J. Phys. Chem. B 2004, 108, 13821-13826
13821
On the Relationship between Thermal Diffusion and Molecular Interaction Energy in Binary Mixtures Hang Li† and Laosheng Wu* Department of EnVironmental Sciences, UniVersity of California, RiVerside, California 92521 ReceiVed: May 19, 2004; In Final Form: June 28, 2004
Characterizing thermal diffusion processes caused by a temperature gradient in binary mixtures is difficult. The classic theory employed to interpret the experimental data does not fully explain the physical meaning of the Soret coefficient and its sign change. In this study, a new thermal diffusion equation is derived based on thermal dynamic theory. The new equation relates thermal diffusion to the net molecular interaction energy of the components in a mixture. It was also demonstrated that the Soret coefficient is a simple function of the difference between the enthalpy of a component in the mixture and its enthalpy at reference state in the mixture. Theoretical analysis of experimental data obtained by other researchers shows that the new equations quantitatively explain the influence of molecular interactions on the Soret effect. Additionally, they qualitatively explain why the Soret coefficient changes its sign as molar fraction increases from 0 to 1 in the mixtures.
1. Introduction Mass diffusion in a homogeneous solution can be induced by imposing a temperature gradient. This process results in the establishment of a concentration gradient in the mixture when the nonequilibrium stationary state is reached. The process is traditionally referred to as thermal diffusion or the Soret effect. Even though the pure Soret effect is considered to be small, it is an important process in nature. In industry, thermal diffusion has been used to effectively separate compounds in a gas or solution mixture.1,2 Costese`que et al. (1987),3,4 Montel (1994),5 and Legros et al. (1994)6 reported that the Soret effect played an important role in the distribution of components along a geothermal gradient in natural reservoirs of oil. The Soret coefficient is a basic parameter for characterizing thermal diffusion, and it can be experimentally determined. There are many experimental methods available to determine the Soret coefficient, including the thermo-gravitational method,7-9 the forced Rayleigh scattering method,10-14 the optical beamdeflection method,15-18 the microgravity method,19 the Thomaes cells method,20 and the Be´nard cells method.21,22 At present, a large amount of thermal diffusion data is available for various liquid mixture systems.1,17,18,23-27 Experimental data show that in dilute solutions Soret coefficients are very small, but they become large for solutions containing compounds of very different molecular weight or for highly nonideal conditions.1,28 Additionally, the Soret coefficient increases as the viscosity of a mixture increases.18 All of these findings imply that the Soret coefficient is related to molecular interactions in mixture.1,18,29 For some solution mixtures, the Soret coefficient may remain either positive or negative as the molar fraction of a component increases from 0 to 1;18 while for other mixtures, it may change from positive to negative as the molar fraction of a component increases from 0 to 1.17,18,27 The sign change of the Soret coefficients in different mixtures once again implies that the Soret coefficient is sensitive to molecular interactions in a mixture. * To whom correspondence may be addressed. E-mail: Laowu@ mail.ucr.edu. Tel.: (951) 827-4664. Fax: (951) 827-4664. † Formerly the Institute of Soil Science, Chinese Academy of Sciences, Nanjing 210008; and College of Resources and Environment, Southwest Agricultural University, Chongqing 400716, P.R. China.
