*Ab initio*statistical mechanics of surface adsorption and desorption. I. on MgO (001) at low coverage

^{1}and M. J. Gillan

^{2}

### Abstract

We present a general computational scheme based on molecular dynamics (MD) simulation for calculating the chemical potential of adsorbed molecules in thermal equilibrium on the surface of a material. The scheme is based on the calculation of the mean force in MD simulations in which the height of a chosen molecule above the surface is constrained and subsequent integration of the mean force to obtain the potential of mean force and hence the chemical potential. The scheme is valid at any coverage and temperature, so that in principle it allows the calculation of the chemical potential as a function of coverage and temperature. It avoids all statistical mechanical approximations, except for the use of classical statistical mechanics for the nuclei, and assumes nothing in advance about the adsorption sites. From the chemical potential, the absolute desorption rate of the molecules can be computed, provided that the equilibration rate on the surface is faster than the desorption rate. We apply the theory by *ab initio*MD simulation to the case of on MgO (001) in the low-coverage limit, using the Perdew-Burke-Ernzerhof (PBE) form of exchange correlation. The calculations yield an *ab initio* value of the Polanyi-Wigner frequency prefactor, which is more than two orders of magnitude greater than the value of often assumed in the past. Provisional comparison with experiment suggests that the PBE adsorption energy may be too low, but the extension of the calculations to higher coverages is needed before firm conclusions can be drawn. The possibility of including quantum nuclear effects by using path-integral simulations is noted.

The work was supported by an allocation of time on the HPCx service provided by EPSRC through the Materials Chemistry Consortium and by resources provided by UCL Research Computing. The work was conducted as part of a EURYI scheme award as provided by EPSRC (see www.esf.org/euryi).

I. INTRODUCTION

II. THEORY AND TECHNIQUES

A. Calculation of the chemical potential using PMF

B. Thermal desorption rate

C. Practical calculation of PMF

III. SIMULATIONS: ON MgO (001) AT LOW COVERAGE

A. Preliminary tests

B. Results for PMF, chemical potential, and desorption rate

C. Spatial distribution and memory

IV. DISCUSSION

V. CONCLUSIONS

### Key Topics

- Desorption
- 47.0
- Adsorption
- 29.0
- Chemical potential
- 23.0
- Adsorbates
- 22.0
- Density functional theory
- 19.0

## Figures

Top and side views of the “tilted” [panels (a) and (b)] and “flat” [panels (c) and (d)] configurations of the .

Top and side views of the “tilted” [panels (a) and (b)] and “flat” [panels (c) and (d)] configurations of the .

The mean force on the water O atom as function of its height above the surface at , 300, 600, and (solid, dotted, chain, and dashed curves, respectively, are guides to the eyes). Bars on data points show statistical errors. Height is relative to a fixed atom in the center of the slab.

The mean force on the water O atom as function of its height above the surface at , 300, 600, and (solid, dotted, chain, and dashed curves, respectively, are guides to the eyes). Bars on data points show statistical errors. Height is relative to a fixed atom in the center of the slab.

Potential of mean force of the water O atom as function of its height above the surface at , 300, 600, and . Symbols and curves have same meaning as in Fig. 2.

Potential of mean force of the water O atom as function of its height above the surface at , 300, 600, and . Symbols and curves have same meaning as in Fig. 2.

Set of 12 trajectories from simulations used to determine the sticking coefficient . Plots show coordinate (Å units) of water O atom relative to the center of the vacuum gap between slabs.

Set of 12 trajectories from simulations used to determine the sticking coefficient . Plots show coordinate (Å units) of water O atom relative to the center of the vacuum gap between slabs.

Arrhenius plot of the desorption rate of from MgO (001) calculated using the PBE exchange-correlation functional. Bars on calculated values show statistical errors. The straight line is drawn to pass through the calculated values at the two lowest temperatures.

Arrhenius plot of the desorption rate of from MgO (001) calculated using the PBE exchange-correlation functional. Bars on calculated values show statistical errors. The straight line is drawn to pass through the calculated values at the two lowest temperatures.

Contour plot of the spatial probability distribution of the water O atom in the plane at . Bottom right and top left corners of plot are Mg sites; bottom left and top right corners are O sites. Probability density is in arbitary units, with equal spacing between contours.

Contour plot of the spatial probability distribution of the water O atom in the plane at . Bottom right and top left corners of plot are Mg sites; bottom left and top right corners are O sites. Probability density is in arbitary units, with equal spacing between contours.

Contour plot of the probability distribution of angles and specifying the orientation of the bisector of molecule (see text) at . Peaks of the distribution correspond to the four equivalent orientations in which the bisector is nearly parallel to the surface and points along one of the diagonal directions (, , , and ).

Contour plot of the probability distribution of angles and specifying the orientation of the bisector of molecule (see text) at . Peaks of the distribution correspond to the four equivalent orientations in which the bisector is nearly parallel to the surface and points along one of the diagonal directions (, , , and ).

Contour plot of the probability distribution of angle specifying the rotation of molecule about its bisector (see text) at . When the molecular bisector is parallel to the surface, the molecular plane is parallel to the surface when .

Contour plot of the probability distribution of angle specifying the rotation of molecule about its bisector (see text) at . When the molecular bisector is parallel to the surface, the molecular plane is parallel to the surface when .

Time variation of the and coordinates of the water O atom in the course of a MD simulation at . Horizontal lines mark and coordinates of perfect-lattice sites, so that spacing between neighboring lines is .

Time variation of the and coordinates of the water O atom in the course of a MD simulation at . Horizontal lines mark and coordinates of perfect-lattice sites, so that spacing between neighboring lines is .

Time variation of the angles (top), (middle), and (bottom) specifying the orientation of the molecule (see text) during the course of a MD simulation at .

Time variation of the angles (top), (middle), and (bottom) specifying the orientation of the molecule (see text) during the course of a MD simulation at .

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