VO Cluster-Stabilized H2O Adsorption on a TiO2 (110) Surface at Room Temperature.

We probe the adsorption of molecular H2O on a TiO2 (110)-(1 × 1) surface decorated with isolated VO clusters using ultrahigh-vacuum scanning tunneling microscopy (UHV-STM) and temperature-programmed desorption (TPD). Our STM images show that preadsorbed VO clusters on the TiO2 (110)-(1 × 1) surface induce the adsorption of H2O molecules at room temperature (RT). The adsorbed H2O molecules form strings of beads of H2O dimers bound to the 5-fold coordinated Ti atom (5c-Ti) rows and are anchored by VO. This RT adsorption is completely reversible and is unique to the VO-decorated TiO2 surface. TPD spectra reveal two new desorption states for VO stabilized H2O at 395 and 445 K, which is in sharp contrast to the desorption of water due to recombination of hydroxyl groups at 490 K from clean TiO2(110)-(1 × 1) surfaces. Density functional theory (DFT) calculations show that the binding energy of molecular H2O to the VO clusters on the TiO2 (110)-(1 × 1) surface is higher than binding to the bare surface by 0.42 eV, and the resulting H2O-VO-TiO2 (110) complex provides the anchor point for adsorption of the string of beads of H2O dimers.

Nascent Decomposition Pathways of CH4 Pyrolysis in Gas-Phase Metal Halides.

We have performed a combined quantum mechanical and microkinetic modeling study to understand the nascent decomposition pathways of methane pyrolysis, catalyzed by gas-phase ZnCl2, in a constant pressure batch reactor at 1273 K. We find that ZnCl2 catalyzes methane pyrolysis with an apparent activation energy of 227 kJ/mol. We have also performed sensitivity analysis on a reaction network comprising initiation, termination, and primary propagation reactions. The results suggest that the whole reaction network can be simplified to four reactions, which contributes to the initial rate of methane decomposition. Based on these insights, we have also explored the catalyzing effects of gas-phase AlCl3, CoCl2, CuCl2, FeCl2, and NiCl2 for methane decomposition. Our calculations suggest that gas-phase CuCl2 and NiCl2 are the most active catalysts among the metal halides studied in this work.

Rates of adsorption and desorption: Entropic contributions and errors due to mean-field approximations.

We have performed exact classical rate calculations to compute adsorption and desorption rate constants with a model representative of a real system. We compute the desorption rate using transition-state theory by taking the dividing-surface far from the surface of the solid. We find that using a mean-field assumption, i.e., applying potential of mean force to transition state theory, could lead to two orders-of-magnitude errors in the rate constant owing to large fluctuations in the desorption barrier. Furthermore, we compute the adsorption rate by including a dynamical factor which reflects the probability of sticking to the solid surface. We find that the sticking probability is highly sensitive to the coverage. Also, we find that the adsorption rate computed from the mean-field assumption is not very different from the exact adsorption rate. We also compute entropic contribution to desorption rates and compare it to that obtained from two limiting models of adsorption-2D ideal gas and 2D ideal lattice gas. We show that at high temperatures (700 K), the entropic contribution to desorption rates computed from the exact calculations is very close to that obtained from the 2D ideal gas model. However, for lower to intermediate temperatures from 200 K to 500 K, the entropic contributions cover a wide range which lies in between the two limiting models and could lead to over two-orders-of-magnitude errors in the rate coefficient.

Catalytic molten metals for the direct conversion of methane to hydrogen and separable carbon.

Metals that are active catalysts for methane (Ni, Pt, Pd), when dissolved in inactive low-melting temperature metals (In, Ga, Sn, Pb), produce stable molten metal alloy catalysts for pyrolysis of methane into hydrogen and carbon. All solid catalysts previously used for this reaction have been deactivated by carbon deposition. In the molten alloy system, the insoluble carbon floats to the surface where it can be skimmed off. A 27% Ni-73% Bi alloy achieved 95% methane conversion at 1065°C in a 1.1-meter bubble column and produced pure hydrogen without CO2 or other by-products. Calculations show that the active metals in the molten alloys are atomically dispersed and negatively charged. There is a correlation between the amount of charge on the atoms and their catalytic activity.

Catechol and HCl Adsorption on TiO2(110) in Vacuum and at the Water-TiO2 Interface.