Overall, thermal diffusion is “particularly ill-characterized”14 as a thermophysical property of binary mixtures. The theoretical explanation of thermal diffusion is still ambiguous.1,18,29 For dilute gas mixtures, the thermal diffusion can be well accounted for by the Chapman-Enskog solution of the Boltzmann equation. However, this approach and the subsequent revised Enskog Theory (RET) are not favorable quantitative tools when applied to dense gas and liquid mixtures.25,30,31 A few new approaches for calculating the Soret coefficient based on the molecular dynamic methods have been developed since the 1980s, and some reasonable results have been obtained.25,31,32 Nevertheless, overall these new calculation methods are far from satisfactory.29,34-36 Macroscopically, for a binary mixture of two components “1” and “2”, the classic expression of mass flux due to thermal diffusion for component “1” is1,18
j1 ) -Dc∇X1 + DRcX1(1 - X1)
1 ∇T T
(1)
or
j1 ) -Dc∇X1 - DSTcX1(1 - X1)∇T
(2)
where j1 is the mass flux, D is the diffusion coefficient, c is the total mole concentration of component “1” plus component “2” (mol/L), X1 is the mole fraction, T is the absolute temperature, R is the dimensionless thermal diffusion factor, and ST is the Soret coefficient with unit K-1. Obviously, ST ) -R/T. The classic theory for dealing with the Soret coefficient in eq 2 (or the thermal diffusion factor in eq 1) in macroscopic theory is the linear phenomenological theory of irreversible processes. For a binary mixture, in the absence of external forces and chemical reactions, the entropy production can be expressed as2
σ)-
( )
1 ∂µ′1 1 j ∇T 2 q w2T ∂X1 T
j1∇X1
(3)
T,P
where σ is the entropy production, jq is heat flux, w2 is the mass
10.1021/jp0478357 CCC: $27.50 © 2004 American Chemical Society Published on Web 08/13/2004
13822 J. Phys. Chem. B, Vol. 108, No. 36, 2004
Li and Wu
fraction of component 2, and µ′1 is the chemical potential of component 1. In some literature, either the mass fraction or the molar concentration was used for X1 instead of the mole fraction as in eq 3.2,29 The corresponding linear phenomenological equations are
( ) ( )
1 1 ∂µ′1 jq ) -Lqq 2∇T - Lq1 w2T ∂X1 T
T,P
1 1 ∂µ′1 j1 ) -L1q 2∇T - L11 w2T ∂X1 T
T,P
∇X1
(4)
∇X1
(5)
A comparison between eq 5 and eq 2 gives
ST )
1 L1q DcX1(1 - X1)T2
(6) 2. Theory
From eqs 2 and 5, D can be expressed as
D)
( )
L11 ∂µ′1 cw2T ∂X1
(7)
ST )
L1q L11
( )
∂µ′1 TX1(1 - X1) ∂X1
µ1 ) µ10 + RT ln X1
(8)
T,P
Tyrrell (1961)7 established a macroscopic thermodynamic relationship for calculating the Soret coefficient, in which the heat of transfer (QT) at a constant temperature replaced the phenomenological coefficients in eq 8. According to the Onsager reciprocity relations, under constant temperature, from eqs 4 and 5, we have
()
jq Lq1 L1q ) ) L11 L11 j1
) QT
(9)
T
where QT is the heat of transfer per mole; it can be considered as the reduced heat transferred across a reference plane in an isothermal system by the passage of one mole of component 1 across the plane.20 Thus eq 8 can be rewritten as
w2QT
( )
ST )
TX1(1 - X1)
∂µ′1 ∂X1
(10)
∂µ′1 ∂X1
) RT
T,P
(
)
1 1 ∂γ1 + X1 γ1 ∂X1
( )
j1 ) c1V ) cX1V
(14)
where c1 is the mole concentration of component “1”, c is the total molar concentration of the mixture, and V is the velocity of diffusion (cm/s). According to eq 13, the mole fraction X1 can be expressed as
X1 ) eµ1-µ1 /RT 0
(15)
The corresponding diffusion velocity can be expressed as2
(
)
0 D µ1 - µ 1 V)- ∇ R T
(16)
If there exists a potential action on the system (e.g., interaction potential between molecules or external electric potential or gravitational potential), eq 16 can be written
)
0 D µ1 + E1 - µ1 V)- ∇ R T
(11)
(17)
where E1 is the molar potential energy of component “1”. Correspondingly, eq 13 can be transformed to
µ′1 ) µ1 + E1 ) µ10 + RT ln X1 + RT ln γ1 )
This allows eq 10 to be written as
w2 ST ) ∂ ln γ1 1+ ∂ ln X1
(13)
where µ1 is the chemical potential and µ10 is the reference chemical potential at reference state. Accordingly, the diffusion flux of component “1” can be expressed as
(
T,P
Since µ′1 ) µ10 + RT ln γ1X1, then
( )
2.1. A General Expression of the Diffusion Equation. In an ideal system where the net interaction energy of a component among molecules can be neglected, the chemical potential is related to the mole fraction of component “1” (X1) by
T,P
Introducing eq 7 into eq 6, one can then obtain
w2
From the above discussions, it is apparent that: (1) although it has been known for a long time that the Soret coefficient or the thermal diffusion is sensitive to molecular interaction, and the most direct way for characterizing the molecular interaction in macroscopic is the net interaction energy for a component in a mixture, the classic equations for calculating the Soret coefficient (eqs 6, 8, 10, and 12) do not explicitly relate thermal diffusion to net interaction energy and that (2) none of the classic macroscopic equations (eqs 6, 8, 10, and 12) for calculating ST explains why the sign of ST changes as the mole fraction of a component in a mixture changes. Thus the objectives of this paper are to develop a new thermal diffusion equation based on the net molecular interaction energy and to use the new equation to explain/evaluate the sign change for ST in different mixtures.
