Wynand S Verwoerd

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Abstract: The stochastic solute dispersion model studied in the previous article, can be applied to more realistic velocity variations by approximating them as piecewise constant. This requires treatment by a boundary value formulation, which raises problems connected with entropy considerations. A method is developed to deal with these by the introduction of a specially designed compensator function into the boundary value probability integral for calculating solute concentration. Applying this even for a single velocity step yields an intractable integration, but a suitable approximation is constructed that allows it to be evaluated in analytical form. The result is that a Gaussian solute plume impinging on a velocity step is transmitted as a modulated and compressed or dilated quasi-Gaussian. Plume dispersion is encapsulated in an enhancement factor F that multiplies the diffusive, linear time, dispersion. F is also time dependent; at the time of step penetration it equals kinematical dilation, but anneals away non-linearly so that a length scale can be established over which downstream effects of a velocity step on the dispersion extends. With multiple steps, the cumulative effect is expressed by a product of enhancement factors, each related to a single step. While kinematical compression and dilation effects cancel out across each stepped fluctuation, stochastic dispersion exceeds diffusive values for any plausible combination of fluctuation amplitude and length. A simple formula is obtained for the effective enhancement factor of a fluctuation in terms of its length and amplitude. Moreover, the algebra leads to the definition of a natural length scale [Lambda] related to the Peclet number of the flow. Algebraic evaluation of the cumulative dispersion enhancement by a sequence of equally spaced identical fluctuations, leads to an expression for dispersivity as a function of the distance traversed by a solute plume. Key features of the model in agreement with observations are that the dispersivity behaves differently for traversal lengths above and below [Lambda], and that above this transition it is proportional to a fractional power of the traversal length. Simple stepwise models do not produce the full extent of the observed exponential increase below the transition, but modifying F by introduction of an adjustable parameter addresses this and leads to good agreement with experimental values over a range of 5 orders of magnitude in both traversal length and dispersivity.

Abstract: BACKGROUND:Compared to more general networks, biochemical networks have some special features: while generally sparse, there are a small number of highly connected metabolite nodes; and metabolite nodes can also be divided into two classes: internal nodes with associated mass balance constraints and external ones without. Based on these features, reclassifying selected internal nodes (separators) to external ones can be used to divide a large complex metabolic network into simpler subnetworks. Selection of separators based on node connectivity is commonly used but affords little detailed control and tends to produce excessive fragmentation.The method proposed here (Netsplitter) allows the user to control separator selection. It combines local connection degree partitioning with global connectivity derived from random walks on the network, to produce a more even distribution of subnetwork sizes. Partitioning is performed progressively and the interactive visual matrix presentation used allows the user considerable control over the process, while incorporating special strategies to maintain the network integrity and minimise the information loss due to partitioning.RESULTS:Partitioning of a genome scale network of 1348 metabolites and 1468 reactions for Arabidopsis thaliana encapsulates 66% of the network into 10 medium sized subnets. Applied to the flavonoid subnetwork extracted in this way, it is shown that Netsplitter separates this naturally into four subnets with recognisable functionality, namely synthesis of lignin precursors, flavonoids, coumarin and benzenoids. A quantitative quality measure called efficacy is constructed and shows that the new method gives improved partitioning for several metabolic networks, including bacterial, plant and mammal species.CONCLUSIONS:For the examples studied the Netsplitter method is a considerable improvement on the performance of connection degree partitioning, giving a better balance of subnet sizes with the removal of fewer mass balance constraints. In addition, the user can interactively control which metabolite nodes are selected for cutting and when to stop further partitioning as the desired granularity has been reached. Finally, the blocking transformation at the heart of the procedure provides a powerful visual display of network structure that may be useful for its exploration independent of whether partitioning is required.

