Alternant systems and their properties

Neutral, ionic, and complete alternant systems are studied using the alternancy symmetry adapted (ASA ) approach. This approach is based on an explicit construction of ASA operators that are up to the sign invariant with respect to the particle-hole symmetry transformation. These operators serve as building blocks of alternant systems, and they determine their characteristic properties. All Hamiltonians describing neutral alternant systems are explicitly constructed. Up to some minor restrictions, all Hamiltonians describing ionic and complete alternant systems are also explicitly constructed. Inversely, given a Hamiltonian ? in a second quantization notation, one can easily check whether or not this Hamiltonian describes a neutral (ionic, complete) alternant system. All linear properties characteristic to neutral (ionic, complete) alternant systems are obtained. In particular, all one- and two-particle properties are derived in an explicit form. The properties obtained substantially generalize “classical” properties of alternant systems such as, in the case of neutral alternant systems, uniform charge density distribution, vanishing bond orders between atomic sites of the same parity, and alternancy selection rules for the electric dipole transitions.     

Intensities of spin-forbidden transitions in molecular oxygen and selective heavy-atom effects

Intensities of a¹Δg←X ³∑_g^? and b ¹∑_g⁺ ← X ³∑_g^? transitions in molecular oxygen have been calculated on the basis of the INDO method taking into account spin-orbit coupling by perturbation theory. The transitions are magnetic dipole in nature. The first of them (a ? X) steals its intensity from ³Π_g-³∑_g^? and ¹Π_g-¹Δ_g transitions, which are determined by the orbital angular-momentum operator. This source is not the principal one for the intensity of the second (b-X) transition. Its intensity is stolen principally from microwave transitions between spin sublevels of the ground ³∑_g^? state. The last source explains the large difference in intensities of the a-X and b-X transitions. Calculated oscillator strengths are in a good agreement with experiment. The same integrals that determine the intensity also determine the parameters of the spin Hamiltonian for the ground ³∑_g^? state. These parameters are in a good agreement with experiment also, showing the validity of the whole calculation. In a condensed phase the investigated transitions are enhanced by intermolecular exchange interaction. It is known that an external heavy atom (EHA ) enhances the b-X transition of oxygen in solution, but does not influence the a-X transition. In the collision complex O₂-EHA , which has a geometry without inversion symmetry, the microwave transitions between spin sublevels of the “³∑_g^?” state obtain electric-dipole moments, which are stolen from the charge-transfer transition. This mechanism explains the selective effect of EHA .     

Lie algebraic approach to the scattering system He + H2

In this article, the Hamiltonian for the scattering of the H_e + H₂ system is given by using the interaction potential V(X, Y, Θ) determined by experiments and the semiclassical method. From this Hamiltonian we find a dynamical algebra h₆. The statistical expectation of the energy and the transition probability of H₂, P_n→m, are derived; therefore, selection rules have been found easily.     

2-甲基-2-丁醇的低温热容和热力学性质

本文用精密自动绝热量热仪测定了2-甲基-2-丁醇在80～305 K温区的热容,从热容曲线(Cp-T) 发现三个固－固相变和一个固－液相变, 其相变温度分别为T = 146.355, 149.929, 214.395, 262.706 K。从实验热容数据用最小二乘法得到以下四个温区的热容拟合方程。在80～140K温区, Cp,m = 39.208 + 8.0724X - 1.9583X2 + 10.06X3 + 1.799X4 - 7.2778X5 + 1.4919X6, 折合温度X = (T –110) / 30; 在 155 ～ 210 K温区, Cp,m = 70.701 + 10.631X + 12.767X2 + 0.3583X3 - 22.272X4 - 0.417X5 + 12.055X6, X = (T –182.5) /27.5; 在220 ～ 250 K温区, Cp,m = 99.176 + 7.7199X - 26.138X2 + 28.949X3 + 0.7599X4 - 25.823X5 + 21.131X6, X = (T – 235)/15; 在 270～305 K温区, Cp,m =121.73 + 16.53 X- 1.0732X2 - 34.937X3 - 19.865X4 + 24.324X5 + 18.544X6, X = (T –287.5)/17.5。从实验热容计算出相变焓分别为0.9392, 1.541, 0.6646, 2.239 kJ×mol-1; 相变熵分别为6.417, 10.28, 3.100, 8.527 J×K-1×mol-1。根据热力学函数关系式计算出80～305 K温区每隔5 K的热力学函数值 [HT –H298.15]和 [ST –S298.15]。     

