Some simplification in spin-free configuration interaction studies

A simplification has been attempted in the procedures for determining the matrix elements of the generators of the unitary group U(n) over a tensor basis spanning the irreducible representation 2 ^N/2–S , 1 ^2S for an N-electron system. It has been shown that these matrix elements require, for their determination, only the corresponding representation matrices of cyclic permutations of the group S _N . A viable algorithm has been obtained for determining these representation matrices.     

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.     

Structural diversity in imidazolidinone organocatalysts: a synchrotron and computational study

(S)‐1‐(Methylaminocarbonyl)‐3‐phenylpropanaminium chloride (S2·HCl), C₁₀H₁₅N₂O⁺·Cl⁻, crystallizes in the orthorhombic space group P2₁2₁2₁ with a single formula unit per asymmetric unit. (5R/S)‐5‐Benzyl‐2,2,3‐trimethyl‐4‐oxoimidazolidin‐1‐ium chloride (R3 and S3), C₁₃H₁₉N₂O⁺·Cl⁻, crystallize in the same space group as S2·HCl but contain three symmetry‐independent formula units. (R/S)‐5‐Benzyl‐2,2,3‐trimethyl‐4‐oxoimidazolidin‐1‐ium chloride monohydrate (R4 and S4), C₁₃H₁₉N₂O⁺·Cl⁻·H₂O, crystallize in the space group P2₁ with a single formula unit per asymmetric unit. Calculations at the B3LYP/6–31G(d,p) and B3LYP/6–311G(d,p) levels of the conformational energies of the cation in R3, S3, R4 and S4 indicate that the ideal gas‐phase global energy minimum conformation is not observed in the solid state. Rather, the effects of hydrogen‐bonding and van der Waals interactions in the crystal structure cause the molecules to adopt higher‐energy conformations, which correspond to local minima in the molecular potential energy surface.     

N-Representability conditions generated by a one-particle grassmann factor of an antisymmetric function

N-representability conditions for a two-particle density operator implied by positive-semidefiniteness of the projection operator P^N+1(?¹ Λ Ψ^N) are derived and discussed. The operator P^N+1(?¹ Λ Ψ^N) projects onto an (N + 1)-particle antisymmetric function ?¹ Λ Ψ^N, the Grassmann product of a one-particle factor ?¹ and an N-particle factor Ψ^N. The polar subcone ??²_N(g, q) to the set of N-representable two-particle density operators ??²_N which corresponds to these conditions is found. It is shown that its extreme rays belong to two orbits for the action of the unitary group of transformations in one-particle Hilbert space. The facial structure of the convex set ??²_N exposed by elements of ??²_N(g, q) is analyzed. An example of the operator that changes the structure of its bottom eigenspace when the number of fermions N surpasses a certain value is noted. A new approach to the diagonal conditions for N-representability is found. It consists of the decomposition of the N-particle antisymmetric identity operator onto the mutually orthogonal projection operators.     

Vibrational Spectrum and Gas‐Phase Structure of Disulfur Dinitride (S2N2)

Gas‐phase FTIR spectra of the ν₆ (B‐type) and the ν₄ (C‐type) fundamental bands of S₂N₂ (D_2h) were recorded with a resolution of ≤0.004 cm^?1 and the vibrational spectrum of S₂N₂ (D_2h) in solid Ar has been revisited. All IR‐active fundamentals and four combination bands were assigned in excellent agreement with calculated values from anharmonic VPT2 and VCI theory based on (explicitly correlated) coupled‐cluster surfaces. Accurate experimental vibrational ground‐ and excited‐state rotational constants of ³²S₂¹⁴N₂ are obtained from a rovibrational analysis of the ν₆ and ν₄ fundamental bands, and precise zero‐point average r_z (R_z(SN)=1.647694(95) Å, α_z(NSN)=91.1125(33)°) and semi‐experimental equilibrium structures (R_e(SN)=1.64182(33) Å, α_e(NSN)=91.0716(93)°) of S₂N₂ have been established. These are compared to the solid‐state structure of S₂N₂ and structural properties of related sulfur nitrogen compounds and to results of ab initio structure calculations.     

