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Molecular Dynamics Simulations with ab Initio Force Fields: A Review of Case Studies on CH4, CCl4, CHF3, and CHCl3 Dimers

  • Arvin H.-T. Li
  • Yi-Siang Wang
  • Sheng D. Chao
Review
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Abstract

Recent progress in our group on quantum chemistry calculated intermolecular interaction potential energy functions, or ab initio force fields, for use in molecular dynamics simulations is reviewed. These ab initio force fields have been calibrated by the current state-of-the-art computational techniques with respect to the correlation-method versus basis-set combinations. The case studies of CH4, CCl4, CHF3, and CHCl3 molecular dimeric systems are presented and compared. The simulation scheme can serve as a modern paradigm before a full-blown quantum mechanical molecular simulation can be achieved. It is our hope that this review can help stimulating interests among computational scientists in further exploring this important field of multiscale science and engineering.

Keywords

Intermolecular interaction ab initio force field Molecular dynamics simulation Molecular dimer 

Introduction

From the fundamental point of view of mechanics, the most important task in particle dynamics is to construct the interaction potential forces, or force fields, among members involved in the physical systems. Once the force fields are known, the laws of mechanics or the equations of motion can then be solved numerically to make useful predictions on the temporal evolution of the system’s dynamical variables. These predicted responses to the interaction forces can be represented as some measurable quantities awaiting experimental verifications. In this sense we are done on the technical tasks and the following comparisons with experimental observations and the analyses based on the theoretical and experimental data can further advance our understanding of the physical systems we are interested in. Therefore, a considerable amount of research efforts has been spent on this specific topic in multiscale molecular simulations.

Intermolecular interaction potentials, or non-covalent bonded interactions, play an important role in many scientific disciplines such as condensed matter physics, materials chemistry, and structural biology. These secondary interactions are about one or two orders of magnitude weaker than the primary or covalent bonds, but they are crucial for obtaining accurate thermodynamic properties and energy transfers among molecular complexes [1, 2]. In particular, the conformational tertiary structures of macromolecules such as protein and DNA are determined by the involved van der Waals interactions [3]. Physically, intermolecular forces do not originate from electron sharing. Instead, they actually arise from simultaneous electron correlation of the separated subsystems [4].

The first quantitative studies of intermolecular interactions were performed in earlier last century [5], but validation of the constructed force fields by experimental means is very challenging. The main difficulty is due to limited samplings of the potential energy surface in most experimental studies [6]. For example, using X-ray crystallography and laser spectroscopy mainly explore the equilibrium regions of the potential, while thermodynamic measurements in condensed phases often yield highly averaged potential data. Worse, the experimentally determined potentials are highly sensitive to the thermodynamic conditions employed. Two models of force fields cannot be discerned from the experimental data, thus yielding ambiguity in the force fields.

Traditionally for simulating macroscopic systems, to obtain the microscopic force fields among the involved atoms is a very challenging task. Usually one resorts to experimental researches. Based on the experimental results one can construct simple and physically motivated force fields which contain a number of parameters determined by the specific experiment conditions and the selection of sample molecules, often called the training set. These empirical force fields have served the community of computational mechanics for many decades and have done a credible job in advancing our understanding for microscopic mechanics. However, once applying these empirical force fields to systems outside the training set, the predictive power deteriorates and the calculated results quickly become uncertain and unreliable. It requires a more fundamental study on the background quantum theory to construct a reliable interaction force field. Quantum chemistry calculated intermolecular interaction energies and forces, or ab initio force fields, are the most accurate data which can be used in molecular dynamics simulations [7, 8, 9, 10, 11]. These quantum mechanics based potentials are requested by ab initio molecular dynamics simulations [12] and by classical molecular simulations using force field constructions [13].

Among the components of an intermolecular interaction, the London dispersion force is the most difficult to calculate. The reason is that dispersion interactions arise from the non-local “dynamic” correlations [14]. This non-locality demands full exploration of the time-dependent perspective of quantum mechanics. Often a high-level electron correlation method and a large basis set are required to obtain accurate dispersion forces [15]. Also a practical note is in order. Most present implementations of quantum chemistry programs utilize Gaussian type functions to fasten the calculations of Coulomb repulsion integrals. Because Gaussian type functions are local functions, a large basis set is indispensable in order to perform an accurate correlation energy calculation. Moreover, these functions do not have the correct asymptotic behavior as the intermolecular separation becomes large. Therefore, the complete basis set limit of the calculated potential must be estimated so as to be consistent with the conventional perturbation theory.

