A new module for constrained multi‐fragment geometry optimization in internal coordinates implemented in the MOLCAS package

A parallel procedure for an effective optimization of relative position and orientation between two or more fragments has been implemented in the MOLCAS program package. By design, the procedure does not perturb the electronic structure of a system under the study. The original composite system is divided into frozen fragments and internal coordinates linking those fragments are the only optimized parameters. The procedure is capable to handle fully independent (no border atoms) fragments as well as fragments connected by covalent bonds. In the framework of the procedure, the optimization of relative position and orientation of the fragments are carried out in the internal “Z‐matrix” coordinates using numerical derivatives. The total number of required single points energy evaluations scales with the number of fragments rather than with the total number of atoms in the system. The accuracy and the performance of the procedure have been studied by test calculations for a representative set of two‐ and three‐fragment molecules with artificially distorted structures. The developed approach exhibits robust and smooth convergence to the reference optimal structures. As only a few internal coordinates are varied during the procedure, the proposed constrained fragment geometry optimization can be afforded even for high level ab initio methods like CCSD(T) and CASPT2. This capability has been demonstrated by applying the method to two larger cases, CCSD(T) and CASPT2 calculations on a positively charged benzene lithium complex and on the oxygen molecule interacting to iron porphyrin molecule, respectively. © 2013 Wiley Periodicals, Inc.     

Structural parameter relaxation and pucker mode—internal mode coupling in the four-ring molecules cyclobutanol,cyclobutylfluoride and cyclobutylchloride

The present work is directed towards the investigation of the pucker—internal rotation modes of cyclobutanol and cyclobutanol-OD by the aid of a semirigid two-dimensional model. Based on spherical four-ring coordinates and general geometrical relations expressing wagging, twisting and rocking coordinates for CH₂ and CHX groups, experimental rotational constants of cyclobutanol, cyclobutylfluoride and cyclobutylchloride in the ground and excited pucker states are analyzed in terms of a two-dimensional four- ring pucker model with structural relaxation of CH₂ and CHX groups concerted with the ring puckering angles. The rotational data are found to be satisfactorily described by a one-dimensional ring pucker model including structural relaxation of the rocking angles of the CHX and the CH₂ groups. Then results of extended SCF computations (4-31G level) of the puckering—internal rotation potential of cyclobutanol are presented. These emphasize again the importance of structural relaxation of bond length and rocking type coordinates of the corner groups concerted with the puckering angle and serve as a basis for numerical solution of the two-dimensional pucker—internal rotation problem within the framework of a semirigid model. Predicted transitions and energy eigenstates will be correlated with far i.r. spectral data of cyclobutanol and cyclobutanol-OD observed in the 30–300 cm⁻¹ range and will be interpreted in terms of an alternative assignment, whose relation to an earlier analysis will finally be discussed.     

PathOpt—A global transition state search approach: Outline of algorithm

We propose a new algorithm to determine reaction paths and test its capability for Ar₁₂ and Ar₁₃ clusters. Its main ingredient is a search for the local minima on a (n?1) dimensional hyperplane (n = dimension of the complete system in Cartesian coordinates) lying perpendicular to the straight line connection between initial and final states. These minima are part of possible reaction paths and are, hence, used as starting points for an uphill search to the next transition state. First, path fragments are obtained from subsequent relaxations starting from these transition states. They can be combined with information from the straight line connection procedure to obtain complete paths. Our test computations for Ar₁₂ and Ar₁₃ clusters prove that PathOpt delivers several reaction paths in one round. © 2013 Wiley Periodicals, Inc.     

Exploiting space‐group symmetry in fragment‐based molecular crystal calculations

Recent developments in fragment‐based methods make it increasingly feasible to use high‐level ab initio electronic structure techniques to molecular crystals. Such studies remain computationally demanding, however. Here, we describe a straightforward algorithm for exploiting space‐group symmetry in fragment‐based methods which often provides computational speed‐ups of several fold or more. This algorithm does not require a priori specification of the space group or symmetry operators. Rather, the symmetrically equivalent fragments are identified automatically by aligning the individual fragments along their principle axes of inertia and testing for equivalence with other fragments. The symmetry operators relating equivalent fragments can then be worked out easily. Implementation of this algorithm for computing energies, nuclear gradients with respect to both atomic coordinates and lattice parameters, and the nuclear hessian is described. © 2014 Wiley Periodicals, Inc.     

