A method for the construction of orthogonal spin eigenfunctions

A method for the construction of the essentially idempotent and Hermitian diagonal elements of the matric algebra of the permutation group S_n is proposed. For the irreducible representation [λ] = [λ₁, λ₂] characterising a spin state S of an n-electron system, it is found that this method generates the complete set of spin projections from the appropriate primitive spin functions. The method is applied to a 7-electron system in the spin state S = M_S = 1/2 and the results are listed in the Appendix.     

Density functional study of structural and electronic properties of maximum‐spin n+1Aun−1Ag clusters

The structures and relative stabilities of high‐spin ⁿ⁺¹Au_n?1Ag and ⁿAu_n?1Ag⁺ (n = 2–8) clusters have been studied with density functional calculation. We predicted the existence of a number of previously unknown isomers. Our results revealed that all structures of high‐spin neutral or cationic Au_n?1Ag clusters can be understood as a substitution of an Au atom by an Ag atom in the high‐spin neutral or cationic Au_n clusters. The properties of mixed gold–silver clusters are strongly sized and structural dependence. The high‐spin bimetallic clusters tend to be holding three‐dimensional geometry rather than planar form represented in their low‐spin situations. Silver atom prefers to occupy those peripheral positions until to n = 8 for high‐spin clusters, which is different from its position occupied by light atom in the low‐spin situations. Our theoretical calculations indicated that in various high‐spin Au_n?1Ag neutral and cationic species, ⁵Au₃Ag, ³AuAg and ⁵Au₄Ag⁺ hold high stability, which can be explained by valence bond theory. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Charge densities for singlet and triplet electron pairs

Analytic properties of charge densities associated with singlet and triplet electron pairs, ρ₀( r ) and ρ₁( r ), are presented. In an N‐electron system with total spin S, distributions ρ₀( r ) and ρ₁( r ) are independent of the spin projection quantum number M (spin rotation invariance), as opposed to the usual spin‐up and spin‐down electron densities, ρ_α( r ) and ρ_β( r ). We derive equations showing that in the case of a wave function which is a spin‐eigenfunction, ρ₀( r ) and ρ₁( r ) are linear combinations of the total charge density ρ( r ) and the uncompensated spin density ρ_s( r )=[ρ_α( r )−ρ_β( r )]/2M. For a wave function which is not an eigenfunction of $\mathcal{S}^{2}$, no such relationship exists. In a related discussion, a definition of the high‐spin solution corresponding to a given spin‐unrestricted Hartree–Fock wave function is proposed, and a notion of effectively unpaired electrons is introduced. The distributions ρ₀( r ) and ρ₁( r ) are shown not to be invariant under spin coupling between isolated systems. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 651–660, 2000     

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.     

Semiempirical local spin: Theory and implementation of the ZILSH method for predicting Heisenberg exchange constants of polynuclear transition metal complexes

The local spin formalism ( 3 ) for computing expectation values 〈S_A · S_B〉 that appear in the Heisenberg spin model has been extended to semiempirical single determinant wave functions. An alternative derivation of expectation values in restricted and unrestricted cases is given that takes advantage of the zero differential overlap (ZDO) approximation. A formal connection between single determinant wave functions (which are not in general spin eigenfunctions) and the Heisenberg spin model was established by demonstrating that energies of single determinants that are eigenfunctions of the local spin operators with eigenvalues corresponding to high‐spin radical centers are given by the same Heisenberg coupling constants {J_AB} that describe the true spin states of the system. Unrestricted single determinant wave functions for transition metal complexes are good approximations of local spin eigenfunctions when the metal d orbitals are local in character and all unpaired electrons on each metal have the same spin (although spins on different metals might be reversed). Good approximations of the coupling constants can then be extracted from local spin expectation values 〈S_A · S_B〉 energies of the single determinant wave functions. Once the coupling constants are obtained, diagonalization of the Heisenberg spin Hamiltonian provides predictions of the energies and compositions of the spin states. A computational method is presented for obtaining coupling constants and spin‐state energies in this way for polynuclear transition metal complexes using the intermediate neglect of differential overlap Hamiltonian parameterized for optical spectroscopy (INDO/S) in the ZINDO program. This method is referred to as ZILSH, derived from ZINDO, Davidson's local spin formalism, and the Heisenberg spin model. Coupling constants and spin ground states obtained for 10 iron complexes containing from 2 to 6 metals are found to agree well with experimental results in most cases. In the case of the complex [Fe₆O₃(OAc)₉(OEt)₂(bpy)₂]⁺, a priori predictions of the coupling constants yield a ground‐state spin of zero, in agreement with variable‐temperature magnetization data, and corroborate spin alignments proposed earlier on the basis of structural considerations. This demonstrates the potential of the ZILSH method to aid in understanding magnetic interactions in polynuclear transition metal complexes. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

DFT+U Analysis on Stability of Low-Index Facets in Hexagonal LaCoO3 Perovskite:Effect of Co3+ Spin States

By the first-principles calculations,most studies indicated that the (11102)-CoO₂ termination of LaCoO₃ cannot be stabilized,which disagrees with the experimental observation.Besides the crystal structure,we found that the spin states of Co³⁺ ions could affect surface stability,which previously were not well considered.By examining the different states of Co³⁺ ions in hexagonal-phase LaCoO₃,including low spin,intermediate spin,and high spin states,the surface grand potentials of these facets are calculated and compared.The results show that the spin states of Co³⁺ ions have an important influence on stability of the LaCoO₃ facets.Different from the previous results,the stability diagrams demonstrate that the (11102)-CoO₂ termination can stably exist under O-rich condition,which can get an agreement with the experimental ones.Furthermore,the surface oxygen vacancy formation energies (E_Ov) of stable facets are computed in different spin states.The E_Ov of these possible exposed terminations strongly depend on the spin state of Co³⁺ ions:in particular,the E_Ov of the HS states is lower than that of other spin states.This indicates that one can tune the properties of LaCoO₃ by directly tuning the spin states of Co³⁺ ions.     

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.     

Studies of a Large Odd‐Numbered Odd‐Electron Metal Ring: Inelastic Neutron Scattering and Muon Spin Relaxation Spectroscopy of Cr8Mn

The spin dynamics of Cr₈Mn, a nine‐membered antiferromagnetic (AF) molecular nanomagnet, are investigated. Cr₈Mn is a rare example of a large odd‐membered AF ring, and has an odd‐number of 3d‐electrons present. Odd‐membered AF rings are unusual and of interest due to the presence of competing exchange interactions that result in frustrated‐spin ground states. The chemical synthesis and structures of two Cr₈Mn variants that differ only in their crystal packing are reported. Evidence of spin frustration is investigated by inelastic neutron scattering (INS) and muon spin relaxation spectroscopy (μSR). From INS studies we accurately determine an appropriate microscopic spin Hamiltonian and we show that μSR is sensitive to the ground‐spin‐state crossing from S=1/2 to S=3/2 in Cr₈Mn. The estimated width of the muon asymmetry resonance is consistent with the presence of an avoided crossing. The investigation of the internal spin structure of the ground state, through the analysis of spin‐pair correlations and scalar‐spin chirality, shows a non‐collinear spin structure that fluctuates between non‐planar states of opposite chiralities.     

Coordination‐Induced Spin‐State Switching of an Aminyl‐Radical‐Bridged Nickel(II) Porphyrin Dimer between Doublet and Sextet States

A bis(Ni^II‐porphyrinyl)aminyl radical with meso‐C₆F₅ groups was prepared as a spin‐delocalized stable aminyl radical with a doublet spin state. Upon addition of pyridine, both Ni^II centers became hexacoordinated by accepting two axial pyridines, which triggered a spin‐state change of the Ni^II centers from diamagnetic (S=0) to paramagnetic (S=1). The resulting high‐spin Ni^II centers interact with the aminyl radical ferromagnetically to give rise to an overall sextet state (S=5/2). Importantly, this coordination‐induced spin‐state switching can be conducted in a reversible manner, in that washing of the high‐spin radical with aqueous hydrochloric acid regenerates the original doublet radical in good yield.     

Some developments in the spin-extended Hartree-Fock method

All the second-order density matrix spin components for the spin-extended Hartree-Fock method are obtained. The coefficients in the final formulae are only ω_sM, ω_sM±1, ω_sM±2, where ω_sM are the weights of pure states of spin s in the initial unprojected determinant with spin projection M. The eigenvalue problem for the best electron density natural orbitals in the spin-extended method is formulated. All the second-order transition density matrix spin components between pure spin basis functions built of orthogonal orbitals and distinguished by different core choice are also found. This basis may be used on CI calculations.     

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).     

Ultrafast Two‐Dimensional NMR Relaxometry for Investigating Molecular Processes in Real Time

Nuclear spin–lattice (T₁) and spin–spin (T₂) relaxation times provide versatile information about the dynamics and structure of substances, such as proteins, polymers, porous media, and so forth. Multidimensional experiments increase the information content and resolution of NMR relaxometry, but they also multiply the measurement time. To overcome this issue, we present an efficient strategy for a single‐scan measurement of a 2D T₁–T₂ correlation map. The method shortens the experimental time by one to three orders of magnitude as compared to the conventional method, offering an unprecedented opportunity to study molecular processes in real‐time. We demonstrate that, despite the tremendous speed‐up, the T₁–T₂ correlation maps determined by the single‐scan method are in good agreement with the maps measured by the conventional method. The concept of the single‐scan T₁–T₂ correlation experiment is applicable to a broad range of other multidimensional relaxation and diffusion experiments.     

