Extension of the PDDG/PM3 and PDDG/MNDO semiempirical molecular orbital methods to the halogens

The new semiempirical methods, PDDG/PM3 and PDDG/MNDO, have been parameterized for halogens. For comparison, the original MNDO and PM3 were also reoptimized for the halogens using the same training set; these modified methods are referred to as MNDO' and PM3'. For 442 halogen-containing molecules, the smallest mean absolute error (MAE) in heats of formation is obtained with PDDG/PM3 (5.6 kcal/mol), followed by PM3' (6.1 kcal/mol), PDDG/MNDO (6.6 kcal/mol), PM3 (8.1 kcal/mol), MNDO' (8.5 kcal/mol), AM1 (11.1 kcal/mol), and MNDO (14.0 kcal/mol). For normal-valent halogen-containing molecules, the PDDG methods also provide improved heats of formation over MNDO/d. Hypervalent compounds were not included in the training set and improvements over the standard NDDO methods with sp basis sets were not obtained. For small haloalkanes, the PDDG methods yield more accurate heats of formation than are obtained from density functional theory (DFT) with the B3LYP and B3PW91 functionals using large basis sets. PDDG/PM3 and PM3' also give improved binding energies over the standard NDDO methods for complexes involving halide anions, and they are competitive with B3LYP/6-311++G(d,p) results including thermal corrections. Among the semiempirical methods studied, PDDG/PM3 also generates the best agreement with high-level ab initio G2 and CCSD(T) intrinsic activation energies for S(N)2 reactions involving methyl halides and halide anions. Finally, the MAEs in ionization potentials, dipole moments, and molecular geometries show that the parameter sets for the PDDG and reoptimized NDDO methods reduce the MAEs in heats of formation without compromising the other important QM observables.     

Comparison of SCC-DFTB and NDDO-based semiempirical molecular orbital methods for organic molecules

Extensive testing of the SCC-DFTB method has been performed, permitting direct comparison to data available for NDDO-based semiempirical methods. For 34 diverse isomerizations of neutral molecules containing the elements C, H, N, and O, the mean absolute errors (MAE) for the enthalpy changes are 2.7, 3.2, 5.0, 5.1, and 7.2 kcal/mol from PDDG/PM3, B3LYP/6-31G(d), PM3, SCC-DFTB, and AM1, respectively. A more comprehensive test was then performed by computing heats of formation for 622 neutral, closed-shell H, C, N, and O-containing molecules; the MAE of 5.8 kcal/mol for SCC-DFTB is intermediate between AM1 (6.8 kcal/mol) and PM3 (4.4 kcal/mol) and significantly higher than for PDDG/PM3 (3.2 kcal/mol). Similarly, SCC-DFTB is found to be less accurate for heats of formation of ions and radicals; however, it is more accurate for conformational energetics and intermolecular interaction energies, though none of the methods perform well for hydrogen bonds with strengths under ca. 7 kcal/mol. SCC-DFTB and the NDDO methods all reproduce MP2/cc-pVTZ molecular geometries with average errors for bond lengths, bond angles, and dihedral angles of only ca. 0.01 A, 1.5 degrees , and 3 degrees . Testing was also carried out for sulfur containing molecules; SCC-DFTB currently yields much less accurate heats of formation in this case than the NDDO-based methods due to the over-stabilization of molecules containing an SO bond.     

Improved semiempirical heats of formation through the use of bond and group equivalents

Deficiencies in energetics obtained using the common semiempirical methods, AM1, PM3, and MNDO, may partly be traced to the use of pseudoatomic equivalents for conversion of molecular energies to heats of formation at 298 K. We present an alternative scheme based on the use of bond and group equivalents. Values for the 61 bond and group equivalents necessary for treatment of molecules containing the common organic elements, hydrogen, carbon, nitrogen, and oxygen have been derived. For a set of 583 neutral, closed-shell molecules mean absolute errors in AM1, PM3, and MNDO heats of formation are reduced from 6.6, 4.2, and 8.2 kcal/mol to 2.3, 2.2, and 3.0 kcal/mol, respectively. Several systematic problems are overcome in the present scheme including relative stabilities of branched hydrocarbons, energetics of conjugated systems, heats of formation of long chain hydrocarbons, and enthalpies of molecules containing multiple heteroatoms. Although the approach is restricted to molecules with well-defined functional groups, the equivalents are easy to incorporate and are chemically relevant. This revised procedure allows semiempirical methods to be used for far more reliable evaluations of heats of reactions. Estimates are made of the errors inherent in these semiempirical formalisms, arising from integral approximations and the neglect of explicit treatment of electron correlation effects, while excluding those from inadequate parameterization.     

