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Significance of Group Frequency Distributions for Group Additivity Gustav Bojesen* epiZell Aps, Gydevang 39-41, 3450 Allerød, Denmark S Supporting Information *

ABSTRACT: The following hypothesis is proposed: When additivity is valid, the molecular frequency distribution (MFD) is a weighted sum of group frequency distributions (GFDs). On the basis of the transformation rules for distribution functions from statistical theory and the rigid-rotator harmonic oscillator approximation, it is shown analytically that the hypothesis leads to group additivity of vibrationally dependent thermochemical parameters. Graph theory has been used to find the structure of the matrices used to determine group additivity values, and the results show that, within the rigid-rotator harmonic oscillator approximation, additivity of the vibrational contributions leads to the experimentally observed group additivity of the total group contributions. Support for the hypothesis is obtained from ab initio and DFT frequency calculations of a total 182 different compounds from seven polymeric series. In agreement with the hypothesis, remarkable similarities of the MFDs are observed throughout each series. Molecular frequencies can be satisfactorily calculated from model calculations based on the hypothesis, and temperature dependent heat capacities for the monomeric units can be derived that are in agreement with experimental values.

I. INTRODUCTION Group additivity schemes are a well-established method for the estimation of thermochemical properties of organic molecules in the gas phase.1−7 Computational methods have to be done at a fairly high level to provide numbers of comparable accuracy, and consequently addivitity schemes remain an important source for thermochemical values of large molecules. The “atoms in molecules theory” provides an explanation for the group additivity of the electronic energy but group additivity schemes are valid also for properties such as the heat capacity which are not given by the electronic energy of the unperturbed molecule.8−11 The heat of formation also includes terms such as the zero-point energy and the vibrational excitation energy which are not directly dependent on the electronic energy. Additivity not only is a convenient method for estimation of thermodynamic properties but also provides an explantion for linear-energy relationships which lies as the foundation for the general analysis of reactivities in terms of functional groups.12−18 Despite the fundamental importance of the group concept and the strong experimental support for group additivity schemes, there appear to have been few attempts to explain the validity of these schemes in fundamental terms. To explain such a general and fundamental property of organic molecules by “cancellation of errors” or similar concepts is clearly unsatisfactory. The present work, and that previously published, takes its origin in the following considerations: (1) For all but very small molecules, at temperatures where additivity is valid (ambient and above), the largest contribution to the heat capacity comes from the vibrational modes. (2) The heat capacity contribution of a molecular oscillator is a decreasing function of its frequency and consequently the normal modes at low frequencies (skeletal vibrations) must provide a sizable fraction, if not the © 2014 American Chemical Society

largest, of the total vibrational heat capacity. (3) Consequently there must be some systematic variation of vibrational frequencies with the groups beyond the well-known spectroscopic group frequencies.19 Such systematic variation can be investigated statistically and this lead to the central hypothesis: When addivity is valid, the molecular f requency distribution (MFD) is a weighted sum of group f requency distributions (GFDs). In the present work it will be shown that, on the basis of this hypothesis and the rigid rotator-harmonic oscillator approximation, group additivity of the heat capacity and other vibrationally dependent thermochemical properties follows directly from the relevant formulas for the contribution of harmonic oscillators to the thermochemical properties, fundamental statistical theory, linear algebra, and graph theory. The hypothesis is subsequently tested on several series of polymers. It has already been shown that the hypothesis directly leads to the group additivity of the zero-point energy.19 Since the work of Debye,20 frequency distributions have played an important role for the understanding of the properties of solids. They have also been used in analyses of the properties of liquid and solid polymers.21−25 A better understanding of the frequency distribution of polymers in the gas phase will be of value in this work, and as will be shown below, it can lead to a very fast method for estimation of vibrational frequencies that is sufficiently accurate to calculate gas-phase heat capacities of organic molecules and their temperature dependence. Received: March 26, 2014 Revised: June 21, 2014 Published: June 26, 2014 5508

dx.doi.org/10.1021/jp503002p | J. Phys. Chem. A 2014, 118, 5508−5519

The Journal of Physical Chemistry A

Article

The paper is organized as follows. In the theoretical section it is proven that the central hypothesis presented above and the rigid rotator-harmonic oscillator approximation leads to addivitity of the vibrational contribution to molecular heat capacity, the molecular energy, and the logarithm of the molecular partition function. It is subsequently shown that the rigid rotator-harmonic oscillator approximation leads to group additivity when the vibrational contribution is additive in the groups. The section with results presents the outcome of theoretical calculation of molecular frequencies and analyses of their distributions in support of the central hypothesis.

C FM (κ ,T ) =

FM(v )̃ =

3NM − 6

∑ njwfj j (v)̃ j

(hcv /̃ kBT )2 exp[hcv /̃ kBT ] (exp[hcv /̃ kBT ] − 1)2

= κ (T )

=

(4)

∑ njwj⟨∂f jC (κ ,T )/∂κ ⟩

(6a) (6b)

j

The term wj⟨∂f jC(κ,T)/∂κ⟩ is equal to the vibrational contribution to the heat capacity of the jth group and (6) can be rewritten as (7), where hjC(T) is equal to wj⟨∂f Cj (κ,T)/ ∂κ⟩.

(1)

Cvib(T ) =

∑ njhCj (T ) (7)

j

This proves that the construction of molecular frequency distributions from linear combinations of GFDs leads to group additivity for the vibrational contribution to the heat capacity. The important requirement for application of the transformation rule that leads to the group additivity expressed in eq 7 is the one-to-one mapping relationship expressed by eq 2, which makes the inverse function *vib−1(κ ,T ) well-defined. Because the expressions for the vibrational zero-point energy (9), the vibrational excitation energy (10), and the logarithm of the vibrational partition function (11) have the same property, they will also be addititive in the groups.

