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# Psi4: an open-source quantum chemistry software package
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import collections
import itertools
import math
import sys
from typing import Dict, List, Tuple, Union
import numpy as np
from qcelemental import Datum
import psi4 # for typing
from .constants import constants
from .libmintsmolecule import compute_atom_map
LINEAR_A_TOL = 1.0E-2 # tolerance (roughly max dev) for TR space
__all__ = ["compare_vibinfos", "filter_nonvib", "filter_omega_to_real", "harmonic_analysis", "hessian_symmetrize", "print_molden_vibs", "print_vibs", "thermo"]
[docs]
def compare_vibinfos(expected: Dict[str, Datum], computed: Dict[str, Datum], tol: float, label: str, verbose: int = 1, forgive: List = None, required: List = None, toldict: Dict[str, float] = None) -> bool:
"""Returns True if two dictionaries of vibration Datum objects are equivalent within a tolerance.
Parameters
----------
expected
Reference value against which `computed` is compared.
computed
Input value to compare against `expected`. Must contain all fields of `expected`.
tol
Absolute tolerance.
label
Label for passed and error messages.
verbose
Control printing.
forgive
Keys in top level which may change between `expected` and `computed` without triggering failure.
required
Keys in top level which must be present in `computed`. ("omega" recc. for vibs.)
toldict
Tolerances for specific keys.
Returns
-------
allclose : bool
Returns True if `expected` and `computed` are equal within tolerance; False otherwise.
"""
np.set_printoptions(formatter={'float': '{: 0.4f}'.format})
def _success(label):
"""Function to print a '*label*...PASSED' line to screen.
Used by :py:func:`util.compare_values` family when functions pass.
"""
msg = f'\t{label:.<66}PASSED'
print(msg)
sys.stdout.flush()
def print_stuff(asp, same, ref, val, space=''):
if verbose >= 1:
print(asp, ':', same)
if (verbose >= 2) or (not same and verbose >= 1):
print('\texp:', space, ref)
print('\tobs:', space, val)
if verbose >= 1:
if not same:
try:
print('\tdif:', space, val - ref)
except TypeError:
print('\tdif: Different, inspect arrays')
if forgive is None:
forgive = []
summsame = []
if required is None:
checkkeys = []
else:
checkkeys = required
checkkeys.extend(expected.keys())
svdtol = 1.e-6 if toldict is None else toldict.get("svd", 1.e-6)
for asp in checkkeys:
if asp not in computed and asp in forgive:
continue
if toldict is not None and asp in toldict:
ktol = toldict[asp]
else:
ktol = tol
if asp in 'qwx':
ccnc = _phase_cols_to_max_element(computed[asp].data)
eenc = _phase_cols_to_max_element(expected[asp].data)
ccnc = _check_degen_modes(ccnc, computed['omega'].data)
eenc = _check_degen_modes(eenc, expected['omega'].data)
same = np.allclose(eenc, ccnc, atol=ktol)
print_stuff(asp=asp, same=same, ref=eenc, val=ccnc, space='\n')
same = _check_rank_degen_modes(ccnc, computed["omega"].data, eenc, difftol=ktol, svdtol=svdtol)
elif asp in ['gamma', 'TRV']:
same = all([computed[asp].data[idx] == val for idx, val in enumerate(expected[asp].data)])
print_stuff(asp=asp, same=same, ref=expected[asp].data, val=computed[asp].data)
elif isinstance(expected[asp].data, float):
same = abs(expected[asp].data - computed[asp].data) < ktol
print_stuff(asp=asp, same=same, ref=expected[asp].data, val=computed[asp].data)
else:
same = (np.allclose(expected[asp].data, computed[asp].data, atol=ktol) and
(expected[asp].data.shape == computed[asp].data.shape))
print_stuff(asp=asp, same=same, ref=expected[asp].data, val=computed[asp].data)
if asp not in forgive:
summsame.append(same)
passed = all(summsame)
if passed:
_success(label)
return passed
def hessian_symmetrize(hess: np.ndarray, mol: psi4.core.Molecule) -> np.ndarray:
"""Apply Abelian symmetry of `mol` to Hessian `hess`.
Parameters
----------
hess
(3 * nat, 3 * nat) Hessian array perhaps with jitter unbecoming a symmetric molecule.
mol
Molecule at which Hessian computed.
Returns
-------
numpy.ndarray
(3 * nat, 3 * nat) symmetrized Hessian array.
"""
ct = mol.point_group().char_table()
# Obtain atom mapping of atom * symm op to atom
atom_map = compute_atom_map(mol)
syms = []
smap = []
for g in range(ct.order()):
syms.append(np.asarray(ct.symm_operation(g).d))
smap.append([atom_map[at][g] for at in range(mol.natom())])
np.set_printoptions(formatter={'float': '{: 16.12f}'.format})
b_hess = blockwise_expand(hess, (3, 3), False)
bDG = []
nat = b_hess.shape[0]
for iat in range(nat):
for jat in range(nat):
for sym in range(len(syms)):
bDG.append(np.zeros_like(b_hess))
bDG[sym][iat, jat] = syms[sym].dot(b_hess[iat, jat].dot(syms[sym]))
# Note that tested syms all diagonal, so above may be off by some transposes
for sym in range(len(syms)):
bDG[sym] = bDG[sym][:, smap[sym]]
bDG[sym] = bDG[sym][smap[sym], :]
tot = np.sum(bDG, axis=0)
tot = np.divide(tot, len(syms))
print('symmetrization diff:', np.linalg.norm(tot - b_hess))
m_tot = blockwise_contract(tot)
return m_tot
def print_molden_vibs(vibinfo: Dict[str, Datum], atom_symbol: Union[np.ndarray, List[str]], geom: Union[np.ndarray, List[List[float]]], standalone: bool = True) -> str:
"""Format vibrational analysis for Molden.
Parameters
----------
vibinfo
Holds results of vibrational analysis.
atom_symbol
(nat,) element symbols for geometry of vibrational analysis.
geom
(nat, 3) geometry of vibrational analysis [a0].
standalone
Whether returned string prefixed "[Molden Format]" for standalone rather than append.
Returns
-------
str
`vibinfo` formatted for Molden, including FREQ, FR-COORD, & FR-NORM-COORD fields.
