displaced path integral method for computing the โฆlinlin/presentations/displacedpath...ย ยท...
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Lin Lin
Joint work with Roberto Car (Princeton), Joseph Morrone (Columbia) and Michele Parrinello (ETHZ)
Computational Research Division, Lawrence Berkeley National Lab
Displaced Path Integral Method for Computing the Momentum Distribution of Quantum Nuclei
Most molecular dynamics simulation treats nuclei as classical particles: Is this always a good approximation?
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Classical statistical mechanics โข NVT ensemble, partition function (single particle), ๐ฝ = 1
๐๐ต๐
๐ = โซ ๐๐ ๐๐ ๐โ๐ฝ ๐2
2๐+๐ ๐ฅ
โข Free energy
๐น = โ1๐ฝ
log๐ = โ1๐ฝ
log2๐๐๐ฝ
12โ
1๐ฝ
log โซ ๐๐ ๐โ๐ฝ๐(๐ฅ)
โข Classical statistical mechanics predicts NO isotope effect
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Classical statistical mechanics โข Momentum distribution (Maxwell-Boltzmann form)
๐ ๐โฒ = ๐ฟ ๐ โ ๐โฒ = 1๐โซ ๐๐ ๐๐๐ฟ ๐ โ ๐โฒ ๐
โ๐ฝ ๐๐
2๐+๐ ๐
=๐ฝ
2๐๐
32๐โ
๐ฝ๐22๐
โข Radial momentum distribution
๐ ๐ =๐2
4๐โซ ๐ฮฉ ๐ ๐ =
๐ฝ2๐๐
32๐2๐โ
๐ฝ๐22๐
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Experimental evidence of nuclear quantum effects
Momentum distribution of protons in water at room temperature. Data from [Reiter et al, Braz. J. Phys. 2004]
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Classical statistical mechanics โข Kinetic energy (equi-partition theorem)
๐พ = โซ ๐๐ ๐๐
2๐ ๐ ๐ = 3๐๐ต๐
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โข Prediction from classical statistical mechanics at
273K: K=35.3 meV 270K: K=34.9 meV
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Experimental evidence of nuclear quantum effects
Temperature dependence of kinetic energy of protons in water Prediction from classical statistical mechnics at 273K: 35meV Data from [Pietropolo et al, Phys. Rev. Lett., 2008]
Melting point (273K)
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Hydrogen atoms (and other light atoms) should be treated as quantum particles. How to compute the momentum distribution of quantum nuclei?
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Quantum statistical mechanics โข Single body Hamiltonian for simplicity, extendable to the many
body case
๐ป = ๐ ๐ + ๐ ๐ = โโ2
2๐๐ป2 + ๐ ๐
โข ๐(๐): Only for nuclei; obtained from force-field model, or density functional theory [Hohenberg-Kohn, 1964; Kohn-Sham 1965]
โข Key quantity: single particle density matrix
๐ ๐, ๐โฒ =๐ ๐โ๐ฝ๐ฝ ๐โฒ
๐
โข ๐(๐) and ๐(๐) do not commute.
