class PDAnalyzerTest(unittest.TestCase):
def setUp(self):
module_dir = os.path.dirname(os.path.abspath(__file__))
(elements, entries) = PDEntryIO.from_csv(
os.path.join(module_dir, "pdentries_test.csv"))
self.pd = PhaseDiagram(entries)
self.analyzer = PDAnalyzer(self.pd)
def test_get_e_above_hull(self):
for entry in self.pd.stable_entries:
self.assertLess(
self.analyzer.get_e_above_hull(entry), 1e-11,
"Stable entries should have e above hull of zero!")
for entry in self.pd.all_entries:
if entry not in self.pd.stable_entries:
e_ah = self.analyzer.get_e_above_hull(entry)
self.assertGreaterEqual(e_ah, 0)
self.assertTrue(isinstance(e_ah, Number))
def test_get_equilibrium_reaction_energy(self):
for entry in self.pd.stable_entries:
self.assertLessEqual(
self.analyzer.get_equilibrium_reaction_energy(entry), 0,
"Stable entries should have negative equilibrium reaction energy!"
)
def test_get_decomposition(self):
for entry in self.pd.stable_entries:
self.assertEqual(
len(self.analyzer.get_decomposition(entry.composition)), 1,
"Stable composition should have only 1 decomposition!")
dim = len(self.pd.elements)
for entry in self.pd.all_entries:
ndecomp = len(self.analyzer.get_decomposition(entry.composition))
self.assertTrue(
ndecomp > 0 and ndecomp <= dim,
"The number of decomposition phases can at most be equal to the number of components."
)
#Just to test decomp for a ficitious composition
ansdict = {
entry.composition.formula: amt
for entry, amt in self.analyzer.get_decomposition(
Composition("Li3Fe7O11")).items()
}
expected_ans = {
"Fe2 O2": 0.0952380952380949,
"Li1 Fe1 O2": 0.5714285714285714,
"Fe6 O8": 0.33333333333333393
}
for k, v in expected_ans.items():
self.assertAlmostEqual(ansdict[k], v)
def test_get_transition_chempots(self):
for el in self.pd.elements:
self.assertLessEqual(
len(self.analyzer.get_transition_chempots(el)),
len(self.pd.facets))
def test_get_element_profile(self):
for el in self.pd.elements:
for entry in self.pd.stable_entries:
if not (entry.composition.is_element):
self.assertLessEqual(
len(
self.analyzer.get_element_profile(
el, entry.composition)), len(self.pd.facets))
expected = [{
'evolution': 1.0,
'chempot': -4.2582781416666666,
'reaction': 'Li2O + 0.5 O2 -> Li2O2'
}, {
'evolution': 0,
'chempot': -5.0885906699999968,
'reaction': 'Li2O -> Li2O'
}, {
'evolution': -1.0,
'chempot': -10.487582010000001,
'reaction': 'Li2O -> 2 Li + 0.5 O2'
}]
result = self.analyzer.get_element_profile(Element('O'),
Composition('Li2O'))
for d1, d2 in zip(expected, result):
self.assertAlmostEqual(d1['evolution'], d2['evolution'])
self.assertAlmostEqual(d1['chempot'], d2['chempot'])
self.assertEqual(d1['reaction'], str(d2['reaction']))
def test_get_get_chempot_range_map(self):
elements = [el for el in self.pd.elements if el.symbol != "Fe"]
self.assertEqual(len(self.analyzer.get_chempot_range_map(elements)),
10)
def test_getmu_vertices_stability_phase(self):
results = self.analyzer.getmu_vertices_stability_phase(
Composition("LiFeO2"), Element("O"))
self.assertAlmostEqual(len(results), 6)
test_equality = False
for c in results:
if abs(c[Element("O")]+7.115) < 1e-2 and abs(c[Element("Fe")]+6.596) < 1e-2 and \
abs(c[Element("Li")]+3.931) < 1e-2:
test_equality = True
self.assertTrue(test_equality,
"there is an expected vertex missing in the list")
def test_getmu_range_stability_phase(self):
results = self.analyzer.get_chempot_range_stability_phase(
Composition("LiFeO2"), Element("O"))
self.assertAlmostEqual(results[Element("O")][1], -4.4501812249999997)
self.assertAlmostEqual(results[Element("Fe")][0], -6.5961470999999996)
self.assertAlmostEqual(results[Element("Li")][0], -3.6250022625000007)
def test_get_hull_energy(self):
for entry in self.pd.stable_entries:
h_e = self.analyzer.get_hull_energy(entry.composition)
self.assertAlmostEqual(h_e, entry.energy)
n_h_e = self.analyzer.get_hull_energy(
entry.composition.fractional_composition)
self.assertAlmostEqual(n_h_e, entry.energy_per_atom)
def test_1d_pd(self):
entry = PDEntry('H', 0)
pd = PhaseDiagram([entry])
pda = PDAnalyzer(pd)
decomp, e = pda.get_decomp_and_e_above_hull(PDEntry('H', 1))
self.assertAlmostEqual(e, 1)
self.assertAlmostEqual(decomp[entry], 1.0)
def test_get_critical_compositions_fractional(self):
c1 = Composition('Fe2O3').fractional_composition
c2 = Composition('Li3FeO4').fractional_composition
c3 = Composition('Li2O').fractional_composition
comps = self.analyzer.get_critical_compositions(c1, c2)
expected = [
Composition('Fe2O3').fractional_composition,
Composition('Li0.3243244Fe0.1621621O0.51351349'),
Composition('Li3FeO4').fractional_composition
]
for crit, exp in zip(comps, expected):
self.assertTrue(crit.almost_equals(exp, rtol=0, atol=1e-5))
comps = self.analyzer.get_critical_compositions(c1, c3)
expected = [
Composition('Fe0.4O0.6'),
Composition('LiFeO2').fractional_composition,
Composition('Li5FeO4').fractional_composition,
Composition('Li2O').fractional_composition
]
for crit, exp in zip(comps, expected):
self.assertTrue(crit.almost_equals(exp, rtol=0, atol=1e-5))
def test_get_critical_compositions(self):
c1 = Composition('Fe2O3')
c2 = Composition('Li3FeO4')
c3 = Composition('Li2O')
comps = self.analyzer.get_critical_compositions(c1, c2)
expected = [
Composition('Fe2O3'),
Composition('Li0.3243244Fe0.1621621O0.51351349') * 7.4,
Composition('Li3FeO4')
]
for crit, exp in zip(comps, expected):
self.assertTrue(crit.almost_equals(exp, rtol=0, atol=1e-5))
comps = self.analyzer.get_critical_compositions(c1, c3)
expected = [
Composition('Fe2O3'),
Composition('LiFeO2'),
Composition('Li5FeO4') / 3,
Composition('Li2O')
]
for crit, exp in zip(comps, expected):
self.assertTrue(crit.almost_equals(exp, rtol=0, atol=1e-5))
# Don't fail silently if input compositions aren't in phase diagram
# Can be very confusing if you're working with a GrandPotentialPD
self.assertRaises(ValueError, self.analyzer.get_critical_compositions,
Composition('Xe'), Composition('Mn'))
# For the moment, should also fail even if compositions are in the gppd
# because it isn't handled properly
gppd = GrandPotentialPhaseDiagram(self.pd.all_entries, {'Xe': 1},
self.pd.elements + [Element('Xe')])
pda = PDAnalyzer(gppd)
self.assertRaises(ValueError, pda.get_critical_compositions,
Composition('Fe2O3'), Composition('Li3FeO4Xe'))
# check that the function still works though
comps = pda.get_critical_compositions(c1, c2)
expected = [
Composition('Fe2O3'),
Composition('Li0.3243244Fe0.1621621O0.51351349') * 7.4,
Composition('Li3FeO4')
]
for crit, exp in zip(comps, expected):
self.assertTrue(crit.almost_equals(exp, rtol=0, atol=1e-5))
# case where the endpoints are identical
self.assertEqual(self.analyzer.get_critical_compositions(c1, c1 * 2),
[c1, c1 * 2])
def test_get_composition_chempots(self):
c1 = Composition('Fe3.1O4')
c2 = Composition('Fe3.2O4.1Li0.01')
e1 = self.analyzer.get_hull_energy(c1)
e2 = self.analyzer.get_hull_energy(c2)
cp = self.analyzer.get_composition_chempots(c1)
calc_e2 = e1 + sum(cp[k] * v for k, v in (c2 - c1).items())
self.assertAlmostEqual(e2, calc_e2)
class PDAnalyzerTest(unittest.TestCase):
def setUp(self):
module_dir = os.path.dirname(os.path.abspath(__file__))
(elements, entries) = PDEntryIO.from_csv(
os.path.join(module_dir, "pdentries_test.csv"))
self.pd = PhaseDiagram(entries)
self.analyzer = PDAnalyzer(self.pd)
def test_get_e_above_hull(self):
for entry in self.pd.stable_entries:
self.assertLess(
self.analyzer.get_e_above_hull(entry), 1e-11,
"Stable entries should have e above hull of zero!")