Notes
-----
Molden format spec from http://www.cmbi.ru.nl/molden/molden_format.html
Specifies "atomic coordinates x,y,z and atomic displacements dx,dy,dz are all in Bohr (Atomic Unit of length)"
Despite it being quite wrong, imaginary modes are represented by a negative frequency.
"""
nat = int(len(vibinfo['q'].data[:, 0]) / 3)
active = [idx for idx, trv in enumerate(vibinfo['TRV'].data) if trv == 'V']
text = ''
if standalone:
text += """[Molden Format]\n"""
text += """\n[FREQ]\n"""
for vib in active:
if vibinfo['omega'].data[vib].imag > vibinfo['omega'].data[vib].real:
freq = -1.0 * vibinfo['omega'].data[vib].imag
else:
freq = vibinfo['omega'].data[vib].real
text += """ {:20.10f}\n""".format(freq)
text += """\n[FR-COORD]\n"""
for at in range(nat):
text += ("""{:3}""" + """{:20.10f}""" * 3 + '\n').format(atom_symbol[at], *geom[at])
text += """\n[FR-NORM-COORD]\n"""
for idx, vib in enumerate(active):
text += """vibration {}\n""".format(idx + 1)
for at in range(nat):
text += (' ' + """{:20.10f}""" * 3 + '\n').format(*(vibinfo['x'].data[:, vib].reshape(nat, 3)[at].real))
# text += """\n[INT]\n"""
# for vib in active:
# text += """1.0\n"""
return text
def _check_rank_degen_modes(arr, freq, ref, difftol, svdtol, verbose=1):
dfreq, didx, dinv, dcts = np.unique(np.around(freq, 1), return_index=True, return_inverse=True, return_counts=True)
normco_ok = True
for idegen, istart in enumerate(didx):
degree = dcts[idegen]
cvecs = arr[:, istart:istart + degree]
evecs = ref[:, istart:istart + degree]
cevecs = np.concatenate((cvecs, evecs), axis=1)
diff_ok = np.allclose(evecs, cvecs, atol=difftol)
rank_cvecs = np.linalg.matrix_rank(cvecs)
rank_evecs = np.linalg.matrix_rank(evecs)
CE = np.linalg.svd(cevecs, compute_uv=False) # hermitian=False
rank_cevecs = np.count_nonzero(CE > svdtol, axis=-1)
# expected normal coordinates and computed normal coordinates span the same space
ranks_ok = rank_cvecs == rank_evecs == rank_cevecs
if degree == 1:
normco_ok = normco_ok and diff_ok
else:
normco_ok = normco_ok and ranks_ok
if verbose >= 2 or not normco_ok:
with np.printoptions(precision=4):
print(f"degree={degree} difftol={difftol} {diff_ok} svdtol={svdtol} {rank_cvecs} == {rank_evecs} == {rank_cevecs} {rank_cvecs == rank_evecs == rank_cevecs} svd={CE}")
return normco_ok
def _check_degen_modes(arr, freq, verbose=1):
"""Use `freq` to identify degenerate columns of eigenvectors `arr` and
sort into std order for comparison. Returns eigenvectors back sorted.
"""
arr2 = np.zeros_like(arr) # lgtm [py/multiple-definition]
dfreq, didx, dinv, dcts = np.unique(np.around(freq, 1), return_index=True, return_inverse=True, return_counts=True)
# judging degen normco to only 2 decimals is probably sign need to resolve evec
idx_max_elem_each_normco = np.argmax(np.around(arr, 2), axis=0)
max_elem_each_normco = np.amax(np.around(arr, 2), axis=0)
idx_vib_reordering = np.empty_like(idx_max_elem_each_normco)
for idegen, istart in enumerate(didx):
degree = dcts[idegen]
# sort degen evec
# primarily (last arg) by value of extreme element
# (sep evec that in this coord sys have diff elements)
# & secondarily (2nd-to-last arg) by index of extreme element
# (order evec with same elements in diff (xyz) arrangements)
idx_sort_wi_degen = np.lexsort(
(idx_max_elem_each_normco[istart:istart + degree], max_elem_each_normco[istart:istart + degree]))
idx_vib_reordering[istart:istart + degree] = np.arange(istart, istart + degree)[idx_sort_wi_degen]
arr2 = arr[:, idx_vib_reordering]
reorderings = ['{}-->{}'.format(i, v) for i, v in enumerate(idx_vib_reordering) if (i != v)]
if reorderings and verbose >= 2:
print('Degenerate modes reordered:', ', '.join(reorderings))
return arr2
def _phase_cols_to_max_element(arr, tol=1.e-2, verbose=1):
"""Returns copy of 2D `arr` scaled such that, within cols, max(fabs)
element is positive. If max(fabs) is pos/neg pair, scales so first
element (within `tol`) is positive.
"""
arr2 = np.copy(arr)
rephasing = []
for v in range(arr.shape[1]):
vextreme = 0.0
iextreme = None
# find most extreme value
for varr in arr[:, v]:
vextreme = max(np.absolute(varr), vextreme)
# find the first index whose fabs equals that value, w/i tolerance
for iarr, varr in enumerate(arr[:, v]):
if (vextreme - np.absolute(varr)) < tol:
iextreme = iarr
break
sign = np.sign(arr[iextreme, v])
if sign == -1.:
rephasing.append(str(v))
arr2[:, v] *= sign
if rephasing and verbose >= 2:
print('Negative modes rephased:', ', '.join(rephasing))
return arr2
def harmonic_analysis(hess: np.ndarray, geom: np.ndarray, mass: np.ndarray, basisset: psi4.core.BasisSet, irrep_labels: List[str], dipder: np.ndarray = None, project_trans: bool = True, project_rot: bool = True) -> Tuple[Dict[str, Datum], str]:
"""Extract frequencies, normal modes and other properties from electronic Hessian. Like so much other Psi4 goodness, originally by @andysim
Parameters
----------
hess
(3*nat, 3*nat) non-mass-weighted Hessian in atomic units, [Eh/a0/a0].
geom
(nat, 3) geometry [a0] at which Hessian computed.
mass
(nat,) atomic masses [u].
basisset
Basis set object (can be dummy, e.g., STO-3G) for SALCs.
irrep_labels
Irreducible representation labels.
dipder
(3, 3 * nat) dipole derivatives in atomic units, [Eh a0/u] or [(e a0/a0)^2/u]
project_trans
Idealized translations projected out of final vibrational analysis.
project_rot
Idealized rotations projected out of final vibrational analysis.