A density functional representation of quantum chemistry. I. Motivation and general formalism

The molecular N-electron problem is formulated in terms of the particle density ??( r ) = ∑σ( r ? q _n) and its canonically conjugated field. Weyl's representation of the canonical commutation relations is used and a characteristic functional of the molecular system is introduced as the expectation value of Weyl's functional exp {i(α, Q) + i(β, P)}. The Hamiltonian equation of motion for the characteristic functional is derived and its cumulant approximation is discussed. The first paper is restricted to a general discussion, independent of a special choice of the Hamiltonian. The density functional representation may be considered as a step towards a formulation of a renormalized quantum chemistry. The lowest order cumulants of the characteristic functional are accessible to actual precision measurements; on the other hand, most experiments are correctly represented by a characteristic functional of the Gaussian type. The significance of the higher order correlations is neither theoretically nor experimentally disclosed yet, their possible importance for biological systems is mentioned, and the question is raised of how far quantum mechanics is empirically confirmed for extremely complex systems.     

The AMO method at small internuclear distances

The AMO function of the hydrogen molecule ψ = ψ_c + η ψ_i, where ψ_c is the covalent part and ψ_i the ionic part, is investigated for small internuclear distances R. We found η → ?1 as R →,?1 as R → 0, contrary to the intuitively expected limit η → 1. However, near R = 0 an analytical expression of ψ is derived, showing that ψ reduces to the helium ground state as R → 0. We have proved that the empirical concept ?covalent and ionic character”? should be replaced by the symmetry argument in the case of small R.     

Unraveling σ and π Effects on Magnetic Anisotropy in cis‐NiA4B2 Complexes: Magnetization,HF‐HFEPR Studies,First‐Principles Calculations,and Orbital Modeling

By using complementary experimental techniques and first‐principles theoretical calculations, magnetic anisotropy in a series of five hexacoordinated nickel(II) complexes possessing a symmetry close to C_2v, has been investigated. Four complexes have the general formula [Ni(bpy)X₂]ⁿ⁺ (bpy=2,2′‐bipyridine; X₂=bpy ( 1 ), (NCS^?)₂ ( 2 ), C₂O₄^2? ( 3 ), NO₃^? ( 4 )). In the fifth complex, [Ni(HIM₂‐py)₂(NO₃)]⁺ ( 5 ; HIM₂‐py=2‐(2‐pyridyl)‐4,4,5,5‐tetramethyl‐4,5‐dihydro‐1H‐imidazolyl‐1‐hydroxy), which was reported previously, the two bpy bidentate ligands were replaced by HIM₂‐py. Analysis of the high‐field, high‐frequency electronic paramagnetic resonance (HF‐HFEPR) spectra and magnetization data leads to the determination of the spin Hamiltonian parameters. The D parameter, corresponding to the axial magnetic anisotropy, was negative (Ising type) for the five compounds and ranged from ?1 to ?10 cm^?1. First‐principles SO‐CASPT2 calculations have been performed to estimate these parameters and rationalize the experimental values. From calculations, the easy axis of magnetization is in two different directions for complexes 2 and 3 , on one hand, and 4 and 5 , on the other hand. A new method is proposed to calculate the g tensor for systems with S=1. The spin Hamiltonian parameters (D (axial), E (rhombic), and g_i) are rationalized in terms of ordering of the 3 d orbitals. According to this orbital model, it can be shown that 1) the large magnetic anisotropy of 4 and 5 arises from splitting of the e_g‐like orbitals and is due to the difference in the σ‐donor strength of NO₃^? and bpy or HIM₂‐py, whereas the difference in anisotropy between the two compounds is due to splitting of the t_2g‐like orbitals; and 2) the anisotropy of complexes 1 – 3 arises from the small splitting of the t_2g‐like orbitals. The direction of the anisotropy axis can be rationalized by the proposed orbital model.     

Rotational diffusivity of rodlike molecules in amorphous polymeric matrices

In this note, scaling laws for rotational diffusivity of dilute monodisperse rigid-rod molecules (guest rods) in semidilute amorphous polymer solutions (host molecules) are derived. The coillike matrix molecules are modeled as a collection of flexibly connected rigid subunits. This allows an analogy with the Doi-Edwards theory for monodisperse rigid rods in semidilute solutions to be used in the analysis. Very strong dependencies are predicted for the rotational diffusivity of the rods on host polymer volume fraction and rod length. In semidilute polymer solutions the coils dramatically hinder the rotational freedom of the rods for r² ? ψ_p^?1, r being the rod aspect ratio and ψ_p the polymer volume fraction.     

