1. Introduction of the project

In this project, I calculated the optimized geometry of H2O. To get it, first, a cubic supercell (side length L angstroms) that is mostly empty space is built. Then the oxygen atom is placed in the supercell with fractional coordinates (0,0,0) and two hydrogen atoms are placed in the super cell with fractional coordinates (a/L, b/L,0) and (-a/L, b/L,0), shown in Figure 1. Next, by calculating the geometry optimization using CASTEP [1], the optimized geometry can be obtained, which includes the bond length and the bond angle.

Figure 1 The initial supercell structure. The red atoms are the oxygen atoms and the grey ones are hydrogen atoms.

During the calculation, the energy cutoff, the k point grid, the supercell lattice L can be changed to obtain most efficient and accurate results. The initial fractional coordinates of H can be changed to analyze if multiple initial geometries for each molecule converge to the same final state.

2. The parameters for the calculations

Some important parameters and inputs for the calculation are listed as following:
Atomic and pseudo atomic structure for H: 1s1
Atomic structure for O: 1s2 2s2 2p4
pseudo atomic structure for O: 2s2 2p4
Functional: GGA Perdew Burke Ernzerhof (PBA) functional
Pseudopotential: OTFG ultrasoft
The k points grid: investigated in the section 3
The energy cutoff: investigated in the section 4

3. The energy cut off

We can start with the k point grid of 3*3*3 and the supercell lattice L=10 Å. The initial fractional coordinates of H are chosen as (0.1, 0.05, 0) and (-0.1, 0.05, 0). From the calculation results shown in table 1, the energy cutoff can be chosen as 700eV.

Table 1 The calculation of the energy cut off.

4. The k point grid

We start with the energy cut off =700eV and the supercell lattice L=10 Å. The initial fractional coordinates of H are chosen as (0.1, 0.05, 0) and (-0.1, 0.05, 0). From the calculation results shown in table 2, the k point grid can be chosen as 1*1*1. It is easy to understand because the supercell is too large and we only consider the isolated molecule instead of the periodic system.

Table 2 The calculation results of the k point grid.

5. The supercell length L

Since we want to get the optimized geometry for an isolated molecule, the supercell lattice L need to be large enough to separate neighbouring molecules. As shown in the Table 3, the supercell length is varied, and the final supercell lengths, final bond length, bond angle are also changed. When initial supercell length is bigger than 10 Å, the final supercell lengths, final bond length and bond angle will become stable, which means the molecules are separated and they cannot feel the forces from neighbouring molecules. So we can choose the supercell length as 10 Å.

Table 3 The calculation results of the supercell length.

6. The initial hydrogen atom positions

The initial and final bond angle and bond length are changed as shown in Table     . As seen in table if we choose the initial bond angle equals 180°, the final bond angle will also be 180° . Because in this situation, the force along atoms direction is already balanced so the angle will not change. But if the initial bond angle is not 180°, the final bond length and angle are always 0.970 Å and around 104.5°.

Table 4 The calculation results for different initial hydrogen atom positions.

7. Conclusion

As shown in Figure 2, under the condition that the energy cut off is 700eV, the k points grid is 1*1*1, the supercell lattice length is 10Å, the initial bond length is 1.118 Å and the initial bond angle is 126.9°, the final results for bond length and angle are 0.970 Å and  104.582°. The experimental results for water molecule are: bond length is 0.958 Å bond angle is 104.45°. So the error for the bond length is about 1.2% and the error for the bond angle is about 0.1%.

Figure 2 The final water molecule structure for initial bond length is 1.118 Å and initial bond angle is 126.9°.

8. Discussion and future works

Although the results are very accurate with small errors, there are still some ways to further increase the accuracy. For example, during the calculation of geometry optimization, the parameters of convergence tolerance, such as Max. force, Max displacement, can be changed to increase the accuracy.

For some future works: Hydrogen bonding exists between water molecules, which will affect the water molecule structure, so it is worthwhile to try to calculate the Hydrogen bonding. Also since the water molecule has the dipole momentum, we can also calculate the dipole momentum of water molecules and compare it with the experimental results.

Reference

[1] S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson, M. C. Payne, “First principles methods using CASTEP”, Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005)

[2] The water molecule structure. https://www.worldofmolecules.com/solvents/water.htm