Dipole oscillator strength distributions, properties, and dispersion energies for ethylene, propene, and 1-butene (1).

Abstract: A recommended isotropic dipole oscillator strength distribution (DOSD) has been constructed for the ethylene molecule through the use of quantum mechanical constraint techniques and experimental dipole oscillator strength (DOS) data; the DOS data employed are recent experimental results not available at the time of the original constrained DOSD analysis of this molecule. The constraints are furnished by molar refractivity data and the Thomas-Reiche-Kuhn sum rule. The DOSD is used to evaluate a variety of isotropic dipole oscillator strength sums, logarithmic dipole oscillator strength sums, and mean excitation energies for ethylene. Pseudo-DOSDs for this molecule, and for propene and 1-butene, which are based on an earlier constrained DOSD analysis for these molecules, are developed. They are used to obtain reliable results for the isotropic dipole-dipole dispersion-energy coefficients [C.sub.6], for the interactions of the alkenes with each other and with 47 other species, and the triple-dipole dispersion-energy coefficients [C.sub.9] for interactions involving any triple of molecules taken from ethylene, propene, and 1-butene.

Key words: alkenes, dipole properties, pseudo-states, dipole-dipole and triple-dipole dispersion energies, long-range additive, non-additive interaction energies.

Resume : Faisant appel a des techniques de contrainte de mecanique quantique et des donnees experimentales de force d'un oscillateur dipolaire (FOD), on a developpe une distribution de la force d'un oscillateur dipolaire isotrope (DFOD) recommandee pour la molecule d'ethylene; les donnees de force d'un oscillateur dipolaire utilisees sont des resultats experimentaux recents qui n'etaient pas disponibles lors de l'analyse originale contrainte de la DFOD de cette molecule. Les contraintes sont fournies par les donnees de refractivite molaire et la regle des sommes de Thomas-Reiche-Kuhn. On utilise la DFOD pour evaluer des sommes de forces d'oscillateurs dipolaires isotropes, des sommes logarithmiques de force d'oscillateur dipolaire et des energies d'excitation moyenne de l'ethylene. On a developpe des valeurs pseudo-DFOD pour cette molecule, pour le propene et le but-1-ene qui sont basees sur une analyse anterieure de ces molecules par une methode contrainte de type DFOD. Les resultats obtenus sont utilises pour obtenir des resultats fiables pour les coefficients [C.sub.6] de l'energie de dispersion dipole-dipole isotrope, pour les interactions des alcenes les uns avec les autres et avec 47 autres especes ainsi que pour les coefficients [C.sub.9] d'energie de dispersion triple-dipole pour les interactions impliquant tout triplet de molecules impliquant de l'ethylene, du propene et du but-1-ene.

Mots-cles : alcenes, proprietes dipolaires, pseudo etats, energies de dispersion dipole-dipole et triple-dipole, energies d'interaction additives, non-additives a longue portee.

[Traduit par la Redaction]

Introduction

One of the purposes of this work is to assess the reliability of the results for a wide variety of the isotropic dipole properties of the ground-state ethylene, propene, and 1-butene molecules evaluated previously (1) by using the constrained dipole oscillator strength distribution (DOSD) technique (2-7). The original constrained DOSD calculations (1) were based on the then rather sparse and disjointed (as a function of photon energy) experimental dipole oscillator strength (DOS) data for these alkenes, which were drawn from the work of several different experimental groups. Our new calculations for ethylene are based on much more recent, extensive, and reliable experimental DOS data (8). They can be used to directly check the reliability of the original DOSD calculations for this molecule and, in conjunction with a recent study (9) for methanol, ethanol, and propan-1-ol, which involves analogous input DOS data sets for these molecules, to indirectly assess the reliability of the analogous results for propene and 1-butene. We also present new reliable results for the isotropic dipole-dipole dispersion energies for the interaction of the three alkenes with each other and with a variety of other species. Discrete pseudo-state representations (10, 11) of the recommended constrained dipole oscillator strength distributions for the alkenes are constructed, which are useful for the efficient evaluation (10-12) of the dipolar dispersion energies, particularly the triple-double dispersion energies, for interactions involving these molecules; explicit reliable results for the triple-dipole dispersion-energy coefficients are included for all triples of the alkene molecules.

The relationship between molecular dipole oscillator strength distributions (DOSDs) and the isotropic dipole properties of molecules, and the dipole-dipole and the triple-dipole dispersion energies for interactions involving molecules, is well known (2, 13-19). The molecular dipole properties explicitly considered here are the dipole oscillator strength sums [S.sub.k], the logarithmic dipole sums [L.sub.k], the mean excitation energies [I.sub.k], and the molar refractivity [R.sub.[lambda]] as a function of wavelength [lambda]. The properties [S.sub.k], [L.sub.k], and [I.sub.k], which depend on the value of the index k, and the dipolar dispersion energies find application in many research areas (2, 13-16, 20-32).

A quantum mechanical constraint technique (2-7) is used to construct the DOSDs for ethylene from a base of experimental dipole oscillator strength input data (1, 8) using constraints comprised of the molar refractivity (1, 33, 34) of the molecule for two well-separated wavelengths and the Thomas-Reiche-Kuhn sum rule (14, 35) for the DOSD. Specific results for integrated dipole oscillator strengths for ethylene, over various energy regions, are computed. Recommended values for the dipole sums, [S.sub.k], k = -10(2) -4(1) - 3(1/2) 0, 1, 2; the logarithmic dipole sums [L.sub.k] and mean excitation energies [I.sub.k], k = -2(1) 2; and the molar refractivities of the molecule are tabulated; those for propene and 1-butane can be found in the literature (1). A ten pseudo-state representation of the recommended DOSD for each of the three alkenes is also constructed and tabulated, and these are used to evaluate the results for the dipole-dipole dispersion-energy coefficients for a variety of interactions involving these molecules, and the triple-dipole dispersion-energy coefficients for the interaction of any combination of the three alkenes. The estimated uncertainties in our results for the dipole properties and dispersion energies, and a comparison with literature values, are included in the discussion. The values obtained from the constrained DOSD approach, either the original (1) or the present calculations, are either the only values or the only reliable values available for most of the dipole molecular properties considered in this paper. The results for the dispersion energies obtained herein are generally the only available reliable values for the interactions considered in this paper.