Magnetochemistry of dimers with iron group ions

General theory of matrix interactions of dimers of iron group ions is considered. Equations for calculation of matrix elements of the exchange energy of dimers and equations for calculation of the magnetic moments of dimers of different d ⁿ configurations and different exchange parameters are reported. The energies of the spin states of dimers are calculated with inclusion of exchange interactions of the first J ₀ and second J ₂ orders, and energy state diagrams for dimers with different spins and exchange energies are reported. For heterospin dimers, equations for calculation of the magnetic moment ?? and magnetic susceptibility ?? are introduced, and for homospin dimers, analytical formulas of calculation are presented. The tables of magnetic moments are calculated and their diagrams as a function of J ₀ for homospin dimers with spins S = 5/2, 2, 3/2, 1, and 1/2 are given. The experimental data are interpreted for Co²⁺ dimers as an example: the experimental and theoretical dependences of ?? and ?? on temperature are considered.     

Activating Bi p-orbitals in Dispersed Clusters of Amorphous BiOx for Electrocatalytic Nitrogen Reduction

Catalytic strategies based on main group metals are significantly less advanced than those of transition metal catalysis, leaving untapped areas of potentially fruitful research. We here demonstrate an effective approach for the modulation of Bi 6p energy levels during the construction of atomically dispersed clusters of amorphous BiO_x. Bi oxidation state is proposed to strongly affects the nitrogen fixation activity, with the half-occupied p_z orbitals of the Bi²⁺ ions being highly efficient toward electron injection into the inert N₂ molecule. With sufficient catalytic sites to adsorb and activate N₂, the bonding between N₂ and catalyst is able to be in situ identified. The catalyst shows an outstanding Faraday efficiency (≈30 %) and high yield (≈113 μg h⁻¹ mg⁻¹_cat) in NH₃ production, outperforming most of the existing catalysts in aqueous solution. These results lay the basis for developing the potential of p-block elements for catalysis of multi-electron reactions.     

Spin-Free configuration-interaction study of molecules using single-parameter alternat molecular orbitals

A procedure is outlined for a programmable spin-free configuration-interaction (CI ) study in molecules using single-parameter alternant molecular orbitals for generating various configurations. The configurations were chosen to form bases for the irreducible representation {2^N/2–2, 1^2S} of the general linear group GL(n). Using a transformation to biorthogonal space the CI matrix elements of a spin-free Hamiltonian were generated. The procedure has been used to obtain the π-electron energies for the ^3,1A_g and ^3,1B_u states of cis- and trans-butadiene.     

Theoretical prediction of ionization potential,electron affinity,and electronic spectrum of the S2N radical

Ab initio MRD –CI calculations using a basis set of near Hartree–Fock quality have been carried out to calculate the ground-state electronic structure of S₂N⁺, S₂N, and S₂N^? and the ionization potential, electron affinity, and vertical electronic spectrum of S₂N. At the highest level of theory (estimated full CI or FCI ), S₂N⁺ is predicted to have a linear structure with r(N? S) = 1.51 Å. For S₂N and S₂N^?, the minimum in energy at the FCI level corresponds to a quasi-linear [with a barrier height to linearity of about 2.0 kcal mol^?1, ] and a bent structure , respectively. The adiabatic/vertical ionization potential and electron affinity of S₂N are predicted to be 7.26/7.82 and 1.60/0.79 eV, respectively. Of the several electronic transitions in S₂N considered, the ones with the excitation energy of 1.87 eV (X² A₁ → ²B₂) and 2.87 eV (X²A₁ → ²B₂) are somewhat intense (? = 0.005 and 0.002) and likely to be observed.     

