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Geometric isotopic effect in hydrogen-containing crystals
Journal of Molecular Structure 613 (2002) 55±59
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Geometric isotopic effect in hydrogen-containing crystals q E.A. Shadchin, A.I. Barabash* Institute of Physics, National Academy of Sciences of Ukraine, 46, Prospekt Nauki, 03039 Kiev, Ukraine Received 7 November 2000; revised 19 February 2002; accepted 19 February 2002
Abstract The geometric isotopic effect in hydrogen-containing crystals of the KDP-type is investigated. The mechanism of changes in the hydrogen bond lengths under deuteron substitution is supposed. It is shown that (in the framework of a three-dimensional model for hydrogen bonds based on the Einstein approximation for anharmonic oscillators) the correspondence between experimental and theoretical data can be achieved if one takes into account the interactions between the neighboring H-bonded chains in the crystal. q 2002 Elsevier Science B.V. All rights reserved. Keywords: Hydrogen bond; Geometric isotopic effect
1. Introduction It is known that KDP-type crystals with hydrogen bonds undergo structural ferroelectric or antiferroelectric phase transitions. The temperatures of these transitions essentially depend on the deuteron substitution action. To explain the microscopic mechanism of phase transitions in hydrogen-containing crystals, the quantum tunneling model was successfully applied [1]. This model, however, does not explain some experimental results connected with the isotopic proton substitution [2,3]. In particular, this concerns the observed concentration dependencies of the Curie temperature (Tc) for phase transition of the order± disorder type. Later it was noticed [4±7] that the linear dependence of Tc as a function of hydrogen bond length R (so called geometric isotopic effect) q This work was presented on three-dimensional Vibrational Spectroscopy in Materials Science, September 2000, Krakow, Poland. * Corresponding author. Tel.: 1380-44-265-0837; fax: 1380-44265-1589. E-mail address: [email protected] (A.I. Barabash).
can be explained in the same manner. To describe the isotopic changes in the structure and dynamics of hydrogen bonds, the problem of isotopic substitution was considered (see, for example Ref. [8]) within the approach of the model crystal built from noninteracted in®nite hydrogen-bonded chains. For the analysis of real crystal structures, however, it is necessary to take into account three-dimensional structure of hydrogen bonds. In particular, in the case of real crystals it is necessary to take into account both stretching and bending vibrations of H-bonds as well interactions between the hydrogen-bonded chains. The present paper aims at the elucidation of the isotopic changes of hydrogen bond structure in the three-dimensional approximation with account to the interactions between the H-bonded chains in the crystal.
2. Three-dimensional model for H-bond According to Ref. [9] the energy of interaction between the proton (deuteron) and the oxygen
0022-2860/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0022-286 0(02)00068-6
56
E.A. Shadchin, A.I. Barabash / Journal of Molecular Structure 613 (2002) 55±59
atoms, can be approximated by three-dimensional Morse potential U r D0 {exp 22a r 2 r0 2 2 exp 2a r 2 r0 }; where r uru characterizes the proton (or deuteron) position with respect to the oxygen atom; D0, a and r0 are potential parameters. According to Ref. [10], the energy of proton (deuteron), located on the symmetric H-bond, describes the double-minimum Morse potential R R V r U x 1 ; y; z 1 U x 2 ; y; z 2 2
D0 {exp aR0 exp 22l1 1 exp 22l2 4 2 4 exp
aR0 exp 2l1 1 exp 2l2 }; 2
1
where s R 2 l^ a2 y2 1 z2 1 a2 x ^ ; 2 1 R0 2 r0 1 ln 2 : a and R is the distance between two neighboring oxygen atoms. In Eq. (1), x, y and z are the coordinates of proton (or deuteron) with respect to the center of the hydrogen bond. The shape of the double-minimum Morse potential (Eq. (1)) essentially depends on the H-bond length. The x0, y0 and z0 are coordinates of two position for proton (deuteron) on the doubleminimum Morse potential for cases, if R . R0 1 a 21 x0 ^ cosh exp R 2 R0 ; a 2 y0 z0 0: One can see that the double-minimum proton potential transforms into the one-minimum potential x0 y0 z0 0; if R # R0 : 3. Free energy of hydrogen bond chains Let us use the method developed in Ref. [8] for the quantum-statistical averaging of the free energy F.