Relationship between DOSDs and dipole properties and dispersion energies

The differential dipole oscillator strength distribution (DOSD) for a molecule is the differential dipole oscillator strength df/dE as a function of photon energy E from the electronic absorption threshold [E.sub.0] for the molecule to very high photon energies. Generally, the dipole properties and dispersion-energy coefficients evaluated using DOSDs are isotropic results, since only orientationally averaged input dipole oscillator strengths are available over the required wide range of photon energies. In some instances (36-40), i.e., CO, [H.sub.2], [N.sub.2], NO, and [O.sub.2], sufficient anisotropic constraints are available to permit the construction of anisotropic DOSDs and hence the evaluation of anisotropic molecular properties.

A variety of important isotropic molecular properties can be evaluated as integrals whose integrands involve the isotropic DOSD weighted by simple functions of the photon energy. These include the dipole oscillator strength sums [S.sub.k], the logarithmic dipole sums [L.sub.k], and the mean excitation energies [I.sub.k], defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [1]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [2]

[I.sub.k] = [E.sub.H] exp([L.sub.k]/[S.sub.k]) [3]

where [E.sub.H] [approximately equal to] 4.35975 x [10.sup.-18] J ([approximately equal to] 27.21 eV) is the Hartree of energy (2, 13, 14). The index k can take on various integer and half-integer values, and each choice defines a different dipole property via eqs. [1-3]. For example, the straggling ([I.sub.1] and [S.sub.1]), stopping ([I.sub.0]), and the total ([I.sub.-1], [S.sub.-1]) inelastic scattering cross-sections of fast charged particles in matter (13, 20, 21); [S.sub.2] and [I.sub.2] determine (2, 13, 14) charge densities at the nucleus and Lamb shifts, while [S.sub.-1], [S.sub.-3], [S.sub.-3/2], [S.sub.-2], and [L.sub.-2] can be used to obtain estimates for the dipole-dipole dispersion energies between molecules (10, 15, 22, 23). An important property, which is involved as a constraint in the construction of DOSDs, is the molar refractivity [R.sub.[lambda]] of a dilute molecular gas. It is related (2, 14-16) to the refractive index n([lambda]) of the gas at wavelength [lambda] and to an integral involving the DOSD

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [4]

where [a.sub.0] [approximately equal to] 5.29177 x [10.sup.-11] m = Bohr, [rho] is the molar density of the gas, h is Planck's constant, [N.sub.A] is Avogadro's constant, c is the speed of light, and [alpha]([lambda])is the wavelength-dependent electronic dipole polarizability of the molecule. The static dipole polarizability is [[alpha].sub.d] = [alpha]([lambda] = [infinity]) = [a.sup.3.sub.0][S.sub.-2]. The [S.sub.k], k = -4, -6, -8, ... are other moments of the DOSD that occur in the Cauchy expansion of [alpha]([lambda]).

The dipole-dipole dispersion energy is the dominant interaction energy at long range for interactions of spherically symmetric systems including freely tumbling molecules. Its importance lies in its use for representing long-range interactions between two atoms or molecules and in constructing potential-energy models that are valid for all intermolecular distances (24-30). The orientationally averaged dipole-dipole dispersion energy (16, 17) for the interaction of molecules A and B is given by -[C.sub.6](A, B)[R.sup.-6.sub.AB] where [R.sub.AB] is the distance between species A and B and [C.sub.6](A, B) is the dipole-dipole dispersion-energy coefficient for the interaction

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [5]

where E(A), df (A)/dE(A), and [E.sub.0](A) are the excitation energy, the differential dipole oscillator strength, and the electronic absorption threshold for molecule A, respectively.

Analogously, the triple-dipole dispersion energy gives the dominant non-additive interaction energy for an assembly of well-separated spherical or "tumbling" species and can be used to help model non-additive effects for other molecular configurations as well (26, 31, 32). For the interaction of three molecules A, B, and C, the triple-dipole dispersion energy is given by (16, 18, 19)

[W.sup.(3)] = (3 cos[[theta].sub.A] cos[[theta].sub.B] cos[[theta].sub.C] + 1)

x [C.sub.9](A, B, C)[R.sup.-3.sub.AB][R.sup.-3.sub.BC][R.sup.-3.sub.AC] [6]

where [[theta].sub.A] is the angle between [R.sub.AB] and [R.sub.AC]. In terms of the original continuous DOSDs for the interacting molecules A, B, and C, the triple-dipole dispersion-energy coefficient [C.sub.9] for "tumbling" molecules is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [7]

Clearly, many important molecular properties and dispersion interaction energies can be evaluated if molecular dipole oscillator strength distributions (DOSDs) are available. The basic procedure used here for constructing molecular dipole oscillator strength distributions has been discussed in detail in the literature (2, 6, 7). Briefly, the initial DOS input data is divided, from the UV-absorption threshold [E.sub.0] to very large values of the photon energy, into [N.sub.0] energy intervals as suggested by the structure of the input DOS data and by the photon-energy regions associated with the individual sources of the DOS data. The number of different initial DOSDs that can be obtained by taking all possible combinations of the

input DOS data is given by [N.sub.D] [[N.sub.0].summation over (j=1)][N.sub.j] where [N.sub.j] is the number of independent sources of DOS data used for the jth input spectral region (41). Each of the initial DOSDs considered is modified by requiring satisfaction of the constraints via application of a constrained least-squares technique (2, 6, 7). The input initial oscillator strength database, [(df /dE).sub.initial] vs. E, for each spectral region i, is modified via

[(df/dE).sup.i.sub.constrained] = (1 + [a.sub.i])[(df/dE).sup.i.sub.initial], i = 1,2, ..., [N.sub.0] [8]

so that the total constrained DOSD satisfies the imposed constraints through the choice of the [a.sub.i]. The degree of modification of the initial DOSD data required to satisfy the constraints can be represented by the standard deviation (STD) defined by

STD = [[N.sub.0].summation over (i=1)][([a.sub.i] - [bar.a]).sup.2] / [N.sub.0]].sup.1/2] [9]

where [bar.a] is the average of all the [a.sub.i] values (6, 41). The recommended DOSD is represented by a set of data points [E.sub.j] and [(df/dE).sub.j], for j = 1, 2, ..., [N.sub.p] >> [N.sub.0], and a set of interpolating functions to connect the points. For very high photon energies, E [greater than or equal to] [10.sup.8] eV, (df/dE) is represented by the Born dipole formula (35) A[E.sup.-2.5]. The dipole properties of molecules are readily evaluated (2) using such a representation of DOSDs.

Construction of the DOSD for ethylene

The experimental dipole oscillator strength (DOS) data used by Jhanwar, Meath, and MacDonald (1) for constructing the original constrained DOSDs for ethylene, propene, and 1-butene have been reviewed by these authors. Generally, relatively few sources of DOS data were available at that time, the experimental DOS data were sparse as a function of photon energy and from a variety of sources, and no direct experimental data was available for E > 70 eV for ethylene, for E > 27 eV for propene, and for E > 24 eV for 1-butene. We base our new constrained DOSD calculations for ethylene on the extensive DOS data of Cooper, Olney, and Brion (8), which are available from the electronic absorption threshold to 200 eV; both high- and low-resolution measurements are available for E [less than or equal to] 50 eV, while for higher photon energies, only low-resolution measurements were made. Analogous experimental studies have unfortunately not been carried out for propene and 1-butene, and so, for these molecules, our recommended DOSD results remain those of the earlier work (1). For ethylene, these more-recent DOS results are part of an extensive collection of experimental DOS data from the same laboratory that in the past has produced DOS data for other molecules that have proved to be very reliable in the sense that the modifications in the data induced by our constraint procedures have often been less than or on the order of the estimated experimental errors of [approximately equal to] 5% (to 10%) (see for example, refs. 9, 42-49).

For higher photon energies, for which the recent experimental DOS data are not available, we use, as input DOS values for ethylene, the recommended values of Jhanwar et al. (1). These were originally based on additivity rules and then modified by the constraints imposed on the overall initial DOS data collection. For large photon energies, the modifications were generally quite small (1).

Following earlier work (1), the constraints for constructing the DOSDs for ethylene are provided by the Thomas-Reiche-Kuhn sum rule [S.sub.0] = 16 and by experimental molar-refractivity data. The molar-refractivity constraints are obtained by using refractive-index data for ethylene, measured by Watson and Ramaswamy (33) and Friberg (34) as normalized to Watson and Ramaswamy's value at 5462 [Angstrom]. The results at two well-separated wavelengths, namely, R([lambda]= 6440 [Angstrom]) = 10.6362 [cm.sup.3] [mol.sup.-1] and R([lambda]= 2302 [Angstrom]) = 13.7115 [cm.sup.3] [mol.sup.-1], are employed as constraints. The percentage difference between these two values of [R.sub.[lambda]], namely, [DELTA][R.sub.[lambda]]= 200([R.sub.[lambda]2] - [R.sub.[lambda]1])/([R.sub.[lambda]1] + [R.sub.[lambda]2]) = 25.3%, is very much larger than the expected experimental error in the refractive-index data, and hence, the least-squares constraint procedure can be used with confidence with this refractivity data to construct the DOSDs for ethylene (41, 50).

The following DOSDs have been constructed for ethylene: (i) DOSD1: 6.02 - 50.0 eV (high-resolution data of Cooper et al. (8)), 50.0 - 200.0 eV (low-resolution data of Cooper et al. (8)), 200 eV - [infinity](recommended DOSD of Jhanwar et al. (1)); STD = 2.11. (ii) DOSD2: as in DOSD1 except the high-resolution data are replaced by the low-resolution data; STD = 5.14.

DOSD1, with the smallest STD for the input data considered here, is chosen as the recommended DOSD for ethylene. The integrated dipole oscillator strengths for the recommended DOSD, together with those calculated from the initial DOS data used to construct the DOSDs for ethylene, are given in Table 1 together with the corresponding integrated oscillator strengths from the original (1) recommended DOSD for ethylene. The approach used here removes errors in the initial DOSD data in a global sense and is capable of yielding reliable dipole molecular properties and very reasonable integrated oscillator strengths. It does not necessarily lead to constrained DOSDs that are accurate on a point-by-point basis (51).

The integrated oscillator strengths evaluated using the initial high-resolution DOS data of Cooper et al. (8) are higher than our recommended integrated oscillator strengths for 6.02 eV [less than or equal to] E [less than or equal to] 20.7 eV and are lower than our results for 20.7 eV [less than or equal to] E [less than or equal to] 50 eV. The maximum modification in the high-resolution data induced by our fitting procedure is about 6.5% in the 6.8-8.0 eV energy region; generally, the modifications in the initial input data in the other energy regions is much smaller than this except for 12.3-18.0 eV where it is 4.7%. Generally, the integrated DOSs obtained from the unmodified experimentral high-resolution DOS data (8) agree with our recommended results to within the estimated experimental error.

The initial low-resolution integrated DOSs are significantly different from the recommended results, and with the integrated DOS's obtained from the unmodified high-resolution data, for E [less than or equal to] 12.3 eV. The low-resolution integrated DOS is high from 6.02 to 6.8 eV, and the disagreements with the recommended (initial high-resolution) results are 13.7% (19.0%), -6.5 (-5.0), -26.5 (-25.8), 4.0 (6.4), -12.1 (-11.7), -18.4 (-17.1), and 13.1% (13.8%) for the energy regions 6.8-8.0 eV, 8.0-8.6 eV, 8.6-9.0 eV, 9.0-10.4 eV, 10.4-10.7 eV, 10.7-11.8 eV, and 11.8-12.3 eV, respectively. For higher excitation energies, the agreement between the various integrated oscillator strengths is within 5% except for the 26.7-35.0 eV and the 35.0 eV-50 eV energy regions where the low-resolution result is 6.7% and 5.1% and 9.7% and 7.9% lower than the recommended and high-resolution values, respectively.

The discrepancies between our integrated oscillator strengths and those corresponding to the original (1) DOSD for ethylene are often less than 5% and generally less than 10% with the exception of the 8.0-8.6 eV, 8.6-9.0 eV, and the 20.7-26.7 eV energy intervals where the original integrated DOSs are 12.7, 28.7, and 12.4% lower than our recommended results, respectively, and the 70.0-100.0 eV and the 100.0-200.0 eV energy regions where the original results are 15.2 and 21.8% higher than the new results.

