Bakowies, D. ,
" ATOMIC-2 protocol for thermochemistry "
J. Chem. Theory Comput. 2022, 18, 4142-4163.
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Computer code available.
Bakowies D., "Ab initio thermochemistry with ATOMIC-2"     DOI: 10.5281/zenodo.5780172

ATOMIC is a midlevel thermochemistry protocol that uses Pople's concept of bond separation reactions (BSRs) as theoretical framework to reduce computational demands in the evaluation of atomization energies and enthalpies of formation. Various composite models are available that approximate bond separation energies at the complete basis set limit of all-electron CCSD(T), each balancing computational cost with achievable accuracy. Evaluated energies are then combined with very high-level, precomputed atomization energies of all auxiliary molecules appearing in the BSR to obtain the atomization energy of the molecule under study. ATOMIC-2 is a new version of the protocol that retains the overall concept and all previously defined composite models, but improves on ATOMIC-1 in various other ways: Geometry optimization and zero-point-energy evaluation are performed at the density functional level (PBE0-D3/6-311G(d)), which shows significant computational savings and better accuracy than the previously employed RI-MP2/cc-pVTZ. The BSR framework is improved, using more accurate CBS extrapolations toward the Full CI limit for the atomization energies of all auxiliary molecules. Finally, and most importantly, an error and uncertainty model termed ATOMIC-2_um is added that estimates average bias and uncertainty for each of the atomization energy contributions that arise from simplified treatment of some contributions to bond separation energies (CCSD(T)) and neglect of others (such as higher order, scalar relativistic or diagonal Born-Oppenheimer corrections) or from residual error in the energies of auxiliary molecules. Large and diverse benchmarks including up to 1179 molecules are used to evaluate necessary reference data and to correlate observed error for each of the contributions with appropriate proxies that are available without additional quantum-chemical calculation for a particular molecule and represent its size and type. The implementation of ATOMIC-2 considers neutral, closed-shell molecules containing H, C, N, O, and F atoms; compared to ATOMIC-1 the framework has been extended to cover a few challenging but rare bond topologies. In comparison to highly accurate reference data for 184 molecules taken from the ATcT database (V. 1.122r), regular ATOMIC-2 shows noticable underbinding, but the bias-corrected protocol ATOMIC-2_um is found to be more accurate than either ATOMIC-1 or standard Gaussian-4 theory, and the uncertainty model is consistent with statistics of actually observed errors. Problems arising from ambiguous or challenging Lewis-valence structures defining BSRs are discussed, and computational efficiency is demonstrated. Computer code is made available to perform ATOMIC-2um analyses.

Bakowies, D.; von Lilienfeld, O. A. ,
" Density functional geometries and zero-point energies in ab initio thermochemical treatments of compounds with first-row atoms (H, C, N, O, F) "
J. Chem. Theory Comput. 2021, 17(8), 4872-4890.
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Density functionals are often used in ab initio thermochemistry to provide optimized geometries for single-point evaluations at high level and to supply estimates of anharmonic zero-point energies (ZPEs). Their use is motivated by relatively high accuracy at modest computational expense, but a thorough assessment of geometry-related error seems to be lacking. We have benchmarked 53 density functionals, focusing on approximations of the first four rungs and on relatively small basis sets for computational efficiency. Optimized geometries of 279 neutral first-row molecules (H, C, N, O, F) are judged by energy penalties relative to the best available geometries, using the composite model ATOMIC / B5 as energy probe. Only hybrid functionals provide good accuracy with root mean square errors around 0.1 kcal/mol and maximum errors below 1.0 kcal/mol, but not all of them do. Conspicuously, first-generation hybrids with few or no empirical parameters tend to perform better than highly parameterized ones. A number of them show good accuracy already with small basis sets (6-31G(d), 6-311G(d)). As is standard practice, anharmonic ZPEs are estimated from scaled harmonic values. Statistics of the latter show less performance variation among functionals than observed for geometry-related error, but they also indicate that ZPE error will generally dominate. We have selected PBE0-D3/6-311G(d) for the next version of the ATOMIC protocol (ATOMIC-2) and studied it in more detail. Empirical expressions have been calibrated to estimate bias corrections and 95% uncertainty intervals for both geometry-related error and scaled ZPEs.

Bakowies, D. ,
" Estimating systematic error and uncertainty in ab initio thermochemistry: II. ATOMIC(hc) enthalpies of formation for a large set of hydrocarbons "
J. Chem. Theory Comput. 2020, 16(1), 399-426.
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ATOMIC is a thermochemistry protocol geared toward larger molecules with first-row atoms. It implements Pople's concept of bond separation reactions in an ab initio fashion and so enhances the accuracy of midlevel composite models for atomization energies. Recently we have introduced ATOMIC(hc), a model for applications to hydrocarbons, that estimates bias and uncertainty for each of the components contributing to the ATOMIC bottom-of-the-well atomization energy (Bakowies, D. J. Chem. Theory Comput. 2019, 15, 5230-5251). Here we scrutinize the remaining components of the ATOMIC protocol, including midlevel composite models to approximate the complete-basis set (CBS) limit of CCSD(T) as well as zero-point energies (ZPEs) and thermal enthalpy increments that are evaluated from scaled harmonic MP2 frequencies. Potential errors relating to imperfections in MP2 geometries and ZPEs are estimated using auxiliary information obtained from geometry optimizations and frequency calculations at the density functional (B3LYP) level. Overall corrections to and uncertainties of enthalpies of formation are obtained from summation and error propagation, respectively. The error and uncertainty model is validated with accurate data from the Active Thermochemical Tables (ATcT) and compared to earlier statistical assessments for the G3/99 benchmark. The proposed model is a welcome alternative to statistical assessment, first because it does not depend on comparison with experiment, second because it recognizes the expected scaling of error with system size, and third because it provides a detailed account of the importance of various contributions to overall error and uncertainty. The evaluation of ZPEs from scaled harmonic frequencies expectedly emerges as the leading source of uncertainty if highly accurate composite models are used to treat the electronic problem, but uncertainties are usually balanced with those arising from computationally more attractive B level (B1...B6) models to estimate the CBS limit of CCSD(T). ATOMIC(hc) enthalpies of formation, complete with uncertainty estimates, are reported for 161 hydrocarbons ranging in size from methane (CH4) to [8]circulene (C32H16) and tetra-tert-butyltetrahedrane (C20H36). Experimental data are available for 127 molecules but cannot be reconciled with theory in 37 cases. Theory helps to identify the more accurate among conflicting experimental values in 11 cases and emerges as a valuable complement to experiment also for larger molecules, provided that fair estimates of uncertainty are available.

