Summary of Ab Initio Free
Energy Calculation Methods
Jizhou Wu
07/23/2020
Historical Methods
1.Lindemann Law
2.Born Criterion
3.Free Energy Approach(FE)
4.Coexistence Approach
5.Z method
6.Two Phase Thermodynamic Method using memory function formalism
(2PT-MF Model)
After 2010s
• upsampled thermodynamic integration using Langevin dynamics (UP-TILD)
method
• two-stage upsampled thermodynamic integration using Langevin dynamics
(TU-TILD)
• two-optimized references thermodynamic integration using Langevin
dynamics (TOR-TILD) method (This Work!)
Lindemann’s Law
F. A. Lindemann, Phys. Z. 11, 609 (1910)
2
u ≥ 10 % a
• Empirical
• Restraint
Born Creterion
M. Born, J. Chem. Phys. 7, 591 (1939)
shearmodulus : ν = 0
• Empirical
• Restraint
Free Energy Approach
O. Sugino and R. Car, Phys. Rev. Lett. 74, 1823 (1995)
D. Alfè, M. J. Gillan, and G. D. Price, J. Chem. Phys. 116, 6170 (2002)
• high computational cost
ΔF2<−1 =
∫0
• dependency of efficiency on the reference system:
•
eg.Stillinger-Weber potential
•
Lennard-Jones fluid
•
inverse-power potentials
•
EAM
•
MEAM
1
dλ < U2 − U1 >λ
Coexistence Approach
A. A.B. Belonoshko, R. Ahuja, and B. Johansson, Phys. Rev. Lett. 84, 3638 (2000)
B. J. R. Morris, C. Z. Wang, K. M. Ho, and C. T. Chan, Phys. Rev. B 49, 3109 (1994)
C. A. Laio, S. Bernard, G. L. Chiarotti, S. Scandolo, and E. Tosatti, Science 287, 1027(2000)
• NVE(NVT/NPT) Ensemble
• Two-phase interface remains
• very low convergence
• Gamma Point Only
Z method
A. B. Belonoshko, L. Burakovsky, S. P. Chen, B. Johansson,S. Mikhaylushkin, D. L. Preston, S. I. Simak, and D. C. Swift, Phys. Rev. Lett. 100, 135701 (2008)
A. B. Belonoshko, N. V. Skorodumova, A. Rosengren, and B. Johansson, Phys. Rev. B 73, 012201 (2006).
• NVE
• Suddenly Melt
• Size Effect?
• Simulation Length?
2 Phase Thermodynamics - Memory Function
S.-T. Lin, P. K. Maiti, and W. A. Goddard, III, J. Chem. Phys.119, 11792 (2003).
M. P. Desjarlais, Phys. Rev. E 88, 062145 (2013).
• Fourier transform of the velocity autocorrelation function(VACF)
• MD -> DOS
• For liquid with zero frequency: 2 phase model
• Part of Anharmonicity
• limited to light chemical elements
UP-TILD
B. Grabowski, L. Ismer, T. Hickel, and J. Neugebauer, Phys.Rev. B 79, 134106 (2009)
• Speed up MD with low accuracy
• Correction with snapshots
• Improve efficiency by 2 orders of magnitude
ΔF =
1
∫0
dλ < ΔUhigh >λ =
1
∫0
dλ < ΔUlow >λ + < ΔE >λ
TU-TILD
A. I. Duff, T. Davey, D. Korbmacher, A. Glensk, B. Grabowski, J. Neugebauer, and M. W. Finnis, Phys. Rev. B 91, 214311 (2015)
• Divide into 2 stages
• DFT —> classical(MEAM)
• Classical(MEAM) —> Einstein
• Minimize the variance
ΔF =
1
∫0
dλ1[ < E
Low computational expense
pot
−E
Ein
>λ1 ] +
1
∫0
dλ2[ <
DFT
Elow
−E
pot
>λ2 + < ΔE
UP
>λ2
]
TOR-TILD
Phys. Rev. B 96, 224202 – Published 13 December 2017
• Fit a second optimized reference potential for liquid
•
Coexist Approach —> TMref1(Solid Reference), VM,liqref1,VM,solref1
•
Perform TDI from refrence1(Solid) to reference2(Liquid) at T= TMref1,V=VM,liqref1
•
Correct Fliqref2(T= TMref1,V=VM,liqref1) to any random Volume V and Temperature T
Fliqref2(V,T)
•
Perform a TDI from Fliqref2(V,T) to FliqDFT-low(V,T)
• Snapshot Correction
The Schematic representation
Size Effect
Efficiency
Thank you ~~~