Clebsch–Gordan coefficients for the group chains O ⊃ T ⊃ D2 and Td ⊃ T ⊃D2 by the method of projective representations

The spinor representations for the groups forming the chains O ? T ? D ₂ and T _d ? T ? D ₂ are constructed as projective representations. The Clebsch–Gordan coefficients are then calculated using a standard method. Projective factor systems, irreducible representations, canonical bases, and tables of Clebsch–Gordan coefficients are given. The subduction from O to D ₃ is discussed.     

Projection operators and Clebsch–Gordan coefficients for projective representations

Previous discussions of the bases and projection operators for projective representations are analyzed and alternatives are proposed. Detailed proofs are provided for a result which is often assumed or proved within unacceptable limitations, namely that the form of the projection operators which is standard for vector representations is also valid for unitary projective representations. These proofs provide necessary conditions for this result, and they are constructed in terms of the definition given for the bases of projective representations. The calculation of Clebsch–Gordan coefficients for projective representations by means of the projection operators is discussed. Whereas in the method of Dirl the work is carried out entirely in terms of the matrix representations, and the symmetrization of the bases has to be considered in a second step, all the work of this paper is conducted starting from the symmetrization of the bases of the projective representations, so that those two steps are carried out simultaneously.     

Intrakonfigurations‐, Trip‐Multiplett‐ sowie anomal polarisierte A1gund A2g‐Übergänge in elektronischen und vibratorischen Resonanz‐Raman‐Spektren von (spinentarteten) trans‐Di(cyano)phthalocyaninatorhenaten

Intraconfigurational, Trip‐Multiplet, and Anomalously Polarised A_1g and A_2g Transitions in Electronic and Vibrational Resonance Raman Spectra of (Spin‐Degenerate) trans ‐Di(cyano)phthalocyaninatorhenates Brown bis(tetra(n‐butyl)ammonium) trans‐di(cyano)phthalocyaninato(2‐)rhenate(II) ( 1 ) is prepared by melting bis(phthalocyaninato(2‐)rhenium(II)) with tetra(n‐butyl)ammonium cyanide. According to electrochemical data, 1 is oxidised by iodine to yield blue tetra(n‐butyl)ammonium trans‐di(cyano)phthalocyaninato(2‐)rhenate(III) ( 2 ), whose cation exchange in the presence of bis(triphenylphosphine)iminium salts has been confirmed by x‐ray structure determination. 1 and 2 dissolve without dissociation of the cyano ligands in conc. sulfuric acid. Dilution with cold water precipitates blue trans‐di(cyano)phthalocyaninato(2‐)rhenium(III) acid. 1 and 2 are oxidised by bromine yielding violet trans‐di(cyano)phthalocyaninato(1‐)rhenium(III). Oxidation of 2 with dibenzoylperoxide and N‐chlorsuccinimide is described. 1 and 2 are characterised by polarised resonance Raman(RR) spectra, FIR/MIR spectra, and UV‐Vis‐NIR spectra. Due to a Kramers degenerate ground electronic state of low‐spin Re^II, a polarisation anomaly of the totally symmetric vibrations a_1g at 598 and 672 cm^–1 with depolarisation ratios ρ_l > 3 is observed in the RR spectra of 1 . Weak bands in the unusual UV‐Vis‐NIR spectrum of 1 , starting at 10200 cm^–1, are attributed to trip‐multiplet (TM) transitions. An electronic RR effect is detected for 2 . The selectively enhanced anomalously polarised line at 1009 cm^–1 with ρ_l ≈ 15 and the (de)polarised lines between 1688 and 2229 cm^–1 are attributed to intraconfigurational transitions A_1g → A_2g > A_1g, B_1g, B_2g, E_g arising from the ³T_1g ground electronic state of low‐spin Re^III split by spin‐orbit coupling and low symmetry (D ). Some of their vibronic bands are detected in the IR spectrum between 1900 and 4000 cm^–1. B and Q transitions of 2 at 16700 and 31900 cm^–1, respectively, as well as eight weak TM transitions are observed between 5050 and 26100 cm^–1.     