Bakowies, D. ,
" Estimating systematic error and uncertainty in ab initio thermochemistry. I. Atomization energies of hydrocarbons in the ATOMIC(hc) protocol "
J. Chem. Theory Comput. 2019, 15(10), 5230-5251.
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Research in ab initio quantum chemistry has produced an increasing number of thermochemistry protocols, serving different needs from benchmark-level accuracy for small molecules to "chemical accuracy" for larger molecules. While in experimental thermochemistry it is accepted standard to report results complete with intervals of 95% confidence, so far few of the most advanced theoretical approaches have followed suit, based on either statistical comparison to well-established experimental data or careful assessment of high level theoretical results for individual molecules. Here we report on the development of intrinsic uncertainty estimates for the ATOMIC protocol in applications to hydrocarbons. ATOMIC is a theoretical procedure geared toward larger molecules and based on the ab initio implementation of bond separation reactions (BSRs) to reduce errors of midlevel composite approaches. Each of the components contributing to the bottom-of-the-well atomization energy (EA,e) is scrutinized for possible error by comparison to a large number of very high-level results, including complete-basis-set estimates of CCSDT(Q) bond separation energies for 83 hydrocarbons up to the size of naphthalene. Some of the observations are the following: Post-CCSD(T) effects are sizable even for saturated aliphatic compounds but well-represented in a BSR model summing over bond contributions, while conjugated systems pose more problems. Another significant source of error is the complete-basis-set extrapolation of all-electron CCSD(T) contributions, which still carries an uncertainty of a few tenths of a kcal/mol for midsize molecules like benzene, even if based on large basis-set calculations (cc-pCV5Z, cc-pCV6Z). Scalar relativistic terms and diagonal Born-Oppenheimer corrections are of less concern, the former because they are well represented in a BSR model and the latter because they are small in general. Observations are cast into simple expressions that separate obvious bias from nonsystematic error, formulating the former as correction to and the latter as uncertainty of an ATOMIC result. The updated protocol, complete with uncertainties and termed ATOMIC(hc) ("hc" for hydrocarbons), is validated in comparisons with both experimental data from the Active Thermochemical Tables and high-level theoretical data generated in this work. Analysis of lower-level ATOMIC models and of further components needed to convert EA,e into enthalpies of formation will be reported separately.

Tahchieva, D. N.; Bakowies, D.; Ramakrishnan R.; von Lilienfeld, O. A. ,
" Torsional potentials of glyoxal, oxalyl halides, and their thiocarbonyl derivatives: Challenges for popular density functional approximations "
J. Chem. Theory Comput. 2018, 14(9), 4806-4817.
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The reliability of popular density functionals was studied for the description of torsional profiles of 36 molecules: glyoxal, oxalyl halides, and their thiocarbonyl derivatives. HF and 18 functionals of varying complexity, from local density to range-separated hybrid approximations and double-hybrid, have been considered and benchmarked against CCSD(T)-level rotational profiles. For molecules containing heavy halogens, most functionals fail to reproduce barrier heights accurately and a number of functionals introduce spurious minima. Dispersion corrections show no improvement. Calibrated torsion-corrected atom-centered potentials rectify the shortcomings of PBE and also improve on sigma-hole based intermolecular binding in dimers and crystals.

Bakowies, D. ,
" Simplified wave function models in thermochemical protocols based on bond separation reactions "
J. Phys. Chem. A 2014, 118(50), 11811-11827.
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The ATOMIC protocol is a quantum-chemical thermochemistry protocol designed to obtain accurate atomization energies and derived heats of formation. It reduces errors of computationally tractable composite schemes through the use of bond separation reactions, which are implemented in a consistent ab initio framework. The present work explores possible simplification of previously introduced ATOMIC models. While coupled cluster calculations with singles and doubles excitations and perturbational treatments of connected triples excitations [CCSD(T)] are still required for high accuracy, basis-set truncations are possible in the CCSD-MP2 and CCSD(T)-CCSD components. The resulting models B4, B5, and B6 show root-mean-square (RMS) errors of only 0.21 to 0.46 kcal/mol for the AE set, which is a benchmark comprising complete-basis-set CCSD(T)(full) atomization energies of 73 neutral, closed-shell molecules composed of H, C, N, O, and F atoms. The evaluation of connected triples excitations can be avoided at medium levels of accuracy if the complete-basis-set MP2 energy is augmented with an empirically calibrated fraction of the difference between MP3 (or CCSD) and MP2 energies, calculated with small basis sets. The corresponding EMP3 and ECCSD models show RMS errors of 1.01 and 0.70 kcal/mol, respectively. Spin-component scaling is an option to rely entirely on the MP2 level of theory and still cut the RMS error of 4.38 kcal/mol by roughly a factor of 2 and achieve an accuracy comparable to accurate density functionals, such as M05-2X. The proposed new models are additionally tested with the HOF benchmark, a subset of G3/99 heats of formation that includes only neutral closed-shell molecules composed of H, C, N, O, and F atoms. The assessment shows that a number of experimental reference values are in error and should be replaced with more recent data. Results obtained with the new models are compared to original HOF (G3/99) reference data, to updated reference data, and to accurate ATOMIC/A theoretical data.

Bakowies, D. ,
" Assessment of density functional theory for thermochemical approaches based on bond separation reactions "
J. Phys. Chem. A 2013, 117(1), 228-243.
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The recently proposed ATOMIC protocol is a fully ab initio thermochemical protocol that rests upon the concept of bond separation reactions (BSRs) to correct for systematic errors of composite wave function approaches. It achieves high accuracy for atomization energies and derived heats of formation if basis-set requirements for all contributing components are balanced carefully. The present work explores the potential of density functionals as possible replacements of composite wave function approaches in the ATOMIC protocol. Twenty density functionals are examined for their accuracy in thermochemical predictions based on calculated bond- separation energies and precomputed high-level data for the small parent molecules entering BSRs. The best density functionals outperform CCSD (coupled cluster with singles and doubles excitations), but none reaches the accuracy of well-balanced composite wave function approaches that consider quasiperturbational connected triples excitations at least with small basis sets. Some functionals show unexpected problems with bond separation reactions and are analyzed further with a model of empirically calibrated bond additivity corrections. Finally, the benefit of adding empirical dispersion terms to common density functionals is analyzed in the context of BSR-corrected thermochemistry.

