Maximum Spin Multiplicity Research Articles

Natural orbital functional for multiplets

A natural orbital functional for electronic systems with any value of the spin is proposed. This energy functional is based on a new reconstruction for the two-particle reduced density matrix (2RDM) of the multiplet, that is, of the mixed quantum state that allows all possible spin projections. The mixed states of maximum spin multiplicity are considered. This approach differs from the methods routinely used in electronic structure calculations that focus on the high-spin component or break the spin symmetry. In the ensemble, there are no interactions between electrons with opposite spins in singly occupied orbitals, as well as new inter-pair alpha-beta-contributions are proposed in the 2RDM. The proposed 2RDM fulfills (2,2)-positivity necessary N-representability conditions and guarantees the conservation of the total spin. The NOF for multiplets is able to recover the non-dynamic and the intrapair dynamic electron correlation. The missing dynamic correlation is recovered by the NOF-MP2 method. Calculation of ionization potentials of the first-row transition-metal atoms is presented as test case. The values obtained agree with those reported at the coupled cluster singles and doubles level of theory with perturbative triples and experimental data.

Magnetism in Simple Metal and 4d Transition Metal Clusters

In this review article I discuss two aspects of magnetism in small metal clusters. The first question discussed is whether simple metal clusters, that obey electronic shell models and mimic properties of elemental atoms, also obey Hund’s rule of maximum spin multiplicity. The second question is whether small clusters of 4d transition metal atoms, that are non-magnetic in the bulk, have magnetic ground states. The question arises because calculations showed that small V clusters are magnetic although the bulk metal is not. We discuss known results on Rh clusters in detail to show that small clusters are generally magnetic, but it is difficult to unequivocally identify the ground state due to the presence of many isomers and spin states that are very close in energy.

Theoretical Study of Inverted Sandwich Type Complexes of 4d Transition Metal Elements: Interesting Similarities to and Differences from 3d Transition Metal Complexes

Inverted sandwich type complexes (ISTCs) of 4d metals, (μ-η(6):η(6)-C(6)H(6))[M(DDP)](2) (DDPH = 2-{(2,6-diisopropylphenyl)amino}-4-{(2,6-diisopropylphenyl)imino}pent-2-ene; M = Y, Zr, Nb, Mo, and Tc), were investigated with density functional theory (DFT) and MRMP2 methods, where a model ligand AIP (AIPH = (Z)-1-amino-3-imino-prop-1-ene) was mainly employed. When going to Nb (group V) from Y (group III) in the periodic table, the spin multiplicity of the ground state increases in the order singlet, triplet, and quintet for M = Y, Zr, and Nb, respectively, like 3d ISTCs reported recently. This is interpreted with orbital diagram and number of d electrons. However, the spin multiplicity decreases to either singlet or triplet in ISTC of Mo (group VI) and to triplet in ISTC of Tc (group VII), where MRMP2 method is employed because the DFT method is not useful here. These spin multiplicities are much lower than the septet of ISTC of Cr and the nonet of that of Mn. When going from 3d to 4d, the position providing the maximum spin multiplicity shifts to group V from group VII. These differences arise from the size of the 4d orbital. Because of the larger size of the 4d orbital, the energy splitting between two d(δ) orbitals of M(AIP) and that between the d(δ) and d(π) orbitals are larger in the 4d complex than in the 3d complex. Thus, when occupation on the d(δ) orbital starts, the low spin state becomes ground state, which occurs at group VI. Hence, the ISTC of Nb (group V) exhibits the maximum spin multiplicity.

Properties of the molecular oxygen trimer from pairwise additive interactions

We report a theoretical study of the properties of the molecular oxygen trimer in its state of maximum spin multiplicity. The most stable structures for the complex were found using an accurate pair potential. Independent ab initio calculations on the trimer for selected geometries demonstrate the validity of this approximation and thus the small contribution of three-body effects. The three lowest structures found can be rationalized on the basis of the most stable dimer structures and possess C2, D3 and C2v point symmetries. The two lowest-lying structures are separated by only 0.6meV and zero-point energy effects, which are sizable, reduce even further the gaps among the three structures studied. Diffusion Monte Carlo calculations indicate that the ground state vibrational wavefunction is mainly located around the C2 structure but also explores the D3 minimum, yielding a near oblate symmetric top with a binding energy of 32.1meV.

A new series of homoleptic, paramagnetic organochromium derivatives: synthesis, characterization, and study of the magnetic properties.

