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A novel partition function for partially asymmetrical internal rotation
Chemical Physics Letters 368 (2003) 473–479 www.elsevier.com/locate/cplett
A novel partition function for partially asymmetrical internal rotation Gernot Katzer *, Alexander F. Sax Institut f€ur Chemie, Karl-Franzens-Universit€at Graz, Strassoldogasse 10, 8010 Graz, Austria Received 9 October 2002; in final form 25 November 2002
Abstract A novel partition function for one isolated internal rotation degree of freedom is presented. Our partition function is designed for torsion potentials with one antiperiplanar and two isoenergetic synclinal (gauche) minima, as in 1,2dichloroethane or butane. Calculating thermodynamic functions (U ; S; Cv ) for the internal rotation in a number of small carbon and silicon compounds, we compare the results to those obtained with a symmetrical internal rotation partition function. The relative energy of the conformers affects the heat capacity most strongly, and gives an additive increment to the internal energy at high temperatures. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction To understand the thermodynamics and kinetics of chemical reactions, precise information of the thermodynamic state functions of all species involved is needed. This is a difficult task as it requires a realistic description of all electronic and nuclear degrees of freedom. The problem can, however, be considerably simplified if the individual degrees of freedom are assumed independent. Furthermore, for most molecules, temperature-dependent contributions of electronic degrees of freedom can be neglected, and the only relevant term is the electronic energy of the ground state obtained with any suitable wave-
*
Corresponding author. Fax: +43-316-380-8993. E-mail address: [email protected] (G. Katzer).
function. Temperature-dependent contributions arise only from nuclear motion, which separates into translation, rotation and vibration; the first two can easily be treated by the methods of statistical thermodynamics, but the vibration term becomes easily tractable only if harmonic vibrations are assumed. In two recent publications [1,2], we demonstrated the influence of anharmonicity on the thermodynamic state functions of silicon hydride compounds, and showed that a purely harmonic treatment often is inadequate. The largest anharmonic corrections arise from internal rotations, which are very common in carbon and silicon chemistry. Most investigations on the thermodynamic treatment of internal rotation concentrated on symmetrical internal rotation, e.g., the internal rotation of a terminal CH3 group. Comparatively
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 1 9 0 7 - 3
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G. Katzer, A.F. Sax / Chemical Physics Letters 368 (2003) 473–479
little attention has been paid to internal rotations of lower symmetry, as exemplified in the internal rotation about the central C–C bond in butane [3–5]. In this publication, we derive a partition function that describes internal rotation for a torsion potential that has three minima in the interval [0; 2p], only two of which are isoenergetic. Although this is still not fully asymmetrical, this novel partition function gives an improved description of internal rotations in systems like 1,2dichloroethane or butane.
metrical potential becomes identical to the symmetrical one.
2. Potential Symmetrical internal rotation is conventionally [1,6–9] identified with motion in a periodic cosine potential of barrier height V0 Vs ðx; V0 Þ ¼
V0 ð1 þ cos nxÞ; 2
Fig. 1. Asymmetrical potential for internal rotations.
0 6 x < 2p:
ð1Þ
For the rest of this work, we will restrict ourselves to n ¼ 3; then, the potential has three minima at x ¼ p=3; p; 5p=3 and three maxima at x ¼ 0; 2p=3; 4p=3 which is typical for internal rotations about single bonds formed by C, N, O, and Si atoms. The potential in Eq. (1) is called symmetrical because all the maxima and minima lie at the same energies, respectively. We construct a more asymmetrical potential by generalization of Eq. (1). In the vicinity of the antiperiplanar minimum (2p=3 < x < 4p=3), we leave the potential unchanged, but apply modifications for the synclinal and synperiplanar conformations: V ðxÞ ¼ 8 jx pj6 p=3; < V1 ðxÞ ¼ Vs ðx; V0 Þ; V2 ðxÞ ¼ Vs ðx; V0 dÞ þ d; p=3 < jx pj6 2p=3; : V3 ðxÞ ¼ Vs ðx; V0 d þ eÞ þ d; 2p=3 < jx pj: ð2Þ The shape of this new potential is shown in Fig. 1; compared to the symmetrical potential, Eq. (1), the abscissa positions of the minima are retained, but the minima and maxima now have in part different energies. For d ¼ 0 and e ¼ 0, the asym-
3. Partition function As in previous work [1], we will assume that the molecular partition function factorizes into independent contributions of translation, rotation and vibration. This is clearly an approximation for asymmetrical internal rotation, as the moment of inertia for figure axis rotation depends on the instantaneous dihedral angle. The moment of inertia does not influence the internal energy in the hightemperature limit, but it enters into the calculation of entropy and free energy. Furthermore, we will assume separation among the individual vibrational degrees of freedom so that the internal rotation is a one-dimensional motion. The quantum mechanical partition function of a one-dimensional oscillator can be constructed by summation over all energy levels: X qðbÞ ¼ expðbei Þ: ð3Þ i
For the harmonic oscillator, the energy levels are given by ei ¼ Eði þ 1=2Þ, and the sum can be evaluated analytically qðbÞ ¼
expðbE=2Þ ; 1 expðbEÞ
ð4Þ
where b ¼ 1=RT is the inverse reduced temperature and E ¼ hNA x is the first excitation energy of the harmonic oscillator in molar units.
