Local spin

Analysis of wave functions for open-shell molecules

During the past decade we have looked at several ways to track the distribution of unpaired electrons during chemical reactions and in different spin states. These methods were inspired by our previous work on singlet di-radicals where the spin density is zero yet there are clearly singly occupied orbitals. More recently we have been concerned with analysis of wave functions for single molecule magnets. This review discusses the mathematical framework by which open-shell systems can be described, in addition to methods that extract the effectively unpaired electron density, the spin state of atoms in a molecule, and other useful properties from a molecular wave function. Some of the difficulties associated with using broken spin Slater determinants to evaluate the exchange coupling parameters in the Heisenberg Hamiltonian are also mentioned.

On the definition of local spin in relativistic and nonrelativistic quantum chemistry

In this work, a survey is given on the definition of electron spin as a local property in nonrelativistic and relativistic quantum chemistry and in wave-function- and density-based theories. In particular, the transition from four-component theory to the two- and one-component Douglas-Kroll-Hess framework with respect to the electronic density is considered. Local spin is an important concept in chemistry comparable to the partial charge concept. It is often applied in transition metal chemistry and especially needed for the description and understanding of the electronic structure of spin-spin interactions in polynuclear clusters. The relevance of spin contamination and the effect of the approximate nature of contemporary density functionals is discussed in the light of results obtained for dinuclear manganese and rhenium model clusters.

DFT Studies on the Magnetic Exchange Across the Cyanide Bridge

Exchange coupling across the cyanide bridge in a series of novel cyanometalate complexes with CuII-NC-MIII (M = Cr and low-spin Mn, Fe) fragments has been studied using the broken-symmetry DFT approach and an empirical model, which allows us to relate the exchange coupling constant with sigma-, pi-, and pi*-type spin densities of the CN- bridging ligand. Ferromagnetic exchange is found to be dominated by pi-delocalization via the CN- pi pathway, whereas spin polarization with participation of sigma orbitals (in examples, where the dz2 orbital of MIII is empty) and pi* orbitals of CN- yields negative spin occupations in these orbitals, and reduces the CuII-MIII exchange coupling constant. When the dz2 orbital of MIII is singly occupied, an additional positive spin density appears in the sigma(CN) orbital and leads to an increase of the ferromagnetic Cu-NC-M exchange constant. For low-spin [MIII(CN)6]3- complexes, the dz2 orbital occupancy results in high-spin metastable excited states, and this offers interesting aspects for applications in the area of molecular photomagnetism. The DFT values of the exchange coupling parameters resulting from different occupations of the t2g orbitals of low-spin (t2g5) FeIII are used to discuss the effect of spin-orbit coupling on the isotropic and anisotropic exchange coupling in linear Cu-NC-Fe pairs.

Trinuclear Copper(II) Complexes Derived from Schiff-Base Ligands Based on a 6-Amino-6-deoxyglucopyranoside: Structural and Magnetic Characterization

The trinuclear copper(II) complexes ([CuL1)(mu-ac)Cu(mu-ac)CuL1) (1) and ([CuL2)(mu-ac)Cu(mu-ac)CuL2) (2) of the tridentate aminosaccharide-derived Schiff-base ligands H2L1 [6-N-(salicylidene)amino-6-deoxy-1,2,3-tri-O-methyl-alpha-D-glucopyranoside] and H2L2 [6-N-(3,5-di-tert-butylsalicylidene)amino-6-deoxy-1,2,3-tri-O-methyl-alpha-D-glucopyranoside] were synthesized and structurally characterized. The trinuclear complex units can be described as two terminal copper-ligand moieties bridged by a central copper acetate moiety, with the Cu centers arranged in a triangular fashion. IR and UV/vis spectroscopic studies strongly indicate that the trinuclear structure is maintained in a methanolic solution. The temperature dependence of the magnetic susceptibility of both complexes shows a moderate antiferromagnetic coupling and can be well interpreted by applying a symmetric Cua-Cub-Cua' model with linear spin topology. The fit of the magnetic data affords coupling constants J of -34 and -24 cm(-1) for 1 and 2, respectively [H = -J(SaSb + SbSa')]. For mu-alkoxo-mu-acetato-bridged copper(II) complexes with a large dihedral angle between the adjacent coordination planes, as found in 1 and 2, such an antiferromagnetic coupling is unusual. However, density functional theory calculations of 2 using BP86, B3LYP*, and B3LYP density functionals confirmed a symmetric doublet ground state.

