We will use the quasi-harmonic approximation (via PyIron's implementation of the popular phonopy package) to evaluate look at thermal expansion and self-diffusion in Aluminium
# Generic imports
from pyiron.project import Project
import numpy as np
%matplotlib inline
import matplotlib.pylab as plt
import seaborn as sns
pr = Project("PhonopyExample")
pot = 'Al_Mg_Mendelev_eam'
pr.remove_jobs(recursive=True)
Because repeating code is evil.
def make_phonopy_job(template_job, name):
"""
Create a phonopy job from a reference job.
Args:
template_job (pyiron job): The job to copy.
name (str): What to call this new job.
Returns:
A new phonopy job.
"""
project = template_job.project
# What I want:
# job_type = template_job.job_type
# What I have to do instead:
job_type = pr.job_type.Lammps
ref = project.create_job(job_type, name + "_ref")
ref.structure = template_job.get_final_structure().copy()
ref.potential = template_job.potential
phono = project.create_job(pr.job_type.PhonopyJob, name)
phono.ref_job = ref
return phono
def scale_structure(struct, scale):
"""
Rescale the atomic positions and cell of a structure simultaneously.
Accepts rescaling by an arbitrary real-valued 3x3 numpy array, but a float can be given
for isotropic rescaling.
Args:
struct (Structure object): The structure to rescale.
scale (float or np.array(3,3)): The matrix to rescale by. (float -> isotropic.)
Returns:
A rescaled copy of the structure.
..TODO: Double check that the scaling matrix still spans 3-space (determinant check?)
"""
if isinstance(scale, float) or isinstance(scale, int):
scale_mat = scale * np.eye(3)
else:
assert(scale.shape == (3,3))
scale_mat = scale.T
new_struct = struct.copy()
new_struct.cell = np.dot(struct.cell, scale_mat)
new_struct.positions = np.dot(struct.positions, scale_mat)
return new_struct
def scale_array(arr, scaler=None, new_range=1.):
"""
Linearly transforms an array so that values equal to the minimum and maximum of the
`scaler` array are mapped to the range (0, `new_range`). Note that rescaled values can
still lie outside this range if the orignal values of `arr` are outside the bounds of
`scaler`.
Args:
arr (np.array): Array to rescale.
scaler (np.array): Array by which to rescale. Default is `arr`.
new_range (float): New value for data which was the size `np.amax(scaler)`.
Default is 1.
"""
if scaler is None:
scaler = arr
return new_range * (arr - np.amin(scaler)) / np.ptp(scaler)
What does the QHA say the lattice constant is as a function of temperature?
pr_te = pr.create_group("ThermalExpansion")
If we were doing VASP instead it would be important to do the least computation as possible, so here we'll start by relaxing a simple unit cell to turn into a supercell later.
job_unit = pr_te.create_job(pr.job_type.Lammps, "UnitCell")
basis = pr_te.create_structure("Al", "fcc", 4.04)
job_unit.structure = basis
job_unit.potential = pot
job_unit.calc_minimize(pressure=0.0)
job_unit.run()
The job UnitCell was saved and received the ID: 3596380
basis_rel = job_unit.get_final_structure()
A relaxation which should take zero steps given our starting position!
job_bulk_1 = pr_te.create_job(pr.job_type.Lammps, "Bulk_1")
# The _1 here refers to the fact that the volume has been rescaled by a factor of "1.0"
# (i.e. it hasn't been rescaled)
n_reps = 3
job_bulk_1.structure = basis_rel.repeat(rep=n_reps)
job_bulk_1.potential = pot
job_bulk_1.structure.plot3d();
_ColormakerRegistry()
job_bulk_1.calc_minimize(pressure=0.0)
job_bulk_1.run()
The job Bulk_1 was saved and received the ID: 3596381
Run phonopy on the bulk supercell
phono_bulk_1 = make_phonopy_job(job_bulk_1, "PhonoBulk_1")
phono_bulk_1.run()
# Run performs a whole bunch of child calculations
# Each one has the positions slightly deformed in the symmetry-appropriate ways needed
# to get the phonon properties
The job PhonoBulk_1 was saved and received the ID: 3596382 The job supercell_phonon_0 was saved and received the ID: 3596383