Application of the computationally efficient self-consistent-charge density-functional tight-binding method to magnesium-containing molecules

The geometric properties, ionization potentials, heats of formation, incremental binding energies, and protonation energies for up to 75 magnesium-containing compounds have been studied using the self-consistent-charge density-functional tight-binding method (SCC-DFTB), the complete-basis set (CBS-QB3) method, traditional B3LYP density-functional theory, and a number of modern semiempirical methods such as Austin Model 1 (AM1), modified neglect of diatomic overlap without and with inclusion of d functions (MNDO, MNDO/d), and the Parametric Method 3 (PM3) and its modification (PM5). The test set contains some widely varying chemical motifs including ionic or covalent, closed-shell or radical compounds, and many biologically relevant complexes. Geometric data are compared to experiment, if available, and otherwise to previous high-level ab initio calculations or the present B3LYP results. SCC-DFTB is found to predict bond lengths to high accuracy, with the root-mean-square (RMS) error being less than half that found for the other semiempirical methods. However, SCC-DFTB performs very poorly for absolute heats of formation, giving an RMS error of 29 kcal mol(-1), but for this property B3LYP and the other semiempirical methods also yield poor but useful results with errors of 12-22 kcal mol(-1). Nevertheless, SCC-DFTB does provide useful results for biologically relevant chemical-process energies such as protonation energies (RMS error 10 kcal mol(-1), with the range 6-19 kcal mol(-1) found for the other semiempirical methods) and ligation energies (RMS error 9 kcal mol(-1), less than the errors of 12-23 kcal mol(-1) found for the other semiempirical methods). SCC-DFTB is shown to provide a computationally expedient means of calculating properties of magnesium compounds, providing results with at most double the inaccuracy of the high-quality but dramatically more-expensive B3LYP method.     

Optimization and application of lithium parameters for PM3

Lithium parameters have been optimized for Stewart's standard PM3 method. The average deviation of the heats of formation calculated for 18 reference compounds is 6.2 kcal/mol from the experimental or high-level ab initio data; the average deviation with Li/MNDO is 18.9 kcal/mol. The average error in bond lengths is also reduced by a factor of two to three. Ionization potentials and dipole moments are reproduced with comparable accuracy than Li/MNDO. However, the mean deviation for the heats of formation of both methods increases when being applied to other systems, especially to small inorgnic molecules. The applicability of the new parameter set is demonstrated further for various compounds not included in the reference set, for the calculation of the activation barriers of several lithiation reactions, as well as for the estimation of oligomerization energies of methyl lithium (including the tetramer). Li/PM3 gives reliable results even for large dimeric complexes, like [{4-(CH₃CR)C₅H₄N}Li]₂, containing TMEDA or THF as coligands and reproduces the haptotropic interaction between Li⁺ and π-systems (e.g., in benzyl lithium) as well as the relative energies and structural features of compounds with “hypervalent” atoms (e.g., in lithiated sulfones). © John Wiley & Sons, Inc.     

Evaluation of MNDO calculated proton affinities

A large set of charged species arising mainly from protonation or deprotonation of hydrocarbons, alcohols, ethers, carboxylic acids, amines, imines, and nitriles has been studied by means of the semiempirical self-consistent-field (SCF ) molecular orbital (MO ) MNDO method. From the calculated heats of formation of such charged species and those of neutral molecules, MNDO -estimated proton affinities have been obtained and the results compared with experimental gas-phase proton affinities. If the small size anions and acetylides, for which the method predicts heats of formation too large, are ruled out, the mean absolute error in calculated proton affinities is ca. 7 kcal/mol for hydrocarbons (22 acid-base pairs) and ca. 8 kcal/mol for oxygen-containing compounds (25 acid-base pairs). For nitrogen-containing molecules it is necessary to discard, in addition, the values corresponding to the protonation of alkylamines and imines in order to achieve a reasonable mean absolute error of 7–8 kcal/mol.     