for entry in self.pd.all_entries:
if entry not in self.pd.stable_entries:
e_ah = self.analyzer.get_e_above_hull(entry)
self.assertGreaterEqual(e_ah, 0)
self.assertTrue(isinstance(e_ah, Number))
def test_get_equilibrium_reaction_energy(self):
for entry in self.pd.stable_entries:
self.assertLessEqual(
self.analyzer.get_equilibrium_reaction_energy(entry), 0,
"Stable entries should have negative equilibrium reaction energy!"
)
def test_get_decomposition(self):
for entry in self.pd.stable_entries:
self.assertEqual(
len(self.analyzer.get_decomposition(entry.composition)), 1,
"Stable composition should have only 1 decomposition!")
dim = len(self.pd.elements)
for entry in self.pd.all_entries:
ndecomp = len(self.analyzer.get_decomposition(entry.composition))
self.assertTrue(
ndecomp > 0 and ndecomp <= dim,
"The number of decomposition phases can at most be equal to the number of components."
)
#Just to test decomp for a ficitious composition
ansdict = {
entry.composition.formula: amt
for entry, amt in self.analyzer.get_decomposition(
Composition("Li3Fe7O11")).items()
}
expected_ans = {
"Fe2 O2": 0.0952380952380949,
"Li1 Fe1 O2": 0.5714285714285714,
"Fe6 O8": 0.33333333333333393
}
for k, v in expected_ans.items():
self.assertAlmostEqual(ansdict[k], v)
def test_get_transition_chempots(self):
for el in self.pd.elements:
self.assertLessEqual(
len(self.analyzer.get_transition_chempots(el)),
len(self.pd.facets))
def test_get_element_profile(self):
for el in self.pd.elements:
for entry in self.pd.stable_entries:
if not (entry.composition.is_element):
self.assertLessEqual(
len(
self.analyzer.get_element_profile(
el, entry.composition)), len(self.pd.facets))
def test_get_get_chempot_range_map(self):
elements = [el for el in self.pd.elements if el.symbol != "Fe"]
self.assertEqual(len(self.analyzer.get_chempot_range_map(elements)),
10)
def test_getmu_vertices_stability_phase(self):
results = self.analyzer.getmu_vertices_stability_phase(
Composition("LiFeO2"), Element("O"))
self.assertAlmostEqual(len(results), 6)
test_equality = False
for c in results:
if abs(c[Element("O")]+7.115) < 1e-2 and abs(c[Element("Fe")]+6.596) < 1e-2 and \
abs(c[Element("Li")]+3.931) < 1e-2:
test_equality = True
self.assertTrue(test_equality,
"there is an expected vertex missing in the list")
def test_getmu_range_stability_phase(self):
results = self.analyzer.get_chempot_range_stability_phase(
Composition("LiFeO2"), Element("O"))
self.assertAlmostEqual(results[Element("O")][1], -4.4501812249999997)
self.assertAlmostEqual(results[Element("Fe")][0], -6.5961470999999996)
self.assertAlmostEqual(results[Element("Li")][0], -3.6250022625000007)
def test_get_hull_energy(self):
for entry in self.pd.stable_entries:
h_e = self.analyzer.get_hull_energy(entry.composition)
self.assertAlmostEqual(h_e, entry.energy)
n_h_e = self.analyzer.get_hull_energy(
entry.composition.fractional_composition)
self.assertAlmostEqual(n_h_e, entry.energy_per_atom)
def test_1d_pd(self):
entry = PDEntry('H', 0)
pd = PhaseDiagram([entry])
pda = PDAnalyzer(pd)
decomp, e = pda.get_decomp_and_e_above_hull(PDEntry('H', 1))
self.assertAlmostEqual(e, 1)
self.assertAlmostEqual(decomp[entry], 1.0)
class PDAnalyzerTest(unittest.TestCase):
def setUp(self):
module_dir = os.path.dirname(os.path.abspath(__file__))
(elements, entries) = PDEntryIO.from_csv(os.path.join(module_dir,
"pdentries_test.csv"))
self.pd = PhaseDiagram(entries)
self.analyzer = PDAnalyzer(self.pd)
def test_get_e_above_hull(self):
for entry in self.pd.stable_entries:
self.assertLess(self.analyzer.get_e_above_hull(entry), 1e-11,
"Stable entries should have e above hull of zero!")
