Classical and QM/MM molecular dynamics simulations of Co2+ in water

March 13, 2007 in Computational Chemistry, Further thesis study, Journal

The publication below is one of the closest research to my thesis research where the system observed involving Co(II) in water. It was performed by one of our lecturer in Chemistry Department, University of Gadjah Mada. However my thesis research emphasized the use of Monte Carlo Simulation, while this PhD research used QM/MM Molecular Dynamics Simulation.

Ria Armunanto¹, Christian F. Schwenk, A. H. Bambang Setiaji¹ and Bernd M. Rode^, Department of Theoretical Chemistry, Institute of General, Inorganic and Theoretical Chemistry, University of Innsbruck, Innrain 52a, A-6020, Innsbruck, AustriaReceived 20 June 2003;  accepted 20 August 2003. ; Available online 22 September 2003.

Abstract

Classical and quantum mechanical/molecular mechanical (QM/MM) molecular dynamics (MD) simulations have been performed to describe structural and dynamical properties of Co²⁺ in water. The most important region, the first hydration shell, was treated by ab initio quantum mechanics at unrestricted Hartree–Fock (UHF) level using the LANL2DZ ECP basis set for Co²⁺ and the double-ζ plus polarization basis set for water. For the rest of the system newly constructed three-body corrected potential functions were used. A well-structured rigid octahedron was observed for the stable first hydration shell showing no first shell water exchange process within a simulation time of 11.9 ps. For second hydration shell ligands, a mean residence time of 28 ps was observed. Librational and vibrational motions as well as the ion–oxygen motion were investigated by means of velocity autocorrelation functions showing significant differences between classical and QM/MM results.

 

Article Outline

1. Introduction

2. Methodology

2.1. Construction of potential functions

2.2. Simulation performance

2.3. QM/MM molecular dynamics simulation

2.4. Velocity autocorrelation functions

2.5. Mean residence times and reorientational times

3. Results and discussion

3.1. Structural data

3.2. Dynamical data

3.2.1. Librational and vibrational motions

3.2.2. Ligand exchange processes

4. Conclusion

Acknowledgements

References

 

1. Introduction

The hydration structure of transition metal ions is of much interest, since they have several key functions in biomolecular systems [1]. The methods used for structural investigations of hydrated metal ions can be classified into three types: scattering methods such as X-ray diffraction (XD) and neutron diffraction (ND), spectroscopic methods such as extended X-ray absorption fine structure (EXAFS) and nuclear magnetic resonance (NMR), and the tools of theoretical chemistry, including a wide variety of different simulation techniques [2]. Simulation methods such as Monte Carlo (MC), classical molecular dynamics (MD) and hybrid quantum mechanical/molecular mechanical (QM/MM) simulations, have proven to be a strong alternative to experiments in particular for investigations where experiments reach their limitations [3, 4, 5, 6, 7, 8 and 9]. Classical molecular dynamics simulation methods have been widely used, inter alia to analyze solutions of alkali metal ions and transition metal ions in water or ammonia [5, 10, 11, 12, 13, 14, 15, 16 and 17], and it has been shown in many cases, that pair potentials are inadequate for such systems [18, 19, 20, 21 and 22]. Non-additive terms (3,4,5,…,n-body) thus play an important role and should, therefore, be included in the potential functions. However, this procedure is usually restricted to three-body terms, as the construction of higher energy surfaces becomes very complicated.

Full ab initio quantum mechanical treatment could include all n-body terms, but is still far beyond current computer capacities. To reduce the time demand without loosing accuracy of the results the system can be partitioned, however, into the region of the ion with its first hydration shell, which is treated quantum mechanically, and the classically described remaining region which uses three-body corrected ab initio evaluated analytical potential functions. This method is referred to as hybrid QM/MM simulation technique.

