Molecular symmetry. IV. The coupled perturbed Hartree–Fock method

Symmetry methods employed in the ab initio polyatomic program HONDO are extended to the coupled perturbed Hartree–Fock (CPHF) formalism, a key step in the analytical computation of energy first derivatives for configuration interaction (CI) wavefunctions, and energy second derivatives for Hartree–Fock (HF) wavefunctions. One possible computational strategy is to construct Fock-like matrices for each nuclear coordinate in which the one- and two-electron integrals of the usual Fock matrix are replaced by the integral first derivatives. “Skeleton” matrices are constructed from the unique blocks of electron-repulsion integral derivatives. The correct matrices are generated by applying a symmetrization operator. The analysis is valid for many wavefunctions, including closed- or open-shell spin-restricted and spin-unrestricted HF wavefunctions. To illustrate the method, we compare the computer time required for setting up the coupled perturbed HF equations for eclipsed ethane using D_3h symmetry point group and various subgroups of D_3h. Computational times are roughly inversely proportional to the order of the point group.     

Adaptation of D2h ab initio computer code to higher-symmetry point groups

The imposition of symmetry in electronic structure calculations can be plagued by artifactual symmetry-breaking in orbital or configuration amplitudes. While most ab initio computer code is well-developed to impose symmetry constraints in D_2h and its subgroups, the problem is not nearly as tractable in higher-symmetry point groups. This article describes the successful modification of existing D_2h ab initio computer code to handle symmetry constraints in higher-symmetry point groups. Prospects for the development of ab initio computer code that runs fully under any point group are also discussed.     

Symmetry Adapted Analysis of Linear Molecules

Symmetry groups of the linear molecules belong to the C_∞v and D_∞h infinite groups. The symmetry adapted analysis of such types of molecule, is usually not systematically performed in the text book or paper. Since the standard formulas of symmetry adapted analysis are usually applicable for the finite groups only, one has to analyze the different subgroups of the linear molecules indirectly and correlates them with the irreducible representation of D_∞h and C_∞v. In this work, a systematic symmetry adapted analysis are introduced for the C_∞v and D_∞h molecules. It is a uniquely convenient way for molecular orbital calculations and vibrational normal mode analysis of the linear molecules.     

Unsymmetry effect of hexa-peri-hexabenzocoronenes on columnar mesophase behaviour

To address the effects of molecular symmetry on the phase of columnar liquid crystals, a series of unsymmetry (C₁) and symmetry (D_3h) hexa-peri-hexabenzocoronene derivatives was prepared. Compounds 3 and 4, as well as a mixture of compounds 3+4, all had a significantly lower melting temperature and clearing temperature than high-symmetry compounds (D_6h), and all results show a strong dependence on molecular symmetry.     

Discrete spatial symmetries in energy derivatives

Based upon the invariance of N-particle systems under discrete operations of reflection, inversion, and rotation by 2π/n, a method for obtaining complete sets of relations among energy derivatives of all orders has been presented. The method is based on the criterion that, for a discrete symmetry operation such as reflection across a plane; the change in energy due to simultaneous arbitrary displacements of each particle is equal to another set of “conjugate” displacements of all particles. Applications of the above relations for particular molecular systems possessing a combination of symmetry operations is also presented. Here, via the row echelon analysis, the number of independent derivatives are found, and simple symmetry relations that allow determination of the remaining derivatives are presented. For example, for a homonuclear diatomic molecule with D_2h symmetry only 1 of the 6 first derivatives and 1 of the 21 second derivatives need to be independently calculated.     

Enantiomeric and Diastereomeric Relationships of Ethylene Derivatives. Restructuring Stereochemistry by a Group-Theoretical and Combinatorial Approach

To restructure stereochemistry into a systematic format, enantiomeric and diastereomeric relationships have been investigated by using ethylene derivatives as examples in the light of a new group-theoretical and combinatorial approach. On one hand, enantiomeric relationship for ethylene derivatives has been characterized by means of a point group of order 8 (D _2h ), where chirality fittingness based on the sphericity concept has been applied to the enumeration of stereoisomers. On the other hand, diastereomeric relationship for ethylene derivatives has been examined by a permutation group of order 8 (S ₉ ^[4]), which is a subgroup of the symmetric group of order 4 (S ^[4]) and isomorphic to a point group D _2d . The subgroups of S ₉ ^[4] have been classified into stereogenic and astereogenic ones. A stereogenic subgroup corresponds to a pair of diastereomers, while an astereogenic subgroup is assigned to a self-diastereomer. The relationship between diastereomers and constitutional isomers have been also discussed.     

