Estimating the hessian for gradient-type geometry optimizations

Optimization methods that use gradients require initial estimates of the Hessian or second derivative matrix; the more accurate the estimate, the more rapid the convergence. For geometry optimization, an approximate Hessian or force constant matrix is constructed from a simple valence force field that takes into account the inherent connectivity and flexibility of the molecule. Empirical rules are used to estimate the diagonal force constants for a set of redundant internal coordinates consisting of all stretches, bends, torsions and out-of-plane deformations involving bonded atoms. The force constants are transformed from the redundant internal coordinates to Cartesian coordinates, and then from Cartesian coordinates to the non-redundant internal coordinates used in the specification of the geometry and optimization. This method is especially suitable for cyclic molecules. Problems associated with the choice of internal coordinates for geometry optimization are also discussed.Fellow of the Alfred P. Sloan Foundation, 1981–83     

Franck-Condon factors within damped displacement harmonic oscillators: Solvent-enhanced absorption and fluorescence spectra

Damped harmonic oscillators that take into account local-mode nuclear vibrations interacting with solvent molecules are developed into Franck-Condon factors within displaced harmonic oscillator approximation. This is practically done by scaling an unperturbed Hessian matrix that represents local modes of force constants for molecules in a gaseous phase, and then by diagonalizing the perturbed Hessian matrix it results in direct modification of Huang–Rhys factors which represent normal modes of solute molecule perturbed by solvent environment. For highly symmetric polycyclic aromatic hydrocarbon molecules in which hydrogen atom vibrations in a solution can be scaled equally, one-set scaling parameters constructed into damped Franck-Condon factors can reproduce solvent-enhanced absorption and fluorescence spectra in solution. However, for low symmetry molecules with atoms other than hydrogen and carbon atoms, multi-set scaling parameters constructed into damped Franck-Condon factors can also reproduce solvent-enhanced absorption and fluorescence spectra in solution. Examples for high symmetry perylene in benzene solution with one-set scaling parameters and for low symmetry carbazole in n-hexane solution with multi-set scaling parameters are given, in both cases, the present damped Franck-Condon simulation can reproduce solvent-enhanced absorption and fluorescence spectra in solution.     

General methods of analysing molecular vibrations

An acute need may arise to develop for the complete analysis of molecular vibrations practically convenient general methods based on coordinates other than “chemical coordinates”. One reason is the proven proposition: Among independent internal coordinates corresponding to a molecule, there cannot be one which describes a small displacement of a chemical group as a whole relative to a certain molecular plane, provided this group contains more than two linearly or three non-linearly arranged atoms. Two methods are presented in some detail. The first is based on the use of X⁰_δ coordinates which are components of “bond vectors” in the “sown” (for each “bond”) Cartesian coordinate system. The second method utilizes X⁰ coordinates, i.e. the components of atomic displacements in the “down” (for each atom) Cartesian coordinate system. Computation of the torsional vibration of transdichloroethane is given as an example illustrating the first method. The Mayants treatment of the symmetry of a molecule, proceeding from elementary considerations which do not use the group theory explicitly and are valid for any coordinates, is expounded in a somewhat improved version. The peculiarities arising when considering the mean-square amplitude matrix, Σ, in X⁰_δ and X⁰ coordinates are also discussed.     

At nonzero temperatures, stacked structures of methylated nucleic acid base pairs and microhydrated nonmethylated nucleic acid base pairs are favored over planar hydrogen-bonded structures: a molecular dynamics simulations study

