The Role of Binary and Many-centre Molecular Interactions in Spin Crossover in the Solid State. Part I. Equation for Free Energy

A formalism has been developed that describes spin crossover equilibrium in the solid state by taking into account the effects of n nearest neighbours of a given molecule on its partition function. In this way binary and many-body interactions of the order n + 1 are included into the theoretical model and represented by non-ideality parameters connected with the splitting of free energy levels. Binary interactions are characterised by the main splittings whereas higher order interactions manifest themselves in asymmetries of splittings within multiplets. The contribution of molecular interactions can also be written in terms of formal excess free energies of the second, third, fourth and higher orders. Simple relationships between excess free energies and parameters of multiplets have been found for binary, ternary and quaternary interactions. This formalism is reduced to that of the model of binary interactions when effects of surroundings are additive leading to equidistant free energy multiplets. Higher order interactions may cause an abrupt spin crossover but in a limited range of compositions around the transition point. The regression of experimental transition curves of one-step spin crossover may yield estimates of excess energies up to the fifth order.     

The Role of Binary and Many-centre Molecular Interactions in Spin Crossover in the Solid State. Part I. Equation for Free Energy

Summary. A formalism has been developed that describes spin crossover equilibrium in the solid state by taking into account the effects of n nearest neighbours of a given molecule on its partition function. In this way binary and many-body interactions of the order n + 1 are included into the theoretical model and represented by non-ideality parameters connected with the splitting of free energy levels. Binary interactions are characterised by the main splittings whereas higher order interactions manifest themselves in asymmetries of splittings within multiplets. The contribution of molecular interactions can also be written in terms of formal excess free energies of the second, third, fourth and higher orders. Simple relationships between excess free energies and parameters of multiplets have been found for binary, ternary and quaternary interactions. This formalism is reduced to that of the model of binary interactions when effects of surroundings are additive leading to equidistant free energy multiplets. Higher order interactions may cause an abrupt spin crossover but in a limited range of compositions around the transition point. The regression of experimental transition curves of one-step spin crossover may yield estimates of excess energies up to the fifth order.     

The Role of Binary and Many-center Molecular Interactions in Spin Crossover in the Solid State. Part III. Expressing Many-body Interactions

The approach of molecular potentials describing the shape of transition curves of spin crossover in the solid state developed earlier has been extended to many-body interactions characterized by the Axilrod-Teller potential. An improved procedure for the minimization of energy developed for this case is presented. Calculations for systems involving Lennard-Jones, electric dipole–dipole, and dispersive Axilrod-Teller triple interactions yield non-zero asymmetries of splittings in expanded/compressed systems alone. The excess energy is unaffected by the Axilrod-Teller potential. Triple interactions of the Axilrod-Teller type thus increase the sensitivity of a transition curve towards compression. Another approach presented employs the deviations of molecules from positions of mechanical equilibrium set up by the known binary potential. In the approximation of small perturbations these deviations are proportional to the gradients of many-center potentials. This allows one to parametrically define non-ideality parameters as functions of gradients of triple potentials of unknown types. Employing regularization bounds an adequate parameterization of experimental transition curve of spin crossover has been achieved in terms of parameters of Lennard-Jones potential and relative deviations of molecules from the position of mechanical equilibrium.     

The Role of Binary and Many-center Molecular Interactions in Spin Crossover in the Solid State. Part III. Expressing Many-body Interactions

Summary. The approach of molecular potentials describing the shape of transition curves of spin crossover in the solid state developed earlier has been extended to many-body interactions characterized by the Axilrod-Teller potential. An improved procedure for the minimization of energy developed for this case is presented. Calculations for systems involving Lennard-Jones, electric dipole–dipole, and dispersive Axilrod-Teller triple interactions yield non-zero asymmetries of splittings in expanded/compressed systems alone. The excess energy is unaffected by the Axilrod-Teller potential. Triple interactions of the Axilrod-Teller type thus increase the sensitivity of a transition curve towards compression. Another approach presented employs the deviations of molecules from positions of mechanical equilibrium set up by the known binary potential. In the approximation of small perturbations these deviations are proportional to the gradients of many-center potentials. This allows one to parametrically define non-ideality parameters as functions of gradients of triple potentials of unknown types. Employing regularization bounds an adequate parameterization of experimental transition curve of spin crossover has been achieved in terms of parameters of Lennard-Jones potential and relative deviations of molecules from the position of mechanical equilibrium.     