Density functional study of S(N) 2 substitution reactions for CH(3) Cl + CX(1) X(2•-) (X(1) X(2) = HH, HF, HCl, HBr, HI, FF, ClCl, BrBr, and II)

A systematic investigation on the S_N2 displacement reactions of nine carbene radical anions toward the substrate CH₃Cl has been theoretically carried out using the popular density functional theory functional BHandHLYP level with different basis sets 6‐31+G (d, p)/relativistic effective core potential (RECP), 6‐311++G (d, p)/RECP, and aug‐cc‐pVTZ/RECP. The studied models are CX¹X^2?? + CH₃Cl → X²X¹CH₃C^? + Cl^?, with CX¹X^2?? = CH₂^??, CHF^??, CHCl^??, CHBr^??, CHI^??, CF₂^??, CCl₂^??, CBr₂^??, and CI₂^??. The main results are proposed as follows: (a) Based on natural bond orbital (NBO), proton affinity (PA), and ionization energy (IE) analysis, reactant CH₂^?? should be a strongest base among the anion‐containing species (CX¹X^2??) and so more favorable nucleophile. (b) Regardless of frontside attacking pathway or backside one, the S_N2 reaction starts at an identical precomplex whose formation with no barrier. (c) The back‐S_N2 pathway is much more preferred than the front‐S_N2 one in terms of the energy gaps [ΔE(front)?ΔE(back)], steric demand, NBO population analysis. Thus, the back‐S_N2 reaction was discussed in detail. On the one hand, based on the energy barriers (ΔE and ΔE) analysis, we have strongly affirmed that the stabilization of back attacking transition states (b‐TSs) presents increase in the order: b‐TS‐CI₂ < b‐TS‐CBr₂ < b‐TS‐CCl₂ < b‐TS‐CHI < b‐TS‐CHBr < b‐TS‐CHCl < b‐TS‐CF₂ < b‐TS‐CHF < b‐TS‐CH₂. On the other hand, depended on discussions of the correlations of ΔE with influence factors (PA, IE, bond order, and ΔE), we have explored how and to what extent they affect the reactions. Moreover, we have predicted that the less size of substitution (α‐atom) required for the gas‐phase reaction with α‐nucleophile is related to the α‐effect and estimated that the reaction with the stronger PA nucleophile, holding the lighter substituted atom, corresponds to the greater exothermicity given out from reactants to products. © 2012 Wiley Periodicals, Inc. J Comput Chem, 2012     

Symmetric-group algebraic variational solutions for Heisenberg models at finite temperature

The Heisenberg spin Hamiltonian for a collection of N spin-1/2 sites is viewed, as favored by Professor Matsen, to be an element of the group algebra of the symmetric group ??_N. Several computationally tractable, variational group–algebraic approximations for the finite-temperature density matrix are made so as to minimize the Gibb's free–energy functional. Relations to previous quite differently motivated approximations are identified, though improvements are noted with the present approach.     

Collision‐induced dissociation mass spectra of glucosinolate anions

Collision‐induced dissociation (CID) mass spectra of differently substituted glucosinolates were investigated under negative‐ion mode. Data obtained from several glucosinolates and their isotopologues (³⁴S and ²H) revealed that many peaks observed are independent of the nature of the substituent group. For example, all investigated glucosinolate anions fragment to produce a product ion observed at m/z 195 for the thioglucose anion, which further dissociates via an ion/neutral complex to give two peaks at m/z 75 and 119. The other product ions observed at m/z 80, 96 and 97 are characteristic for the sulfate moiety. The peaks at m/z 259 and 275 have been attributed previously to glucose 1‐sulfate anion and 1‐thioglucose 2‐sulfate anion, respectively. However, based on our tandem mass spectrometric experiments, we propose that the peak at m/z 275 represents the glucose 1‐thiosulfate anion. In addition to the common peaks, the spectrum of phenyl glucosinolate (β‐D ‐Glucopyranose, 1‐thio‐, 1‐[N‐(sulfooxy)benzenecarboximidate] shows a substituent‐group‐specific peak at m/z 152 for C₆H₅‐C(?NOH)S^?, the CID spectrum of which was indistinguishable from that of the anion of synthetic benzothiohydroxamic acid. Similarly, the m/z 201 peak in the spectrum of phenyl glucosinolate was attributed to C₆H₅‐C(?S)OSO₂^?. Copyright © 2009 John Wiley & Sons, Ltd.     

A programmable spin-free method for configuration interaction

In this note a method is presented for quick implementation of configuration interaction (CI) calculations in molecules. A spin-free Hamiltonian for anN electron system in a spin stateS, expressed in terms of the generators for the unitary group algebra, is diagonalized over orbital configurations forming a basis for the irreducible representation [2^1/2N-S 1^2S ] of the permutation group S _N . It has been found that the basic algebraic expressions necessary for the CI calculation involve a limited category of permutations. These have been displayed explicitly. On leave from the Indian Institute of Technology, Bombay, India.     