Abstract: BACKGROUND:The numerous diverse metabolic pathways by which plant compounds can be produced make it difficult to predict how colour pigmentation is lost for different tissues and plants. This study employs mathematical and in silico methods to identify correlated gene targets for the loss of colour pigmentation in plants from a whole cell perspective based on the full metabolic network of Arabidopsis. This involves extracting a self-contained flavonoid subnetwork from the AraCyc database and calculating feasible metabolic routes or elementary modes (EMs) for it. Those EMs leading to anthocyanin compounds are taken to constitute the anthocyanin biosynthetic pathway (ABP) and their interplay with the rest of the EMs is used to study the minimal cut sets (MCSs), which are different combinations of reactions to block for eliminating colour pigmentation. By relating the reactions to their corresponding genes, the MCSs are used to explore the phenotypic roles of the ABP genes, their relevance to the ABP and the impact their eliminations would have on other processes in the cell.RESULTS:Simulation and prediction results of the effect of different MCSs for eliminating colour pigmentation correspond with existing experimental observations. Two examples are: i) two MCSs which require the simultaneous suppression of genes DFR and ANS to eliminate colour pigmentation, correspond to observational results of the same genes being co-regulated for eliminating floral pigmentation in Aquilegia and; ii) the impact of another MCS requiring CHS suppression, corresponds to findings where the suppression of the early gene CHS eliminated nearly all flavonoids but did not affect the production of volatile benzenoids responsible for floral scent.CONCLUSIONS:From the various MCSs identified for eliminating colour pigmentation, several correlate to existing experimental observations, indicating that different MCSs are suitable for different plants, different cells, and different conditions and could also be related to regulatory genes. Being able to correlate the predictions with experimental results gives credence to the use of these mathematical and in silico analyses methods in the design of experiments. The methods could be used to prioritize target enzymes for different objectives to achieve desired outcomes, especially for less understood pathways.

Abstract: A new model of solute dispersion in porous media that avoids Fickian assumptions and that can be applied to variable drift velocities as in non-homogeneous or geometrically constricted aquifers, is presented. A key feature is the recognition that because drift velocity acts as a driving coefficient in the kinematical equation that describes random fluid displacements at the pore scale, the use of Ito calculus and related tools from stochastic differential equation theory (SPDE) is required to properly model interaction between pore scale randomness and macroscopic change of the drift velocity. Solute transport is described by formulating in integral version of the solute mass conservation equations, using a probability density. By appropriate linking of this to the related but distinct probability density arising from the kinematical SPDE, it is shown that the evolution of a Gaussian solute plume can be calculated, and in particular its time-dependent variance and hence dispersivity. Exact analytical solutions of the differential and integral equations that this procedure involves, are presented for the case of it constant drift velocity, its well its for it constant velocity gradient. In the former case, diffusive dispersion as familiar from the advection-dispersion equation is recovered. However, in the latter case, it is shown that there are not only reversible kinematical dispersion effects, but also irreversible, intrinsically stochastic contributions ill excess of that predicted by diffusive dispersion. Moreover, this intrinsic contribution has a non-linear time dependence and hence opens up the way for all explanation of the strong observed scale dependence of dispersivity. (C) 2007 Elsevier Inc. All rights reserved.

Abstract: In this note a Fredholm integral equation of the first kind with exponential expressions for the xD;kernel and right hand side is considered. The task of finding a practically usable solution to xD;such an equation may need more effort than following a standard procedure, even when xD;such a procedure yields a formal solution. An apparently elegant solution as an orthogonal xD;polynomial expansion is obtained using the standard method based on transformation to a xD;form where the kernel is an orthogonal polynomial generating function, but this is of limited xD;use due to slow convergence. It is shown that this can nevertheless be transformed into a xD;closed form solution that is computationally efficient

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Abstract: TDPAD spectroscopy measurements have been performed on two fullerene samples, C-60 and C-70, with recoil implanted F-19 ions as probe nuclei. Quadrupole coupling frequencies of nu(Q) = 54(1) MHz for C-60 and nu(Q) = 57(1) MHz for C-70 were observed. These frequencies are associated with a C-F molecular complex formation. The C-70 sample was also examined for possible effects due to variation of temperature. No significant changes were observed. An additional frequency of nu(Q) = 13(1) MHz registered for both samples has to be attributed to the formation of H-F complexes. Finally, both samples were analysed for proton dose dependent effects, due to the (p, p'gamma) implantation process but no effects have shown up. Specifically, computational studies based on quantum chemical methods were performed to assist in the interpretation of the experimental results, using the code MOPAC.

Abstract: An algorithm (ADUM) is developed to decompose an arbitrary N x N unitary matrix M into 1/2N(N-1) simple factor matrices. Each factor matrix has the form of an N x N unit matrix, except for a 2 x 2 complex rotation submatrix located at an appropriate position on the diagonal. The factor matrices each contain a rotation angle and between 0 and 3 phase angles, adding up to a total of N-2 independent real angles. This can be summarized into an N x N real angle matrix Gamma, containing the same information as the unitary matrix M. The factorisation can be extended to Hermitian or even generally complex matrices by applying an eigenvalue expansion or, alternatively, a singular value decomposition. Several applications to physical problems are discussed, and it is shown that ADUM is a powerful tool in the interpolation of matrices which depend on external parameters because it efficiently represents the degrees of freedom of a matrix while guaranteeing that matrix properties are maintained. (C) 1998 Elsevier Science B.V.