G. Katzer, A.F. Sax / Chemical Physics Letters 368 (2003) 473–479
Quantum mechanical eigenvalues for general potentials are difficult to calculate. Therefore, the quantum mechanical partition function is often replaced by the classical partition function. For one single degree of freedom along a coordinate x, the classical partition function qcl ðbÞ can be written as an integral over phase space Z Z 1 exp ð bH Þdp dx qcl ðbÞ ¼ h Z 1 1 bp2 ¼ exp dp h Z1 2m
Z
2p=3
p expðbdÞ 3 0 bðV0 dÞ bðV0 dÞ exp I0 ; 2 2 expðbV2 ðxÞÞ dx ¼
Z 0
exp ð bV ðxÞÞ dx;
ð5Þ
xl
where h is PlanckÕs constant, H ¼ T þ V is the Hamiltonian function, m is the mass, p is the momentum and V ðxÞ is the potential which is defined in the coordinate range xl 6 x 6 xu . The integral over momentum space in Eq. (5) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi can be easily solved and yields a value of 2pm=b. The solution of the second integral depends on the nature of the vibration potential; for a harmonic oscillator, the classical partition function is 1 : ð6Þ qcl ðbÞ ¼ Eb We now derive the classical partition function for potential Eq. (2). After plugging the potential into Eq. (5) and choosing the integration interval [0; 2p] for the spatial coordinate, we can simplify the expression by splitting the integral at the points of junction (x ¼ p=3; 2p=3; 4p=3; 5p=3) and recombining terms with the same integrand. Because of the periodicity of the cosine function, the integration limits can be chosen from 0 to 2p=3 for all three integrals Z 2p 3 Z 2p=3 X expðbV ðxÞÞ dx ¼ expðbVi ðxÞÞ dx: 0
i¼1
0
ð7Þ The integral involving V1 ðxÞ is well known in the literature [7,10] and can be written in closed form using a modified Bessel function of the first kind, I0 ; the others are easily derived from it: Z 2p=3 expðbV1 ðxÞÞ dx 0 p bV0 bV0 ¼ exp I0 ; ð8Þ 3 2 2
ð9Þ
2p=3
p expðbdÞ 3 bðV0 d þ eÞ bðV0 d þ eÞ exp I0 : 2 2 expðbV3 ðxÞÞ dx ¼
ð10Þ
xu
475
Combining the results of the integrations over momentum and space coordinates, we arrive at the partition function for the asymmetrical potential in Eq. (2). The result can be written in a more convenient form by considering the relation pffiffiffiffiffiffiffiffi ffi E ¼ hNA k=m, where the harmonic force constant k ¼ 9V0 =2 is given by the second derivative of V ðxÞ at the minimum position x ¼ p: sffiffiffiffiffiffiffi 1 pV0 bV0 bV0 qcl ðbÞ ¼ exp I0 E b 2 2 bd bðV0 dÞ I0 þ exp 2 2 bðd þ eÞ bðV0 d þ eÞ þ exp I0 : 2 2 ð11Þ To include quantum effects, which are essential for the behaviour at low and medium temperatures, we employ the simple scheme proposed by Pitzer and Gwinn [6] qðbÞ ¼ qcl ðbÞ
qho ðbÞ ; qho cl ðbÞ
ð12Þ
which uses the ratio of a quantum mechanical and a classical harmonic oscillator partition function as a correction factor (Eqs. (4) and (6), respectively). The validity of the Pitzer–Gwinn correction has already been demonstrated [1,11]. Thermodynamic state functions (internal energy, entropy, isochoric heat capacity) can be calculated from Eq. (11) via the conventional formalism of statistical thermodynamics
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G. Katzer, A.F. Sax / Chemical Physics Letters 368 (2003) 473–479
U ðT Þ ¼
q0 ; q
SðT Þ ¼ R ln q þ
U ðT Þ ; T
2
Cv ¼
1 q00 q q0 ; RT 2 q2
ð13Þ
where no distinction between temperature-dependent and temperature-independent internal energy is made because the zero point of the energy was chosen at the potential minimum.