Bakowies, D. ,
" Ab initio thermochemistry with high-level isodesmic corrections: Validation of the ATOMIC protocol for a large set of compounds with first-row atoms (H, C, N, O, F) "
J. Phys. Chem. A 2009, 113(43), 11517-11534.
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The recently proposed ATOMIC protocol is a fully ab initio thermochemical approach designed to provide accurate atomization energies for molecules with well-defined valence structures. It makes consistent use of the concept of bond-separation reactions to supply high-level precomputed bond increments which correct for errors of lower-level models. The present work extends the approach to the calculation of standard heats of formation and validates it by comparison to experimental and benchmark level ab initio data reported in the literature. Standard heats of formation (298 K) have been compiled for a large sample of 173 neutral molecules containing hydrogen and first-row atoms (C, N, O, F), resorting to several previous compilations and to the original experimental literature. Statistical evaluation shows that the simplest implementation of the ATOMIC protocol (composite model C) achieves an accuracy comparable to the popular Gaussian-3 approach and that composite models A and B perform better. Chemical accuracy (1 - 2 kcal/mol) is normally achieved even for larger systems with about 10 non-hydrogen atoms and for systems with charge-separated valence structures, bearing testimony to the robustness of the bond-separation reaction model. Effects of conformational averaging have been examined in detail for the series of n-alkanes, and our most refined composite model A reproduces experimental heats of formation quantitatively, provided that conformational averaging is properly accounted for. Several cases of larger discrepancy with respect to experiment are discussed, and potential weaknesses of the approach are identified.

Bakowies, D. ,
" Ab initio thermochemistry using optimal-balance models with isodesmic corrections: The ATOMIC protocol "
J. Chem. Phys. 2009, 130, 144113/1-21.
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Free Download Copyright Notice: Copyright (2009) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The article appeared in the Journal of Chemical Physics (citation above) and may be found at the journal webpage .

A theoretical composite approach, termed ATOMIC for "Ab initio Thermochemistry using Optimal-balance Models with Isodesmic Corrections", is introduced for the calculation of molecular atomization energies and enthalpies of formation. Care is taken to achieve optimal balance in accuracy and cost between the various components contributing to high-level estimates of the fully correlated energy at the infinite-basis set limit. To this end, the energy at the coupled-cluster level of theory including single, double, and quasi-perturbational triple excitations is decomposed into Hartree-Fock, low-order correlation (MP2, CCSD) and connected-triples contributions, and into valence-shell and core contributions. Statistical analyses for 73 representative neutral closed-shell molecules containing hydrogen and at least three first-row atoms (CNOF) are used to devise basis set and extrapolation requirements for each of the eight components to maintain a given level of accuracy. Pople's concept of bond-separation reactions is implemented in an ab initio framework, providing for a complete set of high-level precomputed isodesmic corrections which can be used for any molecule for which a valence structure can be drawn. Use of these corrections is shown to lower basis set requirements dramatically for each of the eight components of the composite model. A hierarchy of three levels is suggested for isodesmically corrected composite models which reproduce atomization energies at the reference level of theory to within 0.1 kcal/mol (A), 0.3 kcal/mol (B), and 1 kcal/mol (C). Large-scale statistical analysis shows that corrections beyond the CCSD(T) reference level of theory, including coupled cluster theory with fully relaxed connected triple and quadruple excitations, first-order relativistic and diagonal Born-Oppenheimer corrections can normally be dealt with using a greatly simplified model that assumes thermoneutral bond-separation reactions and that reduces the estimate of these corrections to the simple task of adding up bond increments. Preliminary validation with experimental enthalpies of formation, using the subset of neutral closed-shell (HCNOF) species contained in the the G3/99 test set, indicates that the ATOMIC protocol performs slightly better than the popular G3 approach. The newly introduced protocol does not require empirical calibration, however, and it is still efficient enough to be applied routinely to molecules with ten or twenty non-hydrogen atoms.

Bakowies, D. ,
" Ab initio thermochemistry of large molecules "
Scientific Report of the Swiss National Supercomputing Centre (2006/2007) 2008, 32-35.
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Thermochemistry is a branch of thermodynamics concerned with the energy balance of chemical reactions. The elements in their standard states define the universal reference, establishing heats of formation as the primary quantity relating the heat content of one compound to that of another. Experimental access is usually provided through combustion calorimetry, supplemented by measurements of heats of vaporization or sublimation. On the theoretical side, heats of formation may be obtained from atomization energies, but ab initio predictions to any useful accuracy (say 1-2 kcal/ mol) have not been possible even for the smallest chemical systems until about 20 years ago. This is mainly attributable to the large error incurred if electron correlation is not dealt with accurately, as the total atomization of a molecule involves a significant change in electron correlation. Accurate treatment, however, requires expensive calculations using extended basis sets as the electron correlation energy is known to converge very slowly with the size of orbital-based expansions. Our interest in accurate ab initio thermochemical approaches reflects the need for supplying high-quality reference data for semiempirical method development. Traditionally such reference data have been obtained almost exclusively from experiment, but a reliable ab initio protocol would offer several advantages: (a) Heats of formation, previously used but theoretically not well justified, can be replaced by more appropriate atomization energies which are easily available only from calculation. (b) The accuracy of experimental data is often hard to quantify, and while many data are very well established, occasionally large errors do occur. (c) Experimental data are entirely unavailable for several important classes of molecules, and, in particular for biologically relevant model systems such as peptides. The latter point is of particular importance as one of the most promising fields of application for improved semiempirical methodology is in biochemistry.

Bakowies, D. ,
" Accurate extrapolation of electron correlation energies from small basis sets "
J. Chem. Phys. 2007, 127, 164109/1-12
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Free Download Copyright Notice: Copyright (2007) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The article appeared in the Journal of Chemical Physics (citation above) and may be found at the journal webpage .