According to this method, one should proceed from the dynamics of the isolated H-bond chains to the thermodynamics of simple system of interacting Hbonded chains. In other words, according to this model, the crystal can be presented as the threedimensional aggregate of linear chains of the hydrogen bonds. The well-known crystals CsH2PO4 and KH2PO4 [11] contain the in®nite chains of hydrogen bonds `attached' to the PO4-groups, fully satisfy the requirements of this model. The interaction between the in®nite chains of the H-bonds can be introduced by the replacement l^ to L^ in Eq. (1) s R 2 2 2 L^ a2 x ^ 1b y 1 z2 : 2 The introduction of one additional parameter b ± a; refers to the fact that the interaction between the Hbond chains changes the curvature of the potential curve (Eq. (1)) in the y- and z-directions. Consider the crystal containing the non-interacting protons (deuterons) positioned in the double-minimum Morse potential. In this case, the system can be described by the Einstein model of anharmonic oscillator with the energy H K 1 V 1 W (where K is kinetic energy of the proton (deuteron) and two oxygen atoms, V is interaction energy, determined by Eq. (1). Here, as it follows from Ref. [8], W is the energy of interaction between two oxygen atoms. This energy was chosen in the following form W w0 {exp 22a R 2 Rm 2 2 exp 2a R 2 Rm }: where w0, Rm are the parameters of this potential. As it follows from the perturbation theory [6] for free energy F, let choose trial Hamiltonian H0 K 1 V0 1 W0 for harmonic oscillations of protons (deuterons) with v s and v b frequencies of stretching and bending vibrations on the H-bond and v is the frequency of the oxygen atoms vibration. It should be noted that for the simpli®cation of the following calculations we use approximation in which vb S vb 0 1 vb 00 ; where vb 0 and vb 00 are frequencies of the non-degenerate bending vibrations. We also choose the vibration energy for the oxygen atoms with the frequency v . Then, according to Refs. [8,11], the free energy F of the hydrogen-bonded
E.A. Shadchin, A.I. Barabash / Journal of Molecular Structure 613 (2002) 55±59
chains standardized on the unit chain cell is F F0 1 kV 1 Wl 2
1 2
the free energy F
j akRl;
kH0 l;
where
where k is the Boltzmann constant. Therefore free energy is mv2s l mv2b m mv2 s R 2 2 : 2a2 b2 2a2
hvs hvb F0 1kT ln 2 sinh 1 2kT ln 2 sinh 2kT 2kT hv 1 kT ln 2 sinh : 2kT
l
a2 h hv coth s ; 2mvs 2kT
Here v is the vibrational frequency of oxygen atoms. The averaging action leads to the following equation 1 2
D0 G 1 w0 B;
3
where G
exp 2 j 2 j 0 1 2l { j 2 2 4x2 cosh 2x j 1 4m 2 2 4x2 1 4m j cosh 2x 2 2x sinh 2x} j 2 j0 l 1 exp 2 2 2 24 { j 2 2 4x2 cosh x 2 2 j 1 2m 2 4 x 1 2m j cosh x 2 x sinh x};
j 0 aR0 ;
sR
m
b2 h hv coth b ; 2mvb 2kT
2
a h hv ; coth 2M v 2kT
6
are the square-averaged values of the thermal displacements for hydrogen and oxygen atoms. Note that the Eq. (4) was received in approach, that the Oatom ¯uctuations s R contribution is small in comparison with contributions of the proton ¯uctuations l . The varying parameters v s, v b, v of free energy F can be found using the minimization procedure for the free energy F. According to this minimization action for F 2F=2v s 2F=2vb 0 and, taking into account Eqs. (2) and (6), one can obtain
v2s
1 2
V a2
2G 2a2 D0 ; ; V a2 2l m
v2b
1 4
V b2
2G 2b2 D0 : ; V b2 2m m
2
f kV 1 Wl
and
In Eqs. (4) and (5)
Here the averaging is performed for the Hamiltonian H0, as it was done in Ref. [8] 2H0 sp ¼exp kT k¼l ; 2H0 sp exp kT
Here
x akxl;
j m aRm :
H0 F0 2kT ln sp exp 2 : kT
F F0 1 f 2
57
7