The dynamic structure of all ten possible nucleic acid (NA) base pairs and methylated NA base pairs hydrated by a small number of water molecules (from 1 to 16) was determined by using molecular dynamics simulations in the NVE microcanonical and NVT canonical ensembles with the Cornell force field (W. D. Cornell, P. Cieplak, C. I. Bayly, I. R. Gould, K. M. Merz, D. M. Ferguson, D. C. Spellmeyer, T. Fox, J. E. Caldwell, P. Kollman, J. Am. Chem. Soc. 1995, 117, 5179). The presence of one water molecule does not affect the structure of any hydrogen-bonded (H-bonded) nonmethylated base pair. An equal population of H-bonded and stacked structures of adenine...adenine, adenine...guanine and adenine... thymine pairs is reached if as few as two water molecules are present, while obtaining equal populations of these structures in the case of adenine...cytosine, cytosine...thymine, guanine... guanine and guanine...thymine required the presence of four water molecules, and in the case of guanine...cytosine, six. A comparable population of planar, H-bonded and stacked structures for cytosine...cytosine and thymine... thymine base pairs was only obtained if at least eight water molecules hydrated a pair. Methylation of bases changed the situation dramatically and stacked structures were favoured over H-bonded ones even in the absence of water molecules in most cases. Only in the case of methyl cytosine...methyl cytosine, methyl guanine...methyl guanine and methyl guanine...methyl cytosine pairs were two, two or six water molecules, respectively, needed in order to obtain a comparable population of planar, H-bonded and stacked structures. We believe that these results give clear evidence that the preferred stacked structure of NA base pairs in the microhydrated environment, and also apparently in a regular solvent, is due to the hydrophilic interaction of a small number of water molecules. In the case of methylated bases, it is also due to the fact that the hydrogen atoms most suitable for the formation of H-bonds have been replaced by a methyl group. A preferred stacked structure is, thus, not due to a hydrophobic interaction between a large bulk of water molecules and the base pair, as believed.     

Activation barriers for DNA alkylation by carcinogenic methane diazonium ions

Methylation reactions of the DNA bases with the methane diazonium ion, which is the reactive intermediate formed from several carcinogenic methylating agents, were examined. The SN2 transition states of the methylation reactions at N7, N3, and O6 of guanine; N7, N3, and N1 of adenine; N3 and O2 of cytosine; and O2 and O4 of thymine were calculated using the B3LYP density functional method. Solvation effects were examined using the conductor-like polarizable continuum method and the combined discrete/SCRF method. The transition states for reactions at guanine N3, adenine N7, and adenine N1 are influenced by steric interactions between the methane diazonium ion and exocyclic amino groups. Both in the gas phase and in aqueous solution, the methylation reactions at N atoms have transition states that are looser, and generally occur earlier along the reaction pathways than reactions at O atoms. The forming bonds in the transition states in water are 0.03 to 0.13 A shorter than those observed in the gas phase, and the activation energies are 13 to 35 kcal/mol higher. The combined discrete/SCRF solvation energy calculations using base-water complexes with three water molecules yield base solvation energies that are larger than those obtained from the CPCM continuum method, especially for cytosine. Reactivities calculated using barriers obtained with the discrete/SCRF method are consistent with the experimentally observed high reactivity at N7 of guanine.     

Properties of the optimal environment inducing tautomerization of complementary base pairs

The optimal environment charge configurations are predicted for the tautomerization of complementary base pairs into their corresponding rare forms, and vice versa. Results indicate that cations approaching the N₃ guanine site may induce tautomerization of the normal guanine—cytosine (G---C) base pair into its rare form. The reverse process requires that the cation approach the O₂ thymine site of the rare adenine*—thymine* pair (A*---31T*) or the O₆ guanine site of rare guanine*—cytosine* base pair (G*---C*). Possible mutagenic and antimutagenic roles of metal cations approaching base pairs are also discussed.     