Coadsorbed water is often unavoidable in electrochemistry and low-temperature catalysis. In addition, water influences the adsorption of biomolecules on surfaces. We use ab initio DFT molecular dynamics and ground-state calculations to study the adsorption of HCl and catechol on the rutile TiO2(110) surface and at a water-rutile interface. We find that a coadsorbed water film reduces the adsorption energy of both catechol and HCl significantly because water molecules must be displaced from the surface before catechol or HCl can adsorb. The adsorption energy of catechol (or HCl) at the water-rutile interface can be estimated as the adsorption energy in vacuum minus the energy to remove two water molecules (respectively, one water molecule) from the rutile surface in vacuum and place them in liquid water. This estimate predicts the effect of a surface water film on adsorption without the need of molecular dynamics.

Investigation of the active sites of rhodium sulfide for hydrogen evolution/oxidation using carbon monoxide as a probe.

Carbon monoxide (CO) was observed to decrease the activity for hydrogen evolution, hydrogen oxidation, and H2-D2 exchange on rhodium sulfide, platinum, and rhodium metal. The temperature at which the CO was desorbed from the catalyst surface (detected by recovery in the H2-D2 exchange activity of the catalyst) was used as a descriptor for the CO binding energy to the active site. The differences in the CO desorption temperature between the different catalysts showed that the rhodium sulfide active site is not metallic rhodium. Using density functional theory, the binding energy of CO to the Rh sites in rhodium sulfide is found comparable to the binding energy on Pt. Coupled with experiment this supports the proposition that rhodium rather than sulfur atoms in the rhodium sulfide are the active site for the hydrogen reaction. This would indicate the active sites for hydrogen evolution/oxidation as well as oxygen reduction (determined by other groups using X-ray absorption spectroscopy) may be the same.

Catalytic oxidation of methanol to formaldehyde by mass-selected vanadium oxide clusters supported on a TiO2(110) surface.

We report the results of a systematic study of the catalytic activity of mass-selected vanadium oxide clusters deposited on rutile TiO2 surfaces under ultrahigh vacuum (UHV) conditions. Our results show that supported V, VO, and VO2 clusters are not catalytically active for the oxidative dehydrogenation of methanol to formaldehyde but can be made catalytically active by postoxidation. In addition, we found that the postoxidized VO/TiO2 produces the most formaldehyde. Scanning tunneling microscopy (STM) imaging of the postoxidized VO/TiO2 reveals isolated clusters with height and width indicative of VO3 bound to the TiO2 surface. Our results are consistent with previous density functional theory (DFT) calculations that predict that VO3 will be produced by postoxidation of VO and that VO3/TiO2 is an active catalyst.

Transition metal sulfide hydrogen evolution catalysts for hydrobromic acid electrolysis.

Mixed metal sulfides containing combinations of W, Fe, Mo, Ni, and Ru were synthesized and screened for activity and stability for the hydrogen evolution reaction (HER) in aqueous hydrobromic acid (HBr). Co- and Ni-substituted RuS(2) were identified as potentially active HER electrocatalysts by high-throughput screening (HTS), and the specific compositions Co(0.4)Ru(0.6)S(2) and Ni(0.6)Ru(0.4)S(2) were identified by optimization. Hydrogen evolution activity of Co(0.4)Ru(0.6)S(2) in HBr is greater than RuS(2) or CoS(2) and comparable to Pt and commercial Rh(x)S(y). Structural and morphological characterizations of the Co-substituted RuS(2) suggest that the nanoparticulate solids are a homogeneous solid solution with a pyrite crystal structure. No phase separation is detected for Co substitutions below 30% by X-ray diffraction. In 0.5 M HBr electrolyte, the Co-Ru electrode material synthesized with 30% Co rapidly lost approximately 34% of the initial loading of Co; thereafter, it was observed to exhibit stable activity for HER with no further loss of Co. Density functional theory calculations indicate that the S(2)(2-) sites are the most important for HER and the presence of Co influences the S(2)(2-) sites such that the hydrogen binding energy at sufficiently high hydrogen coverage is decreased compared to ruthenium sulfide. Although showing high HER activity in a flow cell, the reverse reaction of hydrogen oxidation is slow on the RuS(2) catalysts tested when compared to platinum and rhodium sulfide, leaving rhodium sulfide as the only suitable tested material for a regenerative HBr cell due its stability compared to platinum.

The properties of small Ag clusters bound to DNA bases.