QT (1 - X1)RT2
µ10 + RT ln a1 (18) (12)
T,P
An equation similar to eq 12 was obtained by Agar37 for an electrolyte solution based on the “Soret Equilibrium”.
where µ′1 is the chemical potential of component “1” subject to influence of the potential action, γ1 is the activity coefficient of component 1, and a1 is the activity in which
E1 ) RT ln γ1
(19)
Thermal Diffusion and Molecular Interaction Energy
J. Phys. Chem. B, Vol. 108, No. 36, 2004 13823
[ ( )
Introducing eq 17 into eq 14, we have
(
µ 1 + E1 - µ 1 0 1 j1 ) -Dc eµ1-µ1 /RT∇ R T
ST ) - -T
)
0
j1 e
(
)
0 1 µ1-µ10+E1/RT µ1 - µ1 + E1 ) -Dc e ∇ R T
(21)
H1 ) -T
j1 e
) -Dc∇(X1 e
E1/RT
+ E1
]
1 RT (1 - X1) 2
(31)
Equation 31 shows that the Soret coefficient is related to the net molecular interaction energy. Using H1 to denote the molar enthalpy of component “1” in the mixture, according to the thermodynamic relationship, we have
Equation 21 can be rewritten as E1/RT
P,X1
(20)
Multiplying both sides of eq 20 with exp[E1/(RT)], we get E1/RT
∂E1 ∂T
)
( ) ∂µ′1 ∂T
P,X1
+ µ′1
(32)
(22) Combining eqs 18 and 19
Define J1 as
J1 ) j 1 e
E1/RT
µ′1 ) µ1 + E1 ) µ10 + RT ln X1 + E1
(23)
(33)
Substituting eq 33 into eq 32, then
and from eq 18
a1 ) X1 eE1/RT
(24)
Equation 22 can be expressed as
J1 ) -Dc∇a1
(25)
Since a1 is activity, or “apparent concentration”, then J1 can be considered “apparent flux”. Equation 25 is a general expression of the diffusion flux since the present coupling and noncoupling diffusion equations are the special cases of eq 25: (1) If ∇E1 ) 0 (E1 ) constant or 0) and ∇T ) 0, eq 25 becomes the Fick’s diffusion equation
j1 ) -Dc∇X1
( )
H1 ) -T
ZF j1 ) -Dc∇X1 - DcX1 ∇φ RT
(27)
In this study, we will focus on the application of eq 25 to a system where ∇T * 0 and E1 is caused by the molecular interaction, namely, “thermal diffusion system”. 2.2. The Thermal Diffusion Equation and the Soret Coefficient. If ∇T * 0 and no external forces are present, eq 25 becomes
()
1 E1 j1 ) -Dc∇X1 - DcX1 ∇ R T
()
P,X1
∇T
(29)
or
[ ( ) ]
∂E1 1 j1 ) -Dc∇X1 + DcX1 2 E1 - T ∂T RT A comparison of eq 30 with eq 2 gives
P,X1
H10 ) -T
( )
P,X1
∇T (30)
∂E1 ∂T
P,X1
+ E1
(34)
∂µ10 ∂T
P,X1
+ µ10
(35)
Introducing eq 35 into eq 34
( ) ∂E1 ∂T
H1 - H10 E1 )+ P,X1 T T
(36)
Therefore
()
∂ E1 ∂T T
)-
H1 - H10
P,X1
T2
(37)
The substitution of eq 37 into eq 29 gives
j1 ) -Dc∇X1 + DcX1
H1 - H10 ∇T RT2
(38)
This is a different expression of the thermal diffusion equation from eq 30 based on eq 25. A comparison of eq 2 with eq 38 gives
(28)
where E1 is the molecular interaction energy per mole of component 1 defined by eq 19. Equation 28 can be rewritten as
1 ∂ E1 j1 ) -Dc∇X1 - DcX1 R ∂T T
( )
+ µ10 - T
Given that H10 is the molar enthalpy of component “1” at standard state (reference state), thus
(26)
(2) If ∇E1 ) Z1F∇φ (where Z1 is the charge of the ion, F is the Faraday constant, and φ is the electric potential) and ∇T ) 0, eq 25 becomes the Nernst-Planck equation
∂µ10 ∂T
ST ) -
H1 - H10 RT2(1 - X1)
(39)
Equation 39 shows that the Soret coefficient is also related to the molar enthalpy of component “1” in the mixture. 2.3. Application of the New Equations to the Associate Transport. For the associated transport-like electrolyte diffusion in solution, eq 19 can be expressed as
E(1 ) (V- + V-)RT ln γ(1 ) (V- + V-)E1
(40)