For general polar molecular systems, the relatively weak dispersion energy is often masked by the competing electrostatic energies and hydrogen bond interactions. Nonpolar or weakly polar atomic and molecular dimers are usually taken as a prototype case to study the dispersion energy. Many previous studies on dispersion forces have focused on atomic inert gas dimers and several important conclusions have been drawn from the calculations [16]. However, because of the extra degrees of freedom and the stereo-chemical responses, the conclusions about atomic dimers may need to be extended or modified in dealing with molecular dimers. Methane and its halogen-substitutes [17, 18, 19, 20] are non-polar or weakly polar molecules. The higher order electrostatic interactions are relatively weak and decay fast at large intermolecular distances. The dominant long-range attraction for the methane dimer is thus due to the London dispersion force. On the other hand, the strong repulsive force almost comes from the exchange–repulsion interaction due to the overlapping of electron clouds. Because the exchange–repulsion interactions have been incorporated in the Hartree–Fock (HF) self–consistent theory, post-HF methods such as the Møller–Plesset (MP) perturbation theory and the coupled cluster (CC) theory are often used to calculate the correlation effect. Contrasting these sets of calculation helps to delineate the relative importance of the dispersion energy in the overall intermolecular interaction.

Many previous studies on these dimers focused on the equilibrium regions of the potentials, but to construct a reliable force field for molecular simulations, the full potential energy surfaces are required. With current computational powers, a detailed editing of the potential data base can be obtained for medium size molecular clusters. Therefore, we use the state-of-the-art methodology to obtain accurate potential energies for the studied dimers and then construct the globally smooth force fields for use in molecular dynamics simulations. In this review we present our recent efforts in constructing reliable ab initio force fields for use in molecular dynamics simulations [21, 22, 23, 24]. We demonstrate the systematic scheme by using several case studies done in our group in the past decade.

The other parts of this paper are organized as follows. In “Computational Details”, we describe the computational details of these calculations. In “Case Studies of Methane CH4, Fluoroform CHF3, Chloroform CHCl4, and Carbon Tetrachloride CCl4” the results are presented, compared and discussed. A summary and a brief perspective are given in “Conclusion and Perspective”.

Computational Details

All the quantum chemistry calculations were performed by using the Gaussian program package [25]. The geometries of the isolated molecules were first optimized at the MP2 method with a number of basis sets up to the aug-cc-pVQZ level of theory and have been checked with the respective experimental data. The isolated methane molecule was found to be at the tetrahedral configuration (Td symmetry) with the C–H bond length of 1.085 Å. The isolated CHF3 molecule was found to be at the near-tetrahedral configuration (C3v symmetry). The C–F bond length, the C–H bond length, and the ∠F–C–F angle are 1.333 Å, 1.085 Å, and 108.47°, respectively. The isolated CHCl3 molecule was found to be at a low tetrahedral configuration where three chlorines are at the corners of its base plane and the top position is occupied by one hydrogen with the carbon at the center (C3v symmetry). The C–Cl bond length, the C–H bond length, the ∠Cl–C–Cl angle and the ∠H–C–Cl angle are 1.76 Å, 1.08 Å, 108.0˚ and 110.9˚, respectively. The isolated CCl4 molecule was found to be at the tetrahedral configuration (Td symmetry) with the C–Cl bond length of 1.768 Å.

When two monomers approach each other, it depends on the relative orientation to form a stable molecular dimer (conformer). For sampling the directional responses we consider several dimer structures as shown in the respective papers [21, 22, 23, 24]. We have used the MP2 method [26] to account for the correlation effect. Pople’s medium size basis sets [up to 6-311++G(3df, 3pd)] [27] and Dunning’s correlation consistent basis sets (cc-pVXZ and aug-cc-pVXZ, X = D, T, Q) [28] were employed in the calculations. The dimerization energies were calculated using the supermolecular approach and the basis set superposition errors (BSSEs) were corrected by the counterpoise method of Boys and Bernardi [29]. Normally we scanned the relative center-of-mass distance, denoted as R, for a wide spatial range with more than 30 potential grid points for each orientation. At least 360 configuration points have been actually sampled and the corresponding energies calculated. During the scan we fixed the monomer geometry; that is, employing the rigid monomer assumption. To check the rigid monomer assumption, we allowed the monomer to be fully relaxed and repeated the energy calculations. Relaxation of the rigid monomer assumption only increases the binding energy by about 1%.