The parameters V a, V b depend on parameters of double-minimum proton potential only. In Eq. (7) m is the mass of the proton (deuteron). The obtained Eq. (7) determine the dependencies of frequencies of stretching and the bending vibrations of the protons (or deuterons) as a functions of the Hbond length. 4. Equations for temperature averaged varying parameters
(4)
B exp 22 j 2 j m 1 4s R 2 2 exp 2 j 2 j m 1 s R :
5 Here j , x and v s, v b, v are varying parameters of
Taking into account that in Eq. (4), the ratio m=j and x value are small, the free energy f, received accurate to a constant, shall have the following form: f
1 2
D 0 G 1 w0 {exp 22s R } j 2 j 2 ;
where
j j m 1 3s R
8
58
E.A. Shadchin, A.I. Barabash / Journal of Molecular Structure 613 (2002) 55±59
equations for v s j and v b j; if j . j cp ; are as follows
v2s V a2 exp 2l 1 2 A22 j; v b2
V b2 exp 2lA 22 j; j
and, when j # j cp
v2s V a2 exp 2l{A22 j 2 A 21 j}; v b2
Fig. 1. Depenedence D R ; RODO 2 ROHO : The solid line is presenting results of our theoretical calculations. The experimental data was taken from Ref. [7].
Here G A22 jcosh 2x 2 4A 21 jcosh xexp 2l; A j exp
1 4m 1 ln 4 2 3 j 2 a 2r 1 l 1 ; 0 2 a j
the critical length j cp can be estimated approximately as: 1 4m j cp a 2r0 1 ln 4 1 3l 2 ; a jc 9 1 j c a 2r0 1 ln 4 1 3l: a Considering free energy minimum condition, new equilibrium position for the proton (or the deuteron) shell will be:
x j cosh 21 A j j . j cp x j 0
j # j cp
:
One should note that, according to Eq. (9), the H-bond critical length j c determined in Ref. [8] at x 0; decreases. It is fairly clear, as due to transversal degree of freedom for the H-bond the doubleminimum character of the potential (Eq. (1)) remains uncharged at smaller values of the H-bond length. The
V b2 exp 2l{2A 21 j 2 A 22 j}: j
The free energy minimum condition as a function of j , can be written as 2f =2j 0: This new expression for the minimum condition leads to two equations for the equilibrium length j H;D of the H-bond. This length depends on the proton (or the deuteron) position on the H-bond: 2G H; D j H;D j 2 g : 10 2j jj H;D Here non-dimensional parameter
g
D0 exp 22s R ; 2w0
determined as ratio of the energies for H±O and O´ ´´O interactions, gives contribution to the magnitude of the thermodynamically equilibrium H-bond length. The expressions for G H; D; taken from Eq. (8), determines the functional dependence of the equilibrium length j H;D : From two equations of the system (10), one can obtain the value of D ; j D 2 j H : Subtracting the ®rst equation of the system (10) from the second equation and, taking into account, that j D j H 1 D; we shall obtain the following transcendental equations p D g exp 2lH exp 2 j 2 j c;H " !# 4mp p 1 2 exp 22lH 2 D 1 ; 11 j c;H for j . j cp ;
E.A. Shadchin, A.I. Barabash / Journal of Molecular Structure 613 (2002) 55±59
Fig. 2. Dependencies lH R and mH R; obtained from Eqs. (10) and (11).
and
"
(
!#
D 2g exp 2lH exp 2 j 2 " £ 1 2 exp 2
2lpH
!#
12
p 2 j 2 j c;H
2 2 exp "
2
lp D 2mp 1 pH £ 1 2 exp 2 H 2 2 2 j c;H for j # j cp : p 221 lpH lH p ; 2
mpH mH
sion and inaccuracy, one could make a conclusion about the fair agreement between the experimental and theoretical data. One should note that the best agreement was achieved if coef®cient g in Eqs. (11) and (12) is equal to < 1. The other parameters were taken as a3A 21 ; b 3:9 A 21 ; D0 follows: r0 0:95 A; 272 KJ=mol: The dependencies l R and m R are shown in Fig. 2. The presented dependence l R agrees with the same dependence l R inRef. [8]. To the best of our knowledge the dependence m R was not previously published. This dependence accounts for both the threedimensional model of the H-bond, and the interactions between the in®nite chains of the H-bonds in crystal.
