EOS package

EOS.EOS module

Created on Wed Dec 9 10:51:00 2014

@author: Matti Ropo @author: Henrik Levämäki

class EOS.EOS.EOS(name, xc='PBE', method='morse', units='bohr', warnings=True)[source]

Bases: object

Fit equation of state for bulk systems.

The following equations are available:

morse PRB 37, 790 (1988)
sjeos PRB 63, 224115 (2001)
taylor A third order Taylor series expansion about the minimum volume
murnaghan PRB 28, 5480 (1983)
birch Intermetallic compounds: Principles and Practice, Vol I: Principles. pages 195-210
birchmurnaghan PRB 70, 224107
pouriertarantola PRB 70, 224107
vinet PRB 70, 224107
antonschmidt Intermetallics 11, 23-32 (2003)
oldpoly A third order polynomial fit (alternative implementation)

Usage:

eos = EquationOfState(volumes, energies, eos='murnaghan')
v0, e0, B = eos.fit()
eos.plot()
Parameters:
  • name –
  • xc – (Default value = ‘PBE’)
  • method – (Default value = ‘morse’)
  • units – (Default value = ‘bohr’)
Returns:
Return type:
angstrom2bohr(V)[source]
Parameters:V –
Returns:
Return type:
antonschmidt(V, Einf, B, n, V0)[source]

From Intermetallics 11, 23-32 (2003)

Einf should be E_infinity, i.e. infinite separation, but according to the paper it does not provide a good estimate of the cohesive energy. They derive this equation from an empirical formula for the volume dependence of pressure,

E(vol) = E_inf + int(P dV) from V=vol to V=infinity

but the equation breaks down at large volumes, so E_inf is not that meaningful

n should be about -2 according to the paper.

I find this equation does not fit volumetric data as well as the other equtions do.

Parameters:
  • V –
  • Einf –
  • B –
  • n –
  • V0 –
Returns:

Energy
Return type:

float
ascii_plot(x, y, z, title='')[source]
birch(V, E0, B0, BP, V0)[source]

From Intermetallic compounds: Principles and Practice, Vol. I: Principles Chapter 9 pages 195-210 by M. Mehl. B. Klein, D. Papaconstantopoulos paper downloaded from Web

case where n=0

Parameters:
  • V –
  • E0 –
  • B0 –
  • BP –
  • V0 –
Returns:
Return type:
birchmurnaghan(V, E0, B0, BP, V0)[source]

BirchMurnaghan equation from PRB 70, 224107

Parameters:
  • V –
  • E0 –
  • B0 –
  • BP –
  • V0 –
Returns:
Return type:
bohr2angstrom(WSrad)[source]
Parameters:WSrad –
Returns:
Return type:
ca_fit(x, y, n, debug=False, title='', find_best_fit=False)[source]

Fits a polynomial to x vs. y data and calculates xmin and ymin from the curve.

Parameters:
  • x –
  • y –
  • n –
Returns:
Return type:
compute_eos_fit()[source]

Performs the least-squares curve fitting for the chosen EOS function. Ground state quantities are returned.

compute_eos_quality()[source]

Compute different estimates for the quality of the fit.

compute_initial_guess()[source]

Calculates initial guessess for the fitting parameters.

distortion_fit(x, y, num=2, title='', ascii_art=False)[source]

Fits the distortion_poly function to the distortion data.

num : number of variables in the fitting function
1 : E=a2*x**2 2 : E=a2*x**2 + a0 3 : E=a2*x**2 + a1*x + a0

The fit coefficient(s) and r-squared describing the accuracy of the fit are returned.

Parameters:
  • x –
  • y –
  • num – (Default value = 2)
Returns:
Return type:
distortion_poly1(x, a2)[source]
One variable form of the 2nd order polynomial
which is used to fitting distortion vs. energy data in order to find elastic constants.
Parameters:
  • x –
  • a2 –
  • a1 –
Returns:
Return type:
distortion_poly2(x, a2, a0)[source]
Two variable form of the 2nd order polynomial
which is used to fitting distortion vs. energy data in order to find elastic constants.
Parameters:
  • x –
  • a2 –
  • a0 –
Returns:
Return type:
distortion_poly3(x, a2, a1, a0)[source]
Three variable form of the 2nd order polynomial
which is used to fitting distortion vs. energy data in order to find elastic constants.
Parameters:
  • x –
  • a2 –
  • a1 –
  • a0 –
Returns:
Return type:
eos_output()[source]

Construct a string for the EOS output message.

fit(swses, energies, shift=False, show_plot=False, show_output=True, pressure=False)[source]

Calculate volume, energy, and bulk modulus.

