3.2.3. `horton.correlatedwfn.restricted_ap1rog` – Correlated wavefunction implementations¶

This module contains restricted AP1roG

Variables used in this module:

nocc:	number of (active) occupied orbitals in the principle configuration
pairs:	number of electron pairs
nvirt:	number of (active) virtual orbitals in the principle configuration
nbasis:	total number of basis functions

Indexing convention:

i,j,k,..:	occupied orbitals of principle configuration
a,b,c,..:	virtual orbitals of principle configuration
p,q,r,..:	general indices (occupied, virtual)

Abbreviations used (if not mentioned in doc-strings):

L_pqrs:	2<pq\|rs>-<pq\|sr>
g_pqrs:	<pq\|rs>

For more information see doc-strings.

class horton.correlatedwfn.restricted_ap1rog.RAp1rog(lf, occ_model, npairs=None, nvirt=None)¶

Bases: horton.correlatedwfn.geminal.Geminal

Arguments:

lf: A LinalgFactory instance.
occ_model: Occupation model

Optional arguments:

npairs: Number of electron pairs, if not specified, npairs = number of occupied orbitals
nvirt: Number of virtual orbitals, if not specified, nvirt = (nbasis-npairs)

__call__(one, two, core, orb, olp, scf, **kwargs)¶

Optimize geminal coefficients and—if required—find optimal set of orbitals.

Arguments:

one, two: One- and two-body integrals (some Hamiltonian matrix elements).
core: The core energy (not included in ‘one’ and ‘two’).
orb: An expansion instance. It contains the MO coefficients (orbitals).
olp: The AO overlap matrix. A TwoIndex instance.
scf: A boolean. If True: Initializes orbital optimization.
Keywords:: See RAp1rog.solve() and RAp1rog.solve_scf()

__init__(lf, occ_model, npairs=None, nvirt=None)¶

Arguments:

lf: A LinalgFactory instance.
occ_model: Occupation model

Optional arguments:

npairs: Number of electron pairs, if not specified, npairs = number of occupied orbitals
nvirt: Number of virtual orbitals, if not specified, nvirt = (nbasis-npairs)

check_convergence(e0, e1, gradient, thresh)¶

Check convergence.

Arguements:

e0, e1: Used to calculate energy difference e0-e1
gradient: The gradient, a OneIndex instance
thresh: Dictionary containing threshold parameters (‘energy’, ‘gradientmax’, ‘gradientnorm’)
Returns:: True if energy difference, norm of orbital gradient, largest element of orbital gradient are smaller than some threshold values.

check_keywords(guess, solver, maxiter, dumpci, thresh, printoptions)¶

Check dictionaries if they contain proper keys.

Arguments:

guess, solver, maxiter, dumpci, thresh, printoptions: See RAp1rog.solve()

check_keywords_scf(guess, solver, maxiter, dumpci, thresh, printoptions, stepsearch)¶

Check dictionaries if they contain proper keys.

Arguments:

guess, solver, maxiter, dumpci, thresh, printoptions, stepseach: See RAp1rog.solve_scf()

check_stepsearch(linesearch)¶: Check trustradius. Abort calculation if trustradius is smaller than 1e-8

clear()¶: Clear all wavefunction information

clear_auxmatrix()¶: Clear auxiliary matrices

clear_dm()¶: Clear RDM information

clear_geminal()¶: Clear geminal information

clear_lagrange()¶: Clear lagrange information

compute_1dm(dmout, mat1, mat2, factor=1.0, response=True)¶

Compute 1-RDM for AP1roG

Arguments:

mat1, mat2: A DenseTwoIndex instance used to calculated 1-RDM. For response RDM, mat1 is the geminal matrix, mat2 the Lagrange multipliers