4. Results and discussion 4.1. Variation of parameters d and e Thermodynamic functions corresponding to partition function Eq. (11) depend on all four parameters E, V0 , d and e. The first two parameters are essentially the same as for the well-studied symmetrical internal rotation, and we will not consider their influence on the thermodynamic quantities explicitly. Instead, we will concentrate on the importance of the asymmetry parameters d and e for given E and V0 . A large e means that the synperiplanar conformer is energetically much more unfavourable than the two anticlinal ones; consequently, a part of the configuration space may become energeti-
cally unavailable, and the entropy drops slightly. On the other hand, the values of the internal energy and the heat capacity are slightly increased because the free rotor limit is not reached at temperatures RT ’V0 , but only at RT ’V0 þ d. Fig. 2 visualizes the effect of e on thermodynamic state functions for two different values of V0 (4 and 20 kJ mol1 ). In both cases, variation of e from 0 (symmetric case) to 100 kJ mol1 (synperiplanar conformer extremely disfavoured) results only in minor effect on the calculated energy, entropy and heat capacity. For realistic molecles, e would be much smaller than 100 kJ mol1 , and would thus influence the thermodynamics even less. The parameter d, which determines the energetic difference between the antiperiplanar minimum and the two synclinal minima, affects the thermodynamic results much more. In most molecules, d is greater or equal to zero, implying that the antiperiplanar structure is the global minimum [12]. Fig. 3 demonstrates that, depending on its magnitude, d may exert considerable influence on internal energy and heat capacity. In the top row (V0 ¼ 4 kJ mol1 ), the curves for symmetric and asymmetric internal rotation are almost identical, indicating a negligible correction due to asymmetry for low-barrier internal rotations. In the middle
Fig. 2. Influence of e on thermodynamic functions. Influence of e on internal energy U (in kJ mol1 ), absolute entropy S and isochorous heat capacity Cv (in J mol1 K1 ) of an asymmetrical internal rotor, for d ¼ 0 and two different values of V0 (4 kJ mol1 in the top row and 20 kJ mol1 in the bottom row).
G. Katzer, A.F. Sax / Chemical Physics Letters 368 (2003) 473–479
477
Fig. 3. Influence of d on thermodynamic functions. Influence of d on internal energy U (in kJ mol1 ), absolute entropy S and isochorous heat capacity Cv (in J mol1 K1 ) of an asymmetrical internal rotor, for e ¼ 0 and two different values of V0 (4 kJ mol1 in the top row and 20 kJ mol1 in the middle and bottom rows).
row, the barrier was chosen to V0 ¼ 20 kJ mol1 , and we find that asymmetry influences the state functions more significantly. Effects of asymmetry are most prominent in the heat capacity Cv , which shows a high maximum at a temperature Tmax . Note that a maximum in the heat capacity at RT /V0 is a regular feature of any hindered internal rotor, indicating the increase in the density of states for energies above V0 ; in the asymmetrical internal rotor, there will be a second, higher maximum at lower temperature RT /d, indicating the population of the synclinal minima. These two maxima usually coalesce unless V0 d. In the high-temperature domain, the asymmetrical internal rotor approaches the symmetrical internal rotor limit (and later the free rotor limit) as d (and later V0 ) vanishes compared to RT . In the bottom row of Fig. 3, the shape of the curves is shown in more detail for low and intermediate temperature. Shape and position of the Cv peak depend strongly on d, and even the small
change von 3–5 kJ mol1 changes the curve significantly. For small d, the Cv peak shifts not only to deeper temperature, but also becomes much narrower. A shoulder in the dotted Cv curve (d ¼ 3 kJ mol1 ) indicates that the Cv maximum arises from two sources, namely population of the higher minima and overcome of barrier. In the internal energy, asymmetry manifests itself in a vertical shift DU of the U ðT Þ curve which is approximately constant in the high-temperature limit. This shift corresponds to the area under the Cv peak. For V0 ¼ 20 kJ mol1 , we find DU values of 1.5, 2.3 and 5.2 kJ mol1 for d ¼ 3, 5 and 15 kJ mol1 , respectively. The shift depends not only on d but also on V0 ; for the top row of Fig. 3, its value is only 1.1 kJ mol1 (V0 ¼ 4 kJ mol1 and d ¼ 3 kJ mol1 ). In real molecules, we often observe high barriers that prohibit interconversion of the rotamers at reasonable temperature, e.g., in 1,2-dichlorethene. As a results, the system behaves nonergodically,
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G. Katzer, A.F. Sax / Chemical Physics Letters 368 (2003) 473–479
i.e., it does not populate all energetically available states. By construction, our partition function (Eq. (11)) cannot reproduce the nonergodicity found in systems with very high V0 . Thermodynamically, such systems usually behave like a harmonic oscillator, and should be treated with the corresponding partition function. 4.2. Sample molecules We apply now our partition function, Eq. (11), to the internal rotations of a couple of selected organic or silicon containing molecules. It is not our aim to discuss all aspects of anharmonic corrections [1], but we restrict ourselves to the differences between the results for symmetrical and asymmetrical internal rotors. The free parameters of the partition functions for asymmetrical internal rotations were obtained by fitting ab initio calculated torsion potential curves by the potential defined in Eq. (2). These curves had been calculated using the restricted Hartree–Fock (RHF) method, basis sets of double or triple f quality, and pseudopotentials for all atoms heavier than fluorine. All calculations were performed with GAMESS [13]. Table 1 presents the fitted parameters and calculated values of Tmax , Cv;max , and DU at three different temperatures.