In order to calibrate the electron correlation effect beyond the MP2 level of theory, single point energy calculations at important geometries were performed by using the CCSD(T) [30] method with the aug-cc-pVDZ and aug-cc-pVTZ basis sets. Overall the CCSD(T) method and the MP2 method yield similar results as long as a large basis set has been used. The CCSD(T) potential data extrapolated to the complete basis set limit are currently the “gold standard” for intermolecular interaction energies.

In the molecular dynamics simulations, a system of 512 molecules were initially arranged in random configurations in a cubic cell with the periodic boundary conditions imposed on the three coordinate directions. A canonical ensemble (constant-NVT) was employed where the temperature control was achieved by rescaling the center-of-mass velocities every 1000 time steps. Newton’s equations of motion for the center-of-mass positions and Euler equations for rotations were solved using the velocity-Verlet algorithm with a time step of 1 fs. The studied molecule was modeled as a rigid body using the quaternion representations. The cut-off radii for the inter-atomic energies were set to be half of the box size, depending on the densities simulated, in order to avoid long range correlations. The system was checked to reach equilibrium after running 20 ps and allowed 200 ps for data collection. The simulated thermodynamic conditions cover a wide density versus temperature range where the experimental data are available. All the calculated results have been checked convergent with respect to the simulation parameters such as box sizes, time steps, etc. For example, no significant differences in the calculated properties of the simulated system were observed by doubling the box size (less than 1% error). Normally the total simulation time was 1 ns but for better convergence on the transport properties, we extended the simulation time to 10 ns. The statistical errors for all the calculated properties were limited to be less than 5% of the mean values.

Case Studies of Methane CH4, Fluoroform CHF3, Chloroform CHCl4, and Carbon Tetrachloride CCl4

The MP2 calculated interaction energies of the studied dimers at their respective several sampled orientations are shown in Fig. 1. Clearly we see significant anisotropy in the orientation sampling. In Table 1 we present the basis set dependence of the binding energies. We can see that adding both the polarization and diffuse functions was required to obtain precise energy values. It requires the aug-cc-pVTZ basis set to converge the binding energy to a chemical precision (e.g., within 10% of the complete basis set limit value). The most stable conformers are shown in Fig. 2.
Fig. 1

The MP2 calculated intermolecular interaction potential energy curves for the sampled conformers. Here the designation letters are used to distinguish the conformers [21, 22, 23, 24]

Table 1

The basis set dependence of the MP2 binding energies (in kcal/mol) for the minimum energy conformers

 

CH4

CCl4

CHF3

CHCl3

cc-pVDZ

− 0.150

− 0.581

− 1.020

− 1.450

cc-pVTZ

− 0.317

− 1.722

− 1.420

− 2.519

cc-pVQZ

− 0.433

− 2.581

− 1.660

− 3.251

aug-cc-pVDZ

− 0.395

− 2.310

− 1.600

− 2.892

aug-cc-pVTZ

− 0.453

− 2.968

− 1.800

− 3.502

aug-cc-pVQZ

− 0.464

− 3.220

− 1.830

− 3.732

Basis set limit

− 0.502

− 3.523

− 1.850

− 3.868

The complete basis set binding energies are also shown as reference data

Fig. 2

The most stable molecular structures of CH4, CCl4, CHF3, CHCl3 dimer. The designation letters for the conformers are J, D, N, and D, respectively

