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.     

PDDG/PM3 and PDDG/MNDO: improved semiempirical methods

Two new semiempirical methods employing a Pairwise Distance Directed Gaussian modification have been developed: PDDG/PM3 and PDDG/MNDO; they are easily implemented in existing software, and yield heats of formation for compounds containing C, H, N, and O atoms with significantly improved accuracy over the standard NDDO schemes, PM5, PM3, AM1, and MNDO. The PDDG/PM3 results for heats of formation also show substantial improvement over density functional theory with large basis sets. The PDDG modifications consist of a single function, which is added to the existing pairwise core repulsion functions within PM3 and MNDO, a reparameterized semiempirical parameter set, and modified computation of the energy of formation of a gaseous atom. The PDDG addition introduces functional group information via pairwise atomic interactions using only atom-based parameters. For 622 diverse molecules containing C, H, N, and O atoms, mean absolute errors in calculated heats of formation are reduced from 4.4 to 3.2 kcal/mol and from 8.4 to 5.2 kcal/mol using the PDDG modified versions of PM3 and MNDO over the standard versions, respectively. Several specific problems are overcome, including the relative stability of hydrocarbon isomers, and energetics of small rings and molecules containing multiple heteroatoms. The internal consistency of PDDG energies is also significantly improved, enabling more reliable analysis of isomerization energies and trends across series of molecules; PDDG isomerization energies show significant improvement over B3LYP/6-31G* results. Comparison of heats of formation, ionization potentials, dipole moments, isomer, and conformer energetics, intermolecular interaction energies, activation energies, and molecular geometries from the PDDG techniques is made to experimental data and values from other semiempirical and ab initio methods.     

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.     

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.     

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.     

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.     

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.     

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*).     

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.     

RM1: a reparameterization of AM1 for H, C, N, O, P, S, F, Cl, Br, and I

Twenty years ago, the landmark AM1 was introduced, and has since had an increasingly wide following among chemists due to its consistently good results and time-tested reliability--being presently available in countless computational quantum chemistry programs. However, semiempirical molecular orbital models still are of limited accuracy and need to be improved if the full potential of new linear scaling techniques, such as MOZYME and LocalSCF, is to be realized. Accordingly, in this article we present RM1 (Recife Model 1): a reparameterization of AM1. As before, the properties used in the parameterization procedure were: heats of formation, dipole moments, ionization potentials and geometric variables (bond lengths and angles). Considering that the vast majority of molecules of importance to life can be assembled by using only six elements: C, H, N, O, P, and S, and that by adding the halogens we can now build most molecules of importance to pharmaceutical research, our training set consisted of 1736 molecules, representative of organic and biochemistry, containing C, H, N, O, P, S, F, Cl, Br, and I atoms. Unlike AM1, and similar to PM3, all RM1 parameters have been optimized. For enthalpies of formation, dipole moments, ionization potentials, and interatomic distances, the average errors in RM1, for the 1736 molecules, are less than those for AM1, PM3, and PM5. Indeed, the average errors in kcal x mol(-1) of the enthalpies of formation for AM1, PM3, and PM5 are 11.15, 7.98, and 6.03, whereas for RM1 this value is 5.77. The errors, in Debye, of the dipole moments for AM1, PM3, PM5, and RM1 are, respectively, 0.37, 0.38, 0.50, and 0.34. Likewise, the respective errors for the ionization potentials, in eV, are 0.60, 0.55, 0.48, and 0.45, and the respective errors, in angstroms, for the interatomic distances are 0.036, 0.029, 0.037, and 0.027. The RM1 average error in bond angles of 6.82 degrees is only slightly higher than the AM1 figure of 5.88 degrees, and both are much smaller than the PM3 and PM5 figures of 6.98 degrees and 9.83 degrees, respectively. Moreover, a known error in PM3 nitrogen charges is corrected in RM1. Therefore, RM1 represents an improvement over AM1 and its similar successor PM3, and is probably very competitive with PM5, which is a somewhat different model, and not fully disclosed. RM1 possesses the same analytical construct and the same number of parameters for each atom as AM1, and, therefore, can be easily implemented in any software that already has AM1, not requiring any change in any line of code, with the sole exception of the values of the parameters themselves.     

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.     

Theoretical studies on heats of formation for cubylnitrates using density functional theory B3LYP method and semiempirical MO methods

The heats of formation (HOF) have been calculated for all the 21 cubylnitrate compounds using the semiemprical molecular orbital (MO) methods (MINDO/3, MNDO, AM1, and PM3) and for 8 of 21 cubylnitrates containing 1–4 ? ONO₂ groups using the density functional theory (DFT) method at the B3LYP/6‐31G* level by means of designed isodesmic reactions. The cubane cage skeletons in cubylnitrate molecules have been kept in setting up isodesmic reactions to produce more accurate and reliable results. It is found that there are good linear relationships between the HOFs of the 8 cubylnitrates calculated using B3LYP/6‐31G* and two semiempirical MO (PM3 and AM1) methods, and the linear correlation coefficients of PM3 and AM1 methods are 0.9901 and 0.9826, respectively. Subsequently, the accurate HOFs at B3LYP/6‐31G* level of other 13 cubylnitrates containing 4–8 ? ONO₂ groups are obtained by systematically correcting their PM3‐calculated HOFs. Compared with noncaged nitrates, all the 21 cubylnitrates have high heats of formation implying that they may be very powerful energetic materials and have highly exploitable value. The relationship between the HOFs and the molecular structures of cubylnitrates has been discussed. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

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.     

