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.     

Synthesis of high energy nitroguanidine derivatives

Russian Chemical Bulletin - New energy-rich derivatives of 2-nitroguanidine were obtained. The synthesis was based on the reaction of 1-chloromethyl-2-nitroguanidine, which was obtained by...     

Variational energy derivatives and perturbation theory

In this note we discuss the variational forms of the energy derivatives and the method of obtaining them. We show that perturbation theory can be formulated in terms of stationary energy derivatives and that this formulation extends the idea of the Hylleraas functional to infinite order and excited states.     

Configuration-interaction energy derivatives in a fully variational formulation

A configuration-interaction energy function (Lagrange) which is variational in all variables, including the orbital rotational parameters, is constructed. When this Lagrangian is used for obtaining configuration-interaction derivatives, all the important simplifications which occur for derivatives of variational wave functions carry over in a straightforward way. In particular, the state and orbital rotational response parameters obey the 2n+1 rule and the Lagrange multipliers obey the somewhat stronger 2n+2 rule. The simplifications which are normally obtained by invoking the Handy-Schaefer technique are automatically incorporated to all orders. Simple expressions for energy derivatives up to third order are presented. The relationship between the numerical errors in the variational parameters and the errors in the calculated energy derivatives is discussed.     

Analytic second-order energy derivatives in natural orbital functional theory

The analytic energy gradients in the atomic orbital representation have recently been published (Mitxelena and Piris in J Chem Phys 146:014102, 2017) within the framework of the natural orbital functional theory (NOFT). We provide here an alternative expression for them in terms of natural orbitals, and use it to derive the analytic second-order energy derivatives with respect to nuclear displacements in the NOFT. The computational burden is shifted to the calculation of perturbed natural orbitals and occupancies, since a set of linear coupled-perturbed equations obtained from the variational Euler equations must be solved to attain the analytic Hessian at the perturbed geometry. The linear response of both natural orbitals and occupation numbers to nuclear geometry displacements need only specify the reconstruction of the second-order reduced density matrix in terms of occupation numbers.     

Analytical second derivatives of the energy in MNDO methods

Analytical second derivatives of the energy are derived and efficiently implemented for semiempirical MNDO-type methods including AM1, PM3, and MNDO/d. A new algorithm for the simultaneous solution of several CPHF equations is proposed in which the amount of memory required is independent of the number of iterations. The analytical approach is faster than the numerical approach typically by a factor of 5 and exhibits a reliable convergence over a wide range of molecules. The asymptotic memory and secondary storage requirements of the reported procedure can be as low as O(N²) without significant degradation of the performance. © 1996 by John Wiley & Sons, Inc.     

Higher molecular-deformation derivatives of the configuration-interaction energy

We derive expressions for the first through fourth derivatives of the configuration-interaction (CI) electronic energy with respect to molecular deformation. By using unitary exponential parameterizations of the wavefunction's orbital and configuration amplitude response together with a power-series expansion of the geometry dependence of the hamiltonian, a computationally attractive expression for the CI energy derivatives is obtained. The use of so-called direct methods in evaluating the CI derivatives is discussed as are the relative efforts involved in using our CI-based energy-derivative expressions and those which we obtained earlier for derivatives of the multiconfigurational self-consistent-field energy. The power-series expansion of the geometry dependence of the hamiltonian that we have derived may be used for evaluating molecular-deformation derivatives for any approximate wavefunction constructed from a set of orthonormal orbitals.     

An efficient approach for self-consistent-field energy and energy second derivatives in the atomic-orbital basis

Based on self-consistent-field (SCF) perturbation theory, we recast the SCF and the coupled-perturbed SCF (CPSCF) equations for time-independent molecular properties into the atomic-orbital basis. The density matrix and the perturbed density matrix are obtained iteratively by solving linear equations. Only matrix multiplications and additions are required, and this approach can exploit sparse matrix multiplications and thereby offer the possibility of evaluating second-order properties in computational effort that scales linearly with system size. Convergence properties are similar to conventional molecular-orbital-based CPSCF procedures, in terms of the number of derivative Fock matrices that must be constructed. We also carefully address the issue of the numerical accuracy of the calculated second derivatives of the energy, in order to specify the minimum precision necessary in the CPSCF procedure. It is found that much looser tolerances for the perturbed density matrices are adequate when using an expression for the second derivatives that is correct through second order in the CPSCF error.     