The synthesis and characterization of the triad of organochromium derivatives [Cr(C(6)Cl(5))(4)](n-) (n=0, 1, 2) are described. By treating [CrCl(3)(thf)(3)] with LiC(6)Cl(5) in 1:5 molar ratio, the salt [Li(thf)(4)][Cr(III)(C(6)Cl(5))(4)] (1) was obtained as a violet solid in 57 % yield. Oxidation of 1 with [N(4-BrC(6)H(4))(3)][SbCl(6)] yielded the neutral complex [Cr(IV)(C(6)Cl(5))(4)] (2) as a brown solid in 71 % yield. The arylation of [CrCl(2)(thf)] with LiC(6)Cl(5) under similar conditions as above gave [[Li(thf)(3)](2)(mu-Cl)](2)[Cr(II)(C(6)Cl(5))(4)] (3) as an extremely air- and water-sensitive red solid in 47 % yield. The crystal and molecular structures of 1 and 3 have been established by X-ray diffraction methods. Complex 3 contains the unusual cation [[Li(thf)(3)](2)(mu-Cl)](+) with an almost linear Li-Cl-Li unit (174.2(6)degrees). All four C(6)Cl(5) groups are sigma-bonded to the Cr(II) center, which is located in a square-planar environment. The local geometry around the Cr(III) center in 1 is, in turn, pseudo-octahedral, since two of the C(6)Cl(5) groups act as standard sigma-bonded monodentate ligands, while the other two act as small-bite didentate ligands coordinated through both the ipso-C and one of the ortho-Cl atoms. Compounds 1-3 are paramagnetic with maximum spin multiplicity each (EPR and magnetization measurements).

On the nature of the spin exchange interaction in poly ( m-aniline)

The energy spectra of two infinite 1D models of the poly(meta-aniline) (PMA) namely the neutral and cationic radical forms are theoretically investigated. The band structure is characterized by a wide energy gap with a half-filled band of nearly degenerate molecular orbitals. Different contributions (potential, kinetic, and indirect exchange) to the effective spin exchange between the unpaired electrons in the half-filled band are calculated. The PMA exhibits a net spin exchange of ferromagnetic nature, i.e. the ground state of the PMA is characterized by maximum spin multiplicity. The ferromagnetic interaction is due to the predominant contribution of the direct (potential) Coulomb exchange.

Local-density approximation: Cohesion in the transition metals and s-->d promotion in the transition-metal atoms.

Systematics in cohesive energies ({ital H}{sub coh}) and {ital d}--non-{ital d} atomic promotion energies have been examined for the transition elements treated within the local-density approximation (LDA). Cohesive energies involve the energy of the solid as compared with that of a reference state in the free atom; going from one atomic configuration to another (e.g., {ital d}{sup {ital n}{minus}2}{ital s2}{r arrow}{ital d}{sup {ital n}{minus}1}{ital s}) for that reference state involves a promotion energy. Errors in the LDA's ability to calculate promotion energies are then translated into changes in calculated {ital H}{sub coh}. Employing the local-spin-density approximation (LSDA), scalar-relativistic values of the promotion energies have been obtained for atomic states of maximum spin multiplicity of the neutral atoms in the 3{ital d}, 4{ital d}, and 5{ital d} rows for those configurations for which experimental spectral data are available for comparison. The intent is that by scanning all three rows and those cases for which there are experimental data that those factors contributing to the LDA's shortcomings in describing electron-electron interactions in the transition elements may become better defined.

Structure and properties of nonclassical polymers. VII. Polymers with heteroatoms

AbstractIt is shown that the Coulson–Rushbrooke theorem for odd alternant hydrocarbons [15] can be generalized for some classes of nonclassical systems containing heteroatoms. The energy spectrum of such conjugated polymers containing heteroatomic groups, in particular a nitroxide group >NO, is analogous to that studied [1–4] in homonuclear alternant nonclassical polymers, whose ground state is characterized by maximum spin multiplicity. Their band structure is characterized by a wide energy gap, in the middle of which there is a band of degenerate or nearly degenerate quantum states.The presence of a narrow band of degenerate quantum states, the Hund rule, and the alternant character of the polymers of this group determine the maximum multiplicity of their ground state and may give rise to the creation of magnetic order in such compounds, if interchain correlations occur. This is in line with the investigations in Refs. 1–11.

An extension of the Coulson-Rushbrooke theorem

It is shown that the Coulson-Rushbrooke theorem for odd alternant hydrocarbons and polymers can be extended to some classes of non-classical (non-Kekulé) systems containing heteroatoms. The extension is only applicable to non-bonding molecular orbitals. The band structure of such polymers is analogous to that of odd alternant non-classical polymers, whose ground state is characterized by maximum spin multiplicity.

