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Thermodynamic functions of a morse oscillator
Volume
S2, number
3
CHEMICAL
THERMODYNAMIC
FUNCTIONS
V.T. A&IOREBIETA
and A.J. COLUSSI
PHYSICS
15 September
LETTERS
1981
OF A MORSE OSCILLATOR
Dcpartamen to de Qnbnica, Universrdad de llfar dei Plata, Wfar del Platg Argentina Received
12 March 1981;
in final form 9 June 1981
The partition function Q. entropy S and heat capacity C of a Morse oscillator are numerically evaluated from its overtone vldratlonal spectrum for typical values of the experimentally accesible fundamental frequency v and dissociation energy D. We show that significant departures from hmiting harmomc oscillator values are to be expected for shallow potential wells.
1. Introduction Application of statistical mechanics to the evaluatlon of thermodynamrc properties of gases usually proceeds from known or estimated structura! and spectroscopic parameters through the analytical expression of a factorized partition function m terms of the several degrees of freedom involved [I] _ In some cases, however, the assumed rigid-rotorharmonic-oscillator approximation is insufficlent and more general procedures are required. Thus, with the exception of hindered internal rotation, a motion arising very often in moderately complex molecules and which has been exactly solved [2], small rovlbrational couplings and anharmonic corrections have been handled by perturbation techmques, at least for diatomics and simple triatomics [3,4]. Closed expressions of the semiclassical partition function for the particular case of an even-power oscillator: H = p2/2m + AXIS, k > 1 are also available [5,6] _ Both methods are naturally restricted, however, to deal with small deviations or rather unrealistic potentials. In this paper we try to assess the accuracy of the harmonic-oscillator approximation in the extreme case of a weakly bound dimer in which the potential function V(r) is fully explored at accessible temperatures. Here, we have chosen the Morse potential, V(r) = D { 1 - exp [fz(ro - r)] }? 530
(1)
to model a one-dimensional anharmonic oscillator, since its Schradinger equation is amenable to analytical solution. Its energy levels, originally obtained by Morse [7], are given by E,, = D - LI-~(A- - rz - $)2,
(3
where A = 20 1/2/hv
(3)
and hr;=_a’l’
(4)
3
with v the harmonic frequency and IZ= 0, 1, 2, ___izbt
0 009-2614/81/0000-0000/S
02.75
hvx,(n
+
i)‘.
0 1981 North-Holland
Volume82,number
identities (3) and (4) reduce to A = 2D1/2/v’ and K = 2D/v’. Defming now the adrmensrona1 parameters x = pv’ and y = PO, we obtain a general expression for BE, :
2. Results and discussion By definition, given by
the canonical partition
function is
HM
/3E, =y (5)
Q=,go exp(--P&l-
tree that in this case E = 0 corresponds to the bottom of V(T). eq. (I), since E. =D - Av2(K
S=kbQ
expressions
for the entropy
[ 1]
-wQ)aQ/wl
-
Notice that since the number of bound levels is given by the integer portion of nM, Q as well as S and C are essentially drscontinuous functions of (x,y)_ While this fact does not affect the data presented in tables 1-3, it has to be borne in mind in relation to figs. 1 and 2, particularly for s-mall values of the ratioylx, where continuous curves have been drawn for convenience. It is apparent that for a given s all functions approach monotonously their harmonic limits at large values ofy, as expected_ More strikingly, the ratios sM/sH and Cbr /CH display local maxima at y = 3 and y = 6 respectively, reflecting the competition between denser vibrational manifolds and lower cut-offs nhl The range of (0, v) values where eq. (2) is a valid approximation to the energy eigenvalues of a Morse
(61
- Q-‘(aQ/ap)2]
(7)
contam the first and second derivatives of Q wrth respect to p. Thus, the t!uee finite sums (5) (8) and (9): ‘1M aQia0 = - ?gu E,, exp( -WJT
(8)
“bl aamv2
(9)
= ,go EZ exp( - W,,)
have to be numerically evaluated for each set {E,,). Wrth D and v both measured in enre,T units, the basrc Table 1 Partition functionofa Morse 0sciIlator _~_________---~Y
s 0.10
-_--__-
0.20
030
2.604 3.282 3.885 5.028 5.446 5.815 6.590 6.833 7.046 8.187 8572 7.710 7.065 6.729 6.660
l-745 2.458 3.078 3.616 4.081 4.482 4.827 5.121 5.372 6.137 6.392 5.777 5.295 5.043 4.992
1.632 2268 2.202 2.714 3.146 3.061 3.407 3.697 4.084 4.209 3.842 3.523 3.355 3.321
0.2 0.3 0.4 0.5 06 0.7 08 0.9 1.0 1.5 2.0 5.0 10.0 50.0 harm.