Optimization of parameters for semiempirical methods II. Applications

MNDO/AM1-type parameters for twelve elements have been optimized using a newly developed method for optimizing parameters for semiempirical methods. With the new method, MNDO-PM3, the average difference between the predicted heats of formation and experimental values for 657 compounds is 7.8 kcal/mol, and for 106 hypervalent compounds, 13.6 kcal/mol. For MNDO the equivalent differences are 13.9 and 75.8 kcal/mol, while those for AM1, in which MNDO parameters are used for aluminum, phosphorus, and sulfur, are 12.7 and 83.1 kcal/mol, respectively. Average errors for ionization potentials, bond angles, and dipole moments are intermediate between those for MNDO and AM1, while errors in bond lengths are slightly reduced.     

Evaluation of PM3, AM1, and MNDO for calculation of higher energy ionization potentials

Higher ionization energies were calculated with PM3, AM1, and MNDO for three series of molecules, representative small molecules, molecules containing heteroatoms, and sterically congested alkenes. Values from PM3, AM1, and MNDO were compared to experimental values. In most instances, the semiempirical calculations correctly predict the ordering of higher ionization energies. In the absence of steric hindrance, MNDO is the method of choice. Within groups of molecules, AM1 performs better on hydrocarbons, especially twisted hydrocarbons, than PM3. PM3 commonly gives sigma orbitals which are too high in energy compared to related pi orbitals. PM3 performed better than AM1 with molecules containing oxygen, but failed to give the correct geometry for hydrogen peroxide.     

A HAM/3 study of substituted benzenes

The semiempirical HAM /3 method is used to study ionization potentials, electron affinities, heats of formation, stabilization energies, dipole moments, and charge of mono- and disubstituted benzenes. Ionization potentials and electron affinities are calculated with good accuracy. The ground state properties are generally not well calculated by HAM /3. Errors in heats of formation and dipole moments are up to 50 kcal/mol and 2.4 D .     

Improved prediction of heats of formation of energetic materials using quantum mechanical calculations

We present simple atom and group-equivalent methods that will convert quantum mechanical energies of molecules to gas phase heats of formation of CHNO systems. In addition, we predict heats of sublimation and vaporization derived from information obtained from the quantum-mechanically calculated electrostatic potential of each isolated molecule. The heats of sublimation and vaporization are combined with the aforementioned gas phase heats of formation to produce completely predicted condensed phase heats of formation. These semiempirical computational methods, calibrated using experimental information, were applied to a series of CHNO molecules for which no experimental information was used in the development of the methods. These methods improve upon an earlier effort of Rice et al. [Rice, B. M.; Pai, S. V.; Hare, J. Combust. Flame 1999, 118, 445] through the use of a larger basis set and the application of group equivalents. The root-mean-square deviation (rms) from experiment for the predicted group-equivalent gas phase heats of formation is 3.2 kcal/mol with a maximum deviation of 6.5 kcal/mol. The rms and maximum deviation of the predicted liquid heats of formation are 3.2 and 7.4 kcal/mol, respectively. Finally, the rms and maximum deviation of predicted solid heats of formation are 5.6 and 12.2 kcal/mol, respectively, an improvement in the rms of approximately 40% compared to the earlier Rice et al. predictions using atom equivalents and a smaller basis set (B3LYP/6-31G*).     

Comparison of semiempirical MO methods applied to large molecules

The heats of formation for 19 molecules have been calculated with PM3 and AM1 semiempirical methods. The values obtained have been compared with experimental heats of formation. With PM3 and AM1 the average differences between calculated and experimental heats of formation are 8.45 and 12.34 kcal mol^?1 respectively. There are significant differences when large molecules are considered: this suggests that the parameterization should be done including larger molecules.     

Testing electronic structure methods for describing intermolecular H...H interactions in supramolecular chemistry