Optional arguments:

factor: A scalar factor
select: Switch between response (True) and PS2 (False) 1-RDM

compute_2dm(dmout, dm1, mat1, mat2, select, response=True)¶

Compute response 2-RDM for AP1roG

** Arguments **

one_dm: A 1DM. A OneIndex instance.
mat1, mat2: TwoIndex instances used to calculate the 2-RDM. To get the response DM, mat1 is the geminal coefficient matrix, mat2 are the Lagrange multipliers
select: Either ‘ppqq’ or ‘pqpq’. Note that the elements (iiii), i.e., the 1DM, are stored in pqpq, while the elements (aaaa) are stored in ppqq.
response: If True, calculate response 2-RDM. Otherwise the PS2 2-RDM is calculated.

compute_correlation_energy(arg=None)¶

Get correlation energy of restricted AP1roG

Optional arguments:

arg: The geminal coefficient matrix (np.array or TwoIndex instance). If not provided, the correlation energy is calculated from self.geminal (default None)

compute_objective_function(coeff=None)¶

Objective function for line search optimization

Optional arguments:

coeff: See RAp1rog.compute_total_energy()

compute_orbital_gradient(*args, **kwargs)¶

Determine orbital gradient for all non-reduntant orbital rotations (oo,vo,vv).

Optional arguments:

ps2c: (boolean) If True, switches to PS2c orbital optimization (default False)

compute_orbital_hessian(lshift=1e-08, pos=False, ps2c=False)¶

Construct diagonal approximation to Hessian for orbital optimization of AP1roG.

Optional arguments:

lshift: Level shift (float) (default 1e-8)
pos: Set all elements positive (boolean) (default False)
ps2c: (boolean) If True, switches to PS2c orbital optimization (default False)

compute_reference_energy()¶: Get energy of reference determinant for restricted AP1roG

compute_rotation_matrix(coeff)¶

Determine orbital rotation matrix for (oo), (vo), and (vv) blocks

Arguments:

coeff: The nonreduntant orbital rotations k_pq (1-dim np.array). The elements are sorted w.r.t. lower triangular indices (p>q).

compute_total_energy(coeff=None)¶

Get total energy (reference+correlation) including nuclear-repulsion/ core energy for restricted AP1roG.

Optional arguments:

arg: The geminal coefficient matrix (np.array or TwoIndex instance). If not provided, the correlation energy is calculated from self.geminal (default None)

do_checkpoint(orb, olp, checkpoint_fn)¶

Dump orbitals for restart

Arguments:

orb: An expansion instance. AO/MO coefficients stored to disk
olp: A TwoIndex instance. The AO overlap matrix. Required for restart at different molecular geometry.
checkpoint_fn: The filename of the checkpoint file.

dump_final(orb, olp, printoptions, dumpci, checkpoint, checkpoint_fn=’checkpoint.h5’)¶

Dump final solution (orbitals, wavefunction amplitudes, geminal coefficients)

Arguments:

orb: An expansion instance. AO/MO coefficients stored to disk
olp: A TwoIndex instance. The AO overlap matrix. Required for restart
printoptions, dumpci: A dictionary. See RAp1rog.solve_scf()
checkpoint:: An integer. See RAp1rog.solve_scf()
checkpoint_fn: The filename of the checkpoint file.

generate_guess(guess, dim=None)¶

Generate a guess of type ‘guess’.

Arguments:

guess: A dictionary, containing the type of guess.

Optional arguments:

dim: Length of guess.

get_auxmatrix(select)¶

Get an auxiliary matrix from cache.

Arguments:

select: ‘t’, ‘gpqpq’, ‘gpqqp’, ‘lpqpq’, ‘pqrq’, ‘gpqrr’, ‘fock’.

get_exact_hessian(*args, **kwargs)¶

Construct exact Hessian for orbital optimization of restricted OAP1roG.

The usage of the exact orbital Hessian for the orbital optimization is currently not supported. The exact Hessian can only be evaluated a posteriori.

