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

Abstract
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.


Molecules with two electronic energy levels: coupling between the molecules in the solid state via the optical and acoustic phonon branches Jamil A. Nasser 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 , which is of the order of 300K. Such molecules can be found in various complexes of transition metal ions [1]. 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 fondamental and the excited electronic levels, respectivement. 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 e k when it is in the (b) level .
ii) the spring constant of the spring which links two molecules nearest neighbours has three values: when they are both in the (a) level , when they are both in the (b) level and when they are not in the same electronic level.
Ronayne et al. [2] 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 [3] 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 (2) The case where the values k and e k are equal while the values , and verify relation (2) looks like the case studied by Nasser [4] who has found a coupling between the molecules.
So, one of aims of this study is to see if there is a coupling when the values , and are equal while the values k and e k verify relation (1).
The present study has never been done before.
The free energy of an harmonic oscillator is where = 1 k B T and k B is the Boltzmann constant. It is clear that a decrease in !, the oscillator frequency value, at constant temperature lows the value of the free energy.
From this result, one can say that the assumptions done in relations (1) and (2) 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 …rst order phase transition. It is worth to emphasize that the lattice vibrations of the crystal have to be studied by using quantum statistical mechanics [5].
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.

A. 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 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 they 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 where b i is a …ctitious spin associated to 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 e k when it is in the (b) level. This assumption can be written iii) we assume that the spring constant e i;i+1 is equal to when the molecules A i B i and A i+1 B i+1 are both in the (a) level, to when they are both in the (b) level and to when they are not in the same level. Those assumptions can be written The chain hamiltonian is

B. Phonons-molecules interactions
By inserting equations (9) and (10) in the right hand of equation (5) 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 …elds-like acting on the spin b i are and Both …elds-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 b i = +1, because the coe¢ cients ( e k k) and ( ) are negative (here the Zeemann-like term is written h i b i ).
The …eld-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 …eld-like h ilat comes 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 (b i b 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 (b i b 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 [6] and to this end we introduce three parameters, h, K and E. The …rst one h describes an uniform, e¤ective …eld, the second one K is an e¤ective spring constant that replaces the spring constants k i and the last one E is an e¤ective 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 With this spin hamiltonian all the …ctitious spins b i have the same mean value m given by Equation (24) 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 the Appendix.
With the hamiltonian H 0ph (K; E), the mean values (u i w i ) 2 and (u i+1 w i ) 2 , calculated with the matrix density of H 0ph (K; E), are independent on i and are equal to and where P ! q 0 is the sum over the two phonon branches and @! @K and @! @E are the partial derivatives versus K and E, respectively, of the dispersion relations ! 1 ( ! q ) and ! 2 ( ! q ). The expressions of the partial derivatives are given in the Appendix. The free energy related to H 0ph (K; E) The free energy associated to the variational Hamiltonian H 0 is At the …rst order perturbation calculation, and 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
2. Discussion i) We have veri…ed that the value of the spring constant E is always positive. It decreases from to when m increases from 1 to 1.
ii) From relation (28), the thermal mean value of the potential energy E p0 (K; E) is given by By inserting equations (29) and (30) in equation (42), we can verify the virial theorem iii) In relation (38) the exchange-like interaction is studied as a mean …eld approximation. This study method could be improved for a 1D system by using the transfert matrix method, if it is possible. In the antiferro-like case it could be interesting to introduce two sublattices.
We assume for relation (38) that the exchange-like interaction is small compared to the Zeemann-like interaction, that is where j 2 + j is the absolute value of the parameter 2 + . With this condition the parameter h phlat is always positive.
iiii) It is worth to notice that, in the expression of the uniform …eld h, the electronic term is negative while the phonon term h ph is positive.

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

A. Reduced Parameters
We take as the unit of elastic force constant and~! M ( ) as the unit of energy with The value of~! M ( ) is roughly estimated to 1000K or 695cm 1 [5].
We introduce the following reduced parameters: the reduced temperature the reduced electronic energy gap the elastic force constants ratios and From condition (1) and from condition (2) 0 x 1 It is interesting to introduce the parameter y by the relation The value varies from to 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 (53) we have So condition (44) can be written We assume that m A + m B , the mass of the molecule, is 500g mol 1 [7]. So, the ratio m B =m A is near 0:002 if the atom B is an Hydrogen atom, and to 0:02 if the atom B is a Carbone 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.

C. Study of the Self-Consistent Equation
We 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 C and t C obtained in this study are somewhat smaller than the exact values.
We have studied for di¤erent 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 to dm, a small variation of m: We have veri…ed that this ratio is positive for all the solutions except the one previously called unstable solution.
We recall that, in the theory of the …rst order phase transition, the unstable solution plays role in the thermal hysteresis loop which appears around the transition temperature.

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

Case = = :
In that case the …eld-like h phlat and the exchange-like interaction are equal to zero. Only the …eld-like h phmol created by the springs contained in the molecules plays role. This case has never been studied before.

Relation (37) can be written
From relation (57) the …eld-like h phmol depends on the parameters xint and z, and also on the mean value (u i w i ) 2 0 . This mean 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 at the transition.
The (b) phase -(a) phase coexistence curve is ended by the critical point C with the coordinates ( C ; t C ). Its slope is positive in agreement with the Clapeyron equation [4]. 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 …rst order phase transitions and that of the related e¤ects 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 an Hydrogen atom, for p = 0:02 it is a Carbone 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 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), e k = k. So the …eld-like h phmol is equal to zero and the e¤ective …eld h is equal to . There is not an interaction between molecules. From Eq.(24), we deduce that m = 0 when t ln(r) = . In that case n b is equal to 0:5. With r = 5 and = 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), e k = 0:4 k. So the …eld-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 compared to the critical value C = 1:454. In curve (2) the value of being lower than C , there is a discontinuity; in curve (3) this value being higher than 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 of the sound velocity is the same for the three curves at 0K. This result is due to the fact that, at 0K, 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 an 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, its temperature increases and it passes successively in the di¤erent 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 …nd similar behaviour.
In that case the …eld-like created by phonons is the sum of the two …eld-like h phmol and h phlat .
Using the reduced parameters and relation (54) in relation (38) the …eld-like h phlat can be written This …eld-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 veri…ed that the value of the critical temperature due to both …elds-like is higher than the sum of the values due to each …eld-like. We have also studied, in this case, the variations of the critical temperature value with the model parameters. This study is necessary because the …rst order phase transition and the related e¤ects 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 three time higher when the atom B is an 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 …rst neighbours depend on the electronic states of both molecules linked. In that case, phonons create un new …eld-like which also favours the (b) level and also create an exchange-like interaction between molecules …rst neighbours. The interesting result is that the presence of both …elds-like increases a lot the critical temperature value.
Some studies [5,8,9]  For two atoms per cell we …nd two dispersion curves. The acoustic branch has a frequency equal to zero when the wave vector ! q is equal to zero, while the optic branch has a …nite frequency for this value of ! q . The atoms are displaced longitudinally along the chain from their equilibrium positions.
The dispersion relation for the optic 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 veri…es the relation qb = n ph 2 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 ! 2 ac ' where V s is the sound velocity.
When K E, the dispersion relations become      (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 is lower than the critical value C = 1:454. In this Figure z = 1.  (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 is lower than the critical value C = 1:454.