Copper(II) complexes of potentially terdentate N2-[(R)-2-hydroxypropyl]- and N2-[(S)-2-hydroxypropyl]-(S)-phenylalaninamide for chiral recognition: Synthesis of the Ligands and Formation Constants

Copper(II) complexes of the ligands N²-[(R)-2-hydroxypropyl]- and N²-[(S)-2-hydroxypropyl]-(S)-phenylalaninamide performed chiral separation of N-dansyl-protected and unmodified amino acids in HPLC (reversed phase). With the aim of investigating which species are potentially involved in the discrimination mechanism, the two ligands were synthesized and their complexation equilibria with Cu²⁺ studied by potentiometry and spectrophotometry in aqueous solution up to pH 11.7. The formation constants of the species observed, [CuL]²⁺, [CuL₂]²⁺, [CuLH_–1]⁺, [CuL₂H_–1]⁺, [CuL₂H_–2], and [CuL₂H_–3]^?, were quite similar for both compounds and were compared to those of (S)-phenylalaninamide. Most probably, in [CuL₂H_–3]^? the ligands behave as terdentate, with the deprotonated OH group occupying an apical position.     

Elastic behavior of single polymer chains adsorbed on rough surfaces

In this paper, elastic behaviors of single polymer chains adsorbed on the rough surfaces with a substrate and some periodically tactic pillars are investigated by the pruned-enriched-Rosenbluth method (PERM). In our simulation, a single polymer chain is firstly adsorbed on the substrate and then pulled along the z-axis direction, which is vertical to the substrate. We investigate the chain size and shape of polymer chains, such as mean-square radii of gyration per bond 〈S²〉_xy/N, 〈S²〉_z/N and shape factor 〈δ^∗〉 in order to show how the size and shape of adsorbed polymer chains change during the desorption process. Due to the occurrences of separation of the chains from the substrate, farther adsorption on the upper surfaces of pillars and complete separation from the whole rough surfaces in the elastic process, the changes of 〈S²〉_xy/N, 〈S²〉_z/N and 〈δ^∗〉 during the process are complicated. On the other hand, some thermodynamic properties such as average energy per bond, average Helmholtz free energy per bond, elastic force f are investigated, and our aim is to study the elastic behaviors of polymer chains adsorbed on the rough surface during the elasticity process. Elastic force f has some plateaus during the desorption process for strong adsorption interaction. If there is no adsorption interaction, the chains can get away from the rough surfaces spontaneously. These investigations can provide some insights into the elastic behaviors of polymer chains adsorbed on the rough surface.     

Chiral Diaminodiamide Copper(II) Complexes for the Enantioselective Recognition of Amino Acids: Synthesis of the Ligands and Formation Constants

(S,S)-N,N′ -Bis(aminoacyl)ethane- and (S,S)-N,N′ -bis(aminoacyl)propanediamines (AA-NN-2 and AA-NN-3, respectively, AA = alanine, phenylalanine, valine) were synthesized as the dihydrochlorides, and their complexes with Cu(II) studied potentiometrically. Since these ligands in the presence of Cu(II) are able to perform chiral resolution of D ,L -dansylamino acids in HPLC (reversed phase), in a certain pH range (6.5–8.5), it is important to know the equilibria existing between ligands and copper in aqueous solution. For AA-NN-2, four species, CuLH³⁺, CuL²⁺, Cu₂L₂H, and CuLH_?2, were detected, whereas for AA-NN-3, only CuLH³⁺, CuL²⁺, and CuLH_?2 were found. The aim is to find out which complexes may be involved in the recognition process.     