Raman spectroscopy of the [TeX6]2− ions (X  Cl or Br) at resonance with their lowest 3T1u and 1T1u states: Evidence for tetragonal distortion in these excited states

The observation that the v₂(e_g) band is the most enchanced Raman band at resonance with the bands assigned to both the ³T_1u ← ¹A_1g and ¹T_1u ← ¹A_1g transitions of the [TeX₆]^2? ions indicates that the ions are tetragonally distorted in these excited states. The depolarisation ratio of 2v₂ band of [TeBr₆]^2? at resonance with the ¹T_1u ← ¹A_1g transition is found to be 0.18, in close agreement with that expected (3/14) for the first overtone of a doubly degenerate vibration coupled to a triply degenerate excite state.     

Microwave—optical double resonance in the A1A2 excited state of thioformaldehyde

Microwave—optical double resonance experiments have been carried out on the 4_o¹ band of the $\text{A}$¹A₂$\text{X}$¹A₁ system of thioformaldehyde (H₂CS). More than 100 microwave and radiofrequency transitions have been observed in the $\text{A}$¹A₂ excited state. Many of these transitions are magnetically sensitive. Some of the excited state levels are perturbed by triplet levels and others by high vibrational levels of the ground state.     

Construction of orbital angular momentum eigenfunctions for many-particle systems by harmonic projection

Formulas are derived which allow the direct construction of total orbital angular momentum eigenfunctions for many-particle systems without the use of Clebsch–Gordan coefficients. One of the equations is closely analogous to Dirac' identity for the total spin operator. This equation describes the action of L² on a function of the particle coordinates in terms of a class operator of the symmetric group and a "contraction operator." A general projection operator for constructing symmetric eigenfunctions of L² is presented.     

Clebsch–Gordan coefficients for chains of groups of interest in quantum chemistry. III. The point symmetry groups

This work amalgamates some basic elements defined in the first paper of this series and in related papers with the theory of coupling coefficients for an arbitrary group with the view of generating the Clebsch–Gordan coefficients and V symbols for point symmetry groups. The connection between Clebsch–Gordan coefficients and V symbols is established for an arbitrary group in a form that reduces to the one known for the chain SU(2) ? U(1). The Clebsch–Gordan coefficients and V symbols of any point symmetry group G are shown to be obtainable from Clebsch–Gordan coefficients and \documentclass{article}\pagestyle{empty}\begin{document}$ {\bar f} $\end{document} symbols of the chain SU(2) ? G through the resolving of a system of nonlinear equations.     

Light absorption by manganese pairs in KMgF3:Mn2+ through a vibronically induced exchange mechanism

Double excitations to ⁴E_g and ⁴A_1g states in manganese pairs of KMgF₃:Mn²⁺ have been studied spectroscopically. Three prominent sharp bands are observed in the low temperature absorption spectrum. These bands are displaced by about 400–500 cm^?1 to higher energies from the expected electronic origins. It is proposed that the observed pair transitions gain their intensity through a vibronically induced exchange mechanism. With this mechanism the symmetric double excitations ⁶A_1g⁶A_1g → ⁴E_gu⁴E_gu, ⁴E_gv⁴E_gv and ⁴A_1g⁴A_1g become allowed in addition to the transition ⁶A_1g⁶A_1g → ⁴E_gu⁴A_1g. Analogy to the spectrum of the linear dinuclear chromium complex [(NH₃)₅CrOCr(NH₃)₅]⁴⁺, where the same mechanism has been postulated, is demonstrated.     

Electronic absorption spectrum of Fe3+ doped in ammonium chloride single crystal

The electronic absorption spectrum of the Fe²⁺ ion doped in ammonium chloride has been studied at room and liquid air temperatures. The observed bands have been assigned transitions from the ground ⁶A_1g(S) state to the excited ⁴A_1g(⁴E_g), ⁴T_1g(G) and ⁴T_2g(G) states. The cubic field approximation with D_q = 675 cm^?1, B = 645 cm^?1 and C = 4.4 B is found to give a good fit to the observed band positions.It is further concluded that the site symmetry of the Fe³⁺ ion in the crystal is lowered from O_h to C_4v symmetry at liquid air temperature.     

