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Home Literature Literature - related Structural investigations g(r) D Author: apsi apsi'at'iki.fi Last revision: 13. août 2008

ab initio simulations on water Ref. [Todorova 2006] 32 molecules, ρ = 0.975 g/cm3 CP dynamics, Δt = 5 atu, μ = 600 au Nosé-Hoover chain (length = 4): ions 350 K, 2400 cm-1; electrons 0.015 au, 10000 cm-1 70 Ry TM: O, lmax = p, rc = 1.12 Bohr, H, lmax = s, rc = 0.5 Bohr Equilibration 10, production 10 ps (hybrids/HF: 5, 5 ps) CPMD mH = 2 Note: Change in BLYP trajectory at ca 10 ps Tables 4&5: Position, height and full width at half-maximum (FWHM) height of the first peak of the partial oxygen-oxygen radial distribution function for different density functionalsa. Self-diffusion constant D functional rmax gOO(rmax) FWHM rmin gOO(rmin) D (Å2/ps) LDA 259(2) 5.07(0.07) 21 312 0.23 0.013 BLYP 279(4) 3.00(0.20) 37 331 0.48 0.048 XLYP 277(2) 3.21(0.11) 34 332 0.40 0.028 PBE 270(5) 2.99(0.08) 37 329 0.47 0.047 rPBE 281(2) 2.29(0.09) 47 334 0.80 0.18 TPSS 271(2) 3.40(0.05) 32 329 0.33 0.032 B3LYP 279(2) 2.48(0.10) 45 340 0.81 0.30 X3LYP 276(6) 2.75(0.10) 40 336 0.62 0.17 PBE0 274(6) 2.58(0.08) 44 335 0.73 0.28 HF 297(6) 2.35(0.13) 78 0.47 TIP4P 278 2.64 44 352 0.90 experiment46 (45oC) 0.2979 a Values in parentheses are maximum deviations over the sampling period, calculated from a sliding window of half width. Values are given in picometers. Table 6: Distribution of the Number of Hydrogen Bonds for Different Density Functionalsb   number of hydrogen bonds functional 1 2 3 4 5 average LDA 0 3 16 75 6 3.84 BLYP 0 6 19 66 7 3.68 XLYP 0 6 19 69 6 3.75 PBE 0 6 24 66 6 3.78 rPBE 1 11 33 51 6 3.56 TPSS 0 4 20 71 6 3.82 B3LYP 0 6 27 61 6 3.67 X3LYP 0 6 34 51 8 3.58 PBE0 0 11 26 57 4 3.48 HF 3 11 40 39 6 3.31 b The values given are percentages of molecules with the given number of hydrogen bonds. Hydrogen bonds are defined by a geometrical criteria. The oxygen-oxygen distance is smaller than 3.5 Å, and the O-H-O angle is larger than 135o. Ref. [Lee 2006b] 32 molecules, L = 18.6226 au CP dynamics, Δt = 0.05 fs, μ = 500 au Nosé-Hoover: each degree of freedom (massive) DVR Production 10+30 ps Tave = 298 K PINY_MD mH = 2 amu TABLES Table I. Simulation parameters used in the recent ab initio MD simulations of liquid water: simulation type (BO=Born-Oppenheimer and CP=Car-Parrinello), ensemble type, fictitious mass (µ), length of the equilibration run (teq), length of the production run (tprod), average temperature during the production run (T), thermostat used in the equilibration run (VS=velocity scaling, NH=Nosé-Hoover, and NHC=Nosé-Hoover chain), exchange-correlation functional (PBE=Perdew-Burke-Ernzerhof (Ref. 60), BLYP=Becke-Lee-Yang-Parr, (Refs. 58,59), and PW91=Perdew-Wang, (Ref. 94)), and the pseudopotential type (PAW=projected augmented wave, (Ref. 95) H=Hamann, (Ref. 96) TM=Troullier-Martins, (Ref. 66) US=Vanderbilt's ultrasoft, (Refs. 