Spin projections for even electron systems

A new method is presented for obtaining the spin eigenfunctions of 2n electron systems in the spin state S and M_S = 0. Using a modified Young operator the function for the state S, M_S ( = S) of the system is first projected out. The projected function is then symmetrized over the last n particles and the weight lowering operator ? is then applied to it, resulting in a projection of the state S, 0. To within a multiplicative factor, the resulting function is identical with the one resulting from the vector coupling methods.     

Extended Hartree–Fock calculations for the helium ground state

The extended Hartree–Fock (EHF) wave function of an n-electron system is defined (Löwdin, Phys. Rev. 97 , 1509 (1955)) as the best Slater determinant built on one-electron spin orbitals having a complete flexibility and projected onto an appropriate symmetry subspace. The configuration interaction equivalent to such a wavefunction for the ¹S state of a two-electron atom is discussed. It is shown that there is in this case an infinite number of solutions to the variational problem with energies lower than that of the usual Hartree–Fock function, and with spin orbitals satisfying all the extremum conditions. Two procedures for obtaining EHF spin orbitals are presented. An application to the ground state of Helium within a basic set made up of 4(s), 3(p₀), 2(d₀) and 1 (f₀) Slater orbitals has produced 90% of the correlation energy.     

A Redox‐Active Bridging Ligand to Promote Spin Delocalization,High‐Spin Complexes,and Magnetic Multi‐Switchability

A dinuclear Co^II complex, [Co₂(tphz)(tpy)₂]ⁿ⁺ (n=4, 3 or 2; tphz: tetrapyridophenazine; tpy: terpyridine), has been assembled using the redox‐active and strongly complexing tphz bridging ligand. The magnetic properties of this complex can be tuned from spin‐crossover with T_1/2≈470 K for the pristine compound (n=4) to single‐molecule magnet with an S_T=5/2 spin ground state when once reduced (n=3) to finally a diamagnetic species when twice reduced (n=2). The two successive and reversible reductions are concomitant with an increase of the spin delocalization within the complex, promoting remarkably large magnetic exchange couplings and high‐spin species even at room temperature.     

A Redox‐Active Bridging Ligand to Promote Spin Delocalization,High‐Spin Complexes,and Magnetic Multi‐Switchability

A dinuclear Co^II complex, [Co₂(tphz)(tpy)₂]ⁿ⁺ (n=4, 3 or 2; tphz: tetrapyridophenazine; tpy: terpyridine), has been assembled using the redox‐active and strongly complexing tphz bridging ligand. The magnetic properties of this complex can be tuned from spin‐crossover with T_1/2≈470 K for the pristine compound (n=4) to single‐molecule magnet with an S_T=5/2 spin ground state when once reduced (n=3) to finally a diamagnetic species when twice reduced (n=2). The two successive and reversible reductions are concomitant with an increase of the spin delocalization within the complex, promoting remarkably large magnetic exchange couplings and high‐spin species even at room temperature.     

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.     

NMR Reveals That a Highly Reactive Nonheme FeIV=O Complex Retains Its Six-Coordinate Geometry and S=1 State in Solution

The [Fe^IV(O)(Me₃NTB)]²⁺ (Me₃NTB=tris[(1-methyl-benzimidazol-2-yl)methyl]amine) complex 1 has been shown by Mössbauer spectroscopy to have an S=1 ground state at 4 K, but is proposed to become an S=2 trigonal-bipyramidal species at higher temperatures based on a DFT model to rationalize its very high C−H bond-cleavage reactivity. In this work, ¹H NMR spectroscopy was used to determine that 1 does not have C₃-symmetry in solution and is not an S=2 species. Our results show that 1 is unique among nonheme Fe^IV=O complexes in retaining its S=1 spin state and high reactivity at 193 K, providing evidence that S=1 Fe^IV=O complexes can be as reactive as their S=2 counterparts. This result emphasizes the need to identify factors besides the ground spin state of the Fe^IV=O center to rationalize nonheme oxoiron(IV) reactivity.     