3503 4.928 6.168 7.244 8.174 8.976 9.666 10.255 10.757 12.286 12.796 11.566 10.603 10.099 9.996
-.-.___-__
______~_____
____0 15
(10)
[31-
and heat capacity: C= k@ [Q-‘$Q/ap*
- (x2/4>j)(2y/x--n -+)2.
As a consequence. it is possible to report numerrcal values of Q(x,J~), S(x ,JJ) and C(x,~) for d Morse oscillator, tables 1,2 and 3 and figs. 1 and 2, very much hke Einstein functions for the harmonic oscillator are tabulated m terms of the single variable x
P=(kT’)-l.No
-i)>O_ The statistical
15September1981
CHEMICALPHYSICSLETTJZRS
3
0.40 ---
1.527 2.099 2.028 2.477 2.401 2.752 2.674 3.168 3.183 2.876 2.635 2.509 2.483
0.50
060
0.80
1.00
2.00
5.00
1.429 1.396 1.874 1.824 2.194 2.132 2.438 2.539 2.293 2.101 2.000 1.979 -
1.339 1.308 1.736 1.686 2.016 2.024 2.154 1.906 1.743 1.659 1.642
1.177 1.150 1.497 1.560 1.568 1.417 1.293 1.230 1.217
1.037 1.190 1.240 1.117 1.020 0.970 0.960
0.570 0.506 0.457 0.431 0.426
0.121 0.098 0.085 0.083
531
3
Volume82,number
CHEMICAL
Table 2 Entropy of a Morse osdator ______-_-I~____--I_-_~ 5 ~_
Y
_-_
02
1151
9.12
03 0.4 05
14.86 17.23
11.49 13.32
19.05 20 52 21.75
16.08 17.15 18-08 19.68
0.6
0.7 0.8 0.9 1.0 1.5 2.0 3.0 4.0 5.0 60 7.0 10.0 50.0 harm.
---
22.80 23.71 2450 27.34 28.99 30.40 30.52 30.16 29.70 29.29 2858 27.66 27.45
18.74 21.59 23.22 24.64 24.77 24.39 23.93 23.56
22.80 21.88 21 72
- .---.
.
Table 3 Heat capacity ofa Morse oscillator (J K-’
-----
---~-___-._--.~--I_-__---~
_v
-___
5.73 9.04 9.00 11.34 11.30 13.05 13.01 16.36 17.49 18.91 19.04 18.70 18.24 17.82 17.07 16.15 15 98
11.40 13.20 13.15 14.59 15.78 18.23 19.70 21 29 21.47 21.03 20 59 20.19 19.45 18 53 18.37
1.5 2.0 3.0 4.0 5.0 60 7.0 10.0 50.0 harm.
-_._--
5.70 5 68 8.97 8.93 11.23 11.16 14.01 15.67 17 09 17.22 16 86 16.40 16.00 15.26 14.34 14.16
5.69 5 -65 8.91 8.87 11.17 12.51 14.56 15.61 15.65 15 -44 14.94 14.52 13.77 12.84 12.68
.-- _ ._ _I.___
-.--.--
0.80
1.00
2.00
5 00
5.60 5.56 8.83 10.67 11.84 13.67 13.43 13.18 12.64 12.26 11.46 10.54 10.38
5.52 8 37 10.08 11.55 11.72 11.34 10 88 10-50 9-75 8.83 8.66
4 85 6.49 6.74 6.44 6.02 5.61 4.90 398 3 81
2.22 1.67 1.80 1.05 0.42 0.33
____..-- --__-_-
mof')
0.10
0.60
0.80
1.00
003 0.29 053 0.84
0.17 0.42 0.75
0.32 0.67
0.50
1.91 3.34 6.55 9.16 10 57 10.98 10.84 9.83 8.49 8.28
--
1.72 3.26 6.32 8.74 10.46 10.75 10 59 9.62 8.24 8.08
1.67 2.97 6.44 8.70 10.38 10 54 10.50 9.41 8.08 7.87
- are -
-----___-
-_-____ -_-_-
oscillator is given by ter Haar’s condition
1.46 2.76 5.86 8.45 9.87 10.33 10.21 9.20 7.82 7.66
[8] :
11 2 1_
For typical values of a = l--2 a-l, r. = 24 A and a lower limit for the bracketed factor of =4. it turns 532
0.60 __
0.50
s
0.2 0.6 0.8 1.0
K [exp@ro)
O-40
5.74 9.08 9.06
13.31 14.77 15.98 17.03 17.95
-
~- l_____--
0.30 -___--
5.73 9.12 11.46
15 September 1981
LETTERS
__________--~-
_-______ 0.20
20.34 20.93 23.97 25.72 27.03 27.13 26.81 26.33 25.92 25.19 24.27 24.10
_-- ----.-
mol-‘)
_ -__-_-
0.15
0.10
__-_I--
(J K’
PHYSICS
~___
---.