97,98), and GTH=Goedecker-Teter-Hutter (Ref. 88)) and the basis set type (PW=plane wave, GPW=hybrid gaussian/plane wave, NAO=numerical atomic orbital, DVR=discrete variable representation, the size of basis set in the parentheses). The number in parentheses in the second column is the number of water molecules used in the simulation. The last column shows the maximum value of oxygen-oxygen radial distribution function obtained from each simulation. Reference Dynamics µ System Ensemble teq (ps) tprod (ps) T (K) Thermo XC PP Basis gOOmax 18 BO(32) ... D2O NVE 10.4 4.4 334 VS PW91 PAW 3.50 -''- BO(32) ... D2O NVE 2.7 3.6 337 VS PBE PAW 3.7 22 CP(32) 340 D2O NVE 2 20 300 VS BLYP H PW(85) 3.6 -''- CP(32) 760 D2O NVE 2 20 300 VS BLYP H PW(85) 3.46 57 CP(54) H2O NVE 3 19.8 296 VS PBE H PW(85) 3.65 -''- BO(64) H2O NVE 3 20.5 306 VS PBE H PW(85) 3.83 24 CP(64) 700 H2O NVE 1.5 4 343 PBE US PW(25) 2.5 26 BO(32) ... H2O NVE 15 20 305 VS BLYP GTH GPW(TZV2P) 3.35 61 BO(64) ... H2O NVE 5 10 315 BLYP TM PW(85) 3.04a -''- CP(64) 400 H2O NVE 12 20 315 BLYP TM PW(85) 2.92a -''- CP(64) 400 H2O NVT 5 10 315 NHC BLYP TM PW(85) 2.96a 62 CP(64) 340 H2O NVE 4 32 300 VS BLYP TM PW(80) 3.24 23 BO(64) ... D2O NVE 4 20 300 VS BLYP TM NAO(DZP) 3.20 43 CP(64) 600 D2O NVT 2 10 300 NHC BLYP TM PW(70) 3.10 25 CP(32) 300 D2O NVE 25 22.1 325 NH PBE TM PW(70) 3.25 30 CP(32) 400 O2Oa NVT 14 60 300 NHC BLYP TM PW(70) 3.20 This work CP(32) 500 D2O NVT 10 30 300 NHC BLYP TM DVR(0.249) 2.90 aThe notation O2O indicates that, in the study referenced, all hydrogens were given the mass of oxygen. Table II. Binding energies of water dimer in kcal/mol computed from the plane-wave basis sets with the kinetic energy cutoffs between 70 and 300  Ry and the DVR basis sets with the grid spacing between h=0.249 and h=0.191  a.u.. For comparison, binding energies from other recent work (Refs. 16,23,57,79) computed with plane-wave basis sets and Gaussian basis sets are included. Reference Basis set Eb (kcal/mol) Reference Basis set Eb (kcal/mol) 16 PW(70  Ry) 4.3 This work PW(70  Ry) 4.27/4.40a 57 aug-cc-pVTZ 4.18 PW(90  Ry) 4.26/4.33 23 DZP(0.5)b 4.76 PW(120  Ry) 4.25/4.33 DZP(0.2)b 4.68 PW(200  Ry) 4.26/4.34 DZP(0.0)b 4.01 PW(300  Ry) 4.25/4.33 79 PW(70  Ry) 4.34 DVR(0.249)c 4.30/4.23 6-311++G 4.17 DVR(0.229) 4.26/4.22 DVR(0.216) 4.25/4.28 DVR(0.201) 4.30/4.30 DVR(0.191) 4.32/4.32 aThe first number corresponds to the binding energy obtained by full geometry optimization while the second corresponds to the binding energy obtained for fixed geometries of the dimer and monomer. bNumerical atomic orbitals with double-ζ (DZP) level. The number in parentheses is the pressure parameter defining the cutoff radii of the support regions of the orbitals. cThe number in parentheses refers to the grid spacing in bohrs. Table III. Percentage of water molecules having the specified number of donor and acceptor hydrogen bonds obtained from the 30  ps production run of