A Brønsted‐Ligand‐Based Iron Complex as a Molecular Switch with Five Accessible States

A mononuclear Fe^II complex, prepared with a Brønsted diacid ligand, H₂L (H₂L=2‐[5‐phenyl‐1H‐pyrazole‐3‐yl] 6‐benzimidazole pyridine), shows switchable physical properties and was isolated in five different electronic states. The spin crossover (SCO) complex, [Fe^II(H₂L)₂](BF₄)₂ ( 1_A ), exhibits abrupt spin transition at T_1/2=258 K, and treatment with base yields a deprotonated analogue [Fe^II(HL)₂] ( 1_B ), which shows gradual SCO above 350 K. A range of Fe^III analogues were also characterized. [Fe^III(HL)(H₂L)](BF₄)Cl ( 1_C ) has an S=5/2 spin state, while the deprotonated complexes [Fe^III(L)(HL)], ( 1_D ), and (TEA)[Fe^III(L)₂], ( 1_E ) exist in the low‐spin S=1/2 state. The electronic properties of the five complexes were fully characterized and we demonstrate in situ switching between multiple states in both solution and the solid‐state. The versatility of this simple mononuclear system illustrates how proton donor/acceptor ligands can vastly increase the range of accessible states in switchable molecular devices.     

Cyano-bridged Ln^3＋-Cr^3＋ Binuclear Complexes [Ln（L）x（H2O）y^- Cr（CN）6]·mL·nH2O （Ln=La-Nd,x=5, y=2, m=1 or 2, n=2 or 2.5; Ln-Sm-Dy,Er, x=4, y=3, m=0, n=1.5 or 2.0; L= 2-pyrrolidinone）： Structure,Magnetism and Spin Density Map

In order to shed light upon the nature and mechanism of 4f-3d magnetic exchange interactions, a series of binuclear complexes of lanthanide（3＋） and chromium（3＋） with the general formula [Ln（L）5（H2O）2Cr（CN）6]·mL· nH2O （Ln=La （1）, Ce （2）, Pr （3）, Nd （4）; x=5, y=2, m=1 or 2, n=2 or 2.5; L=2-pyrrolidinone） and [Ln（L）4（H2O）3Cr（CN）6] ·nH2O （Ln=Sm （5）, Eu （6）, Gd （7）, Tb （8）, Dy （9）, Er （10）; x=4, y=3, m=0, n= 1.5 or 2.0; L=2-pyrrolidinone） were prepared and the X-ray crystal structures of complexes 2, 6 and 7 were determined. All the compounds consist of a Ln-CN-Cr unit, in which Ln^3＋ in a square antiprism environment is bridged to an octahedral coordinated Cr^3＋ ion through a cyano group. The magnetic properties of the complexes 3 and 6-10 show an overall antiferromagnetic behavior. The fitting to the experimental magnetic susceptibilities of 7 give g= 1.98, J=0.40 cm^-1, zJ＇= -0.21 cm^-1 on the basis of a binuclear spin system （Scd=7/2, Scr=3/2）, revealing an intra-molecular Gd^3＋-Cr^3＋ ferromagnetic interaction and an inter-molecular antiferromagnetic interaction. For 7 the calculation of quantum chemical density functional theory （DFT）, combined with the broken symmetry approach, showed that the calculated spin coupling constant was 20.3 cm^-1, supporting the observation of weak ferromagnetic intra-molecular interaction in 7. The spin density distributions of 7 in both the high spin ground state and the broken symmetry state were obtained, and the spin coupling mechanism between Gd^3＋ and Cr^3＋ was discussed.     