We study the binding of neutral silver clusters, Ag(n) (n=1-6), to the DNA bases adenine (A), cytosine (C), guanine (G), and thymine (T) and the absorption spectra of the silver cluster-base complexes. Using density functional theory (DFT), we find that the clusters prefer to bind to the doubly bonded ring nitrogens and that binding to T is generally much weaker than to C, G, and A. Ag(3) and Ag(4) make the stronger bonds. Bader charge analysis indicates a mild electron transfer from the base to the clusters for all bases, except T. The donor bases (C, G, and A) bind to the sites on the cluster where the lowest unoccupied molecular orbital has a pronounced protrusion. The site where cluster binds to the base is controlled by the shape of the higher occupied states of the base. Time-dependent DFT calculations show that different base-cluster isomers may have very different absorption spectra. In particular, we find new excitations in base-cluster molecules, at energies well below those of the isolated components, and with strengths that depend strongly on the orientations of planar clusters with respect to the base planes. Our results suggest that geometric constraints on binding, imposed by designed DNA structures, may be a feasible route to engineering the selection of specific cluster-base assemblies.

O2 evolution on a clean partially reduced rutile TiO2(110) surface and on the same surface precovered with Au1 and Au2: the importance of spin conservation.

We have used spin-polarized density functional theory (DFT) to study O(2) evolution on a clean partially reduced rutile TiO(2)(110) surface (i.e., a surface having oxygen vacancies) and its interaction with Au(1) or Au(2) cluster adsorbed on it. We assume that the total spin of the electronic wave function is related to the number of unpaired spins (N(s)) and calculate the binding and the activation energies involved in O(2) evolution for fixed values of N(s). In addition to keeping N(s) constant, we assume that reactions in which the N(s) of the reactants differs from that of the products are very slow. The potential energy surfaces obtained in this way depend strongly on N(s). For example, O(2) dissociation at the vacancy site on a clean partially reduced TiO(2)(110) surface is exothermic by 0.85 eV in the triplet state and the highest activation energy in the chain of reactions leading to the O(2) dissociation is 0.67 eV. In the singlet state, O(2) dissociation is endothermic by 0.11 eV and the activation energy leading to dissociation is 1.30 eV. These observations are in qualitative agreement with scanning tunneling microscopy experiment in which O(2) dissociation on a partially reduced rutile TiO(2)(110) surface is observed at temperature as low as 120 K. In contrast, O(2) dissociation is predicted to be endothermic and is prevented by an activation barrier larger than 1 eV in all the previous DFT calculations, in which the DFT program varies N(s) to get the lowest energy state. We find that on a partially reduced rutile TiO(2)(110) with Au(1) and Au(2) preadsorbed on its surface, O(2) dissociates at the vacancy site: One oxygen atom fills the oxygen vacancy and the other becomes available for oxidation chemistry. This means that Au(1) and Au(2) supported on a partially reduced TiO(2)(110) surface is not an oxidation catalyst since the presence of oxygen turns it into a stoichiometric Au(n)/TiO(2)(110) surface. Finally, we find that the evolution of oxygen on Au(1) and Au(2) in the gas phase is very different from the evolution on the same clusters supported on the partially reduced TiO(2)(110) surface. For example, the molecular adsorption of O(2) is favored in the gas phase (except on Au(1) (-) and Au(2) (-) in the quartet state), while the dissociative adsorption is favored by more than 1 eV when Au(1) and Au(2) are supported on the partially reduced TiO(2)(110). Furthermore, the activation energies associated with O(2) dissociation in the gas phase (DeltaE(act)>2.4 eV) are reduced by at least a factor of 2 when the clusters are supported on TiO(2)(110).

Enhanced adsorption energy of Au1 and O2 on the stoichiometric TiO2(110) surface by coadsorption with other molecules.

