Calculations of Jahn-Teller coupling constants in the weak-field basis via the angular overlap model: a simple operator equivalent technique

Within the framework of the angular overlap model the matrix elements of the linear Jahn-Teller operator may easily be calculated in the weak-field basis in terms of simple operator equivalents. The method is applied to the calculation of the 〈M_J|?V/?Q|M_j〉 matrix elements for the |LSJMj〉 ground states of t^x species (x = 1) ? 13) in )^* symmetry.     

A variation principle for energy differences between states of two different Hamiltonians

A modification of a variation principle due to Delves, is derived which permits the direct calculation of energy differences between states of two different Hamiltonians: [Δ ??] = 〈X₀| ??_x – W_x|X₁〉 – 〈Y₀|??_y – W_y|y₁〉 + 〈X₀| Δ ??|Y₀〉 · 〈X₀| Y₀〉^?1. Δ ?? = ??_y – ??_x, |X₀〉 and |Y₀〉 are the wave functions for the X and Y states and |X₁〉 and |Y₁〉 are functions defined in the text. The principle is applied to a few simple examples.     

Hypervirial theorem and matrix elements for the Coulomb potential

It is well known that hydrogen like atoms may be represented as Morse oscillators. Our aim in this article was to show how this result and the hypervirial theorem lead to a computational method with which the matrix elements for the Coulomb potential can be easily calculated. Also, a closed analytical expression for 〈n₂l₂|r^k|n₁l₁〉 is obtained. © 1995 John Wiley & Sons, Inc.     

Ultrasonic method of determination of stability constants of charge transfer complexes of certain carbonyl compounds and diethylamine in n -hexane

The ultrasonic velocities (U), densities (ρ) and viscosities (η) were measured for solutions containing equimolar concentrations of diethylamine (donor), nine aldehydes and nine ketones (acceptors) in n-hexane at 303 K. Acoustical parameters such as adiabatic compressibility (β), free length (L _f), viscous relaxation time (τ), and molecular interaction parameter (χ_U) have been computed. These values indicate the formation of charge transfer complexes between carbonyl compounds and amine. Formation constant (K) values of the complexes have been evaluated using the equation proposed by Kannappan. The constant values of free energy of activation (ΔG ) and relaxation time indicate the formation of similar charge transfer complexes in these systems. However, the variation in free energy of formation (ΔG°_F) values suggests that their thermodynamic stability depends on the structure of donor and acceptor.     

Partitioning technique,perturbation theory,and rational approximations

Instead of the Schródinger equation ??Ψ = EΨ subject to the boundary condition 〈φ|Ψ〉 = 1, where φ is a normalized reference function in the Hilbert space, one studies the inhomogeneous equation (?? ? ?)Ψ_? = aφ, where ? is a complex variable, with the same boundary condition, which gives a = 〈φ|??|Ψ_?〉 ? ? = ?₁ ? ?. Introducing the projector P = 1 ? |φ〉〈φ| for the complement to O = |φ〉〈φ|, one finds easily the explicit solution Ψ_? = (1 ? P??/?)^?1φ = (1 + T_???)φ, where T_? = (? ? P??)^?1P = P(? ? P??P)^?1P is the reduced resolvent associated with the auxiliary Hamiltonian H? = P??P. The existence of these operators is discussed. It is shown that, if the parameter ? is real in the “discrete part” of the spectrum to ??, then ? and ?₁ = 〈φ|??|Ψ_?〉 = 〈φ|?? + ??T_???|Φ〉 ≡f(?) bracket a true eigenvalue E satisfying the relation E = f(E). The Newton-Raphson solution to the equation F(?) = ? ? f(?) = 0 is related to the variation principle. Putting ?? = ??₀ + V and expanding the inverse (? ? P??₀ ? PV)^?1 in terms of powers of V or (V ? α), one gets various expansions relating to finite-order perturbation theory. Exact expressions for the ordinary wave and reaction operators are obtained. If A is an arbitrary nonsingular operator and h = {h₁,h₂,…,h_n} is a linearly independent set, the inner projection Á_n = | h 〉 〈 h |A^?1| h 〉^?1〈 h | is a “rational approximation” to the operator A which converges toward A when n→∞ and the set h becomes complete. If A is positive (or has a finite negative part) the convergence is from below. Applying this principle to the partitioning technique, one gets rational perturbation approximations instead of the standard power series, similar to the Padé approximants but derived in a different way with the remainder term under control. The method has been used to calculate lower bounds to eigenvalues.     