__-__I_
1.50
2.00
3.00
4.00
5.00
10.00
1.02 2.48 5.16 7.28 9.46 9.61 9.38 8.49 7.10 6.92
1.63 4.35 6.82 8.28 8.74 8.66 7.66 6.23 6.02
2.79 5.78 5.87 7.04 7.24 6.00 4.36 4.13
3.49 6.12 555 6.08 4.53 2.79 2.53
3.64 3.43 4.69 3.47 1.67 1.42
1.38 0.08 0.04
into K 2 l/4, which is always satisfied whenever rzhI Z 1. The blank spaces at the upper right portions of the numerical
matrices
in tables
l-3
simply
reflect
this criterion.
Some systems actually fzIllwithin the limits where
Volume
82, number
3
CHEMICAL
PHYSICS
LETTERS
15 September
1981
related to the heat of vaporization L,, D * ; L,, = 200-1000 cm-1 [13] and V z 20-100 cm-l_ In this context, our calculations show how the contrrbutions made by the lattice modes to the heat capacity of liquads can be appreciably larger than the harmonic values without invoking configurational effects. Conversely, smaller values do not necessarily call for the onset of free rotation in molecular liquids [14,15]. Finally, the present results represent a second-order approximation to the thermodynamic functions of a generalized oscillator_
Acknowledgement Fig. 1. SM and SH, the entropies of Morse and harmonic cillators, as a function ofx = plzu and y = p0.
os-
This work was supported by the Consejo National de Investigaciones Cientificas y TCcnicas of Argentina_
References [l] [2] [3] [4] [5 ] [6]
Fig. 2. CM and CH, the heat capacities of Morse and harmonic oscillators, as a function of x = phv and y = pO_
171 181 191 [lOI
large anharmonic effects are predicted such as: (1) hydrogen-bonded dirners, whose characteristic pararnecm-l, fi = SOters are in the rangesD * 1000-4800 200 cm- 1 [lo] a(2) van der WaaIs moIecuIes with D = 20-500 cm-l and V = 5-50 cm-l [l l] and (3) vibrations of the lattice modes in solids and liquids [12] _Assuming that D can be identified with the energy required to create a hole in the liquid phase it is
1111 r121 r131 1141
P51
J.E. Mayer and M.G. Mayer, Statistical mechanics, 2nd Ed. (WiIey, New York, 1977). J-E. Kilkpatrick and KS PItzer, J. Chem. Phys. 17 (1949) 1064. K S. Pitzer and L Brewer, Thermodynamics (McGrawHill, New York, 1961). H.W. WooIIey, J. Res. Natl. Bur. Srd. US 56 91956) 10.5. IV_ Wtschel, Chem. Phys. Letters 71 (1980) 131. hi. Schwarz, J. Stat. Phys. 15 (1976) 255. P.M. Morse, Phys Rev. 34 (1929) 57. D. ter Haar, Phys. Rev. 70 (1946) 222. 1-N Levine, MolecuIar spectroscopy OVrley, New York, 1975). P. Schuster, G. Zundel and C. Sandorfy, eds., The hydrogen bond, Vols. 2 and 3 (North-Holland, Amsterdam, 1976). G-E. Ewing, Accounts Chem. Res. 8 (1975) 185. T.L. HII& An introduction to statistical thermodynamics (Addison-Wesley, Reading, 1960). R.S. Berry, S-A. Rice and J. Ross, Physical chemistry (WiIey, New York, 1980) ch. 29. E-A- Moehvyn-Hughes, Physical chemistry, 2nd Ed. (Pergamon, London, 1965). S-W. Benson, J. Chem. Sot. 100 (1978) 5640.