About the units:

Input “swses” should be the WS-radii in bohr. Input “energies” should be the energies in Ry.

Volumes are always in Angstrom**3 Bulk moduli are always in GPa

Parameters:
  • swses (list(float)) – List of WS-radii
  • energies (list(float)) – List of energies
  • shift (True or False) – Shift the energies so that the lowest energy is zero. Improves fit quality. Sometimes instable.
Returns:

Eq. WS-rad, energy, bulk modulus, Gruneisen parameter
Return type:

float, float, float, float
fit_eval(sws)[source]

Evaluate the fitting function at given points

Parameters:sws –
Returns:
Return type:
morse(sws, a0, b0, c0, l0)[source]
Parameters:
  • w –
  • a0 –
  • b0 –
  • c0 –
  • l0 –
murnaghan(V, E0, B0, BP, V0)[source]

From PRB 28,5480 (1983)

Parameters:
  • V –
  • E0 –
  • B0 –
  • BP –
  • V0 –
Returns:
Return type:
oldpoly(V, c0, c1, c2, c3)[source]

polynomial fit, 3rd order

Parameters:
  • V –
  • c0 –
  • c1 –
  • c2 –
  • c3 –
Returns:

Energy
Return type:

float
parabola(x, a, b, c)[source]

Parabola polynomial function

This function is used to fit the data to get good guesses for the equation of state fits.

A 4th order polynomial fit to get good guesses for was not a good idea because for noisy data the fit is too wiggly 2nd order seems to be sufficient, and guarentees a single minimum.

Parameters:
  • x –
  • a –
  • b –
  • c –
Returns:

Energy
Return type:

float
plot(filename=None, show=True)[source]

Plot fitted energy curve.

Uses Matplotlib to plot the energy curve. Use show=True to show the figure and filename=’abc.png’ or filename=’abc.eps’ to save the figure to a file.

Parameters:
  • filename – (Default value = None)
  • show – (Default value = None)
Returns:
Return type:
static polyfit(x, y, n)[source]
pouriertarantola(V, E0, B0, BP, V0)[source]

Pourier-Tarantola equation from PRB 70, 224107

Parameters:
  • V –
  • E0 –
  • B0 –
  • BP –
  • V0 –
Returns:
Return type:
predicted()[source]

Evaluates the EOS function using the calc. EOS params.

Returns:EOS function
Return type:func
pressure(sws)[source]

Pressure

print_eos_output()[source]
relax_fit(x, y, n, debug=False, title='')[source]

Fits a polynomial to x vs. y data and calculates xmin and ymin from the curve.

Parameters:
  • x –
  • y –
  • n –
Returns:
Return type:
sjeos(V, a, b, c, d)[source]

Stabilized jellium (SJEOS) fitting function from PRB 63, 224115 (2001).

Note: EOS parameters here differ from the PRB definitions
by having V0 being included in them. a(PRB) = a/V0 b(PRB) = b/V0**(2/3) c(PRB) = c/V0**(1/3)
taylor(V, E0, beta, alpha, V0)[source]

Taylor Expansion up to 3rd order about V0

Parameters:
  • V –
  • E0 –
  • beta –
  • alpha –
  • V0 –
Returns:
Return type:
vinet(V, E0, B0, BP, V0)[source]

Vinet equation from PRB 70, 224107

Parameters:
  • V –
  • E0 –
  • B0 –
  • BP –
  • V0 –
Returns:
Return type:
vol2wsrad(V)[source]
Parameters:V –
Returns:
Return type:
wsrad2vol(WSrad)[source]
Parameters:WSrad –
Returns:
Return type:
EOS.EOS.curve_fit(f, x, y, p0)[source]

Fits an arbitraty function to data (x,y).

This part comes from: http://projects.scipy.org/scipy/browser/trunk/scipy/optimize/minpack.py

Parameters:
  • f –
  • x –
  • y –
  • p0 –
Returns:
Return type: