Simple posterior frequency correction for vibrational spectra from moleculardynamicsDenis S. Tikhonov Citation: The Journal of Chemical Physics 144, 174108 (2016); doi: 10.1063/1.4948320 View online: http://dx.doi.org/10.1063/1.4948320 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/144/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Simulations of vibrational spectra from classical trajectories: Calibration with ab initio force fields J. Chem. Phys. 127, 084502 (2007); 10.1063/1.2756837 Dynamical effects on vibrational and electronic spectra of hydroperoxyl radical water clusters J. Chem. Phys. 123, 084310 (2005); 10.1063/1.2006674 Hydrogen bonding in supercritical tert-butanol assessed by vibrational spectroscopies and molecular-dynamics simulations J. Chem. Phys. 122, 174512 (2005); 10.1063/1.1886730 Vibrational spectra from atomic fluctuations in dynamics simulations. II. Solvent-induced frequencyfluctuations at femtosecond time resolution J. Chem. Phys. 121, 12247 (2004); 10.1063/1.1822915 Non-Gaussian statistics of amide I mode frequency fluctuation of N-methylacetamide in methanol solution:Linear and nonlinear vibrational spectra J. Chem. Phys. 120, 1477 (2004); 10.1063/1.1633549

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 129.70.91.129 On: Tue, 03 May

2016 14:50:46

THE JOURNAL OF CHEMICAL PHYSICS 144, 174108 (2016)

Simple posterior frequency correction for vibrational spectrafrom molecular dynamics

Denis S. Tikhonov1,2,a)1Universität Bielefeld, Lehrstuhl für Anorganische Chemie und Strukturchemie, Universitätsstrasse 25,33615 Bielefeld, Germany2M. V. Lomonosov Moscow State University, Department of Physical Chemistry, GSP-1,1-3 Leninskiye Gory, 119991 Moscow, Russia

(Received 18 March 2016; accepted 18 April 2016; published online 3 May 2016)

Vibrational spectra computed from molecular dynamics simulations with large integration time stepssuffer from nonphysical frequency shifts of signals [M. Praprotnik and D. Janežic, J. Chem. Phys.122, 174103 (2005)]. A simple posterior correction technique was developed for compensation of thisbehavior. It performs through replacement of abscissa in the calculated spectra using following for-

mula: νcorrected =

√2·(1−cos(2π ·∆t ·νinitial))

2π ·∆t , where ν are initial and corrected frequencies and ∆t is the MDsimulation time step. Applicability of this method was tested on gaseous infrared spectra of hydrogenfluoride and formic acid. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4948320]

I. INTRODUCTION

Molecular dynamic (MD) simulations are widely used forcalculation of different molecular properties.1–10 Particularly,they can give vibrational spectra.1,2,4–10 In case of infraredabsorption spectra, the results of MD simulation are convertedinto the frequency (ν) vs. intensity (I) curve by the followingequation:6–8

I(ν) ∝ν tanh

(hν

2kBT

)n(ν) Re

∞

0e−i2πνt⟨d(t),d(0)⟩dt, (1)

where d is dipole moment of the system, ⟨d(t),d(0)⟩ = f (t)is autocorrelation function of the dipole moment, and n(ν)is refractive index. In case of gaseous spectra n ≈ 1.4 It wasfound earlier that large time steps in the MD simulationslead to an overestimation of vibrational frequencies.9 Thus arecommendation was given to use small integration time stepsin order to sample high frequency vibrations correctly.9,10 Thissufficiently increases computational cost of MD simulationsfor calculation of vibrational spectra. Several solutions forthis problems were suggested.8,11 They focus on modificationof integration methods for equations of motion. For example,larger time steps can be used if high frequency bond stretchvibrations are treated analytically.8 However, these methodsimply that some changes should be made in the softwarecarrying MD simulations.

In the present work, we have tried to develop a simpletechnique for posterior correction of the spectra obtained fromMD trajectories with different time steps. This correction wasdesigned for classical Verlet (Störmer–Verlet) method.12,13

But other numerical integration techniques, such as velocityVerlet14 and leap-frog,15,16 are used more frequently for MDsimulations. It is due to their better evaluation of atoms’velocities: they are calculated directly during the simulationand their values are characterized by smaller errors.15,16 It can

a)Electronic mail: denis.tikhonov@uni-bielefeld.de

be shown that after elimination of velocities, these methods arealgebraically equivalent to the Verlet integration.16 This factwas assumed to be rationale for transferability of the designedcorrection. Numerical tests of the obtained correction weredone for the gaseous infrared spectra of hydrogen fluoride andformic acid.

II. DERIVATION OF THE CORRECTION

Let us consider a simple system: a one dimensionalharmonic oscillator with the mass µ and potential V (x)= µω2x2/2, where ω is angular frequency and x is coordinate.Note, that experimental and calculated data are usually given interms of frequencies ν = ω/2π. The solution for this system inthe classical mechanics is x = A cos(ωt + φ0), where A is theamplitude of the vibration, φ0 is phase.17 Both parameters Aand φ0 are determined by initial conditions. Verlet integrationis given by the following formula:13,15,18

xn+1 = 2xn − xn−1 + an∆t2, (2)

where xn denotes position, an is acceleration on the nthstep, and ∆t is the time increment. Obviously, in our casea = −V ′x/µ = −ω2x. Therefore Eq. (2) takes form

xn+1 = xn(2 − ω2∆t2) − xn−1. (3)

Let us assume that the result of integration can beexpressed as xn = A cos(ω∆tn + φ0), where ω is the effectivefrequency observed in the simulation. Obviously, xn±1 =

A (cos(ω∆tn + φ0) cos(ω∆t) ∓ sin(ω∆tn + φ0) sin(ω∆t)). Sub-stitution of the xn and xn±1 into Eq. (3) leads to the followingexpression: cos(ω∆t) = 1 − ω2∆t2/2. ω stands here for thetrue frequency of the system, while ω represents effectivefrequency that is observed from numerical integration. Notethat this equation is valid only if the ω∆t 6 2, since absolutevalue of cosine function cannot exceed 1. Therefore the

0021-9606/2016/144(17)/174108/4/$30.00 144, 174108-1 Published by AIP Publishing.

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 129.70.91.129 On: Tue, 03 May

2016 14:50:46

174108-2 Denis S. Tikhonov J. Chem. Phys. 144, 174108 (2016)

correction will be given by the following formula:

ω =

2 (1 − cos(ω∆t))

∆t. (4)

This equation defines a nonlinear transformation of theabscissa for the frequency vs. intensity spectra.

Let us use an inverse form of Eq. (4): ω= 1∆t

arccos(1 − ω2∆t2/2) ≈ω∆t→0

ω�1 + ω2∆t2/24 + o(ω2∆t2)�

in order to check properties of this replacement of coordinatesystem. We can see that ω −−−−−−→

ω∆t→0ω. In other words, the

smaller is time step or frequency, the closer is the observedvalue to the real one. And since ω > 0 =⇒ ω > ω, theobserved frequency is always shifted towards higher valueswith respect to the real one. These facts are in a goodcorrelation with the known results.9

III. APPLICABILITY CHECK

A. Computational details

The chosen objects were hydrogen fluoride (HF)and formic acid (HCOOH). A cheap quantum chemicalapproximation PBE0/def2-SVP was used for numericaltests. Geometry optimizations for both molecules as wellas calculation of Hessian and one dimensional potentialenergy scan for HF molecule were performed using Fireflyprogram.19 Ab initio MD simulations, VPT2-QFF (vibrational2-nd order perturbation theory based on the expansionof potential energy surface up to quartic force field)calculations,20 and computation of Hessian in the case ofHCOOH were done using GAMESS US software.21 PBE0functional,22 VPT2-QFF anharmonic model,20 and Nose-Hoover thermostat23,24 were used as implemented in theseprograms. def2-SVP basis set25 was taken from EnvironmentalMolecular Science Laboratory (EMSL) Basis Set ExchangeLibrary.26,27 Integration of MD trajectories was performedusing velocity Verlet method.14 A numerical solution of thetime-independent one-dimensional Schrödinger equation forHF molecule was obtained using finite difference methodas implemented in our in-house software. Conversion ofthe results of MD simulations into absorption spectra wasperformed using in-house AWK scripts. Smoothing of resultsof Fourier transformation given by Eq. (1) was performed withhelp of Gaussian window function w(t) = exp(−t2/τ2). τ was50 fs and 1 ps in cases of HF and HCOOH, respectively.Results of harmonic and VPT2-QFF approximations forHCOOH were converted into the intensity curve usingLorentzian profile function with half-width of 10 cm−1. Thisprocedure was performed using Gabedit software.28

B. Hydrogen fluoride: One dimensional vibration

Hydrogen fluoride (HF) molecule was chosen as a simpleexample. Only the bond stretch vibration was considered forthis molecule and thus classical and quantum equations ofmotion that were used were one dimensional.

MD simulations with constant energy were performed.Three series of trajectories were obtained. They differed byinitial conditions. Starting velocities of atoms were always

set to zero, but the initial distance between the atoms wasvaried: 0.91, 0.89, and 0.87 Å. Therefore these series haddifferent total energies. Time steps in each batch were 0.1,0.5, 1.0, and 2.0 fs. It is important to note that 1.0 and 2.0 fsare inappropriate time steps. They violate the basic rule thatrequires time increment to be at least 10 times smaller thanthe period of the vibration (the period of harmonic vibrationfor this system is 8 fs).18,29 Total length of every simulationwas 1 ps. Absence of equilibration phase in this simple casewas assumed. Translational and rotational degrees of freedomwere eliminated by initial conditions.

Positions of the peaks’ maxims of the obtained spectrabefore and after the application of the correction definedby Eq. (4) can be found in Table I. There also arestarting conditions and average temperature of simulationTav =

mi⟨v2

i ⟩/kBNf , where mi and ⟨v2i ⟩ are the mass and

average velocity of ith nucleus and Nf is the number ofdegrees of freedom (in our case Nf = 1).16,18

It can be seen that in each series of simulations, theuncorrected frequency ν increases with the time step. Thisis in a good agreement with previous results9 and theoreticalmodel. The corrected frequencies νcorr are always closer toeach other than the uncorrected ones. In other words, thediscrepancies in the νcorr are insignificant compared to theerror arising from large time increment. It is interesting tonote that Tav drops down with increase of ∆t in each batch.This effect is caused by failure of energy conservation.

The more energy the system has, the more correctfrequencies (ν for ∆t = 0.1 fs or νcorr for ∀ ∆t) are shiftedtowards lower values. It is the result of anharmonicity:the larger is the amplitude of vibration, the larger is theanharmonic frequency shift.10,17

External standard values for the comparison wereharmonic vibrational frequency (νharmonic = 3927 cm−1) andenergy difference between the zeroth and first vibrationallevels (ν0→1 = 3772 cm−1). ν0→1 was obtained from numericalsolution of Schrödinger equation on the computed potentialenergy curve. Thus this value represents real anharmonic

TABLE I. Positions of the absorption intensities maxims for HF moleculeobtained from MD-NVE simulations.

∆Ee,a rHF,b ∆t , ν, νcorr, Tav,c

K Å fs cm−1 cm−1 K

145 0.910

0.1 3915 3914 1450.5 3952 3929 1401.0 4021 3926 1242.0 4408 3919 63

583 0.890

0.1 3917 3916 5820.5 3928 3906 5621.0 4008 3913 4972.0 4420 3927 259

1363 0.870

0.1 3896 3895 13530.5 3922 3900 13051.0 3995 3901 11462.0 4384 3902 722

aEnergy difference between equilibrium and starting geometries of simulation.bDistance between H and F atoms at the beginning of simulation.cAverage temperature of the simulation (see text for explanation).