Arguements

mo1, mo2: The 1- and 2-el integrals in the MO basis (TwoIndex and FourIndex instances)

get_four_dm(select)¶

Get a density matrix (4-RDM). If not available, it will be created (if possible)

Arguments:

select

get_one_dm(select)¶

Get a density matrix (1-RDM). If not available, it will be created (if possible)

Arguments:

select: ‘ps2’, or ‘response’.

get_slater_determinant(*indices)¶

Return excited Slater Determinant.

Arguments:

indices: A (list of) multi-index. First element contains occupied index w.r.t. reference determinant, second element contains virtual index w.r.t. reference determinant.

get_three_dm(select)¶

Get a density matrix (3-RDM). If not available, it will be created (if possible)

Arguments:

select

get_two_dm(select)¶

Get a density matrix (2-RDM). If not available, it will be created (if possible)

Arguments:

select: ‘(r(esponse))ppqq’, or ‘(r(esponse))pqpq’.

init_auxmatrix(select)¶

Initialize auxiliary matrices

Arguments:

select: One of t, fock, gppqq, gpqpq, gpqqp, lpqpq, lpqrq, gpqrr

init_four_dm(select)¶

Initialize 4-RDM

Arguments

select

init_one_dm(select)¶

Initialize 1-RDM as OneIndex object

The 1-RDM expressed in the natural orbital basis is diagonal and only the diagonal elements are stored.

Arguments

select: ‘ps2’ or ‘response’.

init_three_dm(select)¶

Initialize 3-RDM

Arguments

select

init_two_dm(select)¶

Initialize 2-RDM as TwoIndex object

Only the symmetry-unique elements of the (response) 2-RDM are stored. These are matrix elements of type

\[Gamma_{p\bar{q}p\bar{q}}\]

(spin-up and spin-down (bar-sign)) or

\[Gamma_{p\bar{p}q\bar{q}}\]

and are stored as elements \({pq}\) of two_dm_pqpq, and two_dm_ppqq.

Arguments

select: ‘(r(esponse))ppqq’, or ‘(r(esponse))pqpq’.

jacobian_ap1rog(coeff, miiaa, miaia, one, fock)¶: Construct Jacobian for optimization of geminal coefficients.

jacobian_lambda(lagrange, gmat, miiaa, miaia, one, diagfock)¶: Construct Jacobian for optimization of Lagrange multipliers for restricted AP1roG.

orbital_rotation_step(lshift=1e-08, pos=True, optimizer=’variational’)¶

Get orbital rotation step (Newton–Raphson)

Arguments:

Optional arguments:

lshift: Level shift (float) for approximate Hessian added to small elements (default 1e-8)
pos: (boolean) Make all elements of Hessian positive if set to True (default True)
optimizer: Orbital otimization method (str) (default ‘variational’)

perm(a)¶

Calculate the permament of a matrix

Arguements

a: A np array

print_final(s=’Final’)¶

Print energies

Optional arguments:

s: A string.

print_geminal_coefficients()¶: Print geminal coefficients

print_options(guess, solver, maxiter, thresh, printoptions, indextrans)¶

Print optimization options.

Arguments:

guess, solver, maxiter, thresh, printoptions, indextrans: See RAp1rog.solve()

print_options_scf(guess, solver, maxiter, lshift, stepsearch, thresh, printoptions, checkpoint, checkpoint_fn, indextrans, orbitaloptimizer, sort)¶

Print optimization options.

Arguments:

See RAp1rog.solve_scf()

print_solution(cithresh=0.01, excitationlevel=2, amplitudestofile=False, filename=’./ap1rog_amplitudes’)¶

Print coefficients \({ci}\) and Slater Determinant if \(|ci|\) > threshold. Prints up to hextuply excited pairs.

