Global geometry optimization of Na⁺(H₂O)_m clusters using a hybrid algorithm

Cao Yilin, Wang Yansong
（College of Chemistry and Environmental Science, Henan Normal University, Xinxiang 453002）

  Supported by the Natural Science Foundation of Henan Province (No.532221).

Abstract A hybrid algorithm, based on genetic algorithm (GA), fast simulated annealing (FSA) and conjugated gradient algorithm (CGA) was proposed and used to optimize the structure of Na⁺(H₂O)_m (m=1-11) clusters. The empirical potential energy functions employed here included SPC, TIP, TIP3P, TIP4P and TIPS2. The optimized structure of Na⁺(H₂O)_m clusters by using these different potential energy functions were found to share the same configurations.
Key words Hybrid optimization algorithm, potential function, Na⁺(H₂O)_m clusters

Many recent developments in the computational technology for random search have been made in the area of atomic and molecular cluster calculations. For example, Niesse used a modified GA method to optimize the geometry of atomic and molecular clusters^[1]. Greguick proposed a modified deterministic/stochastic GA method for the determination of the global minimum of (Ar)_n and B(Ar)_n clusters. The deterministic / stochastic GA converged more quickly to the global minimum than other random search procedures^[2]. As a global optimization algorithm, the simulated annealing algorithm has been widely used to optimize the molecular clusters too^[3]. But, applying all these methods to larger systems is limited. To make the random search more efficient, based on GA, FSA and CGA we have presented a hybrid optimization algorithm and applied it to water clusters in the previous works. The results showed the structures of water clusters with lower potential energy were found and a more quick optimization strategy than using other was got. So, this method is proved more effective and it can be used to optimize the structure of water clusters in the electrostatic field of sodium ion in this paper.

1 POTENTIAL FUNCTION
The most stable structure of a cluster is usually the geometry with the lowest potential energy calculated through some potential functions. In the optimization of Na⁺(H₂O) _m clusters, the interaction energy between water molecule and sodium ion can be computed using the following equation^[4]:

(1)

Where the index I denotes the sodium ion, the charge q_I is +1, and the parameters A_I, B_I are 447.69 kJ Å ⁶ /mol and 19.25 kJ Å ³ /mol, respectively. The interaction energy between water molecules will be calculated according to equation 2:

         (2)

The parameters of the potential functions are listed in Table 1. In this table, SPC, TIP, TIP3P, TIP4P and TIPS2 are five kinds of potential energy functions which have the same literal expression but with different parameters. The first 3 are three-site models, and the last 2 are four-site models. The geometries of the models are shown in Figures 1. In the four-site model, the negative charge is moved off the oxygen atom and towards the hydrogen at a point (M) on the bisector of the HOH angle.

Table 1 Water monomer parameters for potential functions^[5,6]

             ^{(a)                                                          (b)}
Figure 1 3-site (a) and 4-site (b) water models

2 METHOD
In this paper, the water molecule was treated as a rigid molecule and could be completely described by six coordinates (Rx, Ry, Rz, q, F, f). The molecule's initial coordinate (Rx, Ry, Rz) were generated randomly within a cubic box of size L³ centered at the origin. We took R = L (d-0.5), where L = 4.0 *(6*n)^1/3, n is the number of water molecules and ion³. d is a random number between 0 and 1. q, F and f are generated randomly on the interval [0, 2p]. In addition, we put the sodium ion at the center of the box with the coordinate of (0.0, 0.0, 0.0, 0.0, 0.0, 0.0). In our procedure, 30 populations with 30 individuals each were produced. Then the best individual in each population was selected to form the initial population. In our hybrid optimization algorithm, the basic structure of GA was used, but the search strategy was improved. With regard to crossover, geometric mean, arithmetic mean, two-point crossover, one-point crossover and n-point crossover were employed; as regard to mutation, inversion and FSA operators were involved. In every generation, the crossover and mutation were performed first to give a coarse solution to the problem, and then the CGA was performed to have local search for the better solution. The process was performed iteratively. The program was terminated while the change in the potential energy was smaller than 0.04 kJ/mol in 500 generations.

