Construction of linearly independent relativistic symmetry orbitals for finite double-point groups including time reversal symmetry

A formalism is developed for the construction of relativistic symmetry-adapted molecular basis functions under consideration of time reversal invariance. The theory is applicable to the finite double point groups C_n, C_nh, S_n, C_nv, D_n, D_nd, D_nh, T, T_h, T_d, O, and O_h. It is based on the LCAO method. A projection operator technique is employed to construct molecular symmetry orbitals from atomic orbitals. The search for linearly independent basis function is simplified by means of group theoretical considerations.     

Molecular and electronic structure of some binary polyhedral clusters

The molecular and electronic structures of some polyhedral alternate molecules X_nY_n, where n=12, 16; X=B, Al, Si; Y=N,P,C, and of homoatomic clusters C₂₄, Si₂₄, C₃₂, and Si₃₂ are calculated in a valence approximation by the MNDO method. It is suggested that the σ-frameworks of these molecules are formed of four- and six-membered rings, with each X atom having only Y atoms as neighbors. The singlet states of all these systems have local minima on the corresponding potential energy surfaces with T_h symmetry for n=12 and T_d symmetry for n=16. The main structural parameters, heats of formation, ionization potentials, and effective charge distributions are given. It is concluded that the X_nY_n heteroatomic clusters can exist when X and Y are atoms of Group III and V elements, respectively, or both are atoms of Group IV elements. A. N. Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 6, pp.976–982, November–December, 1995. Translated by L. Smolina     

A D4D structure for [8]-prismane

[n]-Prismanes, n > 7,need not belong to D_nh point groups. Ab initio SCFMO Calculations on the model C₂H₆, and molecular mechanics calculations on [n]-prismanes indicate that [8]-prismane may adopt a D_4D, structure, 1.     

Structural stability and energetics of C,Si, and Ge microclusters: empirical many-body potential energy function calculation

The structural stability and energetics of carbon, silicon, and germanium microclusters containing 3?7 atoms have been investigated by using a recently developed empirical many-body potential energy function (PEF), which comprises two- and three-body atomic interactions. The PEF satisfies both bulk cohesive energy per atom and bulk stability exactly. It has been found that the most stable C_3?4 microclusters are linear withD _∞h symmetry but C_5?7 microclusters are planar withD _nh symmetry. Silicon and germanium microclusters show similar structural stability. TheX _n (X=Si, Ge;n=3?7) microclusters are found to be most stable in the following forms:X ₃ is triangular withD _3h symmetry,X ₄ is tetragonal withT _d symmetry,X ₅ is square pyramidal withD _4h symmetry,X ₆ is bipyramidal square withO _h symmetry, and finallyX ₇ is square pyramidal having two atoms underneath withD _2h symmetry.     

Hydrogenation of B 120/? : A Planar-to-Icosahedral Structural Transition in B12H n0/? (n?=?1–6) Boron Hydride Clusters

A systematic density functional theory and wave function theory investigation performed in this work reveals a planar-to-icosahedral structural transition between n = 4–5 in the partially hydrogenated B₁₂H _n ^0/− clusters (n = 1–6) upon hydrogenation of all-boron B₁₂^0/−. Coupled cluster calculations with triple excitations (CCSD(T)) indicate that a distorted icosahedral B₁₂H₆ cluster with C₂ symmetry is overwhelmingly favored (by 35 kcal/mol) over the recently proposed perfectly planar borozene (D_3h B₁₂H₆) (Szwacki et al., Nanoscale Res Lett 4:1085, 2009) which proves to be a high-lying local minimum. A similar 2D–3D structural transition occurs to the corresponding boron boronyl analogues of B₁₂(BO) _n with n –BO terminals. Detailed adaptive natural density partitioning (AdNDP) analyses reveal the bonding patterns of these quasi-planar or cage-like clusters which are characterized with delocalized σ and π molecular orbitals. The electron detachment energies of the concerned anions and excitation energies of the neutrals are also predicted to facilitate their future experimental characterizations.     

