def logm(P):
"""Returns logarithm of a matrix.
This method should work if the matrix is positive definite and
diagonalizable.
"""
roots, ev = eigenvectors(P)
evI = inverse(ev.T)
evT = ev
if not allclose(P, innerproduct(evT * roots, evI)):
raise ArithmeticError("eigendecomposition failed")
log_roots = log(roots)
return innerproduct(evT * log_roots, evI)
def logm(P):
"""Returns logarithm of a matrix.
This method should work if the matrix is positive definite and
diagonalizable.
"""
roots, ev = eigenvectors(P)
evI = inverse(ev.T)
evT = ev
if not allclose(P, innerproduct(evT * roots, evI)):
raise ArithmeticError("eigendecomposition failed")
log_roots = log(roots)
return innerproduct(evT * log_roots, evI)
def dot(self, other):
"Returns the contraction with |other|."
if isTensor(other):
a = self.array
b = Numeric.transpose(other.array, range(1, other.rank)+[0])
return Tensor(Numeric.innerproduct(a, b), 1)
else:
return Tensor(self.array*other, 1)
def dot(self, other):
"Returns the contraction with |other|."
if isTensor(other):
a = self.array
b = Numeric.transpose(other.array, range(1, other.rank) + [0])
return Tensor(Numeric.innerproduct(a, b), 1)
else:
return Tensor(self.array * other, 1)
def __mul__(self, other): if isTensor(other): a = self.array[self.rank*(slice(None),)+(Numeric.NewAxis,)] b = other.array[other.rank*(slice(None),)+(Numeric.NewAxis,)] return Tensor(Numeric.innerproduct(a, b), 1) elif VectorModule.isVector(other): return other.__rmul__(self) else: return Tensor(self.array*other, 1)
def __mul__(self, other):
if isTensor(other):
a = self.array[self.rank * (slice(None), ) + (Numeric.NewAxis, )]
b = other.array[other.rank * (slice(None), ) + (Numeric.NewAxis, )]
return Tensor(Numeric.innerproduct(a, b), 1)
elif VectorModule.isVector(other):
return other.__rmul__(self)
else:
return Tensor(self.array * other, 1)
def logm(P):
"""Returns logarithm of a matrix.
This method should work if the matrix is positive definite and
diagonalizable.
"""
roots, ev = eigenvectors(P)
evI = inverse(ev.T)
evT = ev
log_roots = log(roots)
return innerproduct(evT * log_roots, evI)
def logm(P):
"""Returns logarithm of a matrix.
This method should work if the matrix is positive definite and
diagonalizable.
"""
roots, ev = eigenvectors(P)
evI = inverse(ev.T)
evT = ev
log_roots = log(roots)
return innerproduct(evT * log_roots, evI)
def main(molecules, moleculedata, results, Ea):
for formula, molecule in molecules.items():
reference = moleculedata[formula]
result = results[formula]
if result['Em0'] is None:
continue
E0 = 0.0
ok = True
for atom in molecule:
symbol = atom.GetChemicalSymbol()
if Ea[symbol] is None:
ok = False
break
E0 += Ea[symbol]
if ok:
result['Ea'] = E0 - result['Em0']
if len(molecule) == 2:
d = result['d0'] + dd
M = np.zeros((4, 5))
for n in range(4):
M[n] = d**-n
a = solve(np.innerproduct(M, M), np.dot(M, result['Em'] - E0))
dmin = 1 / ((-2 * a[2] +
sqrt(4 * a[2]**2 - 12 * a[1] * a[3])) / (6 * a[3]))
#B = xmin**2 / 9 / vmin * (2 * a[2] + 6 * a[3] * xmin)
dfit = np.arange(d[0] * 0.95, d[4] * 1.05, d[2] * 0.005)
emin = a[0]
efit = a[0]
for n in range(1, 4):
efit += a[n] * dfit**-n
emin += a[n] * dmin**-n
result['d'] = dmin
result['Eamin'] = -emin
pylab.plot(dfit, efit, '-', color='0.7')
if ok:
pylab.plot(d, result['Em'] - E0, 'g.')
else:
pylab.plot(d, result['Em'] - E0, 'ro')
pylab.text(dfit[0], efit[0], fix(formula))
pylab.xlabel(u'Bond length [Å]')
pylab.ylabel('Energy [eV]')
pylab.savefig('molecules.png')
o = open('molecules.txt', 'w')
print >> o, """\
.. contents::
==============
Molecule tests
==============
Atomization energies (*E*\ `a`:sub:) and bond lengths (*d*) for 20
small molecules calculated with the PBE functional. All calculations
are done in a box of size 12.6 x 12.0 x 11.4 Å with a grid spacing of
*h*\ =0.16 Å and zero-boundary conditions. Compensation charges are
expanded with correct multipole moments up to *l*\ `max`:sub:\ =2.