(2a) (2b)

for v ̃ = *vib−1(κ ,T )

j

C Cvib(T ) = (3NM − 6)⟨∂FM (κ ,T )/∂κ ⟩

, zpe(v )̃ = NA

At any given temperature, *vib(v ̃,T )/∂v ̃ is always negative and *vib(v ̃,T ) is a strictly decreasing function of the wavenumber. Consequently, for any wavenumber (ṽ), the transforming function will give an associated temperature dependent heat capacity number κ(T), and the inverse function *vib−1(v ̃,T ) is well-defined (at constant temperature *vib(v ̃,T )is bijective). As the frequency distribution covers the range of wavenumbers from 0 cm−1 to ∞, the heat capacity numbers varies between 8.31 and 0 J K−1 mol−1. With a frequency distribution F(ṽ), the temperature dependent distribution of the heat capacity numbers, FC(κ,T) will, according to the transformation rule, be given by (Appendix A) F C(κ ,T ) = 1 − F(v )̃

∑ njwfj jC (κ ,T )

It follows from the definition of a mean that for a given molecule, the vibrational heat capacity is equal to the mean of its heat capacity numbers times the number of its vibrational modes. This is shown in eq 6a, where use of (5) leads to (6b)

In eq 1, NM is the number of atoms in the molecule and nj is the number of the jth group. The weighting factor wj is the number of vibrational modes associated with each group and is equal to 3 times the number of atoms in the group. The significance of the linear combination of the frequency distributions (1) for the additivity of vibrationally dependent thermochemical properties can be derived from the rules concerning transformation of distributions.26−28 With the vibrational heat capacity as an example, the transforming function is given by the usual expression for the wavenumber dependency of the heat capacity contribution of a single oscillator: *vib(v ̃,T ) = NAkB

3NM − 6

The linear relationship between the distributions of molecular heat capacity numbers FCM(κ,T) and of group heat capacity numbers f Cj (κ,T), expressed in eq 4, can be extended to the C means of their density functions: ⟨∂FM (κ,T)/∂κ⟩ and C ⟨∂f j (κ,T)/∂κ⟩. This leads to 1 C ⟨∂FM (κ ,T )/∂κ ⟩ = ∑ njwj⟨∂f jC (κ ,T )/∂κ ⟩ 3NM − 6 j (5)

II. THEORETICAL BACKGROUND Frequency Distributions. The central hypothesis can be formulated as shown in eq 1 where FM(ṽ) is a cumulative molecular frequencies distribution (MFD) and f j(ṽ) are functions (GFDs) that describe the characteristic contribution of each group to the MFD. 1

1

hc ṽ 2

(9)

,*(v )̃ = NAkB

hcv ̃ exp[hcv /̃ kBT ] − 1

(10)

ln q(v ̃,T ) = ln

1 1 − exp[−hcv /̃ kBT ]

(11)

The additivity of the zero-point energy and the vibrational excitation energy are important for the additivity of the total energy of a molecule, whereas the additivity of the logarithm of the vibrational partition function is necessary to ensure the additivity of the molecular entropy. It is important that the only requirement for the validity of the proof given above is that the functions describing the relationship between the thermochemical value and the wavenumber provide a one-to-one mapping relationship, which ensures that the inverse function is welldefined. Any function that fulfills this criterion will lead to group additivity of the vibrational contribution to the chosen thermochemical value. This is particularly relevant for the vibrational excitation energy and the vibrational entropy. The functions given in eqs 10 and 11 go to infinity when the wavenumber goes toward zero, and refinements of the rigid

(3)

In eq 3, *vib−1(κ ,T )is the inverse of the function for the vibrational heat capacity of a single oscillator given in (2a). The transformation rule in eq 3 is valid for both molecular and group frequency distribution, and consequently the distribution of molecular heat capacity numbers can be expressed as a function of the group contributions as shown (Appendix A): 5509



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rotator-harmonic oscillator approximation will typically involve changes in the functional relationships shown above between the frequency of an single oscillator and its contribution molecular property in question. The statistical analysis above provides an explanation for the additivity of the vibrational contributions to the heat capacity and other thermochemical properties. It relies on the physical significance of the group frequency distribution, and to examine this question, and test the central hypothesis, several series of polymers (oligomers) have been investigated. For a polymer P, eq 1 can be rewritten as FP(v )̃ =

1 (wefe (v )̃ + nwmfm (v )) ̃ we + nwm − 6

Cp ,i(T ) = 4R +

j

∑ ni ,jg jC (T ) j

(14)

In eq 14, 4R is the sum of the translational and rotational contribution to the heat capacity. With the coefficients ni,j in the two equations being identical, a set of heat capacity values or other numbers determined from a set of equation of the form shown in (13) cannot in general be a solution to a set of equations with the form as shown in (14). A close fit might be obtained if the constant 4R were insignificant compared to the vibrational contribution but for the heat capacity of a typical small molecule at 300 K this is clearly not the case. Another possibility for the simultaneous solution of eqs 13 and 14 is that the matrix given by the coefficients ni,j allows it. For convenience, a matrix of this kind can be called a Benson matrix. The elements in each row are the number of groups in a molecule and the columns are the groups. The number of groups in organic molecules cannot be chosen freely, and consequently, it can be expected that Benson matrices will have some kind of structure with certain ratios between the elements being possible and others not. In more general terms, the question is whether consistent g-values can be found by solving the system of equation expressed in eq 15 (the temperature dependencies have been omitted) where k is a constant which may or may not depend on the temperature:

(12)

In eq 12, we and wm are the weighting factors for the end group and the monomeric unit, and fe(ṽ) and f m(ṽ) are their GFDs. To illustrate this, in a polyethylene, fe(ṽ) is the frequency distribution of the two terminal methyl groups, we and wm are the numbers of vibrational modes associated with respectively the end groups and the monomeric unit. A polyethylene molecule can be regarded as an ethane molecule combined with a number of methylene groups and the number of vibrational modes contributed by the end groups we is then equal to 18, and wm is equal to 9. n is the degree of polymerization and f m(ṽ) is the GFD of a methylene group. It is clear from eq 12 that in the limit of an infinitely long chain (n →∞) FP(ṽ) will be equal to f m(ṽ). Accordingly, in this limit the MFD of the polymer is a continuous function equal to the GFD of the monomeric unit, and for any polymer the frequency distribution should approch a characteristic distribution as the degree of polymerization increases. That polymer chains have a characteristic frequency distribution with only limited dependence on the degree of polymerization has been included as an assumption in work on the thermochemical properties of polymers.21−25 However, a polymer can only have a characteristic frequency distribution independent of the chain length if the monomer unit gives a characteristic contribution to the MFD, i.e., that the GFD of the monomer unit is a well-defined and characteristic function independent of the molecular structure of which this group forms a part. Rigid Rotator-Harmonic Approximation and Additivity. Any physical justification for overall group additivity, which is based on the rigid rotator-harmonic oscillator approximation and group addivity for the vibrational contributions to thermochemical properties, must be able to reconcile two different forms for calculations of thermochemical properties. With the molecular heat capacity as the example, overall group additivity is based on equations of the form Cp ,i(T ) =

∑ ni ,jhCj (T )

M[g1 , g2 , g3 , ..., gj]T = M k [h1 , h2 , h3 , ..., hj , k]T

(15)

In eq 15, the g- and h-values are respectively the total and the vibrational contribution to whatever vibrationally dependent thermochemical property is chosen and M is a Benson matrix based on the coefficients ni,j from (13) and (14). Mk is an augmented version of M that includes a column for the constant term. In this latter column all the elements are equal to 1. With the limitations imposed by elementary graph theory on the structure of Benson matrices, it can be shown that eq 15 will be valid for a wide range of groups and that k will be a free variable (Appendix B).29,30 Apart from showing that the rigid rotator-harmonic oscillator approximation is compatible with group additivity, the justification for this analysis is supported by the answer it provides to a conundrum regarding the variation of heat capacity of various groups with their ratios of light to heavy atoms. Benson and Buss2 described group additivity as a secondorder approximation following after atomic and bond additivity schemes. The idea of group additivity being a refinement of bond additivity is, however, clearly at odds with the observed decrease of the group value for the heat capacity with the number of hydrogen atoms in a set of groups where the central atom is the same. In bond additivity schemes, the contribution of a C−H bond is lower than that of a C−C bond by 1 J K−1 mol−1 at 25 °C.5 This higher contribution of a C−C bond compared to a C−H bond is in agreement with the high vibrational frequency of the stretch mode in C−H, which is inactive at low temperatures. In this respect, the experimental group contributions of the four different groups in alkanes appear unphysical as their heat capacity contribution is positively correlated with the number of C−H bonds.5 Accepting the rigid rotor-harmonic oscillator approximation, it is possible to use eq 15, as described in Appendix B, to find the relationship between gj-values and hj-values. In the rigid

(13)

In eq 13, Cp,i(T) are the experimental values and the index i runs over the number of compounds that are included in the calculations. gCj (T) are the group additivity values of the various groups, and ni,j is the number of the jth group in the various molecules. In the rigid rotator-harmonic oscillator approximation, the translational and rotational contributions to the heat capacity are constants and when this is combined with group additivity of the vibrational contribution, e.g., eq 7, the total heat capacity for a series of molecules is given by 5510



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contribution with factors between 3.7 and 6.5. When the hydrogens are exchanged for carbons, the increases lie between 24% and 52%. This difference is in agreement with our general understanding of molecules with partly localized vibrations. The groups consisting of only heavy atoms have all their vibrational modes active also at low temperatures whereas the groups with light atoms require high temperatures before all their vibrational modes are active. This leads to the heat capacity contribution of the groups with light atoms to show a much higher temperature dependency than the groups with only heavy atoms. Overall, the vibrational contributions of the various groups and their temperature dependence are in much better agreement with our general understanding of group properties and how they depend on their atomic composition than the values obtained for the total group contribution. The functional relationships between g- and h-values show that, when a group with two ligands is without a branching point, the vibrational contribution to the heat capacity (hvalue) is equal to the total contribution of the group to the heat capacity. This is important in the analysis of polymeric series described below.

rotator-harmonic oscillator approximation a constant contribution of a given thermochemical property is assigned to the whole molecule. The relationships between gj-values and hjvalues express how this contribution should be partitioned between the groups such that consistent values are assigned to each group independent of the selection of molecules used to estimate the gj-values. From this relationship the vibrational contribution can be calculated and in Table 1 are shown the results for a selection of groups. Table 1. Total and Vibrational Heat Capacity Contribution of Selected Groups (J K−1 mol−1)a func relationb gj

group

300 K

1000 K

gj

hj

gj

hj

C−(C)(H)3 C−(C)2(H)2 C−(C)3(H) C−(C)3

hj + 2R hj hj − 2R hj − 4R

25.9 23.0 19.0 18.3

9.3 23.0 35.6 51.6

61.80 51.63 42.05 36.7

45.2 51.63 58.68 70.0

Cd−(H)2 Cd−(C)(H) Cd−(C)2 C−(Cd)(C)(H)2 C−(Cd)(C)2(H) C−(Cd)3(C)3

hj + 2R hj hj − 2R hj h − 2R hj − 4R

21.3 17.4 17.2 21.4 17.4 16.7

4.7 17.4 33.8 21.4 34.0 50.0

47.15 35.4 25.4 52.22 42.63 37.5

30.52 35.4 42.0 52.22 59.26 70.8

CB−(H) CB−(C) C−(CB)(C)(H)2 C−(CB)(C)2(H) C−(CB)(C)3

hj hj hj hj hj

− 2R − 4R

13.6 11.2 24.4 20.4 18.3

8.1 22.3 24.4 37.0 51.6

35.2 22.8 52.47 42.89 37.5

29.7 33.9 52.47 59.52 70.8

C−(C)(H)2(OH)c C−(C)2(H)(OH)c C−(C)3(OH)c

hj + 2R hj hj − 2R

38.9 38.1 36.1

22.3 38.1 52.7

79.29 71.21 62.2

62.66 71.21 78.8

CO−(H)(C) CO−(C)2 C−(CO)(C)(H)2 C−(CO)(C)2(H) C−(CO)(C)3

hj + 2R hj hj hj − 2R hj − 4R

29.4 23.4 25.9 n.a. n.a.

12.8 23.4 25.9

51.04 40.2 51.04 n.a. n.a.

34.4 40.2 51.04

+ 2R/3 − 4R/3

III. RESULTS In the following section it will be described how the validity of the central hypothesis as formulated for a polymer in eqs 12 has been tested. The tests are based on the frequencies obtained from DFT and ab initio calculations, and the following series have been examined: alkanes, perfluoralkanes, polyethylene glycols, all-cis- and all-trans-polyenes, dendralenes, and polyacenes. To investigate the significance of the calculational level, the alkane frequency distributions have been calculated at three different levels and the all-cis- and all-trans-polyenes at two. The section has been organized as follows: First, the general properties of frequency distributions will be presented and discussed. This will be followed by a detailed presentation and discussion of the alkane results and finally particular results from the other series will be dealt with. In Figure 1 is shown a selection of the molecular frequency distributions that are examined here. Others are shown in Figures S1−S3 (Supporting Information) for each polymeric series between 39 and 21 MFDs are drawn. For each compound the plots are constructed as cumulative distributions known from statistical analyses, such that the ordinate of each point is the fraction of frequencies in the molecule with a wavenumber smaller than or equal to the abscissa. The noticeable feature of these distributions is how well they, within each class of polymer, seem to follow eq 12. The MFDs of the shortest chains are based on relatively few wavenumbers and consequently the distribution functions are very discontinuous. This is particularly noticeable for the alkanes and perfluoralkanes based on the smallest monomers. However, as the lengths increase and the MFDs appear more continuous, the fine structures of the distributions are maintained independently of the chain length. For the larger monomers such as the polyethylene glycols, the fine structure is noticeable already for the dimers and trimers. This constant fine structure is a strong indication for a constant contribution from each a additional group to the MFD. If the contribution of a monomeric unit to the MFD were dependent on the molecular structure, i.e., the chain length, the fine structure of

a Based on experimental group additivity values from ref 7. bThe functional relationship between the total group contribution (gj) and the vibrational; contribution (hj) determined as described in Appendix B. cComposite group.4