Returns
-------
dict, str
Returns dictionary of vibration Datum objects (fields: label units data comment).
Also returns text suitable for printing.
Notes
-----
.. _`table:vibaspectinfo`:
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
| key | description (label & comment) | units | data (real/imaginary modes) |
+===============+============================================+===========+======================================================+
| omega | frequency | cm^-1 | ndarray(ndof) complex (real/imag) |
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
| q | normal mode, normalized mass-weighted | a0 u^1/2 | ndarray(ndof, ndof) float |
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
| w | normal mode, un-mass-weighted | a0 | ndarray(ndof, ndof) float |
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
| x | normal mode, normalized un-mass-weighted | a0 | ndarray(ndof, ndof) float |
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
| degeneracy | degree of degeneracy | | ndarray(ndof) int |
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
| TRV | translation/rotation/vibration | | ndarray(ndof) str 'TR' or 'V' or '-' for partial |
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
| gamma | irreducible representation | | ndarray(ndof) str irrep or None if unclassifiable |
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
| mu | reduced mass | u | ndarray(ndof) float (+/+) |
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
| k | force constant | mDyne/A | ndarray(ndof) float (+/-) |
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
| DQ0 | RMS deviation v=0 | a0 u^1/2 | ndarray(ndof) float (+/0) |
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
| Qtp0 | Turning point v=0 | a0 u^1/2 | ndarray(ndof) float (+/0) |
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
| Xtp0 | Turning point v=0 | a0 | ndarray(ndof) float (+/0) |
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
| theta_vib | char temp | K | ndarray(ndof) float (+/0) |
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
| IR_intensity | infrared intensity | km/mol | ndarray(ndof) float (+/+) |
+---------------+--------------------------------------------+-----------+------------------------------------------------------+
Examples
--------
>>> # displacement of first atom in highest energy mode
>>> vibinfo['x'].data[:, -1].reshape(nat, 3)[0]
>>> # remove translations & rotations
>>> vibonly = filter_nonvib(vibinfo)
"""
if (mass.shape[0] == geom.shape[0] == (hess.shape[0] // 3) == (hess.shape[1] // 3)) and (geom.shape[1] == 3):
pass
else:
raise ValidationError(
f"""Dimension mismatch among mass ({mass.shape}), geometry ({geom.shape}), and Hessian ({hess.shape})""")
def mat_symm_info(a, atol=1e-14, lbl='array', stol=None):
symm = np.allclose(a, a.T, atol=atol)
herm = np.allclose(a, a.conj().T, atol=atol)
ivrt = a.shape[0] - np.linalg.matrix_rank(a, tol=stol)
return """ {:32} Symmetric? {} Hermitian? {} Lin Dep Dim? {:2}""".format(lbl + ':', symm, herm, ivrt)
def vec_in_space(vec, space, tol=1.0e-4):
merged = np.vstack((space, vec))
u, s, v = np.linalg.svd(merged)
return (s[-1] < tol)
vibinfo = {}
text = []
nat = len(mass)
text.append("""\n\n ==> Harmonic Vibrational Analysis <==\n""")
if nat == 1:
nrt_expected = 3
elif np.linalg.matrix_rank(geom) == 1:
nrt_expected = 5
else:
nrt_expected = 6
nmwhess = hess.copy()
text.append(mat_symm_info(nmwhess, lbl='non-mass-weighted Hessian') + ' (0)')
# get SALC object, possibly w/o trans & rot
mints = psi4.core.MintsHelper(basisset)
cdsalcs = mints.cdsalcs(0xFF, project_trans, project_rot)
Uh = collections.OrderedDict()
for h, lbl in enumerate(irrep_labels):
tmp = np.asarray(cdsalcs.matrix_irrep(h))
if tmp.size > 0:
Uh[lbl] = tmp
# form projector of translations and rotations
space = ('T' if project_trans else '') + ('R' if project_rot else '')
TRspace = _get_TR_space(mass, geom, space=space, tol=LINEAR_A_TOL)
nrt = TRspace.shape[0]
text.append(
f' projection of translations ({project_trans}) and rotations ({project_rot}) removed {nrt} degrees of freedom ({nrt_expected})'
)
P = np.identity(3 * nat)
for irt in TRspace:
P -= np.outer(irt, irt)
text.append(mat_symm_info(P, lbl='total projector') + f' ({nrt})')
# mass-weight & solve
sqrtmmm = np.repeat(np.sqrt(mass), 3)
sqrtmmminv = np.divide(1.0, sqrtmmm)
mwhess = np.einsum('i,ij,j->ij', sqrtmmminv, nmwhess, sqrtmmminv)
text.append(mat_symm_info(mwhess, lbl='mass-weighted Hessian') + ' (0)')
pre_force_constant_au = np.linalg.eigvalsh(mwhess)
idx = np.argsort(pre_force_constant_au)
pre_force_constant_au = pre_force_constant_au[idx]
uconv_cm_1 = (np.sqrt(constants.na * constants.hartree2J * 1.0e19) /
(2 * np.pi * constants.c * constants.bohr2angstroms))
pre_frequency_cm_1 = np.lib.scimath.sqrt(pre_force_constant_au) * uconv_cm_1
pre_lowfreq = np.where(np.real(pre_frequency_cm_1) < 100.0)[0]
pre_lowfreq = np.append(pre_lowfreq, np.arange(nrt_expected)) # catch at least nrt modes
for lf in set(pre_lowfreq):
vlf = pre_frequency_cm_1[lf]
if vlf.imag > vlf.real:
text.append(' pre-proj low-frequency mode: {:9.4f}i [cm^-1]'.format(vlf.real, vlf.imag))
else:
text.append(' pre-proj low-frequency mode: {:9.4f} [cm^-1]'.format(vlf.real, ''))
text.append(' pre-proj all modes:' + str(_format_omega(pre_frequency_cm_1, 4)))
# project & solve
mwhess_proj = np.dot(P.T, mwhess).dot(P)
text.append(mat_symm_info(mwhess_proj, lbl='projected mass-weighted Hessian') + f' ({nrt})')
#print('projhess = ', np.array_repr(mwhess_proj))
force_constant_au, qL = np.linalg.eigh(mwhess_proj)