# Let's see what we got...
T_min = 0
T_max = 800 # a bit below melting
T_step = 25
temperatures = np.linspace(T_min, T_max, int((T_max - T_min) / T_step))
tp_bulk_1 = phono_bulk_1.get_thermal_properties(temperatures=temperatures)
# `get_thermal_properties` uses the displacements and forces to generate phonon information
U_bulk_1 = job_bulk_1.output.energy_pot[-1]
Fvib_bulk_1 = tp_bulk_1.free_energies
plt.plot(temperatures, U_bulk_1 + Fvib_bulk_1)
plt.xlabel("Temperature [K]")
plt.ylabel("Free energy ($U+F_{vib}$) [eV]")
Text(0, 0.5, 'Free energy ($U+F_{vib}$) [eV]')
Above we have the (QHA approximation to the) free energy as a function of temperature at a fixed volume. To evaluate the thermal expansivity, we need to create the entire F(V,T) surface. To get this, we just loop over jobs like the above, but scaled to have different lattice constants.
# According to Wikipedia, the thermal expansivity is about 0.0023% / Kelvin
# So at our maximum temperature, we expect around 1.8% expansion
scale_min = 0.995
scale_max = 1.02
scale_step = 0.002
scales = np.linspace(scale_min, scale_max, int((scale_max - scale_min) / scale_step))
# Let's keep things clean by making another sub-directory
pr_scales = pr_te.create_group("ScanScales")
# Loop the phonon calculation over all the volumes
sc_bulk_rel = job_bulk_1.get_final_structure()
bulk_free_energies = np.zeros((len(scales), len(temperatures)))
for i, scale in enumerate(scales):
name_tail = "_{}".format(str(scale).replace(".", "_"))
# Make a bulk job with the rescaled structure
# (already relaxed, by symmetry won't change, calc static will be enough)
job_bulk = pr_scales.create_job(pr.job_type.Lammps, "Bulk" + name_tail)
job_bulk.potential = pot
job_bulk.structure = scale_structure(sc_bulk_rel, scale)
job_bulk.calc_static()
job_bulk.run()
U = job_bulk.output.energy_tot[-1]
# Use that job as a reference for a phonopy job
phono_bulk = make_phonopy_job(job_bulk, "PhonoBulk" + name_tail)
phono_bulk.run()
tp_bulk = phono_bulk.get_thermal_properties(temperatures=temperatures)
Fvib = tp_bulk.free_energies
# Fill in the row of free energies for this volume
bulk_free_energies[i] = Fvib + U
The job Bulk_0_995 was saved and received the ID: 3596385 The job PhonoBulk_0_995 was saved and received the ID: 3596386 The job supercell_phonon_0 was saved and received the ID: 3596387 The job Bulk_0_9972727272727273 was saved and received the ID: 3596388 The job PhonoBulk_0_9972727272727273 was saved and received the ID: 3596389 The job supercell_phonon_0 was saved and received the ID: 3596390 The job Bulk_0_9995454545454545 was saved and received the ID: 3596394 The job PhonoBulk_0_9995454545454545 was saved and received the ID: 3596400 The job supercell_phonon_0 was saved and received the ID: 3596405 The job Bulk_1_0018181818181817 was saved and received the ID: 3596418 The job PhonoBulk_1_0018181818181817 was saved and received the ID: 3596419 The job supercell_phonon_0 was saved and received the ID: 3596420 The job Bulk_1_0040909090909091 was saved and received the ID: 3596434 The job PhonoBulk_1_0040909090909091 was saved and received the ID: 3596436 The job supercell_phonon_0 was saved and received the ID: 3596439 The job Bulk_1_0063636363636363 was saved and received the ID: 3596449 The job PhonoBulk_1_0063636363636363 was saved and received the ID: 3596450 The job supercell_phonon_0 was saved and received the ID: 3596451 The job Bulk_1_0086363636363636 was saved and received the ID: 3596456 The job PhonoBulk_1_0086363636363636 