2.5. Multi-configurations free energy

2.5.2. Same phonon density of states

class qha.multi_configurations.same_phonon_dos.PartitionFunction(temperature, degeneracies, q_weights, static_energies, frequencies, precision=500)

A class that represents the partition function of multiple configurations with the same phonon density of states. In mathematics, it is represented as

\[Z_{\text{all configs}}(T, V) = \sum_{j = 1}^{N_{c}} g_{j} \bigg\{ \exp \Big( -\frac{ E_j(V) }{ k_B T } \Big) \prod_{\mathbf{q}s} \bigg( \tfrac{\exp \big(-\tfrac{ \hbar \omega_{\mathbf{ q }s}^j(V) }{ 2 k_B T }\big)}{1 - \exp \big(-\tfrac{ \hbar \omega_{\mathbf{ q }s}^j(V) }{ k_B T }\big)} \bigg)^{w_{\mathbf{ q }}^j} \bigg\}.\]

Parameters

• temperature (float64) – The temperature at which the partition function is calculated.

• degeneracies (array(float64, 1d, A)) – An array of degeneracies of each configuration, which will not be normalized in the calculation. They should all be positive integers.

• q_weights (array(float64, 1d, A)) – The weights of all q-points that are sampled, should be a vector since all configurations should have the same q-point weights.

• static_energies (array(float64, 2d, A)) – The static energies of each configuration of each volume.

• frequencies (array(float64, 3d, A)) – It is a 3D array that specifies the frequency on each volume, q-point and mode. It is not 4D since we have all configurations sharing the same phonon density of states.

• precision (Optional[int]) – The precision of a big float number to represent the partition function since it is a very large value, by default, is 500.

get_free_energies()

Give the free energy by

\[F_{\text{all configs}}(T, V) = - k_B T \ln Z_{\text{all configs}}(T, V).\]

Returns

The free energy on the temperature-volume grid.

property total

Calculate the total partition function.

Returns

The partition function on the temperature-volume grid.

class qha.multi_configurations.same_phonon_dos.FreeEnergy(temperature, degeneracies, q_weights, static_energies, volumes, frequencies, static_only=False, order=3)

A class that represents the free energy of multiple configurations with the same phonon density of states. In mathematics, it is represented as

\[F_{\text{all configs}}(T, V) = - k_B T \ln Z_{\text{all configs}}(T, V) = - k_B T \ln \bigg( \sum_{j = 1}^{N_{c}} g_{j} \exp \Big( -\frac{ E_j(V) }{ k_B T } \Big) \bigg) + \sum_{\mathbf{ q }s} w_\mathbf{ q } \bigg\{ \frac{ \hbar \omega_{\mathbf{ q }s}(V) }{ 2 } + k_B \ln \bigg( 1 - \exp \Big( -\frac{ \hbar \omega_{\mathbf{ q }s}(V) }{ k_B T } \Big) \bigg) \bigg\}.\]

Parameters

• temperature (float64) – The temperature at which the partition function is calculated.

• degeneracies (array(float64, 1d, A)) – An array of degeneracies of each configuration, which will not be normalized in the calculation. They should all be positive integers.

• q_weights (array(float64, 1d, A)) – The weights of all q-points that are sampled, should be a vector since all configurations should have the same q-point weights.

• static_energies (array(float64, 2d, A)) – The static energy of each configuration of each volume.

• volumes (array(float64, 2d, A)) – A matrix of volumes of each configurations, should have the same values for each configuration.

• frequencies (array(float64, 3d, A)) – It is a 3D array that specifies the frequency on each volume, q-point and mode. It is not 4D since we have all configurations sharing the same phonon density of states.

• static_only (Optional[bool]) – Whether the calculation only takes static contribution and does not consider the vibrational contribution, by default, is False.

• order (Optional[int]) – The order of Birch–Murnaghan equation of state fitting, by default, is 3.

property aligned_static_energies_for_each_configuration

If the input static energies are not aligned for each configuration, then do a fitting to align all static energies.

Returns

A matrix of aligned static energy of each configuration of each volume.

get_free_energies()

If static_only = True is specified in class instantiation, then only the static contribution will be counted. Then it is equivalent to the static_part property. If not, then this is equivalent to the total property.

Returns

The free energy on the temperature-volume grid.

property harmonic_part

Calculate the harmonic contribution to the free energy.

Returns

The harmonic contribution on the temperature-volume grid.

property static_part

Calculate the static contribution to the free energy.

Returns

The static contribution on the temperature-volume grid.

property total

Calculate the total partition function, which is the static part plus the harmonic part.

Returns

Total free energy on the temperature-volume grid.

2.5.2. Different phonon density of states

class qha.multi_configurations.different_phonon_dos.PartitionFunction(temperature, degeneracies, q_weights, static_energies, volumes, frequencies, static_only=False, precision=500, order=3)

A class that represents the partition function of multiple configurations with different phonon density of states. In mathematics, it is represented as

\[Z_{\text{all configs}}(T, V) = \sum_{j = 1}^{N_{c}} g_{j} Z_{j}(T, V),\]

where \(N_{c}\) stands for the number of configurations and \(g_{j}\) stands for degeneracy for the \(j\) th configuration.

Parameters

• temperature (float64) – The temperature at which the partition function is calculated.

• degeneracies (array(float64, 1d, A)) – An array of degeneracies of each configuration, which will not be normalized in the calculation. They should all be positive integers.

• q_weights (array(float64, 2d, A)) – The weights of all q-points that are sampled, can be a 2D matrix so each configuration can have a little bit different q-point weights, but the number of q-points of each configuration must be the same.

• static_energies (array(float64, 2d, A)) – The static energy of each configuration of each volume.

• volumes (array(float64, 2d, A)) – A matrix of volumes of each configuration, should have the same values for each configuration.

• frequencies (array(float64, 4d, A)) – A 4D array that specifies the frequency on each configuration, volume, q-point and mode.

• static_only (Optional[bool]) – Whether the calculation only takes static contribution and does not consider the vibrational contribution, by default, is False.

• precision (Optional[int]) – The precision of a big float number to represent the partition function since it is a very large value, by default, is 500.

• order (Optional[int]) – The order of Birch–Murnaghan equation of state fitting, by default, is 3.

property aligned_free_energies_for_each_configuration

Then do a fitting to align all these free energies.

Returns

A matrix, the aligned free energy of each configuration of each volume.

get_free_energies()

The free energy calculated from the partition function \(Z_{\text{all configs}}(T, V)\) by

\[F_{\text{all configs}}(T, V) = - k_B T \ln Z_{\text{all configs}}(T, V).\]

Returns

The free energy on a temperature-volume grid.

property partition_functions_for_all_configurations

Sum the partition functions for all configurations.

\[Z_{\text{all configs}}(T, V) = \sum_{j} Z_{j}(T, V).\]

Returns

A vector, the partition function of each volume.

property partition_functions_for_each_configuration

Inversely solve the free energy to get partition function for \(j\) th configuration by

\[Z_{j}(T, V) = \exp \bigg( -\frac{ F_{j}(T, V) }{ k_B T } \bigg).\]

Returns

A matrix, the partition function of each configuration of each volume.

property unaligned_free_energies_for_each_configuration

If the input free energy is not aligned for each configuration, first just calculate the “raw” free energy on each volume and each configuration.

Returns

A matrix, the “raw” free energy of each configuration of each volume.