Molecular model with quantum mechanical bonding information

The molecular structure can be defined quantum mechanically thanks to the theory of atoms in molecules. Here, we report a new molecular model that reflects quantum mechanical properties of the chemical bonds. This graphical representation of molecules is based on the topology of the electron density at the critical points. The eigenvalues of the Hessian are used for depicting the critical points three-dimensionally. The bond path linking two atoms has a thickness that is proportional to the electron density at the bond critical point. The nuclei are represented according to the experimentally determined atomic radii. The resulting molecular structures are similar to the traditional ball and stick ones, with the difference that in this model each object included in the plot provides topological information about the atoms and bonding interactions. As a result, the character and intensity of any given interatomic interaction can be identified by visual inspection, including the noncovalent ones. Because similar bonding interactions have similar plots, this tool permits the visualization of chemical bond transferability, revealing the presence of functional groups in large molecules.     

Chemical potential inequality principle

Regional density functional theory has been extended to treat irreversible thermodynamic electronic processes for application to adiabatic electron-transfer processes of chemical reactions. Onsager's local equilibrium hypothesis is slightly modified to take into account the quantum mechanical nature of the electron. The quantum mechanical interference effect has been demonstrated to be included in the entropy production rate formula associated with electron transfer through an interface. A new formula for the determination of the transition state of a chemical reaction has been postulated that corresponds to the maximum of the regional electron transferability. A quantum mechanical law of mass action has been established and applied to prove the regional electrochemical potential inequality principle. Received: 1 July 1998 / Accepted: 2 September 1998 / Published online: 8 February 1999     

Derivation of class II force fields: V. Quantum force field for amides,peptides, and related compounds

As the field of biomolecular structure advances, there is an ever-growing need for accurate modeling of molecular energy surfaces to simulate and predict the properties of these important systems. To address this need, a second generation amide force field for use in simulations of small organics as well as proteins and peptides has been derived. The critical question of what accuracy can be expected from calculations in general, and with this class II force field in particular, is addressed for structural, dynamic, and energetic properties. The force field is derived from a recent methodology we have developed that involves the systematic use of quantum mechanical observables. Systematic ab initio calculations were carried out for numerous configurations of 17 amide and related compounds. Relative energies and first and second derivatives of the energy of 638 structures of these compounds resulted in 140,970 ab initio quantum mechanical observables. The class II peptide quantum mechanical force field (QMFF), containing 732 force constants and reference values, was parameterized against these observables. A major objective of this work is to help establish the role of anharmonicity and coupling in improving the accuracy of molecular force fields, as these terms have not yet become an agreed upon standard in the ever more extensive simulations being used to probe biomolecular properties. This has been addressed by deriving a class I harmonic diagonal force field (HDFF), which was fit to the same energy surface as the QMFF, thus providing an opportunity to quantify the effects of these coupling and anharmonic contributions. Both force field representations are assessed in terms of their ability to fit the observables. They have also been tested by calculating the properties of 11 stationary states of these amide molecules. Optimized structures, vibrational frequencies, and conformational energies obtained from the quantum calculations and from both the QMFF and the HDFF are compared. Several strained and derivatized compounds including urea, formylformamide, and butyrolactam are included in these tests to assess the range of applicability (transferability) of the force fields. It was found that the class II coupled anharmonic force field reproduced the structures, energies, and vibrational frequencies significantly more faithfully than the class I harmonic diagonal force field. An important measure, rms energy deviation, was found to be 1.06 kcal/mol with the class II force field, and 2.30 kcal/mol with the harmonic diagonal force field. These deviations represent the error in relative configurational energy differences for strained and distorted structures calculated with the force fields compared with quantum mechanics. This provides a measure of the accuracy that might be expected in applications where strain may be important such as calculating the energy of a system as it approaches a (rotational) barrier, in ligand binding to a protein, or effects of introducing substituents into a molecule that may induce strain. Similar results were found for structural properties. Protein dynamics is becoming of ever-increasing interest, and, to simulate dynamic properties accurately, the dynamic behavior of model compounds needs to be well accounted for. To this end, the ability of the class I and class II force fields to reproduce the vibrational frequencies obtained from the quantum energy surface was assessed. An rms deviation of 43 cm⁻¹ was achieved with the coupled anharmonic force field, as compared to 105 cm⁻¹ with the harmonic diagonal force field. Thus, the analysis presented here of the class II force field for the amide functional group demonstrates that the incorporation of anharmonicity and coupling terms in the force field significantly improves the accuracy and transferability with regard to the simulation of structural, energetic, and dynamic properties of amides. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 430–458, 1998     