Optional arguments:

cithresh: Upper threshold (a float) for CI coefficients to be reconstructed. (Default 0.01)
excitationslevel: The maximum number of substitutions w.r.t. the reference determinant for which the CI coefficients are reconstructed. (Default 2)
amplitudestofile: A boolean. If True, the CI coefficients are stored in a separate file. (Default False)
filename: The file name for the wfn amplitudes. (Default ap1rog_amplitudes)

prod(lst)¶

solve(*args, **kwargs)¶

Optimize AP1roG coefficients for some Hamiltonian. For restricted, closed-shell AP1roG.

Arguments:

one, two: One- (TwoIndex instance) and two-body integrals (FourIndex or Cholesky instance) (some Hamiltonian matrix elements)
core: The core energy (not included in ‘one’ and ‘two’); a float.
orb: An expansion instance. It contains the MO coefficients.
olp: The AO overlap matrix. A TwoIndex instance.

Keywords:

indextrans:
4-index Transformation (str). Choice between tensordot (default) and einsum

warning:
Print warnings (boolean) (default False)

guess:
initial guess (dictionary) containing:

type: guess type (str). One of random (random numbers, default), const (a constant scaled by a factor)

factor: a scaling factor (float) (default -0.1)

geminal: external guess for geminal coefficients (1-d np.array); if provided, ‘type’ is ignored (default None)

solver:
wfn solver (dictionary) containing:

wfn: wavefunction solver (str) (default krylov)

maxiter:
max number of iterations (dictionary) containing:

wfniter: maximum number of iterations (int) for wfn solver (default 200)

dumpci:
dump ci coefficients (dictionary):

amplitudestofile: write wfn amplitudes to file (boolean) (default False)

amplitudesfilename: (str) (default ap1rog_amplitudes.dat)

thresh:
optimization thresholds (dictionary) containing:

wfn: threshold for geminal coefficients (float) (default 1e-12)

printoptions:
print level (dictionary) containing:

geminal: (boolean), if True, geminal matrix is printed (default True)

ci: threshold for CI coefficients (float) (requires evaluation of a permanent), all coefficients (for a given excitation order) larger than ‘ci’ are printed (default 0.01)

excitationlevel: number of excited pairs w.r.t. reference determinant for which wfn amplitudes are reconstructed (int) (default 1)

swapa:
swap orbitals (numpy 2-dim array), each row contains 2 orbital indices to be swapped (default np.array([[]]))

solve_geminal(*args, **kwargs)¶

Solves for geminal matrix

Arguments:

guess: The initial guess. A 1-dim np array.
solver: The solver used. A dictionary with element ‘wfn’.
wfnthreshold: The optimization threshold. A float.
wfnmaxiter: The maximum number of iterations. An integer.

solve_lagrange(*args, **kwargs)¶

Solves for Lagrange multipliers

Arguments:

guess: The initial guess. A 1-dim np array.
solver: The solver used. A dictionary with element ‘lagrange’.
wfnthreshold: The optimization threshold. A float.
wfnmaxiter: The maximum number of iterations. An integer.

solve_model(one, two, orb, **kwargs)¶

Solve for geminal model.

Arguments:

one, two

One- and two-body integrals (some Hamiltonian matrix elements). A TwoIndex and FourIndex/Cholesky instance

orb

An expansion instance which contains the MO coefficients.

Keywords:

guess: initial guess for wfn (1-dim np.array) guesslm: initial guess Lagrange multipliers (1-dim np.array) solver: wfn/Lagrange solver (dictionary) indextrans: 4-index Transformation (str). maxiter: maximum number of iterations (dictionary) thresh: thresholds (dictionary) orbitaloptimizer: orbital optimization method (str)

For more details, see RAp1rog.solve_scf()

solve_scf(*args, **kwargs)¶

Find Geminal expansion coefficient for some Hamiltonian. For restricted, closed-shell AP1roG. Perform orbital optimization.

Arguments:

one, two: One- (TwoIndex instance) and two-body integrals (FourIndex or Cholesky instance) (some Hamiltonian matrix elements)
core: The core energy (not included in ‘one’ and ‘two’).
orb: An expansion instance. It contains the MO coefficients.
olp: The AO overlap matrix. A TwoIndex instance.

