Molecules with two electronic energy levels: coupling between the molecules in the solid state via the optical and acoustic phonon branches

In the adiabatic approximation the values of the spring constants of the springs contained in a molecule depend on its electronic state. We consider molecules with two electronic energy levels separated by Δ. For a crystal of such molecules, the phonon branches depend therefore on the electronic states of the molecules. One can ask if that dependence does not introduce a coupling between the molecules via the optical and the acoustic branches. It is known that for a one-dimensional chain of N identical diatomic molecules there are two phonon branches, an optical branch and an acoustic one. In this study we introduce in the hamiltonian of the chain two assumptions: (i) each molecule has two electronic energy levels separated by Δ and the spring constant of the spring contained in the molecule has a value which depends on its electronic state; (ii) the spring constant of the spring which links two molecules nearest neighbours has a value which depends on the electronic states of both molecules linked. One can show that phonons create on each molecule a field-like which favours the excited level and create between two molecules nearest neighbours an exchange-like interaction which can be ferro-like, antiferro-like and which can be equal to zero. For some values of T and Δ, the chain can display a first-order phase transition with the presence of a thermal hysteresis loop. The phase transition takes place between the phase where all the molecules are in the fundamental level and that where they are in the excited one. The parameters of the model can be expressed in function of the applied pressure and of the volume of the crystal.