A new two-point scheme is proposed for the extrapolation of electron correlation energies obtained with small basis sets. Using the series of correlation-consistent polarized valence basis sets, cc-pVXZ, the basis set truncation error is expressed as δE_X∝(X+ ξ_i)^-γ. The angular momentum offset ξ_i captures differences in effective rates of convergence previously observed for first-row molecules. It is based on simple electron counts and tends to values close to 0 for hydrogen-rich compounds and values closer to 1 for pure first-row compounds containing several electronegative atoms. The formula is motivated theoretically by the structure of correlation-consistent basis sets which include basis functions up to angular momentum L=X-1 for hydrogen and helium and up to L=X for first-row atoms. It contains three parameters which are calibrated against a large set of 105 reference molecules (H, C, N, O, F) for extrapolations of MP2 and CCSD valence-shell correlation energies from double- and triple-zeta (DT) and triple- and quadruple-zeta (TQ) basis sets. The new model is shown to be three to five times more accurate than previous two-point schemes using a single parameter, and (TQ) extrapolations are found to reproduce a small set of available R12 reference data better than even (56) extrapolations using the conventional asymptotic limit formula δE_X∝X ^-3. Applications to a small selection of boron compounds and to neon show very satisfactory results as well. Limitations of the model are discussed.

Bakowies, D. ,
" Extrapolation of electron correlation energies to finite and complete basis set targets "
J. Chem. Phys. 2007, 127, 084105/1-23.
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Free Download Copyright Notice: Copyright (2007) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The article appeared in the Journal of Chemical Physics (citation above) and may be found at the journal webpage .

The electron correlation energy of two-electron atoms is known to converge asymptotically as (L+1) ^-3 to the complete basis set limit, where L is the maximum angular momentum quantum number included in the basis set. Numerical evidence has established a similar asymptotic convergence X ^-3 with the cardinal number X of correlation-consistent basis sets cc-pVXZ for coupled cluster singles and doubles (CCSD) and second order perturbation theory (MP2) calculations of molecules. The main focus of this article is to probe for deviations from asymptotic convergence behavior for practical values of X by defining a trial function X ^{- β} that for an effective exponent β=β_eff(X,X+1,X+N) provides the correct energy E_X+N, when extrapolating from results for two smaller basis sets, E_X and E_{X+1}. This analysis is first applied to "model" expansions available from analytical theory, and then to a large body of finite basis set results (X=D,T,Q,5,6) for 105 molecules containing H, C, N, O, and F, complemented by a smaller set of 14 molecules for which accurate complete basis set limits are available from MP2-R12 and CCSD-R12 calculations.  β_eff  is generally found to vary monotonically with the target of extrapolation, X+N, making results for large but finite basis sets a useful addition to the limited number of cases where complete basis set limits are available. Significant differences in effective convergence behavior are observed between MP2 and CCSD (valence) correlation energies, between hydrogen-rich and hydrogen-free molecules, and, for He, between partial-wave expansions and correlation-consistent basis sets. Deviations from asymptotic convergence behavior tend to get smaller as X increases, but not always monotonically, and are still quite noticeable even for X=5. Finally, correlation contributions to atomization energies (rather than total energies) exhibit a much larger variation of effective convergence behavior, and extrapolations from small basis sets are found to be particularly erratic for molecules containing several electronegative atoms. Observed effects are discussed in the light of results known from analytical theory. A carefully calibrated protocol for extrapolations to the complete basis set limit is presented, based on a single "optimal" exponent β_opt(X,X+1,∞) for the entire set of molecules, and compared to similar approaches reported in the literature.

van Gunsteren, W. F.; Bakowies, D.; Baron, R. et al.,
" Biomolecular modeling: Goals, problems, perspectives "
Angew. Chem. Int. Ed. 2006, 45, 4064-4092, Angew. Chem. 2006, 118, 4168-4198.
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Computation based on molecular models is playing an increasingly important role in biology, biological chemistry, and biophysics. Since only a very limited number of properties of biomolecular systems is actually accessible to measurement by experimental means, computer simulation can complement experiment by providing not only averages, but also distributions and time series of any definable quantity, for example, conformational distributions or interactions between parts of systems. Present day biomolecular modeling is limited in its application by four main problems: 1) the force-field problem, 2) the search (sampling) problem, 3) the ensemble (sampling) problem, and 4) the experimental problem. These four problems are discussed and illustrated by practical examples. Perspectives are also outlined for pushing forward the limitations of biomolecular modeling.

Bakowies, D. ,
" Atomization energies from ab initio calculations without empirical corrections "
Scientific Report of the Swiss National Supercomputing Centre 2005, 14-17.
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Accurate thermochemistry has turned out to be a major challenge for standard ab initio quantum chemistry. High levels of electron correlation combined with very large basis sets are required to adequately treat the formation of a molecule from its constituent atoms. The application of current approaches is thus limited to very small molecules unless empirical corrections are permitted to account for the average effects of basis set incompleteness and higher order electron correlation. Here we test economical compound methods based entirely on ab initio electronic structure theory and void of empirical corrections, taking advantage of the observations that (a) higher-order electron correlation corrections are much less basis- set dependent than low-order (MP2) correlation energies, and (b) basis set incompleteness errors can largely be eliminated through extrapolation techniques. These compound methods should be accurate enough even for larger molecules so that they provide useful references for the parametrization of more approximate methods, particularly in semiempirical quantum chemistry.

Christen, M.; Hünenberger, P. H.; Bakowies, D. et al.,
" The GROMOS software for biomolecular simulation: GROMOS05 "
J. Comput. Chem. 2005, 26, 1719-1751.
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We present the latest version of the Groningen Molecular Simulation program package, GROMOS05. It has been developed for the dynamical modelling of (bio)molecules using the methods of molecular dynamics, stochastic dynamics, and energy minimization. An overview of GROMOS05 is given, highlighting features not present in the last major release, GROMOS96. The organization of the program package is outlined and the included analysis package GROMOS++ is described. Finally, some applications illustrating the various available functionalities are presented.

Chandrasekhar, I.; Bakowies, D.; Glättli, A.; Hünenberger, P.; Pereira, C.; van Gunsteren, W. F.,
" Molecular dynamics simulation of lipid bilayers with GROMOS96: Application of surface tension "
Mol. Simul. 2005, 31, 543-548.
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The GROMOS96 force fields 45A3 and 53A5, when applied to dipalmitoylphosphatidylcholine (DPPC) membranes, have a tendency to result in a reduced area per lipid in constant pressure simulations. The application of surface tension is effective in increasing the area per lipid, a measure of the phase of the membrane, but only if the area is already close to the experimental range. Therefore the surface tension cannot compensate for strong inadequacies in the force-field parameters. The behaviour of the 45A3 force field from long NPnγT simulations of tens of nanoseconds is analysed over a range of different surface tensions. Comparisons are made with the corresponding NPnAT simulations.