Spectrophotometric and spectroscopic studies of charge transfer complexes of p-toluidine as an electron donor with picric acid as an electron acceptor in different solvents

The charge transfer complexes of the donor p-toluidine with π-acceptor picric acid have been studied spectrophotometrically in various solvents such as carbon tetrachloride, chloroform, dichloromethane acetone, ethanol, and methanol at room temperature using absorption spectrophotometer. The results indicate that formation of CTC in non-polar solvent is high. The stoichiometry of the complex was found to be 1:1 ratio by straight-line method between donor and acceptor with maximum absorption bands. The data are discussed in terms of formation constant (K_CT), molar extinction coefficient (?_CT), standard free energy (ΔG^o), oscillator strength (f), transition dipole moment (μ_EN), resonance energy (R_N) and ionization potential (I_D). The results indicate that the formation constant (K_CT) for the complex was shown to be dependent upon the nature of electron acceptor, donor and polarity of solvents that were used.     

Spectrophotometric study of charge transfer complex between 2,6-diaminopyridine and 2,5-dihydroxy-p-benzoquinone

Charge transfer (CT) complex formation between 2,6-diaminopyridine (2,6-DAP) as the electron donor with 2,5-dihydroxy-p-benzoquinone (DHBQ) as the electron acceptor has been studied spectrophotometrically in different polar solvents at room temperature. A new absorption band due to CT complex formation was observed near 490?nm. The stoichiometric ratio of the complex has been identified by Job's, photometric and conductometric titration methods to be 1?:?1. Benesi–Hildebrand equation has been applied to estimate the formation constant (K _CT) and molecular extinction coefficient (ε). They recorded high values confirming high stability of the formed complex. The physical parameters, oscillator strength (f), transition dipole moment (μ), ionisation potential (I _D), resonance energy (R_N ) and standard free energy change (ΔG°) of the formed complex were determined and evaluated in the different solvents. The solid complex between 2,6-DAP and DHBQ has been isolated and characterised using elemental analysis, FT-IR and ¹H-NMR measurements.     

Hypervirial theorem and ladder operators. Recurrence relations for harmonic oscillator integrals

A second-quantization formalism combined with a hypervirial theorem is used to derive new recurrence relations for one-dimensional harmonic oscillator matrix elements. The most general case of 〈m|f(â, â⁺)|n〉 is considered, and the recurrence relations forf(â, â) = X^k, exp(?βX), and exp(?X²) are given as examples. The relations obtained are considerably simpler than those derived by using only the hypervirial theorem; comparatively, the recurrence relations presented here have the advantage of avoiding the use of the quantum mechanical sum-rules when determining initial matrix elements. The proposed procedure can be used to determine the recurrence relations for other potentials as well as to evaluate the two-center integrals.     

Intensity parameters,radiative transitions and non-radiative relaxations of Ho3+ in various tellurite glasses

The intensity parameters of holmium in sodium, barium and zinc tellurite glasses were obtained from the absorption spectra. Using these parameters and the matrix elements of Ho³⁺, transition probabilities and branching ratios from the excited states (⁵F₄, ⁵S₂), ⁵F₅, ⁵I₄, ⁵I₅ and ⁵I₆ were calculated. Quantum efficiencies for the (⁵F₄, ⁵S₂) state were obtained and the multiphonon transition rates calculated at room temperature. Non-radiative multiphonon relaxations were obtained from the (⁵F₄, ⁵S₂) level. The relaxation increased in the order of zinc, barium, sodium, and is assumed to be dependent on the local site symmetry of Ho³⁺ in glass.     