In this article a wide variety of computational approaches (molecular mechanics force fields, semiempirical formalisms, and hybrid methods, namely ONIOM calculations) have been used to calculate the energy and geometry of the supramolecular system 2-(2'-hydroxyphenyl)-4-methyloxazole (HPMO) encapsulated in beta-cyclodextrin (beta-CD). The main objective of the present study has been to examine the performance of these computational methods when describing the short range H. H intermolecular interactions between guest (HPMO) and host (beta-CD) molecules. The analyzed molecular mechanics methods do not provide unphysical short H...H contacts, but it is obvious that their applicability to the study of supramolecular systems is rather limited. For the semiempirical methods, MNDO is found to generate more reliable geometries than AM1, PM3 and the two recently developed schemes PDDG/MNDO and PDDG/PM3. MNDO results only give one slightly short H...H distance, whereas the NDDO formalisms with modifications of the Core Repulsion Function (CRF) via Gaussians exhibit a large number of short to very short and unphysical H...H intermolecular distances. In contrast, the PM5 method, which is the successor to PM3, gives very promising results. Our ONIOM calculations indicate that the unphysical optimized geometries from PM3 are retained when this semiempirical method is used as the low level layer in a QM:QM formulation. On the other hand, ab initio methods involving good enough basis sets, at least for the high level layer in a hybrid ONIOM calculation, behave well, but they may be too expensive in practice for most supramolecular chemistry applications. Finally, the performance of the evaluated computational methods has also been tested by evaluating the energetic difference between the two most stable conformations of the host(beta-CD)-guest(HPMO) system.     

PM3-compatible zinc parameters optimized for metalloenzyme active sites

Recent studies have shown that semiempirical methods (e.g., PM3 and AM1) for zinc-containing compounds are unreliable for modeling structures containing zinc ions with ligand environments similar to those observed in zinc metalloenzymes. To correct these deficiencies a reparameterization of zinc at the PM3 level was undertaken. In this effort we included frequency corrected B3LYP/6-311G* zinc metalloenzyme ligand environments along with previously utilized experimental data. Average errors for the heats of formation have been reduced from 46.9 kcal/mol (PM3) to 14.2 kcal/mol for this new parameter set, termed ZnB for "Zinc, Biological." In addition, the new parameter sets predict geometries for the Bacillus fragilis active site model and other zinc metalloenzyme mimics that are qualitatively in agreement with high-level ab initio results, something existing parameter sets failed to do.     

Theoretical studies of [n]paracyclophanes and their valence isomers. I. Geometries,strain energies,and enthalpies of the inter-conversions of [n]paracyclophanes and their Dewar benzee isomers

Four semiempirical methods (AM1, MNDO, PM3, and MINDO/3) are used to calculate the deformation angles of [n]paracyclophanes and their Dewar benzene isomers for n = 3… 10. The results obtained by all these methods are in good agreement with data from X-ray studies. We have determined the strain energies that, in both series of compounds, are due to two components: (1) the strain energy of deformation of the cycle (aromatic or Dewar Benzene skeletons) and (2) the strain energy of the oligomethylene chain. In [6]paracyclophane, the strain energy [SE_ring(MNDO) ≈? 32.9 kcal/mol] almost compensates the resonance energy (E_resonance ≈ 36 kcal/mol) so that its chemical properties are closer to alkenes than to benzenic compounds. To better reproduce the enthalpy of the valence isomerization [n]Dewar bezene → [n]paracyclophane, which is poorly calculated with these methods, a correction is proposed and the reaction enthalpy of [6]paracyclophane is estimated to be about ΔH_r ≈ 15 ± 15 kcal/mol. It is found that MNDO and MINDO/3 need the smallest corrections, but MNDO leads to better geometries than MINDO/3. In conclusion, MNDO seems to be the best technique for further studies of these compounds. © 1992 by John Wiley & Sons, Inc.     

The implementation of a fast and accurate QM/MM potential method in Amber

Version 9 of the Amber simulation programs includes a new semi-empirical hybrid QM/MM functionality. This includes support for implicit solvent (generalized Born) and for periodic explicit solvent simulations using a newly developed QM/MM implementation of the particle mesh Ewald (PME) method. The code provides sufficiently accurate gradients to run constant energy QM/MM MD simulations for many nanoseconds. The link atom approach used for treating the QM/MM boundary shows improved performance, and the user interface has been rewritten to bring the format into line with classical MD simulations. Support is provided for the PM3, PDDG/PM3, PM3CARB1, AM1, MNDO, and PDDG/MNDO semi-empirical Hamiltonians as well as the self-consistent charge density functional tight binding (SCC-DFTB) method. Performance has been improved to the point where using QM/MM, for a QM system of 71 atoms within an explicitly solvated protein using periodic boundaries and PME requires less than twice the cpu time of the corresponding classical simulation.     