Algebraic solutions for point groups: Cubic groups G in the group chain G⊃T⊃D2⊃C2

Concise algebraic expressions of the symmetry‐adapted functions (SAFs) for both single‐valued and double‐valued representations are derived for the group chain O⊃T⊃D₂⊃C₂ and O⊃D₄⊃D₂⊃C₂, which are functions of only the quantum numbers of the respective group chain without involving any irreducible matrix elements. It is shown that the SAFs of the cubic groups G=O,T_d,T_h,O_h can be expressed in a simple way in terms of the SAFs of the group T. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 585–599, 2000     

Carbometallic derivatives of hydroborons C2h and D5d: A list of substitution isomers

The problem of determining the number and type of X-substituted (where X represents certain substituents) carbometallic derivatives of hydroborons [Co(C₂B₉H₁₁)₂]⁻ C _2h and [Co(C₅B₆H₁₁)₂]⁻ D _5d (from apexes) was solved on the basis of G. Pólya’s theorem by means of combinatory analysis. The formulae of symmetry Z and generating functions of the number of achiral substitution isomers were determined. The family distributions of isomers (depending on the type and number of substituents) and the sites of possible substitution depending on the number m were determined. Mono and di-X-substituted isomers of [Co(C₅B₆H₁₁)₂]⁻ C _2h , as well as mono and di-X-substituted isomers of [Co(C₅B₆H₁₁)₂]⁻ D _5d , were identified. The procedure for plotting an additive model of calculating the properties of isomers of apical substitution of [Co(C₅B₆H₁₁)₂]⁻ D _5d was described on the basis of splitting of polygonal numbers (K ₃, K _TE, K ₅, and so on) of Pascal triangle, upon use of which there is no randomization in the choice of parameters of calculation model. The additive schemes containing 7 and 25 parameters for the calculation of properties of X-substituted carbometallic derivative of hydroboron [Co(C₅B₆H₁₁)₂]⁻ D _5d to various approximations were obtained.     

The Stabilization of Bridged Structures of Ethanes

The source of the instability of D_2h, bridged, diborane type geometries of ethanes is traced to a single molecular orbital, b_2g or π* in symmetry. The energy of this orbital can be lowered, and the barrier to bridging accordingly decreased, by appropriately placed acceptor substituents. Low bridging barriers, relative to ethane, are predicted for cyclopentenyl cation, and cyano or nitro substituted ethanes.     

Theoretical investigations on structure,density, detonation properties,and sensitivity of the derivatives of PYX

The ? NH₂, ? NO₂, ? N₃, ? NHNO₂, and ? ONO₂ substitution derivatives of PYX (2,6‐bis(picrylamino)‐3,5‐dinitropyridine) were studied at the B3LYP/6‐31G** level of density functional theory. The sublimation enthalpies and heats of formation (HOFs) in gas phase and solid state of these compounds were calculated. The theoretical predicted density (ρ), detonation pressure (P), and detonation velocity (D) showed that these derivatives have better detonation performance than PYX. The effects of substituent groups on HOF, ρ, P, and D were discussed. The order of contribution of various groups to P and D was ? ONO₂ > ? NO₂ > ? NHNO₂ > ? N₃ > ? NH₂. Sensitivity was evaluated using the frontier orbital energies, bond orders, bond dissociation enthalpies (BDEs), and characteristic heights (h₅₀). The trigger bonds in the pyrolysis process for these PYX derivatives may be Ring‐NO₂, NH? NO₂, or O? NO₂ varying with the substituents. The h₅₀ of most compounds are larger than that of CL‐20, and those of ? NH₂, ? NO₂, and most ? ONO₂ derivatives are larger than that of RDX. The BDEs of the trigger bonds of all but the ? ONO₂ derivatives are sufficiently large. Taking both detonation performance and sensitivity into consideration, some derivatives of PYX may be good candidates of explosives. © 2012 Wiley Periodicals, Inc.     