Investigations on the local structure and spin Hamiltonian parameters for the orthorhombic Cu<Superscript>2+</Superscript> center in Ca(OD)<Subscript>2</Subscript>

The local structure and the spin Hamiltonian parameters (the g factors g _i (i = x, y, z) and the hyperfine structure constants A _i ) for Cu²⁺-doped Ca(OD)₂ are theoretically investigated from the perturbation formulas of these parameters for a 3d ⁹ ion in an orthorhombically elongated octahedron. From the studies, the planar Cu²⁺-OD⁻ bonds are found to experience the relative variation ΔR (≈0.014 ?) along the X and Y axes, while those parallel to the Z axis may undergo the relative elongation ΔZ (≈0.25 ?) due to the Jahn-Teller effect. The theoretical spin Hamiltonian parameters based on the above local lattice distortions agree well with the experimental data. As compared with the previous treatments, the improvements of the theoretical spin Hamiltonian parameters are achieved in this work by adopting the uniform calculation formulas and the tetragonal field parameters based on the superposition model.     

A generalized formula for the energies of alternant molecular orbitals. II. Heteronuclear molecules

Using the method of alternant molecular orbitals (AMO ), it is shown that the energies of AMOS (E_kσ) for an arbitrary heteronuclear alternant system, having a singlet ground state, are connected with the energies of MOS (e_{k(k )}) obtained by means of the conventional Hartree–Fock (HF ) method (SCF -LCAO -MO -PPP ) via the formula: In the general case, the determination of the correlation corrections δ_i,kσ is connected with the solving of a complicated system of integral equations, which is considerably simplified if the Hubbard approximation is accepted for the electron interaction. The energy spectrum of a chain with two atoms in the elementary cell (AB)_n is considered as an example. It is shown that if nontrivial solutions exist (δ_i,kσ ≠ 0), the correlation correction for AMOS of different spin are different (δ_i,kσ ≠ δ_i,kβ), from which it follows, that the width of the energy gap ΔE_∞ for AMOS with different spin is different: ΔE_∞,α ≠ ΔE_∞,β.     

Twiston vs. random fluctuation as the source of the rapid motion of polyethylene in the inclusion complex with perhydrotriphenylene

Recently molecular dynamics simulations were performed for polyethylene in the inclusion complex with perhydrotriphenylene. The system contained ninety molecules of perhydrotriphenylene, arranged in six stacks of fifteen molecules each, and one molecule of n-tetracontane, C₄₀H₈₂. The internal CH₂-CH₂ bonds in n-tetracontane have a very strong preference for the trans state. Nevertheless, the chain exhibits a high degree of internal flexibility. This motion produces a characteristic pattern in δ|ψ N + i ψi| vs. N, where ψ_i describes the instantaneous angle of a C-H bond vector at carbon atom i about the axis defined by the channel, and δ denotes the fluctuation. The pattern expected for δ|ψ N + i ψi| vs. N is derived for the case where the rapid internal motion is produced by a twiston. It is compared with the results from the simulation and from the expectation for the case where the rapid motion arises from uncorrelated internal fluctuations within the trans state at each CH₂-CH₂ bond.     

Hylleraas method for many‐electron atoms. I. The Hamiltonian

A general expression for the nonrelativistic Hamiltonian for n‐electron atoms with the fixed nucleus approximation is derived in a straightforward manner using the chain rule. The kinetic energy part is transformed into the mutually independent distance coordinates r_i, r_ij, and the polar angles θ_i, and φ_i. This form of the Hamiltonian is very appropriate for calculating integrals using Slater orbitals, not only of states of S symmetry, but also of states with higher angular momentum, as P states. As a first step in a study of the Hylleraas method for five‐electron systems, variational calculations on the ²P ground state of boron atom are performed without any interelectronic distance. The orbital exponents are optimized. The single‐term reference wave function leads to an energy of ?24.498369 atomic units (a.u.) with a virial factor of η = 2.0000000009, which coincides with the Hartree–Fock energy ?24.498369 a.u. A 150‐term wave function expansion leads to an energy of ?24.541246 a.u., with a factor of η = 1.9999999912, which represents 28% of the correlation energy. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Bemerkung zur Gleichheit der Aufspaltungen ΔI (zwischen den ersten beiden π-Ionisationspotentialen) und ΔE (zwischen den entsprechenden π* ← π Übergangsenergien) des Spiro[4,4]nonatetraens. Vorläufige Mitteilung