Computational studies on optoelectronic and charge transfer properties of some perylene‐based donor‐π‐acceptor systems for dye sensitized solar cell applications

We report DFT studies on some perylene‐based dyes for their electron transfer properties in solar cell applications. The study involves modeling of different donor‐π‐acceptor type sensitizers, with perylene as the donor, furan/pyrrole/thiophene as the π‐bridge and cyanoacrylic group as the acceptor. The effect of different π‐bridges and various substituents on the perylene donor was evaluated in terms of opto‐electronic and photovoltaic parameters such as HOMO‐LUMO energy gap, λ_max, light harvesting efficiency(LHE), electron injection efficiency (Ø_inject), excited state dye potential (E^dye*), reorganization energy(λ), and free energy of dye regeneration (). The effect of various substituents on the dye–I₂ interaction and hence recombination process was also evaluated. We found that the furan‐based dimethylamine derivative exhibits a better balance of the various optical and photovoltaic properties. Finally, we evaluated the overall opto‐electronic and transport parameters of the TiO₂‐dye assembly after anchoring the dyes on the model TiO₂ cluster assembly.     

Ionization Constants of o-Phthalic Acid and Standard pH Values of Potassium Hydrogen Phthalate Buffer Solutions in Glycerol + Water Solvent Mixtures at Normal and Subzero Temperatures

Determination of primary standards for pH measurements in glycerol + water solventmixtures has been carried out based on reversible emf measurements of the cellPt|H₂|KHPh (m _PS) + KCl (m _Cl)|AgCl|Ag|Ptwhere KHPh denotes the potassium hydrogen phthalate buffer solution of molalitym _PS = 0.05 mol-kg^–1, at glycerol mass fractions w _G = 0.2 and 0.4, within thetemperature range –10 to 40°C. A multilinear regression procedure as a functionof electrolyte molality, glycerol mass fraction w _G, and temperature T has beenapplied for the data processing leading to the values of primary standards pH_PS.These can be represented by the following regression equationpH_PS = (4.007037±0.001113) + (3.55844±0.01776)x _G+(0.39622±0.01410)z + (4.3084±0.3377)z ²– (50.66±10.53)x _G z ² + (457.10±78.48)x _G ² z ²where z = (T – 298.15)/298.15. Parallel values of the first ionization constantof o-phthalic acid (H₂Ph; benzene-1,2-dicarboxylic acid, the parent acid of KHPh),which are essential for the above calculations, have been determined fromreversible emf measurements of the cellPt|H₂|H₂Ph (m ₁) + KHPh (m ₂) + KCl (m ₃)|AgCl|Ag|Ptover the range of solvent composition and temperatures mentioned above.     

Spectroscopic studies of multiple charge transfer complexes of p-toluidine with π-acceptor picric acid in different polar solvents

The charge transfer complexes of the donor p-toluidine with π-acceptor picric acid have been studied spectrophotometrically in various solvents such as acetone, ethanol, and methanol at room temperature using absorption spectrophotometer. The results indicate that formation of CTC in less polar solvent is high. The stoichiometry of the complex was found to be 1: 1 ratio by straight line method between donor and acceptor with maximum absorption bands. The data are discussed in terms of formation constant (K _CT), molar extinction coefficient (?_CT), standard free energy (ΔG°), oscillator strength (f), transition dipole moment (μ_EN), resonance energy (R _N) and ionization potential (I _D). The results indicate that the formation constant (K _CT) for the complex were shown to be dependent upon the nature of electron acceptor, donor and polarity of solvents which were used.     

500‐Fold Amplification of Small Molecule Circularly Polarised Luminescence through Circularly Polarised FRET

Strongly dissymmetric circularly polarised (CP) luminescence from small organic molecules could transform a range of technologies, such as display devices. However, highly dissymmetric emission is usually not possible with small organic molecules, which typically give dissymmetric factors of photoluminescence (g_PL) less than 10^?2. Here we describe an almost 10³‐fold chiroptical amplification of a π‐extended superhelicene when embedded in an achiral conjugated polymer matrix. This combination increases the |g_PL| of the superhelicene from approximately 3×10^?4 in solution to 0.15 in a blend film in the solid‐state. We propose that the amplification arises not simply through a chiral environment effect, but instead due to electrodynamic coupling between the electric and magnetic transition dipoles of the polymer donor and superhelicene acceptor, and subsequent CP Förster resonance energy transfer. We show that this amplification effect holds across several achiral polymer hosts and thus represents a simple and versatile approach to enhance the g‐factors of small organic molecules.     