Open-shell atoms are treated as non-spherical with integer occupation
numbers, and zero-point energy is not included in the atomization
energies. The numbers are compared to very accurate, state-of-the-art,
PBE calculations (*ref* subscripts).
.. figure:: molecules.png
Bond lengths and atomization energies at relaxed geometries
===========================================================
(*rlx* subscript)
.. list-table::
:widths: 2 3 8 5 6 8
* -
- *d* [Å]
- *d*-*d*\ `ref`:sub: [Å]
- *E*\ `a,rlx`:sub: [eV]
- *E*\ `a,rlx`:sub:-*E*\ `a`:sub: [eV]
- *E*\ `a,rlx`:sub:-*E*\ `a,rlx,ref`:sub: [eV] [1]_"""
for formula, Ea1, Ea2 in Ea12:
reference = moleculedata[formula]
result = results[formula]
if 'Eamin' in result:
print >> o, ' * -', fix2(formula)
print >> o, ' - %5.3f' % result['d']
if 'dref' in reference:
print >> o, (' - ' +
', '.join(['%+5.3f [%d]_' % (result['d'] - dref,
ref)
for dref, ref in reference['dref']]))
else:
print >> o, ' -'
print >> o, ' - %6.3f' % result['Eamin']
if result.get('Ea') is not None:
print >> o, ' - %6.3f' % (result['Eamin'] -
result['Ea'])
else:
print >> o, ' - Unknown'
if formula in atomization_vasp:
print >> o, ' - %6.3f' % (result['Eamin'] -
atomization_vasp[formula][1] /
23.0605)
else:
print >> o, ' -'
print >> o, """\
Atomization energies at experimental geometries
===============================================
.. list-table::
:widths: 6 6 12
* -
- *E*\ `a`:sub: [eV]
- *E*\ `a`:sub:-*E*\ `a,ref`:sub: [eV]"""
for formula, Ea1, Ea2 in Ea12:
reference = moleculedata[formula]
result = results[formula]
print >> o, ' * -', fix2(formula)
if 'Ea' in result:
print >> o, ' - %6.3f' % result['Ea']
if 'Earef' in reference:
print >> o, (' - ' +
', '.join(['%+5.3f [%d]_' % (result['Ea'] -
Ecref, ref)
for Ecref, ref in reference['Earef']]))
else:
print >> o, ' -'
else:
print >> o, ' -'
print >> o, ' -'
print >> o, """
References
==========
.. [1] "The Perdew-Burke-Ernzerhof exchange-correlation functional
applied to the G2-1 test set using a plane-wave basis set",
J. Paier, R. Hirschl, M. Marsman and G. Kresse,
J. Chem. Phys. 122, 234102 (2005)
.. [2] "Molecular and Solid State Tests of Density Functional
Approximations: LSD, GGAs, and Meta-GGAs", S. Kurth,
J. P. Perdew and P. Blaha, Int. J. Quant. Chem. 75, 889-909
(1999)
.. [3] "Comment on 'Generalized Gradient Approximation Made Simple'",
Y. Zhang and W. Yang, Phys. Rev. Lett.
.. [4] Reply to [3]_, J. P. Perdew, K. Burke and M. Ernzerhof
"""
o.close()
os.system('rst2html.py ' +
'--no-footnote-backlinks ' +
'--trim-footnote-reference-space ' +
'--footnote-references=superscript molecules.txt molecules.html')
def main(molecules, moleculedata, results, Ea):
for formula, molecule in molecules.items():
reference = moleculedata[formula]
result = results[formula]
if result['Em0'] is None:
continue
E0 = 0.0
ok = True
for atom in molecule:
symbol = atom.GetChemicalSymbol()
if Ea[symbol] is None:
ok = False
break
E0 += Ea[symbol]
if ok:
result['Ea'] = E0 - result['Em0']
if len(molecule) == 2:
d = result['d0'] + dd
M = np.zeros((4, 5))
for n in range(4):
M[n] = d**-n
a = solve(np.innerproduct(M, M), np.dot(M, result['Em'] - E0))
dmin = 1 / ((-2 * a[2] + sqrt(4 * a[2]**2 - 12 * a[1] * a[3])) /
(6 * a[3]))
#B = xmin**2 / 9 / vmin * (2 * a[2] + 6 * a[3] * xmin)
dfit = np.arange(d[0] * 0.95, d[4] * 1.05, d[2] * 0.005)
emin = a[0]
efit = a[0]
for n in range(1, 4):
efit += a[n] * dfit**-n
emin += a[n] * dmin**-n
result['d'] = dmin
result['Eamin'] = -emin
pylab.plot(dfit, efit, '-', color='0.7')
if ok:
pylab.plot(d, result['Em'] - E0, 'g.')
