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1 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

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1

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 โ€ข NVT ensemble, partition function (single particle), ๐›ฝ = 1

๐‘˜๐ต๐‘‡

๐‘ = โˆซ ๐‘‘๐‘‘ ๐‘‘๐‘‘ ๐‘’โˆ’๐›ฝ ๐‘2

2๐‘š+๐‘‰ ๐‘ฅ

โ€ข Free energy

๐น = โˆ’1๐›ฝ

log๐‘ = โˆ’1๐›ฝ

log2๐‘š๐‘š๐›ฝ

12โˆ’

1๐›ฝ

log โˆซ ๐‘‘๐‘‘ ๐‘’โˆ’๐›ฝ๐‘‰(๐‘ฅ)

โ€ข Classical statistical mechanics predicts NO isotope effect

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Experimental evidence of nuclear quantum effects

Courtesy of Joseph Morrone

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Classical statistical mechanics โ€ข Momentum distribution (Maxwell-Boltzmann form)

๐‘› ๐’‘โ€ฒ = ๐›ฟ ๐’‘ โˆ’ ๐’‘โ€ฒ = 1๐‘โˆซ ๐‘‘๐’‘ ๐‘‘๐’™๐›ฟ ๐’‘ โˆ’ ๐’‘โ€ฒ ๐‘’

โˆ’๐›ฝ ๐’‘๐Ÿ

2๐‘š+๐‘‰ ๐’™

=๐›ฝ

2๐‘š๐‘š

32๐‘’โˆ’

๐›ฝ๐’‘22๐‘š

โ€ข Radial momentum distribution

๐‘› ๐‘‘ =๐‘‘2

4๐‘šโˆซ ๐‘‘ฮฉ ๐‘› ๐’‘ =

๐›ฝ2๐‘š๐‘š

32๐‘‘2๐‘’โˆ’

๐›ฝ๐‘22๐‘š

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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 โ€ข Kinetic energy (equi-partition theorem)

๐พ = โˆซ ๐‘‘๐’‘ ๐’‘๐Ÿ

2๐‘š ๐‘› ๐’‘ = 3๐‘˜๐ต๐‘‡

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โ€ข 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 โ€ข Single body Hamiltonian for simplicity, extendable to the many

body case

๐ป = ๐‘‡ ๐‘‘ + ๐‘‰ ๐‘Ÿ = โˆ’โ„2

2๐‘š๐›ป2 + ๐‘‰ ๐‘Ÿ

โ€ข ๐‘‰(๐‘Ÿ): Only for nuclei; obtained from force-field model, or density functional theory [Hohenberg-Kohn, 1964; Kohn-Sham 1965]

โ€ข Key quantity: single particle density matrix

๐œŒ ๐‘Ÿ, ๐‘Ÿโ€ฒ =๐‘Ÿ ๐‘’โˆ’๐›ฝ๐›ฝ ๐‘Ÿโ€ฒ

๐‘

โ€ข ๐‘‡(๐‘‘) and ๐‘‰(๐‘‘) do not commute.

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Trotter expansion (Strang splitting)

๐‘’โˆ’๐›ฝ๐›ฝ = lim๐‘ƒโ†’โˆž

ฮ ๐‘–๐‘ƒ๐‘’โˆ’๐›ฝ๐‘ƒ๐›ฝ = lim

๐‘ƒโ†’โˆžฮ ๐‘–๐‘ƒ๐‘’

โˆ’ ๐›ฝ2๐‘ƒ๐‘‰ ๐‘Ÿ ๐‘’โˆ’

๐›ฝ๐‘ƒ๐‘‡ ๐‘ ๐‘’โˆ’

๐›ฝ2๐‘ƒ๐‘‰ ๐‘Ÿ

Insert P-1 unity operators ๐ผ = โˆซ ๐‘‘๐‘Ÿ๐‘–|๐‘Ÿ๐‘–โŸฉโŸจ๐‘Ÿ๐‘–| ๐œŒ ๐‘Ÿ, ๐‘Ÿโ€ฒ