Stability of nucleic acid base pairs in organic solvents: molecular dynamics, molecular dynamics/quenching, and correlated ab initio study

The dynamic structure and potential energy surface of adenine...thymine and guanine...cytosine base pairs and their methylated analogues interacting with a small number (from 1 to 16 molecules) of organic solvents (methanol, dimethylsulfoxide, and chloroform) were investigated by various theoretical approaches starting from simple empirical methods employing the Cornell et al. force field to highly accurate ab initio quantum chemical calculations (MP2 and particularly CCSD(T) methods). After the simple molecular dynamics simulation, the molecular dynamics in combination with quenching technique was also used. The molecular dynamics simulations presented here have confirmed previous experimental and theoretical results from the bulk solvents showing that, whereas in chloroform the base pairs create hydrogen-bonded structures, in methanol, stacked structures are preferred. While methanol (like water) can stabilize the stacked structures of the base pairs by a higher number of hydrogen bonds than is possible in hydrogen-bonded pairs, the chloroform molecule lacks such a property, and the hydrogen-bonded structures are preferred in this solvent. The large volume of the dimethylsulfoxide molecule is an obstacle for the creation of very stable hydrogen-bonded and stacked systems, and a preference for T-shaped structures, especially for complexes of methylated adenine...thymine base pairs, was observed. These results provide clear evidence that the preference of either the stacked or the hydrogen-bonded structures of the base pairs in the solvent is not determined only by bulk properties or the solvent polarity but rather by specific interactions of the base pair with a small number of the solvent molecules. These conclusions obtained at the empirical level were verified also by high-level ab initio correlated calculations.     

A method for the calculation of normal-mode vibrational frequencies using symmetry coordinates. Application to the calculation of secondary deuterium isotope effects on carbocations

A method is introduced for the calculation of normal-mode vibrational frequencies of polyatomic molecules based on numerical differencing of analytical gradients in symmetry coordinates. This procedure requires a number of gradient evaluations equal to the largest number of symmetry coordinates belonging to any single irreducible representation of the molecular point group (plus a single gradient evaluation at the equilibrium configuration), which is fewer than the 3N-6 (N atoms) gradient evaluations needed for schemes based on Cartesian or internal coordinates. While the proposed method will not generally be as efficient as procedures which involve the direct calculation of energy second derivatives analytically (as are now available for single-determinant wavefunctions) it appears to be equally accurate, and it should be the method of choice for frequency calculations involving multideterminant wavefunctions for which analytical second-derivative algorithms have yet to be developed. The method is illustrated by the calculation of equilibrium secondary deuterium-isotope effects on a number of reactions involving simple carbocations.     

On the role of mercury in the non-covalent stabilisation of consecutive U-Hg(II)-U metal-mediated nucleic acid base pairs: metallophilic attraction enters the world of nucleic acids

Metal atoms with a closed-shell electronic structure and positive charge as for example the Au(I), Pt(II), Ag(I), Tl(I) or Hg(II) atoms do not in some compounds repel each other due to the so-called metallophilic attraction (P. Pyykk?, Chem. Rev., 1997, 97, 597-636). Here we highlight the role of the Hg(II)Hg(II) metallophilic attraction between the consecutive metal-mediated mismatched base pairs of nucleic acids. Usually, the base stacking dominates the non-covalent interactions between steps of native nucleic acids. In the presence of metal-mediated base pairs these non-covalent interactions are enriched by the metal-base interactions and the metallophilic attraction. The two interactions arising due to the metal linkage of the mismatches were found in this study to have a stabilizing effect on nucleic acid structure. The calculated data are consistent with recent experimental observations. The stabilization due to the metallophilic attraction seems to be a generally important concept for the nucleic acids containing heavy metals with short contacts.     

Oligonucleotide cross-linking with copper ions. Spectral and quantum chemical study

Copper(II) complexes with synthetic oligonucleotides consisting of repeating adenine–thymine and guanine–cytosine complementary base pairs have been studied by UV spectroscopy and simulated by DFT quantum chemical calculations at the B3LYP/6-311G++(d,p) level of theory with inclusion of solvation (hydration) effects. The obtained data suggest selective interaction of copper(II) ions with guanine–cytosine complementary pairs, followed by DNA cross-linking at those sites.     

