Fast density matrix-based partitioning of the energy over the atoms in a molecule consistent with the Hirshfeld-I partitioning of the electron density

For the Hirshfeld-I atom in the molecule (AIM) model, associated single-atom energies and interaction energies at the Hartree-Fock level are efficiently determined in one-electron Hilbert space. In contrast to most other approaches, the energy terms are fully consistent with the partitioning of the underlying one-electron density matrix (1DM). Starting from the Hirshfeld-I AIM model for the electron density, the molecular 1DM is partitioned with a previously introduced double-atom scheme (Vanfleteren et al., J Chem Phys 2010, 132, 164111). Single-atom density matrices are constructed from the atomic and bond contributions of the double-atom scheme. As the Hartree-Fock energy can be expressed solely in terms of the 1DM, the partitioning of the latter over the AIM naturally leads to a corresponding partitioning of the Hartree-Fock energy. When the size of the molecule or the molecular basis set does not grow too large, the method shows considerable computational advantages compared with other approaches that require cumbersome numerical integration of the molecular energy integrals weighted by atomic weight functions.     

Critical analysis and extension of the Hirshfeld atoms in molecules

The computational approach to the Hirshfeld [Theor. Chim. Acta 44, 129 (1977)] atom in a molecule is critically investigated, and several difficulties are highlighted. It is shown that these difficulties are mitigated by an alternative, iterative version, of the Hirshfeld partitioning procedure. The iterative scheme ensures that the Hirshfeld definition represents a mathematically proper information entropy, allows the Hirshfeld approach to be used for charged molecules, eliminates arbitrariness in the choice of the promolecule, and increases the magnitudes of the charges. The resulting "Hirshfeld-I charges" correlate well with electrostatic potential derived atomic charges.     

A Hirshfeld-I interpretation of the charge distribution, dipole and quadrupole moments of the halogenated acetylenes FCCH, ClCCH, BrCCH, and ICCH

We report the dipole and quadrupole moments of the halogenated acetylenes calculated using large basis sets and the SCF, DFT(B3LYP), and CCSD methods, and we analyze the charge density using the Hirshfeld and Hirshfeld-I techniques. The atomic charges, dipoles, and quadrupoles resulting from the Hirshfeld-I analysis are used to interpret the unusually small molecular dipole moments in the sequence as well as the molecular quadrupole moments. The very small dipoles obtain for two reasons. First, the dipole moment associated with the σ and π electron densities is comparable in magnitude and opposite in direction. Second, the charge and induced dipole contributions for ClCCH, BrCCH, and ICCH have opposite signs further reducing the molecular dipoles. The molecular quadrupole moments are the sum of a charge, atomic dipole, and in situ quadrupole terms, and are dominated by the atomic dipoles and in situ quadrupoles with the charge contributions playing an unexpectedly minor role.     

New fragment weighting scheme for the Bayesian inference network in ligand-based virtual screening

Many of the conventional similarity methods assume that molecular fragments that do not relate to biological activity carry the same weight as the important ones. One possible approach to this problem is to use the Bayesian inference network (BIN), which models molecules and reference structures as probabilistic inference networks. The relationships between molecules and reference structures in the Bayesian network are encoded using a set of conditional probability distributions, which can be estimated by the fragment weighting function, a function of the frequencies of the fragments in the molecule or the reference structure as well as throughout the collection. The weighting function combines one or more fragment weighting schemes. In this paper, we have investigated five different weighting functions and present a new fragment weighting scheme. Later on, these functions were modified to combine the new weighting scheme. Simulated virtual screening experiments with the MDL Drug Data Report (23) and maximum unbiased validation data sets show that the use of new weighting scheme can provide significantly more effective screening when compared with the use of current weighting schemes.     