Structures and aromaticity of the planar Al<Subscript>2</Subscript>P<Subscript>2</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis>−</Superscript> (<Emphasis Type="Italic">n</Emphasis>=1–4) clusters

Clusters Al₂P₂ ⁿ⁻ (n = 1–4) were theoretically investigated using density functional theory (DFT) methods at the B3LYP/6-311+G* and B3PW91/6-311+G* levels of theory. The calculated results showed that the planar structure (D _2h symmetry) of Al₂P₂ ⁿ⁻ (n = 1–4) species was the global minimum. And the negative nucleus-independent chemical shift (NICS) value of Al₂P₂ ⁿ⁻ (n = 1–4) species indicated the existence of a ring current in the planar structure (D _2h symmetry). A detailed molecular orbital (MO) analysis revealed that the planar structures (D _2h symmetry) had π aromaticity, which further exhibited the strongly aromatic character for Al₂P₂ ⁿ⁻ (n = 1–4) species.     

Topological structures of tetrahedral (Td and T) fullerenes with hexagonal and triangular faces and their vibration and NMR spectra

On analyzing the topological structures of the three types of tetrahedral fullerenes (which consist only of triangles and hexagons), (1) C _n (T _d ,n=12h ²; h=1,2,…), (2) C _n (T _d ,n=4h ²;h=1,2,…), and (3) C _n (T,n=4(h ²+hk+k ²);h>k,h,k=1,2,…), we have obtained theoretically the Infrared and Raman active modes by means of the derived formulas for the decomposition of their nuclear motions into irreducible representations, and the ¹³C NMR spectra with natural abundance for ¹³C by using the distribution functions for all of the tetrahedral (T _d and T) fullerenes, respectively. Received: 25 May 1998 / Accepted: 30 July 1998 / Published online: 23 November 1998     

Tetra-hydrides of the third-row transition elements: spin–orbit coupling effects on geometrical deformation in WH4 and OsH4

The dissociation energies of MH₄ (M =  La, Hf–Hg) were computed using full optimized reaction space (FORS) multi-configuration self-consistent field (MCSCF) and second-order multi-reference Møller–Plesset perturbation methods with the SBKJC basis sets augmented by a set of polarization functions (SBKJC(f,p)). It was shown that of the molecules examined, only four tetra-hydrides HfH₄, TaH₄, WH₄, and OsH₄ with T_d symmetry are lower in energy than the corresponding dissociation limits. For WH₄ and OsH₄, the potential energy surfaces from the D_4h to the T_d structure were explored from both theoretical calculations and symmetry arguments based on the pseudo-Jahn- Teller effect. As for WH₄, it is found that the ground state could be ³E_g, ³A_2g, or ³B_2g at the D_4h structure. The present calculations suggest that the ground state is ³E_g, and that this state is stabilized by the e_u deformation into a C_2v structure (³B₁) and then sequentially to the most stable T_d structure (³A₂). If the molecular system is promoted to the lowest ³B_2g state, the D_4h structure can directly deform into the most stable T_d structure along the b_2u vibrational mode. For OsH₄, the ground state (⁵B_1g) at the D_4h structure deforms into a D_2d structure and the resulting ⁵B₂ state strongly interacts with the lowest ³E and ¹A₁ states due to the spin-orbit couplings (SOCs). As a result, it was shown that the relativistic potential energy of the lowest spin-mixed state (ground state) monotonically decreases along the D_2d deformation path from the D_4h to the T_d structure.     

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.     

Fourier-transform infrared spectroscopic studies of dithia tetraphenylporphine

We present here infrared absorption spectra of dithia tetraphenylporphine and its cation in the 450–1600 and 2900–3400 cm⁻¹ regions. Most of the allowed IR bands are observed in pairs due to overallD _2h point group symmetry of the molecule. The observed bands have been assigned to the porphyrin skeleton and phenyl ring modes. Some weak bands, which are forbidden underD _2h , also appear in the spectra due to the distortion of the molecule from planarity-caused by the out-of-plane positioned N and S atoms. Increased intensity of some phenyl ring bands compared to free-base tetraphenylporphine is explained on the basis of rotation of phenyl rings towards the mean molecular plane. Contrary to the point group symmetry of cation of dithia tetraphenylporphine, certain bands are observed to be degenerate due to identical bonding arrangements in pyrrole rings of the cation     

