low-temperature isomers of the water hexamer · model potentials q-tip4p/f s. habershon, t.e....

Low-temperature isomers of the

water hexamer

Volodymyr Babin and Francesco Paesani

Department of Chemistry & Biochemistry,University of California, San Diego,La Jolla, CA 92093-0314

Hundreds of yearsThousands of pages

http://www.lsbu.ac.uk/water/anmlies.htmlXSEDE13 ? San Diego CA ? July 25, 2013 1 / 25

Unanswered Questions

• Does supercooled water has two phases and asecond critical point?

• Does normal liquid water consists of smallerclusters of two different states of water?

XSEDE13 ? San Diego CA ? July 25, 2013 2 / 25

Liquid Water


Hydrogen Bond Network


J.A. MORRONE, V. SRINIVASAN, D. SEBASTIANI, AND R. CAR, Proton momentumdistribution in water: an open path integral molecular dynamics study, J. Chem. Phys.126, (2007), pp. 234504.


T.E. MARKLAND, AND B.J. BERNE, Unraveling quantum mechanical effects in waterusing isotopic fractionation, PNAS 109, (2012), pp. 79887991.

αl−v =(xD,l/xH,l)

xD,v/xH,v)


Trimer/Tetramer/Pentamer

Nearly planar


Hexamers

12

3 4

5 6 78

910 11 12


Born-Oppenheimer approximation:

1) Electronic ground state for clamped nuclei.[CCSD(T) is very demanding: memory ∝ O(N4

atom),

time ∝ O(N8atom)]

2) QM treatment of the nuclei.[not necessarily cheap either]


1 2 3 4 5 6 7 8 9 10 11 12 130

1

2

3

4

5

HBB2-pol

WHBB-6WHBBTTM3-Fq-TIP4P/F

MP2/aug-cc-pVTZ

CCSD(T)-F12(b)/VDZ-F12

Isomer index

E–Eprism

(kcal/mol)

Ecage – Eprism = 0.25 kcal/mol

[CCSD(T)/CBS by Tschumper et al]


Model Potentials

• q-TIP4P/FS. HABERSHON, T.E. MARKLAND, AND D.E. MANOLOPOU-LOS, Competing quantum effects in the dynamics of a flexible wa-ter model, J. Chem. Phys. 131, (2009), pp. 024501.

• TTM3-F

G.S. FANOURGAKIS AND S.S. XANTHEAS, Development oftransferable interaction potentials for water. V. Extension of theflexible, polarizable, Thole-type model potential (TTM3-F, v. 3.0)to describe the vibrational spectra of water clusters and liquid wa-ter, J. Chem. Phys. 128, (2008), pp. 074506.

•WHBB

Y. WANG, X. HUANG, B.C. SHEPLER, B.J. BRAAMS, ANDJ.M. BOWMAN, Flexible, ab initio potential, and dipole momentsurfaces for water. I. Tests and applications for clusters up to the22-mer, J. Chem. Phys. 134, (2011), pp. 094509.

• HBB2-polG. R. MEDDERS, V. BABIN, AND F. PAESANI, A Critical Assess-ment of Two-Body and Three-Body Interactions in Water, JCTC9(2), (2013), pp. 1103–1114.


q-TIP4P/F

• Very crude intra-molecular terms.

• Geometry-independent point charges.

• No polarization.

EN = ∑a

Ea+ ∑a<b

Eab


TTM3-F

• Ab initio intra-molecular sector.

• Ab initio geometry-dependent charges.

• Polarizable site.

EN 6= ∑a

Ea+ ∑a<b

Eab


WHBB & HBB2-pol

EN =∑a

V1(xa)

+∑a<b

V2(xa,xb)

+ ∑a<b<c

V3(xa,xb,xc)

+Vind(x1, . . . ,xN)

• Ab initio intra-molecular sector.

• Ab initio 2-/3-body terms.

• Polarizable site(s).


Partition Function

Z = trexp(−βH ) =

= limP→∞

∫dq1 . . .dqP exp(−βEP)

tre−βH = tre−β (K +V ) = limP→∞

tr[e−βK /Pe−βV /P

]P

︸︷︷︸Trotter

〈q1|e−βK /Pe−βV /P|q2〉 ∝ exp[−βmω

2P(q1−q2)

2/2−βV (q2)/P], ω

2P =

Ph̄2

β 2

EP =1P

P

∑a=1

V (qa)+mω2

P

2

P

∑a=1

(qa+1−qa)2


Path Integral Molecular Dynamics

• Linear transformation to “normal modes”.

J. CAO AND B.J. BERNE, A Born–Oppenheimer approximationfor path integrals with an application to electron solvation in polar-izable fluids, J. Chem. Phys. 99, (1993), pp. 2902–2916.

• Fictitious momenta conjugate to these new variables.• Nose-Hoover chains at every degree of freedom.

M.E. TUCKERMAN, B.J. BERNE, G.J. MARTYNA, ANDM.L. KLEIN, Efficient molecular dynamics and hybrid MonteCarlo algorithms for path integrals, J. Chem. Phys. 99, (1993),pp. 2796–2808.


Replica Exchange

• Two chains sampling different (but related) densities π1(x)and π2(x) can be joined into a chain that samples q = (x,y)∼π(q) = π1(x)π2(y).

• The q = (x,y)→ q′ = (y,x) move with Metropolis acceptanceprobability

min{

1,π(q′)π(q )

}= min

{1,

π1(y)π2(x)π1(x)π2(y)

}preserves π(q) = π1(x)π2(y).


Performance on Gordon

10 100 1000 10000100

1000

10000

1e+05

RE-PIMD (32 beads/32 replicas)

PIMD (256 beads, 30K)

# processors

WallTim

e[s]


Results (TTM3-F)

30 60 90 120 1500

0.2

0.4

0.6

0.8

1

T (K)

fraction

TTM3-F (classical)

prismcage

30 60 90 120 1500

0.2

0.4

0.6

0.8

1

T (K)fraction

TTM3-F (PIMD, 32 beads)

prismcage

Nuclear quantum effects favor cage over prism.


Prism Cage


0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

30 60 90 1200

0.2

0.4

0.6

0.8

30 60 90 120 30 60 90 120 30 60 90 120 150

Temperature (K)Temperature (K)Temperature (K)Temperature (K)

Fraction

Fraction

Fraction

q-TIP4P/F TTM3-F WHBB HBB2-pol

Prism (1) Cage (2) Prism-Book (6) Other (7–13)Book (3,4,5)

Classical

Quantum

(D2 O

)Quantum

(H2 O

)


Free Energies

FCage−FPrism =−kBT ln#Cage#Prism

FCage−FPrism ≈ ∆E + kBT ∑n

ln1− e−h̄ω

Cagen /kBT

1− e−h̄ωPrismn /kBT


Free Energies

-0.2

-0.1

0

0.1

0.2

0 30 60 90 120

-0.2

-0.1

0

0.1

0.2

0 30 60 90 120 0 30 60 90 120 150

Temperature (K)Temperature (K)Temperature (K)

FCage–FPrism

(kcal/mol)

TTM3-F WHBB HBB2-pol

D2 O

H2 O


Summary

• There is several nearly iso-energetic non-planar isomersof water hexamer.

• Their thermal equilibrium is due a subtle balance be-tween energetic, entropic, and nuclear quantum effects.

• The relative stability of the prism isomer increases uponsubstitution of H2O with D2O.

CAICE


low-temperature isomers of the water hexamer · model potentials q-tip4p/f s. habershon, t.e....

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