the CPAIMD-BLYP/DVR simulation. Three different definitions of the hydrogen bond geometry are taken from (1) Ref. 93 (rO...H ≤ 2.45 Å, β ≤ 30°), (2) Ref. 20 (1.59 Å < rO...H < 2.27 Å, α > 140°), and (3) Ref. 50 (rO...O ≤ -0.000 44 β2 + 3.3 Å, β in degrees). No. of acceptor (1)93 (2)20 (3)50 No. of donor 0 1 2 0 1 2 0 1 2 0 0 0 1 0 2 2 0 1 0 1 1 8 16 2 13 19 1 12 18 2 0 11 63 1 16 45 1 14 53 Ref. [Mantz 2006] N molecules, ρ = 0.975 g/cm3 CP dynamics, Δt = 5 atu, μ = 600 au Nosé-Hoover: ions massively; electrons NORB scheme (independent-state thermostatting) 70 Ry PINY_MD mH = 15.9994 amu Pre-conditioning of wave functions above μ0 = 400 au AIPI: 64 molecules, P = 16 beads, 80 Ry, BLYP Table 1: Major Simulation Studies of Pure Liquid Water, Excluding Those at 323 K for Brevitya force field NH2O L300 (Å) L353 (Å) T (ps) Δt (fs) BLYP 32 9.865 9.948 60 0.125 BLYPb 64 12.419   10 0.121 BLYP-PIc 64 12.419   15 0.121 TIP3P 256 19.730 19.896 15000 6.0 TIP4P 128 15.660 15.792 1200 6.0 TIP4P-flex 128 15.660 15.792 1200 1.0 TIP4P-flex-PI 128 15.660 15.792 200 1.0 TIP4P-pol-2 500 24.630d 24.972d     TIP5P 500 24.783d 25.206d     a Symbols are defined and force fields are referenced in the text. b Reference 29 c Reference 30 d Average value during a MC simulation in the NPT ensemble Table 2: gOO(r) first peak height at rOOmax and the coordination number, nOO(r), at rOOmin the first local minimum of gOO(r)   TIP3P TIP4P -flex -flex-PI -pol-2 TIP5P AIMD AIPI T (K) 300 353 300 353 300 353 300 353 300 353 300 353 300 323 353 300 rOOmax(Å) 2.78 2.78 2.78 2.78 2.74 2.74 2.74 2.78 2.74 2.78 2.74 2.78 2.74 xxx 2.78 2.74 g(rOOmax) 2.8 2.5 3.0 2.6 3.3 2.8 3.1 2.7 3.0 2.6 2.9 2.4 3.2 2.8 2.6 3.4 rOOmin(Å) 3.34 3.34 3.22 3.30 3.22 3.26 3.22 3.26 3.22 3.30 3.26 3.34 3.22 xxx 3.30 3.22 g(rOOmin) xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx 0.5 0.63 0.73 xxx n(rOOmin) 4.7 4.6 4.1 4.3 4.1 4.1 4.1 4.1 4.0 4.2 4.1 4.2 4.1 xxx 4.2 4.0 Ref. [Lee 2006a] 32 molecules, L = 18.6226 au CP dynamics, Δt = 0.05 fs, μ = 500 au Nosé-Hoover: each degree of freedom DVR Production 10 ps Tave = 298 K PINY_MD mH = ? amu Table xxx: gOO(r) first peak height at rOOmax and the coordination number, nOO(r), at rOOmin the first local minimum of gOO(r)   AIMD T (K) 298 rOOmax(Å) xxx g(rOOmax) 2.95 rOOmin(Å) xxx g(rOOmin) 0.65 Ref. [Sit 2005] 32 molecules, n = 1.0957~g/cm$^3$, bcc cell CP dynamics, Δt = 10 atu, μ = 700 au Single Nosé-Hoover on ions Simulation time 25 ps Tave = 325 K mH = 2 amu Table II. Vibrational frequencies of the water monomer: 1, 2, and 3 are the symmetric stretching, bending, and asymmetric modes, respectively. PBE Expt. (Ref. 34) BLYP (Ref. 33) US NC 1 (cm–1) 3781 3704 3657 3567 2 (cm–1) 1573 1599 1595 1585 3 (cm–1) 3908 3816 3756 3663 Table III. Vibrational frequencies of the water dimer: 1, 2, and 3 are the symmetric stretching, bending, and asymmetric stretching modes, respectively. Proton acceptor and donor molecules