Ligand exchange rates and residence times of ligands in the coordination shell of ions are important dynamical parameters to understand the reactivity of these ions in chemical and biological systems. The exact structure and in particular the dynamics of hydrated transition metal ions are highly sensitive to the accuracy of simulation techniques, and it has been shown in several cases [7, 8, 23, 24, 25 and 26] that only ab initio QM/MM simulations reach a sufficient level of accuracy. In the present work, we have extended these investigation to hydrated Co²⁺, in order to obtain structural and dynamical properties of this ion, which plays a quite significant role in solution chemistry and biomolecules.

2. Methodology

2.1. Construction of potential functions

New potential functions for Co²⁺–H₂O and H₂O–Co²⁺–H₂O interactions were constructed from ab initio quantum mechanical calculations at unrestricted Hartree–Fock (UHF) level using the double-ζ plus polarization basis set for water and the LANL2DZ ECP basis set for Co²⁺ [27 and 28]. These energies were fitted to analytical functions using the Levenberg algorithm. Experimental gas phase values were used for the water geometry (O–H=0.9601 Å, H–O–H=104.47°) and kept constant throughout the energy calculations [29]. Oxygen and hydrogen charges were set to −0.6598 and 0.3299, respectively, in agreement with the BJH-CF2 water model [30, 31 and 32] used for water–water interactions in this work. The basis set superposition error (BSSE) [33] in this system is very small amounting 0.183 kcal mol⁻¹. About 2719 ab initio energies points were calculated using the Turbomole program [34, 35 and 36]. The minimum energy for the Co²⁺–H₂O interaction was found to be −86.5 kcal mol⁻¹ at a distance of 2.02 Å. An MP2 calculation yielded nearly the same Co–O distance (2.00 Å) and a slightly lower energy (2.87 kcal mol⁻¹) showing small electron correlation effects in this system. According to previous results [24 and 37] the usage of UHF calculations with DZP basis set seems a reasonable compromise between accuracy and computational effort, also minimizing possible BSSE errors [37]. The limitation of the method could be seen rather in the size of the QM region and not so much in the QM level of calculation.

The ab initio calculated Co²⁺–H₂O energies were fitted to an analytical function of the following form:
 

(1)

where M denotes Co²⁺ and i water atoms; A, B, C and D are the optimized parameters summarized in Table 1, and q represents the atomic charges.

Table 1. Optimized parameters of the analytical Co²⁺–H₂O pair potential function

A total of 13,631 ab initio energy points were generated to describe the H₂O–Co₂₊–H₂O energy surface and to construct a three-body correction function
 

ΔEcorr3bd=(EabWMW−EabM−2EabW)−ΔE2bdMW(r1)−ΔE2bdMW(r2)−ΔE2bdWW(r3),

(2)

where ab and 2bd denote ab initio and two-body energies; MW and WW indicate ion–water and water–water interactions; r₁, r₂ and r₃ correspond to ion–water(1), ion–water(2) and water(1)–water(2) distances, respectively. The obtained three-body correction function is

ΔE3bdFit=0.54e−0.25(r1+r2)e−0.52r3(CL−r1)2(CL−r2)2,

(3)

where CL, set to 6.0 Å, is the cut-off limit beyond which three-body terms are negligible.

2.2. Simulation performance

The simulations were performed for one Co²⁺ and 499 water molecules in an elementary cubic box of 24.6 Å side length, at 298.16 K, which corresponds to a density of 0.99072 g cm⁻³. Periodic boundary conditions were applied to the simulation box and the temperature was kept constant by the Berendsen algorithm [38 and 39]. The flexible BJH-CF2 water model which includes an intramolecular term was used [30, 31 and 32]. Accordingly, the time step of the simulation was set to 0.2 fs, which allows for explicit movement of hydrogens. A cut-off of 12.0 Å was set except for O–H and H–H non-Coulombic interactions for which it was set to 5.0 and 3.0 Å . The reaction field method was used to account for long-range electrostatic interactions [40].