Introduction
In the adiabatic approximation the values of the spring constants of the springs contained in a molecule depend on its electronic state. We first recall this point.
Let us consider the case of a diatomic molecule. There are three steps in the study of the Hamiltonian of the molecule: i) Movement of the electrons in presence of the fixed nuclei As the masses of the nuclei are much more higher than the mass of the electron and as the forces acting on the nuclei and those acting on the electrons have nearly the same intensity, the speed of the nuclei is much more smaller than that of the electrons. So we can first neglect the kinetic energy of the nuclei. We then obtain the electronic Hamiltonian, H elec , which describes the movement of the electrons in presence of the fixed nuclei. In this a e-mail: jamil.nasser@uvsq.fr Hamiltonian, the nuclei interaction, 1 4π 0 ZZ e 2 R , is constant because R, the distance between the two nuclei, is constant. However, the eigenvalues and the associated eigenfunctions of H elec depend on R as a parameter. We recall that the eigenfunction ϕ p (R, r), with p = 1, 2,. . . , is a quantum state (more precisely a stationary state) of the electrons of the molecule and that the associated eigenvalue, E p (R), is the energy of this stationary state. The parameter r represents the coordinates of all the electrons of the molecule.
ii) The adiabatic approximation If the nuclei move, R becomes a function of time, R (t). It is known that an interaction which (explicitly) depends on time can induce transitions between the stationary states of a system. So, if the stationary state of the electrons is ϕ i at the time t 0 it can be ϕ j at a time t > t 0 . However, as the speeds of the nuclei are very small, we can consider that the stationary state of the electrons does not change with time. That is, the adiabatic approximation [1].
iii) Hamiltonian of the nuclei in the adiabatic approximation In the adiabatic approximation, the Hamiltonian of the nuclei is approximativly if the stationary state of the electrons is ϕ p (R, r). In relation (1), T N is the kinetic energy of the nuclei [1]. As there is a chemical bond between the two atoms, the function E p (R) has a minimum for R = R 0p . The limited development of E p (R) in the vicinity of the minimum is If we neglect terms (R − R 0p ) raised to a power higher than two, we obtain In relation (3), E p (R 0p ) is a constant which modifies the origin of energy and 1 2 E p (R 0p ) (R − R 0p ) 2 is an harmonic oscillator potential. It is the potential energy of a spring which has an equilibrium length R 0p and a spring constant E p (R 0p ). So, one can say that two atoms linked by a chemical bond are linked by a spring-like, the spring constant of which depends on the stationary state of the electrons of two atoms. As shown in relation (3) this spring constant value is the value of the second derivative of the function E p (R) for R = R 0p . From statistical mechanics results it is known that only the fundamental level, E 1 (R), and the first excited levels, E 2 (R), E 3 (R),. . . are to be taken into account in many studies. The points (i) and (ii) of the previous method are also called the Born-Oppenheimer approximation.
The kinetic energy T N contains the translational, the rotational and vibrational kinetic energy of the molecule. We remove the translational kinetic energy by using the frame of reference of center-of-mass. If the variations of R − R 0p are small, the movements of rotation and vibration of the molecule are independent. We then can obtain H vib the Hamiltonian of vibrations of the molecule. The study of H vib in classical mechanics leads to with where m R is the reduced mass of the masses of the two atoms. The study of H vib in quantum mechanics shows that the eigenvalues of this Hamiltonian are where n = 0, 1, 2,. . . In a molecule containing a number N of atoms, the nuclei are linked by spring-like which have different spring constants. That is a system of harmonic oscillators in interaction. In classical mechanics, one can show that all the nuclei can oscillate around their equilibrium position with the same frequency value ω α , with α = 1, 2,. . . , s and s < 3N . A frequency value ω α is called an eigenfrequency of the molecule or the frequency value of the normal vibration mode (α). In quantum mechanics, the values ω α appear in the expression of the eigenvalues of the nuclei Hamiltonian.
In a crystal, the equilibrium positions of the molecules (and of the atoms of the molecules) are periodic in space. There are two kinds of springs: those which are inside the molecules and those which link the molecules. Due to the periodicity, the eigenfrequencies of the crystal are related to the values of the components (q x , q y , q z ) of a vector − → q , called wave vector. But, many different values ω α can be associated to the same value of the wave vector − → q .
However, it is possible to group the values ω α in different branches: the values ω α which are related to the wave vector components by the same mathematical relation belong to the same branch. This mathematical relation is called the dispersion relation of the branch. If the crystal is a linear chain the wave vector has only one coordinate q x or simply q. We consider molecules with two electronic energy levels separated by ∆, which is of the order of 300 K. Such molecules can be found in various complexes of transition metal ions [2]. We therefore can assume that the values of the spring constants of the springs contained in one molecule are not the same in both levels. Consequently, in the solid state, for a crystal of N identical molecules, the phonon branches depend on the electronic states of the molecules, that is on n ex , the fraction of molecules in the excited level. One can ask if that dependence does not introduce a coupling between the molecules via the optical and the acoustic branches. It seems clear that the optical branches which do not display a dispersion cannot contribute to such a coupling. Indeed, they correspond to springs well localized inside the molecules. In this study we consider the case of an optical branch which displays dispersion and the case of an acoustic branch.