Information distance analysis of molecular electron densities

The information‐theoretic basis of the Hirshfeld partitioning of the molecular electronic density into the densities of the “stockholder” atoms‐in‐molecules (AIM) is summarized. It is argued that these AIM densities minimize both the directed divergence (Kullback–Leibler) and divergence (Kullback) measures of the entropy deficiency between the AIM and their free atom analogs of the promolecule. The local equalization of the information distance densities of the Hirshfeld components, at the local value of the corresponding global entropy deficiency density, is outlined and several approximate relations are established between the alternative local measures of the missing information and the familiar function of a difference between the molecular and promolecule densities. Various global (of the system as a whole) and atomic measures of the entropy deficiency or the displacements relative to the isoelectronic promolecule, defined for densities or probabilities (shape functions) in both the local resolution and the Hirshfeld AIM discretization, are introduced and tested. This analysis is performed also for the valence electron (frozen‐core) approximation. Illustrative results for representative linear molecules, including diatomics, triatomics, and tetraatomics, are reported. They are interpreted as complementary characteristics of changes in the net AIM charge distribution and of the displacements in the information content of the electron distributions of bonded atoms. These numerical results confirm the overall similarity of the stockholder AIM to their free atom analogs and reflect the information displacements due to the AIM polarization and charge transfer in molecules. They also demonstrate the semiquantitative nature of the approximate relations established between the entropy deficiency densities and the related functions involving the density difference function. This development extends the range of interpretations based on the density difference diagrams into probing the associated information displacements in a molecule accompanying the formation of the chemical bonds. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2002     

Communication: Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

A double-atom partitioning of the molecular one-electron density matrix is used to describe atoms and bonds. All calculations are performed in Hilbert space. The concept of atomic weight functions (familiar from Hirshfeld analysis of the electron density) is extended to atomic weight matrices. These are constructed to be orthogonal projection operators on atomic subspaces, which has significant advantages in the interpretation of the bond contributions. In close analogy to the iterative Hirshfeld procedure, self-consistency is built in at the level of atomic charges and occupancies. The method is applied to a test set of about 67 molecules, representing various types of chemical binding. A close correlation is observed between the atomic charges and the Hirshfeld-I atomic charges.     

Single electron densities: a new tool to analyze molecular wavefunctions

A new partitioning scheme for the electron density of a many-electron wavefunction into single electron densities is proposed. These densities are based on the most probable arrangement of the electrons in an atom or molecule. Therefore, they contain information about the electron-electron interaction and, most notably, the Fermi hole due to the antisymmetry of the many-electron wavefunction. The single electron densities overlap and can be combined to electron pair distributions close to the qualitative electron pairs that represent, for instance, the basis of the valence shell electron pair repulsion model. Single electron analyses are presented for the water, ethane, and ethene molecules. The effect of electron correlation on the single electron and pair densities is investigated for the water molecule.     

Prediction of densities for solid energetic molecules with molecular surface electrostatic potentials

The densities of high energetic molecules in the solid state were calculated with a simplified scheme based on molecular surface electrostatic potentials (MSEP). The MSEP scheme for density estimation, originally developed by Politzer et al., was further modified to calculate electrostatic potential on a simpler van der Waals surface. Forty-one energetic molecules containing at least one nitro group were selected from among a variety of molecular types and density values, and were used to test the suitability of the MSEP scheme for predicting the densities of solid energetic molecules. For comparison purposes, we utilized the group additivity method (GAM) incorporating the parameter sets developed by Stine (Stine-81) and by Ammon (Ammon-98 and -00). The absolute average error in densities from our MSEP scheme was 0.039 g/cc. The results based on our MSEP scheme were slightly better than the GAM results. In addition, the errors in densities generated by the MSEP scheme were almost the same for various molecule types, while those predicted by GAM were somewhat dependent upon the molecule types.     

Bonding study in all-metal clusters containing Al4 units

The nature of the bonding of a series of gas-phase all-metal clusters containing the Al4 unit attached to an alkaline, alkaline earth, or transition metal is investigated at the DFT level using Mulliken, quantum theory of atoms in molecules (QTAIM), and Hirshfeld iterative (Hirshfeld-I) atomic partitionings. The characterization of ionic, covalent, and metallic bonds is done by means of charge polarization and multicenter electron delocalization. This Article uses for the first time Hirshfeld-I multicenter indices as well as Hirshfeld-I based atomic energy calculations. The QTAIM charges are in line with the electronegativity scale, whereas Hirshfeld-I calculations display deviations for transition metal clusters. The Mulliken charges fail to represent the charge polarization in alkaline metal clusters. The large ionic character of Li-Al and Na-Al bonds results in weak covalent bonds. On the contrary, scarcely ionic bonds (Be-Al, Cu-Al and Zn-Al) display stronger covalent bonds. These findings are in line with the topology of the electron density. The metallic character of these clusters is reflected in large 3-, 4- and 5-center electron delocalization, which is found for all the molecular fragments using the three atomic definitions. The previously reported magnetic inactivity (based on means of magnetic ring currents) of the pi system in the Al42- cluster contrasts with its large pi electron delocalization. However, it is shown that the different results not necessary contradict each other.     