1,2-Difluoroethane differs [12] from all sample molecules by it its d value of almost zero; consequently, the small asymmetry corrections can be attributed to the high e parameter entirely: the maximum in the heat capacity shows a height typical for symmetrical internal rotors, and the DU values increase slightly with the temperature (cf. Fig. 2). The higher 1,2-dihaloethanes are a homogenous group with similarly high Cv;max values. Their asymmetry contributions to the internal energy, DU , have values of about 5 kJ mol1 attributable to the similar d values. The slight temperature dependence of DU is due to their finite e values. In butane, C4 H10 , the parameter d is much smaller than in the heavy dihaloethanes, and consequently, butane shows less corrections due to asymmetry. Ethanol, C2 H5 OH, appears as an almost symmetrical internal rotor with negligible values of DU . For the silicon compounds, the influence of asymmetry on the thermodynamical functions is less pronounced because of the smaller V0 which in turn limits d to low values. The Cv maximum is shifted to deep temperatures, and values of DU are rather small for the dihalodisilanes, and they are almost negligible for tetrasilane. In all silicon compounds, DU depends hardly on the tempera-
Table 1 Parameters and calculated thermochemical data for asymmetrical internal rotors System
CH2 F–CH2 F CH2 Cl–CH2 Cl CH2 Br–CHBr2 CH2 I–CH2 I C2 H5 –C2 H5 C2 H5 –OH SiH2 Cl–SiH2 Cl SiH2 Br–SiH2 Br SiH2 I–SiH2 I Si2 H5 –Si2 H5
E
1.6 1.5 1.3 1.2 0.37 3.7 0.49 0.42 0.39 0.37
V0
11.9 21.7 22.9 24.8 15.8 5.5 5.5 6.3 7.2 3.7
d
)0.02 9.1 11.2 14.2 4.8 1.1 3.5 4.3 5.1 1.2
e
21.0 22.8 22.7 22.9 10.4 1.7 4.4 5.6 7.1 3.4
Tmax
320 424 491 585 230 174 137 164 192 62
Cv;max
9.3 16.2 16.6 16.9 15.7 7.1 17.0 17.0 16.9 15.5
DU at T ¼ 2000 K
3000 K
4000 K
1.0 4.5 5.2 6.0 2.7 0.63 1.8 2.2 2.6 0.89
1.6 5.0 5.7 6.7 2.9 0.64 1.8 2.2 2.7 0.92
2.0 5.4 6.1 7.1 3.0 0.64 1.8 2.3 2.7 0.93
E, V0 , d and e are given in kJ mol1 . Tmax (in K) is the temperature at which the heat capacity assumes its maximum value Cv;max (in J mol1 K1 ). DU , the increase of the internal energy relative to a symmetrical internal rotor, is given in kJ mol1 for three temperatures.
G. Katzer, A.F. Sax / Chemical Physics Letters 368 (2003) 473–479
ture because the free rotor limit is already reached at 2000 K. For all systems shown in Table 1, we observe DU d=2. This finding could be used as a criterion to quickly judge whether or not internal rotor asymmetry might play a role for a given molecule.
5. Conclusions The closed form of the partition function for an asymmetrical hindered rotor allows to evaluate the importance of asymmetry in internal rotations. We demonstrated that the key parameter is the energetic difference between the rotamers, d, and to a lesser extent the barrier height, V0 . The energetic difference between the maxima, e, turned out comparatively unimportant. For small V0 values, asymmetry plays a minor role for all the thermodynamic functions, as the system reaches early the free rotor limit where the influences of all potential parameters on Cv vanish. In silicon, and even more germanium, chemistry, small barriers are the rule, and asymmetry is therefore not an important issue in calculating thermochemical quantities. In carbon chemistry, on the other hand, barriers to internal rotations are typically in the range of 20 kJ mol1 , and the asymmetry corrections to U and
479
Cv may become sizeable, even if they are still a small fraction of the total anharmonicity correction. Explicit treatment as asymmetrical hindered rotor is a necessity to predict the heat capacity correctly at temperatures RT /d. For the internal energy, the corrections relative to a symmetrical internal rotor are smaller; in the high-temperature limit, we observe an approximately temperatureindependent correction to U of the order DU d=2.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
G. Katzer, A.F. Sax, J. Phys. Chem. A 106 (2002) 7204. G. Katzer, A.F. Sax, J. Comp. Chem. (submitted). K.S. Pitzer, J. Chem. Phys. 14 (1946) 239. J. Gang, M.J. Pilling, S.H. Robinson, J. Chem. Soc., Faraday Trans. 92 (1996) 3509. J. Gang, M.J. Pilling, S.H. Robinson, J. Chem. Soc., Faraday Trans. 93 (1997) 1481. K.S. Pitzer, W.D. Gwinn, J. Chem. Phys. 10 (1942) 428. V.D. Knyazev, J. Phys. Chem. A 102 (1998) 3916. P.Y. Ayala, H.B. Schlegel, J. Chem. Phys. 108 (1998) 2314. R.B. McClurg, R.C. Flagan, W.A. Goddard III, J. Chem. Phys. 106 (1997) 6675. V.D. Knyazev, W. Tsang, J. Phys. Chem. A 102 (1998) 9167. A.D. Isaacson, D.G. Truhlar, J. Chem. Phys. 75 (1981) 4090. E. Juaristi, J. Chem. Educ. 56 (1979) 438. M.W. Schmidt, et al. J. Comp. Chem. 14 (1993) 1347.