During heterogeneous catalysis the surface is simultaneously covered by several adsorbed molecules. The manner in which the presence of one kind of molecule affects the adsorption of a molecule of another kind has been of interest for a long time. In most cases the presence of one adsorbate does not change substantially the binding energy of another adsorbate. The calculations presented here show that the stoichiometric rutile TiO(2)(110) surface, on which one of the compounds -OH, Au(3), Au(5), Au(7), Na, K, or Cs or two different gold strips was preadsorbed, behaves differently: the binding energy of Au(1) or O(2) to such a surface is much stronger than the binding to the clean stoichiometric TiO(2)(110) surface. Moreover, the binding energy of Au(1) or O(2) and the amount of charge they take from the surface when they adsorb are the same, regardless of which of the above species is preadsorbed. The preadsorbed species donate electrons to the conduction band of the oxide, and these electrons are used by Au(1) or O(2) to make stronger bonds with the surface. This suggests that adding an electron to the conduction band of the clean stoichiometric TiO(2)(110) slab used in the calculation will affect similarly the adsorption energy of Au(1) or O(2). Our calculations show that it does. We have also studied how the preadsorption of Au(4) or Au(6) affects the binding of Au(1) or O(2) to the surface. These two gold clusters do not donate electrons to the surface when they bind to it and therefore should not influence substantially the binding energy of Au(1) or O(2) to the surface. However, adsorbing O(2) or Au(1) on the surface forces the clusters to change their structure into that of isomers that donate charge to the oxide. This charge is used by Au(1) or O(2) to bind to the surface and the energy of this bond exceeds the isomerization energy. As a result the surface with the isomerized cluster is the lowest energy state of the system. We believe that these results can be generalized as follows. The molecules that we coadsorbed with Au(1) or O(2) donate electrons to the oxide and are Lewis bases. By giving the surface high energy electrons, they turn it into a Lewis base and this increases its ability to bind strong Lewis acids such as Au(1) and O(2). We speculate that this kind of interaction is general and may be observed for other oxides and for other coadsorbed Lewis base-Lewis acid pairs.

Inelastic scattering with Chebyshev polynomials and preconditioned conjugate gradient minimization.

We describe and test an implementation, using a basis set of Chebyshev polynomials, of a variational method for solving scattering problems in quantum mechanics. This minimum error method (MEM) determines the wave function Psi by minimizing the least-squares error in the function (H Psi - E Psi), where E is the desired scattering energy. We compare the MEM to an alternative, the Kohn variational principle (KVP), by solving the Secrest-Johnson model of two-dimensional inelastic scattering, which has been studied previously using the KVP and for which other numerical solutions are available. We use a conjugate gradient (CG) method to minimize the error, and by preconditioning the CG search, we are able to greatly reduce the number of iterations necessary; the method is thus faster and more stable than a matrix inversion, as is required in the KVP. Also, we avoid errors due to scattering off of the boundaries, which presents substantial problems for other methods, by matching the wave function in the interaction region to the correct asymptotic states at the specified energy; the use of Chebyshev polynomials allows this boundary condition to be implemented accurately. The use of Chebyshev polynomials allows for a rapid and accurate evaluation of the kinetic energy. This basis set is as efficient as plane waves but does not impose an artificial periodicity on the system. There are problems in surface science and molecular electronics which cannot be solved if periodicity is imposed, and the Chebyshev basis set is a good alternative in such situations.

Density functional study of the interaction between small Au clusters, Au(n) (n=1-7) and the rutile TiO(2) surface. I. Adsorption on the stoichiometric surface.

This is the first paper in a series of four dealing with the adsorption site, electronic structure, and chemistry of small Au clusters, Au(n) (n=1-7), supported on stoichiometric, partially reduced, or partially hydroxylated rutile TiO(2)(110) surfaces. Analysis of the electronic structure reveals that the main contribution to the binding energy is the overlap between the highest occupied molecular orbitals of Au clusters and the Kohn-Sham orbitals localized on the bridging and the in-plane oxygen of the rutile TiO(2)(110) surface. The structure of adsorbed Au(n) differs from that in the gas phase mostly because the cluster wants to maximize this orbital overlap and to increase the number of Au-O bonds. For example, the equilibrium structures of Au(5) and Au(7) are planar in the gas phase, while the adsorbed Au(5) has a distorted two-dimensional structure and the adsorbed Au(7) is three-dimensional. The dissociation of an adsorbed cluster into two adsorbed fragments is endothermic, for all clusters, by at least 0.8 eV. This does not mean that the gas-phase clusters hitting the surface with kinetic energy greater than 0.8 eV will fragment. To place enough energy in the reaction coordinate for fragmentation, the impact kinetic energy needs to be substantially higher than 0.8 eV. We have also calculated the interaction energy between all pairs of Au clusters. These interactions are small except when a Au monomer is coadsorbed with a Au(n) with odd n. In this case the interaction energy is of the order of 0.7 eV and the two clusters interact through the support even when they are fairly far apart. This happens because the adsorption of a Au(n) cluster places electrons in the states of the bottom of the conduction band and these electrons help the Au monomer to bind to the five-coordinated Ti atoms on the surface.