Interaction of angelicin with DNA‐bases (III) DFT and ab initio second‐order Moeller–Plesset study

Angelicin geometry was optimized at MP2/6‐31+G(d,p) level and compared with X‐ray experimental data. The highest π‐electron density was found to be localized on C1? C2 and on C13? C14 as confirmed by the calculated bond length and bond order values. Spectrophptometric properties of angelicin were measured and compared with the computed within the TD‐DFT. Quantum chemical methods were used to study the interaction of angelicin, as a nonlinear furocoumarin, with DNA bases and base pairs. The interactions with DNA bases and base pairs were studied to shade more light on the nature of the intercalation binding forces between angelicin and DNA. Comparing computed electronic properties of angelicin with that of linear psoralens show that the former is a weaker intercalator. The geometry of complexes of angelicin with adenine, thymine, adenine–thymine base pair, cytocine, guanine as well as cytocine–guanine base pair have been optimized in two main orientations, planar and stacked, at the levels of B3LYP/cc‐pVDZ, MP2/6‐31G(d,p) and MP2/cc‐pVDZ. Effect of vertical distance and rotational angle between the stacked molecules on the interaction energy were investigated by the aforementioned methods in gas phase and water media. It was found that ab initio methods which account for the electron correlation effects are the minimum level for studying the noncovalent interactions. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Cis Watson-Crick/Watson-Crick Base Pairs in DNA: A Density Functional Theory(DFT) Study

The four nucleic acid DNA bases(adenine, thymine, guanine, cytosine) and ten cis Watson-Crick/Watson-Crick(cis WC/WC) DNA base pairs were investigated by density functional theory(DFT) quantum chemical calculations. Geometry optimizations were carried out on the four bases and ten base pairs at the B3LYP level with 6-31G~(**) basis set. All the optimizations were performed within Cs symmetry. The optimum structures for the four bases and seven cis WC/WC base pairs were obtained, and Natural Bond Orbital analysis(NBO) was based on these structures. The possibilities of matches between any two of the four bases through their Watson-Crick(WC) edges were discussed. The structures of seven cis WC/WC base pairs change to a certain extent relative to these of the four bases due to the formation of hydrogen bonds. These base pairs existing in DNA have an important influence on the structural stability of the double helix. The analysis of the electronic structures and molecular orbitals for seven cis WC/WC base pairs can provide significant information about the relationship between charge transfer along the hydrogen bond and the Frontier orbitals of these base pairs.     

Beryllium bonding with noble gas atoms

Quantum chemical calculations were carried out to investigate the nature of the bonding between a neutral Be₃ ring and noble gas atom. Electronic structure calculation for these complexes was carried out at different computational levels in association with natural bond orbital, quantum theory of atoms in molecules, electron localization function, symmetry adapted perturbation theory, and molecular electrostatic potential surface analysis of Be₃ complexes. The Be atoms in the Be₃ moiety are chemically bonded to one another, with the Be Be bond dissociation energy being ~125 kJ mol⁻¹. The Be₃ ring interacts with the noble gases through non-covalent interactions. The binding energies of the noble gas atoms with the Be₃ ring increases with increase in their atomic number. The non-covalent interaction index, density overlap region indicator and independent gradient model analyses reveal the presence of non-covalent inter-fragment interactions in the complexes. Energy decomposition analysis reveals that dispersion plays the major role towards stabilizing these systems.     

The Role of Aromaticity,Hybridization, Electrostatics,and Covalency in Resonance-Assisted Hydrogen Bonds of Adenine–Thymine (AT) Base Pairs and Their Mimics

Invited for this issues cover are Dr. Célia Fonseca Guerra from the VU University of Amsterdam and her collaborators at the University of Girona. The cover picture shows H-bonds in the adenine–thymine Watson–Crick base pair. An essential part of these H-bonds is their covalent component arising from donor–acceptor interactions between N or O lone pairs and the N−H antibonding σ* acceptor orbital. This charge-transfer interaction is represented by green figures walking on the pedestrian crossing, connecting the bases. This covalent component is the reason why H-bonds between DNA and/or unsaturated model bases are significantly stronger than those between analogous saturated bases. This contrasts sharply with the classical picture of predominantly electrostatic H-bonds which is not only incomplete in terms of a proper bonding mechanism, but also fails to explain the trend in stability. For more details, see the Full Paper on p. 318 ff.     