A number of sources of DOS data for ethylene were involved in the construction of the original (1) constrained DOSD for this molecule, and these and other DOSs, and associated integrated DOSs, for the molecule have been discussed extensively earlier (see for example, refs. 1 and 8 and papers cited therein).

Relatively recently, Berkowitz (52) has updated his earlier (53) discussion and compilation of the photoabsorption cross sections for a variety of gaseous atoms and molecules including ethylene. As pointed out by Berkowitz (52), there are several sources of DOS data, in addition to Cooper et al. (8), that were not available at the time of the construction of the original (1) constrained DOSD for ethylene. Those of particular interest in his analysis (52), in addition to Cooper et al. (8), are the measurements of Holland et al. (54) and Kempgens et al. (55) in the energy regions 10.513 eV to 24.8 eV and 284 eV to 340 eV, respectively. Berkowitz presents integrated oscillator strengths for ethylene over different energy intervals than used in Table 1, and so to compare with his results, we have calculated integrated DOSs for our recommended DOSD over appropriate energy regions. Our results are 7.785(-1), 6.189, 4.624, 6.034(-1), 1.287(-1), 8.451(-1), 2.691, 1.364(-1), and 3.448(-3) for the energy regions 6.62-10.513 eV, 10.513-24.8 eV, 24.8-80.0 eV, 80.0-200.0 eV, 200-285 eV, 285-340 eV, 340-1740 eV, 1740-104 eV, and [10.sup.4]-[infinity] eV, respectively. Berkowitz's corresponding results, using his raw experimental DOS input, differ from our results by -6.9%, -0.5%, -5.8%, 14.0%, 39.8%, 19.9%, -2.2%, 2.7%, and -7.2%, respectively. He adjusted the experimental data in the 6.62-10.513 eV and the 10.513-24.8 eV regions to obtain good agreement with the experimental result (see next section) for [S.sub.-2] and the Thomas-Reiche-Kuhn sum rule; the corresponding adjusted integrated DOSs in these regions differ by 0.6% and 1.5%, respectively, with our constrained DOSD results. As we shall see in the next section, the dipole sums obtained by Berkowitz agree very well with those obtained by our constrained DOSD approach even though the integrated DOSs in some energy regions differ appreciably. This arises for two reasons: the relevant energy regions do not contribute significantly to the sums or the discrepancies in the DOSs tend to cancel in the calculation of the dipole sums, since they can vary in sign as E changes. Finally, we point out that our DOSs for ethylene begin with an absorption threshold of 6.02 eV relative to that of 6.62 eV used in ref. 52. The oscillator strength from 6.02 to 6.62 eV is only 1.856(-4) and has no effect on the dipole properties considered by Berkowitz (52).

Dipole properties

Results for the molar refractivity of ethylene evaluated using our recommended DOSD are listed in Table 2 for a variety of wavelengths between 2302 A and 6709 [Angstrom] where they are compared with the experimental results of Watson and Ramaswamy (33), Friberg (34), and Lowery (56); for common wavelengths, the agreement between the results from the various sources is excellent as is the agreement with the molar refractivity results evaluated from the original (1) DOSD for ethylene.

The results for the dipole properties of ethylene, namely, the [S.sub.k], k = -10(2) - 4(1) - 3(1/2) 0, 1, 2; and the logarithmic dipole sums [L.sub.k] and mean excitation energies [I.sub.k], k = -2(1) 2, evaluated using both the constrained and unconstrained ethylene DOSD1, are compared for common properties, with the literature results of Olney et al. (57) in Table 3. Olney et al. (57) computed the dipole properties by employing the photoabsorption spectrum of Cooper et al. (8); the calculations were based on the high-resolution data, augmented by the low-resolution data as needed. In order to obtain absolute values of the DOSs for ethylene, the original experimental data of Cooper et al. (8) were normalized using the valence-shell TKR (VTKR) sum rule (58). Using the normalized data of Cooper et al., Olney et al. obtained [S.sub.-2] = 28.26 and then renormalized the distribution of Cooper et al. by satisfying the experimental result [S.sub.-2] = 27.79 (59). It is this renormalized distribution that was used to evaluate the dipole properties of ethylene listed in Olney et al. (57). The dipole properties [S.sub.k] and [L.sub.k], corresponding to the absolute scale of the original normalized DOS data of Cooper et al. (8), can be obtained from those of Olney et al. (57) by multiplying the latter by the factor 1.0169, which is the ratio of the [S.sub.-2] evaluated from the Cooper et al. data to that finally used by Olney et al.; the mean excitation energies are unaffected by this normalization procedure.

For the dipole properties [S.sub.k], the results of Olney et al. (57) are lower than our values by 2.5% for k = -1 and are higher by about 0.3, 1.3, 1.9, 2.7, 3.2, and 3.6% for k = -2(-1) -10. For [L.sub.-1] and [L.sub.-2], they are 13.5% and 2.4% lower than our recommended results. The differences between the two sets of values for common properties are less than the (rather large) uncertainties estimated by Olney et al. Results for [S.sub.k] with k = -5/2, -3/2, -1/2, k [greater than or equal to] 0, and [L.sub.k] and [I.sub.k] with k [greater than or equal to] 0, are not available in Olney et al. (57); those for [I.sub.-1] and [I.sub.-2], which can be calculated from their corresponding results for the analogous [S.sub.k] and [L.sub.k] sums via eq. [3], are 6.0% and 1.4% lower than our recommended values. For k [approximately equal to] 0 and k > 0, the original DOS data of Copper et al. (8), and hence the DOS results of Olney et al. (57), will lead to poor values for the dipole properties unless extended to higher photon energies and constrained to both refractivity data and the full TRK sum rule [S.sub.0] = 16. For example, the [S.sub.0]'s evaluated from the unconstrained DOSD1 and DOSD2 are 15.98 and 15.75, respectively; using the Cooper et al. data (8) (i.e., up to 200 eV only) yields 12.27 and 12.04, respectively. Using the original DOS data in this way requires an extension of this data to very high photon energies as illustrated in the construction of the DOSDi, i = 1, 2.

The values of the dipole properties evaluated using the original (1) recommended DOSD for ethylene are also included in Table 3. The agreement with our new results is excellent; to within about 1%, and often to significantly less than 1%, except for [S.sub.-10] where the older result is about 2% higher than our current recommended result.