2.3. QM/MM molecular dynamics simulation

A classical molecular dynamics simulation was carried out for 60.0 ps after 60.0 ps of equilibration using the pair plus three-body function. Subsequently, the QM/MM simulation was performed for 11.9 ps after 4 ps of re-equilibration. The ab initio quantum mechanical treatment was applied to the ion and the full first hydration shell, and for the remaining MM region the same 2 + 3-body potential as in the classical simulation was used. According to the Co–O RDF of the classical simulation, the QM radius was set to 3.8 Å to fully include the first hydration shell. A smoothing function was applied to the transition region between the QM and the MM regions [38]. The force of the system, F_system, is defined as
 

Fsystem=FMM+S(FQM−FQM/MM),

(4)

where F_MM is the MM force of the full system, F_QM the QM force in the QM region, F_QM/MM the MM force in the QM region. S denotes the smoothing function [41]. The use of this smoothing function and the algorithm of our QM/MM simulation allows the water ligands to migrate freely between the two regions with a steady transition of forces. In this context, the flexibility of the MM water molecules is another important factor, as this flexibility is thus given for ligands both inside and outside the QM region.

2.4. Velocity autocorrelation functions

The evaluation of spectral properties such as librational and vibrational frequencies of water molecule motions was carried out using velocity autocorrelation functions (VACFs), C(t), defined as
 

(5)

where N is the number of particles, N_t is the number of time origins t_i, and denotes a certain velocity component of particle j. The power spectrum of the VACF was calculated by Fourier transformation. A correlation length of 2.0 ps was used to obtain the power spectra with 4000 (classical) and 2000 (QM/MM) averaged time origins. Librational and vibrational frequencies of water molecules were computed using the approximative normal coordinate analysis [42]. Six scalar quantities Q₁, Q₂, Q₃, R_x, R_y, R_z define the symmetric stretching, bending and asymmetric stretching vibrations, and rotations around the three principal axes of the water molecules.

2.5. Mean residence times and reorientational times

The mean residence time (MRT) of water molecules in the second hydration shell of Co²⁺ was calculated with the following formalism proposed by Impey et al. [43]:
 

(6)

where n_ion(t) is the number of water molecules which lie initially within the coordination shell and are still there after a time t elapsed. The parameter t^* is introduced to avoid counting of water molecules leaving the coordination shell only temporarily and returning to it within t^*. The parameter of t^* was set to 2.0 ps in accordance with Impey [43].Reorientational time correlation functions (RTCFs) of water molecules were calculated as
 

(7)

where P_l is the Legendre polynomial of lth order and is a unit vector along the three principal axes i defined in a fixed coordinate frame as the rotations above.As exponential decay is assumed for the MRTs and RTCFs, an exponential fit was used
 

Cl(t)=aexp(−t/τ),

(8)

where a and τ are the fitting parameters, and τ describes the corresponding relaxation time.

3. Results and discussion

3.1. Structural data

The radial distribution functions (RDFs) of Co²⁺–O and Co²⁺–H together with their integration numbers obtained from the classical and the QM/MM simulations are displayed in Fig. 1. Two well-defined peaks are obtained from both simulations indicating first and second hydration shell. The first QM/MM peak is shifted closer to the ion in comparison with the classical peak, reflecting a remarkable influence of higher n-body effects. The sharpness of the first peak corresponds to a highly structured, rather rigid first hydration shell. The zero-value Co–O RDF between the two peaks indicates that no exchange process occurred within the simulation time. The broad second peaks observed in both classical and QM/MM simulations shows a high flexibility of water molecules in this shell. The first peak obtained from the QM/MM simulation is centered at 2.17 Å, while the classical simulation shifts it to 2.27 Å. The second shell peaks are centered around 4.6 Å in both simulations, but with a broad plateau in the classical case. These results are in good agreement with Co–O distances of the first and second hydration shell obtained by EXAFS, XD and ND experiments [2 and 44]. The average Co–O distance obtained from the QM/MM simulation (2.17 Å) is only slightly higher than XD (2.09 Å) and EXAFS (2.08 Å) data, the difference being probably due to concentration effects [2].

Fig. 1. Co–O and Co–H radial distribution functions and their corresponding integration numbers obtained from QM/MM (solid line) and classical (dotted line) MD simulations.

