# Dfttk postprocessing mechanism

## Flow controls

Depending on the given condition, postprocessing can take data from

The qha-phonon collection
The qha collection (Debye model)
The phonon collection (in case of qha-phonon calculations failed)

Then, invoke Yphon to recalculate the phonon properties based on the force constants obtained from the database

## Scheme to find equilibrium volume

If the data quality is excellent, use central symmetric 7-point difference
If the data quality is very good, fit the free energies use 4-parameter Birch-Murnaghan
If the data quality is good (may use as default), fit the 0-K total energies using 4-parameter Birch-Murnaghan
Fit the finite T part of the free energies by UnivariateSpline

## Scheme to calculate derivative

1. 7-point symmetrical central difference

$Deriv= \frac{1}{3}\sum_{i=1}^{3}{\frac{f(X_{N+i})-f(X_{N-i})}{X_{N+i}-X_{N-i}}}$
1. Birch-Mannhan Euqations of state fitting

$Deriv=-\frac{2}{3}bx^{-\frac{5}{3}}-\frac{4}{3}cx^{-\frac{7}{3}}-\frac{6}{3}dx^{-\frac{9}{3}}$

## Scheme for LTC

1. By entropy derivative

$\alpha =\frac{1}{B_{T}}\frac{\partial S}{\partial V}$

where $$B_{T}$$ is isothermal bulk modulus

1. By internal energy derivative via Birch-Mannhan fitting

$\alpha =\frac{1}{B_{T} T}(\frac{\partial U}{\partial V}+P)$

where $$P$$ is pressure

1. When the data quality is fair (~20% cases)

Fit the 0-K total energies using 4-parameter Birch-Murnaghan; and
fit the finite T part of the free energies by linear function f=a+b*V