Molecular reorientation of liquid benzene. Anisotropic Raman scattering of the CH and CD stretching modes in the deuterated molecules

The anisotropic Raman spectra of the CH and CD stretching modes in seven deuterated benzenes of D_6h, D_3h, D_2h and C_2h symmetry are reported. The reorientational linewidths are interpreted within the model of anisotropic rotational diffusion. The data are consistent with NMR relaxation studies. The study covers the temperature range between T/T_c = 0.49 and T/T_c = 0.97.     

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     

Relativistic symmetry orbitals for the double groups C2v,C∞v,D∞h,and Oh

Relativistic symmetry orbitals are given for molecular LCAO calculations for the double groups: C_2v, C_∞v, D_∞h, and O_h (6-fold) coordination. The atomic orbitals used in the LCAO are of the four component form. A discussion of the comparison between nonre ativistic and relativistic molecular eigenvalues is presented.     

Effect of temperature on the H/D-isotope effects in the enthalpy of hydration of tetramethyl-bis-carbamide

The heats of dissolution of tetramethyl-bis-carbamide (the pharmaceutical Mebicarum) in H₂O and D₂O were measured at 288.15, 298.15, and 318.15 K using a sealed microcalorimeter with an isothermal shell. The error of measurements did not exceed 0.2%. The limiting molar enthalpies of dissolution Δ_sol H _n ^∞ and the H/D-isotope enthalpy effects of hydration δΔ_hydr H _n ^∞(H₂O → D₂O) were determined. Different effects of temperature on the pattern of variation of δΔ _hydr H _n ^∞ were found: when T ≤ 315 K, this value is positive and decreases with T, while for T ≥ 315 K, hydration of tetramethyl-bis-carbamide upon replacement of H₂O by D₂O progressively becomes less endothermic. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 4, pp. 715–717, April, 2006.     

Gertrude Belle Elion: Biochemist,Nobel Prize 1988

Most chemistry courses involving group theory do not treat the infinite point groups C_v or D_h. Even the character tables themselves are not transparent looking, with the profusion of ellipses. Because of this, it is not possible to deduce the vibrational symmetry modes of CO₂, even though the symmetric, asymmetric stretch, and doubly-degenerate bending modes are discussed in almost all physical chemistry courses. In this paper we show how linear molecules such as CO₂, C₂H₂, and HCN can be treated using the point groups C_nv or D_nh for general values of n. When determining which irreducible representations comprise the normal vibrational modes of a linear molecule such as CO₂, we show that the n-dependence vanishes. The calculations presented here do not require advanced mathematical knowledge and could be incorporated into an undergraduate chemistry curriculum in which group theory is presented.     

1A1 5T2 Spin transition in the solid phases of FexNi1-x(ATr)3(NO3)2 (ATr = 4- amino- 1,2,4- triazole)

The effect of Fe²⁺ substitution by Ni²⁺ in the complex of iron(II) nitrate with 4- amino- 1,2,4- triazole Fe(ATr)₃(NO₃)₂ on the character of the¹A₁ ⁵T₂ spin transition (ST) is studied by magnetic susceptibility and calorimetry methods. Solid phases of Fe_xNi_{1- x}(ATr)₃(NO₃)₂ (0.1 ≤ x ≤ 0.9) were synthesized. The temperature dependences of the effective magnetic moment were measured in the range of 78– 360 K. Heat capacities were measured in the range of 210– 340 K for 0.1 ≤ x ≤ 0.5 and in the range of 230– 340 K for 0.6 ≤x ≤ 0.9. As x decreases, the transition temperature (T_C), hysteresis (δT_C, and transition enthalpy (δH) decrease and the ST is leveled. The results are compared with the data obtained previously for the solid phases of Fe_xZn_{1- x}(ATr)₃(NO₃)₂ (0.01 ≤ x ≤ 0.8). The dependence Μ_eff(T) is analyzed theoretically in terms of both the domain model and the spin equilibrium model. Translated fromZhumal Strukturnoi Khimii, Vol. 38, No. 4, pp. 696–703, July–August, 1997.     