=1๐‘

lim๐‘ƒโ†’โˆž

โˆซ ๐‘‘๐‘Ÿ2 โ‹ฏ๐‘‘๐‘Ÿ๐‘ƒ๏ฟฝ๐‘’โˆ’๐›ฝ2๐‘ƒ๐‘‰ ๐‘Ÿ๐‘–

๐‘ƒ

๐‘–=1

โŸจ๐‘Ÿ๐‘–|๐‘’โˆ’๐›ฝ๐‘ƒ๐‘‡ ๐‘ ๐‘Ÿ๐‘–+1 ๐‘’โˆ’

๐›ฝ2๐‘ƒ๐‘‰ ๐‘Ÿ๐‘–+1

๐‘Ÿ1 = ๐‘Ÿ, ๐‘Ÿ๐‘ƒ+1 = ๐‘Ÿโ€ฒ

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Trotter expansion (Strang splitting)

๐‘Ÿ๐‘–| ๐‘’โˆ’๐›ฝ๐‘ƒ๐‘‡(๐‘) |๐‘Ÿ๐‘–+1 โˆ ๐‘’

โˆ’ ๐‘š๐‘ƒ2๐›ฝโ„2 ๐‘Ÿ๐‘–โˆ’๐‘Ÿ๐‘–+1

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๐œŒ ๐‘Ÿ, ๐‘Ÿโ€ฒ โˆ lim๐‘ƒโ†’โˆž

โˆซ ๐‘‘๐‘Ÿ2 โ‹ฏ๐‘‘๐‘Ÿ๐‘ƒ๐‘’โˆ’๐›ฝ๐‘ˆ๐‘’๐‘’๐‘’

๐‘ˆ๐‘’๐‘’๐‘’ = ๏ฟฝ๐‘š๐‘š2๐›ฝโ„2

๐‘Ÿ๐‘– โˆ’ ๐‘Ÿ๐‘–+1 2 +12๐‘š

๐‘‰ ๐‘Ÿ1 + ๐‘‰ ๐‘Ÿ๐‘ƒ+1

๐‘ƒ

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 {๐‘š๐‘–}

lim๐‘ƒโ†’โˆž

โˆซ ๐‘‘๐‘Ÿ1๐‘‘๐‘Ÿ2 โ‹ฏ๐‘‘๐‘Ÿ๐‘ƒ๐‘‘๐‘Ÿ๐‘ƒ+1๐‘’โˆ’๐›ฝ๐‘ˆ๐‘’๐‘’๐‘’

โˆ lim๐‘ƒโ†’โˆž

โˆซ ๐‘‘๐‘‘1๐‘‘๐‘‘2 โ‹ฏ๐‘‘๐‘‘๐‘ƒ๐‘‘๐‘‘๐‘ƒ+1๐‘‘๐‘Ÿ1๐‘‘๐‘Ÿ2 โ‹ฏ๐‘‘๐‘Ÿ๐‘ƒ๐‘‘๐‘Ÿ๐‘ƒ+1๐‘’โˆ’๐›ฝ(โˆ‘

๐‘๐‘–2

2๐‘š๐‘–+๐‘– ๐‘ˆ๐‘’๐‘’๐‘’)

Newtonโ€™s equation (quantum-to-classical isomorphism)

๐‘Ÿ๏ฟฝฬ‡๏ฟฝ =๐‘‘๐‘–๐‘š๐‘–

๐‘‘๏ฟฝฬ‡๏ฟฝ = โˆ’๐œ•๐‘ˆ๐‘’๐‘’๐‘’๐œ•๐‘Ÿ๐‘–

+ ๐‘ก๐‘ก๐‘’๐‘Ÿ๐‘š๐‘ก๐‘ก๐‘ก๐‘ก๐‘ก

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Computing the position distribution and momentum distribution Position distribution