Stockholder projector analysis: a Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

A previously introduced partitioning of the molecular one-electron density matrix over atoms and bonds [D. Vanfleteren et al., J. Chem. Phys. 133, 231103 (2010)] is investigated in detail. Orthogonal projection operators are used to define atomic subspaces, as in Natural Population Analysis. The orthogonal projection operators are constructed with a recursive scheme. These operators are chemically relevant and obey a stockholder principle, familiar from the Hirshfeld-I partitioning of the electron density. The stockholder principle is extended to density matrices, where the orthogonal projectors are considered to be atomic fractions of the summed contributions. All calculations are performed as matrix manipulations in one-electron Hilbert space. Mathematical proofs and numerical evidence concerning this recursive scheme are provided in the present paper. The advantages associated with the use of these stockholder projection operators are examined with respect to covalent bond orders, bond polarization, and transferability.     

Density-difference maps in quantum chemistry

Electron density-difference maps, used to study the changes that occur when a molecule changes its state or when the nuclei of a molecule change their relative positions, are generally useful only if the atomic densities cancel when one molecular density distribution is subtracted from the other. When, as in the case of the nonrigid internal rotation in ethane, such a cancellation of atomic densities is not possible the method of simple subtraction is no longer appropriate. It is shown that useful density-difference maps can nevertheless be obtained, when the changes in geometric parameters are small, by the calculation of two generalized density-difference functions: a point difference function which allows a comparison of the densities at corresponding points in the two systems, and a volume difference function to compare the amounts of charge in corresponding regions. The method is illustrated by consideration of a change in bond length of the nitrogen molecule and by the nonrigid internal rotation in ethane.Presented at the XIVth Quantum Theory Conference in Canterbury, England, September 1981.     

Energy partitioning for "fuzzy" atoms

The total energy of a molecule is presented as a sum of one- and two-atomic energy components in terms of "fuzzy" atoms, i.e., such divisions of the three-dimensional physical space into atomic regions in which the regions assigned to the individual atoms have no sharp boundaries but exhibit a continuous transition from one to another. By proper definitions the energy components are on the chemical energy scale. The method is realized by using Becke's integration scheme and weight function permitting very effective numerical integrations.     

基于电负性均衡理论快速计算多肽分子中原子电荷的新方法

Atomic charges play a crucial role in the understanding and modeling of the chemical behavior of proteins. Fast assessment of atomic charge distributions in larger molecules can be performed by implementing the electronegativity equalization method (EEM). To further improve the accuracy of the EEM approach, a novel and efficient method based on Bader's concept of high degree fragment transferability of atomic charges has been proposed for the parameterization of atoms-in-molecules (AIM) charges of polypeptides or proteins. The EEM parameterization method considers both the factors of connectivity and hybridized states, and the effect of the local chemical environment in fragments or groups. The types of atoms were defined on the basis of the local chemical environments of the fragments or functional groups of these atoms. The fragment transferability feature of QTAIM indicates that the atomic properties for the contributing atoms can be reproduced if the chemical environment is comparable. The constituent fragments or functional groups of macromolecules such as polypeptides and proteins can be utilized as building blocks for the additive generation of their electronic densities. The main peptide group (NH―HαCα―C＝O) of the polypeptide in the backbone was used as a building block to model the EEM parameters for reproducing the atomic charges in the polypeptides. A training set of 20 terminally blocked amino acids (Ac-X-NHMe, X = any neutral residue), which recreated the immediate local environment of the main chain fragments or functional groups of the polypeptides, were chosen for the calibration of AIM charges using the differential evolution (DE) algorithm. The effects of the optimized methods on the results were discussed and it was found that the DE algorithm showed a better performance for the objective function. The quality of the AIM charges obtained from the EEM method presented in this study was evaluated by comparison with those obtained from B3LYP/6-31G+(d, p) calculations for the two test tetrapeptides not contained in the training set. It was found that a remarkable improvement was achieved using the EEM model developed in this study as compared to the previous studies. The introduction of Bader's high fragment charge transfer model into the EEM provided a new scheme for its calibration and parameterization for larger systems such as polypeptides or polynucleotides, which possess highly repetitive segments. Among all types of atomic charges, only the AIM charges showed a significant meaning in experiments and could be obtained by X-ray diffraction experiments. Rapidly reproducing the accurate AIM charge for large systems seems to be more meaningful, especially for the prediction of protein-protein, protein-DNA, and drug-receptor recognition and interactions.     