High symmetries in quantum chemistry

A method of simplifying the solution of secular equations occurring in electronic structure, normal mode and nuclear spin state calculations for molecules possessing symmetry is illustrated by applying it to the π-electrons of the recently discovered C₆₀ cluster. In contrast to the usual procedure, the method of reduction to characters does not require actual matrix realizations of the irreducible representations of the symmetry group but only the information contained in the character and multiplication tables. In the case of C₆₀ it leads to explicit algebraic solutions for all the π-orbital energies.     

Concurrent symmetries: the interplay between local and global molecular symmetries

We analyze in this article the degree to which different groups of atoms retain local symmetries when assembled in a molecule. This study is carried out by applying continuous symmetry measures to several families of mixed sandwiches, a variety of piano-stool molecules, and several organic groups. An analysis of the local symmetry of the electron density shows that, sandwiched between two regions of different symmetry that correspond to the ligand sets, its symmetry is cylindrical at the central metal atom.     

Absolute free energy calculations by thermodynamic integration in four spatial dimensions

An optimized technique for calculating the excess chemical potential of small molecules in dense liquids and the binding affinity of molecular ligands to biomolecules is reported. In this method, a molecular species is coupled to the system of interest via a nonphysical fourth spatial dimension w through which insertion or extraction can be carried out [R. Pomes, E. Eisenmesser, C. B. Post et al., J. Chem. Phys. 111, 3387 (1999)]. Molecular simulations are used to compute the potential of mean force (PMF) acting on the solute molecule in the fourth dimension. The excess chemical potential of that molecule is obtained as the difference in the PMF between fully coupled and fully decoupled systems. The simplicity, efficiency, and generality of the method are demonstrated for the calculation of the hydration free energies of water and methanol as well as sodium, cesium, and chloride ions. A significant advantage over other methods is that the 4D-PMF approach provides a single effective and general route for decoupling all nonbonded interactions (i.e., both Lennard-Jones and Coulombic) at once for both neutral and charged solutes. Direct calculation of the mean force from thermodynamic integration is shown to be more computationally efficient than calculating the PMF from umbrella sampling. Statistical error analysis suggests a simple strategy for optimizing sampling. The detailed analysis of systematic errors arising from the truncation of Coulombic interactions in a solvent droplet of finite size leads to straightforward corrections to ionic hydration free energies.     

Free energy derivatives: A new method for probing the convergence problem in free energy calculations

The convergence behavior of free energy calculations has been explored in more detail than in any previously reported work, using a model system of two neon atoms in a periodic box of water. We find that for thermodynamic integration-type free energy calculations as much as a nanosecond or more molecular dynamics sampling is required to obtain a fully converged value for a single λ point of the integrand. The concept of “free energy derivatives” with respect to the individual parameters of the force field is introduced. This formalism allows the total convergence of the simulation to be deconvoluted into components. A determination of the statistical “sampling ratio” from these simulations indicates that for window-type free energy calculations carried out in a periodic waterbox of typical size at least 0.6 ps of sampling should be performed at each window (0.7 ps if constraint contributions to the free energy are being determined). General methods to estimate and reduce the error in thermodynamic integration and free energy perturbation calculations are discussed. We show that the difficulty in applying such methods is determining a reliable estimate of the correlation length from a short series of data. © 1994 by John Wiley & Sons, Inc.     

New potential high energy density compounds: Oxadiaziridine derivatives

The -CN, -N₃, -NF₂, -NH₂, -NHNO₂, -NO₂, and -ONO₂ derivatives of oxadiaziridine were studied using B3LYP/6-311G** level of density functional theory. The gas phase heats of formation of oxadiaziridine derivatives were calculated by isodesmic reaction. All these compounds have high and positive heats of formation due to strain energies of small ring. Detonation properties were calculated via Kamlet-Jacobes equations and specific impulse. The effects of substituent groups on detonation performance were discussed. The impact sensitivity was estimated according to the “available free space per molecule in unit cell” and “energy gaps” methods. The similar conclusions were given by two different methods. The effects of substituents on impact sensitivity were discussed. According to the given estimations of detonation performance and sensitivity, some oxadiaziridine derivatives may be considered promising high energies materials.     

