Unitary group approach to the many-electron problem. II. Adjoint tensor operators for U(n)

In this paper we present a derivation of the U(n) adjoint coupling coefficients for the representations appropriate to many-electron systems. Since the states of a many-fermion system are to comprise the totally antisymmetric Nth rank tensor representation of U(2n), the work of this paper enables the matrix elements of the U(2n) generators to be evaluated directly in the U(n) × U(2) (i.e., spin orbit) basis using their transformation properties as adjoint tensor operators. A connection between the adjoint coupling coefficients, as derived in this paper, and the matrix elements of certain (spin independent) two-body operators is also presented. This indicates that in CI calculations, one may obtain the matrix elements of spin-dependent operators from the known matrix elements of certain spin-independent two-body operators. In particular this implies a segment-level formula for the matrix elements of the U(2n) generators in the spin-orbit basis.     

Unitary group approach to the many-electron problem. III. Matrix elements of spin-dependent Hamiltonians

This is the final paper in a series of three directed toward the evaluation of spin-dependent Hamiltonians. In this paper we derive the reduced matrix elements of the U(2n) generators in a basis symmetry adapted to the subgroup U(n) × U(2) (i.e., spin-orbit basis), for the representations appropriate to many-electron systems. This enables a direct evaluation of the matrix elements of spin-dependent Hamiltonians in the spin-orbit basis. An alternative (indirect) method, which employs the use of U(2n) ↓ U(n) × U(2) subduction coefficients, is also discussed.     

The evaluation of matrix elements for non-canonical Weyl tableau basis states adapted toU(n 1+n 2)⊃U(n 1)×U(n 2)

Summary This is the first paper of a series of two, which enables the evaluation ofU(n) generator matrix elements in the non-canonical Weyl tableau basis adapted to subgroupU(n ₁)×U(n ₂). In this paper the explicit closed formulae for subduction coefficients are presented. These formulae will become useful through an inductive method to be presented in the second paper.     

Matrix elements of the U(2n) generators in the spin-orbit basis

An algorithm for the evaluation of the matrix elements of the U(2n) generators in the spin-orbit basis [induced by the U(n) ? U(2) subgroup] is proposed. This algorithm can be used for calculations with spin-dependent Hamiltonians for many-electron problems.     

A spin-dependent unitary group approach to many-electron systems

In this article we derive a segment-level formula for the matrix elements of the U(2n) generators in a basis symmetry adapted to the subgroup U(n) × U(2) (i.e., spin-orbit basis), for the representations appropriate to many-electron systems. This enables the direct evaluation of the matrix elements of spin-dependent Hamiltonians.     

Subduction coefficients for 〈2N/2–S, 12S〉 ↓ 〈2, 1〉 ⊗ 〈2, 1〉 of U(n) ↓ U(n1) ⊗ U(n2)

The determination of the subduction coefficients for states of the unitary group U(n) under the restrictions U(n) ↓ U(n₁) ? U(n₂) have been considered for the spin free states of many electron systems. Using the transformation properties of the tensor basis spanning the irreducible representation 〈2^N/2–S, 1^2S〉 of U(n) under the permutations of electron coordinates, a simple programmable procedure has been developed for the determination of these coefficients. The procedure has been illustrated using a simple example.     

Calculation of SN 1 + N 2 ⊃ SN1 ⊗ sn 2 and U(n1 + n2) ⊃ u (n1) ⊗ u (n2) subduction coefficients by using spin graph

By use of the graphical method of spin algebra, simple, and closed expressions for S_N1+_N2 ? S_N1 ? S_N2 and U(n₁ + n₂) ? U(n₁) ? U(n₂), subduction coefficients are derived.     

Unitary group tensor operator algebras for many-electron systems: I. Clebsch-Gordan and Racah coefficients

A basis for the Racah-Wigner algebra of irreducible representations of the unitary group U(n) that are pertinent to quantum chemical models of many-electron systems is developed. Standard Clebsch-Gordan coefficients and isoscalar factors (also called coupling factors or reduced Wigner coefficients) for both symmetric (S _N ) and unitary [U(n)] groups are extended to transformation coefficients and corresponding isoscalar factors relating canonical Young-Yamanouchi or Gel'fand-Tsetlin bases to simple partitioned bases. All these different types of isoscalar factors are interrelated using the well-known reciprocity between the S _N and U(n) tensor representations, and general expressions relating these different factors are given. For the two-column representations characterizing the many-electron theory, detailed explicit expressions are presented for both the above-mentioned relationships and for all relevant U(n) isoscalar factors. Finally, U(n) Racah coefficients are introduced and explicit expressions derived for certain special classes of these coefficients.Killam Research Fellow 1987–89.     

