2.7. Fitting Birch–Murnaghan equation of state

This module is one of the most important part of this package since it implements a robust Birch–Murnaghan EoS fitting for grid_interpolation module’s use.

qha.fitting.polynomial_least_square_fitting(xs, ys, new_xs, order=3)

The algorithm is referenced from the Wolfram MathWorld.

Parameters

• xs – A vector of existing x-coordinates.

• ys – A vector of y-coordinates correspond to the xs.

• new_xs – A new vector of x-coordinates to be applied with the polynomial-fitting result.

• order (Optional[int]) – The order chose to fit the finite strain EoS, the default value is 3, which is, the third-order Birch–Murnaghan EoS.

Returns

A tuple, the polynomial-fitting coefficients and the new vector of y-coordinates.

qha.fitting.apply_finite_strain_fitting(strains_sparse, free_energies, strains_dense, order=3)

Calculate the free energies \(F(T, V)\) for some strains (strains_dense), with the free energies (free_energies) on some other strains (strains_sparse) known already. Do a polynomial curve-fitting the apply the fitted function to the strains_dense.

Parameters

• strains_sparse (array(float64, 1d, A)) – A vector of the Eulerian strains for a sparse set of volumes.

• free_energies (array(float64, 2d, A)) – The free energies correspond to strains_sparse at several temperature.

• strains_dense (array(float64, 1d, A)) – A vector of the Eulerian strains at a denser set of volumes.

• order (Optional[int]) – The order chose to fit the finite strain EoS, the default value is 3, which is, the third-order Birch–Murnaghan EoS.

Returns

The free energies correspond to strains_dense at different temperature.