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Trotter expansion (Strang splitting)
๐โ๐ฝ๐ฝ = lim๐โโ
ฮ ๐๐๐โ๐ฝ๐๐ฝ = lim
๐โโฮ ๐๐๐
โ ๐ฝ2๐๐ ๐ ๐โ
๐ฝ๐๐ ๐ ๐โ
๐ฝ2๐๐ ๐
Insert P-1 unity operators ๐ผ = โซ ๐๐๐|๐๐โฉโจ๐๐| ๐ ๐, ๐โฒ
=1๐
lim๐โโ
โซ ๐๐2 โฏ๐๐๐๏ฟฝ๐โ๐ฝ2๐๐ ๐๐
๐
๐=1
โจ๐๐|๐โ๐ฝ๐๐ ๐ ๐๐+1 ๐โ
๐ฝ2๐๐ ๐๐+1
๐1 = ๐, ๐๐+1 = ๐โฒ
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Trotter expansion (Strang splitting)
๐๐| ๐โ๐ฝ๐๐(๐) |๐๐+1 โ ๐
โ ๐๐2๐ฝโ2 ๐๐โ๐๐+1
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๐ ๐, ๐โฒ โ lim๐โโ
โซ ๐๐2 โฏ๐๐๐๐โ๐ฝ๐๐๐๐
๐๐๐๐ = ๏ฟฝ๐๐2๐ฝโ2
๐๐ โ ๐๐+1 2 +12๐
๐ ๐1 + ๐ ๐๐+1
๐
๐=1
Quantum-Classical isomorphism [Chandler and Wolynes, J Chem Phys,1981] Continuous form: Feynman path along imaginary time
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Molecular dynamics Introduce a set of (fictitious) masses {๐๐}
lim๐โโ
โซ ๐๐1๐๐2 โฏ๐๐๐๐๐๐+1๐โ๐ฝ๐๐๐๐
โ lim๐โโ
โซ ๐๐1๐๐2 โฏ๐๐๐๐๐๐+1๐๐1๐๐2 โฏ๐๐๐๐๐๐+1๐โ๐ฝ(โ
๐๐2
2๐๐+๐ ๐๐๐๐)
Newtonโs equation (quantum-to-classical isomorphism)
๐๏ฟฝฬ๏ฟฝ =๐๐๐๐
๐๏ฟฝฬ๏ฟฝ = โ๐๐๐๐๐๐๐๐
+ ๐ก๐ก๐๐๐๐ก๐ก๐ก๐ก๐ก
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Computing the position distribution and momentum distribution Position distribution
๐ ๐ = โซ ๐๐๐ฟ ๐ โ ๐ ๐(๐,๐) End-to-end distribution
๐๏ฟฝ ๐ = โซ ๐๐๐๐โฒ๐ฟ ๐ โ ๐โฒ โ ๐ ๐(๐,๐โฒ) Momentum distribution
๐ ๐ =1
2๐โ 3 โซ ๐๐ ๐๐โ๐๐ฅ๐๏ฟฝ(๐)
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Open and closed path
โOpenโ path Sample ๐ ๐, ๐โฒ Momentum distribution
โClosedโ path Sample ๐(๐, ๐) Position distribution
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Many particle case
๐๐ ๐ โข Sample one particle at a time: inefficient
procedure and poor statistical accuracy โข Closed loops are important: environment
information
Alternative methods? โข Can we compute the end-to-end distribution only with
closed path integrals? โ Perturbation theory.
โข Naรฏve perturbation method does not work: infinite variance in the continuous limit
Displaced path formulation โข End-to-end distribution
โข Key idea: Add uniform displacement to the whole path ๐ ๐ = 1
2โ ๐
๐ฝโ.
[LL-Morrone-Car-Parrinello, Phys. Rev. Lett. 2010]
Interpretation of the displaced path formulation โข ๐ ๐ = ๐โ
๐๐ฅ2
2๐ฝโ2 ๐ ๐ฅ๐(0)
; ๐ ๐ = โlog ๐ ๐ฅ๐(0)
is precisely the free energy difference corresponding to the order parameter x.
Displaced path + free energy perturbation method
โข SPC/F2 water system [Lobaugh and Voth, JCP, 1996] โข Red: 268 ps open path โข Blue: 12 ps displaced path
Free energy perturbation [Zwanzig, 1954]
Displaced path + thermodynamic integration method
โข 1D Double well potential ๐ ๐ง = 0.023๐ง2 + 0.012๐โ16.000๐ง2
โข Red: exact result โข Black: displaced path
Thermodynamic integration [Kirkwood, 1935]
Environmental part of the end-to-end distribution โข ๐ ๐ = ๐โ
๐๐ฅ2
2๐ฝโ2 ๐ ๐ฅ๐(0)
โก ๐โ๐๐ฅ2
2๐ฝโ2๐๏ฟฝ๐(๐)
โข ๐๏ฟฝ๐ contains all the environmental information of the quantum system.
โข Superposition of ๐๏ฟฝ๐ for all protons in hexagonal ice (log scale), reflecting the symmetry of the underlying oxygen sublattice (experimental verification under process)
โข Classical dynamics: ๐๏ฟฝ๐ โก 1
โข [LL-Morrone-Car-Parrinello, Phys. Rev. B. 2011]
Conclusion โข Displaced path integral method: efficient and accurate
method for estimating the momentum distribution
โข Directional character: useful for crystal system.
โข Factorized free particle and environmental contribution
โข Improve the applicability of the free energy perturbation method using enhanced sampling technique for displaced path method [LL-Quah-Car-Parrinello, in preparation]