The functional relationship between the vibrational group additivity values (hj) and the group additivity values for the total contribution (gj) were found from a matrix with all 23 groups in Table 1 by the method described for alkanes in Appendix B. Vibrational group additivity values found from these relationships are in good agreement with the idea of group additivity being a refinement of bond additivity. For all of the various functional with identical central atoms a negative correlation is observed between the vibrational contribution to the heat capacity and the number of hydrogens. Conversely, for all functional groups the total heat capacity contribution is positively correlated with the number of hydrogens. Furthermore, the temperature dependence of the vibrational contribution is also as one would expect. At a temperature increase from 300 to 1000 K the groups C−(H)3, Cd−(H)2, and CB−(H) show an increase in their heat capacity 5511



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Figure 1. Molecular frequency distributions: (a) alkanes (B3LYP/6-31-G(d)); (b) perfluoralkanes, (B3LYP/6-31-G(d)); (c) dendralenes, (B3LYP/ 6-31-G(d)); (d) polyacenes, (B3LYP/6-31-G(d)).

tetracontane in the all-anti conformation. MFDs of the same class of compounds obtained at other calculational levels are shown in Figure S1 (Supporting Information). For the shortest chains, the MFDs are very discontinuous functions but from around octane and above, the fine structure of the functions can be recognized. As for all the distributions, the fine structure appears to be maintained throughout the homologous series but the distributions are obviously not identical. In ethane, 6 out of 18 vibrational modes are C−H stretch modes, and in the flat region of the ethane MFD between approximately 1500 and 3000 cm−1, the value of the MFD is equal to (18 − 6)/18 (≈0. 67). In tetracontane with 82 C−H stretch modes out of 360, the MFD has a value of (360 − 82)/360 (≈0.772) in the same region. The constant fine-structural elements are particularly noticeable in the region below 1500 cm−1, where the only change observed is the improved features of the fine as the chain length increases and the MFDs is based on more point. The quantitative test of eq 12 relies on the fact that the frequencies of a molecule is easy to calculate when its frequency distribution is known. The test requires the estimation of two MFDs: that of an infinitely long chain and that of the end groups. For the infinitely long methylene chain, the MFD has been estimated from tetracontane. In the tetracontane MFD there are several flat regions, i.e., wavenumber intervals in which there are no frequencies, and all the frequencies fall in five intervals: 0−530 cm−1, 737−1154 cm−1, 1218−1441 cm−1, 1512−1546 cm−1, and above 3008 cm−1. From the whole series it appears that the 9 additional frequencies, from each extension of the chain by a methylene group, are distributed among each of the five intervals as 2, 2, 2, 1, and 2. An estimated MFD of an infinitely long chain was then obtained by adjusting the flat regions of the triacontane MFD to the fractions: 2/9, (2 + 2)/ 9, (2 + 2 + 2)/9, and (2 + 2 + 2 + 1)/9. The resultant distribution was used as GFD for a methylene group: f m(ṽ) in eq 12. For the frequency distribution of the end groups there

the molecular frequency distributions would necessarily change with the chain length. Quantitatively this has been tested by model calculations based on eq 12. This equation shows how the MFDs of the members of polymeric series can be calculated from the GFDs of the end groups and the monomeric unit. For each class of compound, the longest polymer has been used to estimate the GFD of the monomeric unit and the GFD of the end groups has been set equal to the MFD of a short-chained compound in the series. On the basis of eq 12, linear combinations of these two distributions have been used to estimate the MFDs of all the compounds in the polymeric series. From these estimated MFDs the model frequencies of all the compounds have been calculated and compared to the frequencies from the DFT or ab initio calculations. This latter calculation is based on the fact that when the distribution of a set of numbers is known and the length of a set is known, the numbers can be found by a quantile analysis (see below). A close match between the model frequencies and the DFT or ab initio results will support the validity of eq 12, and the results can be used to investigate the influence of the calculational level. The monomeric units have two ligands and are without branching points, and as noticed above, this means that their vibrational contribution to the heat capacity (hC(T)-value) will be equal to their total heat capacity contribution (gC(T)-value). The vibrational heat capacity group additivity value of the monomer: hCm(T) is equal to wm⟨δf Cm(κ,T)/δκ⟩, which then should be equal to the total group contribution. Following eq 3, the weighted average of the density function for the group heat capacity numbers at different temperatures can be found by transformation of the GFD. These numbers have been compared to the experimentally derived total group contributions, gCm(T)-values, from the NIST database. Alkanes. In Figure 1a is shown the MFDs of the homologoues series of linear alkanes from ethane to 5512



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are several possibilities. The ethane MFD is one possibility, but it can be argued that the methyl groups in ethane are poor models for the methyl groups in a long-chained alkane. For this reason it may be appropiate to consider the end groups to be two ethyl groups, in which case the MFD of butane can be used as the fe(ṽ) in eq 12. With knowledge of fe(ṽ) and f m(ṽ), model MFDs for all the alkanes in the series can be calculated, and the model frequencies can then be estimated by inversion of the frequency distribution. This calculation of the frequencies from the model MFDs is equivalent to a quantile analysis where a set of numbers of a given length are found from a known distribution. The procedure is illustrated in Figure 2.

Figure 2. Calculation of the molecular frequencies for pentane from inversion of a model MFD: (a) methylene-GFD estimated from tetracontane; (b) model MFD for pentane; (c) ethane MFD.