Hole-particle characterization of coupled-cluster singles and doubles and related models

The hole-particle analysis introduced in the paper [J. Chem. Phys. 124, 224109 (2006)] is fully described and extended for coupled-cluster models of practical importance. Based on operator renormalization of the conventional amplitudes t(ai) and t(ab,ij), we present a simplified method for estimating the hole-particle density matrices for coupled-cluster singles and doubles (CCSD). With this procedure we convert the first-order density matrix of the configuration interaction (CI) singles and doubles (CISD) model, which lacks size consistency, into an approximately size-consistent expression. This permits us to correctly estimate specific indices for CCSD, including the hole and particle occupation numbers for each atom, the total occupation of holes/particles, and the entropylike measure for effective unpaired geminals. Our calculations for simple diatomic and triatomic systems indicate reasonable agreement with the full CI values. For CCSD and CISD we derive special types of two-center indices, which are similar to the charge transfer analysis of excited states previously given within the CIS model. These new quantities, termed charge transfer correlation indices, reveal the concealed effects of atomic influence on electronic redistribution due to electron correlation.

Density and wave function analysis of actinide complexes: what can fuzzy atom, atoms-in-molecules, Mulliken, Lowdin, and natural population analysis tell us?

Recent advances in computational methods have made it possible to calculate the wave functions for a wide variety of simple actinide complexes. Equally important is the ability to analyze the information contained therein and produce a chemically meaningful understanding of the electronic structure. Yet the performance of the most common wave function analyses for the calculation of atomic charge and bond order has not been thoroughly investigated for actinide systems. This is particularly relevant because the calculation of charge and bond order even in transition metal complexes is known to be fraught with difficulty. Here we use Mulliken, Lowdin, natural population analysis, atoms-in-molecules (AIM), and fuzzy atom techniques to determine the charges and bond orders of UO(2)(2+), PuO(2)(2+), UO(2), UO(2)Cl(4)(2-), UO(2)(CO)(5)(2+), UO(2)(CO)(4)(2+), UO(2)(CN)(5)(3-), UO(2)(CN)(4)(2-), UO(2)(OH)(5)(3-), and UO(2)(OH)(4)(2-). This series exhibits a clear experimental and computational trend in bond lengths and vibrational frequencies. The results indicate that Mulliken and Lowdin populations and bond orders are unreliable for the actinyls. Natural population analysis performs well after modification of the partitioning of atomic orbitals to include the 6d in the valence space. The AIM topological partitioning is insensitive to the electron donating ability of the equatorial ligands and the relative atomic volume of the formally U(VI) center is counterintuitively larger than that of O(2-) in the UO(2)(2+) core. Lastly, the calibrated fuzzy atom method yields reasonable bond orders for the actinyls at significantly reduced computational cost relative to the AIM analysis.

Analysis of multiconfigurational wave functions in terms of hole-particle distributions

A detailed study of hole-particle distributions in many-electron molecular systems is presented, based on a representation of the high-order density matrices obtained by an operator technique reminiscent of Bogolyubov's quantum statistical operator theory. A rigorous definition of density matrices of arbitrary order is given for a composite system of holes and particles. Particular attention is focused on the description of mixed hole-particle distributions. The main results are given as the functionals of excitation operators (generators) that are used in the conventional configuration interaction (CI) and coupled cluster (CC) theories. Local atomic occupation numbers for holes and particles are introduced to provide a measure of the participation of specific atoms in the electron correlation processes. The corresponding total occupations--as well as the hole-hole, particle-particle, and hole-particle mean distances--provide a useful and physically intuitive description of electron correlation. Suitable computational schemes for numerical evaluation of the above characteristics within full CI and typical CC approaches are presented. The insights one can gain with the developed approach into the peculiarities and nuances of the hole-particle picture in typical electronic processes such as excitation and molecular dissociation are illustrated with specific computations on small molecules and closed-shell atoms.

Accurate magnetic exchange couplings in transition-metal complexes from constrained density-functional theory

We demonstrate an accurate method for extracting Heisenberg exchange-coupling constants (J) from density-functional theory (DFT) calculations. We note that the true uncoupled low-spin state of a given molecule should be identified with the ground state of the system subject to a constraint on the spin density of the atoms. Using an efficient optimization strategy for constrained DFT we obtain these states directly, leading to a simple, physically motivated formula for J. Our method only depends on state energies and their associated electron densities and assigns no unphysical meaning to the Kohn-Sham determinant or individual orbitals. We study several bimetallic transition-metal complexes and find that the constrained DFT approach is competitive with, if not better than, the best broken symmetry DFT results. The success of constrained DFT in these cases appears to result from a balanced elimination of self-interaction error and static correlation from the simulation.