Coordination number distributions of hydrated Co²⁺ obtained from the classical and the QM/MM simulation are displayed in Fig. 2. The obtained six-coordinated complex in the first hydration shell (100% occurrence) is in agreement with EXAFS data [2 and 44], whereas the classical simulation gives a slightly lower value (5.9). Classical pair plus three-body simulations often allow a correct description of rough structural data as first shell coordination numbers. However, the too repulsive three-body potential apparently causes a small shift of the first hydration shell to a larger distance. This rather small difference between classical and QM/MM result in the first coordination shell then induces larger deviations in the second shell caused by different ligand orientations in the first shell and polarization effects of first shell ligands not accounted for by the classical potentials. The classical simulation thus strongly overestimates the second shell coordination number yielding a value of 22.7 whereas the QM/MM value of 15.9 is closed to the value of 14.8 estimated from XD [2]. The classical simulation thus also yields a broader coordination number distribution (18–28), while the QM/MM simulation gives values between 11 and 19.

Fig. 2. Coordination number distributions of Co²⁺ in water obtained from (a) QM/MM and (b) classical MD simulations.

The angular distribution function (ADF) of O–Co²⁺–O angles is shown in Fig. 3. The ADF obtained from the QM/MM simulation displays two peaks located at 90° and 180°. The first peak located at 91° is caused by two neighboring oxygens, and is in good agreement with the angle deduced from mass spectroscopic analysis (90°) [2 and 44]. The second peak culminates at 173° indicating an octahedral arrangement of the water molecules in the first shell of Co²⁺ (Fig. 4), in agreement with XD data (see Table 3). The small artificial peak at 70° in the classical simulation is caused by short-lived sevenfold coordinated intermediates which have not been obtained in the QM/MM simulation.

Fig. 3. Angular Distribution Function of O–Co²⁺–O angles observed from classical (dotted line) and QM/MM (solid line) MD simulations.

Fig. 4. Octahedral structure of the first hydration shell of Co²⁺ in water.

Table 3. Structural parameters of the first hydration shell of Co²⁺ in water

Two further angles were defined to describe the orientation of the water molecules relative to the ion (θ and tilt). The θ angle is the angle between the O–Co vector and the water plane, while the tilt angle is defined as the angle between the O–Co vector and dipole vector of the water molecule. The QM/MM simulation yields a θ value of 171° and a tilt value of 8° (see Table 2), whereas the classical simulation shows slightly lower θ angles and nearly no tilt.

Table 2. Hydration parameters for Co²⁺ in aqueous solution obtained from QM/MM and classical MD simulations

The hydration energy values of −550 and −547 kcal mol⁻¹ obtained from classical and QM/MM simulation are considerably lower than the experimentally estimated value of −487 kcal mol⁻¹ [45]. This difference is probably caused by the specific assumptions necessary to assign single-ion values to thermocalorimetric measurements of salts [45].

3.2. Dynamical data

3.2.1. Librational and vibrational motions

The power spectra of VACFs for the librational motions R_x, R_y, R_z and the vibrational motions Q₁, Q₂, Q₃ obtained from the classical and the QM/MM simulations are displayed in Fig. 5 and their frequencies are summarized in Table 4. The order of R_z<R_x<R_y is found in the second hydration shell and the bulk for QM/MM as well as classical simulation. In the first shell the QM/MM order is different (R_z<R_y<R_x), as already observed in previous simulations [7 and 23]. The R_z value in the first hydration shell (QM/MM simulation) is only slightly red-shifted since the rotation around the dipole axis is not energetically restricted. In contrast, the R_x QM/MM frequency is strongly blue-shifted due to the ligand fixation by the ion, as in the case of Ni²⁺ [7], Ca²⁺ [23], Fe²⁺ and Fe³⁺ [52]. The R_y value is nearly unchanged. The order R_x>R_y obtained from the QM/MM simulation contradicts the classical result, revealing some inadequacy of the 2 + 3-body function. In the second hydration shell, the librational motions of the classical and the QM/MM simulation are rather similar, showing slightly higher values in the QM/MM case. The frequencies of the bulk phase are in good agreement with the values obtained for pure liquid water (BJH model) as listed in Table 4.