A graphical comparison of DFT and model calculations are shown in Figure 3 for a few of the alkanes. For all but the shortest chains the agreement is excellent. Some of deviation, i.e., around 800 cm−1, appears to be systematic and is likely to arise from the GFD of the end group. When ethane or butane frequencies are used as the end-group GFD, some of the frequencies in the shortest chains will be identical to the ethane and butane frequencies as can be seen in Figure 2. This is unphysical and may be prevented by use of an estimated GFD for the end group rather than take the MFD of a short chain. The alkane MFDs have also been calculated at the B3LYP/6311G(d,p) level and the series from from ethane to docosane at the MP/6-31G(d) level as well. The results are shown in Figures S1 (Supporting Information). A summary of model calculations for the alkanes are shown for all calculational levels in Table 2. All the error distributions are far from Gaussian (Figure S5, Supporting Information, as an example), and the standard deviations of the errors are only used as convenient numbers by which the different calculations can be compared. The best agreement between the ab initio and DFT frequencies and those obtained by the model calculations is found with the model calculations in which the methylene GFD is estimated from the longest chain (tetracontane). This is in agreement with eq 12 because the longer chain is closer to the infinitely long chain and it will consequently give a better estimate for the methylene GFD. The choice of calculational level and endgroup GFDs appear to have much less significance. From the summary of the data in Table 2 it is clear that the model frequencies show the largest deviation from the DFT or ab initio results for wavenumbers below 700 cm−1. This is a general feature for all the series examined here (see also Table 3). At higher wavenumbers where characteristic group frequencies are common in infrared spectra, the agreement is generally much

Figure 3. Comparison of model frequencies with DFT results for a selection of alkanes. The full line is drawn with a slope of one.

better, but these vibrational modes have a smaller influence on the heat capacity at low temperatures. As shown in Table 1, the methylene group vibrational heat capacity contribution (hC(T)-values) should be equal to the total heat capacity contribution (gC(T)-values). In the last column of Table 2 the vibrational heat capacity contribution from the methylene GFD is compared to the total methylene group additivity value derived from experimental measurements. When it is considered that the model frequency distributions are based on unscaled frequencies, the agreement appears very satisfactory. Perfluoroalkanes. Computational investigations have shown that perfluoroalkanes prefer a helical conformation that is also that obtained in the present calculations.31,32 The MFDs of the 21 perfluoroalkanes from perfluoroethane to perfluorodocosane are shown in Figure 1b. As for the alkanes, the fine structure of the distributions is maintained throughout the series with only a gradual change as the chain length increases. A curious indication of how well the fine structure is maintained throughout the series is the complete absence of frequencies in the narrow region between 269 and 288 cm−1 for all the compounds in the series. Each additional CF2 group contributes with three vibrational modes below and six above this interval. For this series there seems to be a significant improvement when the perfluorobutane MFD rather than the perfluoroethane MFD is used as a model for the end-group GFD (Table 3). This is in contrast to the alkanes where the MFD of butane did not appear to provide a more realistic GFD 5513



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Table 2. Summary of Results from Model Calculations on Alkanes

a

The alkane from which the CH2 group frequency distribution was estimated. bThe alkanes from which the frequency distributions of the end groups was taken. cThe mean of the absolute relative error (%) and the standard deviation of the absolute error for different model calculations of frequencies compared to the DFT and ab initio results. dMethylene group additivity values for the heat capacity (J K−1 mol−1) at the temperaures 300, 400, 500, 600, 800, 1000, and 1500 K: Absolute difference (pct. difference) between the value obtained from ref 7 and the value of wm⟨δf Cm(κ,T)/δκ⟩ calculated from the model GFD.

Table 3. Summary of Results from Model Calculations

a The molecule from which the monomeric group frequency distribution was estimated. bThe molecule from which the frequency distributions of the end groups was taken. cThe mean of the absolute relative error (%) and the standard deviation of the absolute error for different model calculations of frequencies compared to the DFT results. dMonomer group additivity values for the heat capacity (J K−1 mol−1) at the temperaures 300, 400, 500, 600, 800, 1000, and 1500. Absolute difference (pct. difference) between the value obtained from ref 7 and the value of wm⟨∂f Cm(κ,T)/∂κ⟩ calculated from the model GFD.

cm−1 (1), 998−1267 cm−1 (5), 1309−1313 cm−1 (1), 1341− 1571 cm−1 (4), 2988−3049 cm−1 (4). The numbers in parentheses indicate the number of additional wavenumber when the chain is lengthened by one monomeric unit. The good agreement shown in Table 3 between the model and the DFT frequencies is overall the best obtained for the polymers examined here. all-cis- and all-trans-Polyenes. The MFDs of the all-cisand all-trans-polyenes from C2 to C50 and calculated with B3LYP/6-31-G(d) and B3LYP/6-311-G(d,p) are shown in Figures S2 and S3 (Supporting Information). Also for these compounds the fine structures of the distributions are maintained throughout the series. The GFDs of the monomeric

for two ethyl groups than the ethane MFD does for two methyl GFDs. Generally, the agreement between the model frequencies and the DFT frequencies is better for this series than for the comparable alkanes. The heat capacity calculations are also closer to the experimentally derived heat capacity group additivity values. Polyethylene Glycols. The calculated molecular frequency distribution of the polyethylene glycols from ethanediol to docosaethylene glycol are shown in Figure S3c (Supporting Information). The monomeric unit has the formula C2H4O and at the B3LYP/6-31G(d) level, the 21 additional frequencies with each unit falls in the intervals: 0−592 cm−1 (6) 835−855 5514



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Polyacenes. The molecular frequency distributions for polyacenes from the monomer benzene to docosacene are shown in Figure 1d. The monomeric unit for polyacenes has the composition C4H2 and at this calculational level each additional monomer gives 16 additional vibrational modes below 1700 cm−1 and two in the region between 3170 and 3210 cm−1. As for the other distributions, the fine structure is maintained throughout the series and, although the results of detailed analysis in Table 3 show agreement between the wavenumbers from model calculations and from the DFT results, it is poorer than for the aliphatic compounds. The heat capacity values obtained from the model distributions for the C4H2 unit are also in good agreement with the experimental group additivity values.

units cis- and trans-ethenediyl were found from all-cis- and alltrans-pentacontapentacosaene. In the all-cis distributions (B3LYP/6-31-G(d)) the wavenumbers occur in the regions 0−253 cm−1 (2), 303−844 cm−1 (3), 920−1225 cm−1 (2), 1284−1340 cm−1 (1), 1373−1698 cm−1 (2), and above 3151 cm−1 (2), where the numbers in parentheses indicate the number of additional wavenumber when the chain is lengthened by one monomeric unit. The all-trans distributions are different from the all-cis distributions at the same calculational level, and the wavenumbers occur in the regions 0−663 cm−1 (4), 836−1200 cm−1 (2), 1210−1703 cm−1 (6), and above 3140 cm−1 (2). The estimated GFDs for the cis- and trans-C2H2 groups obtained from the two different C50polyenes are shown in Figure S4 (Supporting Information). At the temperatures listed in Table 3 the average contributions to the heat capacity for these groups calculated from the distributions shown in Figure S4 (Supporting Information) are (J K−1 mol−1): 32.9 (33.5), 42.4 (42.7), 50.6 (50.9, 57.4 (57.6), 67.6 (67.7), 74.7 (74.8), 85.3 (85.4) with the all-trans values shown in parentheses. Although not identical, the differences are small and close to the experimentally derived identical group additivity values. For both the all-cis and the all-trans series, the agreement between the wavenumbers from the model calculations and from the DFT results is satisfactory (Table 3) and as for the alkanes with a slightly better agreement with the larger basis set. Dendralenes. In Figure 4 are shown the conformations of [5]dendralene and [6]dendralene. These conformations are in

