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A practical method for finding free energy barriers for transitions in high-dimensional classical and quantum systems is presented and used to calculate the dissociative sticking probability of H 2 on a metal surface within transition state theory. The reversible work involved in shifting the system confined to a hyperplane from the reactant region towards products is evaluated directly. Quantum mechanical degrees of freedom are included by using Feynman Path Integrals with the hyperplane constraint applied to the centroid of the cyclic paths. An optimal dividing surface for the rate estimated by transition state theory is identified naturally in the course of the reversible work evaluation. The free energy barrier is determined relative to the reactant state directly so that an estimate of the transition rate can be obtained without requiring a solvable reference model for the transition state. The method has been applied to calculations of the sticking probability of a thermalized hydrogen gas on a Cu(110) surface. The two hydrogen atoms and eight surface Cu atoms were included quantum mechanically and over two hundred atoms in the Cu crystal where included classically. The activation energy for adsorption and desorption was determined and found to be significantly lowered by tunneling at low temperature. The calculated values agree quite well with experimental estimates for adsorption and desorption. Dynamical corrections to the classical transition state theory rate estimate were evaluated and found to be small.

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It has been shown recently that while bulk gold is chemically inert, small Au clusters are catalytically active. The reasons for this activity and its dramatic dependence on cluster size are not understood. We use density functional theory to study O2 binding to Au clusters and to a Au(111) surface modified by adsorption of Au clusters on it. We find that O2 does not bind to a flat face of a planar Au cluster, even though it binds well to its edge. Moreover, O2 binds to Au clusters deposited on a Au(111) surface, even though it does not bind to Au(111). This indicates that a band gap is not an essential factor in binding O2, but surface roughness is. Adding electrons to the surface of a Au(111) slab, on which one has deposited a Au cluster, increases the binding energy of O2. However, adding electrons to a flat Ausurface has no effect on O2binding energy. These observations have a simple explanation: in clusters and in the rough surface, the highest occupied molecular orbital (HOMO) is localized and its charge density sticks out in the vacuum. This facilitates charge transfer into the π * orbital of O2, which induces the molecule to bind to gold. A flat face of a cluster or a flat bulk surface tends to delocalize the HOMO, diminishing the ability of the surface to bind O2. The same statements are true for the LUMO orbital, which is occupied by the additional electron given to the system to charge the system negatively. It has been shown recently that while bulk gold is chemically inert, small Au clusters are catalytically active. The reasons for this activity and its dramatic dependence on cluster size are not understood. We use density functional theory to study O 2 binding to Au clusters and to a Au͑111͒ surface modified by adsorption of Au clusters on it. We find that O 2 does not bind to a flat face of a planar Au cluster, even though it binds well to its edge. Moreover, O 2 binds to Au clusters deposited on a Au͑111͒ surface, even though it does not bind to Au͑111͒. This indicates that a band gap is not an essential factor in binding O 2 , but surface roughness is. Adding electrons to the surface of a Au͑111͒ slab, on which one has deposited a Au cluster, increases the binding energy of O 2 . However, adding electrons to a flat Au surface has no effect on O 2 binding energy. These observations have a simple explanation: in clusters and in the rough surface, the highest occupied molecular orbital ͑HOMO͒ is localized and its charge density sticks out in the vacuum. This facilitates charge transfer into the * orbital of O 2 , which induces the molecule to bind to gold. A flat face of a cluster or a flat bulk surface tends to delocalize the HOMO, diminishing the ability of the surface to bind O 2 . The same statements are true for the LUMO orbital, which is occupied by the additional electron given to the system to charge the system negatively.

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