Clustering of nucleobases with alkali metals studied by electrospray ionization tandem mass spectrometry: implications for mechanisms of multistrand DNA stabilization

Self-clustering of the five common nucleobases was investigated by electrospray ionization tandem mass spectrometry and shown to provide insight into the non-covalent interactions between identical bases. Alkali and ammonium cations significantly increase self-aggregation of the nucleobases and lead to the formation of uniquely stable magic number clusters. Sodium adducts of guanine, thymine and uracil preferentially take the form of tetrameric (quartet) clusters. This gas-phase result correlates with previously reported solution-phase data on sodium cation stabilized guanosine, thymine and uracil quartet structures believed to be responsible for telomere stabilization. In the presence of potassium, cesium or ammonium cations, pentameric magic number clusters are formed from thymine and uracil, while in solution the nucleoside isoguanosine yields clusters of this favored size. The formation of magic number metaclusters occurs for thymine and uracil in the presence of ammonium cations. These doubly charged 10- and 15-mers are tentatively attributed to the formation of pentamer/ammonium cation/ pentamer sandwich structures.     

Glycan–protein cross-linking mass spectrometry reveals sialic acid-mediated protein networks on cell surfaces

A cross-linking method is developed to elucidate glycan-mediated interactions between membrane proteins through sialic acids. The method provides information on previously unknown extensive glycomic interactions on cell membranes. The vast majority of membrane proteins are glycosylated with complicated glycan structures attached to the polypeptide backbone. Glycan–protein interactions are fundamental elements in many cellular events. Although significant advances have been made to identify protein–protein interactions in living cells, only modest advances have been made on glycan–protein interactions. Mechanistic elucidation of glycan–protein interactions has thus far remained elusive. Therefore, we developed a cross-linking mass spectrometry (XL-MS) workflow to directly identify glycan–protein interactions on the cell membrane using liquid chromatography-mass spectrometry (LC-MS). This method involved incorporating azido groups on cell surface glycans through biosynthetic pathways, followed by treatment of cell cultures with a synthesized reagent, N-hydroxysuccinimide (NHS)–cyclooctyne, which allowed the cross-linking of the sialic acid azides on glycans with primary amines on polypeptide backbones. The coupled peptide–glycan–peptide pairs after cross-linking were identified using the latest techniques in glycoproteomic and glycomic analyses and bioinformatics software. With this approach, information on the site of glycosylation, the glycoform, the source protein, and the target protein of the cross-linked pair were obtained. Glycoprotein–protein interactions involving unique glycoforms on the PNT2 cell surface were identified using the optimized and validated method. We built the GPX network of the PNT2 cell line and further investigated the biological roles of different glycan structures within protein complexes. Furthermore, we were able to build glycoprotein–protein complex models for previously unexplored interactions. The method will advance our future understanding of the roles of glycans in protein complexes on the cell surface.

The cell surface glycocalyx is highly interactive defined by extensive covalent and non-covalent interactions. A method for cross-linking and characterizing glycan–peptide interactions in situ is developed.     

Second derivatives in generalized Born theory

Generalized Born solvation models offer a popular method of including electrostatic aspects of solvation free energies within an analytical model that depends only upon atomic coordinates, charges, and dielectric radii. Here, we describe how second derivatives with respect to Cartesian coordinates can be computed in an efficient manner that can be distributed over multiple processors. This approach makes possible a variety of new methods of analysis for these implicit solvation models. We illustrate three of these methods here: the use of Newton-Raphson optimization to obtain precise minima in solution; normal mode analysis to compute solvation effects on the mechanical properties of DNA; and the calculation of configurational entropies in the MM/GBSA model. An implementation of these ideas, using the Amber generalized Born model, is available in the nucleic acid builder (NAB) code, and we present examples for proteins with up to 45,000 atoms. The code has been implemented for parallel computers using both the OpenMP and MPI environments, and good parallel scaling is seen with as many as 144 OpenMP processing threads or MPI processing tasks.     