Algorithm for variational calculations of molecular symmetry in solving anharmonic vibrational problems

A universal program for variational calculations of molecular symmetry in solving anharmonic vibrational problems, realized by the author, is described. The program uses the group-theoretical method. Symmetrized basis wave functions are constructed with the aid of the generalized KJebsch-Gordan series suggested by the author. The method of constructing symmetrized basis wave functions and the program for adequate calculations of molecular symmetry were verified for many molecules of different symmetry groups: O_h, O, T_d, T_h, T, D_∞h, C_t8v, D_nd, D_nh, D_n, C_nv, C_nh, S_2n, C_n, C_i, C_s, and C₁ where 2 ≤n ≤6. It was confirmed that the program provides correct results and high-speed operation. Translated fromZhurnal Strukturnoi Khimii, Vol. 38, No. 6, pp. 1146–1153, November–December, 1997.     

Approximate symmetry-breaking in the independent particle model of monocyclic completely conjugated polyenes

We examine the singlet stability of symmetry adapted, restricted Hartree-Fock (RHF) solutions and the implied symmetry breaking for various planar, π-electron systems with conjugated double bonds as described by the semiempirical Pariser-Parr-Pople Hamiltonian. In particular, we explore the energy and charge- density waves (CDWs) in various real and hypothetical structures that result by a systematic deformation of the nuclear framework: we start with a highly symmetric cyclic polyene C _N H _N having a nondegenerate ground state (N = 2n = 4ν + 2, ν = 1, 2,...), whose sites form a regular N-gon (D _Nh point group), and proceed to structures with lower symmetry (D _6h , D _3h , D _2h point groups), or with only the planar symmetry of the conjugated π-electron system (C _1h ). The objective of this study is to explore the phenomenon that could be referred to as a breaking of an approximate symmetry or an implied symmetry breaking.     

Substitution symmetry

Pólya's enumeration theory and its generalizations are refined to count derivatives of symmetrical parent compounds with any specified subsymmetry. Equivalently, enumeration of orbits of mappings, upon which a group acts by acting on their domain and their range, is refined to count orbits with stabilizers in any specified conjugacy class of subgroups.     

The combinatorics of cation-deficient close-packed structures

Pólya's enumeration theorem is used to derive an algorithm for counting all hexagonal close-packed structures with a unit cell of given size, having composition MX₂, where X is an hcp anion, M is an octahedrally coordinated cation, and no face sharing is permitted between octahedra. Generalizations of this algorithm to enumerate ordered derivatives of these structures, hcp structures with tetrahedral instead of octahedral cations, and similar structures having different stacking sequences among the close-packed layers are sketched.     

Control of Chromophore Symmetry by Positional Isomerism of Peripheral Substituents

The first example of the control of porphyrinoid chromophore symmetry based on the positional isomerism of peripheral substituents has been achieved by preparing tetraazaporphyrins (TAPs) with C_4h, D_2h, C_2v, and C_s symmetry due to the relative arrangement of peripheral tert‐butylamino and cyano groups as push and pull substituents, respectively. The four structural isomers were successfully isolated and characterized by ¹H NMR spectroscopy and X‐ray crystallography. The band morphology in the Q‐band region varies depending on the molecular symmetry due to the significant perturbation introduced into the chromophore by the push and pull substituents. The C_4h and C_2v isomers exhibit a single Q band, whereas the Q bands of the D_2h and C_s isomers show a marked splitting. The magnetic circular dichroism spectra indicate that the push–pull TAPs retain the properties of the 16‐membered 18π‐electron perimeter generally observed for porphyrinoids. Theoretical calculations have demonstrated that the perturbation introduced by the substituents lowers the D_4h symmetry of the parent TAP π‐conjugated system, and this results in significant spectral changes. A novel approach to the fine‐tuning of the spectral properties of porphyrinoids based on changes in the chromophore symmetry is described.     