Non-Heme FeII Diastereomeric Complexes Bearing a Hexadentate Ligand: Unexpected Consequences for the Spin State and Catalytic Oxidation Properties

The reactivity and selectivity of non-heme Fe^II complexes as oxidation catalysts can be substantially modified by alteration of the ligand backbone or introduction of various substituents. In comparison with the hexadentate ligand N,N,N′,N′-tetrakis(pyridin-2-ylmethyl)ethane-1,2-diamine (TPEN), N,N′-bis[1-(pyridin-2-yl)ethyl]-N,N′-bis(pyridin-2-ylmethyl)ethane-1,2-diamine (_2MeL₆²) has a methyl group on two of the four picolyl positions. Fe^II complexation by _2MeL₆² yields two diastereomeric complexes with very similar structures, which only differ in the axial/equatorial positions occupied by the methylated pyridyl groups. In solution, these two isomers exhibit different magnetic behaviors. Whereas one isomer exhibits temperature-dependent spin-state conversion between the S=0 and S=2 states, the other is more reluctant towards this spin-state equilibrium and is essentially diamagnetic at room temperature. Their catalytic properties for the oxidation of anisole by H₂O₂ are very different and correlate with their magnetic properties, which reflect their lability/inertness. These different properties most likely depend on the different steric constraints of the methylated pyridyl groups in the two complexes.     

Ion‐Pair SN2 Reaction of OH− and CH3Cl: Activation Strain Analyses of Counterion and Solvent Effects

We have theoretically studied the non‐identity S_N2 reactions of M_nOH^(n?1)+CH₃Cl (M⁺=Li⁺, Na⁺, K⁺, and MgCl⁺; n=0, 1) in the gas phase and in THF solution at the OLYP/6‐31++G(d,p) level using polarizable continuum model (PCM) implicit solvation. We want to explore and understand the effect of the metal counterion M⁺ and solvation on the reaction profile and the stereoselectivity of these processes. To this end, we have explored the potential energy surfaces of the backside (S_N2‐b) and frontside (S_N2‐f) pathways. To explain the computed trends, we have carried out analyses with an extended activation strain model (ASM) of chemical reactivity that includes the treatment of solvation effects.     

Products of class operators of the symmetric group

A combinatorial derivation of the product of the class of three cycles, [(1)^N?3(3)]_N with an arbitrary class operator of the symmetric group S_N is presented. The form of this result suggests a conjecture concerning the expression of the general class operator product in terms of a relatively small number of reduced class coefficients. The conjecture is applied to the determination of the products of [(1)^N?4(4)]_N, [(1)^N?4(2)²]_N, and [(1)^N?5(5)]_N with arbitrary class operators. General expressions for the reduced class coefficients of the simplest type are obtained.     

Diagnostic cyclisation reactions which follow phosphate transfer to carboxylate anion centres for energised [M-H]- anions of pTyr-containing peptides

The low‐energy negative ion phosphoTyr to C‐terminal ‐CO₂PO₃H₂ rearrangement occurs for energised peptide [M–H]^– anions even when there are seven amino acid residues between the pTyr and C‐terminal amino acid residues. The rearranged C‐terminal ‐CO₂PO₂H(O^–) group effects characteristic S_Ni cyclisation/cleavage reactions. The most pronounced of these involves the electrophilic central backbone carbon of the penultimate amino acid residue. This reaction is aided by the intermediacy of an H‐bonded intermediate in which the nucleophilic and electrophilic reaction centres are held in proximity in order for the S_Ni cyclisation/cleavage to proceed. The ΔG_reaction is +184 kJ mol^?1 with the barrier to the S_Ni transition state being +240 kJ mol^?1 at the HF/6‐31 + G(d)//AM1 level of theory. A similar phosphate rearrangement from pTyr to side chain CO₂^– (of Asp or Glu) may also occur for energised peptide [M–H]^– anions. The reaction is favourable: ΔG_reaction is ?44 kJ mol^?1 with a maximum barrier of +21 kJ mol^?1 (to the initial transition state) when Asp and Tyr are adjacent. The rearranged species R¹‐Tyr‐NHCH(CH₂CO₂PO₃H^–)COR² (R¹ = CHO; R² = OCH₃) may undergo an S_Ni six‐centred cyclisation/cleavage reaction to form the product anion R¹‐Tyr(NH^–). This process has a high energy requirement [ΔG_reaction = +224 kJ mol^?1, with the barrier to the S_Ni transition state being +299 kJ mol^?1]. Copyright © 2011 John Wiley & Sons, Ltd.