are denoted as (A) and (D). (Hb) are the two libration modes between molecules and (O–O) is the hydrogen-bond stretching mode. PBE Expt. (Refs. 38,39) BLYP (Ref. 33) US NC 1(A) (cm–1) 3778 3695 3622 3577 2(A) (cm–1) 1570 1596 1600 1593 3(A) (cm–1) 3901 3804 3714 3675 1(D) (cm–1) 3601 3532 3548 3446 2(D) (cm–1) 1593 1616 1618 1616 3(D) (cm–1) 3871 3781 3698 3647 (Hb) (cm–1) 666 644 520 600 (Hb) (cm–1) 379 378 320 333 (O–O) (cm–1) 202 196 243 214 Table IV. Structural and dynamical parameters before and after the 10  ps (see Fig. 1) compared with the experimental results at 298  K. g(r)max D (cm2/s) Before 2.82 1.5×10–5 After 3.21 0.14×10–5 Expt. (Refs. 40,3) 2.75 2.0×10–5 Table V. Simulation details for our production runs. Pseudopotentials T (K) Density (g/cm3) µ (a.u.) t (a.u.) Production time (ps) US 325 1.0957 450 7 37.6 NC 325 1.0957 300 5 22.1 US 350 1.0815 450 7 22.9 US 350 1.0815 450 7 22.9 US 400 1.0554 450 7 32.5 NC 400 1.0554 300 5 20.2 Table VI. Summary of the structural and dynamical properties of water. Dself is the self-diffusion coefficient, gmax is the height of the first peak of the O–O radial distribution function, and R[gmax] is the location of the first peak. The last column shows the number of hydrogen bonds per molecule. Dself 10–5  cm2/s gmax R[gmax] No. of H bonds per molecule 325  K (US) 0.07 3.38 2.67 3.86 325  K (NC) 0.16 3.25 2.73 3.79 350  K (US) 0.25 3.25 2.73 3.77 375  K (US) 0.26 3.10 2.71 3.72 400  K (US) 2.03 2.50 2.73 3.45 400  K (NC) 1.66 2.55 2.75 3.37 Expt. (Refs. 40,46) at 300  K 1.80 2.75 2.80 3.58 Ref. [Kuo 2004] N molecules CP+BO dynamics, Δt = xxx fs, μ = xxx au Nosé-Hoover: Production 10 ps Tave = xxx K CPMD, CP2K mH = ? amu Table 3: Comparison of the Simulation Resultsa   Tion (K) g(rOOmax) r(rOOmax) (Å) coordination dOH (Å) αHOH(deg) CVclass [J/(mol K)] Dself (Å2/ps) CPMD-NVE-BO 323 3.0 2.76 4.0 0.99 106   0.04 CP2K-MD-NVE-1 330 3.0 2.75 4.1 1.00 105   0.03 CP2K-MD-NVE-2 343 3.2 2.75 4.0 1.00 106   0.02 CP2K-MD-NVE-3 334 3.1 2.75 4.0 1.00 106   0.03 CPMD-NVE-400 314 2.9 2.76 4.0 0.99 106   0.06 CPMD-NVE-400-128 319 2.8 2.76 4.0 0.99 106   0.06 CPMD-NVT-i-400 312 3.0 2.75 3.9 0.99 106 59   CPMD-NVT-ie-800 310 2.8 2.75 4.0 0.99 106 65   CP2K-MC-NVT-1   2.9 2.75 3.9 1.00 104 73   CP2K-MC-NVT-2   3.0 2.77 4.1 1.00 106 89   a The ensemble averages of the ionic kinetic temperature, the height and position of the first maximum in the oxygen-oxygen radial distribution function, the oxygen-oxygen coordination number, the OH bond length, the HOH bond angle, the classical constant-volume heat capacity, and the self-diffusion constant are listed. Ref. [Grossman 2004a] N molecules, L = 9.86 Å for 32-water runs and L = 11.74 Å for the 54-water run CP dynamics, Δt = 0.075 fs, μ = xxx au 85 Ry, Hamann pseudo potentials NVE Production xxx ps Tave = xxx K jeep? mH = ? amu Table I. For each simulation performed in this work, the number of molecules N (32 or 54), molecule