was saved and received the ID: 3596458 The job supercell_phonon_0 was saved and received the ID: 3596461 The job Bulk_1_010909090909091 was saved and received the ID: 3596473 The job PhonoBulk_1_010909090909091 was saved and received the ID: 3596475 The job supercell_phonon_0 was saved and received the ID: 3596478 The job Bulk_1_0131818181818182 was saved and received the ID: 3596488 The job PhonoBulk_1_0131818181818182 was saved and received the ID: 3596490 The job supercell_phonon_0 was saved and received the ID: 3596494 The job Bulk_1_0154545454545454 was saved and received the ID: 3596505 The job PhonoBulk_1_0154545454545454 was saved and received the ID: 3596507 The job supercell_phonon_0 was saved and received the ID: 3596510 The job Bulk_1_0177272727272728 was saved and received the ID: 3596521 The job PhonoBulk_1_0177272727272728 was saved and received the ID: 3596523 The job supercell_phonon_0 was saved and received the ID: 3596526 The job Bulk_1_02 was saved and received the ID: 3596537 The job PhonoBulk_1_02 was saved and received the ID: 3596539 The job supercell_phonon_0 was saved and received the ID: 3596542
# The lattice constant is probably a more informative value than the 0K-relative strain
latts = basis_rel.cell[0][0] * scales
# At each temperature, find the optimal volume by a simple quadratic fit
# ...Wait, which order fit will be good enough? Let's just spot-check
free_en = bulk_free_energies[:, -1]
plt.plot(latts, free_en, color='b', label='data')
# We'll plot the fit on a much denser mesh
fit_deg = 4
p = np.polyfit(latts, free_en, deg=fit_deg)
dense_latts = np.linspace(np.amin(latts), np.amax(latts), 1000)
#dense_latts = np.linspace(0, 10, 1000)
plt.plot(dense_latts, np.polyval(p=p, x=dense_latts), color='r', label='fit')
plt.xlabel('Lattice constant [$\mathrm{\AA}$]')
plt.ylabel('Bulk free energy [eV/supercell]')
plt.legend()
# Ok, a fourth-order fit seems perfectly reasonable
<matplotlib.legend.Legend at 0x2b61a42bf278>
# Now find optimal temperatures
best_latts = np.zeros(len(temperatures))
best_latt_guess = basis_rel.cell[0][0]
for i, T in enumerate(temperatures):
free_en = bulk_free_energies[:, i]
p = np.polyfit(latts, free_en, deg=fit_deg)
extrema = np.roots(np.polyder(p, m=1)).real # Find where first-derivative is zero
best_latts[i] = extrema[np.argmin(np.abs(extrema - best_latt_guess))]
# Check that they're resonable
print(best_latt_guess, '\n', best_latts)
4.045270475668763 [4.05946291 4.05949371 4.05987201 4.06083718 4.06226186 4.06396024 4.06579637 4.06768361 4.06956868 4.0714193 4.07321635 4.07494901 4.07661182 4.07820271 4.07972185 4.08117076 4.08255184 4.08386799 4.08512237 4.08631821 4.08745877 4.0885472 4.08958656 4.09057975 4.0915295 4.09243842 4.09330892 4.09414326 4.09494357 4.09571182 4.09644984 4.09715935]
# Let's look at the landscape
fig, ax = plt.subplots()
sns.heatmap(bulk_free_energies, ax=ax, cmap="coolwarm",
xticklabels=['{:,.0f}'.format(T) for T in temperatures],
yticklabels=['{:,.2f}'.format(a) for a in latts])
ax.set_xlabel("Temperature [K]")
ax.set_ylabel("Lattice constant [$\mathrm{\AA}$]")
# Overlaying the optimal path takes a couple changes of variables
# since the heatmap is plotting integer cells
ax.plot(scale_array(temperatures, new_range=len(temperatures)),
scale_array(best_latts, scaler=latts, new_range=len(latts)),
color='k')
[<matplotlib.lines.Line2D at 0x2b61d9e8ee80>]
Another common use of QHA is to calculate the pre-factor for migration in a diffusion event.