# expected order for vibrations is steepest downhill to steepest uphill
idx = np.argsort(force_constant_au)
force_constant_au = force_constant_au[idx]
qL = qL[:, idx]
qL = _phase_cols_to_max_element(qL)
vibinfo['q'] = Datum('normal mode', 'a0 u^1/2', qL, comment='normalized mass-weighted')
# frequency, LAB II.17
frequency_cm_1 = np.lib.scimath.sqrt(force_constant_au) * uconv_cm_1
vibinfo['omega'] = Datum('frequency', 'cm^-1', frequency_cm_1)
# degeneracies
ufreq, uinv, ucts = np.unique(np.around(frequency_cm_1, 1), return_inverse=True, return_counts=True)
vibinfo['degeneracy'] = Datum('degeneracy', '', ucts[uinv])
# look among the symmetry subspaces h for one to which the normco
# of vib does *not* add an extra dof to the vector space
active = []
irrep_classification = []
for idx, vib in enumerate(frequency_cm_1):
if vec_in_space(qL[:, idx], TRspace, 1.0e-4):
active.append('TR')
irrep_classification.append(None)
else:
active.append('V')
for h in Uh.keys():
if vec_in_space(qL[:, idx], Uh[h], 1.0e-4):
irrep_classification.append(h)
break
else:
irrep_classification.append(None)
# catch partial Hessians
if np.linalg.norm(vib) < 1.e-3:
active[-1] = '-'
vibinfo['TRV'] = Datum('translation/rotation/vibration', '', active, numeric=False)
vibinfo['gamma'] = Datum('irreducible representation', '', irrep_classification, numeric=False)
lowfreq = np.where(np.real(frequency_cm_1) < 100.0)[0]
lowfreq = np.append(lowfreq, np.arange(nrt_expected)) # catch at least nrt modes
for lf in set(lowfreq):
vlf = frequency_cm_1[lf]
if vlf.imag > vlf.real:
text.append(' post-proj low-frequency mode: {:9.4f}i [cm^-1] ({})'.format(vlf.imag, active[lf]))
else:
text.append(' post-proj low-frequency mode: {:9.4f} [cm^-1] ({})'.format(vlf.real, active[lf]))
text.append(' post-proj all modes:' + str(_format_omega(frequency_cm_1, 4)) + '\n')
if project_trans and not project_rot:
text.append(f' Note that "Vibration"s include {nrt_expected - 3} un-projected rotation-like modes.')
elif not project_trans and not project_rot:
text.append(
f' Note that "Vibration"s include {nrt_expected} un-projected rotation-like and translation-like modes.')
# general conversion factors, LAB II.11
uconv_K = (constants.h * constants.na * 1.0e21) / (8 * np.pi * np.pi * constants.c)
uconv_S = np.sqrt((constants.c * (2 * np.pi * constants.bohr2angstroms)**2) /
(constants.h * constants.na * 1.0e21))
# normco & reduced mass, LAB II.14 & II.15
wL = np.einsum('i,ij->ij', sqrtmmminv, qL)
vibinfo['w'] = Datum('normal mode', 'a0', wL, comment='un-mass-weighted')
reduced_mass_u = np.divide(1.0, np.linalg.norm(wL, axis=0)**2)
vibinfo['mu'] = Datum('reduced mass', 'u', reduced_mass_u)
xL = np.sqrt(reduced_mass_u) * wL
vibinfo['x'] = Datum('normal mode', 'a0', xL, comment='normalized un-mass-weighted')
# IR intensities, CCQC Proj. Eqns. 15-16
uconv_kmmol = (constants.get("Avogadro constant") * np.pi * 1.e-3 * constants.get("electron mass in u") *
constants.get("fine-structure constant")**2 * constants.get("atomic unit of length") / 3)
uconv_D2A2u = (constants.get('atomic unit of electric dipole mom.') * 1.e11 /
constants.get('hertz-inverse meter relationship') /
constants.get('atomic unit of length'))**2
if not (dipder is None or np.array(dipder).size == 0):
qDD = dipder.dot(wL)
ir_intensity = np.zeros(qDD.shape[1])
for i in range(qDD.shape[1]):
ir_intensity[i] = qDD[:, i].dot(qDD[:, i])
# working but not needed
#vibinfo['IR_intensity'] = Datum('infrared intensity', 'Eh a0/u', ir_intensity)
#ir_intensity_D2A2u = ir_intensity * uconv_D2A2u
#vibinfo['IR_intensity'] = Datum('infrared intensity', '(D/AA)^2/u', ir_intens_D2A2u)
ir_intensity_kmmol = ir_intensity * uconv_kmmol
vibinfo['IR_intensity'] = Datum('infrared intensity', 'km/mol', ir_intensity_kmmol)
# force constants, LAB II.16 (real compensates for earlier sqrt)
uconv_mdyne_a = (0.1 * (2 * np.pi * constants.c)**2) / constants.na
force_constant_mdyne_a = reduced_mass_u * (frequency_cm_1 * frequency_cm_1).real * uconv_mdyne_a
vibinfo['k'] = Datum('force constant', 'mDyne/A', force_constant_mdyne_a)
force_constant_cm_1_bb = reduced_mass_u * (frequency_cm_1 * frequency_cm_1).real * uconv_S * uconv_S
Datum('force constant', 'cm^-1/a0^2', force_constant_cm_1_bb, comment="Hooke's Law")
# turning points, LAB II.20 (real & zero since turning point silly for imag modes)
nu = 0
turning_point_rnc = np.sqrt(2.0 * nu + 1.0)
with np.errstate(divide='ignore'):
turning_point_bohr_u = turning_point_rnc / (np.sqrt(frequency_cm_1.real) * uconv_S)
turning_point_bohr_u[turning_point_bohr_u == np.inf] = 0.
vibinfo['Qtp0'] = Datum('Turning point v=0', 'a0 u^1/2', turning_point_bohr_u)
with np.errstate(divide='ignore'):
turning_point_bohr = turning_point_rnc / (np.sqrt(frequency_cm_1.real * reduced_mass_u) * uconv_S)
turning_point_bohr[turning_point_bohr == np.inf] = 0.
vibinfo['Xtp0'] = Datum('Turning point v=0', 'a0', turning_point_bohr)
rms_deviation_bohr_u = turning_point_bohr_u / np.sqrt(2.0)
vibinfo['DQ0'] = Datum('RMS deviation v=0', 'a0 u^1/2', rms_deviation_bohr_u)
# characteristic vibrational temperature, RAK thermo & https://en.wikipedia.org/wiki/Vibrational_temperature
# (imag freq zeroed)
uconv_K = 100 * constants.h * constants.c / constants.kb
vib_temperature_K = frequency_cm_1.real * uconv_K
vibinfo['theta_vib'] = Datum('char temp', 'K', vib_temperature_K)
return vibinfo, '\n'.join(text)
def _br(string):
return '[' + string + ']'
def _format_omega(omega, decimals):
"""Return complex frequencies in `omega` into strings showing only real or imag ("i"-labeled)
to `decimals` precision.