The Role of Binary and Many-centre Molecular Interactions in Spin Crossover in the Solid State. Part II. Non-ideality Parameters Defined via Binary Molecular Potentials

Summary. Parameters of the formalism [1–6] describing spin crossover in the solid state have been defined via molecular potentials in model systems of neutral and ionic complexes. In the first instance Lennard-Jones and electric dipole–dipole potentials have been used whereas in ionic systems Lennard-Jones and electric point-charge potentials have been used. Electric dipole–dipole interaction of neutral complexes brings about a positive excess energy controlled by the difference of electric dipole moments of HS and LS molecules. Differences of the order of Δμ = 1–2 D cause an abrupt spin crossover in systems with T_1/2 = 100–150 K. Magnetic coupling contributes both to the excess energy and excess entropy, however the overall effect is equivalent to a modest positive excess energy. Ionic systems in the absence of specific interactions are characterised by very small excess energies corresponding to practically linear van’t Hoff plots. Detectable positive and negative excess energies in these systems may arise from interactions of ligands belonging to neighbouring complexes. The HOMO–LUMO overlap in HS–LS pairs can bring about a nontrivial variation of the shape of transition curves. Examples of regression analysis of experimental transition curves in terms of molecular potentials are given.     

The Role of Binary and Many-centre Molecular Interactions in Spin Crossover in the Solid State. Part II. Non-ideality Parameters Defined via Binary Molecular Potentials

Parameters of the formalism [1–6] describing spin crossover in the solid state have been defined via molecular potentials in model systems of neutral and ionic complexes. In the first instance Lennard-Jones and electric dipole–dipole potentials have been used whereas in ionic systems Lennard-Jones and electric point-charge potentials have been used. Electric dipole–dipole interaction of neutral complexes brings about a positive excess energy controlled by the difference of electric dipole moments of HS and LS molecules. Differences of the order of Δμ = 1–2 D cause an abrupt spin crossover in systems with T_1/2 = 100–150 K. Magnetic coupling contributes both to the excess energy and excess entropy, however the overall effect is equivalent to a modest positive excess energy. Ionic systems in the absence of specific interactions are characterised by very small excess energies corresponding to practically linear van’t Hoff plots. Detectable positive and negative excess energies in these systems may arise from interactions of ligands belonging to neighbouring complexes. The HOMO–LUMO overlap in HS–LS pairs can bring about a nontrivial variation of the shape of transition curves. Examples of regression analysis of experimental transition curves in terms of molecular potentials are given.     

Molecular statistical model of spin crossover equilibrium in the crystal state taking into account the phenomenon of ordering

Formalism is developed in which contributions of binary and ternary interactions towards free energy of a mixture of low-spin (A) and high-spin (B) isomers of spin crossover compounds as well as effects of ordering are taken into account. Parameters characterising non-ideality in this formalism are the excess free energy (ΔF _ex) and absolute asymmetries (Δ ^A , Δ ^B ) of splittings of free energy levels. The excess free energy characterises the effects of binary interactions whereas asymmetries arise from ternary interactions. According to this model, the plateau in the spin crossover transition curve originates from the phenomenon of ordering taken into account in the Gorsky-Bragg-Williams approximation.     

Why the model of non-interacting chains yields adequate description of spin crossover in space structures?

Considering many-body interactions in tetrahedral structures as perturbations of binary potentials by third bodies yields a free energy functional of the binary mixture equivalent to one earlier derived for spin crossover equilibrium in one-dimensional chains. Formal non-ideality parameters of this functional, the excess energy and asymmetries of splittings can be expressed in terms of molecular parameters based on binary potentials.     

Free energy functional of the Ising-like model of spin crossover

Abstract  

The model of spin crossover based on the Ising-like Hamiltonian (IHM) has been analysed by deriving the functional of free energy from the mean-field solutions of this Hamiltonian. The contribution of the configurational entropy was found to be identical to that in the functional of the molecular statistical model (MSM) of spin crossover. However, the polynomial expansion over composition (x _B) and degree of order (s _B) in these functionals differ fundamentally due to different ways of accounting for the effects of molecular interactions. It was found that IHM takes into account next-to-nearest neighbour interactions by introducing affinities of sublattices towards molecules of given kinds. This yields a term proportional to the first power of the degree of order in the functional of IHM, whereas the MSM free energy is only proportional to s _B². The choice of formal independent variables does not affect the results of simulations of transition curves provided the functional remains unaltered. This provides for more flexibility in numerical simulations of transition curves.     