Baron, R.; Bakowies, D.; van Gunsteren, W. F.,
" Principles of carbopeptoid folding: A molecular dynamics simulation study "
J. Peptide Sci. 2005, 11, 74-84.
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The conformational spaces of five oligomers of tetrahydrofuran-based carbopeptoids in chloroform and dimethyl sulfoxide were investigated through nine molecular dynamics simulations. Prompted by nuclear magnetic resonance experiments that indicated various stable folds for some but not all of these carbopeptoids, their folding behaviour was investigated as a function of stereochemistry, chain length and solvent. The conformational distributions of these molecules were analysed in terms of occurrence of hydrogen bonds, backbone torsional-angle distributions, conformational clustering and solute configurational entropy. While a cis-linkage across the tetrahydrofuran ring favours right-handed helical structures, a trans-linkage results in a larger conformational variability. Intra-solute hydrogen bonding is reduced with increasing chain length and with increasing solvent polarity. Solute configurational entropies confirm the picture obtained: they are smaller for cis- than for trans-linked peptides, for chloroform than for dimethyl sulfoxide as solvent and for shorter peptide chains. The simulations provide an atomic picture of molecular conformational variability that is consistent with the available experimental data.

Baron, R.; Bakowies, D.; van Gunsteren, W. F.,
" Carbopeptoid folding: Effects of stereochemistry, chain length, and solvent "
Angew. Chem. Int. Ed. 2004, 43, 4055-4059, Angew. Chem. 2004, 116, 4147-4151.
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No abstract.

Daura, X.; Bakowies, D.; Seebach, D.; Fleischhauer, J.; van Gunsteren, W. F.; Krüger, P.,
" Circular dichroism spectra of β-peptides: Sensitivity to molecular structure and effects of motional averaging "
Eur. Biophys. J. 2003, 32, 661-670.
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Circular dichroism spectra of two β-peptides, i.e. peptides composed of β-amino acids, calculated using ensembles of configurations obtained by molecular dynamics simulation are presented. The calculations were based on 200 ns simulations of a β-heptapeptide in methanol at 298 K and 340 K and a 50 ns simulation of a β-hexapeptide in methanol at 340 K. In the simulations the peptides sampled both folded (helical) and unfolded structures. Trajectory structures with common backbone conformations were identified and grouped into clusters. The CD spectra were calculated for individual structures, based on peptide-group dipole transition moments obtained from semi-empirical molecular orbital theory and using the so-called matrix method. The single-structure spectra were then averaged over entire trajectories and over clusters of structures. Although certain features of the experimental CD spectra of the β-peptides are reproduced by the trajectory-average spectra, there exist clear differences between the two sets of spectra in both wavelength and peak intensities. The analysis of individual contributions to the average spectra shows that, in general, the interpretation of a CD signal in terms of a single structure is not possible. Moreover, there is a large variation in the CD spectra calculated for a set of individual structures that belong to the same cluster, even when a structurally tight clustering criterion is used. This indicates that the CD spectra of these peptides are very sensitive to small local structural differences.

Bakowies, D.
" Biomolekulare Reality-Simulationen "
Nachr. Chem. 2003, 51, 788-793.
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"Observe while it happens": Während man früher häufig auf vereinfachte Kraftfelddarstellungen zurückgriff und Energiebarrieren künstlich durch Anlegen von Zwangskräften überwand, sehen wir dank schnellerer Rechner heute schon die ersten Simulationen der Membranaggregation, des Wassertransports durch Membrankanäle und der Peptid- und Proteinfaltung im vollen atomaren Detail.

"Observe while it happens": While we have been used to simplified force field descriptions and artificial bias forces enabling us to surmount energy barriers, we are now seeing - thanks to faster computers - the first simulations providing full atomistic detail of membrane aggregation, of water transport through membrane channels and of peptide and protein folding.

Bakowies, D.
" Trendbericht Theoretische Chemie 2002: Kraftfelder für biomolekulare Simulationen "
Nachr. Chem. 2003, 51, 325-327.
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No abstract.

Bakowies, D.
" Analyzing solvent in protein cavities: Methods and application to fatty acid binding protein "
Annual Report of the Competence Center for Computational Chemistry, ETH Zürich 2003, 24-43.
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(Technical Report on a Feature Project)

In this paper we present a method to generate a closed surface which embraces a set of predefined atoms. The central idea is to construct a polyhedron whose vertices represent the set of chosen atoms and to find all solvent molecules enclosed by the polyhedron. The proposed procedure entails five steps: (1) Choice of a set of protein atoms which define the boundaries of the protein interior, the protein cavity, or more generally the protein region of interest. (2) Projection of this set of atoms onto a sphere around their geometrical center. (3) Triangulation of the spherical surface. (4) Back transformation to the original coordinate system. (5) Identification of solvent molecules inside the so-generated polyhedron. This procedure is applied to 5 ns long MD trajectories of apo- and holo-fatty acid binding protein (FABP). The beta-barrel type structure of FABP encompasses a large solvent-filled cavity, and only some of the internal water is expelled upon introduction of the palmitate ligand. In this report we concentrate on structural and dynamical aspects of internal water and on differences between the apo and holo forms of the protein.

Baron, R.; Bakowies, D.; van Gunsteren, W. F.; Daura, X.,
" β-peptides with different secondary-structure preferences: How different are their conformational spaces? "
Helv. Chim. Acta 2002, 85, 3872-3882.
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The conformational spaces accessible to two beta-hexapeptides in MeOH at 298 K and 340 K are investigated by molecular-dynamics simulation with an atomistic model of both solute and solvent. The structural properties of these peptides have been previously studied by NMR in MeOH at room temperature. The experimental data could be fitted to a model (P)-12/10-helix for one of the peptides and a model hairpin with a ten-membered H-bonded turn for the other. The goal of the present work is to determine whether the conformational spaces accessible to these two peptides of seemingly different conformational properties contain any common regions. In other words, to what extent are the evident differences found at the macroscopic level also present at the microscopic structural level? It is found that, for the two peptides studied, the conformational spaces sampled in the respective simulations show significant overlap.