Abstract: The ternary system B-C-N has attracted considerable attention in recent years, as a result of potentially interesting properties of the phases in this system. A non-self-consistent, ab initio based tight-binding molecular-dynamics method, has been used to investigate the stability, structural and electronic properties of BC2N-crystals with zinc-blende structure. We investigate cubic crystals with different BN-C, stacking sequences in the [100] direction. The competition between structural stress and chemical bond strength can be seen as crucial for the formation of real ternary crystalline structures or a segregation into two cubic phases: diamond and c-BN. (C) 1998 Elsevier Science S.A. All rights reserved.

Abstract: We have performed a high resolution, high sensitivity study of the electron-positron momentum distribution in both natural and synthetic diamond, using the two-dimensional Doppler broadening technique of positron annihilation spectroscopy. The measurements were carried out with the <100>, <110> and <111> axes each separately aligned collinearly with the detectors so that the anisotropy of the momentum distribution could be mapped out. Small differences in the momentum profiles were amplified with subtraction spectra and compare well with results of a two-component density functional theory calculation of the electron-positron momentum distribution in the diamond lattice. There has been much debate regarding previously unexplained anomalies in the annihilation characteristics of the bulk positron configuration in diamond. These new results strongly support a model describing the bulk positron as delocalised and inhomogeneous, but with sufficient amplitude at the intra-bond region such that the annihilation characteristics are determined by the bond electron density and momentum distribution.

Abstract: We have measured the momentum distribution of the electron-positron annihilating pair in diamond as a function of crystallographic orientation, using the Doppler broadening technique of positron annihilation spectroscopy. A 1% variation in the longitudinal projection of the momentum distribution was observed as the sample was rotated about the [110] axis. This anisotropic distribution shows an excellent agreement with a theoretical calculation performed based on two component density functional theory in the local density approximation. The results suggest strongly that the bulk positron density, though delocalized and inhomogeneous, has sufficient amplitude at the intra-bond regions such that the annihilation characteristics are dominated by the bond electron density and momentum distribution.

Abstract: We have performed a high-resolution, high-sensitivity study of the electron-positron momentum distribution in diamond, using the two-detector coincidence Doppler broadening technique of positron annihilation spectroscopy. The measurements were carried out with the (100), (110), and (111) axes each separately aligned collinearly with the detectors so that the anisotropy of the momentum distribution could be mapped out. Small differences in the momentum profiles were enhanced with difference spectra, and compare well with results of a density-functional theory calculation of the electron-positron momentum distribution in the diamond lattice. There has been much debate regarding previously unexplained anomalies in the annihilation characteristics of the bulk positron configuration in diamond. These new results strongly support a model describing the bulk: positron as delocalized and inhomogeneous, but with sufficient amplitude at me intra-bond region such that the annihilation characteristics are determined by the bond electron density and momentum distribution.

Abstract: Quantum chemical calculations based on the PM3 molecular Hamiltonian are reported investigating the optimum geometry for the implantation of a fluorine atom in a C-60 molecule. Of he various sites investigated we find the fluorine attached to the outside of the spherical shell energetically most favourable. It is shown that in this configuration the F-atom partially destroys the delocalized electronic state of the fullerene in its immediate vicinity creating a locally diamond-like geometry with C-C bonds of 1.53 Angstrom. Three other possible equilibrium sites are also found; F occupying a substitutional site, F in the centre of the C-60 molecule, and F at the centre of a C-C bond. Those implantation sites, however, all have substantially higher total energies. Investigating higher addition products of the form C60Fi, 1 less than or equal to i less than or equal to 60, we find an energy minimum at i approximate to 24. The total energy curve E-tot as a function of i is qualitatively and to a large extent even quantitatively correctly reduced by a simple model calculation containing only the C-F binding energy and the nearest neighbour interactions. (C) 1997 Elsevier Science Ltd.