In particular, the diffusion jump barrier looks like $\omega_0 = \nu_0^\star \exp(-H_\mathrm{m} / k_\mathrm{B} T)$, where $\nu_0^\star = \prod_{i=1}^{3N-3} \nu_i^\mathrm{IS} / \prod_{i=1}^{3N-4} \nu_i^\mathrm{TS}$, with IS and TS indicating the initial and transition states, respectively. Note that the transition state is missing a single frequency, which is from the instability of the transition state. It's either an imaginary mode, which I think means a negative frequency. Meanwhile, $H_\mathrm{m}$ is the enthalpic barrier (difference between the initial and transition states) and $k_\mathrm{B} T$ is the usual thermal energy term.
Typically, these sorts of investigations use the nudged elastic band (NEB) to find the 0K transition state. You can do that with our new flexible jobs, but we'll save that for later. For now we'll just "approximate" the transition state with a simple linear interpolation.
Let's start by generating and relaxing the initial and final states
pr_vac = pr.create_group("Vacancies")
# Find two adjacent sites
print(job_bulk_1.structure.positions[0])
print(job_bulk_1.structure.positions[1])
# Yep, 1 and 2 will do
[0. 0. 0.] [2.02263524 2.02263524 0. ]
job_vac_i = pr_vac.create_job(pr.job_type.Lammps, "VacancyInitial")
job_vac_f = pr_vac.create_job(pr.job_type.Lammps, "VacancyFinal")
job_vac_i.potential = pot
job_vac_f.potential = pot
sc_vac_i = sc_bulk_rel.copy()
sc_vac_i.pop(0)
job_vac_i.structure = sc_vac_i
sc_vac_f = sc_bulk_rel.copy()
sc_vac_f.pop(1)
job_vac_f.structure = sc_vac_f
# Relax the new vacancy structures
job_vac_i.calc_minimize(pressure=0.0)
job_vac_i.run()
job_vac_f.calc_minimize(pressure=0.0)
job_vac_f.run()
The job VacancyInitial was saved and received the ID: 3596547 The job VacancyFinal was saved and received the ID: 3596549
The PyIron implementation of phonopy makes it very easy to look at the DOS. Let's see what the effect is of introducing a vacancy, and confirm that our two vacancies are equivalent.
