds

alias of IMDS

kernel()

Calculates GW roots.

Returns:

GW roots.

class pyscf.gw.gw_slow.IMDS(td, eri=None)

Bases: AbstractIMDS

construct_tdm()

property entire_space

The entire orbital space. Returns:

An iterable of the entire orbital space.

get_rhs(p)

The right-hand side of the quasiparticle equation. Args:

p (int, tuple): the orbital;

Returns:

Right-hand sides of the quasiparticle equation

get_sigma_element(omega, p, eta, vir_sgn=1)

The diagonal matrix element of the self-energy matrix. Args:

omega (float): the energy value; p (int, tuple): the orbital;

Returns:

The diagonal matrix element.

initial_guess(p)

Retrieves the initial guess for the quasiparticle energy for orbital p. Args:

p (int, tuple): the orbital;

Returns:

The value of initial guess (float).

class pyscf.gw.gw_slow.LoggingFunction(m)

Bases: object

plot_call_history(title='')

Plots calls to this function. Args:

title (str): plot title;

property x

property y

pyscf.gw.gw_slow.corrected_moe(eri, p)

Calculates the corrected orbital energy. Args:

eri (PhysERI): a container with electron repulsion integrals; p (int): orbital;

Returns:

The corrected orbital energy.

pyscf.gw.gw_slow.kernel(imds, orbs=None, linearized=False, eta=0.001, tol=1e-09, method='fallback')

Calculates GW energies. Args:

imds (AbstractIMDS): GW intermediates; orbs (Iterable): indexes of MO orbitals to correct; linearized (bool): whether to apply a single-step linearized correction to energies instead of iterative procedure; eta (float): imaginary energy for the Green’s function; tol (float): tolerance for the search of zero; method (str): ‘bisect’ finds roots no matter what but, potentially, wrong ones, ‘newton’ finding roots close to the correct one but, potentially, failing during iterations, or ‘fallback’ using ‘newton’ and proceeding to ‘bisect’ in case of failure;

Returns:

Corrected orbital energies.

pyscf.gw.rpa module

Spin-restricted random phase approximation (direct RPA/dRPA in chemistry) with N^4 scaling

Method:

Main routines are based on GW-AC method descirbed in: T. Zhu and G.K.-L. Chan, J. Chem. Theory. Comput. 17, 727-741 (2021) X. Ren et al., New J. Phys. 14, 053020 (2012)

class pyscf.gw.rpa.RPA(mf, frozen=None, auxbasis=None)

Bases: StreamObject

ao2mo(mo_coeff=None)

dump_flags()

get_frozen_mask()

Get boolean mask for the restricted reference orbitals.

In the returned boolean (mask) array of frozen orbital indices, the element is False if it corresonds to the frozen orbital.

get_nmo()

get_nocc()

kernel(mo_energy=None, mo_coeff=None, Lpq=None, nw=40, x0=0.5)

Args:

mo_energy : 1D array (nmo), mean-field mo energy mo_coeff : 2D array (nmo, nmo), mean-field mo coefficient Lpq : 3D array (naux, nmo, nmo), 3-index ERI nw: integer, grid number x0: real, scaling factor for frequency grid

Returns:

self.e_tot : RPA total eenrgy self.e_hf : EXX energy self.e_corr : RPA correlation energy

property nmo

property nocc

pyscf.gw.rpa.get_rho_response(omega, mo_energy, Lpq)

Compute density response function in auxiliary basis at freq iw.

pyscf.gw.rpa.get_rpa_ecorr(rpa, Lpq, freqs, wts)

Compute RPA correlation energy

pyscf.gw.rpa.kernel(rpa, mo_energy, mo_coeff, Lpq=None, nw=40, x0=0.5, verbose=3)

RPA correlation and total energy

Args:

Lpq : density fitting 3-center integral in MO basis. nw : number of frequency point on imaginary axis. x0: scaling factor for frequency grid.

Returns:

e_tot : RPA total energy e_hf : EXX energy e_corr : RPA correlation energy

pyscf.gw.ugw_ac module

Spin-unrestricted G0W0 approximation with analytic continuation

This implementation has N^4 scaling, and is faster than GW-CD (N^4) and analytic GW (N^6) methods. GW-AC is recommended for valence states only, and is inaccuarate for core states.