Studies on Equation-of-State Parameters and Linear Polymer Dimensions of Poly(p-chlorostyrene) in Diethylene Glycol Monobutyl Ether by Intrinsic Viscosity Measurements

Intrinsic viscosities, [η], of poly(p-chlorostyrene) (PPCS) in diethylene glycol monobutyl ether (DGMBE) which exhibits an exothermic solubility behavior with the polymer were measured using an Ubbelohde type capillary viscometer between 25 and 85°C. Polymer solvent interaction parameters at infinite dilution (χ₁), exchange energy parameter ([`(X)]₁₂)(\bar{X}_{12}) , exchange enthalpy (X₁₂), and entropy parameters (Q₁₂), of the PPCS/DGMBE pair were found at studied temperature range according to equation-of-state theory. In the blob theory, dependence of [η] on temperature can be scaled by a master curve in a plot of α_η³|τ|M^1/2 versus |τ|M^1/2 as the polymer coil contracts below the Θ-point, however, it can be scaled by a master curve in a plot of α_η⁻⁵|τ|M^1/2 versus |τ|M^1/2 as the polymer coil expands above the Θ-point in an endothermic solubility behavior. Since the studied PPCS/DGMBE system represents exothermic solubility behavior, the master curves of the system were plotted in α_η³|τ|M^1/2 versus |τ|M^1/2 as the polymer coil contracts above the Θ-point and in α_η⁻⁵|τ|M^1/2 versus |τ|M^1/2 as the polymer coil expands below the Θ-point. The universal plots of α_η(N/N_c)^1/6 versus N/N_c and α_η(N/N_c)^−1/10 versus N/N_c were plotted above and below Θ-point, respectively.     

Predissociation of the C2Σu+ state of N2+

Two mechanisms for the predissociation of the C²Σ_u⁺ state of N₂⁺ are discussed - the accidental mechanism and a direct, homogeneous process C → B²Σ_u⁺. The matrix elements for the latter channel are dominated by a contribution from the nuclear kinetic energy operator, containing a Franck-Condon integral of form 〈?|?/?R|υ′〉.     

Photoelectrochemical behavior of coupled SnO2|CdSe nanocrystalline semiconductor films

A photoelectrochemical cell with a coupled SnO₂|CdSe nanocrystalline semiconductor electrode has been prepared by sequential deposition of SnO₂ and CdSe films onto an optically transparent electrode (OTE), and its photoelectrochemical behavior has been studied. The results show that the coupling of CdSe with SnO₂ leads to an improvement in the performance of OTE|SnO₂|CdSe over OTE|CdSe cells in terms of increased incident photon-to-current conversion efficiency, increased stability and smaller reversal of current. The favorable positioning of the energy bands of SnO₂ and CdSe is responsible for the above observations. Various photoelectrochemical parameters of the OTE|SnO₂|CdSe cell obtained for an incident light power of 0.31 mW cm⁻² at 470nm, are as follows: I_sc ≈ 25–30 μA cm⁻², V_oc ≈ 0.5–0.6 V, ƒƒ = 0.47 and a power conversion efficiency of about 2.25%.     

Photoelectrochemical properties of MoO2 thin films

Thin MoO₂ films were electrodeposited on a selenium pre-deposited SnO₂|glass plate. The photoelectrochemical properties of MoO₂ films were investigated in 0.1 M Na₂SO₄ solution by the ultraviolet–visible spectrophotometry, linear sweep voltammetry, and altering current impedance measurement techniques. It was found that under illumination with the incident light of λ?=?366 nm, the photo response of the MoO₂|SnO₂|glass electrode resulted from the MoO₂ layer, while the SnO₂ layer served as a sink for photogenerated charge carriers. The MoO₂ film exhibited n-type conductivity. A schematic band structure diagram of MoO₂ in 0.1 M Na₂SO₄ solution was constructed. The flat band potential (E _fb), the donor concentration (N _D), the photogeneration current efficiency depended on MoO₂ film thickness. The [Fe(CN)₆]^4?/3? redox PEC cell with MoO₂|SnO₂|glass plate as a photoanode was constructed. Power output characteristics such as the open circuit voltage (V _OC), short circuit current (I _SC), the fill factor (FF), and the light-to-electrical conversion efficiency (η) were determined. The maximum light-to-electrical conversion efficiency exhibited by the PEC cell was 0.94 %.     