phon_vac_i = make_phonopy_job(job_vac_i, "PhonoVacInitial")
phon_vac_f = make_phonopy_job(job_vac_f, "PhonoVacFinal")
phon_vac_i.run()
tp_vac_i = phon_vac_i.get_thermal_properties(temperatures=temperatures)
phon_vac_f.run()
tp_vac_f = phon_vac_i.get_thermal_properties(temperatures=temperatures)
# Note that the vacancy structures spawn many more child processes
# This is because the vacancy structure has lower symmetry
The job PhonoVacInitial was saved and received the ID: 3596552 The job supercell_phonon_0 was saved and received the ID: 3596554 The job supercell_phonon_1 was saved and received the ID: 3596556 The job supercell_phonon_2 was saved and received the ID: 3596559 The job supercell_phonon_3 was saved and received the ID: 3596561 The job supercell_phonon_4 was saved and received the ID: 3596564 The job supercell_phonon_5 was saved and received the ID: 3596566 The job supercell_phonon_6 was saved and received the ID: 3596569 The job supercell_phonon_7 was saved and received the ID: 3596571 The job supercell_phonon_8 was saved and received the ID: 3596573 The job supercell_phonon_9 was saved and received the ID: 3596576 The job supercell_phonon_10 was saved and received the ID: 3596578 The job supercell_phonon_11 was saved and received the ID: 3596580 The job supercell_phonon_12 was saved and received the ID: 3596582 The job supercell_phonon_13 was saved and received the ID: 3596585 The job supercell_phonon_14 was saved and received the ID: 3596587 The job supercell_phonon_15 was saved and received the ID: 3596589 The job supercell_phonon_16 was saved and received the ID: 3596592 The job supercell_phonon_17 was saved and received the ID: 3596594 The job supercell_phonon_18 was saved and received the ID: 3596597 The job supercell_phonon_19 was saved and received the ID: 3596599 The job supercell_phonon_20 was saved and received the ID: 3596601 The job PhonoVacFinal was saved and received the ID: 3596613 The job supercell_phonon_0 was saved and received the ID: 3596616 The job supercell_phonon_1 was saved and received the ID: 3596618 The job supercell_phonon_2 was saved and received the ID: 3596620 The job supercell_phonon_3 was saved and received the ID: 3596623 The job supercell_phonon_4 was saved and received the ID: 3596625 The job supercell_phonon_5 was saved and received the ID: 3596628 The job supercell_phonon_6 was saved and received the ID: 3596630 The job supercell_phonon_7 was saved and received the ID: 3596632 The job supercell_phonon_8 was saved and received the ID: 3596635 The job supercell_phonon_9 was saved and received the ID: 3596637 The job supercell_phonon_10 was saved and received the ID: 3596640 The job supercell_phonon_11 was saved and received the ID: 3596642 The job supercell_phonon_12 was saved and received the ID: 3596644 The job supercell_phonon_13 was saved and received the ID: 3596646 The job supercell_phonon_14 was saved and received the ID: 3596649 The job supercell_phonon_15 was saved and received the ID: 3596651 The job supercell_phonon_16 was saved and received the ID: 3596653 The job supercell_phonon_17 was saved and received the ID: 3596655 The job supercell_phonon_18 was saved and received the ID: 3596658 The job supercell_phonon_19 was saved and received the ID: 3596659 The job supercell_phonon_20 was saved and received the ID: 3596660
fig, ax = plt.subplots()
phono_bulk_1.plot_dos(ax=ax, color='b', label='bulk')
phon_vac_i.plot_dos(ax=ax, color='r', label='vac_i')
phon_vac_f.plot_dos(ax=ax, color='orange', label='vac_f')
plt.legend()
<matplotlib.legend.Legend at 0x2b61da20bcc0>
Now we get the attack frequency by comparing the individual phonon spectra of initial and transition states
# Interpolate initial and final positions to guesstimate the transition state
sc_vac_ts = sc_vac_i.copy()
sc_vac_ts.positions = 0.5 * (sc_vac_i.positions + sc_vac_f.positions)
job_vac_ts = pr_vac.create_job(pr.job_type.Lammps, "VacancyTransition")
