A theoretical study on the temperature dependence of zero-field splitting for the tetragonal Cr3+ center in MgO crystal

This paper reports a detailed theoretical calculation of the temperature dependence of zero-field splitting D (characterized by ΔD(T)=D(T)-D(0)) for the tetragonal Cr3+ center in MgO crystal by considering both the static contribution due to the thermal expansion of Cr3+ center and the vibrational contribution caused by electron-phonon (including the acoustic and optical phonons) interaction. The vibrational contribution due to the acoustic phonon is calculated using the long-wave approximation similar to the study on the specific heat of crystals and that due to optical phonon is estimated using the single-phonon model. The calculated results are in reasonable agreement with the experimental values. From the calculation, it is found that the static contribution ΔDstat(T) (which is often regarded as very small and is neglected in the previous papers) is larger than the vibrational contribution ΔDvib(T) and so the reasonable studies of temperature dependence of zero-field splitting should take both the static and the vibrational contributions into account.     

Calculation of two-center zero-field splitting integrals

Two-center zero-field splitting (ZFS) integrals have been calculated by numerical integration of Coulomb repulsion integrals which are evaluated over basic charge distributions as defined by Roothaan in terms of Slater atomic orbitals. The method is applied to the calculation of the ZFS integrals for -, - and - electron interactions on C, N and N⁺ centers. Numerical results are given.

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Zusammenfassung Zweizentren ZFS-Integrale sind mittels numerischer Integration von Coulombintegralen berechnet worden, und zwar die -, -- und --Integrale an C-, N- und N⁺-Zentren. Die numerischen Resultate werden mitgeteilt.

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Résumé Les intégrales bi-centriques de séparation à champ nul (ZFS) ont été calculées par intégration numérique des intégrales de répulsion coulombiennes évaluées pour les distributions de charge fondamentales définies par Roothan en termes d'orbitales atomiques de Slater. La méthode est appliquée au calcul des intégrales ZFS pour les interactions -, - et - sur les centres C, N et N⁺. Résultats numériques.

     

First-principles calculations of zero-field splitting parameters

In this work, an implementation of an approach to calculate the zero-field splitting (ZFS) constants in the framework of ab initio methods such as complete active space self-consistent field, multireference configuration interaction, or spectroscopy oriented configuration interaction is reported. The spin-orbit coupling (SOC) contribution to ZFSs is computed using an accurate multicenter mean-field approximation for the Breit-Pauli Hamiltonian. The SOC parts of ZFS constants are obtained directly after diagonalization of the SOC operator in the basis of a preselected number of roots of the spin-free Hamiltonian. This corresponds to an infinite order treatment of the SOC in terms of perturbation theory. The spin-spin (SS) part is presently estimated in a mean-field fashion and appears to yield results close to the more complete treatments available in the literature. Test calculations for the first- and second-row atoms as well as first-row transition metal atoms and a set of diatomic molecules show accurate results for the SOC part of ZFSs. SS contributions have been found to be relatively small but not negligible (exceeding 1 cm(-1) for oxygen molecule). At least for the systems studied in this work, it is demonstrated that the presented method provides much more accurate estimations for the SOC part of ZFS constants than the emerging density functional theory approaches.     

The use of improved atomic orbitals in the evaluation of zero-field splitting integrals

The effect of improving the 2p _z-atomic orbital representation on the values of molecular zero-field splitting integrals is assessed on the example of the two-center Coulomb integral involving the (r ²–3z²)/r⁵ operator in the cases of nitrogen and carbon. The results suggest that the use of the classical Slater orbital or its Gaussian equivalent may be misleading.

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Zusammenfassung Ein spezielles, bei der Berechnung der molekularen Nullfeldaufspaltung von Tripletts auftretendes Zweizentrenintegral wird in AbhÄngigkeit von der Güte der 2p _z-Orbitale des Kohlenstoff- bzw. Stickstoffatoms untersucht. Offenbar sind die üblichen Slaterorbitale und deren Approximation durch Gau\funktionen ungeeignet.

