Density of states of substoichiometric TiN1−x

The electronic states of substoichiometric TiN_1–x are studied by the CPA method and both the partial and total densities of states (DOS) are calculated. The results reveal that with increasing vacancy concentrationx the Fermi levelE _F moves to higher energies and the DOS atE _F,g(E _F), increases linearly. The calculated L_{II, III} X-ray spectra of Ti compare well with experiment.Work supported by the Research Institute for Powder Metallurgy, Sumperk, Czechoslovakia.The author is indebted to Dr. B. Velický, CSc for suggestion of the method to solve the Soven's equation. The financial assistance of the Research Institute for Powder Metallurgy, umperk, Czechoslovakia is gratefully acknowledged.     

Anharmonic corrections for linear triatomic molecules subject to the Rennet-Teller effect

The effects of vibrational anharmonic terms and of the g_K -correction on the energy levels of a triatomic molecule in a degenerate electronic state are considered. The electronic wavefunctions are described using the approach first suggested in the original paper of Renner. Formulae for the anharmonic corrections in a number of different situations are derived. For an electronic Π state the corrections are given in the form where i runs over the various contributions and x ₁, x ₂, … depend on the anharmonic force constants. The functions F_i can be determined numerically (see equations (4.3) and 4.7)). For the case without spin-orbit interaction the F_i s are given explicitly to first order in ∈ in table 1. Further-more, the same results apply for levels with |K|=v ₂+1 even if the spinorbit interaction is not negligible. Explicit results for levels with |K|<v ₂ including spin-orbit interaction are given in tables 2 and 3. The cases with larger values for Λ (2, 3, …) are also considered. The energy level formulae for a Δ state differ from those derived earlier by Merer and Travis. In particular, the small separation between the vibronic Φ and Π levels with v ₂=1 is now found to be 4g_K . The additional terms that arise from end-over-end rotation of the molecule are discussed in § 6.     

具有D4h对称性构型的C2+4分子的Jahn-Teller效应与能级分裂

依据量子理论与配位场理论,利用群论和对称性分析的方法探讨了C²⁺₄分子在具有D_4h对称性构型时,E×(b_1g+b_2g)系统的Jahn-Teller效应中的相关问题.研究了C²⁺₄分子的电子态与声子态的对称性及其活跃声子态,讨论了系统声子间的耦合与CG系数,构建了E×(b_1g+b_{2g 关键词： 2+4分子')" href="#">C²⁺4分子 对称性 能级分裂 Jahn-Teller畸变}     

Vibronic sidebands in ruby

An experimental and group-theoretical study of single phonon vibronic sidebands of theR-lines in ruby has been performed. Restrictions in the electron-phonon-spectra due to crystal symmetry occur only inEC-polarization and for phonons withk-vectors zero or parallel to theC-axis. An analysis of the fluorescence and excitation spectra led to an identification of part of the lines in the vibronic sidebands with normal modes of the lattice. In the regime of acoustical phonons the intensity of the sideband increases with frequency following a potential law with an exponent of slightly less than 5. A strong enhancement of the vibronic transition probability was found on the high frequency side of theR-lines due to a resonance with the electronic transitions⁴ A ₂-² T ₁. Crosssections of the vibronic transitions in the spectral range of Stokes- and anti-Stokessidebands are given in a diagram. For acoustical phonons with energies up to 6 THz they are also expressed in a formula.     

Pseudo-Jahn–Teller effect in Si4X4 (X = F,Cl, Br,I) molecules: a theoretical investigation

ABSTRACT

The variability of planar rings in Si₄X₄ (X?=?F, Cl, Br, I) molecules caused by the pseudo-Jahn–Teller impact (PJTE) was evaluated as an original PJTE work. Optimisation and the following frequency calculations in these molecules illustrated that in high-symmetry planar (with D_4h symmetry) geometry, all of these compounds were unstable and their structures were puckered to lower C_2h symmetry stable geometry. Furthermore, the vibronic coupling interaction between ¹A_1g ground and the first ¹E_g excited states through (¹A_1g?+?¹E_g) ? e_g PJTE problem was the cause of non-planarity of the four-member ring and the symmetry breaking phenomenon in those series. The calculated gaps (Δ) between the ground state and the E_g excited state, the vibronic coupling (F) and ground state primary force constant values (k₁) were obtained from the numerical fitting of the ground state adiabatic potential energy surface with the analytical expressions of these molecules. Finally, natural bond analysis (NBO) was used for the design of the strongest interaction and natural atomic charges of these structures.     

Localization-delocalization transition in one-dimensional system with long-range correlated diagonal disorder

We show that the electronic states in a one-dimensional (1D) Anderson model of diagonal disorder with long-range correlation proposed by de Moura and Lyra exhibit localization-delocalization phase transition in varying the energy of electrons. Using transfer matrix method, we calculate the average resistivity and investigate how it changes with the size of the system N. For given value of α (> 2) we find critical energies E_c1 and E_c2 such that the resistivity decreases with N as a power law ∝ N ^{- γ} for electron energies within the range of [E _c1, E _c2], and exponentially grows with N outside this range. Such behaviors persist in approaching the transition points and the exponent γ is in the range from 0.92 to 0.96. The origin of the delocalization in this 1D model is discussed. Received 18 December 2001 / Received in final form 2 May 2002 Published online 14 October 2002 RID="a" ID="a"e-mail: sjxiong@nju.edu.cn     

Two Components of the Total Ionization Cross Sections of Atoms and Molecules by Electron Impact: Resonant Excitation and Decay of the Fermi Electron-Hole System and the Dipole Electron Transition from a Bound to a Free State at a Binary Collision

Large volume of the experimental data on the ionization cross sections of atoms and molecules by electron impact obtained by various authors using different methods is analyzed. The dependence of the ionization cross section Q on the energy of the incident electron E is described by a curve with a maximum. For E I, where I is the ionization potential, the cross section naturally vanishes. For E > I, it first increases fast, passes through a high maximum, and then monotonically decreases with increasing E. For comparatively large energies E > 100 I, the Bethe formula describes the experimental data almost exactly, but in the region of maximum I < E < 10 I, it deviates significantly from the measurement data. In the present paper it has been established that the experimental dependence Q(E) in the region of maximum is well described by a resonant curve similar to Lorentz distribution. It is assumed that the main contribution to the atomic ionization by a slow electron comes from the resonant excitation and the decay of the Fermi electron-hole system. An empirical formula for the cross section of atom ionization by electron impact Q(E) is suggested which takes into account resonance for incident electrons of small energies and is transformed into the Bethe formula for large E. The parameters of the formula for the ionization cross section are calculated by the least-squares method for H, He, Ne, Ar, C, N, O, Li, Na, H₂, N₂, O₂, K-shells of C, N, Ne, Ar, K, Ca, Rb, and Sr atoms and molecules. A comparison of the experimental dependences of the ionization cross section on the energy of the incident electron with the Bethe theoretical formula and empirical formulas suggested by Lotz, Alkhazov, Kim and Rudd, and Povyshev et al. demonstrates that the formula suggested in the present paper describes the available experimental data better than others.     

Electronic properties of orthorhombic LiGaS<Subscript>2</Subscript> and LiGaSe<Subscript>2</Subscript>

We report theoretical calculations of the band structure and density of states for orthorhombic LiGaS₂ (LGS) and LiGaSe₂ (LGSe). These calculations are based on the full potential linear augmented plane wave (FP-LAPW) method within a framework of density functional theory. Our calculations show that these crystals have similar band structures. The valence band maximum (VBM) and the conduction band minimum (CBM) are located at Γ, resulting in a direct energy band gap. The VBM is dominated by S/Se-p and Li-p states, while the CBM is dominated by Ga-s, S/Se-p and small contributions of Li-p and Ga-p. From the partial density of states we find that Li-p hybridizes with Li-s below the Fermi energy (E _F), while Li-s/p hybridizes with Ga-p below and above E _F. Also, we note that S/Se-p hybridizes with Ga-s below and above E _F.     

The correcting method for the estimation of correlation energies of MF2 (M = Be,Mg, Ca) set molecules

The intrapair and interpair correlation energies of MF₂ (M = Be, Mg, Ca) set molecules are calculated and analysed, and the transferability of inner core correlation effects of M^δ+ are investigated. A detailed analysis of the comparison of correlation energies of neutral atoms with their corresponding ions of M^δ+ and F^δ/2 is given in terms of the correlation contribution of this component. The study reveals that the total correlation energy of MF₂ molecules can be obtained by summing the correlation contributions of M^δ+ and two Fδ-/2 components. This simple estimation method does shed light on the importance of searching useful means for the calculation of electron correlation energy for large biological systems.     

Analysis of the rotational spectrum of C3v molecules by using factorization and diagonalization of the energy matrix: Application to CH3C15N

A method is given for the analysis of the rotational spectrum in the ground and excited states of C_3v molecules; it consists in a direct diagonalization of the energy matrix including all elements whose contribution can become significant for the analysis up to the sixth order of approximation.The method of factoring the energy matrix into four submatrices A₁, A₂, E, E, according to the symmetry species of the full point group C_3v, is given. The programm enables the calculation of the rotational frequencies and also carries out by a least-squares method the fitting of the molecular constants for vibrational states v = 0 (ground state) and v_E = 1, 2, 3, and 4, separately or simultaneously over several of these states.The analysis of the rotational spectrum of CH₃C¹⁵N in the v₈ = 0, 1, 2 states is given as an example.     