Synthesis and X-Ray Crystal Structures of Fe2Ru(CO)12−n (CNBu t ) n and FeRu2(CO)12−n (CNBu t ) n (n=1, 2)

The clusters Fe₂Ru(CO)_12–n (CNBu ^t ) _n (3, n=1; 4, n=2), FeRu₂(CO)_12–n (CNBu ^t ) _n (5, n=1, 6, n=2) and FeRu₂(CO)₁₁(CNCy) (5a) have been prepared by direct substitution from the parent carbonyl precursors Fe₂Ru(CO)₁₂ (1) and FeRu₂(CO)₁₂ (2). All compounds have been characterized spectroscopically and clusters 3, 4, 5, and 6 by single crystal X-ray determinations. In all cases, the isonitrile ligands adopt axial or pseudo-axial positions on a ruthenium atom. The structures of 3–5 are very similar to their parent clusters, but the extent of metal framework disorder is significantly less. Cluster 6 adopts the same C _2v Fe₃(CO)₁₂ type structure as 4, and thus differs markedly from the parent compound 2, which has a D ₃ structure .     

Photochemical reactions of Re(CO)5Br with Ph2P(CH2) n PPh2 (n = 1, dppm; 2, dppe; 3, dppp)

The new complexes fac-[Re(CO)₃Br{Ph₂P(CH₂) _n PPh₂}] (1a–3a) [(1a), n = 1; (2a), n = 2; (3a), n = 3] and [Re₂(CO)₈Br₂{-Ph₂P(CH₂) _n PPh₂}] (1b–3b) [(1b), n = 1; (2b), n = 2; (3b), n = 3] have been prepared by the photochemical reaction of Re(CO)₅Br with Ph₂P(CH₂) _n PPh₂ (n = 1, dppm; 2, dppe; 3, dppp). The complexes have been characterized by elemental analysis, mass spectroscopy, f.t.-i.r. and ³¹P-[¹H]-n.m.r. spectrometry. The spectroscopic studies suggest cis-chelate bidentate coordination of the ligand in fac-[Re(CO)₃Br{Ph₂P(CH₂) _n PPh₂}] (1a–3a) and cis-bridging bidentate coordination of the ligand between two metals in [Re₂(CO)₈Br₂{cis--Ph₂P(CH₂) _n PPh₂}] (1a–3a).     

Highly excited vibrational levels of triatomic molecule N2O: U(4) algebraic model

In the U(4) algebraic framework, the triatomic molecules are of U₁(4) ? U₂(4) dynamical symmetry. A molecular Hamiltonian is constructed including the third‐order conbination of the invariant operators. Within this framework, the highly vibrational energy levels of the linear triatomic nitrous oxide molecule, including both bending and stretching vibrations, are studied. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Density Functional Theory Study on (Cl2GaN3)n(n=1–4) Clusters

The molecular geometries, vibrational properties, and thermodynamic properties of the clusters (Cl₂GaN₃)_n(n=1–4) have been predicted at the B3LYP/6‐311+G* level. The optimized clusters (Cl₂GaN₃)_n (n=2–4) all possess cyclic structures containing Ga N_α Ga linkages. The relationships between geometrical parameters and oligomerization degree n are discussed. The gas‐phase structures of the trimers prefer to exist in boat‐twisting conformation. As for the tetramer, the S₄ symmetry structure is the most stable. The infrared spectra are obtained and assigned by vibrational analysis. Thermodynamic properties derived from the infrared spectra on the basis of statistical thermodynamic principles are linearly correlated with the oligomerization degree n as well as the temperature. Meanwhile, thermodynamic analysis of the gas‐phase reaction suggests that the oligomerization is exothermic and favorable under high temperature.     

Unitary group approach to general system partitioning. II. U(n) matrix element evaluation in a composite basis

An explicit segment level formalism is derived for the matrix elements of the U(n) generators in an arbitrary (multishell) composite basis. The results of this paper, which contain the usual (spin-independent) unitary calculus approach as a limiting case, yield a more powerful and versatile algorithm than the traditional (spin-independent) unitary group formalism.     

Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S2 eigenfunctions

We present an implementation of the spin‐dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry‐adapted functions. Using S² eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin‐independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one‐ and two‐electron charge densities.     

Syntheses and crystal structures of two cadmium coordination polymers [Cd(2-mBIM)(NCS)(SCN)] n and [Cd2(2-mBIM)2(NO3)2(C4H4O4)(H2O)5] n based on bis(2-methylimidazol-1-yl)methane

Two cadmium(II) coordination polymers, [Cd(2-mBIM)(NCS)(SCN)] _n (1) and [Cd₂(2-mBIM)₂(NO₃)₂(C₄H₄O₄)(H₂O)₅] _n (2) (2-mBIM = bis(2-methylimidazo-1-yl)methane, C₄H₄O₄= succinate), have been synthesized and characterized by X-ray diffraction. Complex 1 crystallizes in the triclinic space group P 1 with a = 9.0770(5) Å, b = 9.4043(4) Å, c = 19.8720(9) Å, α = 101.551(1)°, β = 93.498(1)°, γ = 108.484(1)°, V = 1562.02(13) ų, and Z = 2. Each Cd(II) is octahedrally coordinated and connected with two adjacent Cd(II)'s by double end-to-end thiocyanate bridges, resulting in the formation of 1-D zigzag chains, linked to each other via bridging 2-mBIM giving a 2-D supramolecular framework. Complex 2 crystallizes in the monoclinic space group P2(1)/n with a = 12.6543(6) Å, b = 7.7128(4) Å, c = 17.3089(9) Å, β = 109.3980(10)°, V = 1593.45(14) ų, and Z = 2. Cd(II) is coordinated with oxygen and nitrogens from two independent 2-mBIM, in a cis-configuration to form a 1-D helical structure. A 3-D supramolecular network comprised of succinate anion bridged 1-D helical chains, and weak hydrogen bonds between dimer waters gave 2-D layers.     

Darstellung der Halogeno-Pyridin-Rhenate(III), [ReX6−n (Py)n](3−n)− (X = Br,Cl; n = 1−3), sowie Kristallstrukturen von trans-[(C4H9)4N][ReBr4(Py)2], mer-[ReCl3(Py)3] und mer-[ReBr3(Py)3]

Preparation of Halogeno Pyridine Rhenates(III), [ReX_6?n(Py)_n]^(3?n)? (X = Br, Cl; n = 1?3) Crystal Structures of trans-[(C₄H₉)₄N][ReBr₄(Py)₂], mer-[ReCl₃(Py)₃], and mer- [ReBr₃(Py)₃] The mixed halogeno-pyridine-rhenates(III), [ReX_6?n(Py)_n]^(3?n)? (X = Br, Cl), n = 1?3, have been prepared for the first time by reaction of the tetrabutylammoniumsalts (TBA)₂[ReX₆] (X = Br, Cl) in pyridine with (TBA)BH₄ and separation by chromatography on Al₂O₃. Apart from the monopyridine complexes only the trans and mer isomers are formed from the bis-and tris-pyridine compounds. The X-ray structure determinations of the isotypic neutral complexes mer- [ReX₃(Py)₃] (monoclinic, space group P 2₁/n, Z = 4; for X = Cl: a = 9,1120(8), b = 12,5156(14), c = 15,6100(13) Å, β = 91,385(7)°; for X = Br: a = 9,152(5), b = 12,852(13), c = 15,669(2) Å, β = 90,43(2)°) reveal, due to the stronger trans influence of pyridine compared with Cl and Br, that the Re? X distances in asymmetric Py? Re? X3 axes with ReCl3 = 2,397 Å and ReBr3 = 2,534 Å are elongated by 1,3 and 1% in comparison with symmetric X1? Re? X2 axes with ReCl1 = ReCl2 = 2,367 Å and ReBr1 = 2,513 and ReBr2 = 2,506 Å, respectively. The Re? N bond lengths are roughly equal with 2,12 Å. Trans-(TBA)[ReBr₄(Py)₂] crystallizes triclinic, space group P1 , a = 9,2048(12), b = 12,0792(11), c = 15,525(2) Å, α = 95,239(10), β = 94,193(11), γ = 106,153(9)°, Z = 2. The unit cell contains two independent but very similar complex anions with approximate D_2h(mmm) point symmetry.     