Bakowies, D.; van Gunsteren, W. F.,
" Water in protein cavities: A procedure to identify internal water and exchange pathways and application to fatty acid binding protein "
Proteins 2002, 47, 534-545.
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A computational approach based on Delaunay triangulation is presented to identify internal water molecules in proteins and to capture pathways of exchange with the bulk. The implemented procedure is computationally efficient and can easily be applied to long molecular dynamics trajectories of protein simulations. In an application to fatty acid-binding protein in apo-form and with bound palmitate, several protein orifices known from crystal structures have been confirmed to be major portals of solvent exchange. Differences between the two forms of the protein are observed and discussed.

Bakowies, D.; van Gunsteren, W. F.,
" Simulations of apo and holo-fatty acid binding protein: Structure and dynamics of protein, ligand and internal water "
J. Mol. Biol. 2002, 315, 713-736.
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Two molecular dynamics simulations of 5 ns each have been carried out for rat intestinal fatty acid binding protein, in apo-form and with bound palmitate. The fatty acid and a number of water molecules are encapsulated in a large interior cavity of the barrel-shaped protein. The simulations are compared to experimental data and analyzed in terms of root mean square deviations, atomic β-factors, secondary structure elements, hydrogen bond patterns, and distance constraints derived from nuclear Overhauser experiments. Excellent agreement is found between simulated and experimental solution structures of holo-FABP, but a number of differences are observed for the apo-form. The ligand in holo-FABP shows considerable displacement after about 1.5 ns and displays significant configurational entropy. A novel computational approach has been employed to identify internal water and to capture exchange pathways. Orifices in the portal and gap regions of the protein, discussed in the experimental literature, have been confirmed as major openings for solvent exchange between the internal cavity and bulk water. A third opening on the opposite side of the barrel experiences significant exchange but it does not provide a pathway for further passage to the central cavity. Internal water is characterized in terms of density distributions, interaction energies, mobility, protein contact times, and water molecule coordination. A number of differences are observed between the apo and holo-forms and related to differences in the protein structure. Solvent inside apo-FABP, for example, shows characteristics of a water droplet, while solvent in holo-FABP benefits from interactions with the ligand headgroup and slightly stronger interactions with protein residues.

Gunsteren, W. F.; Bakowies, D.; Bürgi, R. et al.,
" Molecular dynamics simulation of biomolecular systems "
Chimia 2001, 55, 856-860.
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The group for computer-aided chemistry at the ETH Zurich focuses its research on the development of methodology to simulate the behavior of biomolecular systems and the use of simulation techniques to analyze and understand biomolecular processes at the atomic level. Here, the current research directions are briefly reviewed and illustrated with a few examples.

van Gunsteren, W. F.; Bakowies, D.; Damm, W.; Hansson, T; Stocker, U.; Daura, X.,
" Practical aspects of simulation studies of biomolecular systems "
in Dynamics, structure and function of biological macromolecules,
edited by Jardetzky, O. and Finucane, M., NATO ASI Series A315, IOS Press, Amsterdam, 2001, pp. 1-26.
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With the ever-continuing increase of the power of computers, simulation of biomolecular systems in atomic detail has come of age. It is nowadays possible to simulate the classical dynamics of a protein solvated in aqueous solution over time periods of several nanoseconds. The classical equations of motion for the 10⁴-10⁵ atoms in the system can be integrated forward in time for millions of time steps of the order of 1 fs. This allows the study of the conformational equilibrium and dynamics of a variety of biomolecular systems, such as membranes, micelles, protein or DNA complexes. Once the reliability of the molecular models, force fields and computational procedures has been established by comparison of simulated properties with known experimental ones, molecular dynamics (MD) simulation can be a very powerful method to predict molecular properties that are inaccessible to experimental probes. It provides a microscopic picture of, in principle, unlimited resolution in time, space and energy. Secondly, it allows for the study of systems or processes that are impossible to create in reality. System parameters can be changed at will to study particular cause-effect relationships, leading to an enhanced understanding of biomolecular systems, which are generally of complex nature. A biomolecular simulation study possesses a number of practical aspects, which require choices of parameters, procedures and approximations to be made. 1. Molecular model, choice of spatial degrees of freedom, spatial boundary conditions, system sizes and equations of motion. 2. Interaction between the particles or atoms, or along the degrees of freedom. 3. Simulation set-up, treatment of long-range forces, coupling to temperature or pressure baths, choice of initial structure and time step. 4. Simulation software, its reliability, efficiency and capabilities. 5. Convergence, statistics and sampling of molecular or system properties. 6. Comparison of simulated properties with experimental data, (in)sensitivity of properties to molecular conformation. 7. Analysis and interpretation of simulated trajectories or ensembles. In this paper, these seven aspects of biomolecular simulation are briefly discussed and illustrated with examples taken from the literature.

Kollman, P. A.; Kuhn, B.; Donini, O.; Peräkylä, M.; Stanton, R.; Bakowies, D.,
" Elucidating the nature of enzyme catalysis utilizing a new twist on an old methodology: Quantum mechanical - free energy calculations on chemical reactions in enzymes and in aqueous solution "
Acc. Chem. Res. 2001, 34, 72-79.
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How do enzymes achieve very large rate enhancements compared to corresponding uncatalyzed reactions in solution? We present a computational approach which combines high-level ab initio quantum mechanical calculations with classical free energy calculations to address this question. Our calculations lead to accurate estimates of ΔG^&Dagger for both trypsin and catechol O-methyltransferase-catalyzed and reference uncatalyzed reactions and give new insights into the nature of enzyme catalysis. The same methodology applied to steps in the catalytic mechanism of citrate synthase further supports the conclusion that one need not invoke special concepts such as "low-barrier hydrogen bonds" or "pK_a matching" to explain enzyme catalysis.

Bakowies, D.; Kollman, P. A.,
" Theoretical study of base-catalyzed amide hydrolysis: Gas and aqueous phase hydrolysis of formamide "
J. Am. Chem. Soc. 1999, 121, 5712-5726.
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Base-catalyzed hydrolysis of formamide in the gas phase and in aqueous solution has been studied using a combination of quantum chemical and statistical mechanical methods. A three-step procedure has been applied which comprises the determination of a gas-phase reaction path by high-level ab initio calculations, the calibration of empirical solute−solvent potentials, and classical Monte Carlo simulations of the solute immersed in a bath of solvent molecules. These simulations yield the solvent effect as a potential of mean force along the predetermined reaction coordinate. Each of the three consecutive steps of base-catalyzed hydrolysis has been analyzed in detail: the formation of a tetrahedral intermediate, its conformational isomerization, and the subsequent breakdown to products. The reaction is very exothermic in the gas phase and involves only moderate barriers for the latter two steps. Aqueous solvent, however, induces a significant barrier toward formation of the intermediate. On the other hand, it also facilitates conformational isomerization and produces a more product-like transition state for the breakdown step. Solvent effects, as expressed by differences in free energy of solvation, are found to reflect variations in the solute's charge distribution and are readily explained by the analysis of hydrogen bond patterns. The calculated free energy profile is in satisfactory agreement with available experimental data for the solution-phase reaction.