job_vac_ts.potential = pot
job_vac_ts.structure = sc_vac_ts
# We _don't_ relax this job, or it would fall into the initial or final state!
job_vac_ts.calc_static()
job_vac_ts.run()
The job VacancyTransition was saved and received the ID: 3596670
phon_vac_ts = make_phonopy_job(job_vac_ts, "PhonoVacTransition")
phon_vac_ts.run()
tp_vac_ts = phon_vac_ts.get_thermal_properties(temperatures=temperatures)
The job PhonoVacTransition was saved and received the ID: 3596673 The job supercell_phonon_0 was saved and received the ID: 3596676 The job supercell_phonon_1 was saved and received the ID: 3596678 The job supercell_phonon_2 was saved and received the ID: 3596681 The job supercell_phonon_3 was saved and received the ID: 3596683 The job supercell_phonon_4 was saved and received the ID: 3596686 The job supercell_phonon_5 was saved and received the ID: 3596688 The job supercell_phonon_6 was saved and received the ID: 3596691 The job supercell_phonon_7 was saved and received the ID: 3596693 The job supercell_phonon_8 was saved and received the ID: 3596695 The job supercell_phonon_9 was saved and received the ID: 3596698 The job supercell_phonon_10 was saved and received the ID: 3596700 The job supercell_phonon_11 was saved and received the ID: 3596703 The job supercell_phonon_12 was saved and received the ID: 3596705 The job supercell_phonon_13 was saved and received the ID: 3596707 The job supercell_phonon_14 was saved and received the ID: 3596709 The job supercell_phonon_15 was saved and received the ID: 3596712 The job supercell_phonon_16 was saved and received the ID: 3596714 The job supercell_phonon_17 was saved and received the ID: 3596716 The job supercell_phonon_18 was saved and received the ID: 3596719 The job supercell_phonon_19 was saved and received the ID: 3596721 The job supercell_phonon_20 was saved and received the ID: 3596723 The job supercell_phonon_21 was saved and received the ID: 3596726 The job supercell_phonon_22 was saved and received the ID: 3596728 The job supercell_phonon_23 was saved and received the ID: 3596731 The job supercell_phonon_24 was saved and received the ID: 3596733 The job supercell_phonon_25 was saved and received the ID: 3596735 The job supercell_phonon_26 was saved and received the ID: 3596738 The job supercell_phonon_27 was saved and received the ID: 3596740 The job supercell_phonon_28 was saved and received the ID: 3596742 The job supercell_phonon_29 was saved and received the ID: 3596744 The job supercell_phonon_30 was saved and received the ID: 3596746 The job supercell_phonon_31 was saved and received the ID: 3596749 The job supercell_phonon_32 was saved and received the ID: 3596751 The job supercell_phonon_33 was saved and received the ID: 3596754 The job supercell_phonon_34 was saved and received the ID: 3596756 The job supercell_phonon_35 was saved and received the ID: 3596758 The job supercell_phonon_36 was saved and received the ID: 3596760 The job supercell_phonon_37 was saved and received the ID: 3596762 The job supercell_phonon_38 was saved and received the ID: 3596765 The job supercell_phonon_39 was saved and received the ID: 3596767 The job supercell_phonon_40 was saved and received the ID: 3596769 The job supercell_phonon_41 was saved and received the ID: 3596772 The job supercell_phonon_42 was saved and received the ID: 3596774 The job supercell_phonon_43 was saved and received the ID: 3596776 The job supercell_phonon_44 was saved and received the ID: 3596778 The job supercell_phonon_45 was saved and received the ID: 3596781 The job supercell_phonon_46 was saved and received the ID: 3596782 The job supercell_phonon_47 was saved and received the ID: 3596783 The job supercell_phonon_48 was saved and received the ID: 3596784 The job supercell_phonon_49 was saved and received the ID: 3596785 The job supercell_phonon_50 was saved and received the ID: 3596786 The job supercell_phonon_51 was saved and received the ID: 3596788 The job supercell_phonon_52 was saved and received the ID: 3596790 The job supercell_phonon_53 was saved and received the ID: 3596792 The job supercell_phonon_54 was saved and received the ID: 3596794 The job supercell_phonon_55 