Electron spin relaxation time of Ni(II) ion in hexapyrazole zinc(II) dinitrate at 300 K

Understanding the electron spin relaxation properties of paramagnetic species is a fundamental requirement to use them as a probe to measure distances between sites in biomolecules by electron paramagnetic resonance (EPR) spectroscopy. Even though Ni(II) ion is an essential trace element for many species, relaxation properties are not well understood. Herein, the polycrystalline sample of Ni(II) ion magnetically diluted in Zn(Pyrazole)₆(NO₃)₂ (Ni/ZPN) has been studied in detail by EPR spectroscopy to explore the electron spin relaxation time. Progressive continuous-wave (CW) EPR power saturation study on Ni/ZPN at 300 K yielded 907 mW as the P_1/2 value. The cavity constant (K_Q) has been calculated using tempol in PVA-BA glass matrix and the product of electron spin-lattice relaxation time (T₁) and spin–spin relaxation time (T₂) for Ni/ZPN at 300 K has been reported for the first time.     

An Investigation of Photo‐ and Pressure‐Induced Effects in a Pair of Isostructural Two‐Dimensional Spin‐Crossover Framework Materials

Two new isostructural iron(II) spin‐crossover (SCO) framework (SCOF) materials of the type [Fe(dpms)₂(NCX)₂] (dpms=4,4′‐dipyridylmethyl sulfide; X=S ( SCOF‐6(S) ), X=Se ( SCOF‐6(Se) )) have been synthesized. The 2D framework materials consist of undulating and interpenetrated rhomboid (4,4) nets. SCOF‐6(S) displays an incomplete SCO transition with only approximately 30 % conversion of high‐spin (HS) to low‐spin iron(II) sites over the temperature range 300–4 K (T_1/2=75 K). In contrast, the NCSe^? analogue, SCOF‐6(Se) , displays a complete SCO transition (T_1/2=135 K). Photomagnetic characterizations reveal quantitative light‐ induced excited spin‐state trapping (LIESST) of metastable HS iron(II) sites at 10 K. The temperature at which the photoinduced stored information is erased is 58 and 50 K for SCOF‐6(S) and SCOF‐6(Se) , respectively. Variable‐pressure magnetic measurements were performed on SCOF‐6(S) , revealing that with increasing pressure both the T_1/2 value and the extent of spin conversion are increased; with pressures exceeding 5.2 kbar a complete thermal transition is achieved. This study confirms that kinetic trapping effects are responsible for hindering a complete thermally induced spin transition in SCOF‐6(S) at ambient pressure due to an interplay between close T_1/2 and T(LIESST) values.     

Accurate Determination of the Electronic Structure Parameters of the Spin Ladder Compounds SrCu2O3, Sr2Cu3O5 and CaCu2O3

Ab initio embedded cluster calculations have been employed to calculate a large number of electronic structure parameters of three different spin ladders, namely SrCu₂O₃, CaCu₂O₃ and Sr₂Cu₃O₅. Using the iterative difference dedicated configuration interaction methodology, magnetic couplings J and hopping amplitudes t are determined for first to fourth nearest neighbors. In addition, the four-body cyclic exchange J _ring is extracted and the direct exchange K, the neutral-ionic hopping integral t ₀ and the on-site repulsion U are calculated for first and second nearest neighbor copper ions. The substitution of these parameters in the pertubative superexchange relation J=2K−4t ₀²/U yields magnetic coupling parameters in close agreement with the variational estimates. The spin ladders can be considered as an interpolation between the one-dimensional (1D) spin chains and the 2D antiferromagnets. Hence, results are compared with similar parameters in the spin chain Sr₂CuO₃ and the two-dimensional antiferromagnet La₂CuO₄.     

Rational Design of [13C,D14]Tert‐butylbenzene as a Scaffold Structure for Designing Long‐lived Hyperpolarized 13C Probes

Dynamic nuclear polarization (DNP) is a technique to polarize the nuclear spin population. As a result of the hyperpolarization, the NMR sensitivity of the nuclei in molecules can be dramatically enhanced. Recent application of the hyperpolarization technique has led to advances in biochemical and molecular studies. A major problem is the short lifetime of the polarized nuclear spin state. Generally, in solution, the polarized nuclear spin state decays to a thermal spin equilibrium, resulting in loss of the enhanced NMR signal. This decay is correlated directly with the spin‐lattice relaxation time T₁. Here we report [¹³C,D₁₄]tert‐butylbenzene as a new scaffold structure for designing hyperpolarized ¹³C probes. Thanks to the minimized spin‐lattice relaxation (T₁) pathways, its water‐soluble derivative showed a remarkably long ¹³C T₁ value and long retention of the hyperpolarized spin state.     