IV. DISCUSSION The proposition that, for molecules where group additivity can be applied, the molecular frequency distribution (MFD) is a weighted sum of group frequency distributions (GFDs) has two parts: first, the claim that MFDs are a sum of GFDs and, second, the claim that this leads to additivity. The analysis based on transformation rules for statistical distributions given above, together with the special property of Benson matrices, shows that within the rigid-rotator harmonic oscillator approximation the second part is true. The analysis of model calculations, based on the hypothesis and summarized in Tables 2 and 3, provides strong support for the first part. In relation to providing an explanation for the experimentally observed validity of additivity schemes, it is important to understand to what extent the limitations imposed by the rigid-rotator harmonic oscillator approximation affect the derivation leading to eq 7. Refinements of the rigid rotator-harmonic oscillator approximation is done by including effects of anharmonicity and ro-vibrational coupling. When we try to assess the significance of these effects, two things are relevant. First, the transformation rule leading to additivity is not dependent on the exact form of the transforming function as long as it provides a one-to-one mapping relationship between the frequency (wavenumber) of a vibrational mode and contribution of the particular vibrational mode to the thermochemical parameter in question. Second, from the validity of eq 15, it is clear that the constant term for the translational and rotational contributions to thermochemical properties, which in the rigid rotator-harmonic oscillator approximation for the heat capacity is equal to 4R, can have any magnitude and any temperature dependence without this affecting the validity of the additivity scheme. These two factors, both of which follow from mathematical relations, seem to give ample leeway for refinement and modifications of the rigid rotator-harmonic oscillator approximation without affecting the additivity of the results. In view of the unphysical aspects of that particular approximation, this is an attractive feature of the central hypothesis and the analysis presented here. The vibrational contribution to the heat capacity, the heat of formation, and the molecular entropy increases as the wavenumber decreases. The vibrational modes at low wavenumbers include large portions of the molecule and this illustrates how long distance interactions in a molecule are important for its thermochemical properties. This fact is directly included in the present hypothesis through the GFDs, which forms the basis for the vibrational contributions to the thermochemical properties.

Figure 4. Dendralene structures.

agreement with those obtained by others.33,34 All the evennumbered dendralenes have similar conformations in which consecutive s-trans-1,3-butadiene units are orientated orthogonally to each other. For the uneven dendralenes, the conformations obtained have an ethylenyl moiety that is not conjugated to a neighboring double-bond as shown for [5]dendralene in Figure 4. The U-conformationa lie slightly lower in energy than the D-conformations. For the [5]dendralene the numbers are HF = −388.1803193 Eh vs HF = −388.1814311 Eh at the calculational level used here. For all the uneven dendralenes in the series, the vibrational frequencies have been calculated for both conformers. The molecular frequency distributions for all the investigated dendralenes, from butadiene to [22]dendralene are shown in Figure 1c. The two different conformations of the uneven dendralenes give MFDs, which are essentially identical. The NIST database does not include additivity values for the ethene-1,1-diyl group and consequently the heat capacity contribution from the frequency distribution cannot be compared with an experimental value. As for the other polymers, the model calculations of the frequencies are in satisfactory agreement with the DFT results. 5515



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heat capacity numbers FC(κ,T) can be expressed as integrals of the density functions p(ṽ) and pC(κ,T):

V. CONCLUSION Application of statistical theory for transformation of cumulative distribution functions shows that within the rigidrotator harmonic oscillator approximation additivity of the vibrational contribution to thermodynamic properties can be derived directly from the central hypothesis that molecular frequency distributions are a weighted sum of group frequency distributions. Graph theoretical analyses of Benson matrices furthermore shows that within the rigid rotor harmonic oscillator approximation, group additivity of vibrational contributions to a thermochemical property leads directly to overall group additivity when the translational and rotational contributions are given by a constant. With a range of different polymers as examples, further support for the central hypothesis is supplied by the calculational results summarized in Tables 2 and 3. A secondary outcome of the analysis is that it provides a very fast method for calculations of molecular frequencies once the frequency distributions of the constituent groups are known. For groups that can form polymers, the GFDs can be established as shown here, by simply calculating the MFDs of the appropiate polymer. For other groups, i.e., methyl, an appropiate method for estimating an optimzed GFD will have to be developed. The analysis of the different frequency distributions of the all-cis- and all-trans-polyenes raises the question of whether form elements such as conformational differences can be associated with specific changes in molecular frequency distributions. This is presently under investigation. Schrödinger35 thought that “a small molecule might be called ‘the germ of a solid’.” In view of how important frequency distributions are for the understanding of the thermal properties of solids, it is only fitting if they can contribute to the understanding of molecular properties.

ṽ

p(v )̃ dv ̃

F C(κ ,T ) =

κ

∫0

(A1)

pC (κ ,T ) dκ

(A2)

−1

Setting *vib (κ ,T ) = φ(κ ,T ), the relation between wavenumbers and heat capacity numbers is given by ṽ = φ(κ,T). The rule for transformation of distributions is based on the usual analytical formula for changing variables, which when applied to eq A1 leads to

∫φ

F(v )̃ =

φ−1(v )̃

p(φ(κ ,T )) φ′(κ ,T ) dκ

−1

(0)

(A3)

φ′(κ,T) is always negative and −p(φ(κ,T))φ′(κ,T) defines the density function, pC(κ,T), for heat capacity numbers at the temperature T. When the heat capacity number κ varies between 0 and R, the wavenumber ṽ varies between infinity and 0, and consequently eq A3 can be rewritten as F(ν)̃ = − =

∫R

∫κ

R

κ

pC (κ ,T ) dκ

(A4a)

pC (κ ,T ) dκ

(A4b)

From the definition of a density function and elementary analysis it follows that

VI. COMPUTATIONAL METHODS Molecular frequencies were calculated with the Gaussian suite of programs at the B3LYP/6-31G(d), B3LYP/6-311G(d,p), and MP2/6-31G(d) levels.36 The choice of these levels was determined by the fact that no electronic energies but only frequencies were required. The optimization criterion was set at “Tight” or “VeryTight”, and in the DFT-based calculations, the grid was specified as “UltraFine”. Because the main purpose of the study is to investigate fundamental properties of the frequency distributions, uncorrected frequencies have been used throughout. All other calculations were done with Mathematica versions 7.0 or 8.0.37 The calculations of the group heat capacity values in Tables 2 and 3 by transformation of a GFD requires inversion of the function *vib(v ̃,T ) from eq 2a. This cannot be done analytically. Inversion of polynomial fits appeared to be unstable for low values of T, and consequently, the heat capacity group values were calculated by averaging the heat capacity numbers obtained from a large set of wavenumbers determined from the appropiate GFD.