Bichromophoric perylene derivatives: energy transfer from non-fluorescent chromophores

The energy of excitation at non-fluorescent chromophores such as fluorenone and anthraquinone has been trapped by a fast energy transfer to the highly fluorescent perylene bisimides. To this end, anthraquinone, fluorenone and anthracene derivatives have been linked to the perylene bisimides by non-conjugating spacers and fluorescence quantum yield of such assemblies have been determined as a function of the wavelengths of excitation. Energy transfer in such assemblies is strongly influenced by the orientation of the two chromophores. This is of interest for the construction of fluorescence switches.     

Molecular symmetries and motions in crystalline silane

Infrared spectra of isotopically isolated SiH₃D molecules in polycrystalline SiH₄ have been recorded at temperatures between 4.2 and 70 K. In the high-temperature phase SiH₄(I) the observed absorption is indicative of rapid molecular reorientation, in agreement with ¹H NMR results. In phase II the molecular symmetry is concluded to be C₃, which appears to be incompatible with previous X-ray diffraction work. Analogous infrared spectra of isotopically dilute SiHD₃ molecules in polycrystalline SiD₄ are very similar to those of SiH₃ D in phases I and II of SiH₄, although there is apparently no phase in solid SiH₄ corresponding to phase III of SiD₄. Possible structural relationships between the various phases of SiH₄, SiD₄. GeH₄ and GeD₄ are suggested on the basis of the available infrared spectra of appropriate isotopic probe molecules.     

Identification of symmetries in molecules and complexes

An algorithm is presented that quickly detects local and global symmetries of single molecules and complexes. Based upon the Morgan Naming Algorithm, the algorithm involves traversing the molecule from a starting atom and building up a molecule name based upon the names of the atoms encountered along the traversal. Additional molecule names are generated from other starting atoms, and name-name matches are identified as corresponding to symmetry operations. A number of enhancements relative to prior methods yield increased efficiency and extended functionality. In particular, the present method detects not only global symmetries but also local symmetries associated with bond rotations as well as symmetries that are only apparent when alternate resonance forms are considered. Importantly, the present method works not only for single molecules but also for multimolecular complexes. As a consequence, it is well, and perhaps uniquely, suited to applications in supramolecular and host-guest chemistry. Applications include filtering out redundant conformations during conformational searching and free energy calculations; accelerating ligand-receptor docking calculations by reducing the sampling ranges of rotatable bonds linked to locally symmetric groups, such as phenyls; and automating the calculation of symmetry numbers for thermochemical applications.     

Effect of spatial configuration on adhesion of 1,2-disubstituted cyclohexane derivatives

Spatial configuration has a significant effect on chemical self-assembly. However, the importance of spatial configuration in supramolecular adhesive materials has been frequently ignored. In this study, the effects of the spatial configuration on cohesion and adhesion were investigated. Owing to the diversities of the chemical structures and assembly patterns, 1,2-disubstituted cyclohexane derivatives were used in this combined experimental and theoretical investigation. The self-sorting assembly of enantiopure isomers improved cohesion but had a negative effect on adhesion. In contrast, racemic mixtures displayed stronger adhesion effects. Moreover, it was proven that the cis-configuration was more favorable for supramolecular adhesion than the trans-counterpart. In addition, the influence of the spatial configuration of 1,2-disubstituted cyclohexane derivatives could be effectively mitigated by hydrogen bond donors or acceptors. The addition of natural acids yielded three-dimensional polymeric networks, in which the spatial configuration was not the decisive factor for supramolecular adhesion.     

A constrained variational approach for energy derivatives in Fock-space multireference coupled-cluster theory

In this paper, we present a formulation based on constrained variational approach to enable efficient computation of energy derivatives using Fock-space multireference coupled-cluster theory. Adopting conventional normal ordered exponential with Bloch projection approach, we present a method of deriving equations when general incomplete model spaces are used. Essential simplifications arise when effective Hamiltonian definition becomes explicit as in the case of complete model spaces or some special quasicomplete model spaces. We apply the method to derive explicit generic expressions upto third-order energy derivatives for [0,1], [1,0], and [1,1] Fock-space sectors. Specific diagrammatic expressions for zeroth-order Lagrange multiplier equations for [0,1], [1,0], and [1,1] sectors are presented.