In this project, the optimal geometry of individual water molecules was found using CASTEP geometry optimization calculations [1]. These calculations utilized the GGA Perdew Burke Ernzerhof (PBE) functional and the Koellig-Harmon atomic solver, which was used to find  psuedopotentials for the hydrogen 1s1 electrons and on the oxygen 1s2 2s2 2p4 electrons. In these calculations, the H atoms had the 1s1 electrons used as valence electrons and the 2s2 2p4 electrons of the O atoms were used as the valence electrons. Additionally, the geometry optimization calculations were performed with a 0.01 eV/Å force tolerance.

Fig. 1. Water Molecule

Determination of Lattice Size

Because CASTEP relies on plane waves, one must construct a lattice to find a molecule’s geometry. Additionally, this lattice must be large enough that the neighboring molecules do not interact in a way that would significantly alter their geometry.  Thus, in performing these calculations, one must ensure that their chosen lattice constant is sufficiently large that the molecule geometry converges past the desired tolerance. In this case, the atomic distances converged to 0.01Å with a 9Å lattice constant, which was used for the primary calculation.

(a)

(b)

Fig. 2. Graph of atomic distances vs. Lattice Constant for (a) OH (b) HH.

Table 1. Effects of lattice constant on bond lengths and angles.

Determination of Energy Cutoff

In order to find the desired energy cutoff for our calculations, the energy cutoff was varied until the distances between atoms converged to 0.01Å. A 435eV cutoff energy was found to be sufficient, and a cutoff of 630eV was used for the main calculation.

(a)

(b)

Fig. 3. Graph of atomic distances vs. cutoff for (a) OH (b) HH.

Table 2. Effects of energy cutoff on atomic distances

Determination of k-space Sampling

To ensure that our k-space sampling was large enough, we checked that the atomic distances converged to 0.01Å at the desired sampling. A sampling of 2x2x2 points, which, after symmetry considerations, resulted in a 4 point total sampling, was found to be sufficient.

(a)

(b)

Fig. 4. Graph of atomic distances vs. cutoff for (a) OH (b) HH.

Table 3. Comparison of total k-space sampling and atomic distances. A single k point was found to be sufficient, and a 4 point (2x2x2) sampling was used in the primary calculation.

Symmetry Considerations

When performing geometry optimization calculations, one must carefully consider the symmetry of the system – in the event that the molecule contains certain symmetries, the forces between atoms may cancel and result in a calculation-breaking local minimum. In this case, when the calculation is performed with the hydrogen atoms perfectly symmetric about an axis (see fig. 5), the calculation fails; the geometry remains symmetric, with OH bond lengths of 0.943Å. However, by starting from slightly modified, symmetry breaking starting positions, we can find values that properly converge.

Fig. 5. A crystal of symmetric H2O molecules with a 9Å lattice spacing. The symmetry about the central oxygen atom causes the calculation to converge to the wrong value.

Conclusion

Utimately, these calculations found that H2O has an OH bond length of 0.97Å with a bond angle of 104.2 degrees. These closely match previously found experimental data  of 0.9572Å and 104.52 degrees [2].

References