Molecular mechanism for H2 release from BH3NH3, including the catalytic role of the Lewis acid BH3

Electronic structure calculations using various methods, up to the coupled-cluster CCSD(T) level, in conjunction with the aug-cc-pVnZ basis sets with n = D, T, and Q, extrapolated to the complete basis set limit, show that the borane molecule (BH3) can act as an efficient bifunctional acid-base catalyst in the H2 elimination reactions of XHnYHn systems (X, Y = C, B, N). Such a catalyst is needed as the generation of H2 from isoelectronic ethane and borane amine compounds proceeds with an energy barrier much higher than that of the X-Y bond energy. The asymptotic energy barrier for H2 release is reduced from 36.4 kcal/mol in BH3NH3 to 6.0 kcal/mol with the presence of BH3 relative to the molecular asymptote. The NH3 molecule can also participate in a similar catalytic process but induces a smaller reduction of the energy barrier. The kinetics of these processes was analyzed by both transition-state and RRKM theory. The catalytic effect of BH3 has also been probed by an analysis of the electronic densities of the transition structures using the atom-in-molecule (AIM) and electron localization function (ELF) approaches.     

Finite element interpolation for combined classical/quantum mechanical molecular dynamics simulations

A method is presented to interpolate the potential energy function for a part of a system consisting of a few degrees of freedom, such as a molecule in solution. The method is based on a modified finite element (FE) interpolation scheme. The aim is to save computer time when expensive methods such as quantum-chemical calculations are used to determine the potential energy function. The expensive calculations are only carried out if the molecule explores new unknown regions of the conformation space. If the molecule resides in regions previously explored, a cheap interpolation is performed instead of an expensive calculation, using known neighboring points. We report the interpolation techniques for the energies and the forces of the molecule, the handling of the FE mesh, and an application to a simple test example in molecular dynamics (MD) simulations. Good performance of the method was obtained (especially for MD simulations with a preceding Monte Carlo mesh generation) without losing accuracy. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1484–1495, 1997     

Quantifying the symmetry content of the electronic structure of molecules: molecular orbitals and the wave function

A scheme to quantify the symmetry content of the electronic wave function and molecular orbitals for arbitrary molecules is developed within the formalism of Continuous Symmetry Measures (CSMs). After defining the symmetry operation expectation values (SOEVs) as the key quantity to gauge the symmetry content of molecular wavefunctions, we present the working equations to be implemented in order to carry out real calculations using standard quantum chemistry software. The potentialities of a symmetry analysis using this new method are shown by means of some illustrative examples such as the changes induced in the molecular orbitals of a diatomic molecule by an electronegativity perturbation, the breaking of orbital symmetry along the dissociation path of the H(2) molecule, the changes in the molecular orbitals upon a geometrical distortion of the benzene molecule, and the inversion symmetry content in the different spin states of the [Fe(CH(3))(4)](2-) complex.     

Can ring strain be realized in momentum space?

An increased electron momentum density (EMD) at low momentum is proposed to be an indicator of ring strain, with the nature of the function tending toward a maximum. A p-space Hirshfeld atomic partitioning scheme is applied for analyzing the effect of strain on molecular EMDs. The Hirshfeld momentum densities for a strained system show an increase in the population for the carbons with the hydrogens becoming more positive in comparison with an unstrained reference molecule. The manifestation of strain in cage-like hydrocarbons such as tetrahedrane, cubane, prismane, etc. as well as their nitrogen-substituted analogues is clearly seen in terms of EMDs.     