533
S2, number
3
CHEMICAL
THERMODYNAMIC
FUNCTIONS
V.T. A&IOREBIETA
and A.J. COLUSSI
PHYSICS
15 September
LETTERS
1981
OF A MORSE OSCILLATOR
Dcpartamen to de Qnbnica, Universrdad de llfar dei Plata, Wfar del Platg Argentina Received
12 March 1981;
in final form 9 June 1981
The partition function Q. entropy S and heat capacity C of a Morse oscillator are numerically evaluated from its overtone vldratlonal spectrum for typical values of the experimentally accesible fundamental frequency v and dissociation energy D. We show that significant departures from hmiting harmomc oscillator values are to be expected for shallow potential wells.
1. Introduction Application of statistical mechanics to the evaluatlon of thermodynamrc properties of gases usually proceeds from known or estimated structura! and spectroscopic parameters through the analytical expression of a factorized partition function m terms of the several degrees of freedom involved [I] _ In some cases, however, the assumed rigid-rotorharmonic-oscillator approximation is insufficlent and more general procedures are required. Thus, with the exception of hindered internal rotation, a motion arising very often in moderately complex molecules and which has been exactly solved [2], small rovlbrational couplings and anharmonic corrections have been handled by perturbation techmques, at least for diatomics and simple triatomics [3,4]. Closed expressions of the semiclassical partition function for the particular case of an even-power oscillator: H = p2/2m + AXIS, k > 1 are also available [5,6] _ Both methods are naturally restricted, however, to deal with small deviations or rather unrealistic potentials. In this paper we try to assess the accuracy of the harmonic-oscillator approximation in the extreme case of a weakly bound dimer in which the potential function V(r) is fully explored at accessible temperatures. Here, we have chosen the Morse potential, V(r) = D { 1 - exp [fz(ro - r)] }? 530
(1)
to model a one-dimensional anharmonic oscillator, since its Schradinger equation is amenable to analytical solution. Its energy levels, originally obtained by Morse [7], are given by E,, = D - LI-~(A- - rz - $)2,
(3
where A = 20 1/2/hv
(3)
and hr;=_a’l’
(4)
3
with v the harmonic frequency and IZ= 0, 1, 2, ___izbt
0 009-2614/81/0000-0000/S
02.75
hvx,(n
+
i)‘.
0 1981 North-Holland
Volume82,number
identities (3) and (4) reduce to A = 2D1/2/v’ and K = 2D/v’. Defming now the adrmensrona1 parameters x = pv’ and y = PO, we obtain a general expression for BE, :
2. Results and discussion By definition, given by
the canonical partition
function is
HM
/3E, =y (5)
Q=,go exp(--P&l-
tree that in this case E = 0 corresponds to the bottom of V(T). eq. (I), since E. =D - Av2(K
S=kbQ
expressions
for the entropy
[ 1]
-wQ)aQ/wl
-
Notice that since the number of bound levels is given by the integer portion of nM, Q as well as S and C are essentially drscontinuous functions of (x,y)_ While this fact does not affect the data presented in tables 1-3, it has to be borne in mind in relation to figs. 1 and 2, particularly for s-mall values of the ratioylx, where continuous curves have been drawn for convenience. It is apparent that for a given s all functions approach monotonously their harmonic limits at large values ofy, as expected_ More strikingly, the ratios sM/sH and Cbr /CH display local maxima at y = 3 and y = 6 respectively, reflecting the competition between denser vibrational manifolds and lower cut-offs nhl The range of (0, v) values where eq. (2) is a valid approximation to the energy eigenvalues of a Morse
(61
- Q-‘(aQ/ap)2]
(7)
contam the first and second derivatives of Q wrth respect to p. Thus, the t!uee finite sums (5) (8) and (9): ‘1M aQia0 = - ?gu E,, exp( -WJT
(8)
“bl aamv2
(9)
= ,go EZ exp( - W,,)
have to be numerically evaluated for each set {E,,). Wrth D and v both measured in enre,T units, the basrc Table 1 Partition functionofa Morse 0sciIlator _~_________---~Y
s 0.10
-_--__-
0.20
030
2.604 3.282 3.885 5.028 5.446 5.815 6.590 6.833 7.046 8.187 8572 7.710 7.065 6.729 6.660
l-745 2.458 3.078 3.616 4.081 4.482 4.827 5.121 5.372 6.137 6.392 5.777 5.295 5.043 4.992
1.632 2268 2.202 2.714 3.146 3.061 3.407 3.697 4.084 4.209 3.842 3.523 3.355 3.321
0.2 0.3 0.4 0.5 06 0.7 08 0.9 1.0 1.5 2.0 5.0 10.0 50.0 harm.