5. Conclusions
p j c;H
4mp 2 A 1 pH j c;H !
59
!#)
p 221 p : 2
The values of l H and m H are determined by Eq. (6), if m is the mass of the proton. In Eqs. (11) and (12) 1 lD ù lH p ; 2 since the following relations are true v vD pH hvH;D q kT; and mD 2mH : 2 The dimensional value D=a ; kRD 2 RH l as a function of the H-bond kRl length is shown in Fig. 1. There is satisfactory agreement between the experimental results [4] and our theoretical data. One should note that, in the range R , Rc ; the theoretical curve (solid line in Fig. 1) lies below zero, as the experimental data do. Taking into account that the experimental points presented for the range R . Rc are characterized by signi®cant disper-
In this work we propose the approach to explain the geometric isotopic effect based on the three-dimensional model of hydrogen bond. The free energy for some model crystal composed from the interacting in®nite hydrogen-bonded chains is obtained in the Einstein anharmonic oscillator approximation. It is shown that the critical length of the hydrogen bond (the length at which the hydrogen bond is characterized by the one-minimum potential for proton) determined by the contribution of the longitudinal and the transversal ¯uctuations of the protons (or the deuterons), and the contribution from the transversal ¯uctuations is negative. It was also shown that the obtained curve D R (solid line in Fig. 1) lies below zero in the range R , Rc ; and has minimum.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
R. Blinc, S. Svetina, Phys. Rev. 147 (1966) 430. M. Ichikawa, K. Motida, J. Phys. Soc. Jpn 56 (1987) 3750. K. Fujii, J. Phys. Soc. Jpn 61 (1992) 342. M. Ichikawa, Acta Crystallogr. B 34 (1978) 2074. M. Ichikawa, Chem. Phys. Lett. 79 (1981) 583. M. Ichikawa, K. Motida, N. Yamada, Phys. Rev. B 36 (1987) 874. E. Schadchin, A. Barabash, J. Mol. Struct. 325 (1994) 59. S. Tanaka, Phys. Rev. B 42 (1990) 10488. E. Matsushita, T. Matsubara, Prog. Theor. Phys. 67 (1982) 1. E.D. Isacs, A. Shukla, P.M. Platzman, D.R. Hamann, B. Barbiellini, C.A. Tulk, Phys. Rev. Lett. 82 (1999) 600. [11] R.P. Feynman, Statistical Mechanics, Institute of Technology, W.A. Benjamin Inc, California, 1972 p. 404.
www.elsevier.com/locate/molstruc
Geometric isotopic effect in hydrogen-containing crystals q E.A. Shadchin, A.I. Barabash* Institute of Physics, National Academy of Sciences of Ukraine, 46, Prospekt Nauki, 03039 Kiev, Ukraine Received 7 November 2000; revised 19 February 2002; accepted 19 February 2002
Abstract The geometric isotopic effect in hydrogen-containing crystals of the KDP-type is investigated. The mechanism of changes in the hydrogen bond lengths under deuteron substitution is supposed. It is shown that (in the framework of a three-dimensional model for hydrogen bonds based on the Einstein approximation for anharmonic oscillators) the correspondence between experimental and theoretical data can be achieved if one takes into account the interactions between the neighboring H-bonded chains in the crystal. q 2002 Elsevier Science B.V. All rights reserved. Keywords: Hydrogen bond; Geometric isotopic effect
1. Introduction It is known that KDP-type crystals with hydrogen bonds undergo structural ferroelectric or antiferroelectric phase transitions. The temperatures of these transitions essentially depend on the deuteron substitution action. To explain the microscopic mechanism of phase transitions in hydrogen-containing crystals, the quantum tunneling model was successfully applied [1]. This model, however, does not explain some experimental results connected with the isotopic proton substitution [2,3]. In particular, this concerns the observed concentration dependencies of the Curie temperature (Tc) for phase transition of the order± disorder type. Later it was noticed [4±7] that the linear dependence of Tc as a function of hydrogen bond length R (so called geometric isotopic effect) q This work was presented on three-dimensional Vibrational Spectroscopy in Materials Science, September 2000, Krakow, Poland. * Corresponding author. Tel.: 1380-44-265-0837; fax: 1380-44265-1589. E-mail address: [email protected] (A.I. Barabash).