Use of spherical coordinates for the 4-ring rotation-puckering problem

The rotational constants of eight typical 4-ring molecules represented by the semi-rigid model of type C_2v/C_s are analysed in terms of the kinetic coefficients of the rotation-puckering model formulated with spherical coordinates θ, φ as internal dynamic coordinates. It is shown that practically all 4-ring molecules may be considered as slightly perturbed D_4h/D_2d-type (cyclobutane, φ = π/4) semi-rigid models. The coordinate φ is practically independent of the puckering state and always lies in the range 40° < φ < 45°. The coordinate θ almost completely describes the puckering mode and its θ_v-values depend either monotonically (low barrier) or zig-zag (high barrier) on v.     

Half-chair conformations of unsaturated heterocyclic compounds

The vicinal proton coupling constants were obtained for some 6-membered ring unsaturated heterocyclic compounds. The R values and ring dihedral angles were determined and found to be consistent with half-chair conformations. Relative to saturated heterocyclic compounds, the effect of the sulfur atom on ring puckering was attenuated. However, an increase in ring puckering resulted from the sulfone group and was attributed to torsional energy.     

Coordinate reduction for exploring chemical reaction paths

The potential energy surface for the reaction of a typical molecular system composed of N atoms is defined uniquely by 3N-6 coordinates. These coordinates can be defined by the Cartesian coordinates of the atomic centers (minus overall translation and rotation), or a set of internally defined coordinates such as bond stretches, angle bends, and torsions. By applying principal component analysis to the geometries along a reaction path, a reduced set of coordinates, d ≪ 3N-6, can be obtained. This reduced set of coordinates can reproduce the changes in geometry along the reaction path with chemical accuracy and may help improve the efficiency of reaction path optimization algorithms.     

pyEFP: Automatic decomposition of the complex molecular systems into rigid polarizable fragments

We present an open source tool able to describe intermolecular electrostatic interactions within the framework of the effective fragment potential (EFP) method. Complex molecular structure is subdivided into compact rigid fragments and parameters of their interactions are obtained from ab initio calculations. Automatic procedure allows for searching of these parameters into the existing database and merge new fragments into it. A set of standard fragments useful for the studies of organic semiconductors is also provided. Input files both for purely EFP and hybrid QM/MM calculations can be generated. The program is written in python and freely available on GitHub: https://github.com/ale-odinokov/pyEFP © 2017 Wiley Periodicals, Inc.     

Phase diagrams for systems formed by manganese and rare earth metal oxides

We analyze the advantages and disadvantages of various methods for representing P-T-x diagrams of three-component systems and their fragments in various coordinates.     

Discrete spatial symmetries: Redundant coordinates and search for stationary points in reduced spaces

Given the invariance of an N-body system under discrete operations of reflection, inversion, a rotation by 2π/n, and the corresponding relations among the derivatives of energy, we have constructed through an invertible transformation a set of active and redundant coordinates. Movement along the active coordinates preserves all symmetry relations. We show that algorithms for locating stationary points or for calculating reaction paths are exactly separable in these active and redundant coordinates. We further show that this formalism is equally applicable when equations of constraints among coordinates are specified for the movement of particles. This includes geometrical constraints on bond lengths, angles, substituent group internal rotations, etc. This formalism enhances the efficiency since (laborious) cartesian derivatives need to be calculated only for the active variables and that the problem is reduced in term of m(?3N) variables. We apply this procedure to obtain the equilibrium geometry of H₂O molecule within the subspace of C_2v symmetry configurations ab initio derivatives.     

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.     

Long-range 4J coupling constants and conformation of the four-membered heterocycles

The correlation between the puckering angle of the four-membered cycle and ⁴J has been obtained from analysis of the PMR spectra of some 2-substituted azetidines and oxetanes. The absolute values of ⁴J_cis and ⁴J_trans were found to decrease with decreasing of the puckering angle.     

Quantum delocalization of benzene in the ring puckering coordinates

The effect of quantum mechanical delocalization of atomic nuclei on the conformation of the six‐membered ring structure in two hydrocarbons, cyclohexane and benzene, is investigated using ab initio path integral approach. A striking feature of benzene species is revealed using ring puckering coordinate representation, which demonstrates that the zero point motion of the heavy atom skeleton dominates over the out‐of‐plane thermal motions of the ring. Even more unexpected is the fact, that this is true not only at low temperature of 150 K, at which such behavior would not be surprising, but also at room temperature, where the nuclear quantum effects are usually of lesser importance, especially in the case of such heavy nuclei as carbon. In view of this finding the planar conformation of benzene, whose equilibrium (T = 0 K) geometry results from the well‐known properties of the electronic structure, can be elucidated also at nonzero temperature. According to our simulations, it appears as a consequence of quantum delocalization of the carbon nuclei rather than a trivial time average over the classical configurations of the puckered ring. This interesting behavior is contrasted with the clearly nonplanar structure of cyclohexane, whose ring puckering states can be unequivocally assigned even if the nuclear delocalization is taken into account. © 2014 Wiley Periodicals, Inc.     