Keywords:

indextrans:
4-index Transformation (str). Choice between tensordot (default) and einsum

warning:
print warnings (boolean) (default False)

guess:
initial guess (dictionary) containing:

type: guess type (str). One of random (random numbers, default), const (a constant scaled by a factor)

factor: a scaling factor (float) (default -0.1)

geminal: external guess for geminal coefficients (1-dim np.array) (default None)

lagrange: external guess for Lagrange multipliers (1-dim np.array) (default None)

solver:
wfn/Lagrange solver (dictionary) containing:

wfn: wavefunction solver (str) (default krylov)

lagrange: Lagrange multiplier solver (str) (default krylov)

maxiter:
max number of iterations (dictionary) containing:

wfniter: maximum number of iterations for

wfn/lagrange solver (int) (default 200)

orbiter: maximum number of orbital optimization

steps (int) (default 100)

dumpci:
dump ci coefficient (dictionary) containing:

amplitudestofile: write wfn amplitudes to file (boolean) (default False)

amplitudesfilename: (str) (default ap1rog_amplitudes.dat)

thresh:
optimization thresholds (dictionary) containing:

wfn: threshold for geminal coefficients and Lagrange multipliers (float) (defaulf 1e-12)

energy: convergence threshold for energy (float) (default 1e-8)

gradientnorm: convergence threshold for norm of orbital gradient (float) (default 1e-4)

gradientmax: threshold for maximum absolute value of orbital gradient (float) (default 5e-5)

printoptions:
print level; dictionary containing:

geminal: (boolean) if True, geminal matrix is printed (default True)

ci: threshold for CI coefficients (requires evaluation of a permanent) (float). All coefficients (for a given excitation order) larger than ci are printed (default 0.01)

excitationlevel: number of excited pairs w.r.t. reference determinant for which wfn amplitudes are reconstructed (int) (default 1)

stepsearch:
step search options (dictionary) containing:

method: step search method used (str). One of trust-region (default), None, backtracking

alpha: scaling factor for Newton step (float), used in backtracking and None method (default 1.00)

c1: parameter used in backtracking (float) (default 1e-4)

minalpha: minimum scaling factor used in backracking (float) (default 1e-6)

maxiterouter: maximum number of search steps (int) (default 10)

maxiterinner: maximum number of optimization step in each search step (int) (used only in pcg, default 500)

maxeta: upper bound for estimated vs actual change in trust-region (float) (default 0.75)

mineta: lower bound for estimated vs actual change in trust-region (float) (default 0.25)

upscale: scaling factor to increase trustradius in trust-region (float) (default 2.0)

downscale: scaling factor to decrease trustradius in trust-region and step length in backtracking (float) (default 0.25)

trustradius: initial trustradius (float) (default 0.75)

maxtrustradius: maximum trustradius (float) (default 0.75)

threshold: trust-region optimization threshold, only used in pcg (float) (default 1e-8)

optimizer: optimizes step to boundary of trustradius (str). One of pcg, dogleg, ddl (default ddl)

checkpoint:
frequency of checkpointing (int). If > 0, writes orbitals and overlap to a checkpont file (defatul 1)

checkpoint_fn:
filename to use for the checkpoint file (default “checkpoint.h5”)

levelshift:
level shift of Hessian (float) (default 1e-8)

absolute:
(boolean), if True, take absolute values of Hessian (default False)

sort:
(boolean), if True, orbitals are sorted according to their natural occupation numbers. This requires us to solve for the wavefunction again. Works only if orbitaloptimizer is set to variational. (default True)

swapa:
swap orbitals (numpy 2-dim array) each row contains 2 orbital indices to be swapped (default np.array([[]]))

givensrot:
rotate two orbitals (numpy 2-dim array) each row contains 2 orbital indices and rotation angle (default np.array([[]]))

orbitaloptimizer:

(str) switch between variational orbital optimization (variational) and PS2c (ps2c) (default variational).

sort_natural_orbitals(orb)¶

Sort orbitals w.r.t. the natural occupation numbers

Arguments:

orb: The AO/MO coefficients (natural orbitals) and the natural occupation numbers (Expansion instance)

start_up(orb, swapa, givensrot=array([], shape=(1, 0), dtype=float64))¶

Modify orbitals prior to optimization

Arguments:

swapa: An integer numpy array with two columns where every row corresponds to one swap.

Optional arguments:

givensrot: Givens rotation of orbital pairs. A numpy array where every row corresponds to one Givens rotation with elements (index1, index2, angle in deg).

update_auxmatrix(*args, **kwargs)¶

Arguments:

select: scf.
two_mo: two-electron integrals in the MO basis. A FourIndex instance.
one_mo: one-electron integrals in the MO basis. A TwoIndex instance.

update_ecore(new)¶: Update core energy

update_four_dm(select, four_dm=None)¶

Update 4-RDM

Optional arguments:

four_dm: When provided, this 4-RDM is stored.

update_geminal(geminal=None)¶

Update geminal matrix

Optional arguments:

geminal: When provided, this geminal matrix is stored.

update_lagrange(lagrange=None, dim1=None, dim2=None)¶

Update Lagragne multipliers

Optional arguments:

lagrange: When provided, this set of Lagrange multipliers is stored.

update_one_dm(select, one_dm=None)¶

Update 1-RDM

Arguments:

select: One of ps2, response

Optional arguments:

one_dm: When provided, this 1-RDM is stored. A OneIndex instance.

update_three_dm(select, three_dm=None)¶

Update 3-RDM

Optional arguments:

three_dm: When provided, this 3-RDM is stored.

update_two_dm(select, two_dm=None)¶

Update 2-RDM

Arguments:

select: One of ps2, response

Optional arguments:

two_dm: When provided, this 2-RDM is stored.

vector_function_geminal(*args, **kwargs)¶

Construct vector function for optimization of geminal coefficients.

Arguments:

coeff: The geminal coefficients (np.array)
miiaa, miaia: Auxiliary matrices (TwoIndex instances)
one: 1-electron integrals (TwoIndex instance)
diagfock: Diagonal inactive Fock matrix (OneIndex instances)

vector_function_lagrange(*args, **kwargs)¶

Construct vector function for optimization of Lagrange multipliers.

Arguments:

lagrange: The lagrange multipliers (np array)
gmat: The geminal coefficients (TwoIndex instance)
miiaa, miaia: Auxiliary matrices (TwoIndex instances)
one: 1-electron integrals (TwoIndex instance)
diagfock: Diagonal inactive Fock matrix (OneIndex instance)

dimension¶: The number of unknowns (i.e. the number of geminal coefficients)

ecore¶: The core energy

geminal¶: The geminal coefficients

lagrange¶: The Lagrange multipliers

lf¶: The LinalgFactory instance

nbasis¶: The number of basis functions

nocc¶: The number of occupied orbitals

npairs¶: The number of electron pairs

nvirt¶: The number of virtual orbitals

one_dm_ps2¶: Alpha 1-RDM

one_dm_response¶: Alpha 1-RDM

two_dm_ppqq¶: Alpha-beta PS2 (ppqq) 2-RDM

two_dm_pqpq¶: Alpha-beta PS2 (pqpq) 2-RDM

two_dm_rppqq¶: Alpha-beta (ppqq) 2-RDM

two_dm_rpqpq¶: Alpha-beta (pqpq) 2-RDM

3.2.3. horton.correlatedwfn.restricted_ap1rog – Correlated wavefunction implementations¶

3.2.3. `horton.correlatedwfn.restricted_ap1rog` – Correlated wavefunction implementations¶