Stanton, R. V.; Peräkylä, M.; Bakowies, D.; Kollman, P. A.,
" Combined ab initio and free energy calculations to study reactions in enzymes and solution: Amide hydrolysis in trypsin and aqueous solution "
J. Am. Chem. Soc. 1998, 120, 3448-3457.
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We present a new more general way to combine ab initio quantum mechanical calculations with classical mechanical free energy perturbation approach to calculate the energetics of enzyme-catalyzed reactions and the same reaction in solution. This approach, which enables enzyme and solution reactions to be compared without the use of empirical parameters, is applied to the formation of the tetrahedral intermediate in trypsin, but it should be generally applicable to any enzymatic reaction. Critical to the accurate calculation of the reaction energetics in solution is the estimate of the free energy to assemble the reacting groups, where the approach recently published by Hermans and Wang (J. Am. Chem. Soc. 1997, 119, 2707) was used. A central aspect of this new approach is the use of the RESP protocol to calculate the charge distribution of structures along the reaction pathway, which enables us to circumvent problems in partitioning the charge across a residue that is being divided into QM and MM parts. The classical mechanical free energy calculations are implemented with two different approaches, "Cartesian mapping" and "flexible FEP". The similarity of the results found by using these two approaches supports the robustness of the calculated free energies. The calculated free energies are in quite good agreement with available experimental data for the activation free energies in the enzyme and aqueous phase reactions.

Bakowies, D.; Thiel, W.,
" Hybrid models for combined quantum mechanical and molecular mechanical approaches "
J. Phys. Chem. 1996, 100, 10580-10594.
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A hierarchy of three models for combined quantum mechanical (QM) and molecular mechanical (MM) approaches is presented. They simplify the QM description of large molecules by reducing it to the electronically important fragment which interacts with the molecular mechanically treated remainder of the molecule. In the simplest model A, the QM fragments are only mechanically embedded in their MM environment. The more refined models B and C include a quantum mechanical treatment of electrostatic interactions between the fragments and a semiclassical description of polarization. The implementation of models A-C for MNDO type wavefunctions and the MM3 force field is outlined. Selected applications in organic chemistry are discussed, addressing the ability of the proposed models to reproduce substituent effects (MM) on chemical structure and reactivity (QM). These applications include protonations, deprotonations, hydride transfer reactions, nucleophilic additions, and nucleophilic ring cleavage reactions.

Bakowies, D.; Thiel, W.,
" Semiempirical treatment of electrostatic potentials and partial charges in combined quantum mechanical and molecular mechanical approaches "
J. Comput. Chem. 1996, 17, 87-108.
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A semiempirical treatment of electrostatic potentials and partial charges is presented. These are the basic components needed for the evaluation of electrostatic interaction energies in combined quantum mechanical and molecular mechanical approaches. The procedure to compute electrostatic potentials uses AM1 and MNDO wave functions and is based on one previously suggested by Ford and Wang. It retains the NDDO approximation and is thus both easy to implement and computationally efficient. Partial atomic charges are derived from a semiempirical charge equilibration model, which is based on the principle of electronegativity equalization. Large sets of ab initio restricted Hartee-Fock (RHF/6-31G*) reference data have been used to calibrate the semiempirical models. Applying the final parameters (C, H, N, O), the ab initio electrostatic potentials are reproduced with an average accuracy of 20% (AM1) and 25% (MNDO), respectively, and the ab initio potential derived charges normally to within 0.1 e. In most cases our parameterized models are more accurate than the much more expensive quasi ab initio techniques, which employ deorthogonalized semiempirical wave functions and have generally been preferred in previous applications.

Bakowies, D.; Bühl, M.; Patchkovskii, S.; Thiel, W.,
" Theoretical studies on giant fullerenes and on endohedral fullerene complexes "
in Fullerenes: Recent advances in the physics and chemistry of fullerenes and related materials, Vol. 3, edited by Ruoff, R. S. and Kadish, K. M., The Electrochemical Society, Pennington, NJ, 1996, pp. 901-910.
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Ab initio SCF, density functional, and semiempirical methods have been used to study selected topics from fullerene chemistry. These include the structural preferences and the stabilities of large icosahedral fullerenes (C₁₈₀, C₂₄₀, C₅₄₀, C₉₆₀), the mechanism of incorporating helium into C₆₀ to form the endohedral complex He@C₆₀, and the NMR chemical shifts of 3He in endohedral complexes involving fullerenes of different sizes as well as C₆₀H₃₆. Our computational results are discussed in relation to other theoretical and experimental work.

Bakowies, D.; Bühl, M.; Thiel, W.,
" A density functional study on the shape of C₁₈₀ and C₂₄₀ fullerenes "
Chem. Phys. Lett. 1995, 247, 491-493.
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At the gradient-corrected BP86/SV level of density functional theory, the fully optimised, facetted geometry of I_h-C₁₈₀ is 126 kcal/mol lower in energy than an optimised spherical structure where all atoms are constrained to lie on the same sphere. Likewise, using MNDO geometries, facetted I_h-C₂₄₀ is more stable than the constrained spherical form by 203 and 202 kcal/mol at the non-local BP86/SV and the local VWN/SV levels, respectively. These findings are at variance with predictions from local density functional calculations employing the divide-and-conquer approximation and the Harris functional, but confirm the results of recent MNDO and ab initio SCF studies.

Bakowies, D.; Bühl, M.; Thiel, W.,
" Can large fullerenes be spherical ? "
J. Am. Chem. Soc. 1995, 117, 10113-10118.
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MNDO geometry optimizations predict a single energy minimum for each of the Goldberg type (I_h) fullerenes C₁₈₀, C₂₄₀, C₅₄₀, and C₉₆₀ which corresponds to an icosahedrally shaped structure with strong curvature at the 12 pentagons and nearly planar segments composed of hexagons. Constrained geometry optimizations preserving a spherical shape lead to considerably larger energies and show that an evenly distributed curvature is strongly disfavored. The results are confirmed quantitatively by ab initio SCF calculations for C₁₈₀ and C₂₄₀ employing a split valence basis set, but contrast the conclusions from a previous density functional study. The observed trends are discussed on the basis of curvature-corrected Hückel calculations and simple force field estimates.