Assessment of performance of the general purpose polarizable force field QMPFF3 in condensed phase

The recently introduced force field (FF) QMPFF3 is thoroughly validated in gas, liquid, and solid phases. For the first time, it is demonstrated that a physically well-grounded general purpose FF fitted exclusively to a comprehensive set of high level vacuum quantum mechanical data applied as it is to simulation of condensed phase provides high transferability for a wide range of chemical compounds. QMPFF3 demonstrates accuracy comparable with that of the FFs explicitly fitted to condensed phase data, but due to high transferability it is expected to be successful in simulating large molecular complexes.     

Derivation of class II force fields. I. Methodology and quantum force field for the alkyl functional group and alkane molecules

A new method for deriving force fields for molecular simulations has been developed. It is based on the derivation and parameterization of analytic representations of the ab initio potential energy surfaces. The general method is presented here and used to derive a quantum mechanical force field (QMFF) for alkanes. It is based on sampling the energy surfaces of 16 representative alkane species. For hydrocarbons, this force field contains 66 force constants and reference values. These were fit to 128,376 quantum mechanical energies and energy derivatives describing the energy surface. The detailed form of the analytic force field expression and the values of all resulting parameters are given. A series of computations is then performed to test the ability of this force field to reproduce the features of the ab initio energy surface in terms of energies as well as the first and second derivatives of the energies with respect to molecular deformations. The fit is shown to be good, with rms energy deviations of less than 7% for all molecules. Also, although only two atom types are employed, the force field accounts for the properties of both highly strained species, such as cyclopropane and methylcyclopropanes, as well as unstrained systems. The information contained in the quantum energy surface indicates that it is significantly anharmonic and that important intramolecular coupling interactions exist between internals. The representation of the nature of these interactions, not present in diagonal, quadratic force fields (Class I force fields), is shown to be important in accounting accurately for molecular energy surfaces. The Class II force field derived from the quantum energy surface is characterized by accounting for these important intramolecular forces. The importance of each 4.2 to 18.2%. This fourfold increase in the second derivative error dramatically demonstrates the importance of bond anharmonicity in the ab initio potential energy surface. The Class II force field derived from the quantum energy surface is characterized by accounting for these important intramolecular forces. The importance of each of the interaction terms of the potential energy function has also been assessed. Bond anharmonicity, angle anharmonicity, and bond/angle, bond/torsion, and angle/angle/ torsion cross-term interactions result in the most significant overall improvement in distorted structure energies and energy derivatives. The implications of each energy term for the development of advanced force fields is discussed. Finally, it is shown that the techniques introduced here for exploring the quantum energy surface can be used to determine the extent of transferability and range of validity of the force field. The latter is of crucial importance in meeting the objective of deriving a force field for use in molecular mechanics and dynamics calculations of a wide range of molecules often containing functional groups in novel environments. © 1994 by John Wiley & Sons, Inc.     

Force-field parametrization of retro-inverso modified residues: Development of torsional and electrostatic parameters

Summary Torsional and the electrostatic parameters for molecular mechanics studies of retro-inverso modified peptides have been developed using quantum mechanical calculations. The resulting parameters have been compared with those calculated for conventional peptides. Rotational profiles, which were obtained spanning the corresponding dihedral angle, were corrected by removing the energy contributions associated to changes in interactions different from torsion under study. For this purpose, the torsional energy associated to each point of the profiles was estimated as the corresponding quantum mechanical energy minus the bonding and nonbonding energy contributions produced by the perturbations that the variation of the spanned dihedral angle causes in the bond distances, bond angles and the other dihedral angles. These energies were calculated using force-field expressions. The corrected profiles were fitted to a three-term Fourier expansion to derive the torsional parameters. Atomic charges for retro-inverso modified residues were derived from the rigorously calculated quantum mechanical electrostatic potential. Furthermore, the reliability of electrostatic models based on geometry-dependent charges and fixed charges has been examined.     