for entry in self.pd.all_entries:
if entry not in self.pd.stable_entries:
e_ah = self.analyzer.get_e_above_hull(entry)
self.assertGreaterEqual(e_ah, 0)
self.assertTrue(isinstance(e_ah, Number))
def test_get_equilibrium_reaction_energy(self):
for entry in self.pd.stable_entries:
self.assertLessEqual(
self.analyzer.get_equilibrium_reaction_energy(entry), 0,
"Stable entries should have negative equilibrium reaction energy!")
def test_get_decomposition(self):
for entry in self.pd.stable_entries:
self.assertEqual(len(self.analyzer.get_decomposition(entry.composition)), 1,
"Stable composition should have only 1 decomposition!")
dim = len(self.pd.elements)
for entry in self.pd.all_entries:
ndecomp = len(self.analyzer.get_decomposition(entry.composition))
self.assertTrue(ndecomp > 0 and ndecomp <= dim,
"The number of decomposition phases can at most be equal to the number of components.")
#Just to test decomp for a ficitious composition
ansdict = {entry.composition.formula: amt
for entry, amt in
self.analyzer.get_decomposition(Composition("Li3Fe7O11")).items()}
expected_ans = {"Fe2 O2": 0.0952380952380949,
"Li1 Fe1 O2": 0.5714285714285714,
"Fe6 O8": 0.33333333333333393}
for k, v in expected_ans.items():
self.assertAlmostEqual(ansdict[k], v)
def test_get_transition_chempots(self):
for el in self.pd.elements:
self.assertLessEqual(len(self.analyzer.get_transition_chempots(el)),
len(self.pd.facets))
def test_get_element_profile(self):
for el in self.pd.elements:
for entry in self.pd.stable_entries:
if not (entry.composition.is_element):
self.assertLessEqual(len(self.analyzer.get_element_profile(el, entry.composition)),
len(self.pd.facets))
def test_get_get_chempot_range_map(self):
elements = [el for el in self.pd.elements if el.symbol != "Fe"]
self.assertEqual(len(self.analyzer.get_chempot_range_map(elements)), 10)
def test_getmu_vertices_stability_phase(self):
results = self.analyzer.getmu_vertices_stability_phase(Composition("LiFeO2"), Element("O"))
self.assertAlmostEqual(len(results), 6)
test_equality = False
for c in results:
if abs(c[Element("O")]+7.115) < 1e-2 and abs(c[Element("Fe")]+6.596) < 1e-2 and \
abs(c[Element("Li")]+3.931) < 1e-2:
test_equality = True
self.assertTrue(test_equality,"there is an expected vertex missing in the list")
def test_getmu_range_stability_phase(self):
results = self.analyzer.get_chempot_range_stability_phase(
Composition("LiFeO2"), Element("O"))
self.assertAlmostEqual(results[Element("O")][1], -4.4501812249999997)
self.assertAlmostEqual(results[Element("Fe")][0], -6.5961470999999996)
self.assertAlmostEqual(results[Element("Li")][0], -3.6250022625000007)
def test_get_hull_energy(self):
for entry in self.pd.stable_entries:
h_e = self.analyzer.get_hull_energy(entry.composition)
self.assertAlmostEqual(h_e, entry.energy)
n_h_e = self.analyzer.get_hull_energy(entry.composition.fractional_composition)
self.assertAlmostEqual(n_h_e, entry.energy_per_atom)
def test_1d_pd(self):
entry = PDEntry('H', 0)
pd = PhaseDiagram([entry])
pda = PDAnalyzer(pd)
decomp, e = pda.get_decomp_and_e_above_hull(PDEntry('H', 1))
self.assertAlmostEqual(e, 1)
self.assertAlmostEqual(decomp[entry], 1.0)
class PDAnalyzerTest(unittest.TestCase):
def setUp(self):
module_dir = os.path.dirname(os.path.abspath(__file__))
(elements, entries) = PDEntryIO.from_csv(os.path.join(module_dir,
"pdentries_test.csv"))
self.pd = PhaseDiagram(entries)
self.analyzer = PDAnalyzer(self.pd)
def test_get_e_above_hull(self):
for entry in self.pd.stable_entries:
self.assertLess(self.analyzer.get_e_above_hull(entry), 1e-11,
"Stable entries should have e above hull of zero!")