Theoretical study of the π→π* excited states of linear polyenes: The energy gap between 11Bu+ and 21Ag− states and their character

Multireference perturbation theory with complete active space self-consistent field (CASSCF) reference functions is applied to the study of the valence π→π* excited states of 1,3-butadiene, 1,3,5-hexatriene, 1,3,5,7-octatetraene, and 1,3,5,7,9-decapentaene. Our focus was put on determining the nature of the two lowest-lying singlet excited states, 1¹B_u⁺ and 2¹A_g⁻, and their ordering. The 1¹B_u⁺ state is a singly excited state with an ionic nature originating from the HOMO→LUMO one-electron transition while the covalent 2¹A_g⁻ state is the doubly excited state which comes mainly from the (HOMO)²→(LUMO)² transition. The active-space and basis-set effects are taken into account to estimate the excitation energies of larger polyenes. For butadiene, the 1¹B_u⁺ state is calculated to be slightly lower by 0.1 eV than the doubly excited 2¹A_g⁻ state at the ground-state equilibrium geometry. For hexatriene, our calculations predict the two states to be virtually degenerate. Octatetraene is the first polyene for which we predict that the 2¹A_g⁻ state is the lowest excited singlet state at the ground-state geometry. The present theory also indicates that the 2¹A_g⁻ state lies clearly below the 1¹B_u⁺ state in decapentaene with the energy gap of 0.4 eV. The 0–0 transition and the emission energies are also calculated using the planar C_2h relaxed excited-state geometries. The covalent 2¹A_g⁻ state is much more sensitive to the geometry variation than is the ionic 1¹B_u⁺ state, which places the 2¹A_g⁻ state significantly below the 1¹B_u⁺ state at the relaxed geometry. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 157–175, 1998     

Two-photon absorption spectrum of (2,2)-paracyclophane

The two-photon absorption spectrum of crystalline (2,2)-paracyclophane was measured in the regions of 31200–33800 cm^?1, 34700–36500 cm^?1, and 40800–48600 cm^?1 by monitoring the fluorescence intensity using a tunable dye laser as a two-photon excitation source. Two absorption bands in the region 34700–36500 cm^?1 were assigned to the two even-parity allowed ¹B_1g ← ¹A_1g and ¹B_2g ← ¹A_1g transitions.     

1 Bu–2 Ag conical intersection in trans-butadiene: ultrafast dynamics and optical spectra

The potential-energy functions of the 1 ¹B_u and 2 ¹A_g excited valence states of trans-butadiene have been characterised by the CASPT2 method. Based on these ab initio data, a vibronic-coupling model describing the conical intersection of the 1 ¹B_u and 2 ¹A_g states has been constructed. UV resonance-Raman and absorption spectra have been calculated, employing the time-dependent approach. The time-dependent wave-packet calculations reproduce the expected ultrafast (≈30 fs) radiationless decay of the optically bright 1 ¹B_u state into the dark 2 ¹A_g state.     

The “phantom” photochemical singlet state of stilbene and its diphenylpolyene relative

Recently the conception on the cis-trans photoisomerization of stilbene has emerged that this process is governed by a higher excited ¹A_g state which exhibits a minimum at the perpendicular conformation crossing the lowest excited ¹B_u state upon bond rotation. An evaluation of the potential surfaces governing the photoisomerization process by a PPP SCF CI method revealed that there exists indeed such a photochemically active ¹A_g state which in the planar molecule lies about 1 eV above the lowest absorption band and involves the excitations of two electrons from the SCF ground state. Extensions of the calculations to diphenylpolyenes demonstrate that this ¹A_g state is related to the forbidden low-lying doubly excited ¹A_g state observed earlier in these molecules.     

Circularly Polarized Luminescence from Axially Chiral BODIPY DYEmers: An Experimental and Computational Study

With our new home‐built circularly polarized luminescence (CPL) instrument, we measured fluorescence and CPL spectra of the enantiomeric pairs of two quasi‐isomeric BODIPY DYEmers 1 and 2 , endowed with axial chirality. The electronic circular dichroism (ECD) and CPL spectra of these atropisomeric dimers are dominated by the exciton coupling between the main π–π* transitions (550–560 nm) of the two BODIPY rings. Compound 1 has strong ECD and CPL spectra (g_lum=4×10^?3) well reproduced by TD‐DFT and SCS‐CC2 (spin‐component scaled second‐order approximate coupled‐cluster) calculations using DFT‐optimized ground‐ and excited‐state structures. Compound 2 has weaker ECD and CPL spectra (g_lum=4×10^?4), partly due to the mutual cancellation of electric–electric and electric–magnetic exciton couplings, and partly to its conformational freedom. This compound is computationally very challenging. Starting from the optimized excited‐state geometries, we predicted the wrong sign for the CPL band of 2 using TD‐DFT with the most recommended hybrid and range‐separated functionals, whereas SCS‐CC2 or a DFT functional with full exact exchange provided the correct sign.     