The Ni2 + O2 reaction: the IR spectrum and structure of Ni2O2. A combined IR matrix isolation and theoretical study

The formation of Ni2O2 can be observed from the condensation of effusive beams of Ni and O2 in neon or argon matrices. Observation of 58Ni(2)16O2, 58Ni60Ni16O2, 60Ni2(16)O2, Ni(2)18O2 and Ni(2)16O18O isotopic data for five fundamental transitions enable a discussion of structural parameters for matrix-isolated Ni2O2 in its cyclic ground state. Analysis of the nickel isotopic effects on the 58,60Ni2(16)O18O fundamentals suggest an elongated rhombic structure with a Ni-O bond force constant (240+/-10 N m-1) and NiONi bond angles around 79 degrees. The latter points to a Ni-Ni internuclear distance shorter than the O-O one. Low-lying singlet, triplet and quintet states have been studied using density functional theory with an unrestricted wave function and broken symmetry formalism. The high spin states and closed shell singlet states have been also investigated at the CCSD(T) level. The Ni2O2 ground state is calculated to be an antiferromagnetic singlet state with all the hybrid functionals. The first order properties (energies, geometry) calculated with a hybrid functional are very similar when different exchange-correlation functionals with different exact exchange fractions are used and the calculated ground state geometry (NiONi bond angle near 80 degrees, NiO bond distance around 179.5 pm) is in good agreement with the experimental estimate. Nevertheless, a correct reproduction of the experimental vibrational properties is found only when a hybrid functional containing an exact exchange fraction in the 0.4-0.5 range is used. The orbital and topological bonding analyses of Ni2O2 reveal that the relatively short Ni-Ni internuclear distance within the molecule should not be interpreted as a remaining metal-metal bonding interaction, but clearly indicate that the bonding driving force is due to the formation of four strong and highly polarized Ni-O bonds. Even in such an early stage of metal oxidation, the Ni-Ni interaction has virtually disappeared.

Energy partitioning schemes

The paper gives an overview, generalization and systematization of the different energy decomposition schemes we have devised in the last few years by using both the 3-D analysis (the atoms are represented by different parts of the physical space) and the Hilbert space analysis in terms of the basis orbitals assigned to the individual atoms. The so called "atomic decomposition of identity" provides us the most general formalism for analyzing different physical quantities in terms of individual atoms or pairs of atoms. (The "atomic decomposition of identity" means that we present the identity operator as a sum of operators assigned to the individual atoms.) By proper definitions of the atomic operators, both Hilbert-space and the different 3-D decomposition schemes can be treated on an equal footing. Several different but closely related energy decomposition schemes have been proposed for the Hilbert space analysis. They differ by exact or approximate treatment of the three- and four-center integrals and by considering the kinetic energy as a part of the atomic Hamiltonian or as having genuine two-center components, too. (Also, some finite basis correction terms may be treated in different manners.) The exact schemes are obtained by using the "atomic decomposition of identity". In the approximate schemes a projective integral approximation is also introduced, thus the energy components contain only one- and two-center integrals. The diatomic energy contributions have also been decomposed into terms of different physical nature (electrostatic, exchange etc.) The 3-D analysis may be performed either in terms of disjunct atomic domains, as in the case of the AIM formalism, or by using the so called "fuzzy atoms" which do not have sharp boundaries but exhibit a continuous transition from one to another. The different schemes give different numbers, but each is capable of reflecting the most important intramolecular interactions as well as the secondary ones--e.g. intramolecular interactions of type C-H[...]O.

About the calculation of exchange coupling constants using density-functional theory: The role of the self-interaction error

The effect of the correction of the self-interaction error on the calculation of exchange coupling constants with methods based on density-functional theory has been tested in simple model systems. The inclusion of the self-interaction correction cancels the nondynamical correlation energy contributions simulated by the commonly used functionals. Hence, such correction should be important in the accurate determination of exchange coupling constants. We have also tested several recent functionals to calculate exchange coupling constants in transition-metal complexes, such as meta-GGA functionals or new formulations of hybrid functionals. The influence of the basis set and of the use of pseudopotentials on the calculated J values has also been evaluated for a Fe(III) dinuclear complex in which the paramagnetic centers bear several unpaired electrons.