Improving description of hydrogen bonds at the semiempirical level: water–water interactions as test case

Hydrogen bonding is not well described by available semiempirical theories. This is an important restriction because hydrogen bonds represent a key feature in many chemical and biochemical processes, besides being responsible for the singular properties of water. In this study, we describe a possible solution to this problem. The basic idea is to replace the nonphysical gaussian correction functions (GCF) appearing in the core–core repulsion terms of most MNDO‐based semiempirical methods by a simple function exhibiting the correct physical behavior in the whole range of intermolecular separation distances. The parameterized interaction function (PIF) is the sum of atom‐pair contributions, each one having five adjustable parameters. In this work, the approach is used to study water–water interactions. The parameters are optimized to reproduce a reference ab initio intermolecular energy surface for the water–water dimer obtained at the MP2/aug‐cc‐pVQZ level. OO, OH, and HH parameters are reported for the PM3 method. The results of PM3‐PIF calculations remarkably improve qualitatively and quantitatively those obtained at the standard PM3 level, both for water–dimer properties and for water clusters up to the hexamer. For example, the root‐mean‐square deviation of the PM3‐PIF interaction energies, with respect to ab initio values obtained using 700 points of the water dimer surface, is only 0.47 kcal/mol. This value is much smaller than that obtained using the standard PM3 method (4.2 kcal/mol). The PM3‐PIF water dimer energy minimum (−5.0 kcal/mol) is also much closer to ab initio data (−5.0±0.01 kcal/mol) than PM3 (−3.50 kcal/mol). The method is therefore promising for the development of new semiempirical approaches as well as for application of combined quantum mechanics and molecular mechanics techniques to investigate chemical processes in water. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 572–581, 2000     

Extension of MNDO to d orbitals: Parameters and results for the halogens

A recently proposed extension of the MNDO formalism to d orbitals has been parameterized for the halogens CI, Br, and I. Extensive test calculations indicate slight consistent improvements for normalvalent molecules and dramatic improvements for hypervalent molecules, in comparison with established MNDO -type methods without d orbitals. The mean absolute errors in calculated heats of formation are 3.9 kcal/mol for 155 normalvalent compounds and 2.8 kcal/mol for 23 hypervalent compounds. The predicted structures of the hypervalent molecules are qualitatively correct, with a mean absolute error of 2° in 19 bond angles.     

Evaluation of MINDO/3 calculated structures. II. Branching errors in alkanes and cycloalkanes

MINDO/3 calculations have been carried out for a series of branched chain alkanes in order to assess effects of branching on calculated geometries and heats of formation (ΔH_f). With vicinal branching, MINDO/3 calculates the central C? C bond to be too long. Bond angles are also found to be distorted. Errors in calculated heats of formation are large when geminal branching is present and significant with vicinal branching. Branching error corrections for ΔH_f have been derived and applied to a separate series of branched acyclic and cyclic compounds. For the test sample, application of the branching error corrections gave calculated structures of acyclic branched hydrocarbons with heats of formation having an average absolute error of 1.3 kcal/mole rather than 17.3 kcal/mole before correction. Cyclic branched hydrocarbons are shown to be less well corrected. Calculations of heats of reaction have also been carried out for some isomerization and cyclization reactions using the MINDO/3 and MNDO methods. It is clear from the comparisons that MNDO calculations give less severe errors for highly branched compounds but the errors are still substantial. For prediction of heats of reaction, the error-corrected calculations are shown to be superior to the “raw” calculations obtained by MINDO/3 or MNDO.     

Energy correctors for accurate prediction of molecular energies

Energy correctors are introduced for the calculation of molecular energies of compounds containing first row atoms (Li-F) to modify ab initio molecular orbital calculations of energies to better reproduce experimental results. Four additive correctors are introduced to compensate for the differences in the treatment of molecules with different spin multiplicities and multiplicative correctors are also calculated for the electronic and zero-point vibrational energies. These correctors, individually and collectively yield striking improvements in the atomization energies for several ab initio methods. We use as training set the first row subset of molecules from the G1 basis of molecules; when the correctors are applied to other molecules not included in the training set, selected from the G3 basis, similar improvements in the atomization energies are obtained. The special case of the B3PW91/cc-pVTZ yields an average error of 1.2 kcal/mol, which is already within a chemical accuracy and comparable to the Gaussian-n theories accuracy. The very inexpensive B3PW91/6-31G** yields an average error of 2.1 kcal/mol using the correctors. Methods considered unsuitable for energetics such as HF and LSDA yield corrected energies comparable to those obtained with the best highly correlated methods.