A novel partition function for partially asymmetrical internal rotation Gernot Katzer *, Alexander F. Sax Institut f€ur Chemie, Karl-Franzens-Universit€at Graz, Strassoldogasse 10, 8010 Graz, Austria Received 9 October 2002; in final form 25 November 2002
Abstract A novel partition function for one isolated internal rotation degree of freedom is presented. Our partition function is designed for torsion potentials with one antiperiplanar and two isoenergetic synclinal (gauche) minima, as in 1,2dichloroethane or butane. Calculating thermodynamic functions (U ; S; Cv ) for the internal rotation in a number of small carbon and silicon compounds, we compare the results to those obtained with a symmetrical internal rotation partition function. The relative energy of the conformers affects the heat capacity most strongly, and gives an additive increment to the internal energy at high temperatures. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction To understand the thermodynamics and kinetics of chemical reactions, precise information of the thermodynamic state functions of all species involved is needed. This is a difficult task as it requires a realistic description of all electronic and nuclear degrees of freedom. The problem can, however, be considerably simplified if the individual degrees of freedom are assumed independent. Furthermore, for most molecules, temperature-dependent contributions of electronic degrees of freedom can be neglected, and the only relevant term is the electronic energy of the ground state obtained with any suitable wave-
*
Corresponding author. Fax: +43-316-380-8993. E-mail address: [email protected] (G. Katzer).
function. Temperature-dependent contributions arise only from nuclear motion, which separates into translation, rotation and vibration; the first two can easily be treated by the methods of statistical thermodynamics, but the vibration term becomes easily tractable only if harmonic vibrations are assumed. In two recent publications [1,2], we demonstrated the influence of anharmonicity on the thermodynamic state functions of silicon hydride compounds, and showed that a purely harmonic treatment often is inadequate. The largest anharmonic corrections arise from internal rotations, which are very common in carbon and silicon chemistry. Most investigations on the thermodynamic treatment of internal rotation concentrated on symmetrical internal rotation, e.g., the internal rotation of a terminal CH3 group. Comparatively
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 1 9 0 7 - 3
474
G. Katzer, A.F. Sax / Chemical Physics Letters 368 (2003) 473–479
little attention has been paid to internal rotations of lower symmetry, as exemplified in the internal rotation about the central C–C bond in butane [3–5]. In this publication, we derive a partition function that describes internal rotation for a torsion potential that has three minima in the interval [0; 2p], only two of which are isoenergetic. Although this is still not fully asymmetrical, this novel partition function gives an improved description of internal rotations in systems like 1,2dichloroethane or butane.
metrical potential becomes identical to the symmetrical one.
2. Potential Symmetrical internal rotation is conventionally [1,6–9] identified with motion in a periodic cosine potential of barrier height V0 Vs ðx; V0 Þ ¼
V0 ð1 þ cos nxÞ; 2
Fig. 1. Asymmetrical potential for internal rotations.
0 6 x < 2p:
ð1Þ
For the rest of this work, we will restrict ourselves to n ¼ 3; then, the potential has three minima at x ¼ p=3; p; 5p=3 and three maxima at x ¼ 0; 2p=3; 4p=3 which is typical for internal rotations about single bonds formed by C, N, O, and Si atoms. The potential in Eq. (1) is called symmetrical because all the maxima and minima lie at the same energies, respectively. We construct a more asymmetrical potential by generalization of Eq. (1). In the vicinity of the antiperiplanar minimum (2p=3 < x < 4p=3), we leave the potential unchanged, but apply modifications for the synclinal and synperiplanar conformations: V ðxÞ ¼ 8 jx pj6 p=3; < V1 ðxÞ ¼ Vs ðx; V0 Þ; V2 ðxÞ ¼ Vs ðx; V0 dÞ þ d; p=3 < jx pj6 2p=3; : V3 ðxÞ ¼ Vs ðx; V0 d þ eÞ þ d; 2p=3 < jx pj: ð2Þ The shape of this new potential is shown in Fig. 1; compared to the symmetrical potential, Eq. (1), the abscissa positions of the minima are retained, but the minima and maxima now have in part different energies. For d ¼ 0 and e ¼ 0, the asym-
3. Partition function As in previous work [1], we will assume that the molecular partition function factorizes into independent contributions of translation, rotation and vibration. This is clearly an approximation for asymmetrical internal rotation, as the moment of inertia for figure axis rotation depends on the instantaneous dihedral angle. The moment of inertia does not influence the internal energy in the hightemperature limit, but it enters into the calculation of entropy and free energy. Furthermore, we will assume separation among the individual vibrational degrees of freedom so that the internal rotation is a one-dimensional motion. The quantum mechanical partition function of a one-dimensional oscillator can be constructed by summation over all energy levels: X qðbÞ ¼ expðbei Þ: ð3Þ i
For the harmonic oscillator, the energy levels are given by ei ¼ Eði þ 1=2Þ, and the sum can be evaluated analytically qðbÞ ¼
expðbE=2Þ ; 1 expðbEÞ
ð4Þ
where b ¼ 1=RT is the inverse reduced temperature and E ¼ hNA x is the first excitation energy of the harmonic oscillator in molar units.