The study of the thermal variation of n ex gives informations on the presence or not of a coupling. If there is not a coupling, the distance between the two energy levels is constant and the thermal variation of n ex is well known. For the following, we call (a) and (b) the fundamental and the excited electronic levels, respectively. So, n ex is replaced by n b . The levels (a) and (b) are obtained by quantum mechanics calculations.
It is known that for a one-dimensional chain of N identical diatomic molecules there are two phonon branches, an optical branch and an acoustic one. Those results are obtained with the following assumptions: the molecules axes are parallel to the chain, the chain is periodic at the equilibrium, the atoms displacements from their equilibrim positions are longitudinal, along the chain, there is a spring between the two atoms of a molecule and one between two molecules nearest neighbours.
In this article, we study the previous linear chain with the following supplementary assumptions: i) Each molecule has two electronic energy levels separated by ∆ and the spring constant of the spring contained in the molecule has a value which depends on its electronic state. This value is k when the molecule is in the (a) level and k when it is in the (b) level.
ii) The spring constant of the spring which links two molecules nearest neighbours has three values: λ when they are both in the (a) level , ν when they are both in the (b) level and µ when they are not in the same electronic level.
Ronayne et al. [3] have studied the frequency values of the normal modes of vibration of the complex [F e(phen) 2 (N CS) 2 ] which has two electronic energy levels. They found that the majority of the frequency values are lower in the excited level than in the fundamental one. Studying the heat capacity of a crystal of this molecule, Sorai and Seki [4] have concluded that the excitation of phonons is much easier when the crystal is in the (b) phase than when it is in the (a) phase . In the thermodynamic (b) phase (resp. (a)) all the molecules (or the majority of them) are in the (b) level (resp. (a)).
Following those results we assume that and λ ≥ µ ≥ ν.
It is known that a linear chain of N identical atoms have only one phonon branch which is an acoustic branch. This result is obtained with the following assumptions: the chain is periodic at the equilibrium, the atoms displacements from their equilibrium positions are longitudinal, along the chain, there is a spring between two atoms nearest neighbours.
Nasser [5] has studied this chain of atoms with the following supplementary assumptions: i) Each atom has two electronic energy levels separated by ∆. The fundamental and the excited levels are called (a) and (b), respectively.
ii) The spring constant of the spring which links two atoms nearest neighbours has three values: λ when they are both in the (a) level , ν when they are both in the (b) level and µ when they are not in the same electronic level. Moreover, the values λ, µ, ν verify relation (8).
The study of this atom-phonon coupling model shows that the acoustic phonons create a coupling between the atoms of the chain. Due to this coupling the chain can display a first-order phase transition when the chain temperature is lower than the value of the critical temperature T C . Consequently, if the value of T C is too small, the experimental study of this first-order phase transition is not easy and not necessary interesting.
Different studies have been done with this atom-phonon coupling model in 1D or 2D crystal [6][7][8][9]. But in those studies the authors have to take small values, near 0.2, for the ratio ν λ in order to have a critical temperature value not too small.
One of aims of this study is to see if there is a coupling when the values λ, µ and ν are equal while the values k and k verify relation (7). In that case only the value of the spring constant of the springs contained in the molecules varies with the electronic state of the molecules. The other aim is to see how the value of T C varies when the value of the spring constant of all springs varies with the electronic state of the molecules.
The free energy of an harmonic oscillator is where β = 1 k B T and k B is the Boltzmann constant. It is clear that a decrease in ω, the oscillator frequency value, at constant temperature lowers the value of the free energy. From this result, one can say that the assumptions done in relations (7) and (8) imply that phonons favour the (b) level. It is clear that the electronic parameter ∆ favours the (a) level. So, there is a competition between the phonons and the electronic interactions. This competition can lead to a first-order phase transition. It is worth to emphasize that the lattice vibrations of the crystal have to be studied by using quantum statistical mechanics [8].
In Section 2, we present the chain Hamiltonian and the variational method used to study it. In Section 3, we give the results obtained by numerical calculations and the last section is devoted to the conclusion.