"""
freqs = []
for fr in omega:
if fr.imag > fr.real:
freqs.append("""{:.{prec}f}i""".format(fr.imag, prec=decimals))
else:
freqs.append("""{:.{prec}f}""".format(fr.real, prec=decimals))
return np.array(freqs)
def print_vibs(vibinfo: Dict[str, Datum], atom_lbl: List[str] = None, *, normco: str = 'x', shortlong: bool = True, groupby: int = None, prec: int = 4, ncprec: int = None) -> str:
"""Pretty printer for vibrational analysis.
Parameters
----------
vibinfo
Results of a Hessian solution.
atom_lbl
Atomic symbols for printing. If None, integers used.
normco
{'q', 'w', 'x'}
Which normal coordinate definition to print (reduced mass, etc. unaffected by this parameter). Must be
* `q` [a0 u^1/2], the mass-weighted normalized eigenvectors of the Hessian,
* `w` [a0], the un-mass-weighted (Cartesian) of q, or
* `x` [a0], the normalized w.
shortlong
Whether normal coordinates should be in (nat, 3) `True` or (nat * 3, 1) `False` format.
groupby
How many normal coordinates per row. Defaults to 3/6 for shortlong=T/F. Value of `-1` uses one row.
prec
Number of decimal places for frequencies, reduced masses, etc.
ncprec
Number of decimal places for normal coordinates. Defaults to 2 for shortlong=short and 4 for shortlong=long.
Returns
-------
str
String suitable for printing.
"""
def grouper(iterable, n, fillvalue=None):
"Collect data into fixed-length chunks or blocks"
# grouper('ABCDEFG', 3, 'x') --> ABC DEF Gxx"
args = [iter(iterable)] * n
return itertools.zip_longest(*args, fillvalue=fillvalue)
if normco not in ['q', 'w', 'x']:
raise ValidationError("""Requested normal coordinates not among allowed q/w/x: """ + normco)
nat = int(len(vibinfo['q'].data[:, 0]) / 3)
if atom_lbl is None:
atom_lbl = [''] * nat
active = [idx for idx, trv in enumerate(vibinfo['TRV'].data) if trv == 'V']
presp = 2
colsp = 2
if shortlong:
groupby = groupby if groupby else 3
ncprec = ncprec if ncprec else 2
width = (ncprec + 4) * 3
prewidth = 24
else:
groupby = groupby if groupby else 6
ncprec = ncprec if ncprec else 4
width = ncprec + 8
prewidth = 24
if groupby == -1:
groupby = len(active)
omega_str = _format_omega(vibinfo['omega'].data, decimals=prec)
text = ''
for row in grouper(active, groupby):
text += """\n{:{presp}}{:{prewidth}}""".format('', 'Vibration', prewidth=prewidth, presp=presp)
for vib in row:
if vib is None:
# ran out of vibrations in this row
break
text += """{:^{width}d}{:{colsp}}""".format(vib + 1, '', width=width, colsp=colsp)
text += '\n'
text += """{:{presp}}{:{prewidth}}""".format('', 'Freq [cm^-1]', prewidth=prewidth, presp=presp)
for vib in row:
if vib is None:
break
text += """{:^{width}} """.format(omega_str[vib], width=width)
text += '\n'
text += """{:{presp}}{:{prewidth}}""".format('', 'Irrep', prewidth=prewidth, presp=presp)
for vib in row:
if vib is None:
break
val = vibinfo['gamma'].data[vib]
if val is None:
val = ''
text += """{:^{width}}{:{colsp}}""".format(val, '', width=width, colsp=colsp)
text += '\n'
text += """{:{presp}}{:{prewidth}}""".format('',
'Reduced mass ' + _br(vibinfo['mu'].units),
prewidth=prewidth,
presp=presp)
for vib in row:
if vib is None:
break
text += """{:^{width}.{prec}f}{:{colsp}}""".format(vibinfo['mu'].data[vib],
'',
width=width,
prec=prec,
colsp=colsp)
text += '\n'
text += """{:{presp}}{:{prewidth}}""".format('',
'Force const ' + _br(vibinfo['k'].units),
prewidth=prewidth,
presp=presp)
for vib in row:
if vib is None:
break
text += """{:^{width}.{prec}f}{:{colsp}}""".format(vibinfo['k'].data[vib],
'',
width=width,
prec=prec,
colsp=colsp)
text += '\n'
text += """{:{presp}}{:{prewidth}}""".format('',
'Turning point v=0 ' + _br(vibinfo['Xtp0'].units),
prewidth=prewidth,
presp=presp)
for vib in row:
if vib is None:
break
text += """{:^{width}.{prec}f}{:{colsp}}""".format(vibinfo['Xtp0'].data[vib],
'',
width=width,
prec=prec,
colsp=colsp)
text += '\n'
text += """{:{presp}}{:{prewidth}}""".format('',
'RMS dev v=0 ' + _br(vibinfo['DQ0'].units),
prewidth=prewidth,
presp=presp)
for vib in row:
if vib is None:
break
text += """{:^{width}.{prec}f}{:{colsp}}""".format(vibinfo['DQ0'].data[vib],
'',
width=width,
prec=prec,
colsp=colsp)
text += '\n'
if 'IR_intensity' in vibinfo:
text += """{:{presp}}{:{prewidth}}""".format('',
'IR activ ' + _br(vibinfo['IR_intensity'].units),
prewidth=prewidth,
presp=presp)
for vib in row:
if vib is None:
break
text += """{:^{width}.{prec}f}{:{colsp}}""".format(vibinfo['IR_intensity'].data[vib],
'',
width=width,
prec=prec,
colsp=colsp)
text += '\n'
if 'theta_vib' in vibinfo:
text += """{:{presp}}{:{prewidth}}""".format('',
'Char temp ' + _br(vibinfo['theta_vib'].units),
prewidth=prewidth,
presp=presp)
for vib in row:
if vib is None:
break
text += """{:^{width}.{prec}f}{:{colsp}}""".format(vibinfo['theta_vib'].data[vib],
'',
width=width,
prec=prec,
colsp=colsp)
text += '\n'
#text += 'Raman activ [A^4/u]\n'
text += ' ' * presp + '-' * (prewidth + groupby * (width + colsp) - colsp) + '\n'
if shortlong:
for at in range(nat):
text += """{:{presp}}{:5d} {:{width}}""".format('',
at + 1,
atom_lbl[at],
width=prewidth - 8,
presp=presp)
for vib in row:
if vib is None:
break
text += ("""{:^{width}.{prec}f}""" * 3).format(*(vibinfo[normco].data[:, vib].reshape(nat, 3)[at]),
width=int(width / 3),
prec=ncprec)
text += """{:{colsp}}""".format('', colsp=colsp)
text += '\n'
else:
for at in range(nat):
for xyz in range(3):
text += """{:{presp}}{:5d} {} {:{width}}""".format('',
at + 1,
'XYZ' [xyz],
atom_lbl[at],
width=prewidth - 14,
presp=presp)
for vib in row:
if vib is None:
break
text += """{:^{width}.{prec}f}""".format((vibinfo[normco].data[3 * at + xyz, vib]),
width=width,
prec=ncprec)
text += """{:{colsp}}""".format('', colsp=colsp)
text += '\n'
return text
def thermo(vibinfo, T: float, P: float, multiplicity: int, molecular_mass: float, E0: float, sigma: int, rot_const: np.ndarray, rotor_type: str = None) -> Tuple[Dict[str, Datum], str]:
"""Perform thermochemical analysis from vibrational output.