can be explained in the same manner. To describe the isotopic changes in the structure and dynamics of hydrogen bonds, the problem of isotopic substitution was considered (see, for example Ref. [8]) within the approach of the model crystal built from noninteracted in®nite hydrogen-bonded chains. For the analysis of real crystal structures, however, it is necessary to take into account three-dimensional structure of hydrogen bonds. In particular, in the case of real crystals it is necessary to take into account both stretching and bending vibrations of H-bonds as well interactions between the hydrogen-bonded chains. The present paper aims at the elucidation of the isotopic changes of hydrogen bond structure in the three-dimensional approximation with account to the interactions between the H-bonded chains in the crystal.
2. Three-dimensional model for H-bond According to Ref. [9] the energy of interaction between the proton (deuteron) and the oxygen
0022-2860/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0022-286 0(02)00068-6
56
E.A. Shadchin, A.I. Barabash / Journal of Molecular Structure 613 (2002) 55±59
atoms, can be approximated by three-dimensional Morse potential U r D0 {exp 22a r 2 r0 2 2 exp 2a r 2 r0 }; where r uru characterizes the proton (or deuteron) position with respect to the oxygen atom; D0, a and r0 are potential parameters. According to Ref. [10], the energy of proton (deuteron), located on the symmetric H-bond, describes the double-minimum Morse potential R R V r U x 1 ; y; z 1 U x 2 ; y; z 2 2
D0 {exp aR0 exp 22l1 1 exp 22l2 4 2 4 exp
aR0 exp 2l1 1 exp 2l2 }; 2
1
where s R 2 l^ a2 y2 1 z2 1 a2 x ^ ; 2 1 R0 2 r0 1 ln 2 : a and R is the distance between two neighboring oxygen atoms. In Eq. (1), x, y and z are the coordinates of proton (or deuteron) with respect to the center of the hydrogen bond. The shape of the double-minimum Morse potential (Eq. (1)) essentially depends on the H-bond length. The x0, y0 and z0 are coordinates of two position for proton (deuteron) on the doubleminimum Morse potential for cases, if R . R0 1 a 21 x0 ^ cosh exp R 2 R0 ; a 2 y0 z0 0: One can see that the double-minimum proton potential transforms into the one-minimum potential x0 y0 z0 0; if R # R0 : 3. Free energy of hydrogen bond chains Let us use the method developed in Ref. [8] for the quantum-statistical averaging of the free energy F.
According to this method, one should proceed from the dynamics of the isolated H-bond chains to the thermodynamics of simple system of interacting Hbonded chains. In other words, according to this model, the crystal can be presented as the threedimensional aggregate of linear chains of the hydrogen bonds. The well-known crystals CsH2PO4 and KH2PO4 [11] contain the in®nite chains of hydrogen bonds `attached' to the PO4-groups, fully satisfy the requirements of this model. The interaction between the in®nite chains of the H-bonds can be introduced by the replacement l^ to L^ in Eq. (1) s R 2 2 2 L^ a2 x ^ 1b y 1 z2 : 2 The introduction of one additional parameter b ± a; refers to the fact that the interaction between the Hbond chains changes the curvature of the potential curve (Eq. (1)) in the y- and z-directions. Consider the crystal containing the non-interacting protons (deuterons) positioned in the double-minimum Morse potential. In this case, the system can be described by the Einstein model of anharmonic oscillator with the energy H K 1 V 1 W (where K is kinetic energy of the proton (deuteron) and two oxygen atoms, V is interaction