This table also includes the dipole properties of the molecule obtained using the unconstrained DOSD1. The unconstrained dipole sums [S.sub.k] are less than the recommended results by 1%, 1.6%, and (only) 0.1% for k = 2, 1, and 0, respectively. For k < 0, they are systematically greater than the recommended values by from 0.8% to 5.3% as k decreases from k = -0.5 to k = -10 in agreement with the greater values of the integrated oscillator strengths for the input DOS data relative to the recommended integrated DOSs for low excitation energies (E [less than or equal to] 20.7 eV); the dipole sum [S.sub.k] for negative k are dominated by the low-energy part of the DOSD, more so as k decreases (41, 60-62). The unconstrained results for the [L.sub.k] are less than the recommended results by 0.9%, 1.7%, 4.0%, 4.7%, and 3.8% for k = 2(-1) -2, respectively, and the unconstrained average energies [I.sub.k] agree with the recommended values to within about 1% except for [I.sub.0] where the discrepancy is 2.3%.

The results for the dipole properties of ethylene developed here and in the original alkene paper (1), and those for propene and 1-butane obtained earlier by Jhanwar et al. (1), are the most reliable compilation of such properties for these alkenes. References to other literature results for some of these properties have been given previously and other relevant literature can be found in these references. A discussion of a selection of more-recent work follows.

Hohm (59) has determined values of the frequency-dependent dipole polarizability of ethylene and propene by measuring accurate values of the refractive index of these molecules over a limited range of wavelengths relative to those in Table 2. The corresponding molar refractivities (see eq. [4]) for ethylene and propene are higher and lower than the results evaluated from our recommended DOSDs for these molecules by only a maximum of 0.4% and 0.25%, respectively. For the wavelengths 6329.9, 6119.71, 5940.96, 5435.16, and 3251.3 [Angstrom], the DOSD results are 10.6465, 10.6680, 10.6881, 10.7531, and 11.6342 [cm.sup.3] [mol.sup.-1] for ethylene, respectively, and 15.6571, 15.6873, 15.7156, 15.8129, and 17.0459 [cm.sup.3] [mol.sup.-1] for propene, respectively.

Hohm has extracted the dipole sums [S.sub.k], k = -2, -4, -6, and -8 by fitting a Cauchy series to his frequency-dependent polarizability data. For ethylene, the sums so obtained are 0.3%, 1.7%, and 1.7% higher, and 11.5% lower; and for propene, they are 0.3% lower, 0.5% and 5.1% higher, and 9.0% lower, than the recommended DOSD results of this paper and Jhanwar et al.(1). The larger discrepancies as k decreases, given the excellent agreement for the refractivity data, are probably due to difficulties in fitting the Cauchy series to polarizability data over a limited wavelength range; see for example the discussion in Hohm (59). The evaluation of the dipole sums using the DOSD approach involves no such fitting procedure.

As discussed in the previous section, Berkowitz (52) has analyzed the experimental DOSs for the ethylene molecule. He has evaluated the dipole sums [S.sub.k], k = -2(1) 2, using both raw and adjusted experimental DOS data and the value of the [S.sub.-2] used in the adjustment is that due to Holm (59) just discussed. The results obtained for the dipole sums [S.sub.k], k = -2 to 2, respectively, differ from our constrained DOSD results of Table 3 by -3.2%, -2.52%, -0.6%, 1.6%, and -1.6% and by only 0.2%, -0.1%, 0.5%, 1.8%, and -1.6% for the raw vs. the adjusted DOS input, respectively.

Time-dependent Hartree-Fock (TDHF) results for the ethylene [S.sub.k], k = -2, -4, and -6, are available from Spackman (63); he employed an 6-31G(+sd +sp) basis set and an ELP (electrical properties) basis set of Liu and Dykstra (64). The ab initio results are 1.7% lower and 4.8% and 16.1% higher for the 6-31G(+sd +sp) basis and 0.5%, 13.3%, and 34.3% higher for the ELP basis set than our recommended results, which are very similar to those of Jhanwar et al. (1) used in the analysis of Spackman (63). Generally, the ELP basis seems to perform better in the calculation of dipole sums and ethylene is one of the exceptions pointed out by Spackman (63).

About a year after Spackman's paper, Maroulis (65) carried out a study of basis set and electron correlation effects in the ab initio calculation of the dipole polarizability (and hyperpolarizability) of ethylene using the self-consistent field and many body-perturbation theory methods. Eight basis sets were discussed/investigated beginning with that due to Dunning (66) and perturbation theory through fourth order, including contributions from single, double, triple, and quadruple substitutions from the zeroth order function were considered. The Hatree-Fock limit of [S.sub.-2] (the dipole polarizability in units of [a.sup.3.sub.0]) was estimated to be 28.36 [+ or -] 0.14, while the best estimate, including correlation effects, was 27.29 [+ or -] 0.12, which is within about 1.5% of the then, and the current, recommended result of 27.70. Reference 65 has a nice discussion of the then available theoretical and experimental results for the dipole polarizability of ethylene.

Dalskov and Sauer (67) have evaluated the ethylene dipole sums [S.sub.k], k = -2, -4, and -6, using the random phase approximation (RPA), the second-order polarization propagator approximation (SOPPA), the SOPPA with coupled cluster singles and doubles amplitudes (SOPPA(CCSD)), and the coupled cluster singles and doubles linear response function method (CCSDLR); two basis sets were investigated, Sadlej's (68, 69) medium-sized polarization basis set (basis set I) and Dunning's (70) correlation consistent valence triple [zeta] basis set augmented by two diffuse polarization functions for all angular momenta in the basis added in an even tempered fashion (71, 72) (basis set II). For the first three ab initio methods, the dipole sums were directly evaluated using the approach discussed by Fowler et al. (73), while a Cauchy fit to the frequency-dependent polarizability was used in the case of the CCSDLR calculation.