else:
pylab.plot(d, result['Em'] - E0, 'ro')
pylab.text(dfit[0], efit[0], fix(formula))
pylab.xlabel(u'Bond length [Å]')
pylab.ylabel('Energy [eV]')
pylab.savefig('molecules.png')
o = open('molecules.txt', 'w')
print >> o, """\
.. contents::
==============
Molecule tests
==============
Atomization energies (*E*\ `a`:sub:) and bond lengths (*d*) for 20
small molecules calculated with the PBE functional. All calculations
are done in a box of size 12.6 x 12.0 x 11.4 Å with a grid spacing of
*h*\ =0.16 Å and zero-boundary conditions. Compensation charges are
expanded with correct multipole moments up to *l*\ `max`:sub:\ =2.
Open-shell atoms are treated as non-spherical with integer occupation
numbers, and zero-point energy is not included in the atomization
energies. The numbers are compared to very accurate, state-of-the-art,
PBE calculations (*ref* subscripts).
.. figure:: molecules.png
Bond lengths and atomization energies at relaxed geometries
===========================================================
(*rlx* subscript)
.. list-table::
:widths: 2 3 8 5 6 8
* -
- *d* [Å]
- *d*-*d*\ `ref`:sub: [Å]
- *E*\ `a,rlx`:sub: [eV]
- *E*\ `a,rlx`:sub:-*E*\ `a`:sub: [eV]
- *E*\ `a,rlx`:sub:-*E*\ `a,rlx,ref`:sub: [eV] [1]_"""
for formula, Ea1, Ea2 in Ea12:
reference = moleculedata[formula]
result = results[formula]
if 'Eamin' in result:
print >> o, ' * -', fix2(formula)
print >> o, ' - %5.3f' % result['d']
if 'dref' in reference:
print >> o, (' - ' + ', '.join([
'%+5.3f [%d]_' % (result['d'] - dref, ref)
for dref, ref in reference['dref']
]))
else:
print >> o, ' -'
print >> o, ' - %6.3f' % result['Eamin']
if result.get('Ea') is not None:
print >> o, ' - %6.3f' % (result['Eamin'] - result['Ea'])
else:
print >> o, ' - Unknown'
if formula in atomization_vasp:
print >> o, ' - %6.3f' % (
result['Eamin'] - atomization_vasp[formula][1] / 23.0605)
else:
print >> o, ' -'
print >> o, """\
Atomization energies at experimental geometries
===============================================
.. list-table::
:widths: 6 6 12
* -
- *E*\ `a`:sub: [eV]
- *E*\ `a`:sub:-*E*\ `a,ref`:sub: [eV]"""
for formula, Ea1, Ea2 in Ea12:
reference = moleculedata[formula]
result = results[formula]
print >> o, ' * -', fix2(formula)
if 'Ea' in result:
print >> o, ' - %6.3f' % result['Ea']
if 'Earef' in reference:
print >> o, (' - ' + ', '.join([
'%+5.3f [%d]_' % (result['Ea'] - Ecref, ref)
for Ecref, ref in reference['Earef']
]))
else:
print >> o, ' -'
else:
print >> o, ' -'
print >> o, ' -'
print >> o, """
References
==========
.. [1] "The Perdew-Burke-Ernzerhof exchange-correlation functional
applied to the G2-1 test set using a plane-wave basis set",
J. Paier, R. Hirschl, M. Marsman and G. Kresse,
J. Chem. Phys. 122, 234102 (2005)
.. [2] "Molecular and Solid State Tests of Density Functional
Approximations: LSD, GGAs, and Meta-GGAs", S. Kurth,
J. P. Perdew and P. Blaha, Int. J. Quant. Chem. 75, 889-909
(1999)
.. [3] "Comment on 'Generalized Gradient Approximation Made Simple'",
Y. Zhang and W. Yang, Phys. Rev. Lett.
.. [4] Reply to [3]_, J. P. Perdew, K. Burke and M. Ernzerhof
"""
o.close()
os.system('rst2html.py ' + '--no-footnote-backlinks ' +
'--trim-footnote-reference-space ' +
'--footnote-references=superscript molecules.txt molecules.html')