Redox potential of iron–sulfur clusters as a model of the Desulfovibrio vulgaris center

A theoretical study of Heisenberg exchange and double exchange effects in clusters with four and six iron ions has been performed for [Fe₄ S₃ O] ^m+, [Fe₄ S₄]^m+ (where m = 3, 2), and [Fe₆ S₆] ⁿ⁺ (where n = 5, 4) ions as models of the Desulfovibrio vulgaris iron–sulfur centers. Assuming that the redox potential mostly depends on the Heisenberg spin coupling and the resonance delocalization, we performed an analysis of the reduction process for the [Fe₄ S₃ O] ^3+/2+, [Fe₄ S₄] ^3+/2+, and [Fe₆ S₆] ^5+/4+ ions and showed that the redox potential can be calculated as a difference between average spin energies of the tetravalent and pentavalent double cubane superclusters. For the Heisenberg parameter of J₁ = 20 cm^-1, the redox potential amounts to about 0.03 V.It complies with close to zero experimental values of the redox potential.Electronic Supplementary Material: Supplementary material is available in the online version of this article at     

Electronic Structure of Thiolate‐bridged Diiron Complexes and a Single‐electron Oxidation Reaction: A Combination of Experimental and Computational Studies

Single‐electron oxidation of a diiron‐sulfur complex [Cp*Fe(μ‐bdt)FeCp*] ( 1 , Cp*=η⁵‐C₅Me₅; bdt=benzene‐1,2‐dithiolate) to [Cp*Fe(μ‐bdt)FeCp*]⁺ ( 2 ) has been experimentally conducted. The bdt ligand with redox‐active character has been computationally proposed to be a dianion (bdt^2?) rather than previously proposed monoanion (bdt^·?) radical in 1 though it has un‐equidistant aromatic C? C bond lengths. The ground state of 1 is predicted to be two low‐spin ferrous ions (S_Fe=0) and 2 has a medium‐spin ferric ion (S_Fe=1/2) and a low‐spin ferrous center (S_Fe=0), and the oxidation of 1 to 2 is calculated to be a single‐metal‐based process. Both complexes have no significant antiferromagnetic coupling character.     

Electronic structure of paramagnetic clusters of transition metal ions. 3. Magnetic properties and scattered wave description of the electronic structure of the hexanuclear octahedral cluster [Fe6(μ3−S)8(PEt3)6] (BPh4)2

Spin-polarized Xα–SW calculations of [Fe₆(μ₃?S)₈(PH₃)₆]²⁺ as a model of the cluster [Fe₆(μ₃?S)₈(PEt₃)₆] (BPh₄)₂ have been performed. The highest occupied energy levels are well separated from empty levels, and up to a maximum of eight electrons can be unpaired, giving a maximum spin state with S = 4. This electronic state is consistent with the magnetic data of [Fe₆(μ₃?S)₈(PEt₃)₆](BPh ₄)₂, which have been interpreted using the Heisenberg–Dirac–Van Vleck exchange spin Hamiltonian. The S = 4 state arises from the magnetic coupling between five low-spin (S_i = 1/2) and one intermediate-spin (S = 3/2) iron(III) center. © 1994 John Wiley & Sons, Inc.     

Geometric and Electronic Structure of WSiN(N = 1–6, 12) Clusters

Geometry optimizations of WSi _N (n = 1–6, 12) clusters are performed using the B3LYP/ LanL2DZ method for a sequence of different spin states, changing from spin singlet to spin septet conditions. The resulting equilibrium structures are discussed under the aspects of geometric features, cluster internal charge transfer and magnetic properties. It is shown that the W impurity in the Si _N environment generally acts as an electron acceptor. However, the charge on the W atom, as obtained by natural population analysis, can be sensitively tuned through the variation of the spin constraint from S = 0 to S = 3. The resulting geometries of WSi _n (N = 3–6) are compared with the known ground state structures of Si _N+1 (N = 3–6), and substitutional geometries are identified for N = 3 and N = 5. The nonzero spin states of WSi _N are shown to display different patterns of magnetic order, corresponding to uniform and to alternating atomic spin orientations within the cluster. Highly compact O _h and D _6h structures are identified as stable geometries of WSi₆ and of the experimentally detected unit WSi₁₂, respectively. Comparison is made with the cluster series MoSi_N(N = 1–6) and CuSi _N (N = 1–6,12).     