Anisotropic nonadditive ab initio force field for noncovalent interactions of H2

A quantum mechanical polarizable force field (QMPFF) has been applied to the noncovalent interactions of molecular hydrogen as well as closed-shell monoatomic species (CSMS): rare gases, alkali cations, and halide anions. The importance of all the main energy components is demonstrated: electrostatics (including penetration effect), exchange repulsion, dispersion, and induction. As the MP2 level of quantum mechanics, which is used to parametrize QMPFF, significantly underestimates the H2-H2 dimer binding energy, the force field was refined using state-of-the-art CCSD(T) data. The approach demonstrates excellent transferability, which is confirmed by accurate reproduction of mixed H2-CSMS dimers and the second virial coefficient of hydrogen vapor.     

Atomic and bond topological properties of the tripeptide L-alanyl-L-alanyl-L-alanine based on its experimental charge density obtained at 20 K

A 20 K high resolution X-ray data set of L-Ala-L-Ala-L-Ala*1/2 H2O was measured using an ultra-low temperature laboratory setup, that combines area detection and a closed cycle helium cryostat. The charge density determination includes integration of atomic basins and topological analysis according to Bader's quantum theory of atoms in molecules. Two tripeptide units are found in the asymmetric unit, allowing the assessment of transferability of bond topological and atomic properties taking also into consideration previous data of oligopeptides. With respect to invariom modeling the limits of such transferability are investigated and the results of this study show the validity of the nearest/next-nearest neighbour approximation and support the use of database approaches for electron density modeling of macromolecules.     

Derivation of class II force fields. VIII. Derivation of a general quantum mechanical force field for organic compounds

A class II valence force field covering a broad range of organic molecules has been derived employing ab initio quantum mechanical "observables." The procedure includes selecting representative molecules and molecular structures, and systematically sampling their energy surfaces as described by energies and energy first and second derivatives with respect to molecular deformations. In this article the procedure for fitting the force field parameters to these energies and energy derivatives is briefly reviewed. The application of the methodology to the derivation of a class II quantum mechanical force field (QMFF) for 32 organic functional groups is then described. A training set of 400 molecules spanning the 32 functional groups was used to parameterize the force field. The molecular families comprising the functional groups and, within each family, the torsional angles used to sample different conformers, are described. The number of stationary points (equilibria and transition states) for these molecules is given for each functional group. This set contains 1324 stationary structures, with 718 minimum energy structures and 606 transition states. The quality of the fit to the quantum data is gauged based on the deviations between the ab initio and force field energies and energy derivatives. The accuracy with which the QMFF reproduces the ab initio molecular bond lengths, bond angles, torsional angles, vibrational frequencies, and conformational energies is then given for each functional group. Consistently good accuracy is found for these computed properties for the various types of molecules. This demonstrates that the methodology is broadly applicable for the derivation of force field parameters across widely differing types of molecular structures. Copyright 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1782-1800, 2001     

Toward a physical understanding of electron-sharing two-center bonds. II. Pseudo-potential based analysis of diatomic molecules

A dozen homo- and heteropolar bonds in a series of diatomic molecules are analyzed by an energy partitioning based on the concepts developed in part I of this series (J Comput Chem 28:411–422, 2007). The different bond types are characterized by various physical contributions to the bond energy, namely classical potential interaction, quantum–mechanical orbital interference, and bond stabilizations by atomic configurational promotion, radial deformation, and angular polarization. The effects of the atomic cores are accounted for by means of pseudo-potential operators. Different atomic cores cause specific bond differences. The various bonding mechanisms can be characterized by several parameters. They describe the quantum–mechanical reduction in the ‘electronic kinetic energy pressure’ due to delocalization (sharing interference) and the increase in the ‘molecular potential energy pull’ due to overlap of atomic electron densities with adjacent atoms in the molecule. In addition, there are one- and two-electron relaxations whose distinctive features depend on the nature of the cores.     

Population analysis of pair densities: A link between quantum chemical and classical picture of chemical structure

The pairon population analysis based on the geminal expansion of pair densities is introduced and applied. As demonstrated by numerical data calculated for a series of simple molecules by the semiempirical MNDO method, the resulting populations provide a new simple means of visualizing the molecular structure. In addition to the reproduction of classical structural formula including the multiplicity of individual bonds, the resulting populations confirm the transferability of bond energies and also provide a simple interpretation of the concept of quantum chemical valence. © 1994 John Wiley & Sons, Inc.     