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 129.70.91.129 On: Tue, 03 May

2016 14:50:46

174108-3 Denis S. Tikhonov J. Chem. Phys. 144, 174108 (2016)

frequency. It can be seen that the uncorrected frequencies withthe large time steps 0.5, 1.0, and 2.0 can even be larger than theνharmonic. But corrected values tend to be 6νharmonic. However,they are still quite far from the quantum anharmonic frequencyν0→1. This is again the result of the smaller amplitude ofvibration in the classical system comparing to its quantumanalog.10

An illustration of the MD spectra in the case of simulationwith the largest total energy (initial rHF = 0.87 Å) andtheir comparison with the harmonic νharmonic and quantumanharmonic ν0→1 frequencies can be found in Figure 1. It canbe seen from this picture that the discussed correction candeform shape of initial intensity peak with the increase of∆t (particularly, it reduces its width). But in the case of MDsimulations, this width mainly comes from finite trajectorylength and from the used window function. Therefore weconsider this effect to be unimportant for discussion.

C. Formic acid: An example of polyatomic molecule

Formic acid was chosen for demonstration of applicabilityof the discussed correction for polyatomic molecules.In this case, MD-NVT simulations were performed. Asingle medium-length NVT simulation was chosen toavoid averaging of several short NVE trajectories as itwas previously recommended.9,10 The NVT ensemble wassimulated using chain of two Nose-Hoover thermostats. Thechosen∆t were 0.1 and 1.0 fs. 1.0 fs is also a badly chosen timestep. Total length of the simulations was 7 ps. First 2 ps wasignored to account for the equilibration phase. Temperature ofsimulations was 300 K. Unfortunately, in both simulations, theO–H and C–H stretching vibrations were significantly affectedby the “flying ice cube” effect:30 energy from these motionswas drained into other degrees of freedom by the thermostat.As a result, relative intensities of the corresponding peaksin spectra were significantly lower than it was expected.Therefore comparison of the spectra for these vibrations wasperformed independently.

FIG. 1. Vibrational spectra for HF molecule computed from MD simulationsstarting from 0.87 Å between H and F atoms with different time increments.Dashed and solid curves denote initial and corrected spectra, respectively.Solid vertical line represents anharmonic fundamental 0→ 1 transition fre-quency ν0→1. Dashed vertical line stands for harmonic frequency νharmonic.

FIG. 2. Comparison of theoretical and experimental spectra of formic acidfor C–H and O–H stretching vibrations. Dashed and solid curves for “MD”data denote initial and corrected spectra, respectively.

The reference theoretical spectra were harmonic andanharmonic. The last was computed using the VPT2-QFFmodel.20 A comparison with the experimental data from theNIST Chemistry Webbook Database was done as well.31

The illustration of range of frequencies 2800–4000 cm−1

(i.e., bond stretching vibrations with the hydrogens) canbe found in Figure 2, and of the range 450–2100 cm−1 inFigure 3.

It can be seen from these figures that the correctedMD spectra computed from the trajectory with the time step1.0 fs are very similar to those from 0.1 fs step simulation.Minor discrepancies of the peaks shape for the C–H and O–Hstretching vibrations are probably caused by thermostat. Thelower the frequency, the more the MD peaks are shifted fromharmonic to VPT2-QFF ones. This is the result of transitionfrom quantum to classical behavior with the lowering ofenergy. It is interesting to note that peaks of harmonic and MDspectra are aligned at one side with respect to the experimentalones, while VPT2-QFF frequencies can be found before andafter them.

FIG. 3. Comparison of theoretical and experimental spectra of formic acidexcluding C–H and O–H stretching vibrations. Dashed and solid curves for“MD” data denote initial and corrected spectra, respectively.