The First Conducting Spin-Crossover Compound Combining a MnIII Cation Complex with Electroactive TCNQ Demonstrating an Abrupt Spin Transition with a Hysteresis of 50 K

We present herein the synthesis, crystal structure, and electric and magnetic properties of the spin-crossover salt [Mn(5-Cl-sal-N-1,5,8,12)]TCNQ_1.5 ⋅ 2 CH₃CN ( I ), where 5-Cl-sal-N-1,5,8,12=N,N′-bis(3-(2-oxy-5-chlorobenzylideneamino)propyl)-ethylenediamine, containing distinct conductive and magnetic blocks along with acetonitrile solvent molecules. The Mn^III complex with a Schiff-base ligand, [Mn(5-Cl-sal-N-1,5,8,12)]⁺, acts as the magnetic unit, and the π-electron acceptor 7,7,8,8-tetracyanoquinodimethane (TCNQ⁻) is the conducting unit. The title compound ( I ) exhibits semiconducting behavior with room temperature conductivity σ_RT≈1×10⁻⁴ ohm⁻¹ cm⁻¹ and activation energy Δ ≈0.20 eV. In the temperature range 73–123 K, it experiences a hysteretic phase transition accompanied by a crossover between the low-spin S=1 and high-spin S=2 states of Mn^III and changes in bond lengths within the MnN₄O₂ octahedra. The pronounced shrinkage of the basal Mn−N bonds in I at the spin crossover suggests that the d orbital is occupied/deoccupied in this transition. Interestingly, the bromo isomorphic counterpart [Mn(5-Br-sal-N-1,5,8,12)]TCNQ_1.5 ⋅ 2 CH₃CN ( II ) of the title compound evidences no spin-crossover phenomena and remains in the high-spin state in the temperature range 2–300 K. Comparison of the chloro and bromo compounds allows the thermal and spin-crossover contributions to the overall variation in bond lengths to be distinguished. The difference in magnetic behavior of these two salts has been ascribed to intermolecular supramolecular effects on the spin transition. Discrete hydrogen bonding exists between cations and cations and anions in both compounds. However, the hydrogen bonding in the crystals of II is much stronger than in I . The relatively close packing arrangement of the [Mn(5-Br-sal-N-1,5,8,12)]⁺ cations probably precludes their spin transformation.     

A tetranuclear nickel(II) heterocubane complex of a bidentate N,O-hydroxymethyl-oxazoline ligand. Synthesis,characterization, magnetic measurements and DFT investigations

Synthesis, structural, and magnetochemical characterization of the tetranuclear [Ni₄O₄] heterocubane cluster [NiCl(L1)(MeOH)]₄, 1, employing the bidentate N,O-ligand 2-hydroxymethyl-2-oxazoline, HL1, is reported. In the solid state, each nickel(II) is coordinated in a distorted octahedral environment, located on four corners of a [Ni₄(μ₃-O)₄] cubane core motif. Measurements of the magnetic susceptibility in solution (Evans method) as well as in the solid state (magnetic susceptibility balance) gave values of 5.74 and 6.08 unpaired electrons, respectively, indicating a spin ground state of S = 3. At maximum spin degeneracy (S = 4), eight unpaired electrons would be expected. Magnetic properties were further evaluated by SQUID measurements of 1, confirming the spin ground state of 1 to be S = 3. The observed deviation is caused by antiferromagnetic coupling between the four Ni atoms. In addition, broken-symmetry DFT calculations confirmed an overlap of magnetic orbitals resulting in exchange coupling between the four nickel(II) ions of 1.     