A study of the partitioning of the first-order reduced density matrix according to the theory of atoms in molecules

This work describes a simple spatial decomposition of the first-order reduced density matrix corresponding to an N-electron system into first-order density matrices, each of them associated to an atomic domain defined in the theory of atoms in molecules. A study of the representability of the density matrices arisen from this decomposition is reported and analyzed. An appropriate treatment of the eigenvectors of the matrices defined over atomic domains or over unions of these domains allows one to describe satisfactorily molecular properties and chemical bondings within a determined molecule and among its fragments. Numerical determinations, performed in selected molecules, confirm the reliability of our proposal.

Homonuclear transition-metal trimers

Density-functional theory has been used to determine the ground-state geometries and electronic states for homonuclear transition-metal trimers constrained to equilateral triangle geometries. This represents the first application of consistent theoretical methods to all of the ten 3d block transition-metal trimers, from scandium to zinc. A search of the potential surfaces yields the following electronic ground states and bond lengths: Sc3(2A1',2.83 A), Ti3(7E',2.32 A), V3(2E",2.06 A), Cr3(17E',2.92 A), Mn3(16A2',2.73 A), Fe3(11E",2.24 A), Co3(6E",2.18 A), Ni3(3A2",2.23 A), Cu3(2E',2.37 A), and Zn3(1A1',2.93 A). Vibrational frequencies, several low-lying electronic states, and trends in bond lengths and atomization energies are discussed. The predicted dissociation energies DeltaE(M3-->M2+M) are 49.4 kcal mol(-1)(Sc3), 64.3 kcal mol(-1)(Ti3), 60.7 kcal mol(-1)(V3), 11.5 kcal mol(-1)(Cr3), 32.4 kcal mol(-1)(Mn3), 61.5 kcal mol(-1)(Fe3), 78.0 kcal mol(-1)(Co3), 86.1 kcal mol(-1)(Ni3), 26.8 kcal mol(-1)(Cu3), and 4.5 kcal mol(-1)(Zn3).

Theoretical Study on the Spin-State Energy Splittings and Local Spin in Cationic [Re]−Cn−[Re] Complexes

Total spin-state energy splittings are calculated for mono- and dications of the formula {[Re]-Cn-[Re]}z+ where [Re] = eta5-(C5Me5)Re(NO)(PPh3). Cn is an even-numbered carbon chain with n ranging from 4 to 20, and z is 1 or 2. These complexes are experimentally known, and their potential role as molecular electronic devices initiated this work. We have considered the different total spin states monocation/doublet, monocation/quartet, dication/singlet, and dication/triplet. Data obtained for two density functionals BP86 and B3LYP were compared to verify the internal consistency of the results. In both ionization states, the low-spin state is the ground state, but the spin-state splittings decrease as the chain gets longer. For the dications, the splitting reaches a nearly constant value of about 10 kJ/mol with BP86 and about 4 kJ/mol with B3LYP when there are at least 14 carbon atoms in the chain, whereas for the monocations, no constant value appears to be reached asymptotically, not even if 20 carbon atoms are in the chain. For monocations, the splittings range from 138 kJ/mol (n = 4) to 68 kJ/mol (n = 20) with BP86 and from 134 kJ/mol (n = 4) to 73 kJ/mol (n = 20) with B3LYP and are thus considerably higher than those of the dications. The spin-state splittings are qualitatively mirrored by the energy splitting between the highest-occupied molecular orbital with beta spin (HOMObeta) and the lowest-unoccupied molecular orbital with alpha spin (LUMOalpha) as obtained in the low-spin state. Furthermore, the HOMOalpha-LUMOalpha gaps decrease as the carbon chain lengthens. In addition, the local distribution of the ŝz expectation value is analyzed for the monocation/doublet, the monocation/quartet, and the dication/triplet state using a modified Löwdin partitioning scheme. In the monocation/doublet and the dication/triplet state, the electron spin is distributed mainly on the metal centers and slightly delocalized onto the carbon chain. In the monocation/quartet state for chain lengths of more than 8 carbon atoms, the electron spin is mainly localized on selected atoms of the chain and not on the metal centers. In all cases, the spin delocalization onto the chain increases as the chain gets longer.