Study of the occupied and unoccupied electronic states of the Y-substituted (Bi,Pb)-2223 high temperature superconductor

Spectra for the filled and unfilled electronic states of the (Bi,Pb)-2223 high temperature superconductor were recorded by photoemission and fluorescence X-ray absorption in the entire doping range achieved by substitution of bivalent Ca ions with trivalent Y. In photoemission these samples show diminishing spectral intensity near E _F and at 1.5 eV binding energy with increasing Y content. Parallel to the observations for the filled states the O1s X-ray absorption spectra show a decrease and a shift to higher energies of the empty states just above E_F The spectral structures are identified in the framework of the Hubbard model.     

Probing the electronic structure of the nickel monohalides: Spectroscopy of the low-lying electronic states of NiBr and NiCl

Laser induced fluorescence and single vibronic level emission spectroscopy has been used to probe five low-lying electronic states (X²Π_3/2, A²Δ_5/2, X²Π_1/2, A²Δ_3/2 and B²) of NiBr and NiCl that arise from the 3d⁹ configuration of Ni⁺. Of these, for NiBr only the X²Π_3/2 and A²Δ_5/2 states have been previously observed experimentally, and a complete vibrational analysis of the low-lying states has not been reported for either molecule. In this work, term energies and a complete set of vibrational parameters were derived for all five electronic states for both NiBr and NiCl, and these are compared with recent high level ab initio calculations. In contrast to NiI, there are very few perturbations observed in the vibronic structure of these levels for either NiBr or NiCl. The data set derived in this work affords a detailed analysis of periodic trends in the Nickel monohalide series.     

Dynamic correlation

From a knowledge of the Hartree-Fock and exact non-relativistic energies of atoms, the correlation energy E_c, as defined by Lowdin, may be calculated. For atoms this correlation is defined as dynamic correlation. The separate like-spin and unlike-spin contributions, E_cσσ, E_cαβ, may be calculated as a sum of pair energies from quantum chemistry; we have used the unrestricted M?ller-Plesset second-order algorithm, and then scaled them to give E_c. These three values may also be computed using dynamic correlation functionals, with the Stoll partitioning. The VWN, LYP and P91 functionals were studied for the atoms from H to Ar. Although the total correlation energies of LYP and P91 are similar, only P91 gives a semi-sensible breakdown into the E_cσσ, and E_cαβ components. It is immediately apparent that a new functional, OPTC, derived from the P91 components as 0.6625 x E_cσσ, + 1. 1015 x E_cαβ is an improvement (its mean absolute error is only 0.006 E_h). Using the recently introduced improved exchange functional OPTX (obtained through a fit to the HF energies of atoms), Kohn-Sham calculations were performed on the atoms using the OPT(=OPTX + OPTC) functional. The total energies have a mean absolute error of 0.006 E_h. The study then moves to molecules. First it is shown that the dynamic correlation energy contribution to the dissociation energies is very similar (within 2kcalmol^?1 in most cases), whether it is calculated with LYP, P91 or OPTC. A calculation is then made of the HF contribution, the dynamic contribution through OPTC and the left-right contribution through OPTX, to molecular binding. In many cases the sum agrees with the observed value, but in some cases the prediction is significantly in error, e.g. O₂ is overbound by 10 kcal mol^?1. Thus either OPTX or OPTC or both are inadequate. An attempt was made to determine improved local exchange and correlation functionals by fitting to both atomic and molecular data, but this was unsuccessful. The conclusion is that the method is close to the limit of accuracy achievable from separately optimized local exchange and correlation functionals. Finally, a new hybrid functional O3LYP, which is a substantial improvement on B3LYP, is presented.     

Assessment of a composite CC2/DFT procedure for calculating 0–0 excitation energies of organic molecules

The task to assess the performance of quantum chemical methods in describing electronically excited states has in recent years started to shift from calculation of vertical (ΔE_ve) to calculation of 0–0 excitation energies (ΔE₀₀). Here, based on a set of 66 excited states of organic molecules for which high-resolution experimental ΔE₀₀ energies are available and for which the approximate coupled-cluster singles and doubles (CC2) method performs particularly well, we explore the possibility to simplify the calculation of CC2-quality ΔE₀₀ energies using composite procedures that partly replace CC2 with more economical methods. Specifically, we consider procedures that employ CC2 only for the ΔE_ve part and density functional theory methods for the cumbersome excited-state geometry optimisations and frequency calculations required to obtain ΔE₀₀ energies from ΔE_ve ones. The results demonstrate that it is indeed possible to both closely (to within 0.06–0.08 eV) and consistently approximate ‘true’ CC2 ΔE₀₀ energies in this way, especially when CC2 is combined with hybrid density functionals. Overall, the study highlights the unexploited potential of composite procedures, which hitherto have found widespread use mostly in ground-state chemistry, to also play an important role in facilitating accurate studies of excited states.     