Fig. 5. Power spectra of rotational modes R_z, R_x, R_y and vibrational modes Q₂, Q₁, Q₃ for water molecules in the first hydration shell obtained from QM/MM (solid line) and classical (dotted line) MD simulations.

Table 4. Librational and vibrational frequencies of water molecules in the first and second hydration shell of Co²⁺ in water and bulk

In comparison with the bulk, the stretching frequencies Q₁ and Q₃ of the first hydration shell from the QM/MM simulation are blue-shifted, whereas the bending frequency Q₂ is red-shifted in accordance with previous simulations of other hydrated ions [7, 23, 24 and 52]. In contrast, the classical simulation fails to reproduce these effects, giving red-shifted stretching modes Q₁ and Q₃, and a nearly unchanged bending mode Q₂. The frequency difference () between Q₁ and Q₃ is 76 cm⁻¹ in the QM/MM case, similar to Ni²⁺ [7] and V²⁺ [24] shifts.

The RTCF values of first (τ₁) and second (τ₂) order in the dipole moment direction of a water molecule are summarized in Table 5. The correlation function for l=1 is related to infrared line shapes and l=2 to Raman line shapes and NMR relaxation time [2]. The QM/MM results exhibit significantly larger relaxation times in comparison with classical data stressing once more the important role of many-body effects. Strongly increased relaxation times were only observed for the first hydration shell, outside of which the ion’s influence is rather weak. The bulk water relaxation times obtained from both simulations are in good agreement with previous simulations [24 and 51] and experimental data [2].

Table 5. Reorientational times of first and second order of water molecules in the first hydration shell of Co²⁺ in water

The power spectra of the Co²⁺–O stretching motion obtained from classical and QM/MM simulations are displayed in Fig. 6. The stretching frequency from the QM/MM and the classical simulations are 305 and 290 cm⁻¹, respectively. The corresponding QM/MM force constant of 69 N m⁻¹ is similar to the ones obtained from previous QM/MM MD simulations of Mn²⁺ and V²⁺ [24] with values of 59 and 70 N m⁻¹. The peak forms of the stretching vibration reflect some further inappropriate description of the dynamics in the first shell by the classical treatment.

Fig. 6. Power spectra of Co²⁺–oxygen vibrational modes obtained from QM/MM MD (solid line) and classical (dotted line) simulations.

3.2.2. Ligand exchange processes

The mean residence time of water molecules (τ) in the second hydration shell has been calculated from the QM/MM simulation with t^*=2.0 ps according to Impey [43]. A mean residence time of 28 ps in the second hydration shell was found, corresponding to the seven observed exchange processes during the 12 ps simulation. No experimental data for residence times of water ligands in the second hydration shell of Co²⁺ are available. In the first hydration shell no water exchange process was observed as a simulation time of 12 ps is by far too short compared to the experimental residence time of 10⁻⁷ s [2 and 53].

4. Conclusion

The inclusion of many-body effects appears mandatory in order to describe the hydration structure of Co²⁺ and its dynamical properties properly. The structural results clearly demonstrate that although rough structural properties as first shell coordination numbers are reproduced correctly, accurate ligand orientations are only available after including many-body terms through a quantum mechanical treatment. Therefore, the evaluation of the much more sensitive spectroscopic and dynamical data also requires this level of accuracy, and the inclusion of a larger number of water ligands in the surroundings of the ion appears desirable in further simulations.

 

Acknowledgements

Financial support for this work by the Austrian Science Foundation (FWF) (project P16221-N08) and a scholarship of the Austrian Federal Ministry for Foreign Affairs for R.A. are gratefully acknowledged.

 

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Corresponding author. Tel.: +435125075160; fax: +435125072714

¹ Present address: Chemistry Department, Faculty of Mathematics and Natural Sciences, Austrian-Indonesian Center for Computer Chemistry, Gadjah Mada University, Jogjakarta Indonesia.

Chemical Physics
Volume 295, Issue 1 , 15 November 2003, Pages 63-70