Theoretical study 2.1 The model and the chain Hamiltonian
Let us consider a linear chain of N identical molecules each having two atoms A and B. The molecules are numbered along the chain from left to right. The ith molecule is called A i B i , i = 1-N . The axes of the molecules are parallel to the chain. At equilibrium the distance between A i and B i is d 1 and that between B i and A i+1 is d 2 . So the chain is periodic with the period d given by The atoms A i and B i are linked by a spring with spring constant k i , and the atoms B i and A i+1 are linked by a spring with spring constant e i,i+1 . When these atoms are displaced longitudinally along the chain from their equilibrium positions by an amount u i for the atoms A i and w i for the atoms B i the potential energy of the chain is where the sum is over the N molecules. We assume periodic conditions.

The vibrations Hamiltonian of the chain is
where E c is the total kinetic energy of the chain. Now we introduce the following assumptions: i) Each molecule has two electronic energy levels: a fundamental level called (a), with a degeneracy equal to the unit, and an excited one called (b), with a degeneracy equal to r. The separation between both levels is called ∆. So, the electronic Hamiltonian of the molecule i, can be written as where σ i is a fictitious spin associated with the molecule i which has two eigenvalues ±1, the eigenvalue −1 (resp. +1) corresponding to the electronic level (a) (resp. (b)). For the chain, the electronic Hamiltonian is given by ii) We assume that the spring constant k i has the value k when the molecule A i B i is in the (a) level and the value k when it is in the (b) level. This assumption can be written as iii) We assume that the spring constant e i,i+1 is equal to λ when the molecules A i B i and A i+1 B i+1 are both in the (a) level, to ν when they are both in the (b) level and to µ when they are not in the same level. Those assumptions can be written as The chain Hamiltonian is

Phonons-molecules interactions
By inserting equations (15) and (16) in the right hand of equation (11) the chain potential energy can be decomposed into three terms with

Zeeman-like interaction
The energy term V 1 corresponds to a Zeemann-like interaction. The fields-like acting on the spin σ i are and Both fields-like depend on the time, on the temperature and they are not uniform. They have the same sign and favour the (b) level, that is σ i = +1, because the coefficients ( k − k) and (ν − λ) are negative (here the Zeemann-like term is written as h i σ i ).
The field-like h imol comes from the spring contained in the molecule i. But the parameter (u i − w i ) 2 depends on the molecules i − 1, i and i + 1. The field-like h ilat comes from the springs which link the molecule i with the molecules i ± 1.