for entry in self.pd.all_entries:
if entry not in self.pd.stable_entries:
e_ah = self.analyzer.get_e_above_hull(entry)
self.assertGreaterEqual(e_ah, 0)
self.assertTrue(isinstance(e_ah, Number))
def test_get_equilibrium_reaction_energy(self):
for entry in self.pd.stable_entries:
self.assertLessEqual(
self.analyzer.get_equilibrium_reaction_energy(entry), 0,
"Stable entries should have negative equilibrium reaction energy!")
def test_get_decomposition(self):
for entry in self.pd.stable_entries:
self.assertEqual(len(self.analyzer.get_decomposition(entry.composition)), 1,
"Stable composition should have only 1 decomposition!")
dim = len(self.pd.elements)
for entry in self.pd.all_entries:
ndecomp = len(self.analyzer.get_decomposition(entry.composition))
self.assertTrue(ndecomp > 0 and ndecomp <= dim,
"The number of decomposition phases can at most be equal to the number of components.")
#Just to test decomp for a ficitious composition
ansdict = {entry.composition.formula: amt
for entry, amt in
self.analyzer.get_decomposition(Composition("Li3Fe7O11")).items()}
expected_ans = {"Fe2 O2": 0.0952380952380949,
"Li1 Fe1 O2": 0.5714285714285714,
"Fe6 O8": 0.33333333333333393}
for k, v in expected_ans.items():
self.assertAlmostEqual(ansdict[k], v)
def test_get_transition_chempots(self):
for el in self.pd.elements:
self.assertLessEqual(len(self.analyzer.get_transition_chempots(el)),
len(self.pd.facets))
def test_get_element_profile(self):
for el in self.pd.elements:
for entry in self.pd.stable_entries:
if not (entry.composition.is_element):
self.assertLessEqual(len(self.analyzer.get_element_profile(el, entry.composition)),
len(self.pd.facets))
expected = [{'evolution': 1.0,
'chempot': -4.2582781416666666,
'reaction': 'Li2O + 0.5 O2 -> Li2O2'},
{'evolution': 0,
'chempot': -5.0885906699999968,
'reaction': 'Li2O -> Li2O'},
{'evolution': -1.0,
'chempot': -10.487582010000001,
'reaction': 'Li2O -> 2 Li + 0.5 O2'}]
result = self.analyzer.get_element_profile(Element('O'), Composition('Li2O'))
for d1, d2 in zip(expected, result):
self.assertAlmostEqual(d1['evolution'], d2['evolution'])
self.assertAlmostEqual(d1['chempot'], d2['chempot'])
self.assertEqual(d1['reaction'], str(d2['reaction']))
def test_get_get_chempot_range_map(self):
elements = [el for el in self.pd.elements if el.symbol != "Fe"]
self.assertEqual(len(self.analyzer.get_chempot_range_map(elements)), 10)
def test_getmu_vertices_stability_phase(self):
results = self.analyzer.getmu_vertices_stability_phase(Composition("LiFeO2"), Element("O"))
self.assertAlmostEqual(len(results), 6)
test_equality = False
for c in results:
if abs(c[Element("O")]+7.115) < 1e-2 and abs(c[Element("Fe")]+6.596) < 1e-2 and \
abs(c[Element("Li")]+3.931) < 1e-2:
test_equality = True
self.assertTrue(test_equality,"there is an expected vertex missing in the list")
def test_getmu_range_stability_phase(self):
results = self.analyzer.get_chempot_range_stability_phase(
Composition("LiFeO2"), Element("O"))
self.assertAlmostEqual(results[Element("O")][1], -4.4501812249999997)
self.assertAlmostEqual(results[Element("Fe")][0], -6.5961470999999996)
self.assertAlmostEqual(results[Element("Li")][0], -3.6250022625000007)
def test_get_hull_energy(self):
for entry in self.pd.stable_entries:
h_e = self.analyzer.get_hull_energy(entry.composition)
self.assertAlmostEqual(h_e, entry.energy)
n_h_e = self.analyzer.get_hull_energy(entry.composition.fractional_composition)
self.assertAlmostEqual(n_h_e, entry.energy_per_atom)
def test_1d_pd(self):
entry = PDEntry('H', 0)
pd = PhaseDiagram([entry])
pda = PDAnalyzer(pd)
decomp, e = pda.get_decomp_and_e_above_hull(PDEntry('H', 1))
self.assertAlmostEqual(e, 1)
self.assertAlmostEqual(decomp[entry], 1.0)
def test_get_critical_compositions_fractional(self):
c1 = Composition('Fe2O3').fractional_composition
c2 = Composition('Li3FeO4').fractional_composition
c3 = Composition('Li2O').fractional_composition
comps = self.analyzer.get_critical_compositions(c1, c2)
expected = [Composition('Fe2O3').fractional_composition,
Composition('Li0.3243244Fe0.1621621O0.51351349'),
Composition('Li3FeO4').fractional_composition]
for crit, exp in zip(comps, expected):
self.assertTrue(crit.almost_equals(exp, rtol=0, atol=1e-5))
comps = self.analyzer.get_critical_compositions(c1, c3)
expected = [Composition('Fe0.4O0.6'),
Composition('LiFeO2').fractional_composition,
Composition('Li5FeO4').fractional_composition,
Composition('Li2O').fractional_composition]
for crit, exp in zip(comps, expected):
self.assertTrue(crit.almost_equals(exp, rtol=0, atol=1e-5))
def test_get_critical_compositions(self):
c1 = Composition('Fe2O3')
c2 = Composition('Li3FeO4')
c3 = Composition('Li2O')
comps = self.analyzer.get_critical_compositions(c1, c2)
expected = [Composition('Fe2O3'),
Composition('Li0.3243244Fe0.1621621O0.51351349') * 7.4,
Composition('Li3FeO4')]
for crit, exp in zip(comps, expected):
self.assertTrue(crit.almost_equals(exp, rtol=0, atol=1e-5))
comps = self.analyzer.get_critical_compositions(c1, c3)
expected = [Composition('Fe2O3'),
Composition('LiFeO2'),
Composition('Li5FeO4') / 3,
Composition('Li2O')]
for crit, exp in zip(comps, expected):
self.assertTrue(crit.almost_equals(exp, rtol=0, atol=1e-5))
# Don't fail silently if input compositions aren't in phase diagram
# Can be very confusing if you're working with a GrandPotentialPD
self.assertRaises(ValueError, self.analyzer.get_critical_compositions,
Composition('Xe'), Composition('Mn'))
# For the moment, should also fail even if compositions are in the gppd
# because it isn't handled properly
gppd = GrandPotentialPhaseDiagram(self.pd.all_entries, {'Xe': 1},
self.pd.elements + [Element('Xe')])
pda = PDAnalyzer(gppd)
self.assertRaises(ValueError, pda.get_critical_compositions,
Composition('Fe2O3'), Composition('Li3FeO4Xe'))
# check that the function still works though
comps = pda.get_critical_compositions(c1, c2)
expected = [Composition('Fe2O3'),
Composition('Li0.3243244Fe0.1621621O0.51351349') * 7.4,
Composition('Li3FeO4')]
for crit, exp in zip(comps, expected):
self.assertTrue(crit.almost_equals(exp, rtol=0, atol=1e-5))
# case where the endpoints are identical
self.assertEqual(self.analyzer.get_critical_compositions(c1, c1 * 2),
[c1, c1 * 2])
def test_get_composition_chempots(self):
c1 = Composition('Fe3.1O4')
c2 = Composition('Fe3.2O4.1Li0.01')
e1 = self.analyzer.get_hull_energy(c1)
e2 = self.analyzer.get_hull_energy(c2)
cp = self.analyzer.get_composition_chempots(c1)
calc_e2 = e1 + sum(cp[k] * v for k, v in (c2 - c1).items())
self.assertAlmostEqual(e2, calc_e2)