type (H2O or D2O), density functional employed (PBE or BLYP), fictitious mass parameter µ (a.u.), diffusion coefficient D (cm2/s), position (Å), and value of first maximum and minimum in the gOO(r), average temperature (K), and average coordination of the water molecules are listed. For each calculation, quantities are averaged over the entire production run (20 ps), with the exception of the 54 H2O simulation which was run for 12 ps. The last two rows contain measured diffusion coefficientsa and structural datab for water at ambient conditions. N Molecule DFT µ D R[g(r)max] g(r)max R[g(r)min] g(r)min Tavg Coordination 32 H2O PBE 340 1.2×10–6 2.71 3.46 3.30 0.41 290.8 4.3 32 H2O PBE 760 3.1×10–5 2.75 2.62 3.45 0.71 285.9 4.9 32 H2O BLYP 340 1.1×10–6 2.73 3.65 3.32 0.40 292.9 4.4 32 H2O BLYP 760 2.3×10–5 2.76 2.59 3.52 0.75 281.3 4.9 32 D2O PBE 340 1.6×10–6 2.72 4.11 3.30 0.35 294.7 4.2 32 D2O PBE 760 1.5×10–6 2.75 3.46 3.35 0.48 293.0 4.4 32 D2O BLYP 340 5.5×10–7 2.73 3.60 3.33 0.39 297.5 4.3 32 D2O BLYP 760 1.5×10–6 2.74 3.46 3.34 0.42 291.4 4.3 32 D2O BLYP 1100 1.0×10–5 2.77 2.84 3.46 0.66 283.9 4.8 54 H2O PBE 340 3.7×10–6 2.71 3.27 3.32 0.46 294.5 4.3 Exp. H2O — — 2.2×10–5 2.73 2.75 3.36 0.78 298.0 4.7c Exp. D2O — — 1.8×10–5 — — — — 298.0 — a See Ref. 57. b See Ref. 2. c From Ref. 3. ab initio simulations on liquid water [Lee 2006b]Structure of liquid water at ambient temperature from ab initio molecular dynamics performed in the complete basis set limit Hee-Seung Lee and Mark Tuckerman Journal of Chemical Physics 125, 154507 (2006) [Todorova 2006] Molecular Dynamics Simulation of Liquid Water: Hybrid Density Functionals Teodora Todorova, Ari P Seitsonen, Jürg Hutter, I-Feng W Kuo and Christopher J Mundy Journal of Physical Chemistry B 110, 3685-3691 (2006) [Mantz 2006] Structural Correlations and Motifs in Liquid Water at Selected Temperatures: Ab Initio and Empirical Model Predictions Y. A. Mantz, B. Chen and G. J. Martyna Journal of Physical Chemistry B 110, 3540 (2006) [Lee 2006a] Ab Initio Molecular Dynamics with Discrete Variable Representation Basis Sets: Techniques and Application to Liquid Water H-S Lee and M. E. Tuckerman Journal of Physical Chemistry A 110, 5549 (2006) [Sit 2005] Static and dynamical properties of heavy water at ambient conditions from first-principles molecular dynamics P H-L Sit and Nicola Marzari Journal of Chemical Physics 122, 204510 (2005) [Kuo 2004] Liquid Water from First Principles: Investigation of Different Sampling Approaches I-F. W. Kuo, C. J. Mundy, M. J. McGrath, J. I. Siepmann, J. VandeVondele, M. Sprik, J. Hutter, B. Chen, M. L. Klein, F. Mohamed, M. Krack and M. Parrinello Journal of Physical Chemistry B 108, 12990 (2004) [Grossman 2004a] Towards an assessment of the accuracy of density functional theory for first principles simulations of water J. C. Grossman, E. Schwegler, E. W. Draeger, D. Gygi and G. Galli Journal of Chemical Physics 120, 300 (2004)