๐‘› ๐‘‘ = โˆซ ๐‘‘๐‘‘๐›ฟ ๐‘‘ โˆ’ ๐‘‘ ๐œŒ(๐‘‘,๐‘‘) End-to-end distribution

๐‘›๏ฟฝ ๐‘‘ = โˆซ ๐‘‘๐‘‘๐‘‘๐‘‘โ€ฒ๐›ฟ ๐‘‘ โˆ’ ๐‘‘โ€ฒ โˆ’ ๐‘‘ ๐œŒ(๐‘‘,๐‘‘โ€ฒ) Momentum distribution

๐‘š ๐‘‘ =1

2๐‘šโ„ 3 โˆซ ๐‘‘๐‘‘ ๐‘’๐‘–โ„๐‘๐‘ฅ๐‘›๏ฟฝ(๐‘‘)

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Open and closed path

โ€œOpenโ€ path Sample ๐œŒ ๐‘Ÿ, ๐‘Ÿโ€ฒ Momentum distribution

โ€œClosedโ€ path Sample ๐œŒ(๐‘Ÿ, ๐‘Ÿ) Position distribution

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Many particle case

๐‘Ÿ๐‘™ ๐œ โ€ข Sample one particle at a time: inefficient

procedure and poor statistical accuracy โ€ข Closed loops are important: environment

information

Alternative methods? โ€ข Can we compute the end-to-end distribution only with

closed path integrals? โ‡’ Perturbation theory.

โ€ข Naรฏve perturbation method does not work: infinite variance in the continuous limit

Displaced path formulation โ€ข End-to-end distribution

โ€ข Key idea: Add uniform displacement to the whole path ๐‘‘ ๐œ = 1

2โˆ’ ๐œ

๐›ฝโ„.

[LL-Morrone-Car-Parrinello, Phys. Rev. Lett. 2010]

Interpretation of the displaced path formulation โ€ข ๐‘› ๐‘‘ = ๐‘’โˆ’

๐‘š๐‘ฅ2

2๐›ฝโ„2 ๐‘ ๐‘ฅ๐‘(0)

; ๐‘ˆ ๐‘‘ = โˆ’log ๐‘ ๐‘ฅ๐‘(0)

is precisely the free energy difference corresponding to the order parameter x.

Displaced path + free energy perturbation method

โ€ข SPC/F2 water system [Lobaugh and Voth, JCP, 1996] โ€ข Red: 268 ps open path โ€ข Blue: 12 ps displaced path

Free energy perturbation [Zwanzig, 1954]

Displaced path + thermodynamic integration method

โ€ข 1D Double well potential ๐‘‰ ๐‘ง = 0.023๐‘ง2 + 0.012๐‘’โˆ’16.000๐‘ง2

โ€ข Red: exact result โ€ข Black: displaced path

Thermodynamic integration [Kirkwood, 1935]

Environmental part of the end-to-end distribution โ€ข ๐‘› ๐‘‘ = ๐‘’โˆ’

๐‘š๐‘ฅ2

2๐›ฝโ„2 ๐‘ ๐‘ฅ๐‘(0)

โ‰ก ๐‘’โˆ’๐‘š๐‘ฅ2

2๐›ฝโ„2๐‘›๏ฟฝ๐‘‰(๐‘‘)

โ€ข ๐‘›๏ฟฝ๐‘‰ contains all the environmental information of the quantum system.

โ€ข Superposition of ๐‘›๏ฟฝ๐‘‰ for all protons in hexagonal ice (log scale), reflecting the symmetry of the underlying oxygen sublattice (experimental verification under process)

โ€ข Classical dynamics: ๐‘›๏ฟฝ๐‘‰ โ‰ก 1

โ€ข [LL-Morrone-Car-Parrinello, Phys. Rev. B. 2011]

Conclusion โ€ข Displaced path integral method: efficient and accurate

method for estimating the momentum distribution

โ€ข Directional character: useful for crystal system.

โ€ข Factorized free particle and environmental contribution

โ€ข Improve the applicability of the free energy perturbation method using enhanced sampling technique for displaced path method [LL-Quah-Car-Parrinello, in preparation]

Acknowledgment Luis Alvarez fellowship supported by LBNL and DOE.

Thank you for your attention!