where V+ is the charges of the positive ion, V- is the charges of the negative ion, and γ( is the mean activity coefficient of the electrolyte. The flux and Soret coefficient for the associated
13824 J. Phys. Chem. B, Vol. 108, No. 36, 2004
Li and Wu
transport can be respectively expressed as
( )
∂ E(1 1 j1 ) -Dc∇X1 - DcX1 (V- + V-)R ∂T T
[ ( )
ST ) - -T
∂E1 ∂T
P,X1
∇T (41)
P,X1
]
1 (42) (V1 + V2)RT2(1 - X1)
+ E1
or
ST ) -
H1 - H10
(43)
(V+ + V-)RT2(1 - X1)
Equations 41, 42, and 43 are the expressions of the new equations as applied to associated transport. 3. Discussion 3.1. Comparison of the Classic Equations with the New Equations. Although it has been known for a long time that the Soret effect is sensitive to molecular interactions within solutions, classic equations (eqs 6, 8, 10, and 12) do not quantify these interactions. The new equations derived in this research (eqs 28 and 31), on the other hand, explicitly show that thermal diffusion is related to the molecular interaction energy in a mixture. If the net molecular interaction energy of a component in a mixture is zero, no thermal diffusion will occur, regardless of how strong the temperature gradient is. These new equations theoretically explain the published experimental results and the molecular dynamic simulation results that the thermal diffusion are sensitive to molecular interaction.1,18,27-29 Since there exists little molecular interaction in a nearly ideal mixture (dilute solutions, X1 f 0, or in a very dense solutions, X1 f 1), the Soret coefficients in such mixtures are very small, as experimentally observed,1,27 because such mixtures have the lowest (but not zero) values of net molecular interaction energy. In the classic approach, the phenomenological coefficient in eqs 6 or 8 (L1q or L11) does not have a clear physical meaning. While the terms of ∂ ln γ/∂ ln X1 and QT in eq 12 have a clearly defined physical meaning, up until now the values of ∂ ln γ/∂ ln X1 could only be theoretically estimated in dilute electrolyte solutions. The calculation of QT is difficult too since QT will be equal to the difference of enthalpy for a component between the end and the initial state of a process only when at equilibrium and under constant pressure conditions. However, the end state of thermal diffusion is not at equilibrium but is instead a nonequilibrium stationary state. In the new equation (eq 39), on the other hand, the ST is a simple function of (H1 - H10), which is the difference of the enthalpy of component “1” between the mixed state and the reference state at equilibrium. The term (H1 - H10) is the heat of mixing (or the heat of solution in liquid mixture) under isobaric conditions, it can be easily measured,38 and the interaction energy E can be estimated from the activity coefficient. Thus eqs 31 and 39 may offer two new ways to determine the Soret coefficients. Comparison between eqs 12 and 39 leads to
1 ∂ ln γ1 1+ ∂ ln X1
(
(H10 - H1) ) QTw2
)
T,P
1 X1 ∂γ1 1+ γ1∂X1
(
) Q Tw 2
)
T,P
(44) Equation 44 shows that there is a complex relationship between the heat of transfer QT and (H1 - H10). For a dilute solution,
Figure 1. Comparison between the theoretical and experimental data of ST (solid line, calculated curves using eq 37; ×, +, 2, O, and 0, experimental data for toluene at mole fractions of 0.05, 0.250, 0.500, 0.750, and 0.950, respectively, in mixture of toluene and n-hexane.