First Step Towards a Devil's Staircase in Spin‐Crossover Materials

The unprecedented bimetallic 2D coordination polymer {Fe[(Hg(SCN)₃)₂](4,4′‐bipy)₂}_n exhibits a thermal high‐spin (HS)?low‐spin (LS) staircase‐like conversion characterized by a multi‐step dependence of the HS molar fraction γ_HS. Between the fully HS (γ_HS=1) and LS (γ_HS=0) phases, two steps associated with different ordering appear in terms of spin‐state concentration waves (SSCW). On the γ_HS≈0.5 step, a periodic SSCW forms with a HS‐LS‐HS‐LS sequence. On the γ_HS≈0.34 step, the 4D superspace crystallography structural refinement reveals an aperiodic SSCW, with a HS‐LS sequence incommensurate with the molecular lattice. The formation of these different long‐range spatially ordered structures of LS and HS states during the multi‐step spin‐crossover is discussed within the framework of “Devil's staircase”‐type transitions. Spatially modulated phases are known in various types of materials but are uniquely related to molecular HS/LS bistability in this case.     

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.     

Spin–Spin Coupling Between Two meta-Benzyne Moieties In a Quinolinium Tetraradical Cation Increases Their Reactivities

The reactivity of a carbon-centered σ,σ,σ,σ-type singlet-ground-state tetraradical containing two meta-benzyne moieties was examined in the gas phase. Surprisingly, the tetraradical showed higher reactivity than its individual meta-benzyne counterparts. The reactivity of meta-benzynes is controlled by their (calculated) distortion energy ΔE_2.3, singlet–triplet spitting ΔE_S–T, and electron affinity (EA_2.3) of the meta-benzyne moiety at the transition state geometry for hydrogen-atom abstraction reactions. The addition of a second meta-benzyne moiety to a meta-benzyne does not significantly change EA_2.3. However, ΔE_2.3 is substantially decreased for both meta-benzyne moieties in the tetraradical, and this explains their higher reactivities. The decrease in ΔE_2.3 for each meta-benzyne moiety in the tetraradical is rationalized by stabilizing spin–spin coupling between one radical site in each meta-benzyne moiety. Therefore, spin–spin coupling between the meta-benzyne moieties in this tetraradical increases its reactivity, whereas spin–spin coupling within each meta-benzyne moiety decreases its reactivity.     

Identification of the Chromophore in the Apatite Pigment [Sr10(PO4)6(CuxOH1−x−y)2]: Linear OCuO− Featuring a Resonance Raman Effect,an Extreme Magnetic Anisotropy,and Slow Spin Relaxation

A new chromophore has been identified in copper‐doped apatite pigments having the general composition [Sr₁₀(PO₄)₆(Cu_xOH_1?x?y)₂], in which x=0.1, 0.3 and y=0.01–0.42. By using X‐ray absorption spectroscopy, low‐temperature magnetization measurements, and synchrotron X‐ray powder structure refinement, it has been shown that the oxygenated compounds contain simultaneously diamagnetic Cu¹⁺ and paramagnetic Cu³⁺ with S=1. Cu³⁺ is located at the same crystallographic position as Cu¹⁺, being linearly coordinated by two oxygen atoms and forming the OCuO^? anion. The Raman spectroscopy study of [A₁₀(PO₄)₆(Cu_xOH_1?x?y)₂,], in which A=Ca, Sr, Ba, reveals resonance bands at 651–656 cm^?1 assigned to the symmetric stretching vibration (ν₁) of OCuO^?. The strontium apatite pigment exhibits a strong paramagnetic anisotropy with an unprecedentedly large negative zero‐field splitting parameter (D) of ≈?400 cm^?1. The extreme magnetic anisotropy causes slow magnetization relaxation with relaxation times (τ) up to 0.3 s at T=2 K, which relates the compounds to single‐ion magnets. At low temperature, τ is limited by a spin quantum‐tunneling, whereas at high temperature a thermally activated relaxation prevails with U_eff≈48 cm^?1. Strong dependence of τ on the paramagnetic center concentration at low temperature suggests that the spin‐spin relaxation dominates in the spin quantum‐tunneling process. The compound is the first example of a d‐metal‐based single‐ion magnet with S=1, the smallest spin at which an energy barrier arises for the spin flipping.