Quantum chemical investigation of hydrogen-bond strengths and partition into donor and acceptor contributions

We present a simple increment model for use in the rapid scoring of hydrogen bond strengths employing 15 chemically diverse donor and 28 acceptor terms. The increments cover a large variety of hydrogen bond donor and acceptor groups and are more specific than SYBYL atom types. The increments have been fitted to quantum chemical ab initio interaction energies of 81 small hydrogen‐bonded complexes determined at the level of second‐order Møller‐Plesset perturbation theory (MP2). The complexes have been chosen such as to represent the most important types of donor‐acceptor pairs found in biological systems. Sulphur is found to be a strong hydrogen bond acceptor while its donor capacities are weak. By taking CH acidic H donors into account, a linear correlation between MP2 energies and the increment model with a coefficient of correlation of r² = 0.994 has been accomplished. The transferability of the fitted parameters has been assessed on a second set of complexes including larger molecules of biological relevance. Very good agreement has been achieved for noncyclic hydrogen bonds. Cooperative effects are not accounted for by the current increment model. For this reason, binding energies of strong cyclic hydrogen bonds, as e.g. present in DNA base pairs, are underestimated by about 30–40%. © 2007 Wiley Periodicals, Inc. J Comput Chem, 2007     

Determining the strengths of hydrogen bonds in solid-state ammonia and urea: insight from periodic DFT calculations

Plane-wave density functional theory has been applied to determine the strengths of hydrogen bonds in the phase I crystal structures of ammonia and urea. For ammonia, each component of the trifurcated hydrogen bond has been found to be almost as strong as a standard N-H.N interaction, and for urea the strengths of the two different N-H.O interactions have been determined by a quantum mechanical technique for the first time.     

Machine learning for atomic forces in a crystalline solid: Transferability to various temperatures

Recently, machine learning has emerged as an alternative, powerful approach for predicting quantum‐mechanical properties of molecules and solids. Here, using kernel ridge regression and atomic fingerprints representing local environments of atoms, we trained a machine‐learning model on a crystalline silicon system to directly predict the atomic forces at a wide range of temperatures. Our idea is to construct a machine‐learning model using a quantum‐mechanical dataset taken from canonical‐ensemble simulations at a higher temperature, or an upper bound of the temperature range. With our model, the force prediction errors were about 2% or smaller with respect to the corresponding force ranges, in the temperature region between 300 K and 1650 K. We also verified the applicability to a larger system, ensuring the transferability with respect to system size.     

Nuclear quantum effects on an enzyme-catalyzed reaction with reaction path potential: proton transfer in triosephosphate isomerase

Nuclear quantum mechanical effects have been examined for the proton transfer reaction catalyzed by triosephosphate isomerase, with the normal mode centroid path integral molecular dynamics based on the potential energy surface from the recently developed reaction path potential method. In the simulation, the primary and secondary hydrogens and the C and O atoms involving bond forming and bond breaking were treated quantum mechanically, while all other atoms were dealt classical mechanically. The quantum mechanical activation free energy and the primary kinetic isotope effects were examined. Because of the quantum mechanical effects in the proton transfer, the activation free energy was reduced by 2.3 kcal/mol in comparison with the classical one, which accelerates the rate of proton transfer by a factor of 47.5. The primary kinetic isotope effects of kH/kD and kH/kT were estimated to be 4.65 and 9.97, respectively, which are in agreement with the experimental value of 4+/-0.3 and 9. The corresponding Swain-Schadd exponent was predicted to be 3.01, less than the semiclassical limit value of 3.34, indicating that the quantum mechanical effects mainly arise from quantum vibrational motion rather than tunneling. The reaction path potential, in conjunction with the normal mode centroid molecular dynamics, is shown to be an efficient computational tool for investigating the quantum effects on enzymatic reactions involving proton transfer.     