New method to measure packing densities of self-assembled thiolipid monolayers

For a monolayer of 2,3-di-phytanyl-sn-glycerol-1-tetraethylene glycol-D,L-a-lipoic acid ester lipid (DPTL) self-assembled (SAM) at a gold electrode surface we propose a new method to determine the charge number per adsorbed molecule and the packing density (area per molecule) in the monolayer. The method relies on chronocoulometry to measure the charge density at the SAM covered gold electrode surface. Two series of measurements have to be performed. In the first series, charge densities are measured for a monolayer transferred from the air-solution to the metal-solution interface using the Langmuir-Blodgett (LB) technique. This series of measurements allows one to determine charge numbers per adsorbed DPTL molecule. The second series is performed using a gold electrode covered with a self-assembled monolayer. The charge densities obtained in this series are then used to calculate the packing density with the help of charge numbers per adsorbed DPTL determined in the first series. The area per adsorbed molecule determined by the new method was compared to the area per molecule determined by the popular reductive desorption method. The molecular area determined with the new method is about 20% larger than the area calculated from the van der Waals model, which is a physically reasonable result. In contrast, the popular reductive desorption method gives an area per molecule 20% lower than the minimum estimated based on a van der Waals model. This is a physically unreasonable result. It is also shown that the charge numbers per adsorbed molecule depend on the electrode potential and may assume values smaller than the number of electrons participating in the reductive desorption step. An explanation of the origin of the "partial charge numbers" is provided. We recommend the new method be used in future studies of thiol adsorption at metal surfaces.     

A reversible minimum-to-minimum mapping method for the calculation of free-energy differences

A general method is introduced for the calculation of the free-energy difference between two systems, 0 and 1, with configuration spaces omega(0), omega(1) of the same dimensionality. The method relies upon establishing a objective mapping between disjoint subsets gamma(i)(0) of omega(0) and corresponding disjoint subsets gamma(i)(1) of omega(1), and averaging a function of the ratio of configurational integrals over gamma(i)(0) and gamma(i)(1) with respect to the probability densities of the two systems. The mapped subsets gamma(i)(0) and gamma(i)(1) need not span the entire configuration spaces omega(0) and omega(1). The method is applied for the calculation of the excess chemical potential mu(ex) in a Lennard-Jones (LJ) fluid. In this case, omega(0) is the configuration space of a (N-1) real molecule plus one ideal-gas molecule system, while omega(1) is the configuration space of a N real molecule system occupying the same volume. Gamma(i)(0) and gamma(i)(1) are constructed from hyperspheres of the same radius centered at minimum-energy configurations of a set of "active" molecules lying within distance a from the ideal-gas molecule and the last real molecule, respectively. An algorithm is described for sampling gamma(i)(0) and gamma(i)(1) given a point in omega(0) or in omega(1). The algorithm encompasses three steps: "quenching" (minimization with respect to the active-molecule degrees of freedom), "mutation" (gradual conversion of the ideal-gas molecule into a real molecule, with simultaneous minimization of the energy with respect to the active-molecule degrees of freedom), and "excitation" (generation of points on a hypersphere centered at the active-molecule energy minimum). These steps are also carried out in reverse, as required by the bijective nature of the mapping. The mutation step, which establishes a reversible mapping between energy minima with respect to the active degrees of freedom of systems 0 and 1, ensures that excluded volume interactions emerging in the process of converting the ideal-gas molecule into a real molecule are relieved through appropriate rearrangement of the surrounding active molecules. Thus, the insertion problem plaguing traditional methods for the calculation of chemical potential at high densities is alleviated. Results are presented at two state points of the LJ system for a variety of radii a of the active domain. It is shown that the estimated values of mu(ex) are correct in all cases and subject to an order of magnitude lower statistical uncertainty than values based on the same number of Widom [J. Chem. Phys. 39, 2808 (1963)] insertions at high fluid densities. Optimal settings for the new algorithm are identified and distributions of the quantities involved in it [number of active molecules, energy at the sampled minima of systems 0 and 1, and free-energy differences between subsets gamma(i)(0) and gamma(i)(1) that are mapped onto each other] are explored.