We then use an analytical site–site model to represent the ab initio potential data. The sites are associated with the hydrogen, halogen, and carbon atoms, respectively, as usually used by other force fields. The site–site interaction is represented by a Lennard–Jones (L–J) function and a Coulomb potential in Eq. (1).
$$ U = \sum\limits_{ij} {\left\{ {4\varepsilon_{ij} \left[ {\left( {\frac{{\sigma_{ij} }}{{r_{ij} }}} \right)^{12} - \left( {\frac{{\sigma_{ij} }}{{r_{ij} }}} \right)^{6} } \right] + \frac{{q_{i} q_{j} }}{{4\pi \varepsilon_{0} r_{ij} }}} \right\}} $$
(1)
where the indices i and j denote the atoms in separated monomers, respectively, and rij represents the atom(i)–atom(j) distance. In this model εij, σij, qi, and qj are the potential parameters to be determined in the numerical regression. No bias weights were put on specific configurations except that we excluded from the non-linear fitting some largest repulsive energy points in the regression to prevent their dominance in the least-squares cost function. The penalty function is shown in Eq. (2).
$$ \sum\limits_{\text{i}} {{\text{w}}_{\text{i}} \varTheta } (E_{\text{i}} - C_{\text{i}} ) $$
(2)
where \( \varTheta \) is the step function, Ei is the energy grid points, Ci is the upper bound for removing the energy grids, and wi is the weighting coefficient. This latter constraint could effectively put more weight on deeper wells. The fitting parameters we obtained are shown in Table 2. Together with the precision errors associated with the quantum chemistry calculations, the overall validity of the model force field is checked by comparison with available experimental data.
Table 2

The fitting potential parameters in the constructed ab initio force fields

CH4

CCl4

CHF3

  

CHCl3

σ

2.67

σ

3.45

σCC

3.10

εCC

0.10

qC

0.12

σCC

3.10

εCC

0.31

qC

1.27

ε

0.054

ε

0.387

σCH

3.32

εCH

0.10

qH

0.15

σCH

2.70

εCH

0.81

qH

− 0.28

    

σCF

3.35

εCF

0.01

qF

− 0.09

σCCl

3.80

εCCl

0.11

qCl

− 0.33

    

σHH

2.15

εHH

0.02

  

σHH

3.00

εHH

0.01

  
    

σHF

2.18

εHF

0.35

  

σHCl

2.30

εHCl

0.41

  
    

σFF

3.05

εFF

0.03

  

σClCl

3.50

εClCl

0.31

  

Here σ is in Å, ε is in kcal/mol, and q is in electron charge unit. There was a misprint of the CHF3 force field potential parameters [23], which has been corrected here

We first calculate the atom(α)–atom(β) radial distribution functions (RDFs) by the definition [31, 32].
$$ g_{\alpha \beta } (r) = \frac{n(r)}{{\rho 4\pi r^{2} \Delta r}} $$
(3)
where \( g_{\alpha \beta } (r) \) is the radial distribution function for the α–β atom pair, \( n(r) \) is the mean number of atoms in a shell of radius \( r \) and thickness \( \Delta r \) surrounding the atom, and \( \rho \) is the mean density for total system. The calculation of \( g_{\alpha \beta } (r) \) consists of about 10,000 trials, with each by selecting an atom as the origin and counting the atoms within the spherical shells of thickness \( \Delta r \) = 0.02 Å using the histogram method.
Figure 3 presents the simulated radial distribution functions. For CH4, the simulated atom–atom radial distribution function is calculated at temperature T = 150 K and density ρ = 0.449 g/cm3. For CCl4, the simulated atom–atom radial distribution functions is calculated at temperature from T = 260.0 K and density ρ = 1.660 g/cm3. For CHF3 the simulated atom–atom radial distribution function is calculated at temperature T = 153.0 K and density ρ = 1.583 g/cm3. For CHCl3 the RDF is calculated at temperature T = 298.0 K and density ρ = 1.480 g/cm3. As we can see from the comparison of the simulated atom–atom radial distribution functions with the experiments, the overall agreement is satisfactory. In particular, the distribution peaks and valleys were well reproduced for the atom-to-atom RDFs except for some subtle differences.
Fig. 3

The calculated atom-wise radial distribution functions gCC, as compared with available experimental data. There are missing experimental data mainly because the neutral scattering experiments are not available in current literature