Parameters
----------
E0
Electronic energy [Eh] at well bottom at 0 [K], :psivar:`CURRENT ENERGY`.
molecular_mass
Mass in [u] of molecule under analysis.
multiplicity
Spin multiplicity of molecule under analysis.
rot_const
(3,) rotational constants in [cm^-1] of molecule under analysis.
sigma
The rotational or external symmetry number determined from the point group.
rotor_type
The rotor type for rotational stat mech purposes: RT_ATOM, RT_LINEAR, other.
T
Temperature in [K]. Psi default 298.15. Note that 273.15 is IUPAC STP.
P
Pressure in [Pa]. Psi default 101325. Note that 100000 is IUPAC STP.
Returns
-------
dict, str
First is every thermochemistry component in atomic units along with input conditions.
Second is formatted presentation of analysis.
"""
sm = collections.defaultdict(float)
# conditions
therminfo = {}
therminfo['E0'] = Datum('E0', 'Eh', E0)
therminfo['B'] = Datum('rotational constants', 'cm^-1', rot_const)
therminfo['sigma'] = Datum('external symmetry number', '', sigma)
therminfo['T'] = Datum('temperature', 'K', T)
therminfo['P'] = Datum('pressure', 'Pa', P)
# electronic
q_elec = multiplicity
sm[('S', 'elec')] = math.log(q_elec)
# translational
beta = 1 / (constants.kb * T)
q_trans = (2.0 * np.pi * molecular_mass * constants.amu2kg /
(beta * constants.h * constants.h))**1.5 * constants.na / (beta * P)
sm[('S', 'trans')] = 5 / 2 + math.log(q_trans / constants.na)
sm[('Cv', 'trans')] = 3 / 2
sm[('Cp', 'trans')] = 5 / 2
sm[('E', 'trans')] = 3 / 2 * T
sm[('H', 'trans')] = 5 / 2 * T
# rotational
if rotor_type == "RT_ATOM":
pass
elif rotor_type == "RT_LINEAR":
q_rot = 1. / (beta * sigma * 100 * constants.c * constants.h * rot_const[1])
sm[('S', 'rot')] = 1.0 + math.log(q_rot)
sm[('Cv', 'rot')] = 1
sm[('Cp', 'rot')] = 1
sm[('E', 'rot')] = T
else:
phi_A, phi_B, phi_C = rot_const * 100 * constants.c * constants.h / constants.kb
q_rot = math.sqrt(math.pi) * T**1.5 / (sigma * math.sqrt(phi_A * phi_B * phi_C))
sm[('S', 'rot')] = 3 / 2 + math.log(q_rot)
sm[('Cv', 'rot')] = 3 / 2
sm[('Cp', 'rot')] = 3 / 2
sm[('E', 'rot')] = 3 / 2 * T
sm[('H', 'rot')] = sm[('E', 'rot')]
# vibrational
vibonly = filter_nonvib(vibinfo)
ZPE_cm_1 = 1 / 2 * np.sum(vibonly['omega'].data.real)
omega_str = _format_omega(vibonly['omega'].data, decimals=4)
imagfreqidx = np.where(vibonly['omega'].data.imag > vibonly['omega'].data.real)[0]
if len(imagfreqidx):
print("Warning: thermodynamics relations excluded imaginary frequencies: {}".format(omega_str[imagfreqidx]))
filtered_theta_vib = np.delete(vibonly['theta_vib'].data, imagfreqidx, None)
filtered_omega_str = np.delete(omega_str, imagfreqidx, None)
rT = filtered_theta_vib / T # reduced temperature
lowfreqidx = np.where(filtered_theta_vib < 900.)[0]
if len(lowfreqidx):
print("Warning: used thermodynamics relations inappropriate for low-frequency modes: {}".format(
filtered_omega_str[lowfreqidx]))
sm[('S', 'vib')] = np.sum(rT / np.expm1(rT) - np.log(1 - np.exp(-rT)))
sm[('Cv', 'vib')] = np.sum(np.exp(rT) * (rT / np.expm1(rT))**2)
sm[('Cp', 'vib')] = sm[('Cv', 'vib')]
sm[('ZPE', 'vib')] = np.sum(rT) * T / 2
sm[('E', 'vib')] = sm[('ZPE', 'vib')] + np.sum(rT * T / np.expm1(rT))
sm[('H', 'vib')] = sm[('E', 'vib')]
assert (abs(ZPE_cm_1 - sm[('ZPE', 'vib')] * constants.R * constants.hartree2wavenumbers * 0.001 /
constants.hartree2kJmol) < 0.1)
#real_vibs = np.ma.masked_where(vibinfo['omega'].data.imag > vibinfo['omega'].data.real, vibinfo['omega'].data)
# compute Gibbs
for term in ['elec', 'trans', 'rot', 'vib']:
sm[('G', term)] = sm[('H', term)] - T * sm[('S', term)]
# convert to atomic units
for term in ['elec', 'trans', 'rot', 'vib']:
# terms above are unitless (S, Cv, Cp) or in units of temperature (ZPE, E, H, G) as expressions are divided by R.
# R [Eh/K], computed as below, slightly diff in 7th sigfig from 3.1668114e-6 (k_B in [Eh/K])
# value listed https://en.wikipedia.org/wiki/Boltzmann_constant
uconv_R_EhK = constants.R / constants.hartree2kJmol
for piece in ['S', 'Cv', 'Cp']:
sm[(piece, term)] *= uconv_R_EhK # [mEh/K] <-- []
for piece in ['ZPE', 'E', 'H', 'G']:
sm[(piece, term)] *= uconv_R_EhK * 0.001 # [Eh] <-- [K]
# sum corrections and totals
for piece in ['S', 'Cv', 'Cp']:
for term in ['elec', 'trans', 'rot', 'vib']:
sm[(piece, 'tot')] += sm[(piece, term)]
for piece in ['ZPE', 'E', 'H', 'G']:
for term in ['elec', 'trans', 'rot', 'vib']:
sm[(piece, 'corr')] += sm[(piece, term)]
sm[(piece, 'tot')] = E0 + sm[(piece, 'corr')]
terms = collections.OrderedDict()
terms['elec'] = ' Electronic'
terms['trans'] = ' Translational'
terms['rot'] = ' Rotational'
terms['vib'] = ' Vibrational'
terms['tot'] = 'Total'
terms['corr'] = 'Correction'
# package results for export
for entry in sm:
if entry[0] in ['S', 'Cv', 'Cp']:
unit = 'mEh/K'
elif entry[0] in ['ZPE', 'E', 'H', 'G']:
unit = 'Eh'
therminfo['_'.join(entry)] = Datum(terms[entry[1]].strip().lower() + ' ' + entry[0], unit, sm[entry])
# display
format_S_Cv_Cp = """\n {:36} {:11.3f} [cal/(mol K)] {:11.3f} [J/(mol K)] {:15.8f} [mEh/K]"""
format_ZPE_E_H_G = """\n {:36} {:11.3f} [kcal/mol] {:11.3f} [kJ/mol] {:15.8f} [Eh]"""
uconv = np.asarray([constants.hartree2kcalmol, constants.hartree2kJmol, 1.])
# TODO rot_const, rotor_type
text = ''
text += """\n ==> Thermochemistry Components <=="""
text += """\n\n Entropy, S"""
for term in terms:
text += format_S_Cv_Cp.format(terms[term] + ' S', *sm[('S', term)] * uconv)
if term == 'elec':
text += """ (multiplicity = {})""".format(multiplicity)
elif term == 'trans':
text += """ (mol. weight = {:.4f} [u], P = {:.2f} [Pa])""".format(molecular_mass, P)
elif term == 'rot':
text += """ (symmetry no. = {})""".format(sigma)
text += """\n\n Constant volume heat capacity, Cv"""
for term in terms:
text += format_S_Cv_Cp.format(terms[term] + ' Cv', *sm[('Cv', term)] * uconv)
text += """\n\n Constant pressure heat capacity, Cp"""
for term in terms:
text += format_S_Cv_Cp.format(terms[term] + ' Cp', *sm[('Cp', term)] * uconv)
del terms['tot']
terms['corr'] = 'Correction'
text += """\n\n ==> Thermochemistry Energy Analysis <=="""
text += """\n\n Raw electronic energy, E_e"""
text += f"""\n Total E_e, Electronic energy at well bottom {E0:15.8f} [Eh]"""
text += """\n\n Zero-point vibrational energy, ZPVE = Sum_i omega_i / 2, E_0 = E_e + ZPVE"""
for term in terms:
if term in ['vib']:
text += format_ZPE_E_H_G.format(terms[term] + ' ZPVE', *sm[('ZPE', term)] * uconv)
text += """ {:15.3f} [cm^-1]""".format(sm[('ZPE', term)] * constants.hartree2wavenumbers)
elif term in ['corr']:
text += format_ZPE_E_H_G.format(terms[term] + ' ZPVE to E_e', *sm[('ZPE', term)] * uconv)
text += """ {:15.3f} [cm^-1]""".format(sm[('ZPE', term)] * constants.hartree2wavenumbers)
text += """\n Total E_0, Enthalpy at 0 [K] {:15.8f} [Eh]""".format(
sm[('ZPE', 'tot')])
text += """\n *** Absolute enthalpy, not an enthalpy of formation ***"""
text += """\n\n Thermal (internal) energy, E (includes ZPVE and finite-temperature corrections)"""
for term in terms:
if term in ['elec']:
text += format_ZPE_E_H_G.format(terms[term] + ' contrib to E beyond E_e', *sm[('E', term)] * uconv)
elif term in ['corr']:
text += format_ZPE_E_H_G.format(terms[term] + ' E', *sm[('E', term)] * uconv)
else:
text += format_ZPE_E_H_G.format(terms[term] + ' contrib to E', *sm[('E', term)] * uconv)
text += """\n Total E, Thermal (internal) energy at {:7.2f} [K] {:15.8f} [Eh]""".format(
T, sm[('E', 'tot')])
text += """\n\n Enthalpy, H_trans = E_trans + k_B * T = E_trans + P * V"""
for term in terms:
if term in ['elec']:
text += format_ZPE_E_H_G.format(terms[term] + ' contrib to H beyond E_e', *sm[('H', term)] * uconv)
elif term in ['corr']:
text += format_ZPE_E_H_G.format(terms[term] + ' H', *sm[('H', term)] * uconv)
else:
text += format_ZPE_E_H_G.format(terms[term] + ' contrib to H', *sm[('H', term)] * uconv)
text += """\n Total H, Enthalpy at {:7.2f} [K] {:15.8f} [Eh]""".format(
T, sm[('H', 'tot')])
text += """\n *** Absolute enthalpy, not an enthalpy of formation ***"""
text += """\n\n Gibbs free energy, G = H - T * S"""
for term in terms:
if term in ['elec']:
text += format_ZPE_E_H_G.format(terms[term] + ' contrib to G beyond E_e', *sm[('G', term)] * uconv)
elif term in ['corr']:
text += format_ZPE_E_H_G.format(terms[term] + ' G', *sm[('G', term)] * uconv)
else:
text += format_ZPE_E_H_G.format(terms[term] + ' contrib to G', *sm[('G', term)] * uconv)
text += """\n Total G, Gibbs energy at {:7.2f} [K] {:15.8f} [Eh]""".format(
T, sm[('G', 'tot')])
text += """\n *** Absolute Gibbs energy, not a free energy of formation ***\n\n"""
return therminfo, text
def filter_nonvib(vibinfo: Dict[str, Datum], remove: List[int] = None) -> Dict[str, Datum]:
"""From a dictionary of vibration Datum, remove normal coordinates.