since X1 f 0, w2 f 1, we have
H10 - H1 ≈ QT
(45)
A comparison of eq 38 with eq 1 gives
R)
H1 - H10 RT(1 - X1)
(46)
Firoozabadi et al.39 established an equation for isobaric condition based on the theory of thermodynamics of irreversible processes and molecular kinetics that is similar to eq 46
R)
Q/1 RT(1 - X1)
(47)
where Q/1 is the net heat of transport of component 1. Comparison between eq 46 and eq 47 shows
Q/1 ) H1 - H10
(48)
Equation 48 implies that the net heat of transport for component 1 under isobaric conditions is equal to the difference of the molar enthalpy between the mixed state and the reference state for component 1. Haase40 and Kempers41 also derived expressions for thermal diffusion factor R. Their expressions, however, are rather complex and involve both enthalpy and chemical potential of the components. Besides, agreement between their theoretical prediction and experimental data shows to be only qualitative.42 Both the new equations (eqs 31 and 39) and the classic equations (eqs 6 and 12) show that ST is explicitly proportional to 1/T2. Some experimental results indeed manifested this relationship. To validate the new equations, the experimental data obtained by Zhang et al.18 were used as an example. By choosing T ) 298 K as a reference point, values of (H10 - H1) were calculated for each mole fraction of toluene at 298 K. By use of the reference (H10 - H1) values, the values of ST at different temperatures were then predicted using eq 39. Comparison between the theoretical curves predicted from eq 39 and the experimental data showed that they matched very well (Figure 1). The new equation, however, like the classic equation (eq 6), does not explicitly explain why in some mixtures ST increases as temperature decreases.17 One possible explanation is that, implicitly, ST is not exactly proportional to 1/T2 since strictly speaking both H10 and H1 are functions of temperature. 3.2. Change of Sign of ST Based on the New Equations. It is well known that, for some mixtures, especially for mixtures
Thermal Diffusion and Molecular Interaction Energy
J. Phys. Chem. B, Vol. 108, No. 36, 2004 13825
Figure 2. The experimental results of E0 vs X1 for some salts, acids, and alkalis.
containing polar molecules, the sign change of ST may take place as the molar fraction increases from 0 to 1 for a component. To show that the sign change of the Soret coefficient as the mole fraction changes for a component is the natural result of the new equations, let us consider a mixture with a temperature gradient. If the average temperature is T0 and the molecular interaction energy corresponding to T0 is E0, application of the Taylor expansion for E1 ≡ E(T) at the point (T0, E0) gives
E1 ) E0 + E′(T0)(T - T0) +
1 E′′(T0)(T - T0)2 + .... (49) 2!
When E′(T0) and/or (T - T0) approach 0, we have E1 ≈ E0. Thus eq 31 can be approximated as
ST = -
E0 RT (1 - X1) 2
(50)
Equation 50 shows that the sign of E0 exclusively determines the sign of ST as E′(T0) f 0 and/or (T - T0) f 0. If E0 > 0, there will be ST < 0; if E0 < 0, there will be ST > 0; and if E0 changes from negative to positive, ST will change from positive to negative. According to eq 19, the sign of E0 exclusively depends on the activity coefficient. If the activity coefficient is greater than 1, E0 > 0; if the activity coefficient lower than 1, E0 < 0. Macroscopically, when considering binary mixtures containing two types of molecules, Component 1 (C1) and Component 2 (C2), if the net attraction (molecular attraction) force between molecules of C1 is greater than the attraction force between molecules of C1 and C2, the activity coefficient for C1 will be greater than 1;43 thus E0 > 0 and ST < 0. However, if the net interaction force between C1 and C2 is stronger than that of between C1 and C1 in the mixture, the activity coefficient for C1 will be smaller than 1,43 and therefore E0 < 0 and ST > 0. As shown by Moore,43 the activity coefficient could remain either greater than 1 or lower than 1 as the molar fraction increases from 0 to 1, which implies the values of ST may remain either negative or positive in mixtures as molar fractions change. Many experimental results showed that the change of sign of ST commonly occurred in polar mixtures as the molar fraction increases from 0 to 1. Figure 2 shows the experimental curves of E0 vs X1 for some salts, acids, and alkalis in aqueous solutions. The values of E0 were calculated from eq 40 using the experimental data of the mean activity coefficients.44 In
Figure 3. The sign change of ST for some salts, acids, and alkalis in aqueous solution calculated from eq 50 based on the experimental data of E0.