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 129.70.91.129 On: Tue, 03 May

2016 14:50:46

174108-4 Denis S. Tikhonov J. Chem. Phys. 144, 174108 (2016)

IV. CONCLUSIONS

A simple posterior correction for MD vibrational spectrawas developed. It accounts for frequency shift that comesfrom usage of large integration time steps. The calculatedspectra should be replotted with new frequencies that aregiven by Eq. (4). Theoretical behavior of this correctionfunction was found to be similar to previous numericalobservations.9 This correction was constructed for the Verletintegration method. Applicability of this correction to otherintegration methods, such as velocity Verlet,14 was postulatedon the basis of their algebraic equivalence.16 This assumptionwas then confirmed by performed numerical simulations thatused velocity Verlet integration. Test objects were hydrogenfluoride (HF) and formic acid (HCOOH). The performedab initio MD computations have shown that the correctionrestores frequency of signal even in the cases of incorrectlychosen time steps. It was also found that this correctiondeforms shape of signals by reducing width of peaks withincrease of the time increment. But this effect was consideredinsignificant since in case of spectra from MD this widthmainly comes the finite length of trajectory.

Note that the developed correction regards only theintegration technique and thus can be used for both force-fieldbased and ab initio MD simulations.

Summing up, we can say that significant computationaltime savings for MD simulations can be achieved with use ofthe developed technique.

ACKNOWLEDGMENTS

The reported study was supported by the DeutscheForschungsgemeinschaft (DFG, Project VI No. 713/1-1). Allcalculations were performed using computational resources ofLaboratory of Inorganic and Structural Chemistry (Universityof Bielefeld). The author would also like to express hisgratitude to Dr. N. B. Tikhonova and B. K. Mesheryakov forvaluable discussions.

1J. S. Tse, “Ab initio molecular dynamics with density functional theory,”Annu. Rev. Phys. Chem. 53, 249–290 (2002).

2D. A. Wann, A. V. Zakharov, A. M. Reilly, P. D. McCaffrey, andD. W. H. Rankin, “Experimental equilibrium structures: Application ofmolecular dynamics simulations to vibrational corrections for gas electrondiffraction,” J. Phys. Chem. A 113, 9511–9520 (2009).

3M. Dracínský, H. M. Möller, and T. E. Exner, “Conformational samplingby ab initio molecular dynamics simulations improves nmr chemical shiftpredictions,” J. Chem. Theory Comput. 9, 3806–3815 (2013).

4S. D. Ivanov, A. Witt, and D. Marx, “Theoretical spectroscopy using molec-ular dynamics: Theory and application to ch+5 and its isotopologues,” Phys.Chem. Chem. Phys. 15, 10270–10299 (2013).

5M. Thomas, M. Brehm, R. Fligg, P. Vohringer, and B. Kirchner, “Computingvibrational spectra from ab initio molecular dynamics,” Phys. Chem. Chem.Phys. 15, 6608–6622 (2013).

6W. B. Bosma, L. E. Fried, and S. Mukamel, “Simulation of the intermolec-ular vibrational spectra of liquid water and water clusters,” J. Chem. Phys.98, 4413–4421 (1993).

7S. Iuchi, A. Morita, and S. Kato, “Molecular dynamics simulation withthe charge response kernel: Vibrational spectra of liquid water and

n-methylacetamide in aqueous solution,” J. Phys. Chem. B 106, 3466–3476(2002).

8M. Praprotnik and D. Janežic, “Molecular dynamics integration and molec-ular vibrational theory. III. The infrared spectrum of water,” J. Chem. Phys.122, 174103 (2005).

9J. Hornícek, P. Kaprálová, and P. Bour, “Simulations of vibrational spectrafrom classical trajectories: Calibration with ab initio force fields,” J. Chem.Phys. 127, 084502 (2007).

10S. A. Fischer, T. W. Ueltschi, P. Z. El-Khoury, A. L. Mifflin, W. P. Hess,H.-F. Wang, C. J. Cramer, and N. Govind, “Infrared and Raman spectroscopyfrom ab initio molecular dynamics and static normal mode analysis: TheC–H region of DMSO as a case study,” J. Phys. Chem. B 120, 1429–1436(2016).