G. Katzer, A.F. Sax / Chemical Physics Letters 368 (2003) 473–479
Quantum mechanical eigenvalues for general potentials are difficult to calculate. Therefore, the quantum mechanical partition function is often replaced by the classical partition function. For one single degree of freedom along a coordinate x, the classical partition function qcl ðbÞ can be written as an integral over phase space Z Z 1 exp ð bH Þdp dx qcl ðbÞ ¼ h Z 1 1 bp2 ¼ exp dp h Z1 2m
Z
2p=3
p expðbdÞ 3 0 bðV0 dÞ bðV0 dÞ exp I0 ; 2 2 expðbV2 ðxÞÞ dx ¼
Z 0
exp ð bV ðxÞÞ dx;
ð5Þ
xl
where h is PlanckÕs constant, H ¼ T þ V is the Hamiltonian function, m is the mass, p is the momentum and V ðxÞ is the potential which is defined in the coordinate range xl 6 x 6 xu . The integral over momentum space in Eq. (5) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi can be easily solved and yields a value of 2pm=b. The solution of the second integral depends on the nature of the vibration potential; for a harmonic oscillator, the classical partition function is 1 : ð6Þ qcl ðbÞ ¼ Eb We now derive the classical partition function for potential Eq. (2). After plugging the potential into Eq. (5) and choosing the integration interval [0; 2p] for the spatial coordinate, we can simplify the expression by splitting the integral at the points of junction (x ¼ p=3; 2p=3; 4p=3; 5p=3) and recombining terms with the same integrand. Because of the periodicity of the cosine function, the integration limits can be chosen from 0 to 2p=3 for all three integrals Z 2p 3 Z 2p=3 X expðbV ðxÞÞ dx ¼ expðbVi ðxÞÞ dx: 0
i¼1
0
ð7Þ The integral involving V1 ðxÞ is well known in the literature [7,10] and can be written in closed form using a modified Bessel function of the first kind, I0 ; the others are easily derived from it: Z 2p=3 expðbV1 ðxÞÞ dx 0 p bV0 bV0 ¼ exp I0 ; ð8Þ 3 2 2
ð9Þ
2p=3
p expðbdÞ 3 bðV0 d þ eÞ bðV0 d þ eÞ exp I0 : 2 2 expðbV3 ðxÞÞ dx ¼
ð10Þ
xu
475
Combining the results of the integrations over momentum and space coordinates, we arrive at the partition function for the asymmetrical potential in Eq. (2). The result can be written in a more convenient form by considering the relation pffiffiffiffiffiffiffiffi ffi E ¼ hNA k=m, where the harmonic force constant k ¼ 9V0 =2 is given by the second derivative of V ðxÞ at the minimum position x ¼ p: sffiffiffiffiffiffiffi 1 pV0 bV0 bV0 qcl ðbÞ ¼ exp I0 E b 2 2 bd bðV0 dÞ I0 þ exp 2 2 bðd þ eÞ bðV0 d þ eÞ þ exp I0 : 2 2 ð11Þ To include quantum effects, which are essential for the behaviour at low and medium temperatures, we employ the simple scheme proposed by Pitzer and Gwinn [6] qðbÞ ¼ qcl ðbÞ
qho ðbÞ ; qho cl ðbÞ
ð12Þ
which uses the ratio of a quantum mechanical and a classical harmonic oscillator partition function as a correction factor (Eqs. (4) and (6), respectively). The validity of the Pitzer–Gwinn correction has already been demonstrated [1,11]. Thermodynamic state functions (internal energy, entropy, isochoric heat capacity) can be calculated from Eq. (11) via the conventional formalism of statistical thermodynamics
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G. Katzer, A.F. Sax / Chemical Physics Letters 368 (2003) 473–479
U ðT Þ ¼
q0 ; q
SðT Þ ¼ R ln q þ
U ðT Þ ; T
2
Cv ¼
1 q00 q q0 ; RT 2 q2
ð13Þ
where no distinction between temperature-dependent and temperature-independent internal energy is made because the zero point of the energy was chosen at the potential minimum.