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Résumé On a étudié l'effet de l'amélioration de la base d'orbitales atomiques 2p _z sur les valeurs des intégrales qui interviennent dans le calcul de la séparation des niveaux d'un triplet moléculaire en l'absence de champ magnétique. L'utilisation de l'orbitale de Slater classique ou de son équivalent en Gaussienne peut conduire à des conclusions appréciablement erronées.

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This work was supported by grant GM 12289-02 of the United States Public Health Service (National Institute of General Medical Sciences).     

An unusually large zero-field splitting in the molecule xanthione

An unusually large zero-field splitting of the lowest triplet state has been observed directly in the phosphorescence spectrum of the xanthione molecule in a xanthone host crystal. There are two xanthione guest sites in the host crystal. In one of the sites the zero-field parameter D is −11 cm⁻¹ and in the other it is −20 cm⁻¹. These results have been confirmed by measurements of the Zeeman splittings. It is argued that the large D values are due to spin-orbit coupling.     

Theoretical determination of the zero-field splitting in copper acetate monohydrate

The zero-field splitting of the copper acetate monohydrate complex is studied using wave function based calculations. The anisotropy parameters extracted from highly correlated methods are in excellent agreement with the most accurate experimental results; in particular, the negative sign of the axial anisotropy parameter D is reproduced. During several decades, the interpretation of experimental data based on an analytical expression derived from perturbation theory led to a positive D-value. Although the validity of this expression is confirmed, it is explained that the incorrect attribution of a positive D is related to the assumption of an antiferromagnetic coupling between excited states. We have found in the present work that this coupling is actually ferromagnetic. The analysis of the various contributions to the anisotropy parameters shows that both spin-spin and spin-orbit couplings participate in the magnetic anisotropy of this complex. Although the anisotropy arising from the spin-spin coupling is essentially independent of the level of calculation, the zero-field-splitting parameters resulting from the spin-orbit coupling are strongly sensitive to the effects of dynamic correlation. This works provides important new insights into the physical origin of the zero-field-splitting parameters in copper dimers.     

Optical determination of the single-ion zero-field splitting in large spin clusters

The dodecametallic Cr(III) cluster has an S = 6 ground state with an axial zero-field splitting (ZFS) of DS=6 = +0.088 cm-1. Analysis of high-resolution optical data (MCD) allows us to determine the single-ion ZFS of the constituent Cr(III) ions directly (D = -1.035 cm-1). A vector coupling analysis demonstrates that the cluster ZFS is almost entirely single-ion in origin. Thus, the relative orientations of the local and cluster magnetic axes can lead to cluster ZFS of opposite sign to the single-ion even when this is the only significant contribution.     

Ab initio studies on the zero-field splitting parameters of manganese porphyrin complexes

The magnetic property of a one-dimensional magnetic chain, 5,10,15,20-tetrakis(4-bromophenyl)porphyrinatomanganese(III) tetracyanoethenide ([MnTBrPP]⁺[TCNE]⁻), is investigated by using model complexes and ab initio calculations. In these models, MnTBrPP is reduced to a manganese porphyrin complex (MnP). The spin-polarized density functional theory (UDFT) and the hybrid UDFT were used to calculate the complexes, and Pederson’s scheme was used to calculate their zero-field splitting (ZFS) parameters. We found from the model calculations that the TCNE coordination hardly affects the magnetic anisotropy of MnP.     

Paramagnetism at ambient temperature, diamagnetism at low temperature in a Ru2(6+) core: structural evidence for zero-field splitting

Variable temperature magnetic studies of the Ru(2)(6+) guanidinate compounds Ru(2)(hpp)(4)Cl(2) (1) and Ru(2)(hpp)(4)(CF(3)SO(3))(2) (2) show that they are paramagnetic with two unpaired electrons at room temperature and that they appear essentially diamagnetic at 2 K. In neither compound do the Ru-Ru distances vary by more than 0.008(1) A from 27 to 296 K. This argues strongly that the ground state electronic configuration remains constant over this temperature range and that the decrease in magnetism as the temperature is lowered must be attributable to zero-field splitting of the (3)A(2g) ground state arising from the electronic configuration sigma(2)pi(4)delta(2)pi(2). The Ru-Ru distance in 1 is about 0.04-0.05 A longer than that in 2 which indicates that the Ru(2)(hpp)(4)(2+) core is quite sensitive to the nature of the axial ligands. The electronic spectra show three absorption bands for each compound.     