energy, determined by Eq. (1). Here, as it follows from Ref. [8], W is the energy of interaction between two oxygen atoms. This energy was chosen in the following form W w0 {exp 22a R 2 Rm 2 2 exp 2a R 2 Rm }: where w0, Rm are the parameters of this potential. As it follows from the perturbation theory [6] for free energy F, let choose trial Hamiltonian H0 K 1 V0 1 W0 for harmonic oscillations of protons (deuterons) with v s and v b frequencies of stretching and bending vibrations on the H-bond and v is the frequency of the oxygen atoms vibration. It should be noted that for the simpli®cation of the following calculations we use approximation in which vb S vb 0 1 vb 00 ; where vb 0 and vb 00 are frequencies of the non-degenerate bending vibrations. We also choose the vibration energy for the oxygen atoms with the frequency v . Then, according to Refs. [8,11], the free energy F of the hydrogen-bonded
E.A. Shadchin, A.I. Barabash / Journal of Molecular Structure 613 (2002) 55±59
chains standardized on the unit chain cell is F F0 1 kV 1 Wl 2
1 2
the free energy F
j akRl;
kH0 l;
where
where k is the Boltzmann constant. Therefore free energy is mv2s l mv2b m mv2 s R 2 2 : 2a2 b2 2a2
hvs hvb F0 1kT ln 2 sinh 1 2kT ln 2 sinh 2kT 2kT hv 1 kT ln 2 sinh : 2kT
l
a2 h hv coth s ; 2mvs 2kT
Here v is the vibrational frequency of oxygen atoms. The averaging action leads to the following equation 1 2
D0 G 1 w0 B;
3
where G
exp 2 j 2 j 0 1 2l { j 2 2 4x2 cosh 2x j 1 4m 2 2 4x2 1 4m j cosh 2x 2 2x sinh 2x} j 2 j0 l 1 exp 2 2 2 24 { j 2 2 4x2 cosh x 2 2 j 1 2m 2 4 x 1 2m j cosh x 2 x sinh x};
j 0 aR0 ;
sR
m
b2 h hv coth b ; 2mvb 2kT
2
a h hv ; coth 2M v 2kT
6
are the square-averaged values of the thermal displacements for hydrogen and oxygen atoms. Note that the Eq. (4) was received in approach, that the Oatom ¯uctuations s R contribution is small in comparison with contributions of the proton ¯uctuations l . The varying parameters v s, v b, v of free energy F can be found using the minimization procedure for the free energy F. According to this minimization action for F 2F=2v s 2F=2vb 0 and, taking into account Eqs. (2) and (6), one can obtain
v2s
1 2
V a2
2G 2a2 D0 ; ; V a2 2l m
v2b
1 4
V b2
2G 2b2 D0 : ; V b2 2m m
2
f kV 1 Wl
and
In Eqs. (4) and (5)
Here the averaging is performed for the Hamiltonian H0, as it was done in Ref. [8] 2H0 sp ¼exp kT k¼l ; 2H0 sp exp kT
Here
x akxl;
j m aRm :
H0 F0 2kT ln sp exp 2 : kT
F F0 1 f 2
57
7
The parameters V a, V b depend on parameters of double-minimum proton potential only. In Eq. (7) m is the mass of the proton (deuteron). The obtained Eq. (7) determine the dependencies of frequencies of stretching and the bending vibrations of the protons (or deuterons) as a functions of the Hbond length. 4. Equations for temperature averaged varying parameters
(4)
B exp 22 j 2 j m 1 4s R 2 2 exp 2 j 2 j m 1 s R :
5 Here j , x and v s, v b, v are varying parameters of
Taking into account that in Eq. (4), the ratio m=j and x value are small, the free energy f, received accurate to a constant, shall have the following form: f
1 2
D 0 G 1 w0 {exp 22s R } j 2 j 2 ;
where
j j m 1 3s R
8
58
E.A. Shadchin, A.I. Barabash / Journal of Molecular Structure 613 (2002) 55±59
equations for v s j and v b j; if j . j cp ; are as follows
v2s V a2 exp 2l 1 2 A22 j; v b2
V b2 exp 2lA 22 j; j
and, when j # j cp
v2s V a2 exp 2l{A22 j 2 A 21 j}; v b2
Fig. 1. Depenedence D R ; RODO 2 ROHO : The solid line is presenting results of our theoretical calculations. The experimental data was taken from Ref. [7].