The discrepancies of the ab initio results with respect to our recommended values for basis set I (and II) are 2.2%, 2.3%, -0.8%, and -0.6% (2.9%, 1.6%, -0.6%, and -0.6%), 15.7%, 13.5%, 6.7%, and -2.4% (15.5%, 9.2%, 4.9%, and -4.7%) and 35.2%, 26.4%, 16.1%, and -6.6% (36.0%, 19.7%, 13.9%, and -9.7%) for [S.sub.-2], [S.sub.-4], and [S.sub.-6], respectively, in the order RPA through CCSDLR. These results are more or less in agreement with the trend discussed in Dalskov and Sauer (67) that the accuracy of these ab initio approaches increases in the order RPA to CCSDLR. Generally, the dipole sums are more difficult to compute as k decreases. Dalskov and Sauer also computed the frequency-dependent dipolar polarizability of ethylene using basis set II and the CCSDLR method over a limited wavelength range (five values from 6329.9 to 3251.3 [Angstrom]). Their corresponding results vary from being 0.7% to 1.5% lower than those evaluated from our recommended DOSD, as the [lambda] decreases; the agreement is well within the estimated errors (67) in their calculations.

The ethylene dipole sums [S.sub.k], k = -2 and -4, have also been evaluated by Caillie and Amos (74) using the self-consistent field (SCF) approach and two variants of density functional theory (DFT), namely, the local density approximation (LDA) and a gradient corrected functional (BLYP); basis sets (see earlier) developed by Sadlej (68, 69) and Woon and Dunning (71, 72) were investigated. The dipole sum [S.sub.-2] was calculated directly, while [S.sub.-4] was determined by a Cauchy fit of the relevant frequency-dependent polarizability through terms varying as [[lambda].sup.-4]. The ab initio results are all greater than our recommended values; for the Sadlej (Woon and Dunning), basis sets by 2.3%, 5.1%, and 3.6% (2.9%, 5.0%, and 3.8%), and 15.5%, 17.8%, and 17.0% (15.5%, 15.6%, and 15.6%) for [S.sub.-2] and [S.sub.-4], respectively, in the order: SCF, LDA, BLYP. The DFT results show no improvement relative to the SCF results and, as in the case of the ab initio methods employed by Dalskov and Sauer (67), the more sophisticated basis set shows little improvement over the results obtained from the Sadlej basis.

A recent discussion of the mean excitation energies for the stopping power of some 32 atoms and molecules can be found in Kamakura et al. (75). This includes the evaluation of [I.sub.0] for these species using, as a basis, the analysis of the relevant oscillator-strength spectra recently given by Berkowitz (52). A detailed comparison with a variety of literature values of [I.sub.0] for the atoms and molecules considered is included in Kamakura et al. (75). Their result for [I.sub.0] for ethylene is 51.3 eV, which is 2.3% higher than our recommended result; the older recommended DOSD result of Jhanwar et al. (1), referred to in ref. 75, is 0.5% lower than our recommended result.

The discussion (76) of the radiation interaction parameters ([S.sub.k], [L.sub.k], [I.sub.k], k = 1,0, -1, [S.sub.1/2]), and their additivity for the alkenes based on the results of ref. 1, is not significantly affected by the results of the current work.

The ab initio quantum mechanically calculated results (63, 65, 67, 74) for the dipole properties discussed earlier do not include vibrational averaging effects, which are included, for example, in the experimentally based results of Hohm (59) and in the DOSD values for these properties. As emphasized, for example, by Russell and Spackman (77), the sizes of these effects are often reasonably small, and so high-quality electronic property calculations are generally required initially before vibrational effects are considered within the ab initio treatment used in the calculation. Ab initio calculations of such properties are complicated by several factors: (i) the selection of the nuclear geometry of the molecule (equilibrium or vibrationally averaged geometries, experimental or calculated geometries, calculated geometries that have been calculated previously or those that are consistent with the theoretical approach used in the property evaluation), (ii) the choice of the ab inito method, (iii) the choice of the basis set used to carry out the calculation, and (iv) whether or not vibrational averging effects are included in the calculation. Each of these factors can introduce error into the final ab initio result for the property.

The molecular property most studied in the context of the last paragraph is the polarizability of molecules. For example, two studies (78, 79) indicate that the effects of vibrational averaging on the isotropic dipole polarizabilities of the first-row hydrides C[H.sub.4], N[H.sub.3], [H.sub.2]O, and HF varies from being 4%-5.5% to 1.5% higher than the unaveraged values; a study (77) of the second-row hydrides shows similar results. As might be expected, molecules with a number of hydrogen atoms bonded to a common atom show the largest vibrational averaging effect as does [H.sub.2] itself. Results for [S.sub.-2], as a function of the bond length of [H.sub.2], are available in the literature (80, 81), and for example, the result at the vibrationally averaged bond length [r.sub.0] = 1.449 [a.sub.0] is 4% higher than at the equilibrium bond length [r.sub.e] = 1.40 [a.sub.0]. Limited results for some of the other dipole properties of [H.sub.2], as a function of bond length, are also available in ref. 82.

Vibrational averaging effects, when available for the molecule of interest, have been used from time to time to help reconcile ab initio and constrained DOSD results for molecular properties (see for example, refs. 37, 42, 48). Also, it is relevant to point out that the effects of molecular geometry, including vibrational averaging, can be relatively more significant in the evaluation of the (intrinsically smaller) anisotropy of molecular properties than it is for the isotropic properties of particular interest in this paper (37, 77-82).

Dispersion-energy coefficients

The dipole dispersion-energy coefficients are generally most conveniently evaluated by using discrete representations (pseudo-DOSDs) of the recommended constrained DOSDs for the interacting species (10, 11). This representation of the original DOSD of the molecule is particularly useful for the efficient evaluation of the triple-double dispersion energy, which is given (eq. [7]) in terms of a triple integral involving the original continuous DOSDs of the interacting species (11, 12). The pseudo-DOSD for a given molecule is determined from known values of the dipole sums [S.sub.k] by requiring

[S.sub.k] = [n.summation over (i=1)] [([E.sub.i] - [E.sub.H]).sup.k] [f.sub.i] ,k = 2,1,0, -1, -2, ..., 3 - 2n [10]

The values of 2n dipole sums generate n pseudo-dipole (excitation energy oscillator strength) pairs ([E.sub.i], [f.sub.i]). The ten pseudo-state representation of the constrained DOSD 1 for ethylene is given in Table 4. It is more than adequate for evaluating the dipole-dipole and the triple-dipole dispersion-energy coefficients, for interactions involving the ethylene molecule, to well within the accuracy ([less than or equal to]1% for dipole-dipole, [less than or equal to]1%-2% for triple-dipole) generally expected (10-12) from using the original constrained DOSD. We have also computed the pseudo-DOSDs for propene and 1-butene using the [S.sub.k] determined from the recommended DOSDs for these molecules constructed by Jhanwar et al. (1); they can be found in Tables 5 and 6.