Parameters
----------
vibinfo
Results of Hessian analysis.
remove
0-indexed indices of normal modes to remove from `vibinfo`. If
None, non-vibrations (R, T, or TR as labeled in `vibinfo['TRV']`)
will be removed.
Returns
-------
dict
Copy of input `vibinfo` with the specified modes removed from all
dictionary entries.
Examples
--------
>>> # after a harmonic analysis, remove first translations and rotations and then all non-A1 vibs
>>> allnormco = harmonic_analysis(...)
>>> allvibs = filter_nonvib(allnormco)
>>> a1vibs = filter_nonvib(allvibs, remove=[i for i, d in enumerate(allvibs['gamma'].data) if d != 'A1'])
"""
work = {}
if remove is None:
remove = [idx for idx, dat in enumerate(vibinfo['TRV'].data) if dat != 'V']
for asp, oasp in vibinfo.items():
if asp in ['q', 'w', 'x']:
axis = 1
else:
axis = 0
work[asp] = Datum(oasp.label, oasp.units, np.delete(oasp.data, remove, axis=axis), comment=oasp.comment, numeric=False)
return work
def filter_omega_to_real(omega: np.ndarray) -> np.ndarray:
"""Returns ndarray (float) of `omega` (complex) where imaginary entries are converted to negative reals."""
freqs = []
for fr in omega:
if fr.imag > fr.real:
freqs.append(-1 * fr.imag)
else:
freqs.append(fr.real)
return np.asarray(freqs)
def _get_TR_space(m: np.ndarray, geom: np.ndarray, space: str = 'TR', tol: float = None, verbose: int = 1) -> np.ndarray:
"""Form the idealized translation and rotation dof from geometry `geom` and masses `m`.
Remove any linear dependencies and return an array of shape (3, 3) for atoms, (5, 3 * nat) for linear `geom`,
or (6, 3 * nat) otherwise. To handle noisy linear geometries, pass `tol` on the order of max deviation.
m1 = np.asarray([1.])
m2 = np.asarray([1., 1.])
m3 = np.asarray([1., 1., 1.])
m4 = np.asarray([1., 1., 1., 1.])
g4 = np.asarray([[ 1., 1., 0.],
[-1., 1., 0.],
[-1., -1., 0.],
[ 1., -1., 0.]])
g2 = np.asarray([[ 1., 1., 0.],
[-1., -1., 0.]])
g3 = np.asarray([[3., 3., 3.],
[4., 4., 4.,],
[5., 5., 5.]])
g3noisy = np.asarray([[3., 3.001, 3.],
[4., 4.001, 4.,],
[5., 5., 5.01]])
g33 = np.asarray([[0., 0., 0.],
[1., 0., 0.],
[-1., 0., 0.]])
g1 = np.asarray([[0., 0., 0.]])
g11 = np.asarray([[1., 2., 3.]])
noise = np.random.normal(0, 1, 9).reshape(3, 3)
noise = np.divide(noise, np.max(noise))
assert(_get_TR_space(m4, g4).shape == (6, 12))
assert(_get_TR_space(m2, g2).shape == (5, 6))
assert(_get_TR_space(m3, g3).shape == (5, 9))
assert(_get_TR_space(m3, g33).shape == (5, 9))
assert(_get_TR_space(m1, g1).shape == (3, 3))
assert(_get_TR_space(m1, g11).shape == (3, 3))
assert(_get_TR_space(m3, g3noisy, tol=1.e-2).shape == (5, 9))
for ns in range(2, 6):
tol = 10. ** -ns
gnoisy = g3 + tol * noise
assert(_get_TR_space(m3, gnoisy, tol=10*tol).shape == (5, 9))
"""
sqrtmmm = np.repeat(np.sqrt(m), 3)
xxx = np.repeat(geom[:, 0], 3)
yyy = np.repeat(geom[:, 1], 3)
zzz = np.repeat(geom[:, 2], 3)
z = np.zeros_like(m)
i = np.ones_like(m)
ux = np.ravel([i, z, z], order='F')
uy = np.ravel([z, i, z], order='F')
uz = np.ravel([z, z, i], order='F')
# form translation and rotation unit vectors
T1 = sqrtmmm * ux
T2 = sqrtmmm * uy
T3 = sqrtmmm * uz
R4 = sqrtmmm * (yyy * uz - zzz * uy)
R5 = sqrtmmm * (zzz * ux - xxx * uz)
R6 = sqrtmmm * (xxx * uy - yyy * ux)
TRspace = []
if 'T' in space:
TRspace.append([T1, T2, T3])
if 'R' in space:
TRspace.append([R4, R5, R6])
if not TRspace:
# not sure about this, but it runs
ZZ = np.zeros_like(T1)
TRspace.append([ZZ])
TRspace = np.vstack(TRspace)
def orth(A, tol=tol):
u, s, vh = np.linalg.svd(A, full_matrices=False)
if verbose >= 2:
print(s)
M, N = A.shape
eps = np.finfo(float).eps
if tol is None:
tol = max(M, N) * np.amax(s) * eps
num = np.sum(s > tol, dtype=int)
Q = u[:, :num]
return Q
TRindep = orth(TRspace.T)
TRindep = TRindep.T
if verbose >= 2:
print(TRindep.shape, '<--', TRspace.shape)
print(np.linalg.norm(TRindep, axis=1))
print('-' * 80)
return TRindep