3503 4.928 6.168 7.244 8.174 8.976 9.666 10.255 10.757 12.286 12.796 11.566 10.603 10.099 9.996
-.-.___-__
______~_____
____0 15
(10)
[31-
and heat capacity: C= k@ [Q-‘$Q/ap*
- (x2/4>j)(2y/x--n -+)2.
As a consequence. it is possible to report numerrcal values of Q(x,J~), S(x ,JJ) and C(x,~) for d Morse oscillator, tables 1,2 and 3 and figs. 1 and 2, very much hke Einstein functions for the harmonic oscillator are tabulated m terms of the single variable x
P=(kT’)-l.No
-i)>O_ The statistical
15September1981
CHEMICALPHYSICSLETTJZRS
3
0.40 ---
1.527 2.099 2.028 2.477 2.401 2.752 2.674 3.168 3.183 2.876 2.635 2.509 2.483
0.50
060
0.80
1.00
2.00
5.00
1.429 1.396 1.874 1.824 2.194 2.132 2.438 2.539 2.293 2.101 2.000 1.979 -
1.339 1.308 1.736 1.686 2.016 2.024 2.154 1.906 1.743 1.659 1.642
1.177 1.150 1.497 1.560 1.568 1.417 1.293 1.230 1.217
1.037 1.190 1.240 1.117 1.020 0.970 0.960
0.570 0.506 0.457 0.431 0.426
0.121 0.098 0.085 0.083
531
3
Volume82,number
CHEMICAL
Table 2 Entropy of a Morse osdator ______-_-I~____--I_-_~ 5 ~_
Y
_-_
02
1151
9.12
03 0.4 05
14.86 17.23
11.49 13.32
19.05 20 52 21.75
16.08 17.15 18-08 19.68
0.6
0.7 0.8 0.9 1.0 1.5 2.0 3.0 4.0 5.0 60 7.0 10.0 50.0 harm.
---
22.80 23.71 2450 27.34 28.99 30.40 30.52 30.16 29.70 29.29 2858 27.66 27.45
18.74 21.59 23.22 24.64 24.77 24.39 23.93 23.56
22.80 21.88 21 72
- .---.
.
Table 3 Heat capacity ofa Morse oscillator (J K-’
-----
---~-___-._--.~--I_-__---~
_v
-___
5.73 9.04 9.00 11.34 11.30 13.05 13.01 16.36 17.49 18.91 19.04 18.70 18.24 17.82 17.07 16.15 15 98
11.40 13.20 13.15 14.59 15.78 18.23 19.70 21 29 21.47 21.03 20 59 20.19 19.45 18 53 18.37
1.5 2.0 3.0 4.0 5.0 60 7.0 10.0 50.0 harm.
-_._--
5.70 5 68 8.97 8.93 11.23 11.16 14.01 15.67 17 09 17.22 16 86 16.40 16.00 15.26 14.34 14.16
5.69 5 -65 8.91 8.87 11.17 12.51 14.56 15.61 15.65 15 -44 14.94 14.52 13.77 12.84 12.68
.-- _ ._ _I.___
-.--.--
0.80
1.00
2.00
5 00
5.60 5.56 8.83 10.67 11.84 13.67 13.43 13.18 12.64 12.26 11.46 10.54 10.38
5.52 8 37 10.08 11.55 11.72 11.34 10 88 10-50 9-75 8.83 8.66
4 85 6.49 6.74 6.44 6.02 5.61 4.90 398 3 81
2.22 1.67 1.80 1.05 0.42 0.33
____..-- --__-_-
mof')
0.10
0.60
0.80
1.00
003 0.29 053 0.84
0.17 0.42 0.75
0.32 0.67
0.50
1.91 3.34 6.55 9.16 10 57 10.98 10.84 9.83 8.49 8.28
--
1.72 3.26 6.32 8.74 10.46 10.75 10 59 9.62 8.24 8.08
1.67 2.97 6.44 8.70 10.38 10 54 10.50 9.41 8.08 7.87
- are -
-----___-
-_-____ -_-_-
oscillator is given by ter Haar’s condition
1.46 2.76 5.86 8.45 9.87 10.33 10.21 9.20 7.82 7.66
[8] :
11 2 1_
For typical values of a = l--2 a-l, r. = 24 A and a lower limit for the bracketed factor of =4. it turns 532
0.60 __
0.50
s
0.2 0.6 0.8 1.0
K [exp@ro)
O-40
5.74 9.08 9.06
13.31 14.77 15.98 17.03 17.95
-
~- l_____--
0.30 -___--
5.73 9.12 11.46
15 September 1981
LETTERS
__________--~-
_-______ 0.20
20.34 20.93 23.97 25.72 27.03 27.13 26.81 26.33 25.92 25.19 24.27 24.10
_-- ----.-
mol-‘)
_ -__-_-
0.15
0.10
__-_I--
(J K’
PHYSICS
~___
---.