Method:

See T. Zhu and G.K.-L. Chan, arxiv:2007.03148 (2020) for details Compute Sigma on imaginary frequency with density fitting, then analytically continued to real frequency

Useful References:

J. Chem. Theory Comput. 12, 3623-3635 (2016) New J. Phys. 14, 053020 (2012)

pyscf.gw.ugw_ac.AC_pade_thiele_diag(sigma, omega)

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

coeff: 2D array (ncoeff, norbs) omega: 2D array (norbs, npade)

pyscf.gw.ugw_ac.AC_twopole_diag(sigma, omega, orbs, nocc)

Analytic continuation to real axis using a two-pole model Returns:

coeff: 2D array (ncoeff, norbs)

class pyscf.gw.ugw_ac.UGWAC(mf, frozen=None)

Bases: StreamObject

ac = 'pade'

ao2mo(mo_coeff=None)

dump_flags()

get_frozen_mask()

Get boolean mask for the unrestricted reference orbitals.

In the returned boolean (mask) array of frozen orbital indices, the element is False if it corresonds to the frozen orbital.

get_nmo()

get_nocc()

kernel(mo_energy=None, mo_coeff=None, Lpq=None, orbs=None, nw=100, vhf_df=False)

Input:

orbs: self-energy orbs nw: grid number vhf_df: whether using density fitting for HF exchange

Output:

mo_energy: GW quasiparticle energy

linearized = False

property nmo

property nocc

pyscf.gw.ugw_ac.get_rho_response(omega, mo_energy, Lpqa, Lpqb)

Compute density response function in auxiliary basis at freq iw

pyscf.gw.ugw_ac.get_sigma_diag(gw, orbs, Lpq, freqs, wts, iw_cutoff=None)

Compute GW correlation self-energy (diagonal elements) in MO basis on imaginary axis

pyscf.gw.ugw_ac.kernel(gw, mo_energy, mo_coeff, Lpq=None, orbs=None, nw=None, vhf_df=False, verbose=3)

GW-corrected quasiparticle orbital energies Returns:

A list : converged, mo_energy, mo_coeff

pyscf.gw.ugw_ac.pade_thiele(freqs, zn, coeff)

pyscf.gw.ugw_ac.thiele(fn, zn)

pyscf.gw.ugw_ac.two_pole(freqs, coeff)

pyscf.gw.ugw_ac.two_pole_fit(coeff, omega, sigma)

pyscf.gw.urpa module

Spin-unrestricted random phase approximation (direct RPA/dRPA in chemistry) with N^4 scaling

Method:

Main routines are based on GW-AC method descirbed in: T. Zhu and G.K.-L. Chan, J. Chem. Theory. Comput. 17, 727-741 (2021) X. Ren et al., New J. Phys. 14, 053020 (2012)

class pyscf.gw.urpa.URPA(mf, frozen=None, auxbasis=None)

Bases: RPA

ao2mo(mo_coeff=None)

dump_flags()

get_frozen_mask()

Get boolean mask for the unrestricted reference orbitals.

In the returned boolean (mask) array of frozen orbital indices, the element is False if it corresonds to the frozen orbital.

get_nmo()

get_nocc()

kernel(mo_energy=None, mo_coeff=None, Lpq=None, nw=40, x0=0.5)

Args:

mo_energy : 2D array (2, nmo), mean-field mo energy mo_coeff : 3D array (2, nmo, nmo), mean-field mo coefficient Lpq : 4D array (2, naux, nmo, nmo), 3-index ERI nw: integer, grid number x0: real, scaling factor for frequency grid

Returns:

self.e_tot : RPA total eenrgy self.e_hf : EXX energy self.e_corr : RPA correlation energy

pyscf.gw.urpa.get_rho_response(omega, mo_energy, Lpqa, Lpqb)

Compute density response function in auxiliary basis at freq iw

pyscf.gw.urpa.get_rpa_ecorr(rpa, Lpq, freqs, wts)

Compute RPA correlation energy

pyscf.gw.urpa.kernel(rpa, mo_energy, mo_coeff, Lpq=None, nw=40, x0=0.5, verbose=3)

RPA correlation and total energy

Args:

Lpq : density fitting 3-center integral in MO basis. nw : number of frequency point on imaginary axis. x0: scaling factor for frequency grid.

Returns:

e_tot : RPA total energy e_hf : EXX energy e_corr : RPA correlation energy

Module contents

G0W0 approximation

pyscf.gw.GW(mf, freq_int='ac', frozen=None, tdmf=None)