3 RESULTS   AND DISCUSSION
We employed five kinds of water potential functions to optimize the structure of Na⁺(H₂O)_m clusters. The results show: the geometries of Na⁺(H₂O)_m clusters for the five potential functions share the same configurations, have a little difference in the distance of Na⁺ and H₂O which is so little that can be ignored. Here, only the geometry structures of Na⁺(H₂O)_m clusters which were optimized by using TIPS2 are given. In Figure 2, the line between sodium ion and oxygen atom in water molecules does not stand for a chemical bond, it is used to help understanding the geometry of the clusters. The dotted line between water molecules represents a hydrogen bond. At the beginning, the water molecules arrange around the Na⁺ symmetrically and form the first shell or the inner wedge into the first shell. With the increasing in m, instead they will be distributing around the first shell of forming hydrogen bond with the water molecules in the first shell. The extrusion of water molecules may be caused by the repulsion interactions between water molecules in the first shell and the formation of hydrogen bond is favourable for decreasing the energy of the clusters while the fifth and the fallowing water molecules are added to the clusters. As a result, the water molecules gradually filled into the second shell and aggregate partly with each other because they can maximize the number of hydrogen bonds and reduce the repulsive interaction in this way. The examination of the results leads us to conclude that the hydration number of Na⁺ is 4, which agrees with the experiments^[7]. The distance between Na⁺ and oxygen atom of water molecule in the first shell was computed by minimizing the TIPS2 potential function and the average value of the distance increased linearly from 2.286 Å to 2.324 Å as we went from the clusters with m=1 to m=7. This phenomenon was also found in other works^[8] and in agreement with that obtained by SCF calculation^[9].

Figure 2 Geometry of the lowest energy structure found for Na⁺(H₂O)_m, m=1-11.

Table 2 The potential energies of the Na⁺(H₂O)_m (kJ/mol )

The energies of Na⁺(H₂O)_m clusters were also computed by minimizing five different potential  functions respectively, as listed in Table 2. Increasing of m in Na⁺(H₂O)_m clusters can be considered as a stepwise hydration process, which can be expressed by Na⁺(H₂O)_m-1 + H₂O =Na⁺(H₂O)_m. Based on the data in Table 2, the stepwise hydration energy can be calculated by the equation: DE_n-1,n =DE_n -DE_n-1. The results are listed in Table 3 with the literature and experiment values. In Gowda's calculation, complicated potential energy functions were used to gain the potential energies of Na⁺(H₂O)_m clusters ,which include 8 terms, such as the bond charge attraction term, the bond charge repulsions, the ion-induced dipole attractions term, the dispersion energy term and so on. By comparison, we can find that the stepwise hydration energies computed by TIP, TIP4P and TIPS2 are generally the same or better than those by SPC, TIP3P and the literature values, even if the simple potential energy functions were employed. The examination provides an energy evidence for optimization of Na⁺(H₂O)_m clusters.

Table 3 The energies (DE_n-1,n) computed employing five kinds of model (kJ/mol)

4 CONCLUSION
We have carried out potential energy minimization searches for the Na⁺(H₂O)_m (m=1-11) clusters using the hybrid algorithm and have succeeded in finding clusters global minimization potential energy structures using SPC, TIP, TIP3P, TIP4P and TIPS2 potential functions. The results show: using the five potential functions all reflect the potential energy properties of Na⁺(H₂O)_m clusters and gain almost the same geometries of Na⁺(H₂O)_m clusters. The hydration number of Na⁺ is 4, which is in agreement with the value of experiment.

Acknowledgments The authors are  grateful to Dr. J.A.Niesse (University of New Hampshire) for helpful suggestions.

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