Design-atom approach for the quantum mechanical/molecular mechanical covalent boundary: a design-carbon atom with five valence electrons

A critical issue underlying the accuracy and applicability of the combined quantum mechanical/molecular mechanical (QM/MM) methods is how to describe the QM/MM boundary across covalent bonds. Inspired by the ab initio pseudopotential theory, here we introduce a novel design atom approach for a more fundamental and transparent treatment of this QM/MM covalent boundary problem. The main idea is to replace the boundary atom of the active part with a design atom, which has a different number of valence electrons but very similar atomic properties. By modifying the Troullier-Martins scheme, which has been widely employed to construct norm-conserving pseudopotentials for density functional calculations, we have successfully developed a design-carbon atom with five valence electrons. Tests on a series of molecules yield very good structural and energetic results and indicate its transferability in describing a variety of chemical bonds, including double and triple bonds.     

Charge-dependent model for many-body polarization, exchange, and dispersion interactions in hybrid quantum mechanical/molecular mechanical calculations

This work explores a new charge-dependent energy model consisting of van der Waals and polarization interactions between the quantum mechanical (QM) and molecular mechanical (MM) regions in a combined QMMM calculation. van der Waals interactions are commonly treated using empirical Lennard-Jones potentials, whose parameters are often chosen based on the QM atom type (e.g., based on hybridization or specific covalent bonding environment). This strategy for determination of QMMM nonbonding interactions becomes tedious to parametrize and lacks robust transferability. Problems occur in the study of chemical reactions where the "atom type" is a complex function of the reaction coordinate. This is particularly problematic for reactions, where atoms or localized functional groups undergo changes in charge state and hybridization. In the present work we propose a new model for nonelectrostatic nonbonded interactions in QMMM calculations that overcomes many of these problems. The model is based on a scaled overlap model for repulsive exchange and attractive dispersion interactions that is a function of atomic charge. The model is chemically significant since it properly correlates atomic size, softness, polarizability, and dispersion terms with minimal one-body parameters that are functions of the atomic charge. Tests of the model are examined for rare-gas interactions with neutral and charged atoms in order to demonstrate improved transferability. The present work provides a new framework for modeling QMMM interactions with improved accuracy and transferability.     

Vibrational spectra and ab initio analysis of tert-butyl, trimethylsilyl, trimethylgermyl, and trimethylstannyl derivatives of 3,3-dimethylcyclopropene VIII. 3,3-Dimethyl-1,2-bis-(trimethylstannyl)cyclopropene

The quantum mechanical force fields of 3,3-dimethyl-1,2-bis-(tert-butyl)cyclopropene (I), 3,3-dimethyl-1,2-bis-(trimethylsilyl)cyclopropene (II), 3,3-dimethyl-1,2-bis-(trimethylgermyl)cyclopropene (III), and 3,3-dimethyl-1,2-bis-(trimethylstannyl)cyclopropene (IV) were calculated at the HF/3-21G*//HF/3-21G* level. The scale factors which were optimized previously for the HF/3-21G*//HF/3-21G* quantum mechanical force field of 3,3-dimethyl-1-(trimethylsilyl)cyclopropene were used for correction of the force fields of these molecules. Good agreement between the frequencies calculated from these scaled force fields and the well-analyzed and assigned experimental frequencies of II and III suggests the transferability of these scale factors and the possibility of the spectroscopically accurate prediction of the vibrational spectrum of IV. Some regularities in the changes of the vibrational frequencies were found for this molecular series.     

Lennard-Jones parameters for the combined QM/MM method using the B3LYP/6-31G*/AMBER potential

A combined DFT quantum mechanical and AMBER molecular mechanical potential (QM/MM) is presented for use in molecular modeling and molecular simulations of large biological systems. In our approach we evaluate Lennard-Jones parameters describing the interaction between the quantum mechanical (QM) part of a system, which is described at the B3LYP/6-31+G* level of theory, and the molecular mechanical (MM) part of the system, described by the AMBER force field. The Lennard-Jones parameters for this potential are obtained by calculating hydrogen bond energies and hydrogen bond geometries for a large set of bimolecular systems, in which one hydrogen bond monomer is described quantum mechanically and the other is treated molecular mechanically. We have investigated more than 100 different bimolecular systems, finding very good agreement between hydrogen bond energies and geometries obtained from the combined QM/MM calculations and results obtained at the QM level of theory, especially with respect to geometry. Therefore, based on the Lennard-Jones parameters obtained in our study, we anticipate that the B3LYP/6-31+G*/AMBER potential will be a precise tool to explore intermolecular interactions inside a protein environment.