Reexamination of the phosphorescence of Ba2Pt2(H2P2O5)4 via thermally modulated emission spectroscopy

Reexamination of the phosphorescence of Ba₂Pt₂(H₂P₂O₅)₄ reveals that the ≈10 K spectrum is a superposition of two electronic transitions [³A₂u(E_u,A_1u → A_1g] separated by ≈40 cm^?1. Each band displays a prominent 110 cm^?1 vibrational progression. Franck-Condon analysis yields a ≈0.25 Å distortion of the PtPt bond in the excited states, interpreted as a contraction.     

Derivation of the Clebsch–Gordan coefficients by means of projection operators

The Clebsch–Gordan coefficients for the coupling of two angular momenta are derived by using the projection operator technique, developed by Löwdin. The derivation is done in two steps; first for the so-called principal case, i.e. k = m, then for the general case with an arbitrary m. Two different derivations are given for the principal case, a direct one and one based on a recursion procedure. The general case is obtained from the principal case with a step-down operator.     

Configuration interaction study of the oscillator strenghts for the A1--- A1 and A1--- A1 transitions of the water molecule

Oscillator strengths for transitions between the ¹A₁ ground state of water and its ¹A₁ and ¹A₁ excited states are computed employing two different theoretical approaches. In one series of calculations a common orthonormal one-electron basis is employed for all of the above states, while in the other type of treatment two different, mutually non-orthogonal, sets are used; the multireference single- and double-excitation (MRD CI) method is employed in each case, with configuration selection, to generate the various electronic wavefunctions. It is found that the use of ground-state SCF MOs leads to poor convergence in the wavefunctions of the (Rydberg-type) ¹A₁ and ¹A₁ excited states and consequently also for the corresponding --- and --- f-values; this behavior is seen to be closely related to the near degeneracy of the two excited states, each of which is a mixture of the 3a₁ → 3sa₁ and 1b₁ → 3pb₁ configurations. Analogous computations with the ¹B₁¹b₁ → 3sa₁ MOs show much better convergence properties, and the resulting f-values compare well with what is obtained when state-specific orbital sets are employed separately for ground and excited states and non-orthonormal techniques are applied to compute the desired transition moments. These results tend to confirm previous findings which indicate conceptual and computational advantages for the calculation of excited-state wavefunctions and properties within the context of a state-specific theory. They also show that although the goal of eliminating the dependence of MRD CI calculations on the choice of MO basis is very nearly approached for energy quantities, it is less satisfactorily achieved for other properties, especially when the existence of nearly degenerate electronic states is a critical factor.     

The valence electronic excited states of trans-1,3-butadiene and trans,trans-1,3,5-hexatriene from generalized valence bond and configuration interact

Self-consistent ab initio generalized valence bond (GVB) and configuration interaction (Cl) calculations are presented for the ground and valence electronic excited states of trans-1,3-butadine and all trans-1,3,5-hexatrine. Previous workers have suggested that (all trans) polyenes exhibit a parity-forbidden valence excited state (2¹ A_g at an energy just below that of the first dipole-allowed (1¹ B_u) state. We find such valence excited electronic states for butadiene (ΔE = 7.06 eV) and hexatriene (ΔE = 5.87 eV), but in both cases the excitation energy is considerably higher than the dipole-allowed transitions (zero-zero transitions at 5.95 eV and 4.95 eV, respectively). The lower two triplet states are found at 3.35 eV and 5.08 eV for butadie and at 2.71 eV and 4.32 eV in hexatrine, in good agreement with experimental values (3.2–3.3 eV and 4.92 eV for butadiene and 2.66 eV and 4.1–4.2 eV for hexatrine). Considering the states formed by removing one electron from the π space we found ion states at 8.95 eV and 11.40 eV for butadiene and at 8.33 eV, 10.53 eV, and 11.60 eV for hexatriene, in godo agreement with experimental results (9.0 eV and 11.5 eV for butadiene and 8.45 eV, 10.43 eV and 11.6 eV for hexatriene).     

Infrared luminescence of the ReBr2−6ion

The well resolved Γ₇ (²T_2g) → Γ₆²T_1g), Γ₈ (²E_1g), Γ₈(²T₈) and Γ₈ (²T_1g) → Γ₈ (⁴A_2g) transitions of the ReBr^r2?₆ion have been observed for the first time.