Exchange-like interaction
The term V 2 corresponds to an exchange-like interaction. The exchange-like constant is This interaction comes from the spring which links the molecules i and i + 1. It depends on the time, on the temperature and it is not uniform. The exchange-like constant is equal to zero when It corresponds to an antiferro-like parameter when Indeed, in that case, the exchange-like interaction favours the situation where two molecules nearest neighbors are not in the same electronic state ( σ i σ i+1 = −1). The exchange-like parameter is a ferro-like one when Indeed, in that case, the exchange-like interaction favours the situation where two molecules nearest neighbors are in the same electronic state ( σ i σ i+1 = 1). In this study the three previous parameters are replaced by parameters which do not depend on time and which are uniform.

Self-consistent equation
We apply now a variational method [10] and to this end we introduce three parameters, h, K and E. The first one h describes a uniform, effective field, the second one K is an effective spring constant that replaces the spring constants k i and the last one E is an effective spring constant that replaces the spring constants e i,i+1 . Those K and E do not depend on the electronic states of the molecules. The variational Hamiltonian H 0 is The spin Hamiltonian H 0sp (h) is With this spin Hamiltonian all the fictitious spins σ i have the same mean value m given by Equation (30) is called self-consistent equation. The free energy related to H 0sp (h) is where the partition function z 0sp is given by The phonon Hamiltonian H 0ph (K, E) is given by where The Hamiltonian H 0ph (K, E) is the phonon Hamiltonian of a linear chain of N identical diatomic molecules, A i B i with i = 1, . . . , N . The atoms A i and B i are linked by a spring with spring constant K and two molecules nearest neighbors are linked by a spring with spring constant E. It is known that there are two dispersion relations which correspond to the optical branch and to the acoustic branch of the chain. In the previous relation the vector − → q is the phonon wave vector. The previous dispersion relations are given in Appendix A.
With the Hamiltonian H 0ph (K, E), the mean values , respectively. We have and where − → q is the sum over the two phonon branches and ∂ω ∂K and ∂ω ∂E are the partial derivatives versus K and E, respectively, of the dispersion relations ω 1 ( − → q ) and ω 2 ( − → q ).
The expressions of the partial derivatives are given in Appendix A. The free energy related to H 0ph (K, E) is The free energy associated with the variational Hamiltonian H 0 is At the first-order perturbation calculation, and with where and The crystal free energy corresponding to the approximation done in this study, is The (b) level fraction is given by It increases from 0 to 1 when the parameter m varies from −1 to 1.
The sound velocity of longitudinal waves is

Discussion
i) We have verified that the value of the spring constant E is always positive. It decreases from λ to ν when m increases from −1 to 1. ii) From relation (34), the thermal mean value of the potential energy E p0 (K, E) is given by By inserting equations (35) and (36) in equation (48), we can verify the virial theorem iii) In relation (44) the exchange-like interaction is studied as a mean field approximation. This study method could be improved for a 1D system by using the transfer matrix method, if it is possible. In the antiferro-like case it could be interesting to introduce two sublattices.
We assume for relation (44) that the exchange-like interaction is small compared to the Zeemann-like interaction, that is where |λ − 2µ + ν| is the absolute value of the parameter λ − 2µ + ν. With this condition the parameter h phlat is always positive. iv) It is worth noticing that, in the expression of the uniform field h, the electronic term −∆ is negative while the phonon term h ph is positive.

Numerical study
The numerical study consists essentially in solving the self-consistent equation. For that, it is interesting to use reduced parameters.

Reduced parameters
We take λ as the unit of elastic force constant and ω M (λ) as the unit of energy with The value of ω M (λ) is roughly estimated to 1000 K or 695 cm −1 [8].
We introduce the following reduced parameters: the reduced temperature the reduced electronic energy gap the elastic force constants ratios and From condition (7) 0 ≤ xint ≤ 1, and from condition (8) 0 ≤ x ≤ 1.
It is interesting to introduce the parameter y by the relation The value µ varies from ν to λ when y varies from −1 to +1. For y = 0 the exchange-like parameter (see V 2 ) is equal to zero. It is an antiferro-like parameter when y < 0 and a ferro-like one when y > 0.
Using relation (59) we have So condition (50) can be written as

Values of the ratio m B /m A
We assume that m A + m B , the mass of the molecule, is 500 g mol −1 [11]. So, the ratio m B /m A is near 0.002 if the atom B is a hydrogen atom, and 0.02 if the atom B is a carbon or a nitrogen one. The good reduced parameter to use for studying the self-consistent equation is where m R is the reduced mass (see Appendix). The parameter p is nearly equal to the ratio m B /m A . In the following, we have studied both cases p = 0.002 and p = 0.02. The parameter p varies but the molecular mass is constant.