4. Results and discussion 4.1. Variation of parameters d and e Thermodynamic functions corresponding to partition function Eq. (11) depend on all four parameters E, V0 , d and e. The first two parameters are essentially the same as for the well-studied symmetrical internal rotation, and we will not consider their influence on the thermodynamic quantities explicitly. Instead, we will concentrate on the importance of the asymmetry parameters d and e for given E and V0 . A large e means that the synperiplanar conformer is energetically much more unfavourable than the two anticlinal ones; consequently, a part of the configuration space may become energeti-
cally unavailable, and the entropy drops slightly. On the other hand, the values of the internal energy and the heat capacity are slightly increased because the free rotor limit is not reached at temperatures RT ’V0 , but only at RT ’V0 þ d. Fig. 2 visualizes the effect of e on thermodynamic state functions for two different values of V0 (4 and 20 kJ mol1 ). In both cases, variation of e from 0 (symmetric case) to 100 kJ mol1 (synperiplanar conformer extremely disfavoured) results only in minor effect on the calculated energy, entropy and heat capacity. For realistic molecles, e would be much smaller than 100 kJ mol1 , and would thus influence the thermodynamics even less. The parameter d, which determines the energetic difference between the antiperiplanar minimum and the two synclinal minima, affects the thermodynamic results much more. In most molecules, d is greater or equal to zero, implying that the antiperiplanar structure is the global minimum [12]. Fig. 3 demonstrates that, depending on its magnitude, d may exert considerable influence on internal energy and heat capacity. In the top row (V0 ¼ 4 kJ mol1 ), the curves for symmetric and asymmetric internal rotation are almost identical, indicating a negligible correction due to asymmetry for low-barrier internal rotations. In the middle
Fig. 2. Influence of e on thermodynamic functions. Influence of e on internal energy U (in kJ mol1 ), absolute entropy S and isochorous heat capacity Cv (in J mol1 K1 ) of an asymmetrical internal rotor, for d ¼ 0 and two different values of V0 (4 kJ mol1 in the top row and 20 kJ mol1 in the bottom row).
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Fig. 3. Influence of d on thermodynamic functions. Influence of d on internal energy U (in kJ mol1 ), absolute entropy S and isochorous heat capacity Cv (in J mol1 K1 ) of an asymmetrical internal rotor, for e ¼ 0 and two different values of V0 (4 kJ mol1 in the top row and 20 kJ mol1 in the middle and bottom rows).
row, the barrier was chosen to V0 ¼ 20 kJ mol1 , and we find that asymmetry influences the state functions more significantly. Effects of asymmetry are most prominent in the heat capacity Cv , which shows a high maximum at a temperature Tmax . Note that a maximum in the heat capacity at RT /V0 is a regular feature of any hindered internal rotor, indicating the increase in the density of states for energies above V0 ; in the asymmetrical internal rotor, there will be a second, higher maximum at lower temperature RT /d, indicating the population of the synclinal minima. These two maxima usually coalesce unless V0 d. In the high-temperature domain, the asymmetrical internal rotor approaches the symmetrical internal rotor limit (and later the free rotor limit) as d (and later V0 ) vanishes compared to RT . In the bottom row of Fig. 3, the shape of the curves is shown in more detail for low and intermediate temperature. Shape and position of the Cv peak depend strongly on d, and even the small
change von 3–5 kJ mol1 changes the curve significantly. For small d, the Cv peak shifts not only to deeper temperature, but also becomes much narrower. A shoulder in the dotted Cv curve (d ¼ 3 kJ mol1 ) indicates that the Cv maximum arises from two sources, namely population of the higher minima and overcome of barrier. In the internal energy, asymmetry manifests itself in a vertical shift DU of the U ðT Þ curve which is approximately constant in the high-temperature limit. This shift corresponds to the area under the Cv peak. For V0 ¼ 20 kJ mol1 , we find DU values of 1.5, 2.3 and 5.2 kJ mol1 for d ¼ 3, 5 and 15 kJ mol1 , respectively. The shift depends not only on d but also on V0 ; for the top row of Fig. 3, its value is only 1.1 kJ mol1 (V0 ¼ 4 kJ mol1 and d ¼ 3 kJ mol1 ). In real molecules, we often observe high barriers that prohibit interconversion of the rotamers at reasonable temperature, e.g., in 1,2-dichlorethene. As a results, the system behaves nonergodically,
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i.e., it does not populate all energetically available states. By construction, our partition function (Eq. (11)) cannot reproduce the nonergodicity found in systems with very high V0 . Thermodynamically, such systems usually behave like a harmonic oscillator, and should be treated with the corresponding partition function. 4.2. Sample molecules We apply now our partition function, Eq. (11), to the internal rotations of a couple of selected organic or silicon containing molecules. It is not our aim to discuss all aspects of anharmonic corrections [1], but we restrict ourselves to the differences between the results for symmetrical and asymmetrical internal rotors. The free parameters of the partition functions for asymmetrical internal rotations were obtained by fitting ab initio calculated torsion potential curves by the potential defined in Eq. (2). These curves had been calculated using the restricted Hartree–Fock (RHF) method, basis sets of double or triple f quality, and pseudopotentials for all atoms heavier than fluorine. All calculations were performed with GAMESS [13]. Table 1 presents the fitted parameters and calculated values of Tmax , Cv;max , and DU at three different temperatures.