Does phenomenological kinetics provide an adequate description of heterogeneous catalytic reactions?

Phenomenological kinetics (PK) is widely used in the study of the reaction rates in heterogeneous catalysis, and it is an important aid in reactor design. PK makes simplifying assumptions: It neglects the role of fluctuations, assumes that there is no correlation between the locations of the reactants on the surface, and considers the reacting mixture to be an ideal solution. In this article we test to what extent these assumptions damage the theory. In practice the PK rate equations are used by adjusting the rate constants to fit the results of the experiments. However, there are numerous examples where a mechanism fitted the data and was shown later to be erroneous or where two mutually exclusive mechanisms fitted well the same set of data. Because of this, we compare the PK equations to "computer experiments" that use kinetic Monte Carlo (kMC) simulations. Unlike in real experiments, in kMC the structure of the surface, the reaction mechanism, and the rate constants are known. Therefore, any discrepancy between PK and kMC must be attributed to an intrinsic failure of PK. We find that the results obtained by solving the PK equations and those obtained from kMC, while using the same rate constants and the same reactions, do not agree. Moreover, when we vary the rate constants in the PK model to fit the turnover frequencies produced by kMC, we find that the fit is not adequate and that the rate constants that give the best fit are very different from the rate constants used in kMC. The discrepancy between PK and kMC for the model of CO oxidation used here is surprising since the kMC model contains no lateral interactions that would make the coverage of the reactants spatially inhomogeneous. Nevertheless, such inhomogeneities are created by the interplay between the rate of adsorption, of desorption, and of vacancy creation by the chemical reactions.

Density functional study of the charge on Aun clusters (n=1-7) supported on a partially reduced rutile TiO2(110): are all clusters negatively charged?

It is widely believed that small gold clusters supported on an oxide surface and adsorbed at the site of an oxygen vacancy are negatively charged. It has been suggested that this negative charge helps a gold cluster adsorb oxygen and weakens the O-O bond to make oxidation reactions more efficient. Given the fact that an oxygen vacancy is electron rich and that Au is a very electronegative element, the assumption that the Au cluster will take electron density from the vacancy is plausible. However, the density functional calculations presented here show that the situation is more complicated. The authors have used the Bader method to examine the charge redistribution when a Aun cluster (n=1-7) binds next to or at an oxygen vacancy on rutile TiO2(110). For the lowest energy isomers they find that Au1 and Au3 are negatively charged, Au5 and Au7 are positively charged, and Au2, Au4, and Au6 exchange practically no charge. The behavior of the Aun isomers having the second-lowest energy is also unexpected. Au2, Au3, Au5, and Au7 are negatively charged upon adsorption and very little charge is transferred when Au4 and Au6 are adsorbed. These observations can be explained in terms of the overlap between the frontier molecular orbitals of the gold cluster and the eigenstates of the support. Aun with even n becomes negatively charged when the lowest unoccupied molecular orbital has a lobe pointing in the direction of the oxygen vacancy or towards a fivefold coordinated Ti (5c-Ti) located in the surface layer; otherwise it stays neutral. Aun with odd n becomes negatively charged when the singly occupied molecular orbital has a lobe pointing in the direction of a 5c-Ti located at the vacancy site or in the surface layer, otherwise it donates electron density into the conduction band of rutile TiO2(110) becoming positively charged.

Dynamics of H2O and Na+ in nafion membranes.

We investigate the transport properties of a model of a hydrated Na-Nafion membrane using molecular dynamics simulations. The system consists of several Nafion chains forming a pore with the water and ions inside. At low water content, the hydrophilic domain is not continuous and diffusion is very slow. The diffusion coefficient of both water and Na+ increases with increasing hydration (more strongly so for Na+). The simulations are in qualitative agreement with experimental results for similar systems. The diffusion coefficient is an average over the motion of ions or water molecules located in different environments. To better understand the role of the environment, we calculate the distribution of the residence times of the ion (or water) at different locations in the system. We discuss the transport mechanism in light of this information.

Nanoscale current imaging of the conducting channels in proton exchange membrane fuel cells.