The higher order electron–electron coalescence condition for the intracule function for states of maximum spin multiplicity

The parallel spin components (hσσ) of the exact intracule function (h) are shown to satisfy the electron–electron coalescence condition (d3hσσ/du3)u=0 =(3/2)(d2hσσ/du2)u=0 for σ=α or β. This implies that the higher order coalescence condition (d3h/du3)u=0 =(3/2)(d2h/du2)u=0 (for S=‖MS‖=N/2) can be used as a nontrivial test of approximate wave functions for states of maximum spin multiplicity. This is particularly useful since the usual cusp condition h′(0)=h(0) is trivially satisfied with both sides equal to zero for such states. The higher order condition is used to test some explicitly correlated wave functions for the lowest triplet S and P states of the helium-like ions.

Hartree‐Fock cluster study of interstitial transition metals in silicon

AbstractResults are presented of a Hartree‐Fock cluster study of interstitial Ti, V, Cr, and Mn impurities in silicon. A Si10 cluster models the nearest Si atoms around a tetrahedral interstitial site in crystalline Si. The dangling bonds of the Si atoms are saturated by hydrogens. The effect of the Si core electrons is represented by an effective potential. Characteristic for the electronic structure of the low‐lying states of the neutral, singly positive, and doubly positive ions in silicon is the presence of fairly delocalized but still predominantly transition‐metal (3d)‐like orbitals of t2 and e symmetry. For all ions the energy of the weighted average of the terms belonging to a configuration is lowest for the configuration with maximum occupation of the t2 orbitals. Ground states with maximum spin multiplicity are found for all ions, except Ti0.

Electronic structure of UH, UF, and their ions

AbstractA relativistic effective core potential (REP) has been generated for the uranium atom and used in self‐consistent‐field calculations of the A states of UH, UF, and their ions. Energy curves were calculated at the base configuration level which ensures the dissociating atoms are described by Hartree–Fock wavefunctions. The electronic bonding of these molecules is found to be similar to that of comparable alkaline–earth hydrides and fluorides. The uranium 6p, 6d, and 5f orbitals retain their atomic character but the orbitals extend into the bonding region and are distorted by overlap repulsion and electrostatic effects. Nonetheless, the atomic energetic coupling determines that low energy states will have the maximum spin multiplicity and maximum orbital angular momentum projection consonant with the charge‐transfer bonding.

Localized orbitals and short-range molecular interactions. IV. Nonadditivity in quartet and quintet states of hydrogen atoms

The separation of the short-range interaction energy into Coulombic and penetration components introduced in Paper I of this series is extended to open-shell systems of maximum spin multiplicity. Ab initio calculations with a minimal set of 1s Slater orbitals are reported for the quartet state of H3 (isosceles triangles) and for the quintet states of tetrahedral and square H4. Nonadditivity of short-range interactions is seen to arise from the penetration of nonorthogonal localized open-shell orbitals into the core provided by nuclei and unperturbed electrons of all other groups, a result already found for closed shells. The validity of representing nonadditive effects in terms of powers of the overlap integrals is shortly discussed.

Simple “local” energy studies on the origin of hund's rule of maximum spin multiplicity and the energies of 1 snlmax configurations for He, Ne 8+ and Ar 16+

For the 2 3P and 2 1P states of the helium atom isoelectronic series, the origin of Hund's rule of maximum spin multiplicity is examied further by calculating stationary values for the “local” energy ( E loc = Ȟψ/ψ). In agreement with Katriel it is found that the interelectronic repulsion is larger for the singlet state oly for ions with atomic number Z > 3. The stationary “local” energies for the 1 snl max configurations of He, Ne 8+ and Ar 16+, with n ⩾ 2 are reported, and compared with the energies of nP and some nD states which have been calculated by other workers.

A simple demonstration of the origin of hund's rule for the helium 2S and 2P states

Using hydrogenic atomic orbitals in the wavefunction 1s(1)2p(2), and simple “local” energy calculations E loc = ▪ ψ/ψ) the origin of Hund's rule of maximum spin multiplicity is examined for the 2 3P) and 2 1P) states of the helium atom. ln agreement with other recent studies, it is found that the greater electron-nuclear attraction stabilises the triplet state relative to the singlet state. Similar conclusions pertain for the (2 3S) and (2 1S) states when Slater 2s orbitals are used in the 1s(1)2s(2) wavefunctions. For each of the calculations, it is not necessary to take account of electron indistinguishability.