was saved and received the ID: 3596796 The job supercell_phonon_56 was saved and received the ID: 3596798 The job supercell_phonon_57 was saved and received the ID: 3596799 The job supercell_phonon_58 was saved and received the ID: 3596801 The job supercell_phonon_59 was saved and received the ID: 3596803 The job supercell_phonon_60 was saved and received the ID: 3596806 The job supercell_phonon_61 was saved and received the ID: 3596808 The job supercell_phonon_62 was saved and received the ID: 3596810 The job supercell_phonon_63 was saved and received the ID: 3596813 The job supercell_phonon_64 was saved and received the ID: 3596815 The job supercell_phonon_65 was saved and received the ID: 3596817 The job supercell_phonon_66 was saved and received the ID: 3596819 The job supercell_phonon_67 was saved and received the ID: 3596822 The job supercell_phonon_68 was saved and received the ID: 3596824 The job supercell_phonon_69 was saved and received the ID: 3596826 The job supercell_phonon_70 was saved and received the ID: 3596829 The job supercell_phonon_71 was saved and received the ID: 3596831 The job supercell_phonon_72 was saved and received the ID: 3596833 The job supercell_phonon_73 was saved and received the ID: 3596836 The job supercell_phonon_74 was saved and received the ID: 3596838 The job supercell_phonon_75 was saved and received the ID: 3596842 The job supercell_phonon_76 was saved and received the ID: 3596846 The job supercell_phonon_77 was saved and received the ID: 3596850 The job supercell_phonon_78 was saved and received the ID: 3596855 The job supercell_phonon_79 was saved and received the ID: 3596858 The job supercell_phonon_80 was saved and received the ID: 3596863 The job supercell_phonon_81 was saved and received the ID: 3596866 The job supercell_phonon_82 was saved and received the ID: 3596869 The job supercell_phonon_83 was saved and received the ID: 3596874 The job supercell_phonon_84 was saved and received the ID: 3596878 The job supercell_phonon_85 was saved and received the ID: 3596881 The job supercell_phonon_86 was saved and received the ID: 3596885 The job supercell_phonon_87 was saved and received the ID: 3596890 The job supercell_phonon_88 was saved and received the ID: 3596894 The job supercell_phonon_89 was saved and received the ID: 3596898 The job supercell_phonon_90 was saved and received the ID: 3596902 The job supercell_phonon_91 was saved and received the ID: 3596907 The job supercell_phonon_92 was saved and received the ID: 3596911 The job supercell_phonon_93 was saved and received the ID: 3596915 The job supercell_phonon_94 was saved and received the ID: 3596920
# The transition state has an imaginary mode (frequency < 0), let's see it
fig, ax = plt.subplots()
phon_vac_i.plot_dos(ax=ax, color='r', label='initial')
phon_vac_ts.plot_dos(ax=ax, color='b', label='transition')
plt.legend()
<matplotlib.legend.Legend at 0x2b61db5474e0>
To calculate the attack frequency, we'll ignore both the negative mode of the transition state (which we were warned about in the equation), as well as the three frequencies which correspond to rigid translation and are very near zero, and sometimes dip to be negative. Phonopy sorts the frequencies by magnitude, so we can just skip the first three and four for the initial and transition states, respectively. We take them at q=0.
freq_i = phon_vac_i.phonopy.get_frequencies(0)[3:]
freq_ts = phon_vac_i.phonopy.get_frequencies(0)[4:]
print(np.prod(freq_i))
6.870675816849329e+236
Recall: $\nu_0^\star = \prod_{i=1}^{3N-3} \nu_i^\mathrm{IS} / \prod_{i=1}^{3N-4} \nu_i^\mathrm{TS}$
# Products are dangerous beasts, so we'll do a little numeric magic
nu = np.prod(freq_i[:-1] / freq_ts) * freq_i[-1]
print("Attack frequency is ", nu, "THz (10^-12 s)")
Attack frequency is 2.6826827779812032 THz (10^-12 s)
Mantina et al. (PRL 2008) report $\nu = 19.3$ THz using DFT and NEB, so our linearly-interpolated "transition state" with EAM is actually not doing so poorly.
There are many more things you can do with phonopy, including looking directly at the force constants, the Hessian matrix, etc. But hopefully this is a useful starting point.