Definitive spectroscopic determination of zero-field splitting in high-spin cobalt(II)

A high-spin Co(II) complex (3d(7), S = 3/2), Co(PPh(3))(2)Cl(2) (Ph = phenyl), has been investigated in the solid state by both high-frequency and -field electron paramagnetic resonance (HFEPR) and by variable-temperature, variable-field magnetic circular dichroism (VTVH-MCD). In HFEPR spectroscopy, the combination of variable sub-THz frequencies generated by backward wave oscillators (150-700 GHz, corresponding to energy 5-23 cm(-1)) and high magnetic fields (0-25 T) constitutes a novel experimental technique allowing accurate determination of a complete set of spin Hamiltonian parameters for this complex: D = -14.76(2) cm(-1), E = 1.141(8) cm(-1), g(x) = 2.166(4), g(y) = 2.170(4), g(z) = 2.240(5). Independent VTVH-MCD studies on multiple absorption bands of the complex yield D = -14(3) cm(-1), E = 0.96(20) cm(-1) (absolute value of E/D = 0.08(2)), g(x) = 2.15(5), g(y) = 2.16(4), and g(z) = 2.17(3). This very good agreement between HFEPR and MCD indicates that there is no inherent discrepancy between these two quite different experimental techniques. Thus, depending on the nature of the sample, either can be reliably used to determine zero-field splitting parameters in high-spin Co(II), with the HFEPR being more accurate but VTVH-MCD being more sensitive.     

Calculations on the zero-field splitting parameters of the phosphrescent triplet states of nitrogen containing aromatics

The semi-empirical method introduced by van der Waals and ter Maten for calculating ZFS parameters of phosphorescent homocyclic aromatic hydrocarbons is extended to nitrogen containing analogs. The theoretically obtained ZFS values are compared with the experimental ones. The reliability of the extended method is briefly discussed.     

Proof of large positive zero-field splitting in a Ru2 5+ paddlewheel

We present the synthesis, as well as the structural and magnetic characterization, of [Ru2(D(3,5-Cl2Ph)F)4Cl(0.5H2O)].C6H14 (D(3,5-Cl2Ph)F = N,N'-di(3,5-dichlorophenyl)formamidinate), a Ru2(5+) compound having a 4B(2u) ground state derived from a sigma2pi4delta2pi2delta electron configuration. The persistence of this configuration from 27 to 300 K is shown by the invariance of the Ru-Ru distance. Orientation-dependent magnetic susceptibility (chiT) and magnetization (M(H)) data are in accord with a spin quartet ground state with large magnetocrystalline anisotropy associated with a large axial zero-field splitting (D) parameter. Theoretical fits to chiT and M(H) plots yielded D/kB = +114 K, implying an S = +/-1/2 Kramers doublet ground state at low temperature. Single-crystal and powder EPR data are consistent with this result, as the only observed transition is between the M(s) = +/1/2 Zeeman levels. The g values are g(perpendicular) = 2.182, g(parallel) = 1.970, and D = 79.8 cm(-1). The totality of the results demands D > 0.     

Calculation of the zero-field splitting tensor on the basis of hybrid density functional and Hartree-Fock theory

The zero-field splitting (ZFS) (expressed in terms of the D tensor) is the leading spin-Hamiltonian parameter for systems with a ground state spin S>12. To first order in perturbation theory, the ZFS arises from the direct spin-spin dipole-dipole interaction. To second order, contributions arise from spin-orbit coupling (SOC). The latter contributions are difficult to treat since the SOC mixes states of different multiplicities. This is an aspect of dominant importance for the correct prediction of the D tensor. In this work, the theory of the D tensor is discussed from the point of view of analytic derivative theory. Starting from a general earlier perturbation treatment [F. Neese and E. I. Soloman, Inorg. Chem. 37, 6568 (1998)], straightforward response equations are derived that are readily transferred to the self-consistent field (SCF) Hartree-Fock (HF) or density functional theory (DFT) framework. The main additional effort in such calculations arises from the solution of nine sets of nonstandard coupled-perturbed SCF equations. These equations have been implemented together with the spin-orbit mean-field representation of the SOC operator and a mean-field treatment of the direct spin-spin interaction into the ORCA electronic structure program. A series of test calculations on diatomic molecules with accurately known zero-field splittings shows that the new approach corrects most of the shortcomings of previous DFT based methods and, on average, leads to predictions within 10% of the experimental values. The slope of the correlation line is essentially unity for the B3LYP and BLYP functionals compared to approximately 0.5 in previous treatments.     