Structures and properties of (H2GaN3)n (n = 1–4) clusters: A DFT study

Computations on the systems of (H₂GaN₃)_n (n = 1–4) are performed using the density functional theory (DFT)/B3LYP method with different basis sets. (H₂GaN₃)₂ possessing D_2h symmetry is found to exhibit the planar Ga₂N₂ ring structure. (H₂GaN₃)₃ involving a six‐membered Ga₃N₃ ring is found to exhibit two minima with very similar binding energies (ca. −235 ∼ −231 kJ · mol⁻¹). One minimum is the newly found boat‐like conformation possessing C_s symmetry. Another minimum possessing C_3v symmetry is the chair‐like conformation. (H₂GaN₃)₄ occurs in several structures with Ga₄N₄ eight‐membered ring structures that correspond to minima with slight energy differences among them. The structural changes of the clusters are large compared with the monomer. Frequency calculations are carried out on each optimized structure, and their infrared (IR) spectra are discussed. Thermodynamic properties demonstrate that the systems of H₂GaN₃ occur at dimer–trimer–tetramer equilibrium, and the trimer is the main component. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Self-consistent Thomas–Fermi–Dirac theory,extended by Gell-Mann and Brueckner correlation,in terms of density n and its two reduced gradients ∇2n/n and ∇n/n

We prove the following results, relevant for the density functional theory: the Thomas–Fermi–Dirac theory, generalized to include the contribution due to the high electron density result of Gell-Mann and Brueckner for the correlation energy, is shown to lead to a differential equation for the self-consistent ground-state density n( r ) in atoms and molecules in the form F(n, { ∇ n/n}², ∇²n/n)=1, where the function F is given explicitly. A straightforward extension yields a similar result for the equation determining the Pauli plus exchange–correlation potential and for the divergence of the many-electron force. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 145–149, 1998     

Spin-free quantum chemistry. XXIII. The generator-state approach

In the unitary-group formulation of quantum chemistry, the spin-projected, configuration-state spaces of quantum chemistry are realized by the irreducible representation spaces (IRS ) of the freeon unitary group U_(n), where n is the number of freeon orbitals. The Pauli-allowed IRS are labeled by the partitions [λ] = [2^(N/2)?s, 1^2S], where N and S are the particle number and the spin, respectively. The generator-state approach (GSA ) to the unitary-group formulation consists of (1) the construction of the overcomplete, nonorthonormal generator basis for each IRS ; (2) the Lie-algebraic computation of matrix elements over generator states; (3) the Moshinsky–Nagel construction of the complete, orthonormal Gel'fand basis in terms of the generator basis; and (4) the computation of matrix elements over Gel'fand states in terms of matrix elements over generator states.     

Kinetics of the reactions of Cl(2PJ) and Br(2P3/2) with O3

A laser flash photolysis-resonance fluorescence technique has been employed to study the kinetics of the important stratospheric reactions Cl(²P_J) + O₃ → ClO + O₂ and Br(²P_3/2) + O₃ → BrO + O₂ as a function of temperature. The temperature dependence observed for the Cl(²P_J) + O₃ reaction is nonArrhenius, but can be adequately described by the following two Arrhenius expressions (units are cm³ molecule^?1 s^?1, errors are 2σ and represent precision only): ??₁(T) = (1.19 ± 0.21) × 10^?11 exp [(?33 ± 37)/T] for T = 189–269K and ??₁(T) = (2.49 ± 0.38) × 10^?11 exp[(?233 ± 46)/T] for T = 269–385 K. At temperatures below 230 K, the rate coefficients determined in this study are faster than any reported previously. Incorporation of our values for ??₁(T) into stratospheric models would increase calculated ClO levels and decrease calculated HCl levels; hence the calculated efficiency of ClO_x catalyzed ozone destruction would increase. The temperature dependence observed for the (²P_3/2) + O₃ reaction is adequately described by the following Arrhenius expression (units are cm³ molecule^?1 s^?1, errors are 2σ and represent precision only): ??₂(T) = (1.50 ± 0.16) × 10^?1 exp[(?775 ± 30)/T] for T = 195–392 K. While not in quantitative agreement with Arrhenius parameters reported in most previous studies, our results almost exactly reproduce the average of all earlier studies and, therefore, will not affect the choice of ??₂(T) for use in modeling stratospheric BrO_x chemistry.