ESR-Spektren von CuN4-Chromophoren II: Komplexe mit Ammoniak, Äthylendiamin,Biguanid, N-Trifluoracetimidoyl-trifluoracetamidin,N-Trichloracetimidoyl-trichloracetamidin,Tetraaza-undecan,Hexamethyl-tetraaza-cyclotetradecan und -cyclotetra decadien in Einkristallen

The ESR spectra of eight copper complexes with two groups of nitrogen ligands have been measured in the solid state. The first group includes the σ-bonding ligands ammonia, ethylenediamine, 1,4,8,11-tetraaza-undecane, and the macrocyclic hexamethyl-1,4,8,11-tetraazacyclotetradecane of CURTIS . Ligands of the second group are biguanide, the CF₃ and CCl₃ derivatives of acetimidoyl-acetamidine, and hexamethyl-1, 4,8,11-tetraazacyclotetradecadiene. Single crystals of the parent Pt^II, Ni^II, or Zn^II compounds were used as host lattices and magnetic diluents. The spectra of the chromophores CuN₄, showing resolved hyperfine structure due to copper and nitrogen nuclear spins, are fitting a Spin-Hamiltonian on the basis of virtual site symmetry D_4h. The parameters g∥, g?, A∥, and A? were used to derive delocalization coefficients in terms of the LCAO-MO model introduced by MAKI and MC GARVEY . The comparative study clearly reveals that no reliable information on π-bonding is provided by application of this model. The results on σ-bonding ligands are particularly valuable in showing that the simple LCAO-MO model cannot account for the attenuation of spin orbit coupling due to σ-bonding in a physically meaningful way. Our results are qualitatively in agreement with AMMETER'S explanation of this phenomenon in terms of a virtual expansion of the d-shell in antibonding orbitals.     

Full Non-Rigid Group and Symmetry of Dimethyltrichlorophosphorus

In this work, a simple method is described, by means of which it is possible to calculate character tables for the symmetry group of molecules consisting of a number of NH3 groups attached to a rigid framework. The full non-rigid group (f-NRG) of dimethyltrichlorophosphorus with the symmetry group D3h was studied. It has been proven that it is a group of order 216 with 27 conjugacy classes and its character table computed. Finally, the Permutation-lnversion group of this molecule was calculated.     

New One‐Pot Procedure for the Synthesis of 2‐Substituted Benzimidazoles

A highly selective synthesis of 2‐substituted benzimidazole derivatives from the reaction of o‐phenylendiamine derivatives and aromatic aldehydes in the presence of an organic salt, NH₄OAc, in absolute ethanol is presented. The products were obtained by evaporating the solvent followed by a simple recrystallization with excellent yields.     

Rapid Access of Some Chloro‐substituted Isocoumarins and Biological Activity Toward Some Pathogenic Systems

Sulphuric acid–catalyzed chloralhydrate condensation with different m‐substituted benzoic acids formed trichlorophthalides 1, from which Zn+AcOH reduction afforded various dichloro derivatives 2. These derivatives 2 on treatment with alkaline Hg(OAc)₂+I₂ furnished different substituted isocoumarins 4.     

Dimerization of C60: Symmetry Considerations

Orbital‐symmetry analysis (OCAMS) of the dimerization of C₆₀ via [2+2] cycloaddition indicates that the reactant monomers should approach one another along a pathway in which C_2h symmetry is conserved. Point‐by‐point computations (AM1/UHF) confirm this prediction: a low‐energy pathway leads to a single‐bonded dimer 2 with C_2h symmetry. Closure to the stable D_2h dimer 1 is effected by relatively facile rotation about the single bond. A similar symmetry analysis was performed on a second isomer 3 with D_2h symmetry, the moieties of which are linked by two two‐atom chains. It raises the possibility that 3 , the so‐called `window' isomer, may be interconvertible with 1 along a pathway that retains C_i (S₂) symmetry. Although the computational results indicate that C₆₀ is in thermal equilibrium with its stable dimer 1 at moderate temperatures, the latter is not observed in the gas phase for thermodynamic reasons. According to THERMO computations (AM1/RHF), the equilibrium is shifted strongly towards the monomer pair at temperatures where vaporization of the solid C₆₀ is observed (>400°).