The temperature dependence of parachor

The results obtained in studying the temperature dependence of parachor are presented. The concepts of the parachor at the critical point and reduced parachor are introduced. The approach based on scaling principles and the assumption of a universal character of changes in crossover functions at critical exponents was used to obtain a universal temperature dependence of the reduced parachor. The dependence of the parachor at the critical point on the critical molar volume was established. The constituents of parachor were determined for halogenated hydrocarbons.     

Magnetism down to 0.90 K, zero-field splitting, and the ligand field in chromocene

The magnetism of chromocene, Cr(cp)₂ where CP = C₅H⁻₅, has been measured as a function of temperature between 0.90 and 303.2 K. The results are reproduced by complete ligand field theory in slightly distorted C_∞v symmetry (Dt = 1153, Ds = 4212, Dq = 28, ζ = 67, B = 553 cm⁻¹, C/B = 4 and k = 0.37). The ground state ³E₂(a₁e³₂) shows a zero-field splitting, D = 15.1 cm⁻¹, E = 0.     

A systematic density functional study of the zero-field splitting in Mn(II) coordination compounds

This work presents a detailed evaluation of the performance of density functional theory (DFT) for the prediction of zero-field splittings (ZFSs) in Mn(II) coordination complexes. Eighteen experimentally well characterized four-, five-, and six-coordinate complexes of the general formula [Mn(L)nL'2] with L' = Cl, Br, I, NCS, or N3 (L = an oligodentate ligand) are considered. Several DFT-based approaches for the prediction of the ZFSs are compared. For the estimation of the spin-orbit coupling (SOC) part of the ZFS, it was found that the Pederson-Khanna (PK) approach is more successful than the previously proposed quasi-restricted orbitals (QRO)-based method. In either case, accounting for the spin-spin (SS) interaction either with or without the inclusion of the spin-polarization effects improves the results. This argues for the physical necessity of accounting for this important contribution to the ZFS. On average, the SS contribution represents approximately 30% of the axial D parameters. In addition to the SS part, the SOC contributions of d-d spin flip (alphabeta) and ligand-to-metal charge transfer excited states (betabeta) were found to dominate the SOC part of the D parameter; the observed near cancellation between the alphaalpha and betaalpha parts is discussed in the framework of the PK model. The calculations systematically (correlation coefficient approximately 0.99) overestimate the experimental D values by approximately 60%. Comparison of the signs of calculated and measured D values shows that the signs of the calculated axial ZFS parameters are unreliable once E/D > 0.2. Finally, we find that the calculated D and E/D values are highly sensitive to small structural changes. It is observed that the use of theoretically optimized geometries leads to a significant deterioration of the theoretical predictions relative to the experimental geometries derived from X-ray diffraction. The standard deviation of the theoretical predictions for the D values almost doubles from approximately 0.1 to approximately 0.2 cm-1 upon using quantum chemically optimized structures. We do not find any noticeable improvement in considering basis sets larger than standard double- (SVP) or triple-zeta (TZVP) basis sets or using functionals other than the BP functional.     

Studies of the electric-field-induced zero-field splitting for a substitution Mn2+ ion in SrCl2 crystal

The zero-field splittings induced by the electric-field have been studied theoretically from the formulas, including three important microscopic mechanisms for a substitutional Mn2+ ion in SrCl2 crystal. From the studies, it is found that the zero-field splitting can be attributed primarily to the electric-field-induced displacement of Mn2+ ion along the electric-field direction and the magnitudes of the displacements at various strengths of electric-field are obtained. The results are comparable with those obtained from the potential-energy curve of Mn2+ center.