Molecular torus group

Generally speaking, the highest symmetry of M?bius cyclacene molecule possesses the C₂ symmetry based on the theory of point group according to the previous works. However, based on the topology principle, the fundamental group of M?buis strip is an infinite continuous cyclic group and its border is made up of twice of the generator. Of course, the M?bius strip-like molecule is associated with a finite discrete cyclic symmetry group. For the cyclacene isomers constructed by several (n) benzene rings, these isomers include: the common cylinder Hückel cyclacene (HC-[n]) molecules, the M?bius cyclacene (MC-[n]) molecules by twisting the linear precursor one time (180°), and the multi-twisted M?bius strip-like cyclacene (M ^m C?[n]) molecules by twisting the linear precursor m times (m × 180°). The relevant results suggest that in addition to the point symmetry, there is a new kind of symmetry called the torus screw rotation (denoted as TSR). In this article, we take the M ^m C?[n] molecules as examples to discuss their TSR group and point group symmetry, and the relative symmetry adaptive atom sets as well as their atomic orbital (AO) sets. Here, the Cartesian coordinates is not quite fit for the investigation of these AOs, so it is replaced by either the torus orthogonal curvilinear coordinates (for M ^m C?[n] molecule) or the cylinder orthogonal curvilinear coordinates (for HC-[n] molecule). According to the features of cyclic group, the character table of the irreducible representation of the TSR group could be constructed easily. Some other relevant point-group symmetries maybe also belong to the molecule, so its symmetry maybe called as the torus group symmetry. For multi-twisted M?bius strip-like molecule, there are some correlations in topology characteristics.     

Intrinsic Dynamic and Static Nature of Halogen Bonding in Neutral Polybromine Clusters,with the Structural Feature Elucidated by QTAIM Dual-Functional Analysis and MO Calculations

The intrinsic dynamic and static nature of noncovalent Br-∗-Br interactions in neutral polybromine clusters is elucidated for Br₄–Br₁₂, applying QTAIM dual-functional analysis (QTAIM-DFA). The asterisk (∗) emphasizes the existence of the bond critical point (BCP) on the interaction in question. Data from the fully optimized structures correspond to the static nature of the interactions. The intrinsic dynamic nature originates from those of the perturbed structures generated using the coordinates derived from the compliance constants for the interactions and the fully optimized structures. The noncovalent Br-∗-Br interactions in the L-shaped clusters of the C_s symmetry are predicted to have the typical hydrogen bond nature without covalency, although the first ones in the sequences have the vdW nature. The L-shaped clusters are stabilized by the n(Br)→σ*(Br–Br) interactions. The compliance constants for the corresponding noncovalent interactions are strongly correlated to the E(2) values based on NBO. Indeed, the MO energies seem not to contribute to stabilizing Br₄ (C_2h) and Br₄ (D_2d), but the core potentials stabilize them, relative to the case of 2Br₂; this is possibly due to the reduced nuclear–electron distances, on average, for the dimers.     

Functional Systems Derived from Nucleobase Self-assembly

Dynamic and reversible non-covalent interactions endow synthetic systems and materials with smart adaptive functions that allow them to response to diverse stimuli, interact with external agents, or repair structural defects. Inspired by the outstanding performance and selectivity of DNA in living systems, scientists are increasingly employing Watson−Crick nucleobase pairing to control the structure and properties of self-assembled materials. Two sets of complementary purine-pyrimidine pairs (guanine:cytosine and adenine:thymine(uracil)) are available that provide selective and directional H-bonding interactions, present multiple metal-coordination sites, and exhibit rich redox chemistry. In this review, we highlight several recent examples that profit from these features and employ nucleobase interactions in functional systems and materials, covering the fields of energy/electron transfer, charge transport, adaptive nanoparticles, porous materials, macromolecule self-assembly, or polymeric materials with adhesive or self-healing ability.