11J. Kubelka and P. Bour, “Simulation of vibrational spectra of large mole-cules by arbitrary time propagation,” J. Chem. Theory Comput. 5, 200–207(2009).

12C. Störmer, “Sur les trajectoires des corpuscules électrisés dans l’espace.Applications à l’aurore boréale et aux perturbations magnétiques,” Radium4, 2–5 (1907).

13L. Verlet, “Computer ‘experiments’ on classical fluids. I. Thermodynam-ical properties of Lennard-Jones molecules,” Phys. Rev. 159, 98–103(1967).

14W. C. Swope, H. C. Andersen, P. H. Berens, and K. R. Wilson, “A computersimulation method for the calculation of equilibrium constants for the forma-tion of physical clusters of molecules: Application to small water clusters,”J. Chem. Phys. 76, 637–649 (1982).

15F. Jensen, Introduction to Computational Chemistry (John Wiley & Sons,Inc., 1999).

16O. M. Becker and M. Watanabe, “Dynamics methods,” in ComputationalBiochemistry and Biophysics, edited by O. M. Becker, J. Alexander, D.MacKerell, B. Roux, and M. Watanabe (Marcel Dekker, Inc., 2001), Chap.3, pp. 39–67.

17Course of Theoretical Physics, Volume I: Mechanics, 3rd ed., edited by L.Landau and E. Lifshits (Butterworth-Heinemann, Oxford, 1976).

18T. Schlick, Molecular Modeling and Simulation: An Interdisciplinary Guide(Springer, New York, Dordrecht Heidelberg London, 2010), Sec. 13.3.

19A. A. Granovsky, Firefly version 8, http://classic.chem.msu.su/gran/firefly/index.html, 2014.

20K. Yagi, K. Hirao, T. Taketsugu, M. W. Schmidt, and M. S. Gordon,“Ab initio vibrational state calculations with a quartic force field: Appli-cations to H2CO, C2H4, CH3OH, CH3CCH, and C6H6,” J. Chem. Phys.121, 1383–1389 (2004).

21M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon,J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. Su, T. L. Windus,M. Dupuis, and J. A. Montgomery, “General atomic and molecular electronicstructure system,” J. Comput. Chem. 14, 1347–1363 (1993).

22C. Adamo and V. Barone, “Toward reliable density functional methodswithout adjustable parameters: The PBE0 model,” J. Chem. Phys. 110,6158–6170 (1999).

23S. Nosé, “A unified formulation of the constant temperature moleculardynamics methods,” J. Chem. Phys. 81, 511–519 (1984).

24W. G. Hoover, “Canonical dynamics: Equilibrium phase-space distribu-tions,” Phys. Rev. A 31, 1695–1697 (1985).

25F. Weigend and R. Ahlrichs, “Balanced basis sets of split valence, triplezeta valence and quadruple zeta valence quality for H to Rn: Design andassessment of accuracy,” Phys. Chem. Chem. Phys. 7, 3297–3305 (2005).

26K. L. Schuchardt, B. T. Didier, T. Elsethagen, L. Sun, V. Gurumoorthi, J.Chase, J. Li, and T. L. Windus, “Basis set exchange: A community databasefor computational sciences,” J. Chem. Inf. Model. 47, 1045–1052 (2007).

27D. Feller, “The role of databases in support of computational chemistrycalculations,” J. Comput. Chem. 17, 1571–1586 (1996).

28A.-R. Allouche, “Gabedit—A graphical user interface for computationalchemistry softwares,” J. Comput. Chem. 32, 174–182 (2011).

29D. Young, Computational Chemistry: A Practical Guide for Applying Tech-niques to Real World Problems (John Wiley & Sons, Inc., 2001).

30S. C. Harvey, R. K.-Z. Tan, and T. E. Cheatham, “The flying ice cube:Velocity rescaling in molecular dynamics leads to violation of energyequipartition,” J. Comput. Chem. 19, 726–740 (1998).

31See http://webbook.nist.gov for NIST Chemistry Webbook.

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 129.70.91.129 On: Tue, 03 May

2016 14:50:46