Phase transitions and dynamical properties of quasi-one-dimensional structures with hydrogen bonds

The frequency dependence of dynamical conductivity of the quasi-one-dimensional structures with hydrogen bonds is studied on the basis of pseudospin-electron model. It takes into account the proton-electron interaction, external longitudinal field h, the tunneling hopping of protons, electron transfer and direct interaction between protons. The dependences of the electron concentration and mean number of protons at the site on temperature and external field are obtained. The phase transition lines from uniform phase into charge ordered phase are determined. The dependence of dynamical conductivity on temperature and field $h$ and its changes at the phase transitions are obtained.


Introduction
The properties of molecular and crystalline structures with hydrogen bonds are mainly determined by the character of proton redistribution on the bonds. We investigate the microscopic mechanisms of charge transfer in such systems on the basis of the proposed pseudospin-electron model [1,2] that takes into account the correlation between the proton displacement and reconstruction of electron states as well as the change of their occupancy. This interaction manifests itself as a cooperative proton-electron transfer (PET) in a number of experimental works [3][4][5][6][7][8][9][10] and it follows also from the results of quantumchemical calculations [1,[11][12][13]. Quantum chemical methods allow us to examine these charge redistributions more in detail. The structural and optical studies of the proton transfer in N -salicylideneaniline [8,10] show that photochromism and thermochromism in these object arise from a proton transfer that is accompanied by a configurational change of electron structure. It was shown that the behaviour of proton dynamics is quite consistent with the temperature dependence of visible absorption spectra of this crystal. If we could construct a molecular conductor based on this type of molecules, the charge transport might strongly be modulated by the proton motion. Photoinduced proton-coupled electron transfer (PCET) is investigated in a number of works [14][15][16][17] as one of the mechanisms of energy transformation in biological and chemical systems. The effect of a such proton-electron coupling plays an important role in passing a proton through the biological membrane in photosynthesis. The design of an electron-proton hybrid system using the elements of one-dimensional metal chains, acceptor (or donor) molecules, and interchain H-bonds are proposed [4]. A new molecular function is expected to be produced in this system, if the motion of proton is closely correlated with the dynamics of the 1D electronic states. A similar effect is observed in the halogen (X)-bridge mixed-valence transition-metal (M) complexes (M-X-complexes) [3].

Hamiltonian
The Hamiltonian of quasi-one-dimensional structures which contains chains with hydrogen bonds are written down in the form [2]: Here, the summation along the chains (indices i , j ) and the summation over the chains (indices l, l ′ ) is performed. Pseudospin operatorŜ i describes the proton position in double potential well on the hydrogen bond. We suppose that the transfer along hydrogen bond is dominant: t = t i(l ),i+1(l ) ; n iσ is operator of electron concentration at i lattice site, σ is electron spin, µ is chemical potential of electrons. The Hamiltonian includes proton-electron interaction (parameter g ), electron transfer (parameter t ), energy of proton tunneling (parameter Ω), asymmetry of the local anharmonic potential (parameter h).
The last term describes proton-proton interaction.
Pseudospin-electron interaction leads to the effective interaction between pseudospins (between protons in our case) and as it is shown in [19][20][21] it can cause the appearance of a modulated phase with doubling of the initial lattice period and can lead to the corresponding charge modulation. The study of this phenomenon is the aim of this paper. In a case of double modulation of the lattice period, the crystal is divided into two sublattices. We introduce the following notations: η α = 〈S z i,α 〉, n α = 〈 σ n i,α,σ 〉, (α = 1, 2 is the sublattice index). In the mean field approximation (MF) and after passing to k-representation the Hamiltonian (1) has a form:

Thermodynamic properties
Using formulae (2)-(5), we can write the equations for electron concentration n α and the average mean of pseudospins η α in sublattices: From all the possible solutions of equations (6)- (7) we choose the ones that correspond to the minimum of grand canonical potential Φ in regime of µ = const or minimum of free energy F = Φ + µN in regime n = const. In the MF-approximation: From the relations (6)- (7) we obtain the equations for δn = n 1 −n 2 and δη = η 1 −η 2 , which can play a role of the order parameter for a modulated phase. Using these equations we obtain the condition of the appearance of nonzero solutions for δn and δη, and the equation for temperature of the second order phase transition to the modulated phase. Here:

Dynamic conductivity of quasi-one-dimensional structures with hydrogen bonds
Calculation of the dynamic conductivity of the structure which possesses the chains with hydrogen bonds was carried out according to Kubo formula [22] σ(ω, whereĵ is the current density operatorĵ that includes electronic and pseudospin (ionic) part. Here δ is the distance between equilibrium positions of a proton on the bond, δ ≈ 0.40 Å. According to quantum-chemical calculations, the effective charge of hydrogen z eff H is equal to z eff In the molecular field approximation, the operator of current density is split into electronic and proton (pseudo-spin) partsĵ =ĵ e +ĵ sp . (13) The following expressions are obtained for these composites: Calculation of correlation functions in the expression (10) with the use of the Wick's theorem yields the following expressions for a real part of conductivity: σ = σ e + σ sp , (16) where the electronic part has a form: Here

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Phase transitions and dynamical properties  For the protonic part of conductivity we obtain:

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Frequency dependence of the electronic part of the dynamical conductivity along the phase transition line is shown in figure 5, curve 1 -before phase transition (uniform phase), curve 2, 3 -after phase transition (modulated structure). At the phase transition from uniform to modulated structure, the conductivity σ e (0), when ω = 0, is abruptly reduced by two to three orders of magnitude at low temperatures and with increasing temperature the value of the jump decreases. Electronic conductivity has one peak (at ω = 0) in uniform phase, (one electronic band is present). We observed the splitting of the electron band in a modulated phase, and electronic conductivity has a broad maximum in the frequency region

Conclusions
The possibility of the first or the second order transitions from uniform phase into phase with doubled lattice period in the quasi-one-dimensional structures with hydrogen bonds is studied in the framework of the proposed pseudospin-electron model. It was shown that pseudospin-electron (proton-electron) interaction may cause the appearance of charge ordered phase in the structures with hydrogen bonds. The electron spectrum is calculated. The dependences of the splitting of the electron spectrum on temperature and asymmetry field are investigated. The dependences of the electron concentration and mean number of protons at the site on temperature and asymmetry field were obtained. It was shown that abrupt changes of these characteristics at the first-order transitions are smaller for the structures with high proton tunneling frequency and stronger direct interaction between protons. The phase transition lines from uniform phase into charge ordered phase are determined. The dependences of the dynamical conductivity on temperature and external field and its changes at the phase transitions are obtained. At the phase transition from uniform to modulated structure the static conductivity σ e (0) is abruptly reduced by two to three orders of magnitude at low temperatures and with increasing temperature the value of the jump decreases. Electronic conductivity has one peak at ω = 0 in a uniform phase. In modulated phase, the dynamical electronic conductivity has a broad maximum as well as a peak at ω = 0. This broad maximum is placed at lower frequencies for the structures with high proton tunneling frequency and stronger direct interaction between protons. It was shown that the frequency dependence of the proton dynamical conductivity has one peak in a uniform phase and two peaks in the charge modulated phase. The model can be applied to a description of quasi-one-dimensional structures, the so-called halogen-bridge mixedvalence transition-metal complexes [3] in which there are charge modulated states.