Abstract: Interface models for matching either w-BN or h-BN layers to an Si(001) substrate, are considered. Aligning the [2 - 1 - 10] direction of h-BN (or equivalently the [- 1 - 120] direction in w-BN) along [1 - 10] of the Si(001), chemical saturation of interface bonds can be achieved. For h-BN, orientations of the basal plane parallel and perpendicular to the surface are both considered. The total energy of atomic clusters constructed to represent these models, are calculated using the AMl quantum chemical Hamiltonian, and atomic geometries optimised allowing full symmetric relaxation of interface atoms. Only the perpendicular h-BN basal plane model is found to be energetically favourable compared to earlier calculations for cubic boron nitride. (C) 1997 Published by Elsevier Science S.A.

Abstract: A previously presented extrapolation method of the density matrix of a standard finite supercell calculation to an infinite supercell is extended to two-dimensional systems. The density matrix of the finite supercell is transformed into q space where it is interpolated and extrapolated in such a way that all its fundamental properties are guaranteed by construction. The resulting modified density matrix contains information from a continuum of q points in the first Brillouin zone which allows for a more realistic calculation of properties like the total energy E-tot per atom and the band structure epsilon(i)(q). It is shown that this more realistic calculation takes better care of the crystal symmetries and is essential in reproducing both important degeneracies in the band structure and rotationally symmetric results. In the special case of pi electrons only, an exact analytical solution for the density matrix of the infinite supercell is presented. (C) 1997 John Wiley & Sons, Inc.

Abstract: Preferential crystallographic growth planes are influenced to a large extent by the shape of the microcrystals that nucleate the growth. In contrast with macroscopic crystals, for nanometer scale microcrystals the energies associated with edges and vertices play a crucial role in determining an energetically favourable shape. We present a model for the computation of edge and vertex energies within the framework of semi-empirical quantum chemical calculations for atomic clusters. For two diamond crystallite shapes, both terminated by hydrogen saturated (111) surfaces, energy coefficients for the bulk and surface as well as two types of edges and vertices are obtained. The values are consistent with measurements. Size dependent effects are investigated by calculating the energy per atom for arbitrary sizes of a given shape, from the coefficients. It is found that even for this simple example, there is an energy cross-over at small crystallite size.

Abstract: Missing dimer models which conform to the electron-hole counting rule have been found energetically favourable for other 3-5 semiconductor materials. We perform molecular cluster calculations using the AM1 :Hamiltonian to study several of these models for the nitrogen-terminated c-BN(001) surface, and compare them with structures terminated with a full coverage of nitrogen dimers or N-N bridges. We find that the energetics is dominated by elastic distortion effects related to the different atomic sizes of boron and nitrogen atoms, and the counting rule is not a reliable guideline for c-BN. The most favourable models are those with a low nitrogen coverage and N-N bridges rather than dimers.

Abstract: We present ab initio pseudopotential calculations on the lifetime and electron-positron momentum density of delocalized positrons in bulk diamond and silicon. Our results compare very well with experimental and earlier theoretical findings available. For diamond we observe a comparatively isotropic momentum distribution with maximum density at the Gamma point which explains naturally recent experimental results.

Abstract: The structures of all III-V and II-VI semiconductor surfaces which have been determined up to now are consistent with the electron-hole counting rule, and the 2 x 4 reconstructions of (001) surfaces with missing dimer rows occur as straightforward consequences of this condition. We perform molecular cluster calculations using the AM1 hamiltonian to study several of these models for the nitrogen-terminated c-BN(001) surface, and compare them with structures terminated with a full coverage of nitrogen dimers or N-N bridges. We find that the energetics is dominated by elastic distortion effects related to the different covalent atomic radii of boron and nitrogen atoms and, as a result, the most favourable models are those with a low nitrogen coverage and N-N bridges rather than dimers. We also observe that the surface energies of some of the structures which do not comply with the electron-hole counting rule are lower than the energies of those which are consistent with it. This can be explained by the fact that the latter contain a larger amount of surface N dimers than the former. Each of these dimers raises the respective surface energy by introducing additional stress into the surface, owing to the relatively short N-N bond length. The surface stress, being the major component of the surface energy, completely counterbalances the energy-reducing electron mechanism of the electron-hole counting rule. Thus this rule is violated in the case of c-BN and cannot serve as a guiding principle for predicting the atomic structure of its surfaces.