Boron rings containing planar octacoordinate iron and cobalt

The boron rings containing planar octacoordinate transition metals, D _8h FeB₈ ²⁻, CoB₈ ⁻ and CoB₈ ³⁺, C _2v FeB₈, D _2h CoB₈ ⁺ and CoB₈, are optimized with all real vibrational frequencies at the B3LYP/6–311+G* level of the theory. The D _8h FeB₈ ²⁻ and CoB₈ ⁻ isomers are global minima, while D _8h CoB₈ ³⁺ is only local minimum. The electronic structure character of these systems is revealed by natural bond orbital (NBO) analysis, showing that the boron rings containing planar octacoordinate transition metals have stability and aromaticity with six π electrons. The aromaticity is confirmed by nucleus independent chemical shifts (NICS) calculations. Supported by the specialized research fund for the doctoral program of higher education (20060007030)     

Flattened fullerenes

Quantum-chemical calculations of giant flattened fullerenes C _n (lentil-shaped) have been carried out. The topology, molecular and electronic structure of these fullerenes have been studied. Such molecules consist of two identical coronenoid fragments of a graphite layer, which are arranged one above the other, and a system of polycondensed five- and six-membered cycles, which form a side surface of the cluster. Polyhedral structures with isolated pentagons of three symmetry types (D _6h ,D _6d , andD _3h ) have been considered. The topology of these structures is described in terms of planar molecular graphs. Electronic structures of eleven flattened lentil-shaped C _n clusters (n = 72–216) have been studied in the π approximation. Most of the considered systems have closed or quasi-closed electron shells (according to Hückel) and rather large energy gaps separating the highest occupied and lowest unoccupied MO, which is indicative of their kinetic stability. Fragments of the potential energy surfaces of the C₇₂ and C₉₆ fullerenes have been studied by the MNDO, AM1, and MNDO/PM3 methods. For the C₉₆ cluster, two local energy minima, which correspond to the lentil-shaped isomers withD _6h andD _6d symmetry, have been determined. As a result of optimization of geometric parameters, it was found that all three methods give close values of heights (H = 6.7 Å) and diameters (D = 9.8 Å) for both isomers. The clusters change to quasi-two-dimensional systems (H«D) with increasing sizes of coronenoid fragments.     

A Density Functional Investigation on C2Aun+ (n?=?1, 3, 5) and C2Aun (n?=?2, 4, 6): from Gold Terminals, Gold Bridges, to Gold Triangles

A systematic density functional theory investigation on C₂Au _n ⁺ (n = 1,3,5) and C₂Au _n (n = 2,4,6) indicates that gold atoms serve as terminals (–Au) in the chain-like C_s C₂Au⁺ (C=C–Au⁺) and D_∞h C₂Au₂ (Au–C≡C–Au) and as bridges (–Au–) in the side-on coordinated C_2v C₂Au₃ ⁺ ([Au–C≡C–Au]Au⁺) and C_s C₂HAu₂ ⁺([H–C≡C–Au]Au⁺). However, when the number of gold atoms reaches four, they form stable gold triangles (–Au₃) in the head-on coordinated C_2v C₂Au₄ (Au–C≡C–Au₃) and the side-on coordinated C_2v C₂Au₅ ⁺ ([Au–C≡C–Au]Au₃ ⁺). Similar –Au₃ triangular units exist in the head-on coordinated C_2v C₂HAu₃ (H–C≡C–Au₃) and D_2d C₂Au₆ (Au₃–C≡C–Au₃). The existence of stable –Au₃ triangular units in small dicarbon aurides is significant and intriguing. The high stability of Au₃ triangles originates from the fact that an equilateral D_3h Au₃ ⁺ cation possesses a completely delocalized three-center-two-electron (3c–2e) σ bond and therefore is σ-aromatic in nature. The extension from H/Au analogy to H/Au₃ analogy established in this work may have important implications in designing new gold-containing catalysts and nano-materials.