Ab initio statistical mechanics
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1 Ab initio statistical mechanics Luca M. Ghiringhelli Fritz-Haber-Institut der MPG, Berlin Hands-on Workshop Density-Functional Theory and Beyond: Accuracy, Efficiency and Reproducibility in Computational Materials Science Humboldt University, Berlin, Germany, July 31 to August 11, 2017
2 Extending the scale
3 Extending the scale
4 Chemical energy conversion: catalysis
5 Entropy?
6 Free energy: one quantity, many definitions
7 Free energy: one quantity, many definitions
8 Free energy: one quantity, many definitions
9 Statistical mechanics: free energy as a probabilistic concept
10 Statistical mechanics, quantities derived from Z
11 Statistical mechanics, quantities derived from Z Evaluation of pressure
12 Ensemble averages on discrete machines
13 The problem of free energy sampling
14 Computational free-energy evaluation: the zoo - Analytic: ab initio atomistic thermodynamics - Canonical sampling: thermodynamic integration - Canonical sampling: thermodynamic perturbation - Generalized sampling: biased sampling / biased dynamics - Unbiased (canonical) sampling re-weighting techniques - Evaluation: Parallel or >>> Serial <<<
15 Computational free-energy evaluation: the zoo - Analytic: ab initio atomistic thermodynamics - Canonical sampling: thermodynamic integration Lecture Lectureof of Sergey SergeyLevchenko, Levchenko,th Tuesday, Tuesday,August August88th,, at at10:00 10:00 - Canonical sampling: thermodynamic perturbation - Generalized sampling: biased sampling / biased dynamics - Unbiased (canonical) sampling re-weighting techniques - Evaluation: Parallel or >>> Serial <<<
16 Computational free-energy evaluation: the zoo - Analytic: ab initio atomistic thermodynamics - Canonical sampling: thermodynamic integration - Canonical sampling: thermodynamic perturbation - Generalized sampling: biased sampling / biased dynamics - Unbiased (canonical) sampling re-weighting techniques - Evaluation: Parallel or >>> Serial <<<
17 Free energy: physical path thermodynamic integration
18 Free energy: unphysical path thermodynamic integration
19 Case study: phase diagram of pure carbon
20 Case study: phase diagram of pure carbon
21 Case study: phase diagram of pure carbon
22 Case study: phase diagram of pure carbon
23 Case study: λ ensemble sampling and integration
24 Case study: integration of P(ρ) equations of state
25 Case study: equating Gibbs free energies
26 Carbon phase diagram
27 Alternative method for finding phase coexistence via F(V)
28 Notable cases (at 0 K): Silicon (1980)
29 Notable cases (at 0 K): Cerium (2013) Casadei et al. PRL (2013)
30 Computational free-energy evaluation: the zoo - Analytic: ab initio atomistic thermodynamics - Canonical sampling: thermodynamic integration - Canonical sampling: thermodynamic perturbation - Generalized sampling: biased sampling / biased dynamics - Unbiased (canonical) sampling re-weighting techniques - Evaluation: Parallel or >>> Serial <<<
31 Thermodynamic perturbation
32 Thermodynamic perturbation
33 Thermodynamic perturbation
34 Thermodynamic perturbation
35 Thermodynamic perturbation
36 Thermodynamic perturbation: recycling data
37 Thermodynamic perturbation: recycling data
38 Computational free-energy evaluation: the zoo - Analytic: ab initio atomistic thermodynamics - Canonical sampling: thermodynamic integration - Canonical sampling: thermodynamic perturbation - Generalized sampling: biased sampling / biased dynamics - Unbiased (canonical) sampling re-weighting techniques - Evaluation: Parallel or >>> Serial <<<
39 Umbrella sampling
40 Umbrella sampling (multiplying by 1 few more times...)
41 Umbrella sampling (multiplying by 1 few more times...) Parallel (over biasing potentials)
42 Umbrella sampling
43 Computational free-energy evaluation: the zoo - Analytic: ab initio atomistic thermodynamics - Canonical sampling: thermodynamic integration - Canonical sampling: thermodynamic perturbation - Generalized sampling: biased sampling / biased dynamics - Unbiased (canonical) sampling re-weighting techniques - Evaluation: Parallel or >>> Serial <<<
44 Parallel tempering: the concept
45 Parallel tempering: the implementation Parallel (over temperatures, and more) To be tuned for efficient sampling: number of temperatures, list of temperatures, attempted swap frequency
46 Parallel tempering: free energy?
47 Au4: coexistence of several isomers
48 Au4: coexistence of several isomers
49 Relative population of larger clusters: Au n, 2D vs 3D
50 The Nested Sampling Obtaining the partition function Consider cumulative density E
51 The Nested Sampling: the main trick Instead of χ(e), compute E(χ) χ(e) E(χ) χ E At E =, we have an ideal gas, χ0 = VN Constrained uniform sampling E(χ) E1 E χ1 χ0 J. Skilling (2006) G. Csányi (2011)
52 The Nested Sampling: the main trick E(χ) E1 χ1 χ0 1. obtain K uniform samples such that E(q) < Elimit 2. compute median: E(χ1) = E1, χ1 χ0/2, Elimit E1 3. repeat... E(χ) E 2 E 1 χ2 χ1 χ0 (linked with LAMMPS), Csányi et al.
53 The Nested Sampling: application to (EAM) Aluminum Temperature [K] Parallel over walkers Pressure [GPa] R. Baldock,, G. Csányi, PRB 93, (2016)
54 Computational free-energy evaluation: the zoo - Analytic: ab initio atomistic thermodynamics - Canonical sampling: thermodynamic integration - Canonical sampling: thermodynamic perturbation - Generalized sampling: biased sampling / biased dynamics - Unbiased (canonical) sampling (replica exchange, nested sampling) - Evaluation: Parallel or >>> Serial <<<
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