Role of Dicarboxylate Linkers in MnIII‐Salicylaldoximate Based Extended Structures: Synthesis,Structures, and Magnetic Behavior

Four new oxo‐centered Mn^III‐salicylaldoximate triangle‐based extended complexes [Mn^III₆O₂(salox)₆(EtOH)₄(phda)]_n?(saloxH₂)_n?(2H₂O)_n ( 1 ), [Mn^III₆O₂(salox)₆(MeOH)₅(5‐I‐isoph)]_n?(3 MeOH)_n ( 2 ), [Mn^III₆O₂(salox)₆(MeOH)₄(H₂O) (5‐N₃‐isoph)]_n?(4 MeOH)_n ( 3 ) and [Mn^III₃NaO(salox)₃(MeOH)₄(5‐NO₂‐isoph)]_n?(MeOH)_n (H₂O)_n ( 4 ) [salox=salicylaldoximate, phda=1,3‐phenylenediacetate, isoph=isophthalate] have been synthesized under similar reaction conditions. Single crystal X‐ray structures show that in 1 , only one type of Mn₆ cluster is arranged in 1 D, whereas in 2 and 3 there are two types of clusters, differing in the way the triangle units are joined and assembled. In complex 4 , however, the basic building structure is heteronuclear and based on Mn₃ units extended in 2 D. Susceptibility measurements (dc and ac) over a wide range of temperatures and fields show that the complexes 1 , 2 , and 3 behave as single molecule magnets (SMMs) with S=4 ground state, while 4 is dominantly antiferromagnetic with a ground spin state S=2. Density functional theory calculations have been performed on model complexes to provide a qualitative theoretical interpretation for their overall magnetic behavior.     

Unitary symmetry and classification of the states of <Emphasis Type="Italic">n</Emphasis>-spin clusters. Magnetic and thermodynamic parameters

The unitary symmetry and classification of spin clusters by spin momenta S are considered on the basis of reduction of the full linear group to unitary groups U _{2s + 1} and orthogonal rotation group R ₃. Reduction of the permutation group P _n of n spins to the point group of the cluster is applied to the classification of the spatial states of a spin cluster with the use of permutation quantum numbers introduced in this work and the Young diagrams of the permutation group P _n . Examples of the classification of spin systems with spins s = 1/2, 1, 3/2, 2, and 5/2 with U _{2s + 1} × P _n groups (n = 5–15) are reported. This classification is common for all spin clusters and is the same for both cyclic clusters and 3D clusters with symmetry groups of a crystal. On the basis of this classification, the magnetic and thermodynamic parameters of a spin system are calculated as a function of the number of spins and temperature. For s = 1/2 clusters, the analytical formulas are derived for magnetic susceptibility, internal energy, heat capacity, and entropy as a function of quantum numbers for a cluster with any number of spins, and their dependences on temperature and the number of atoms are considered.     

Structure determination of a bis[4-(di-n-butylamino)phenyl](pyridin-3-yl)borane tetramer highlighting a unique geometric conformation of the core 16-membered ring

The tetramer of bis(4-di-n-butylaminophenyl)(pyridin-3-yl)borane [systematic name: 2λ⁴,4λ⁴,6λ⁴,8λ⁴-tetrabora-1,3,5,7(1,3)-tetrapyridinacyclooctaphane-1¹,3¹,5¹,7¹-tetrakis(ylium)], C₁₃₂H₁₉₂B₄N₁₂, was synthesized unexpectedly and crystallized. Its structure contains an unusual 16-membered ring core made up of four (pyridin-3-yl)borane groups. The ring adopts a conformation with pseudo-S₄ symmetry that is very different from the two other reported examples of this ring system. Density functional theory (DFT) computations indicate that the stability of the three reported ring conformations is dependent on the substituents on the B atoms, and that the pseudo-S₄ geometry observed in the bis(4-dibutylaminophenyl)(pyridin-3-yl)borane tetramer becomes significantly more stable when phenyl or 2,6-dimethylphenyl groups are attached to the boron centers.     