■

∫0

F(v )̃ =

1=

∫0

=

∫0

R

κ

pC (κ ,T ) dκ pC (κ ,T ) dκ +

∫κ

R

pC (κ ,T ) dκ

(A5)

Insertion of the relevant terms from eqs A2 and A4b yields F C(κ ,T ) = 1 − F(v )̃

for v ̃ = *vib−1(κ ,T )

(A6)

Using this transformation rule on both the molecular and group distributions functions and inserting the results for F(ṽ) and f j(ṽ) into eq 1 of the main text leads to 1 − F C(κ ,T ) =

1 3NM − 6

∑ njwj[1 − f jC (κ ,T )] j

(A7)

Because the total number of oscillators 3NM − 6 is equal to ∑njwj, eq A7 can be reduced to eq 4 in the main text. When the transforming function is a strictly increasing function of the wavenumber such as the vibrational zero-point energy, , zpe(v )̃ = NA

hc ṽ = ξ 2

(A8)

the derivation of the transformation rule follows the same pattern and leads to an expression such as F zpe(ξ) = F(ν)̃

APPENDIX A

for ν ̃ = (, zpe)−1(ξ)

(A9)

A transformation of this form also leads to additivity as has already been shown.19 The transformation rules for the vibrational excitation energy and the logarithm of the vibrational partitition function can be derived in exact analogy to that of the vibrational heat capacity.

Rule for Transformation of Distributions

With the heat capacity as an example, the transformation of frequency distributions can be derived as follows: cumulative frequency distributions F(ṽ) and cumulative distributions of 5516



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APPENDIX B

to be a valid basis for a matrix of all alkanes, the other rows must be formed by linear combinations of these vectors:

Basis of Benson Matrices Augmented with a Constant Vector

wi = c1 w1 + c3 w3 + c3 w3 + c4 w4

By rewriting eq 14 of the main text as Cp ,i(T ) =

ni ,1h1C (T )

+

ni ,2h2C (T )

+ ni ,jhCj (T ) + 4R

+

ni ,3h3C (T )

(B4)

However, to ensure the correct translational and rotational contribution the last column of Mk must only include ones, i.e., ni,5 must be equal to 1. The elements of the wi-vector can be calculated from the four basis vectors as shown:

+ ... (B1)

one can see that translational and rotational contribution can be included as an extra column in the matrix that is defined by the coefficients ni,j. For other thermochemical parameters the vibrational group contributions (h-values) and the constant term is changed as appropiate. However, all the elements in the additional column, which represents the sum of translational and rotational contributions, must be equal to 1. Group additivity schemes depend on the fact that the properties of an infinite number of compounds can be calculated from the properties of few. For the purpose of brevity and clarity, the following analysis is based on alkanes only and uses the small alkanes: ethane, propane, isobutane, and neopentane as basis.6 A general Benson matrix for all alkanes can then be written as shown:

ni,1 = c12 + c 2 2 + c33 + c4 4 ni,2 = c 2 ni,3 = c3

(B5)

ni,4 = c4

ni,5 = c1 + c 2 + c3 + c4

The elements ni,1 to ni,4 are the numbers of the four groups in any alkane whereas the fifth element ni,5 is the coefficient of the constant k. The coefficients c1 to c4 cannot be chosen arbitrarily. In a straight-chained alkane the number of methyl groups is always two, independent of the number of methylene groups. The number of methyl groups will increase by one for every branching point introduced by a tertiary carbon. Introduction of a quaternary carbon leads to two additional methyl groups. The coefficients ni,1 to ni,4 are consequently related as ni ,1 − ni ,3 − 2ni ,4 = 2

(B6)

Insertion of the expression for the coefficients (B5) into (B6) leads to an expression for the interdependency of the coefficients c1 to c4, which after reduction has the form c1 + c 2 + c3 + c4 = 1

(B7)

Comparison of eqs B5 and B7 shows that the coefficient ni,5 for the constant term is always equal to 1 as required, and that the vectors w1 to w4 consequently form a basis for the general Mk matrix for alkanes. It follows from this analysis that a given thermochemical property can be determined from both matrices M and Mk by multiplication with the appropriate vector when the translational and rotational contributions are given by a constant. This leads to eq B8, of which a general version is given as eq 15 in the main text.

The four row vectors v1 to v4 are the basis for any matrix based on alkanes, and the row vector vi, which represents all higher alkanes, can be written as a linear combination of these four vectors. The various thermochemical values for the alkanes can be determined from M by multiplication with an appropiate gvector, which includes the total group additivity values for the chosen property. When a constant term is included in the matrix by augmentation with an additional column, this leads to

M[g1 , g2 , g3 , g4 ]T = M k [h1 , h2 , h3 , h4 , k]T

(B8)

The rank of Mk cannot be higher than that of M, and consequently any solution to systems of equations based on Mk will have k as a free variable. After row reduction of the matrices M and Mk and multiplication with the g- and h-vectors, the relationship between the total group contributions and the vibrational contribution can be read out. When the heat capacity is the thermochemical value of interest, k is equal to 4R and for the four groups of the alkanes this leads to the relationships shown as

The row vectors forming the basis are now w1 to w4, and the last element in these four vectors is equal to 1. Consequently, under the rigid rotator-harmonic oscillator approximation, the thermochemical values of the four alkanes forming the basis is equal to the scalar product of the first four row vectors of Mk with an h-vector. This h-vector must include only the vibrational contribution of each group, followed by a final element k, which is the sum of the translational and rotational contribution to the chosen thermochemical value. For w1 to w4

C C gCH (T ) = hCH (T ) + 2R 3 3

C C gCH (T ) = hCH (T ) 2 2

(B9)