Abstract: Calculations of finite atomic clusters based on the Hartree-Fock self-consistent field theory are modified to model more closely the ideal behavior of the infinite system. The density matrix of the standard finite supercell calculation is extrapolated to an infinite supercell so that it contains information from a continuum of k points in the first Brillouin zone. This modification is incorporated into the self-consistency loop of the MOPAC quantum chemistry program and leads to improved results compared to a standard finite supercell calculation. Heats of formation, bond lengths, and electronic properties converge more quickly to the correct ground-state values. For polyacetylene, we obtain a reduced bond-length alternation of Delta r = 0.084 Angstrom, which is in agreement with more sophisticated calculations containing electron correlation effects. (C) 1994 John Wiley & Sons, Inc.

Abstract: A method is proposed how to calculate the correct density matrix of an infinite polymeric chain from that of a standard finite supercell calculation. The density matrix of the finite supercell is transformed into k-space for all k-values allowed by the periodic boundary conditions. The k-dependent matrices are then unitarily transformed, with each unitary matrix being represented by a set of complex rotation matrices. It is shown that the corresponding angles can be interpolated and extrapolated toward the zone boundaries in a straightforward manner and that this extrapolation can be done from any finite supercell with reasonable accuracy. This gives rise to an infinite system density matrix for which all fundamental properties are guaranteed by construction. This infinite system density matrix may be used to construct a corrected density matrix for the finite supercell calculation. (C) 1994 John Wiley & Sons, Inc.

Abstract: Several models with different topologies for the atomic structure in the interface between a Si(001) substrate and an epitaxial c-BN(001) adfilm are investigated. The models represent 3:2 parallel epitaxy as well as epitaxy associated with an adsorbate structure rotated by 45-degrees relative to the substrate. Variations including hydrogen and oxygen atoms in the interface are also considered. For each model the atomic positions are optimised using a quantum chemical total energy calculation with periodic repetition of clusters, containing up to 8 atomic layers. The most favourable 3:2 model has an interface energy of 2.01 eV/1 x 1 substrate surface unit cell (SSUC), slightly more favourable that the 2.13 eV/SSUC of the best 45-degrees rotated model. This relative advantage increases with the film thickness due to a smaller epitaxial mismatch for 3: 2 models. The calculated energetics indicates that the most favourable of the investigated interface structures should be able to sustain an epitaxially grown adsorbate of up to 90 atomic layers without the need for stress reducing defects.

Abstract: A previous proposal for the atomic structure at the interface of a diamond (001) adfilm growing epitaxially on Si(001) in a 45 degree rotated orientation, is extended by the inclusion of Si atoms in the second adfilm layer. In addition, two new models for growth in 3:2 registry in a parallel orientation are investigated, including versions with and without incorporation of hydrogen atoms into the structure. For each model, a total energy optimisation of the geometry is carried out using quantum chemical calculations with the PM3 Hamiltonian applied to periodically repeated atomic clusters containing up to eight atomic layers. Due to the non-stoichiometric nature of the interface layers, comparisons of the relative energetics of the models are performed by the introduction of carefully defined interface and adhesive bonding energies. It is found that the rotated model has a very low intrinsic interface energy and is favourable for very thin adfilms, but due to the smaller elastic deformation needed for registry the parallel growth mode becomes most favourable for thicker layers. Detailed atomic geometries for the optimised structures of all three models are given.

Abstract: Two new structures for epitaxy of diamond and c-BN on a Si(001) substrate, in a parallel orientation and with 3:2 registry, are proposed and also compared with a structure proposed earlier for a 45-degrees rotated orientation of a diamond overlayer. All the models largely eliminate dangling bonds in the interface, and several variations are considered with and without additional atoms of hydrogen or oxygen incorporated into the structure. The atomic geometry of each model is optimized by minimization of the total energy of periodically repeated atomic clusters, calculated by fully self-consistent semi-empirical quantum chemical methods. It is found that the 45-degrees rotated structure is particularly favourable for very thin diamond layers, but for thicker layers the single bridge 3:2 model that works best for c-BN is also preferred by diamond. The differences in the two materials are interpreted in terms of the way that the parallel model exploits the differences in both the size and atomic valencies of the constituent atoms in c-BN.