Photoisomerization Mechanism of Ruthenium Sulfoxide Complexes: Role of the Metal‐Centered Excited State in the Bond Rupture and Bond Construction Processes

Phototriggered intramolecular isomerization in a series of ruthenium sulfoxide complexes, [Ru(L)(tpy)(DMSO)]ⁿ⁺ (where tpy=2,2’:6’,2’’‐terpyridine; DMSO=dimethyl sulfoxide; L=2,2’‐bipyridine (bpy), n=2; N,N,N’,N’‐tetramethylethylenediamine (tmen) n=2; picolinate (pic), n=1; acetylacetonate (acac), n=1; oxalate (ox), n=0; malonate (mal), n=0), was investigated theoretically. It is observed that the metal‐centered ligand field (³MC) state plays an important role in the excited state S→O isomerization of the coordinated DMSO ligand. If the population of ³MC_S state is thermally accessible and no ³MC_O can be populated from this state, photoisomerization will be turned off because the ³MC_S excited state is expected to lead to fast radiationless decay back to the original ¹GS_S ground state or photodecomposition along the Ru²⁺?S stretching coordinate. On the contrary, if the population of ³MC_S (or ³MC_O) state is inaccessible, photoinduced S→O isomerization can proceed adiabatically on the potential energy surface of the metal‐to‐ligand charge transfer excited states (³MLCT_S→³MLCT_O). It is hoped that these results can provide valuable information for the excited state isomerization in photochromic d⁶ transition‐metal complexes, which is both experimentally and intellectually challenging as a field of study.     

Theoretical studies of spin state‐specific [2 + 2] and [5 + 2] photocycloaddition reactions of n‐(1‐penten‐5‐yl)maleimide

N‐alkenyl maleimides are found to exhibit spin state‐specific chemoselectivities for [2 + 2] and [5 + 2] photocycloadditions; but, reaction mechanism is still unclear. In this work, we have used high‐level electronic structure methods (DFT, CASSCF, and CASPT2) to explore [2 + 2] and [5 + 2] photocycloaddition reaction paths of an N‐alkenyl maleimide in the S₁ and T₁ states as well as relevant photophysical processes. It is found that in the S₁ state [5 + 2] photocycloaddition reaction is barrierless and thus overwhelmingly dominant; [2 + 2] photocycloaddition reaction is unimportant because of its large barrier. On the contrary, in the T₁ state [2 + 2] photocycloaddition reaction is much more favorable than [5 + 2] photocyclo‐addition reaction. Mechanistically, both S₁ [5 + 2] and T₁ [2 + 2] photocycloaddition reactions occur in a stepwise, nonadiabatic means. In the S₁ [5 + 2] reaction, the secondary C atom of the ethenyl moiety first attacks the N atom of the maleimide moiety forming an S₁ intermediate, which then decays to the S₀ state as a result of an S₁ → S₀ internal conversion. In the T₁ [2 + 2] reaction, the terminal C atom of the ethenyl moiety first attacks the C atom of the maleimide moiety, followed by a T₁ → S₀ intersystem crossing process to the S₀ state. In the S₀ state, the second C C bond is formed. Our present computational results not only rationalize available experiments but also provide new mechanistic insights. © 2017 Wiley Periodicals, Inc.     

Traces of spin-adapted reduced density matrices

The traces of the p-order reduced density matrices (p-RDM) split into independent contributions associated to the subsets of p-electron eigenstates of the Ŝ² and Ŝ_z operators. Here, we report the partial traces for the blocks of the low-order RDMs corresponding to pure spin states of an N-electron system. A systematic method for calculating those of higher order RDMs is described and some useful relations are also given. All these relations which must be fulfilled independently by a RDM can be considered as N- and S-representability conditions © 1997 John Wiley & Sons, Inc.