C C gCH (T ) = hCH (T ) − 2R

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gCC (T ) = hCC (T ) − 4R

(4) Cox, J.D.; Pilcher, G. Thermochemistry of Organic and Organometallic Compounds; Academic Press: New York, 1970; Chapter 7. (5) Benson, S. W. Thermochemical Kinetics; J. Wiley: New York, 1976. (6) Cohen, N.; Benson, S. W. Chem. Rev. 1993, 93, 2419−2438. (7) Stein, S.E.; Brown, R.L. Structures and Properties Group Additivity Model; Linstrom, P. J., Mallard, W. G., Eds.; NIST Chemistry WebBook, NIST Standard Reference Database Number 69; National Institute of Standards and Technology: Gaithersburg, MD 20899; June 2005 (http://webbook.nist.gov). (8) Bader, R. F. W. Acc. Chem. Res. 1985, 18, 9−15. (9) Bader, R. F. W. Atoms in Molecules; Clarendon: Oxford, U.K., 1990. (10) Bader, R. F. W.; Popelier, P. L. A.; T.A. Keith. Angew. Chem., Int. Ed. Engl. 1994, 33, 620−631. (11) Bader, R. F. W.; Bayles, D. J. Phys. Chem. A 2000, 104, 5579− 5589. (12) Truong, T. N. J. Chem. Phys. 2000, 113, 4958−4964. (13) Zhang, S.; Truong, T. N. J. Phys. Chem. A 2003, 107, 1138− 1147. (14) Sumathi, R.; Carstensen, H.-H.; Green, W. H. J. Phys. Chem. A 2001, 105, 6910−6925. (15) Sumathi, R.; Carstensen, H.-H.; Green, W. H. J. Phys. Chem. A 2001, 105, 8969−8984. (16) Sumathi, R.; Carstensen, H.-H.; Green, W. H. J. Phys. Chem. A 2002, 106, 5474−5489. (17) Sumathi, R.; Green, W. H. Theor. Chem. Acc. 2002, 108, 187− 213. (18) Saeys, M.; Reyniers, M. F.; van Speybroeck, V.; Waroquier, M.; Marin, G. B. ChemPhysChem 2006, 7, 188−199. (19) Bojesen, G. J. Phys. Org. Chem. 2008, 21, 833−843. (20) Debye, P. Ann. Phys. 1912, 39, 789−839. (21) Bu, H. S.; Cheng, S. Z. D.; Wunderlich, B. J. Phys. Chem. 1987, 91, 4179−4188. (22) Loufakis, K.; Wunderlich, B. J. Phys. Chem. 1988, 92, 4205− 4209. (23) Wunderlich, B. Pure Appl. Chem. 1995, 67, 1019−1026. (24) Wunderlich, B. Thermochim. Acta 1997, 300, 43−65. (25) Pyda, M.; Boller, A.; Grebowicz, J.; Chuah, H.; Lebedev, B. V.; Wunderlich, B. J. Polym. Sci. B 1998, 36, 2499−2511. (26) Hald, A. Statistical Theory With Engineering Applications; J. Wiley and Sons: New York, 1952; Chapter 5. (27) Billingsley, P. Probability and Measure; J. Wiley and Sons: New York, 1995; Section 17. (28) Bickel, P. J.; Doksum, K. A. Mathematical Statistics, Basic Ideas and Selected Topics; Pearson Prentice Hall: Upper Saddle River, NJ, 2007; Appendix B, section 2. (29) Harary, F. Graph Theory; Addison-Wesley: Reading, MA, 1969. (30) Clark, J.; and Holton, D. A. A First Look at Graph Theory; World Scientific: Singapore, 1991. (31) Watkins, E. K.; Jorgensen, W. L. J. Phys. Chem. A 2001, 105, 4118−4125. (32) Golden, W. G.; Zoellner, R. W. C. R. Chim. 2005, 8, 1610−1616. (33) Brain, P. T.; Smart, B. A.; Robertson, H. E.; Davis, M. J.; Rankin, D. W. H.; Henry, W. J.; Gosney, I. J. Org. Chem. 1997, 62, 2767−2773. (34) Palmer, M. H.; Blair-Fish, J. A.; Sherwood, P. J. Mol. Struct. 1997, 412, 1−18. (35) Schrödinger, E. What is Life?; Cambridge University Press: London, 1944. (36) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; G. E..Scuseria, Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A.; , Jr., Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.;

Extension of this analysis to include additional groups depends on constructing an appropiate Mk-matrix and inclusion of the appropiate coefficients in the interdependence expressed in eq B6, which arises from elementary graph theory.29,30 The relevant formulas (both due to Euler) are

∑ deg(nj) = 2E j

(B10)

∑ nj − E + F = 2 j

(B11)

In eqs B10 and B11, deg(nj) is the number of groups bound to the j’th group, E is the total number of bonds between the groups in the molecule, and F is the number of faces in the graph which can be drawn from the molecule when every group is represented as a vertex in the graph. When the molecule is acyclic, F is equal to one and the number of bonds between groups is equal to the number of groups minus one. F increases by one for every ring in the molecule and the number of rings is determined by the number of ring-forming groups, i.e., CB−H and CB−C in the case of benzene rings. For all Mk matrices hitherto investigated, the conditions imposed by eqs B10 and B11 have led to the desired result.38

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ASSOCIATED CONTENT

S Supporting Information *

Figure S1: MFDs of alkanes at various calculational levels and MFDs of polyethylene glycols. Figure S2: MFDs of all-cispolyenes. Figure S3: MFDs of all-trans-polyenes. Figure S4: GFDs of cis-C2H2 and trans-C2H2 estimated from all-cis- and all-trans- pentacontapentacosaen. Figure S5: Examples of error distributions in the model calculations. All calculated frequencies from the ab initio and DFT calculation as well as from the model calculations with the smallest end groups are available for each class of polymers as text files (Tables S1− S11). This material is available free of charge via the Internet at http://pubs.acs.org

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Ph: +45 6169 4501. Fax: +45 4343 4501. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS I am grateful to Jeppe Woetmann Nielsen for discussions on statistical theory and for pointing me toward ref 27. I am grateful to Steen Hammerum, University of Copenhagen, for assistance with the Gaussian calculations and many discussions on group additivity and to the Department of Chemistry, University of Copenhagen, for access to computational resources.

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REFERENCES

(1) Laider, K. J. Can. J. Chem. 1956, 34, 626−648. (2) Benson, S. W.; Buss, J. H. J. Chem. Phys. 1958, 29, 546−572. (3) Benson, S. W.; Cruickshank, F. R.; Golden, D. M.; Haugen, G. R.; O’Neal, H. E.; Rodgers, A. S.; Shaw, R.; Walsh, R. Chem. Rev. 1969, 69, 279−324. 5518



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Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; M. A. Al-Laham, Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, Revision E.01; Gaussian, Inc.: Wallingford, CT, 2004. (37) Mathematica; Wolfram Research: Champaign, IL (http:// wolfram.com). (38) Bojesen, G. Unpublished results.

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NOTE ADDED AFTER ASAP PUBLICATION This article posted ASAP on July 9, 2014. The following have been revised: Tables 1, 2, 3, equations 9, 10, A8, and Results section, paragraph 20, sentence 4. The correct version posted on July 14, 2014.

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