Study of the self-consistent equation
We When there are three solutions, the values of the free energy of the three solutions must be compared. The thermodynamic stable solution is the one with the lowest associated free energy. When the values of the free energy of two solutions are equal both solutions are stable and the chain shows a first, order phase transition. In that case, the third solution corresponds to an unstable state. When there are three solutions, the value of t is the value of the transition temperature and that of δ is the value of δ at the transition.
The two solutions which correspond to the same free energy are not equal. Let us call m b the higher value and m a the other. The discontinuity in m at the transition is defined by ∆m = m b − m a . It is known that the magnitude of this discontinuity decreases as the transition temperature increases and that it disappears at the critical temperature t C . We have used this property for finding the values of the critical point coordinates. In fact it is difficult to obtain near the critical point, the three solutions of the self-consistent equation. In this study, we consider that the critical point is reached when the discontinuity in m is lower than 0.210. So, the values of δ C and t C obtained in this study are somewhat smaller than the exact values.
We have studied for different values of the model parameters the phase diagram of the chain, the thermal variation of the (b) level fraction and that of the chain sound velocity.
For each solution, we have calculated numerically the ratio dF dm where dF is the variation of the free energy associated with dm, a small variation of m. We have verified that this ratio is positive for all the solutions except the one previously called unstable solution.
We recall that, in the theory of the first-order phase transition, the unstable solution plays role in the thermal hysteresis loop which appears around the transition temperature.

Results
In this study the length of the chain is N = 2000 and the degeneracy of the excited level is r = 5.

Case λ = µ = ν
In that case the field-like h phlat and the exchange-like interaction are equal to zero. Only the field-like h phmol created by the springs contained in the molecules plays role. This case has never been studied before.
Relation (43) can be written as From relation (63), the field-like h phmol depends on the parameters xint and z, and also on the mean value . This means value depends on the spring constants E and K which depend on all the model parameters. The chain phase diagram for xint = 0.4 is shown in Figure 1. The coordinates of the points of Figure 1 are the chain transition temperature and the value of δ at the transition. The (b) phase-(a) phase coexistence curve is ended by the critical point C with the coordinates (δ C , t C ). Its slope is positive in agreement with the Clapeyron equation [5]. The (a) phase is stable in the region above the coexistence curve and the (b) phase is stable below it.
The experimental study of the first-order phase transitions and that of the related effects cannot be done if the value of the critical temperature is too small. For this reason we have studied the variations of the reduced critical temperature with the model parameters. The variations of t C with xint is shown in Figure 2. For p = 0.002 the atom B is a hydrogen atom, for p = 0.02 it is a carbon or a nitrogen one. As shown in Figure 2, the t C value increases when xint decreases and it is divided by nearly 2.4 when the mass of the atom B is multiplied by 10.
The variations of t C with the parameter z is shown in Figure 3. In this Figure, xint = 0.4. The value of the ratio ∆t C ∆z is near 0.09 for p = 0.002 and near 0.03 for p = 0.02. In previous ratio, ∆t C is the variation of t C for the variation ∆z of z.
The thermal variation of n b is shown in Figure 4. For curve (1), k = k. So the field-like h phmol is equal to zero and the effective field h is equal to −∆. There is not an interaction between molecules. From equation (30)  deduce that m = 0 when t ln(r) = δ. In that case n b is equal to 0.5. With r = 5 and δ = 1.445, the (b) level fraction is equal to 0.5 for t = 0.9, as shown in Figure 4. For curves (2) and (3), k = 0.4 k. So the field-like h phmol is not equal to zero and it is the same in the curves (2) and (3). The presence or not of a discontinuity depends on the value of δ compared to the critical value δ C = 1.454. In curve (2), the value of δ being lower than δ C , there is a discontinuity; in curve (3) this value being higher than δ C , there is not a discontinuity. The thermal variation of the sound velocity is shown in Figure 5. The sound velocity varies when the spring constants E and K vary. The discontinuity of the sound velocity in the curve (2) corresponds to that of n b in the curve (2) of Figure 4. As shown in Figure 5, the value  In curve (1) there is not an interaction between the molecules and the distance between both energy levels is constant. In curves (2) and (3) there is an interaction between the molecules which leads to a discontinuity when the value of δ is lower than the critical value δC = 1.454. In this figure z = 1.
of the sound velocity is the same for the three curves at 0 K. This result is due to the fact that, at 0 K, all the molecules of the chain are in the fundamental level for the three curves, and consequently, the values of the spring constants E and K are the same for the three curves. The (b) level fraction and the sound velocity display a hysteresis loop around the transition temperature. The hysteresis loop of the sound velocity is shown in Figure 6. In this figure, heating the chain from the (a) phase, increases its temperature and it passes successively in the different states of the (a) phase which are stable. At the transition temperature value, t t , the chain stays in the (a) phase which is now metastable. It can stay in this phase up to the temperature value t up at which it is forced to pass in the (b) phase which is the stable one. In fact, the chain can pass in the stable phase at any temperature comprised between t t and t up . Cooling the chain from the (b) phase we find similar behaviour. (1) Sound velocity (arbitrary unit) Reduced temperature t  (1), the spring constants E and K do not vary with the temperature because there is not an interaction between the molecules. Consequently, the sound velocity is constant. In curves (2) and (3), there is an interaction between the molecules which leads to a thermal variation of E and K, and of the sound velocity.
There is a discontinuity when the value of δ is lower than the critical value δC = 1.454.  In that case the field-like created by phonons is the sum of the two field-like h phmol and h phlat . Using the reduced parameters and relation (60) in relation (44) the field-like h phlat can be written as This field-like depends on the parameters x, y and on the mean value (u i+1 − w i ) 2 0 . This mean value depends on all the model parameters.
We have verified that the value of the critical temperature due to both fields-like is higher than the sum of the values due to each field-like. For example: for (λ = µ = ν; z = 1, p = 0.002, xint = 0.4), the value of the critical temperature is t C (mol) = 0.07175. This value is due to the field-like h phmol alone. For ( x = 0.6 , y = 0; z = 1, p = 0.002, xint = 1), the value of the critical temperature is t C (lat) = 0.02758. This value is due to the field-like h phlat alone. For (x = 0.6, y = 0; z = 1, p = 0.002, xint = 0.4), both field-like are different of zero and the value of the chain critical temperature is 0.22067. This value is higher than the sum t C (mol) + t C (lat) = 0.09933. We can say that the presence of the two fields-like amplifies the value of the chain critical temperature.
The role of the parameter y is shown in Figure 7. The coordinates (δ C , t C ) of the critical point vary with the value of y.