Phase diagrams of binary crystalline-crystalline polymer blends

A thermodynamically self-consistent theory has been developed to establish binary phase diagrams for two-crystalline polymer blends by taking into consideration all interactions including amorphous-amorphous, crystal-amorphous, amorphous-crystal, and crystal-crystal interactions. The present theory basically involves combination of the Flory-Huggins free energy for amorphous-amorphous isotropic mixing and the Landau free energy of polymer solidification (e.g., crystallization) of the crystalline constituents. The self-consistent solution via minimization of the free energy of the mixture affords determination of eutectic, peritectic, and azeotrope phase diagrams involving various coexistence regions such as liquid-liquid, liquid-solid, and solid-solid coexistence regions bound by liquidus and solidus lines. To validate the present theory, the predicted eutectic phase diagrams have been compared with the reported experimental binary phase diagrams of blends such as polyethylene fractions as well as polycaprolactone/trioxane mixtures.     

Interactions between frenkel excitons. I. Formalism and preliminary results

For the tightest-bound Frenkel excitons a many-particle hamiltonian is adopted which includes terms representing interactions between the excitons. The parameters of the model are the excitation energy of an isolated localized excitation, the excitation-transfer matrix elements, and the interaction energies. Several special cases, involving particular relations between the latter two sets of parameters, are treated qualitatively. Equations are translated into the language of spin-$\text{1}\text{2}$ lattices, so that use may be made of the results of that theory. Favorable conditions for observation of polyexcitons and of phase transition to a liquid of excitons are discussed. A standard formalism for the determination of absorption and emission spectra and of their moments is adapted to the present problem. Possible generalizations of the model are briefly discussed.     

Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S2 eigenfunctions

We present an implementation of the spin‐dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry‐adapted functions. Using S² eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin‐independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one‐ and two‐electron charge densities.     

Density and Viscosity of Alkyl Acetates + Aromatic Hydrocarbons at (303.15 and 313.15) K

Densities and viscosities have been determined for binary mixtures of isopropyl acetate or isobutyl acetate with o-xylene, m-xylene, p-xylene and ethyl benzene at (303.15 and 313.15) K for the entire composition range. The excess molar volumes and deviations in viscosity have been calculated from the experimental values. The variations of these parameters, with composition of the mixtures and temperature, have been discussed in terms of molecular interactions occurring in these mixtures. Further, the viscosities of these binary mixtures were calculated theoretically from their corresponding pure component data by using empirical relations, and the results were compared with the experimental findings.     

Construction of Giant‐Spin Hamiltonians from Many‐Spin Hamiltonians by Third‐Order Perturbation Theory and Application to an Fe3Cr Single‐Molecule Magnet

A general giant‐spin Hamiltonian (GSH) describing an effective spin multiplet of an exchange‐coupled metal cluster with dominant Heisenberg interactions was derived from a many‐spin Hamiltonian (MSH) by treating anisotropic interactions at the third order of perturbation theory. Going beyond the existing second‐order perturbation treatment allows irreducible tensor operators of rank six (or corresponding Stevens operator equivalents) in the GSH to be obtained. Such terms were found to be of crucial importance for the fitting of high‐field EPR spectra of a number of single‐molecule magnets (SMMs). Also, recent magnetization measurements on trigonal and tetragonal SMMs have found the inclusion of such high‐rank axial and transverse terms to be necessary to account for experimental data in terms of giant‐spin models. While mixing of spin multiplets by local zero‐field splitting interactions was identified as the major origin of these contributions to the GSH, a direct and efficient microscopic explanation had been lacking. The third‐order approach developed in this work is used to illustrate the mapping of an MSH onto a GSH for an trigonal Fe₃Cr complex that was recently investigated by high‐field EPR spectroscopy. Comparisons between MSH and GSH consider the simulation of EPR data with both Hamiltonians, as well as locations of diabolical points (conical intersections) in magnetic‐field space. The results question the ability of present high‐field EPR techniques to determine high‐rank zero‐field splitting terms uniquely, and lead to a revision of the experimental GSH parameters of the Fe₃Cr SMM. Indeed, a bidirectional mapping between MSH and GSH effectively constrains the number of free parameters in the GSH. This notion may in the future facilitate spectral fitting for highly symmetric SMMs.     

Spin-Spin Interactions of Trapped Radicals Produced in 3-Methylpentane by Gamma-Irradiation at 77°K

EPR microwave power saturation technique has been applied to the systems of radiation-induced radicals in protiated and wholely deuterated 3-methylpentane to obtain the information concerning the high local spin concentration and radiation energy deposition mechanism. Evidences of spin-spin interactions have been given from the studies of isothermal decay of these trapped radicals at 77°K. The annealing of the matrix is shown to have almost no effect on the relaxation times of radicals. The influence of nuclear magnetic moment on the spin-lattice relaxation process of radicals is revealed and discussed on the basis of experimental results from the deuteration of the compound studied.