Large spin differences in structurally related Fe6 molecular clusters and their magnetostructural explanation

The syntheses, crystal structures, and magnetic characterizations of three new hexanuclear iron(III) compounds are reported. Known [Fe(6)O(2)(OH)(2)(O(2)CBu(t))(10)(hep)(2)] (1) is converted to new [Fe(6)O(2)(OH)(O(2)CBu(t))(9)(hep)(4)] (3) when treated with an excess of 2-(2-hydroxyethyl)-pyridine (hepH). Similarly, the new compound [Fe(6)O(2)(OH)(2)(O(2)CPh)(10)(hep)(2)] (2), obtained from the reaction of [Fe(3)O(O(2)CPh)(6)(H(2)O)(3)] with hepH, is converted to [Fe(6)O(2)(OH)(O(2)CPh)(9)(hep)(4)] (4) when treated with an excess of hepH. This can be reversed by recrystallization from MeCN. The cores of the four Fe(6) complexes all comprise two triangular [Fe(3)(mu(3)-O)(O(2)CR)(3)(hep)](+3) units connected at two of their apices by two sets of bridging ligands. However, 1 and 2 differ slightly from 3 and 4 in the precise way the two Fe(3) units are linked together. In 1 and 2, the two sets of bridging ligands are identical, consisting of one mu-hydroxo and two mu-carboxylate groups bridging each Fe(2) pair, i.e., a (mu-OH(-))(mu-O(2)CR(-))(2) set. In contrast, 3 and 4 have two different sets of bridging ligands, a (mu-OH(-))(mu-O(2)CR(-))(2) set as in 1 and 2, and a (mu-OR(-))(2)(mu-O(2)CR(-)) set, where RO(-) refers to the alkoxide arm of the hep(-) chelate. Variable-field and -temperature dc magnetization measurements establish that 1 and 2 have S = 5 ground states and significant and positive zero-field splitting parameters (D), whereas 3 and 4 have S = 0 ground states. This dramatic difference of 10 unpaired electrons in the ground state S values for near-isomeric compounds demonstrates an acute sensitivity of the magnetic properties to small structural changes. The factors leading to this have been quantitatively analyzed. The semiempirical method ZILSH, based on unrestricted molecular orbital calculations, was used to obtain initial estimates of the Fe(2) pairwise exchange interaction constants (J). These calculated values were then improved by fitting the experimental susceptibility versus T data, using a genetic algorithm approach. The final J values were then employed to rationalize the observed magnetic properties as a function of the core topologies and the presence of spin frustration effects. The large difference in ground state spin value was identified as resulting from a single structural difference between the two types of complexes, the different relative dispositions (cis vs trans) of two frustrated exchange pathways. In addition, use of the structural information and corresponding J values allowed a magnetostructural correlation to be established between the J values and both the Fe-O bond distances and the Fe-O-Fe angles at the bridging ligands.

Comparative analysis of local spin definitions

This work provides a survey of the definition of electron spin as a local property and its dependence on several parameters in actual calculations. We analyze one-determinant wave functions constructed from Hartree-Fock and, in particular, from Kohn-Sham orbitals within the collinear approach to electron spin. The scalar total spin operators S2 and Sz are partitioned by projection operators, as introduced by Clark and Davidson, in order to obtain local spin operators SASB and SzA, respectively. To complement the work of Davidson and co-workers, we analyze some features of local spins which have not yet been discussed in sufficient depth. The dependence of local spin on the choice of basis set, density functional, and projector is studied. We also discuss the results of Sz partitioning and show that SzA values depend less on these parameters than SASB values. Furthermore, we demonstrate that for small organic test molecules, a partitioning of Sz with preorthogonalized Lowdin projectors yields nearly the same results as one obtains using atoms-in-molecules projectors. In addition, the physical significance of nonzero SASB values for closed-shell molecules is investigated. It is shown that due to this problem, SASB values are useful for calculations of relative spin values, but not for absolute local spins, where SzA values appear to be better suited.

p-Benzyne derivatives that have exceptionally small singlet-triplet gaps and even a triplet ground state

In an effort to find a p-benzyne (1,4-didehydrobenzene) derivative with a triplet ground state, we have investigated tetrasubstitution by -F, -NH(2), -CH(3), and -NO(2) groups. These were predicted to reduce the singlet-triplet gap, but none led to a triplet ground state because of unexpected destabilization of one of the radical orbitals. This effect is likely the result of rehybridization of the substituted C atom, which has been observed for substituted benzene and perturbs the side sigma and sigma* orbital energies of the phenyl ring. The role of substituent rotation on the energy difference between the two nominally singly occupied orbitals (S and A) was then investigated. The energy of the A radical orbital was found to be much more sensitive to perturbations within the sigma C[bond]C framework than the S MO. Consequently, we believe that rehybridization of the ring carbons destabilizes the A radical orbital and can lead to large singlet-triplet splittings. To test this hypothesis, calculations on a p-benzyne with 2,6 substitution by oxygen were performed. Interestingly, a triplet ground state was predicted. Yet, examination of the geometry and wave function showed that 2,6-quinone p-benzyne is a very twisted molecule with a C3-C4-C5 allene linkage and a C1 triplet carbene center.