Correlation energy functionals dependent on an effective number of electrons: charged species and equilibrium geometries

Recently proposed spin-dependent and spin-independent correlation energy functionals [Perez-Jimenez et al., J. Chem. Phys. 116, 10571 (2002)] based on an effective number of electrons N are extended to deal with charged systems. By introducing the concept of an effective atomic number Z analogous to N, the spin-dependent functional in combination with Becke's exchange [Becke, Phys. Rev. A 38, 3098 (1988)] yields a mean absolute error (MAE) of 5.4 kcal/mol for the 88 ionization potentials and 58 electron affinities included in the extended G2 set, and a MAE of 4.1 kcal/mol for the 312 data comprising the above plus the 148 enthalpies of formation of the extended G2 set and the 18 total energies of the neutral atoms H through Ar. Geometry optimizations performed on the 53 molecules of the G2-1 test set with the above combination of exchange and correlation functionals yield MAEs of 0.017 A and 1.5 degrees for the 68 bond lengths and 29 angles analyzed as compared with the experimental estimates.     

Bond additivity corrections for G3B3 and G3MP2B3 quantum chemistry methods

We have developed bond additivity correction (BAC) procedures for the G3-based quantum chemistry methods, G3B3 and G3MP2B3. We denote these procedures as BAC-G3B3 and BAC-G3MP2B3. We apply the procedures to compounds containing atoms from the first three rows of the periodic table including H, B, C, N, O, F, Al, Si, P, S, and Cl atoms. The BAC procedure applies atomic, molecular, and pairwise bond corrections to theoretical heats of formation of molecules. The BAC-G3B3 and BAC-G3MP2B3 procedures require parameters for each atom type but not for each bond type. These parameters have been obtained by minimizing the error between the BAC-G3B3 and BAC-G3MP2B3 predictions and the experimental heats of formation for a 155 molecule reference set, containing open and closed shell molecules representing various functional groups, multireference configurations, isomers, and degrees of saturation. As compared to former BAC-MP4, BAC-G2, and BAC-hybrid methods, BAC-G3B3 provides better agreement with experiment for a wider range of chemical moieties, including highly oxidized species involving SOx s, NOx s, POx s, and halogens. The BAC-G3B3 and BAC-G3MP2B3 procedures are applied to an extended test suite involving 273 compounds. We assess the overall quality of BAC-G3B3 with experiments and other theoretical approaches. For the reference set, the average error for the BAC-G3B3 results is 0.44 kcal/mol as compared to 0.82 kcal/mol for the raw G3B3. For the extended test set, the average error for the BAC-G3B3 results is 0.91 kcal/mol as compared to 1.38 kcal/mol for the raw G3B3. As compared to the other BAC procedures, the improved predictive capability of BAC-G3B3 and BAC-G3MP2B3 procedures is, to a large extent, due to the improved quality of G3-based methods resulting in much smaller BAC correction terms.     

Energetics and structure of glycine and alanine based model peptides: Approximate SCC-DFTB,AM1 and PM3 methods in comparison with DFT,HF and MP2 calculations

We calculate relative energies and geometries of important secondary structural elements for small glycine and alanine based polypeptides containing up to eight residues. We compare the performance of the approximate methods AM1, PM3 and self-consistent charge, density-functional tight-binding (SCC-DFTB) to density-functional theory (DFT), Hartree–Fock (HF) and MP2. The SCC-DFTB is able to reproduce structures and relative energies of various peptide models reliably compared to DFT results. The AM1 and PM3 methods show deficiencies in describing important secondary structure elements like extended, helical or turn structures. The discrepancies between different ab initio (HF, MP2) and DFT (B3LYP) methods for medium sized basis sets (6-31G*) also show the need for higher level calculations, since systematic errors found for small molecules may add up when investigating longer polypeptides.     

Semiempirical computation of homolytic O-H bond dissociation energies of alcohols: comparison of the AM1 and PM3 methods

The heats of formation of various alcohols and alkoxy radicals were calculated using the AM1 and PM3 semiempirical methods, which were then used to calculate the bond dissociation energies of the alcohols. Both restricted Hartree-Fock (RHF) and unrestricted Hartree-Fock (UHF) calculations were performed to determine which technique was most applicable to the computation of bond dissociation energies within the semiempirical frameworks. It was determined that AM1/RHF calculations gave the most accurate results for O-H bond dissociation energies of alcohols. The effect of using configuration interaction calculations to calculate bond dissociation energies within the semiempirical framework was also examined.     

PM3 study of the proton affinities of 2-, 3-, and 4-monosubstituted pyridines in the gas phase

Heats of formation (ΔH_f) and proton affinities (PA) of 2-, 3-, and 4-monosubstituted pyridines in the gas phase are calculated using the AM1 and PM3 semiempirical methods. The following substitutents are considered: NO₂, CN, CF₃, CHO, F, Cl, COCH₃, H, CH₃, OCH₃, SCH₃, NH₂, and N(CH₃)₂. The results are compared with the experimental data. Both methods reproduce the ΔH_f with comparble accuracy; the rms deviations are 4.1 (AM1) and 4.5 kcal/mol (PM3) for the free bases and 9.5 (AM1) and 9.7 kcal/mol (PM3) for their conjugated acids. The PA are systematically underestimated by both methods, but AM1 appears to be clearly better than PM3 for reproducing the experimental values. The rms deviations for AM1 and PM3 are 5.1 and 9.6 kcal/mol, respectively. This is due to a cancellation of systematic errors in the calculated ΔH_f in the AM1 case and to a summation of the errors in the PM3 case. Both methods correctly reproduce conformations of the molecules under consideration.     

Heats of formation of organic molecules by ab initio methods: Thiaalkanes

The bond energy scheme for calculating heats of formation of organic molecules from ab initio data (6–31G*) has been extended to include 24 compounds containing sulfur in the sulfide oxidation state. The rms deviation from the experimental values for these compounds is 0.54 kcal/mol, which is approximately experimental error.