Treatment of redundancies among internal coordinates in optimizing molecular mechanics force constants

The use of redundant coordinate bases in the construction of molecular mechanics force fields is discussed. It is shown that the intrinsic indeterminacy in a force field in redundant coordinates in general stems from the squares of the first-order redundancy relations. The necessity to use constraints in such a force field is pointed out, and a method to check whether or not a set of constraints makes the force field determinate is described. It is also explained how force fields corresponding to different sets of constraints can be transformed into one another. To facilitate the utilization of ab initio or other spectroscopic force fields, a procedure is given by which force constants pertaining to a nonredundant coordinate basis can be optimized in molecular mechanics calculations where redundant coordinates are used. © 1992 by John Wiley & Sons, Inc.     

A modular strategy for generating starting conformations and data structures of polynucleotide helices for potential energy calculations

We describe a simple and rapid algorithm for generating data structures and starting coordinates of polynucleotides for potential energy calculations. The algorithm is tailored to investigations in cartesian coordinate, rather than dihedral angle, space. First, instead of a tree structure for molecular design, we set up a helix from a simple list of bonds for the basic DNA subunits (sugar, phosphate, and bases). Second, instead of using successive transformations to obtain a set of coordinates in one reference frame, we apply a simple “matching” routine to patch DNA subunits. Third, we avoid ring closure and geometry optimization by allowing deviations from equilibrium values only for P? O3′ bond lengths and O5′? P? O3′ bond angles at the residue connection sites. A double-stranded helix is constructed from duplex building blocks (2 hydrogen-bonded nucleotides) which are in turn built from the basic structural units. Every building block is constructed from two sets of geometric variables: {α, β, γ, χ, P, τ_max}, one for each strand. The building blocks are then assembled into a helix by using the 6 rigid body transformations {Δx, Δy, Δz, Θ_ROLL , Θ_TILT , Θ_TWIST }. For cartesian space programs, generating starting coordinates by this procedure is particularly useful as an alternative to using actual crystal structure coordinates. After describing the algorithm in detail, we illustrate how it was used to generate model A, B, and Z DNA helices. We conclude by suggesting how the algorithm can be used to pursue a build-up technique and to set up a wide range of starting conformations in the goal of locating novel helical structures.     

Regioisomeric 4‐nitroindazole N1‐ and N2‐(β‐d‐ribonucleosides)

The structures of the isomeric nucleosides 4‐nitro‐1‐(β‐d ‐ribo­furan­osyl)‐1H‐indazole, C₁₂H₁₃N₃O₆, (I), and 4‐nitro‐2‐(β‐d ‐ribo­furan­osyl)‐2H‐indazole, C₁₂H₁₃N₃O₆, (II), have been determined. For compound (I), the conformation of the gly­cosylic bond is anti [χ = −93.6 (6)°] and the sugar puckering is C2′‐exo–C3′‐endo. Compound (II) shows two conformations in the crystalline state which differ mainly in the sugar pucker; type 1 adopts the C2′‐endo–C3′‐exo sugar puckering associated with a syn base orientation [χ = 43.7 (6)°] and type 2 shows C2′‐exo–C3′‐endo sugar puckering accompanied by a somewhat different syn base orientation [χ = 13.8 (6)°].     

A formulation of the quantum-mechanical many-body problem in terms of hyperspherical coordinates

Hyperspherical coordinates are used to construct a matrix representation of a general N-particle Hamiltonian in the case where the interaction is electrostatic. The Yukawa interaction can be treated similarly, as is shown in an appendix. The basis functions used to construct the matrix representation of H are mononomials inthe 3N coordinates of the particles, multiplied by functions of the hyperradius. Methods for transforming from this representation to a symmetry-adapted representation are also discussed.     

Constrained optimization in delocalized internal coordinates

Using the recently introduced delocalized internal coordinates, in conjunction with the classical method of Lagrange multipliers, an algorithm for constrained optimization is presented in which the desired constraints do not have to be satisfied in the starting geometry. The method used is related to a previous algorithm by the same author for constrained optimization in Cartesian coordinates [J. Comput. Chem., 13 , 240 (1992)], but is simpler and far more efficient. Any internal (distance or angle/torsion) constraint can be imposed between any atoms in the system whether or not the atoms involved are formally bonded. Imposed constraints can be satisfied exactly. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 :1079–1095, 1997