__-__I_
1.50
2.00
3.00
4.00
5.00
10.00
1.02 2.48 5.16 7.28 9.46 9.61 9.38 8.49 7.10 6.92
1.63 4.35 6.82 8.28 8.74 8.66 7.66 6.23 6.02
2.79 5.78 5.87 7.04 7.24 6.00 4.36 4.13
3.49 6.12 555 6.08 4.53 2.79 2.53
3.64 3.43 4.69 3.47 1.67 1.42
1.38 0.08 0.04
into K 2 l/4, which is always satisfied whenever rzhI Z 1. The blank spaces at the upper right portions of the numerical
matrices
in tables
l-3
simply
reflect
this criterion.
Some systems actually fzIllwithin the limits where
Volume
82, number
3
CHEMICAL
PHYSICS
LETTERS
15 September
1981
related to the heat of vaporization L,, D * ; L,, = 200-1000 cm-1 [13] and V z 20-100 cm-l_ In this context, our calculations show how the contrrbutions made by the lattice modes to the heat capacity of liquads can be appreciably larger than the harmonic values without invoking configurational effects. Conversely, smaller values do not necessarily call for the onset of free rotation in molecular liquids [14,15]. Finally, the present results represent a second-order approximation to the thermodynamic functions of a generalized oscillator_
Acknowledgement Fig. 1. SM and SH, the entropies of Morse and harmonic cillators, as a function ofx = plzu and y = p0.
os-
This work was supported by the Consejo National de Investigaciones Cientificas y TCcnicas of Argentina_
References [l] [2] [3] [4] [5 ] [6]
Fig. 2. CM and CH, the heat capacities of Morse and harmonic oscillators, as a function of x = phv and y = pO_
171 181 191 [lOI
large anharmonic effects are predicted such as: (1) hydrogen-bonded dirners, whose characteristic pararnecm-l, fi = SOters are in the rangesD * 1000-4800 200 cm- 1 [lo] a(2) van der WaaIs moIecuIes with D = 20-500 cm-l and V = 5-50 cm-l [l l] and (3) vibrations of the lattice modes in solids and liquids [12] _Assuming that D can be identified with the energy required to create a hole in the liquid phase it is
1111 r121 r131 1141
P51
J.E. Mayer and M.G. Mayer, Statistical mechanics, 2nd Ed. (WiIey, New York, 1977). J-E. Kilkpatrick and KS PItzer, J. Chem. Phys. 17 (1949) 1064. K S. Pitzer and L Brewer, Thermodynamics (McGrawHill, New York, 1961). H.W. WooIIey, J. Res. Natl. Bur. Srd. US 56 91956) 10.5. IV_ Wtschel, Chem. Phys. Letters 71 (1980) 131. hi. Schwarz, J. Stat. Phys. 15 (1976) 255. P.M. Morse, Phys Rev. 34 (1929) 57. D. ter Haar, Phys. Rev. 70 (1946) 222. 1-N Levine, MolecuIar spectroscopy OVrley, New York, 1975). P. Schuster, G. Zundel and C. Sandorfy, eds., The hydrogen bond, Vols. 2 and 3 (North-Holland, Amsterdam, 1976). G-E. Ewing, Accounts Chem. Res. 8 (1975) 185. T.L. HII& An introduction to statistical thermodynamics (Addison-Wesley, Reading, 1960). R.S. Berry, S-A. Rice and J. Ross, Physical chemistry (WiIey, New York, 1980) ch. 29. E-A- Moehvyn-Hughes, Physical chemistry, 2nd Ed. (Pergamon, London, 1965). S-W. Benson, J. Chem. Sot. 100 (1978) 5640.
533