The discretized version of eq. [7], useable for the evaluation of the dipole-dipole dispersion-energy coefficient [C.sub.6] by employing the pseudo-DOSDs of the interacting species, is given by (10, 14)

[C.sub.6](A, B)

= [E.sub.H] [a.sup.6.sub.0](3/2) [[N.sub.A].summation over (i=1)] [[N.sub.B].summation over (j=1)] [f.sub.i](A)[f.sub.j](B)[([E.sub.H]).sup.3]/ [E.sub.i](A)[E.sub.j](B)[[E.sub.i](A) + [E.sub.j](B)] [11]

The results for the [C.sub.6] coefficients obtained from this result for the interaction of ethylene, propene, and 1-butane with themselves and with some 47 other atoms and molecules, can be found (units of [E.sub.H][a.sup.6.sub.0]) in Tables 7, 8, and 9, respectively. In these calculations, the alkene molecules are represented by the pseudo-states listed in Tables 4, 5, and 6, and the pseudo-states for the other species can be found in the literature (6, 7, 9, 10, 12, 42, 46-49, 51, and refs. cited therein). The uncertainties are [approximately equal to]1%.

The pseudo-DOSD expression for the triple-dipole interaction energy coefficient [C.sub.9] is (11, 14)

[C.sub.9](A, B, C) = [E.sub.H][a.sup.9.sub.0](3/2) [[N.sub.A].summation over (i=1)] [[N.sub.B].summation over (j=1)] [[N.sub.C].summation over (k=1)] [f.sub.i](A)[f.sub.j](B)[f.sub.k](C)/ [E.sub.i](A)[E.sub.j](B)[E.sub.k](C)

x [[E.sub.i](A)[E.sub.j](B) + [E.sub.k](C)] [([E.sub.H]).sup.5]/ [[E.sub.i](A)[E.sub.j](B)][[E.sub.j](B) + [E.sub.k](C)] [[E.sub.k](C) + [E.sub.i](A)] [12]

Using the pseudo-states given in Tables 4-6, the tripledouble dispersion-energy coefficients for the interaction of any three alkenes taken from ethylene, propene, and 1-butene can be evaluated with an estimated error of less than 2%. The results are given, in units of [E.sub.H][a.sup.9.sub.0], in Table 10. The [C.sub.9] coefficients for all the interactions involving the alkenes and any of the other species occurring in the earlier tables can be evaluated by using the pseudo-states for the species referred to previously.

Olney et al. (57) have evaluated two dipole-dipole dispersion-energy coefficients using their DOSs for He, Ne, and [C.sub.2][H.sub.4] and obtained [C.sub.6](He-[C.sub.2][H.sub.4]) = 19.92 and [C.sub.6](Ne-[C.sub.2][H.sub.4]) = 39.96, which are 0.7% and 1.7% lower than our results.

Spackman (83) has carried out time-dependent Hartree-Fock (TDHF) calculations for [C.sub.6] and [C.sub.9], for the dimers and trimers, respectively, of ethylene, propene, and 1-butene, using a 6-31G(+sd +sp) basis set. His results of 285.4, 596.9, 1037.92, 5555.2, 16418.7, and 37286.5, respectively, are 4.9%, 9.8%, 8.2% and 6.6%, 15.2% and 13.0% lower than our recommended results. As thoroughly discussed by Spackman (63, 83, 84), who has used many of our previous constrained DOSD results for [C.sub.6] and [C.sub.9] for a variety of molecules to assess the situation, this level of TDHF calculation cannot be expected to give accurate results for these dispersion-energy coefficients.

Spackman has discussed the systematics of the deviations of 6-31G(+sd +sp) TDHF results of the dispersion-energy coefficients from the constrained DOSD values and suggested how to correct the TDHF results to obtain values that are generally much closer to the DOSD results if the static dipole polarizabilities of the interacting species are known both at the 6-31G(+sd +sp) TDHF level of approximation and reliably (for example, experimentally or through a constrained DOSD analysis). His equations for the predicted values, written for the interaction of like molecules, have the form

[C.sub.6,pre] = c[[[[alpha].sub.d,exp]/[[alpha].sub.d,TDHF].sup.a] [C.sub.6,TDHF] [13]

and

[C.sub.9,pre] = d[[[[alpha].sub.d,exp]/[[alpha].sub.d,TDHF].sup.b] [C.sub.9,TDHF] [14]

where in Spackman's work, the parameters a, b, c, and d are a = 2, b = 3, c = 0.9435, and d = 0.9283. Using Spackman's results (63, 83) for [[alpha].sub.d,TDHF], [C.sub.6,TDHF], and [C.sub.9,TDHF], and our recommended values of [S.sub.-2] as [[alpha].sub.d,exp], one obtains [C.sub.6,pre] = 278.4, 639.5, and 1100.0 and [C.sub.9,pre] = 5421.4, 18440.5, and 41206.8 for the like interactions involving ethylene, propene, and 1-butene; these results differ from our recommended results by -7.3%, -3.4%, and -2.7% (avg. = -4.4%), and -8.8%, -4.7%, and -3.8% (avg. = -5.8%), respectively. The improvement relative to the TDHF values is very good except for ethylene where the correction factors multiplying the TDHF results are less than unity.

More recently, Cybulski and Haley (85) have discussed several new approximations for calculating the dispersion-energy coefficients [C.sub.6] and [C.sub.9], which are based on eqs. [13] and [14] for the interaction of like species. They developed several schemes using our literature DOSD results to determine the parameters a, b, c, and d and tested their approach against our accurate DOSD results for some fourteen interactions involving rare gases, HF, [N.sub.2], CO, [H.sub.2]O, and normal alkanes. Their approximations have been set up using higher level basis set values for the input TDHF quantities than Spackman's scheme, which is designed for the use of 6-31G (+sd +sp) TDHF results for the input polarizabilities and dispersion-energy coefficients; the TDHF input from refs. 63, 83, and 85, for common molecules are significantly different. Thus, as discussed previously (9), the approximations of Cybulski and Haley, which yield an improvement on the Spackman scheme for the fourteen test interactions, do not necessarily improve the situation for other interactions (partly depending on the nature of the TDHF input).