Conclusions
In this study we have considered two kinds of springs: the springs which link the atoms A i and B i of the molecule A i B i , i = 1, . . . N , and the springs which link two molecules nearest neighbours along the chain. We have first studied the case where only the spring constants of the springs contained in the molecules depend on the electronic state of the molecules. We have shown that, in this case, phonons create on each molecule a field-like which favours the excited level, or (b) level. The competition between this field-like and the electronic parameter ∆, which favours the fundamental level, or (a) level, leads, for some values of T and ∆, to a first-order phase transition. This result is important because it is certainly possible to identify in a molecule which springs contained in the molecule can contribute, in the solid state, to the coupling.
We have also studied, in this case, the variations of the critical temperature value with the model parameters. This study is necessary because the first, order phase transition and the related effects cannot be observed if the value of the critical temperature is too small. One interesting result is that the value of the critical temperature is thrice larger when the atom B is a hydrogen atom than when it is a carbon or nitrogen one. This result leads to think that hydrogen bonds can play role in the coupling between the molecules in the solid state.
We have then added the assumption that the spring constants of the springs which link two molecules nearest neighbours depend on the electronic states of both molecules linked. In that case, phonons create a new field-like which also favours the (b) level and furthermore creates an exchange-like interaction between molecules nearest neighbours. The interesting result is that the presence of both fields-like increases a lot the critical temperature value. We think that the studies previously mentioned [6][7][8][9] could be made again in the present model with a value near 0.6 for the ratio ν λ . We recall that λ is the value of the spring constant when both linked molecules are in the (a) level and ν is the value when they are both in the (b) level.
In some compounds, pressure can modify, at a given temperature, the value of n b , the fraction of molecules in the (b) level [12]. In that case the volume have to also vary with n b because the volume is the conjugate parameter of pressure.
We can relate the parameters N m and ∆ of our model with the volume of the crystal and applied pressure. Let us call v a and v b the values of the molecule volume in the fundamental level and in the excited one, respectively. We assume that those values can be obtained by quantum mechanics calculations. They can be different because the quantum state of the molecule is not the same in both levels. They are independent on the temperature.
It is clear that the volume of the crystal is given by with In the above relations, N a and N b are the numbers of molecules in the fundamental and excited levels, respectively. Using relation (66) in relation (65) we obtain At 0 K all the molecules are in the fundamental level and N b is equal to zero. So, the volume of the crystal at 0 K is N v a . When the crystal temperature increases, the parameter N b increases and the volume of the crystal can vary. From relation (67), it is possible to obtain the value of the parameter (v b − v a ) by measuring the thermal variation of the volume and that of N b . This thermal variation of volume is not the well known thermal expansion of the crystal.
There are two cases: , the crystal volume does not vary with n b (and consequently with temperature). In that case, the applied pressure cannot modify the value of n b because pressure is the conjugate parameter of the volume.
ii) If v b − v a = 0, the crystal volume varies with n b and the applied pressure can modify the value of n b at a given temperature. From relation (67) we deduce that the volume displays a discontinuity when the parameter N b displays a discontinuity.
Taking the infinitesimal variation of equation (45) we find So, the parameter N m is the extensive parameter conjugate to the intensive parameter ∆ 2 . By a partial Legendre transformation we introduce the thermodynamic potential A defined by Taking the infinitesimal variation of A we find Using the expression of n b , relation (46), in relation (67), the volume of the crystal can be written as So, and Let us call ∆ 0 the value of the parameter ∆ at the atmospheric pressure P 0 = 1 atm, P the variation of applied pressure and ∆ the corresponding variation of ∆. We have Let us assume that − P is the conjugate parameter of the volume V . So, by definition we have and from relation (74) we deduce We can control the assumption that − P is the conjugate parameter of the volume V .
Using relation (77) we obtain By a Legendre transformation we introduce the thermodynamic potential B If the assumption is correct, the differential dB has to be equal to the differential dF given by equation (68). Taking the infinitesimal variation of B we find Using relations (72) and (78) we obtain As we have and then By comparing relation (85) and relation (68) we deduce Relation (78) must be written as From relation (87), we see that increasing the applied pressure does not modify the value of ∆ if (v b − v a ) = 0. Similarly, increasing the applied pressure increases the value of ∆ if (v b − v a ) is positive. That means that an increase of the applied pressure favours the level (a) which has the lower molecular volume. This result is in agreement with the Le Chatelier's principle.
In our study, ∆ 0 is of the order of 500 K = 4.5 10 −2 eV. Some molecules which have two electronic energy levels are organised around an iron II ion in an octahedral environment. The value of the spin total of the 3d electrons is S = 0 for the fundamental level and S = 2 for the excited one. When the molecule passes from the fundamental level to the excited one the value of its spin changes. For this reason it is called a spin conversion molecule. A crystal of N identical spin conversion molecules is called a spin conversion compound. For some spin conversion compounds, the thermal variation of the fraction of molecules in the excited level displays a hysteresis loop which means that there is a coupling between the molecules in the solid state. Many models have been proposed for the origin of this coupling [5,[14][15][16][17][18]. We propose the present model which has three characteristics: it is found on the adiabactic approximation which is a well admitted approximation, experimental studies and DFT calculations show that the values of the spring constants of the springs contained in the molecule vary with its electronic state [3,[19][20][21] and the exchange-like interaction allows to reproduce many experimental results obtained on spin conversion compounds [6,7].

A.1 Phonon dispersion relations
We consider a chain of 2N atoms with N units cell but with two atoms in each cell so there are 2N atoms. At the equilibrium the period of the translational symmetry is b. For two atoms per cell we find two dispersion curves. The acoustic branch has a frequency equal to zero when the wave vector − → q is equal to zero, while the optical branch has a finite frequency for this value of − → q . The atoms are displaced longitudinally along the chain from their equilibrium positions. The dispersion relation for the optical branch is and that for the acoustic branch is In the above relations, m R is the reduced mass Using the periodic conditions the wave vector q verifies the relation qb = n ph 2π N , with n ph = 0, ±1, ±2, ..., N 2 when N is even.
When q → 0, the above relations become where O q 2 is a function of q 2 and where V s is the sound velocity.
When K E, the dispersion relations become In the above relations, the symbol ± must be replaced by + for the frequency values of the optical branch and by − for those of the acoustic one. So,