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-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-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-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-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.     

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.     

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.     

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.     

Chemical Bonds and Spin State Splittings in Spin Crossover Complexes. A DFT and QTAIM Analysis

Summary. Density functional theory (DFT) calculations have been performed for the high-spin (HS) and low-spin (LS) isomers of a series of iron(II) spin crossover complexes with nitrogen ligands. The calculated charge densities have been analyzed in the framework of the quantum theory of atoms in molecules (QTAIM). For a number of iron(II) complexes with substituted tris(pyrazolyl) ligands the energy difference between HS and LS isomers, the spin state splitting, has been decomposed into atomic contributions in order to rationalize changes of the spin state splitting due to substituent effects.     

Calorimetric measurements of diluted spin crossover complexes [FexM1−x(btr)2(NCS)2]·H2O with M = Zn and Ni

Motivated by the application of the First Order Reversal Curve (FORC) technique to obtain a distribution of like spin domains in the spin crossover complexes [Fe_xM_1−x(btr)₂(NCS)₂]·H₂O with M^II = Zn and Ni, we have performed Differential Scanning Calorimetry (DSC) experiments. We have measured the heat capacity in the heating and cooling modes and then calculated the enthalpy and the entropy variations at the phase transition. In the framework of an Ising-like model, we have converted the experimental thermal hysteresis into terms of physical parameters related to these compounds. We have also performed a statistical analysis of these parameters and concluded that the correlation between them increases with the dilution degree of these spin crossover complexes.     

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.     

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.     

Experimental and Computational Models for Side Chain Discrimination in Peptide–Protein Interactions

A bis(18-crown-6) Tröger's base receptor and 4-substituted hepta-1,7-diyl bisammonium salt ligands have been used as a model system to study the interactions between non-polar side chains of peptides and an aromatic cavity of a protein. NMR titrations and NOESY/ROESY NMR spectroscopy were used to analyze the discrimination of the ligands by the receptor based on the substituent of the ligand, both quantitatively (free binding energies) and qualitatively (conformations). The analysis showed that an all-anti conformation of the heptane chain was preferred for most of the ligands, both free and when bound to the receptor, and that for all of the receptor-ligand complexes, the substituent was located inside or partly inside of the aromatic cavity of the receptor. We estimated the free binding energy of a methyl- and a phenyl group to an aromatic cavity, via CH-π, and combined aromatic CH-π and π-π interactions to be −1.7 and −3.3 kJ mol⁻¹, respectively. The experimental results were used to assess the accuracy of different computational methods, including molecular mechanics (MM) and density functional theory (DFT) methods, showing that MM was superior.