For a given molecule M, the difference ΔI between the first two vertical ionization potentials I_v,2 and I_v,2 (from MOs ψ₁ and ψ₂) and ΔE between the corresponding singlet-singlet excitation energies E₁ and E₂ (transitions ψ?₁ ←₁, ψ?₁ ψ₂) are related by ΔE = ΔI- (J_2,?1?J_1,?1) ?2(K_1,?1 ? K_2,?1), using Koopmans approximation. A simple MO model suggests that under certain conditions of symmetry and quasi-alternancy (e. g. in spiro[4,4]nonatetraene 1 ) the bracketed differences between the Coulomb- and exchange-integrals should vanish to first order, thus leading to the simple (almost) equality ΔE = ΔI. It is shown that the results from a photoelectron- and electron-spectroscopic investigation of 1 support this conclusion i.e. ΔI = 1.23 eV, ΔE = 1.19 to 1.23 eV.     

Über die Eigenschaften des Xp-Teils von AnBXp-Kernresonanzspektren

The subspectral breakdown of the X-part in A_nBX_p-spectra is derived and the general properties of the subspectra are discussed. The nature of the asymmetry of X_p-spectra for n > 1 is explained and conditions for its disappearance are given. The degeneration of the X_p-part into deceptively simple spectra is described and degeneration conditions are derived. The determination of relative signs of coupling constants is discussed.     

Spin symmetry and gel'fand patterns of higher SU2×ℓ n groups

The nature of? ₅ spin algebra is considered together with? ₅↓? ₅↓C _4v mapping in order to specify the dual spin symmetry for MQ-NMR ofnido-¹¹B₅H₉. The forms of Gel'fand shapes for? ₅ spin symmetry are presented to show how they specify thefull range of multiplicities found in higher-n ? _n -partitions. Tuples, or number-partitions and their? _n ↓G invariance sets provide models for both the five-foldI _i =3/2 component, and the two distinct types of spin-1/2 subsystems. The full spin symmetry is derived in terms of the direct product ((? ₅↓C _4v )?(? ₄↓C _4v ))^1/2?(? ₅↓C _4v )^3/2. The concepts used are implicit in the substructure ofp-tuple model invariances over the subduced symmetry, or derive from the inner tensor product algebras under the? _n group. Both as a check on the combinatorially derived multiplicities of [λ]s and for insight into (non-simply-reducible) substructure of number-partitions, the study of mapping from :hrr'.:-tuples onto the? _n -partitional set for higherI _i is invaluable. The motivation for this work lies in its pertinence to the MQ-NMR spin dynamics of clusterlike molecules. The accessible information content of a spin algebra over either form of spin space is bound up with a suitable symmetry partitioning of the problem, as implied by the use of {T ^kq (v:[λ])} bases within higherq subspaces of the Liouville formalism.     

Construction of orthogonal subspaces

An orthogonalization procedure is presented that allows construction of at least (n?m) vectors orthogonal to {X_j}, j equals; 1, m, by linear combinations solely among {η_i}, i equals; 1, n, n>m, and 〈X_j/η_i〉≠0. An important application of the procedure is in effective core potential methods for which valence orbitals can be constructed that are orthogonal to the core orbitals and yet involve no component of the core. Thus, a separate calculation for only the valence electrons can be performed without any explicit reference to the core electrons (orbitals).     

A new operator formulation of the many-electron problem for molecules

An n-electron operatorX _n , called a wave operator, is associated with a 2n-electron molecular wave function. Electron densities and energy are written in terms ofX _n . An equation defining an exact wave operator is found. Thus, a 2n-electron vector problem (for the wave function) is rigorously reduced to an n-electron operator problem. Conditions are formulated which guarantee thatX _n corresponds to a state with a given spin. The configuration-interaction problem is considered and methods of approximate construction ofX _n are discussed. In particular, a matrix algorithm is proposed for calculations in the two-body approximation. A generalizaton of the approach to the case of systems with an odd number of electrons is given. The waveoperator model developed forms a general basis for construction of covariant electron models of molecules.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 25, No. 1, pp. 1–12, January–February, 1989.     