Abstract: An approach to the calculation of the tunneling current in STS and STM experiments from the results of quantum chemical atomic cluster calculations, is presented. Essential features of the model are that the final states involved in the expression derived by Tersoff and Hamann, are excited states rather than ionised states, and furthermore these states have to be calculated from multiconfiguration CI wavefunctions. The method is applied to measurements on the clean and hydrogenated Si(100) surface. Good agreement with measured STS curves is obtained, but only when the dimer geometry is allowed to respond to the biasing voltage to give a zero buckling of the dimer when a positive bias voltage is applied to the surface, and a buckling of 0.41 angstrom for a negative bias. For the hydrogenated surface the tunneling is found to drastically weaken the dimer bond. Good qualitative agreement is also found for calculated and measured topography curves for an STM tip scanning over the dimer in the dimer plane.

Abstract: Dimer- and bridge-type reconstruction on the (001) surface of cubical boron nitride (c-BN) is studied, using AM1 quantum chemical calculations of semi-infinite slabs containing 8-10 atomic layers. The optimum structure is found to be slightly buckled dimers on the boron terminated surface, but two somewhat different structures having N-N bridges spanning across second layer boron dimers are found to be energetically favoured above simple dimerization on the nitrogen terminated surface. Detailed geometries and calculated surface energies are reported for each reconstructed surface, showing that the surface energy of a boron terminated surface is significantly lower than that of the nitrogen terminated surface.

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Abstract: A quantum chemical atomic cluster calculation is reported to investigate the possible stabilization of the DAS (7 x 7) reconstruction of Si(111) by the inclusion of an oxygen atom below the surface near the corner adatom. Several metastable configurations are found, with a minimum barrier to removal of the oxygen (in the form of an SiO molecule) of 4.5 eV. This supports the hypothesis of stabilization by oxygen.

Abstract: We investigated the possibility of epitaxial growth of diamond on an Si(100) substrate in the C(square-root 2 x square-root 2) R45 configuration, for which the epitaxial misfit between the crystals is only 7.6%. In order to take covalent bonding effects fully into account, our study used a quantum chemical calculation of various stages in building up the diamond layer. First we consider a monolayer of carbon. It is found that carbon atoms chemisorb at two-fold bridge positions directly between two silicon atoms and almost in-the surface plane. For the second monolayer of carbon we find two different configurations. Firstly, the carbon atoms bond weakly in a highly stretched linear chain, or secondly, the carbon atoms chemisorb directly between the silicon atoms. The second configuration is the starting point for the C(square-root 2 x square-root 2) R45 configuration. We find that the binding energy progressively increases as subsequent layers are added to produce a diamond (100) layer and that at the interface substantial rearrangements of the bonding structure take place both in the silicon and diamond crystals.

Abstract: A fully symmetrical orthogonalization algorithm is described and applied to the localized orbitals (LMOs) generated by the previously published POPLOC localization program. The results are compared with those of the psi-4 method as implemented in the MOPAC program. The LMOs resulting from both methods are generally quite similar. However, our method is computationally faster, is more successful in separating sigma and pi bonds and gives a better account of the inherent molecular symmetries.

Abstract: Self-consistent quantum-chemical cluster calculations are reported which address the controversy about the presence or absence of dimer buckling in the Si(001)-(2x1) reconstructed surface. The results indicate that biasing the surface, as in the scanning tunneling microscopy (STM) experiment, is likely to produce a relatively symmetric STM image even if dimers in the unbiased surface are buckled, as deduced from scattering experiments. We have also investigated a dimer close to a surface defect, such as a step, and propose a mechanism that makes the dimer there appear buckled to STM.

Abstract: A comparison is made of the performance of the MINDO/3, MNDO, AM1, and PM3 methods in calculating the nature of the dimer reconstruction observed on the silicon (100) crystal surface. Based on this case study we conclude that MINDO/3 gives the most realistic results, with PM3 calculations being quite similar but both MNDO and AM1 missing some key features of this system and giving rather unrealistic charge distributions. Hence use of PM3 is recommended for Si containing molecules where a lack of parameters or other restrictions prevent the use of MINDO/3.

Abstract: Atomic cluster calculations are reported which model adatom generated (2 x 2) and c(2 x 8) reconstruction on the germanium (111) surface, for both the T4 and H3 adsorption sites. The clusters are four atomic layers deep and the positions of atoms in the first three atomic layers are found by minimising the total energy calculated with the MNDO quantum chemical Hamiltonian, according to the prevailing symmetry. Both closed and open shell wavefunctions are considered. We find at least four different electronic states with similar energies but different geometries. The lowest energy is found for the c(2 x 8)H3 configuration with a p-like dangling bond on the rest atom, while for the (2 x 2) structure an s-like dangling bond orbital on the adatom is most favorable.

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