Studies on Equation-of-State Parameters and Linear Polymer Dimensions of Poly(p-chlorostyrene) in Diethylene Glycol Monobutyl Ether by Intrinsic Viscosity Measurements

Summary. Intrinsic viscosities, [η], of poly(p-chlorostyrene) (PPCS) in diethylene glycol monobutyl ether (DGMBE) which exhibits an exothermic solubility behavior with the polymer were measured using an Ubbelohde type capillary viscometer between 25 and 85°C. Polymer solvent interaction parameters at infinite dilution (χ₁), exchange energy parameter , exchange enthalpy (X₁₂), and entropy parameters (Q₁₂), of the PPCS/DGMBE pair were found at studied temperature range according to equation-of-state theory. In the blob theory, dependence of [η] on temperature can be scaled by a master curve in a plot of α_η³|τ|M^1/2 versus |τ|M^1/2 as the polymer coil contracts below the Θ-point, however, it can be scaled by a master curve in a plot of α_η⁻⁵|τ|M^1/2 versus |τ|M^1/2 as the polymer coil expands above the Θ-point in an endothermic solubility behavior. Since the studied PPCS/DGMBE system represents exothermic solubility behavior, the master curves of the system were plotted in α_η³|τ|M^1/2 versus |τ|M^1/2 as the polymer coil contracts above the Θ-point and in α_η⁻⁵|τ|M^1/2 versus |τ|M^1/2 as the polymer coil expands below the Θ-point. The universal plots of α_η(N/N_c)^1/6 versus N/N_c and α_η(N/N_c)^−1/10 versus N/N_c were plotted above and below Θ-point, respectively.     

Reaction of polyamine polymer with molecular iodine via charge-transfer complex formation

The reaction of a Mannich base type-polyamine polymer with iodine (I₂) was studied kinetically and thermodynamically in order to clarify the polymer effects in the formation of triiodide ions (I₃^?). N,N-Dimethyl-p-(4-methylpiperazinomethyl) aniline and 1,4-dimethylpiperazine were used as low molecular weight donor model compounds. Triiodide ions are produced from the polyamine–I₂ system immediately after mixing the two-component solutions, while in the systems with I₂ and N,N-dimethyl-p-(4-methyl-piperazinomethyl)aniline and 1,4-dimethylpiperazine they are obtained only when relatively high concentrations of both donor and acceptor solutions were mixed. This is explained by the entropic contributions of the polymer chain such as the stacking effect of donor nitrogen atoms, i.e., the increment of local donor concentration around I₂ in the reaction field. The relation between the solution behavior of the reaction systems and the rate of formation of I₃^? ions also supports this kind of polymer effect. The effects of neighboring groups and dielectric constant on the reaction are also discussed.     

Synthesis and Characterization of Donor–π‐Acceptor‐Based Porphyrin Sensitizers: Potential Application of Dye‐Sensitized Solar Cells

New porphyrin sensitizers based on donor–π‐acceptor (D‐π‐A) approach have been designed, synthesized, characterized by various spectroscopic techniques and their photovoltaic properties explored. N,N′‐Diphenylamine acts as donor, the porphyrin is the π‐spacer, and either carboxylic acid or cyanoacryclic acid acts as acceptor. All compounds were characterized by using ¹H NMR spectroscopy, ESI‐MS, UV–visible emission spectroscopies as well as electrochemical methods. The presence of aromatic groups between porphyrin π‐plane and acceptor group push the absorption of both Soret and Q‐bands of porphyrin towards the red region. The electrochemical properties suggests that LUMO of these sensitizers above the TiO₂ conduction band. Finally, the device was fabricated using liquid redox electrolyte (I^?/I₃^?) and its efficiency was compared with that of a leading sensitizer.