1,2-Difluoroethane differs [12] from all sample molecules by it its d value of almost zero; consequently, the small asymmetry corrections can be attributed to the high e parameter entirely: the maximum in the heat capacity shows a height typical for symmetrical internal rotors, and the DU values increase slightly with the temperature (cf. Fig. 2). The higher 1,2-dihaloethanes are a homogenous group with similarly high Cv;max values. Their asymmetry contributions to the internal energy, DU , have values of about 5 kJ mol1 attributable to the similar d values. The slight temperature dependence of DU is due to their finite e values. In butane, C4 H10 , the parameter d is much smaller than in the heavy dihaloethanes, and consequently, butane shows less corrections due to asymmetry. Ethanol, C2 H5 OH, appears as an almost symmetrical internal rotor with negligible values of DU . For the silicon compounds, the influence of asymmetry on the thermodynamical functions is less pronounced because of the smaller V0 which in turn limits d to low values. The Cv maximum is shifted to deep temperatures, and values of DU are rather small for the dihalodisilanes, and they are almost negligible for tetrasilane. In all silicon compounds, DU depends hardly on the tempera-
Table 1 Parameters and calculated thermochemical data for asymmetrical internal rotors System
CH2 F–CH2 F CH2 Cl–CH2 Cl CH2 Br–CHBr2 CH2 I–CH2 I C2 H5 –C2 H5 C2 H5 –OH SiH2 Cl–SiH2 Cl SiH2 Br–SiH2 Br SiH2 I–SiH2 I Si2 H5 –Si2 H5
E
1.6 1.5 1.3 1.2 0.37 3.7 0.49 0.42 0.39 0.37
V0
11.9 21.7 22.9 24.8 15.8 5.5 5.5 6.3 7.2 3.7
d
)0.02 9.1 11.2 14.2 4.8 1.1 3.5 4.3 5.1 1.2
e
21.0 22.8 22.7 22.9 10.4 1.7 4.4 5.6 7.1 3.4
Tmax
320 424 491 585 230 174 137 164 192 62
Cv;max
9.3 16.2 16.6 16.9 15.7 7.1 17.0 17.0 16.9 15.5
DU at T ¼ 2000 K
3000 K
4000 K
1.0 4.5 5.2 6.0 2.7 0.63 1.8 2.2 2.6 0.89
1.6 5.0 5.7 6.7 2.9 0.64 1.8 2.2 2.7 0.92
2.0 5.4 6.1 7.1 3.0 0.64 1.8 2.3 2.7 0.93
E, V0 , d and e are given in kJ mol1 . Tmax (in K) is the temperature at which the heat capacity assumes its maximum value Cv;max (in J mol1 K1 ). DU , the increase of the internal energy relative to a symmetrical internal rotor, is given in kJ mol1 for three temperatures.
G. Katzer, A.F. Sax / Chemical Physics Letters 368 (2003) 473–479
ture because the free rotor limit is already reached at 2000 K. For all systems shown in Table 1, we observe DU d=2. This finding could be used as a criterion to quickly judge whether or not internal rotor asymmetry might play a role for a given molecule.
5. Conclusions The closed form of the partition function for an asymmetrical hindered rotor allows to evaluate the importance of asymmetry in internal rotations. We demonstrated that the key parameter is the energetic difference between the rotamers, d, and to a lesser extent the barrier height, V0 . The energetic difference between the maxima, e, turned out comparatively unimportant. For small V0 values, asymmetry plays a minor role for all the thermodynamic functions, as the system reaches early the free rotor limit where the influences of all potential parameters on Cv vanish. In silicon, and even more germanium, chemistry, small barriers are the rule, and asymmetry is therefore not an important issue in calculating thermochemical quantities. In carbon chemistry, on the other hand, barriers to internal rotations are typically in the range of 20 kJ mol1 , and the asymmetry corrections to U and
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Cv may become sizeable, even if they are still a small fraction of the total anharmonicity correction. Explicit treatment as asymmetrical hindered rotor is a necessity to predict the heat capacity correctly at temperatures RT /d. For the internal energy, the corrections relative to a symmetrical internal rotor are smaller; in the high-temperature limit, we observe an approximately temperatureindependent correction to U of the order DU d=2.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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