Slanina, Z.; François, J.-P.; Kolb, M.; Bakowies, D.; Thiel, W.,
" Calculated relative stabilities of C₈₄ "
Fullerene Sci. Techn.  1993, 1, 221-230.
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C₈₄ is treated as a system composed of 24 local minima whose energies, geometries, and vibrational frequencies are obtained from MNDO calculations. The predicted global minimum of D_2d symmetry remains the most abundant species in the equilibrium isomeric mixture only till 276 K, being replaced by a D2 species beyond that point. However, a C₁ structure is prevalent in the high temperature limit. The calculated composition around a temperature of 1000 K is consistent with very recent NMR observations.

Slanina, Z.; François, J.-P.; Bakowies, D.; Thiel, W.,
" Fullerene C₇₈ isomers: Temperature dependence of their calculated relative stabilities "
J. Mol. Struct.: THEOCHEM 1993, 279, 213-216.
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C₇₈ is treated as a system composed of five local minima (D_3h(I), D_3h(II), D₃, D_2v(I) and C_2v(II)) whose energies, geometries and vibrational frequencies are obtained from MNDO calculations. The AM1 and PM3 energy data also considered. The predicted global minimum (C_2v(II)) remains the most stable species of the equilibrium isomeric mixture even at high temperatures, and the D_3n (I) structure is negligible throughout. The remaining three species exhibit comparable stabilities and make up nearly 50% of the high temperature mixture. The recently observed ratio of 1:5 of the D₃ and C_2v (I) structures is reached at temperatures of 1308 K, 734 K and 980 K for the MNDO, AM1 and PM3 energetics respectively.

Bakowies, D.; Kolb, M.; Thiel, W.; Richard, S.; Ahlrichs, R.; Kappes, M. M.,
" Quantum chemical study of C₈₄ fullerene isomers "
Chem. Phys. Lett. 1992, 200, 411-417.
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Semiempirical and ab initio SCF calculations are reported for the C₈₄ fullerenes with isolated pentagons. The optimized geometries and relative stabilities are discussed. All methods applied predict two nearly isoenergetic structures with D₂ and D_2d symmetry to be the most stable of the 24 isomers considered, which is consistent with the experimental observed 13C-NMR spectrum. Infrared spectra are predicted for these D₂ and D_2d isomers. The semiempirical results (MNDO, AM1, PM3) for the geometries and relative energies are in excellent agreement with the ab initio predictions at the split-valence SCF level.

Bakowies, D.; Gelessus, A.; Thiel, W.,
" Quantum chemical study of C₇₈ fullerene isomers "
Chem. Phys. Lett. 1992, 197, 324-329.
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Semi-empirical and ab initio calculations are reported for the five C₇₈ fullerenes with isolated pentagons. The optimized geometries and relative stabilities are discussed. The D_3h structure previously favored on the basis of simple Hückel arguments is found to be the least stable isomer at all theoretical levels applied. The most stable isomer corresponds to a C_2v structure which has recently been observed experimentally together with two other isomers. Infrared spectra are predicted for all five isomers.

Slanina, Z.; Adamowicz, L.; Bakowies, D.; Thiel, W.,
" Fullerene C₅₀ isomers: Temperature-induced interchange of relative stabilities "
Thermochim. Acta 1992, 202, 249-254.
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The species C₅₀ is treated as a system composed of three local energy minima (D_5h, D₃, and C_2v) found in recent modified neglect of diatomic overlap (MNDO) calculations. Although the D_5h species corresponds to the deepest minimum it is the most stable structure only up to about 1390 K. Beyond this temperature the D₃ species becomes relatively more populous.

Bakowies, D.; Thiel, W.,
" Theoretical study of Buckminsterfullerene derivatives C₆₀X_n (X=H, F; n = 2, 36, 60) "
Chem. Phys. Lett. 1992, 192, 236-242.
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Semi-empirical SCF calculations at the MNDO, AM1, and PM3 levels are reported for the title compounds. The predicted relative stabilities are discussed for all 18 clusters studied. The calculated equilibrium geometries and vibrational spectra are presented for C₆₀X₃₆(T_h) and C₆₀X₆₀(I_h). Contrary to a previous suggestion, C₆₀H₆₀ and C₆₀F₆₀ prefer an icosahedral I_h structure over a distorted I structure. The calculated bond dissociation enthalpies, equilibrium bond lengths, and vibrational frequencies indicate a reduced C---X bond strength in C₆₀X₆₀(I_h).

Bakowies, D.; Thiel, W.,
" MNDO study of large carbon clusters "
J. Am. Chem. Soc. 1991, 113, 3704-3714.
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MNDO calculations with complete geometry optimization are reported for 30 polyhedral carbon clusters C_n (20 ≤ n ≤ 540). The MNDO results for a planar graphite sheet are extrapolated from calculations on D_6h hydrocarbons C_nH_m (n=6k², m=6k, k=1...6) and used as reference for discussing the properties of the clusters. The structural features of the clusters are correlated with their stability. The relative MNDO energies with respect to graphite are compared with curvature-corrected Hückel calculations and with force field estimates, and criteria for the stability of the clusters are discussed. Infrared spectra are predicted for six stable clusters. Several cationic lithium complexes and their interconversions are investigated for C₆₀ and C₄₂. Finally, computational aspects and performance data are considered, particularly for the largest clusters studied.

Bakowies, D.; Thiel, W.,
" Theoretical infrared spectra of large carbon clusters "
Chem. Phys. 1991, 151, 309-321.
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Harmonic force constant calculations at the MNDO SCF level are reported for 22 polyhedral carbon clusters C_n (20 ≤ n ≤ 240) and for the reference molecules benzene, coronene, and corannulene. New or revised assignments are suggested for coronene and corannulene. Based on the results for the reference molecules and for C₆₀, MNDO is expected to overestimate the vibrational frequencies of the unknown clusters by 10% and to yield reasonable intensity patterns. Calculated infrared spectra are shown and discussed for 12 carbon clusters, with particular emphasis on those which might be observable spectroscopically. The zero-point vibrational energies and the entropy contributions to the free enthalpy are only of minor importance for the thermodynamic stabilities of different clusters.

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