Ionic bonding of lanthanides,as influenced by d‐ and f‐atomic orbitals,by core–shells and by relativity

Lanthanide trihalide molecules LnX₃ (X = F, Cl, Br, I) were quantum chemically investigated, in particular detail for Ln = Lu (lutetium). We applied density functional theory (DFT) at the nonrelativistic and scalar and SO‐coupled relativistic levels, and also the ab initio coupled cluster approach. The chemically active electron shells of the lanthanide atoms comprise the 5d and 6s (and 6p) valence atomic orbitals (AO) and also the filled inner 4f semivalence and outer 5p semicore shells. Four different frozen‐core approximations for Lu were compared: the (1s²–4d¹⁰) [Pd] medium core, the [Pd+5s²5p⁶ = Xe] and [Pd+4f¹⁴] large cores, and the [Pd+4f¹⁴+5s²5p⁶] very large core. The errors of Lu? X bonding are more serious on freezing the 5p⁶ shell than the 4f¹⁴ shell, more serious upon core‐freezing than on the effective‐core‐potential approximation. The Ln? X distances correlate linearly with the AO radii of the ionic outer shells, Ln³⁺‐5p⁶ and X^?‐np⁶, characteristic for dominantly ionic Ln³⁺‐X^? binding. The heavier halogen atoms also bind covalently with the Ln‐5d shell. Scalar relativistic effects contract and destabilize the Lu? X bonds, spin orbit coupling hardly affects the geometries but the bond energies, owing to SO effects in the free atoms. The relativistic changes of bond energy BE, bond length R_e, bond force k, and bond stretching frequency v_s do not follow the simple rules of Badger and Gordy (R_e～BE～k～v_s). The so‐called degeneracy‐driven covalence, meaning strong mixing of accidentally near‐degenerate, nearly nonoverlapping AOs without BE contribution is critically discussed. © 2015 Wiley Periodicals, Inc.     

Schwingungs- und Elektronenspektren der Dekahalogenodiosmate(IV), [Os2X10]2−, X  Cl,Br

Vibrational and Electronic Spectra of Decahalogenodiosmates(IV), [Os₂X₁₀]^2?, X ? Cl, Br The IR and Raman spectra of the edge-sharing bioctahedral anions [Os₂X₁₀]^2?, X ? Cl, Br, are assigned according to point group D_2h. The bands are found in three characteristic regions; at high wavenumbers stretching vibrations with terminal ligands v(OsCl_t): 365–280, v(OsBr_t): 235–195; in a middle region with bridging ligands v(OsCl_b): 270–240, v(OsBr_b): 175–165 cm^?1; the deformation bands are observed at distinct lower frequencies. The electronic spectra of the dimers show intraconfigurational transitions near 2000, 1000, and 600 nm which by position and intensity correspond to those of the monomeric complexes. They are therefore discussed separately for both metal centers according to C_2v symmetry. Two additional band systems are presumable pair transitions arising from interactions of the central ions within the dimeric complexes. Due to the different bonding strength of terminal or bridging ligands the intensive charge transfer bands are shifted by 3000–4000 cm^?1 bathochromicly or by 2000–3000 cm^?1 hypsochromicly compared with the hexahaloosmates(IV).     

Theoretically derived absorption and MCD spectral data for the cellular species produced in the hydrolysis of cis-diamminedichloroplatinum(II)

The cellular species formed in the hydrolysis of cis-Pt(NH₃)₂Cl₂ (DDP), namely, cis-[Pt(NH₃)₂XY]ⁿ⁺ (X, Y = Cl^?, H₂O, OH^?; n = 0, 1, 2) have been investigated theoretically using the relativistic and nonrelativistic extended Huckel molecular orbital method. Molecular orbital (MO) results for trans-DDP and its hydrolysis products are also reported for comparison. Transition energies, molar absorption coefficients (?), and B terms from magnetic circular dichroism (MCD) derived from theory are presented for each of the species studied. The electronic absorption and MCD spectra of all the complexes are predicted to exhibit ligand field transitions arising primarily from excitations between the occupied Pt 5d orbitals and the unoccupied Pt 5d and 6p_z orbitals, respectively. The 5d → 6p_z transitions are expected to yield intense absorptions in the UV spectral region. Some intensity is generated in the d → d transitions as a result of the low symmetry of these complexes. Correlation of available experimental data with theory allows spectral assignments to be made and predicted. Substituent effects in the cis- and trans-isomeric species are discussed. Finally, the applicability of the EHMO method to these systems is examined.