Figure 2, although the range of molar fractions is relatively small (0 to 0.1),44 a sign change for E0 is noted. If considering a full range of molar fractions from 0 to 1, then sign changes from negative to positive for E0 should be even more widely expected. According to eq 50, the sign change of E0 will lead to the sign change of ST. The corresponding curves of ST vs X1 based on Figure 2 are shown in Figure 3. It shows that the sign change of ST is a natural result of the new thermal diffusion equations. The approximate expression of eq 50 is correct only when (T - T0) and/or (∂E1/∂T)P,X1 approaches 0. If an experiment does not meet the above condition, eq 31 should be used to construct the “ST-X1” curves, and the shape of the curves may be more complex, as observed by Gaeta.45 4. Conclusions It has been long known that the Soret effect is related to molecular interactions in solution mixtures. Nevertheless, no quantitative relationship between the two was documented in the literature. In this study, a new flux equation for thermal diffusion was developed based on thermal dynamics. For the first time, it is theoretically demonstrated that the Soret effect is quantitatively related to molecular interactions. The thermal diffusion flux and thus the Soret coefficient in the new equations are a function of molecular interaction energy. Further analysis showed that the new equations (flux and Soret coefficient) are a function of the difference of the enthalpy in the mixed state
13826 J. Phys. Chem. B, Vol. 108, No. 36, 2004 and the reference state for a component in a mixture. Like the classic equation, the new equation for ST is explicitly proportional to 1/T2, but implicitly the relationship between ST and T could be more complex since the enthalpy is a function of T. The ST and molecular interaction energy are explicitly related in the new equation. On the basis of the new equation, the sign change of ST with molar fractions was qualitatively demonstrated by imposing a small perturbation of temperature gradient to the mixture. Results showed that the sign of the net molecular interaction energy for a component in the mixture determines the sign of ST. For a mixture containing C1 and C2, the ST for C1 may remain either positive, negative, or change from positive to negative or vice versa as the mole fraction of C1 increases from 0 to 1. It depends on the net relative strength of interactions between C1 molecules themselves as compared to between C1 and C2 molecules in the mixture. If C1-C1 interactions are stronger than C1-C2 interactions, the ST value for C1 will be negative; vice versa, if C1-C2 interactions are stronger than C1-C1 interactions, the ST value of C1 will be positive. References and Notes (1) Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems, 2nd ed.; Cambridge University Press: New York, 1997. (2) De Groot, S. R.; Mazur, P. Nonequilibrium Thermodynamics; NorthHolland Publishing Co.: Amsterdam, 1962. (3) Costese`que, P.; Hridabba, M.; Sahores, J. C. R. Acad. Sci. Ser. II. 1987, 304, 1069. (4) Costese`que, P.; Hridabba, M.; Sahores, J. C. R. Acad. Sci. Ser. II. 1987, 305, 1531. (5) Montel, F. Entropie 1994, 184/185, 83. (6) Legros, J. C.; Van Vaerenbergh, S.; Decroly, Y.; Colinet P.; Montel, F. Entropie 1994, 184/185, 38. (7) Tyrrell, H. J. V. Diffusion and Heat Flow in Liquids; Butterworths: London, 1961. (8) Ma, N Y. R.; Beyerlein, A. L. J. Phys. Chem. 1983, 87, 245. (9) Ecenarro, O.; Madariaga, J. A.; Navarro, J.; Santamaria, C. M.; Carrio´n, J. A.; Saviro´n, J. M. Sep. Sci. Technol. 1989, 24, 555. (10) Pohl, D. W.; Schwarz S. E.; Irniger, V. Phys. ReV. Lett. 1973, 31, 32. (11) Eichler, H. J.; Gunter, P.; Pohl, D. W. Laser-Induced Dynamic Gratings; Springer-Verlag: Berlin, 1986. (12) Ko¨hler, W. J. Chem. Phys. 1993, 98, 660. (13) Ko¨hler, W.; Rosenauer C.; Rossmanith, P. Int. J. Thermophys. 1995, 16, 11. (14) Ko¨hler, W.; Mu¨ller, J. Chem. Phys. 1995, 103, 4367.
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