We have also calculated the self-diffusion coefficients using the Green–Kubo formula [33, 34],
$$ D = \frac{1}{3N}\int_{0}^{\infty } {\left\langle {\sum\limits_{i}^{N} {\nu_{i} (t) \cdot \nu_{i} (0)} } \right\rangle } dt $$
(4)
where \( v_{i} \) is the velocity vector of particle i and the statistical average is the velocity autocorrelation functions (VAFs). In Table 3, we present the comparison of the calculated self-diffusion coefficients with experiments. We see the results are generally in good agreement with the experiments. In particular the temperature dependence of the diffusion constant has been well reproduced. At most temperatures, the calculated D values are lower than the experimental values. This might be due to that the fitting models overestimate the interactions of dimers. Overall our simulated D values cover wider experimental data than previous studies. We note that our results are also consistent with previous simulations using polarizable empirical force fields.
Table 3

Comparison of the experimental (EXP) [38, 39] and molecular dynamics (MD) self-diffusion coefficients for a wide range of thermodynamic conditions

 

Temperature (K)

D (EXP) (10−9 m2/s)

D (MD) (10−9 m2/s)

CH4 [38]

112

5.4 ± 0.5b

4.760

207

68 ± 7

58.800

279

87 ± 9

76.800

225

32 ± 3

25.700

306

45 ± 5

31.800

381

53 ± 5

37.000

CCl4

260

0.30

0.706

293

1.284

1.306

328

NAa

1.801

453.2

NAa

6.638

556.5

NAa

40.040

CHF3

142

0.84

1.35

154

1.25

1.73

168

1.9

2.26

177

2.35

2.66

188

2.85

3.2

208

4.4

4.48

222

5.5

5.48

250

8.0

8.58

CHCl3 [39]

217

0.65

0.508

233

0.94

0.698

254

1.35

1.078

262

1.55

1.197

275

1.85

1.471

293

2.40

1.930

313

3.05

2.494

339

4.00

3.330

aNot available

Conclusion and Perspective

To simulate molecular fluid properties, we have employed quantum chemistry calculation to obtain extensive ab initio potential data for the CH4, CCl4, CHF3, and CHCl3 molecular dimers in several orientations. The full potential energy curves for the dimer configurations were calculated by the MP2 method using the basis set up to aug-cc-pVQZ. For these dimers, dispersion interactions contributed more to the stabilization than the electrostatic interactions. The potentials exhibited significant anisotropy, which was then analyzed and considered in the site–site force field model used to fit the potential data. We determined the accuracy of the constructed ab initio force field using molecular dynamics simulations and the results were compared with experiments. Quantitative agreements with the observed radial distribution functions and the self-diffusion coefficients have been achieved for a wide range of thermodynamic conditions. Therefore, these ab initio force fields can serve as a useful tool for studying molecular fluid properties.

After we have published these force fields [21, 22, 23, 24], at that time the first ab initio force fields available, many follow-up studies appeared. For methane, we have refined our first fitted model by using higher level of theory and a five-site model [35]. Our force fields have been compared with existing force fields for this special molecule [36]. Overall, ours are still the only available ab initio force fields which can reproduce the structural, dynamical and thermodynamic properties within the experimental uncertainties. For the other halogen-substituted methane molecules, our force fields have been discussed in a recent review [37] but no quantitative molecular simulation results were compared. As far as our best knowledge extends, the ab initio force fields are still the only available models which can compete with existing empirical force fields. Similar modeling schemes have been employed also by other groups for other molecular fluid systems. It is the most sincere hope that our works can stimulate some interests among computational scientists in further exploring this important field of multiscale science and engineering.

Notes

Acknowledgements

This work was partly supported by National Taiwan University through NTU-CCP-106R891607 and the Ministry of Science and Technology of Taiwan, ROC through MOST 104-2221-E-002-032-MY3. We acknowledge the National Center for High-performance Computing (NCHC) for providing computing resources.

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Copyright information

© Korean Multi-Scale Mechanics (KMSM) 2019

Authors and Affiliations

  1. 1.ARVIN Bio-Medical Devices Co., Ltd.HsinchuTaiwan, ROC
  2. 2.Institute of Applied MechanicsNational Taiwan UniversityTaipeiTaiwan, ROC
  3. 3.Department of ChemistryUniversity of Southern CaliforniaLos AngelesUSA

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