The electronic structure of the nitridoosmate(VIII) ion,OsO3N-

The electronic spectrum of the nitridoosmate(VIII) ion, OsO₃N^-, has been measured at liquid helium temperatures for thin films of the n-tetrabutyl-ammonium salt and for oriented, doped LiClO₄. 3H₂O single crystals. Four structured band systems, all individually composed of superimposed A ₁ ← A ₁ and E←A ₁ transitions, are observed at energies comparable to those of the two lowest ¹ T ₂←¹ A ₁ transitions of OsO₄. Prominent vibronic structure assigned to ν(Os-N) is observed for all of the bands. The bands are assigned to transitions to completely spin-orbit mixed states. The highest filled orbitals of OsO₃N^- are oxygen-localized and π non-bonding. The nitrido ligand appears to ‘dominate’ the π-bonding in the ion.     

The F-spin symmetric limits of the neutron-proton interacting boson model

The F-spin symmetric limits U(5), SU(3), and O(6) of the neutron-proton interacting boson model are studied. The conditions for and consequences of F-spin symmetry are investigated. In each of the three limits closed expressions for the following properties are given: the excitation energies; the M1, E2, and M3 excitation strengths from the ground state to all 1⁺, 2⁺, and 3⁺ states; the M1 and E2 decay of the nonsymmetric 1⁺ and 2⁺ states; and the dipole and quadrupole moments of maximal F-spin states and of the nonsymmetric 1⁺ and 2⁺ states. The predictions for these properties are compared with the experimental data.     

Orbital angular momentum in triatomic molecules

The effects of electronic angular momentum in triatomic linear molecules are considered. An effective vibronic hamiltonian is derived and second order energy level expressions are obtained for the bending levels in an electronic Π state. The formulae allow for the anharmonicity of the bending potentials and for the variation of the expectation value <L_z > with bond angle; effects of electron spins are also included. The vibronic levels predicted by the analytic expressions are compared with those calculated using a full matrix treatment of the orbital angular momentum; it is shown that they are far more accurate than the levels predicted by the formulae currently available in the literature. The relationship between the anharmonic corrections and the deviation of <L_z > from unity is discussed in terms of an electrostatic interaction between linear molecule states of different symmetry.     

Calculation of vibrational energy levels of triatomic molecules with the C2v and Cs symmetries by summing divergent series of the Rayleigh-Schr?dinger perturbation theory

The Rayleigh-Schr?dinger perturbation theory is applied to calculation of vibrational energy levels of triatomic molecules with the C _2v and C _s symmetries: SO₂, H₂S, F₂O, HOF, HOCl, and DOCl. Particular attention is given to the states coupled by anharmonic resonances; for such states, the perturbation theory series diverge. To sum these series, the known methods of Padé, Padé-Borel, and Padé-Hermite and the method of power moments are used. For low-lying levels, all the summation methods give satisfactory results, while the method of quadratic Padé-Hermite approximants appears to be more efficient for high-excited states. Using these approximants, the structure of singularities of the vibrational energy, as a function in the complex plane, is studied.     

The magnetic circular dichroism spectrum of the 1 B 2u ←1 A 1g transition of benzene in the vapour phase

The magnetic circular dichroism (MCD) spectrum of the ¹ B _2u ←¹ A _1g transition of benzene has been measured in the vapour phase. Many of the bands due to transitions between single vibronic levels display A terms. It has been shown that the angular momentum arises by vibronic mixing both of the ¹ E _1u state with the ¹ B _1u state and of the ¹ E _2g states with the ¹ A _1g ground state by e _2g vibrations. The magnitudes and signs of the experimental and calculated ratios, A/D, for the A ₀ ⁰ vibronic origin are in excellent agreement. Two strong MCD progressions of opposite sign with B-term dispersion have been observed in regions of low absorption. These are identified with vibronic origins due to the v ₈ and v ₉ e _2g modes. By contrast the MCD spectrum of hexadeuterobenzene vapour has a much lower magneto-rotational strength and displays none of the striking features of the benzene MCD spectrum.     

Superconformal Indices of $${{\mathcal N}=4}$$ SYM Field Theories

Superconformal indices (SCIs) of 4d N = 4{{\mathcal N} = 4} SYM theories with simple gauge groups are described in terms of elliptic hypergeometric integrals. For F ₄, E ₆, E ₇, E ₈ gauge groups this yields first examples of integrals of such type. S-duality transformation for G ₂ and F ₄ SCIs is equivalent to a change of integration variables. Equality of SCIs for SP(2N) and SO(2N + 1) group theories is proved in several important special cases. Reduction of SCIs to partition functions of 3d N=2{{\mathcal N}=2} SYM theories with one matter field in the adjoint representation is investigated, corresponding 3d dual partners are found, and some new related hyperbolic beta integrals are conjectured.