The electrochemically active area of a proton exchange membrane fuel cell (PEMFC) is investigated using conductive probe atomic force microscopy (CP-AFM). A platinum-coated AFM tip is used as a nanoscale cathode in an operating PEMFC. We present results that show highly inhomogeneous distributions of conductive surface domains at several length scales. At length scales on the order of the aqueous domains of the membrane, approximately 50 nm, we observe single channel electrochemistry. I-V curves for single conducting channels are obtained, which yield insight into the nature of conductive regions across the PEM. In addition, we demonstrate a new characterization technique, phase current correlation microscopy, which gives a direct measure of the electrochemical activity for each aqueous domain. This shows that a large number ( approximately 60%) of the aqueous domains present at the surface of an operating Nafion membrane are inactive. We attribute this to a combination of limited aqueous domain connectivity and catalyst accessibility.

Density functional study of the interaction between small Au clusters, Au(n) (n=1-7) and the rutile TiO2 surface. II. Adsorption on a partially reduced surface.

We use density functional theory to examine the electronic structure of small Au(n) (n=1-7) clusters, supported on a rutile TiO(2)(110) surface having oxygen vacancies on the surface (a partially reduced surface). Except for the monomer, the binding energy of all Au clusters to the partially reduced surface is larger by approximately 0.25 eV than the binding energy to a stoichiometric surface. The bonding site and the orientation of the cluster are controlled by the shape of the highest occupied molecular orbitals (HOMOs) of the free cluster (free cluster means a gas-phase cluster with the same geometry as the supported one). The bond is strong when the lobes of the HOMOs overlap with those of the high-energy states of the clean oxide surface (i.e., with no gold) that have lobes on the bridging and the in-plane oxygen atoms. In other words, the cluster takes a shape and a location that optimizes the contact of its HOMOs with the oxygen atoms. Fivefold coordinated Ti atoms located at a defect site (5c-Ti(*)) participate in the binding only when a protruding lobe of the singly occupied molecular orbital (for odd n) or the lowest unoccupied molecular orbital (for even n) of the free Au(n) cluster points toward a 5c-Ti(*) atom. The oxygen vacancy influences the binding energy of the clusters (except for Au(1)) only when they are in direct contact with the defect. The desorption energy and the total charge on clusters that are close to, but do not overlap with, the vacancy differ little from the values they have when the cluster is adsorbed on a stoichiometric surface. The behavior of Au(1) is rather remarkable. The atom prefers to bind directly to the vacancy site with a binding energy of 1.81 eV. However, it also makes a strong bond (1.21 eV) with any 5c-Ti atom even if that atom is far from the vacancy site. In contrast, the binding of a Au monomer to the 5c-Ti atom of a surface without vacancies is weak (0.45 eV). The presence of the vacancy activates the 5c-Ti atoms by populating states at the bottom of the conduction band. These states are delocalized and have lobes protruding out of the surface at the location of the 5c-Ti atoms. It is the overlap of these lobes with the highest orbital of the Au atom that is the major reason for the bonding to the 5c-Ti atom, no matter how far the latter is from the vacancy. The energy for breaking an adsorbed cluster into two adsorbed fragments is smaller than the kinetic energy of the mass-selected clusters deposited on the surface in experiments. However, this is not sufficient for breaking the cluster upon impact with the surface, since only a fraction of the available energy will go into the reaction coordinate for breakup.

Minimum-error method for scattering problems in quantum mechanics: Two stable and efficient implementations.

We examine here, by using a simple example, two implementations of the minimum error method (MEM), a least-squares minimization for scattering problems in quantum mechanics, and show that they provide an efficient, numerically stable alternative to Kohn variational principle. MEM defines an error-functional consisting of the sum of the values of (HPsi - EPsi)2 at a set of grid points. The wave function Psi, is forced to satisfy the scattering boundary conditions and is determined by minimizing the least-squares error. We study two implementations of this idea. In one, we represent the wave function as a linear combination of Chebyshev polynomials and minimize the error by varying the coefficients of the expansion and the R-matrix (present in the asymptotic form of Psi). This leads to a linear equation for the coefficients and the R-matrix, which we solve by matrix inversion. In the other implementation, we use a conjugate-gradient procedure to minimize the error with respect to the values of Psi at the grid points and the R-matrix. The use of the Chebyshev polynomials allows an efficient and accurate calculation of the derivative of the wave function, by using Fast Chebyshev Transforms. We find that, unlike KVP, MEM is numerically stable when we use the R-matrix asymptotic condition and gives accurate wave functions in the interaction region.