Here G A22 jcosh 2x 2 4A 21 jcosh xexp 2l; A j exp
1 4m 1 ln 4 2 3 j 2 a 2r 1 l 1 ; 0 2 a j
the critical length j cp can be estimated approximately as: 1 4m j cp a 2r0 1 ln 4 1 3l 2 ; a jc 9 1 j c a 2r0 1 ln 4 1 3l: a Considering free energy minimum condition, new equilibrium position for the proton (or the deuteron) shell will be:
x j cosh 21 A j j . j cp x j 0
j # j cp
:
One should note that, according to Eq. (9), the H-bond critical length j c determined in Ref. [8] at x 0; decreases. It is fairly clear, as due to transversal degree of freedom for the H-bond the doubleminimum character of the potential (Eq. (1)) remains uncharged at smaller values of the H-bond length. The
V b2 exp 2l{2A 21 j 2 A 22 j}: j
The free energy minimum condition as a function of j , can be written as 2f =2j 0: This new expression for the minimum condition leads to two equations for the equilibrium length j H;D of the H-bond. This length depends on the proton (or the deuteron) position on the H-bond: 2G H; D j H;D j 2 g : 10 2j jj H;D Here non-dimensional parameter
g
D0 exp 22s R ; 2w0
determined as ratio of the energies for H±O and O´ ´´O interactions, gives contribution to the magnitude of the thermodynamically equilibrium H-bond length. The expressions for G H; D; taken from Eq. (8), determines the functional dependence of the equilibrium length j H;D : From two equations of the system (10), one can obtain the value of D ; j D 2 j H : Subtracting the ®rst equation of the system (10) from the second equation and, taking into account, that j D j H 1 D; we shall obtain the following transcendental equations p D g exp 2lH exp 2 j 2 j c;H " !# 4mp p 1 2 exp 22lH 2 D 1 ; 11 j c;H for j . j cp ;
E.A. Shadchin, A.I. Barabash / Journal of Molecular Structure 613 (2002) 55±59
Fig. 2. Dependencies lH R and mH R; obtained from Eqs. (10) and (11).
and
"
(
!#
D 2g exp 2lH exp 2 j 2 " £ 1 2 exp 2
2lpH
!#
12
p 2 j 2 j c;H
2 2 exp "
2
lp D 2mp 1 pH £ 1 2 exp 2 H 2 2 2 j c;H for j # j cp : p 221 lpH lH p ; 2
mpH mH
sion and inaccuracy, one could make a conclusion about the fair agreement between the experimental and theoretical data. One should note that the best agreement was achieved if coef®cient g in Eqs. (11) and (12) is equal to < 1. The other parameters were taken as a3A 21 ; b 3:9 A 21 ; D0 follows: r0 0:95 A; 272 KJ=mol: The dependencies l R and m R are shown in Fig. 2. The presented dependence l R agrees with the same dependence l R inRef. [8]. To the best of our knowledge the dependence m R was not previously published. This dependence accounts for both the threedimensional model of the H-bond, and the interactions between the in®nite chains of the H-bonds in crystal.
5. Conclusions
p j c;H
4mp 2 A 1 pH j c;H !
59
!#)
p 221 p : 2
The values of l H and m H are determined by Eq. (6), if m is the mass of the proton. In Eqs. (11) and (12) 1 lD ù lH p ; 2 since the following relations are true v vD pH hvH;D q kT; and mD 2mH : 2 The dimensional value D=a ; kRD 2 RH l as a function of the H-bond kRl length is shown in Fig. 1. There is satisfactory agreement between the experimental results [4] and our theoretical data. One should note that, in the range R , Rc ; the theoretical curve (solid line in Fig. 1) lies below zero, as the experimental data do. Taking into account that the experimental points presented for the range R . Rc are characterized by signi®cant disper-
In this work we propose the approach to explain the geometric isotopic effect based on the three-dimensional model of hydrogen bond. The free energy for some model crystal composed from the interacting in®nite hydrogen-bonded chains is obtained in the Einstein anharmonic oscillator approximation. It is shown that the critical length of the hydrogen bond (the length at which the hydrogen bond is characterized by the one-minimum potential for proton) determined by the contribution of the longitudinal and the transversal ¯uctuations of the protons (or the deuterons), and the contribution from the transversal ¯uctuations is negative. It was also shown that the obtained curve D R (solid line in Fig. 1) lies below zero in the range R , Rc ; and has minimum.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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