The alkene ab initio results (83) for the dipole-dipole dispersion-energy coefficients do not contain vibrational averaging effects. Also, all the problems discussed at the end of the last section regarding the ab initio calculation of molecular properties apply to the calculation of the [C.sub.6] coefficients as well; various approximations (15, 16) for [C.sub.6] indicate that this coefficient qualitatively behaves as the product of the dipole polarizabilities of the two interacting molecules. Results for [C.sub.6] for the [H.sub.2]-[H.sub.2] interaction, and for the interaction of [H.sub.2] with all the rare gases ([R.sub.g]), as a function of the [H.sub.2] bond length are available in the literature (80-82). For example, the increase in [C.sub.6] due to increasing the [H.sub.2] bond length from [r.sub.e] = 1.40 [a.sub.0] to [r.sub.0] = 1.449 [a.sub.0] is about 6% for the [H.sub.2]-[H.sub.2] interaction and about 2.8% to 3.2% for the [H.sub.2]-[R.sub.g] interactions. In general, the bond length dependence of the dispersion-energy coefficients, the higher-order coefficients as well as [C.sub.6], is important as input for the construction of multi-dimensional potential-energy surfaces that reflect the relevant vibrational motions in a molecular complex, see for example ref. 86 and citations therein.

Discussion and conclusions

Estimates for the uncertainties in our recommended values for molecular dipole properties, evaluated using the constrained dipole oscillator distribution approach, are based on methods developed and used previously (2, 6, 7, 21, 41). The recommended results are compared with those obtained from alternative DOSDs satisfying the same constraints (i.e., DOSD1 and DOSD2 for ethylene in this paper and the various DOSDs considered for all three alkenes in ref. 1). For the properties that depend significantly (21, 41, 50, 60, 61) on energy regions of the DOSD that dominate the constraints used in constructing the DOSDs, that is the [S.sub.k], [L.sub.k], and [I.sub.k] for -6 [less than or equal to] k [less than or equal to] 0 or 1, the estimated errors are [less than or equal to]1%-2%. The uncertainties will increase, relatively slowly, as k decreases for k < -6 because local errors in the DOSD for small photon energies are magnified for these properties (21, 50, 60, 61); they can be as high as 10% to 15% for [S.sub.-10]. These error estimates assume reliable values (errors of a few tenths of a percent) for the molar refractivities used as constraints. For k [greater than or equal to] 1, and particularly for k = 2, the estimated errors in the dipole properties are related to errors in the high-energy input data used in the construction of the DOSD and generally can be on the order of several percent. These error estimates apply for the three alkene molecules considered in this work and in ref. 1. Because of the analysis of Berkowitz (52), which includes consideration of relatively recent high-energy DOS data for ethylene, and the agreement between his and our results for [S.sub.2] and [S.sub.1], we estimate the errors in our recommended results for the [S.sub.k], [L.sub.k], and [I.sub.k], k = 1, 2, to be 1%-2% for this molecule.

As indicated earlier, part of motivation for this work was to verify the reliability of the results for the dipole properties of the alkenes obtained in the original (1) constrained DOSD analysis for these molecules. That work was carried out in the early 1980s and was based on the then available DOS data for the molecules, which, as discussed earlier in the present paper, were sparse, from various experimental groups, with no direct experimental DOS data being available for photon energies greater than 50 eV for ethylene, 27 eV for propene, and 24 eV for 1-butene (in ref. 1, the higher-energy DOSs were obtained via mixture rules with ethylene playing a relevant role for propene and 1-butene). The constrained DOSD analysis in the current paper for ethylene, which is based on extensive experimental DOS data from a single experimental group (8), indicates that the results for the properties of these molecules obtained originally are indeed reliable and that the present recommended values for ethylene offer only a marginal improvement. Similarly, the values of the dipole properties of ethylene for k [less than or equal to] 1 obtained by Olney et al. (57), which were obtained in 1997 by using the same experimental molecular DOS database as used in the present paper (with no extension beyond 200 eV using mixture rules) and an energy-independent normalization approach, offer little improvement relative to the original (1) evaluation of these properties. This indicates the power of the constraint procedure used in our work, namely, the reproduction of experimental molar refractivity data and the satisfaction of the full Thomas-Reiche-Kuhn sum rule ([S.sub.0] = N). The use of both of these constraints normalizes the complete DOSD, from small to very large E, in a smooth, continuous, and energy-dependent manner. Energy-independent normalization (8, 57, 87) and (or) no extension of the DOS database to very high photon energies can lead to difficulties in determining reliable values of atomic and molecular dipole properties even from extensive experimental DOS data sets; this has been discussed in the literature (see for example, refs. 9, 47, 49, and some of the citations therein) and is also illustrated by the discussion of the dipole properties of ethylene given earlier here.

The results for the dipole-dipole and the triple-dipole dispersion energies for the interactions considered in this paper are new and are apparently (essentially) the only reliable values available for these alkenes. They represent prototype interactions involving prototypical molecules containing the alkene group and therefore should be of considerable interest. The estimated errors for the [C.sub.6] and [C.sub.9] results given in this paper are 1% and 2%, respectively.

Accurate DOSD values of [C.sub.6] for important prototype interactions furnish useful checks of other methods for evaluating dispersion-energy coefficients, for example, empirical, non-empirical, ab initio quantum mechanical, and Pade approximant approaches (10, 11, 15, 28, 29, 51, 83, 88-93). In this context, it is relevant to point out that in the construction of potential-energy models (24-30) (mentioned briefly previously), it is necessary to include the dispersion energy to at least terms that vary as [R.sup.-10] at long range. The terms varying as powers of [R.sup.-1] higher than [R.sup.-6] are not accessible by DOSD techniques due to the lack of (experimental) higher multipole (than dipole) oscillator strengths. The method of choice for determining these higher-order dispersion energies is via ab initio quantum mechanical methods analogous to those used for the dipole-dipole case. The reliability of these methods for these energies can be checked indirectly via the checks on the dipole-dipole dispersion energy referred to earlier, since the ab initio calculation of the higher-order terms is not easier than for the dipole-dipole dispersion energy. In any event, as pointed out earlier, the dipole-dipole dispersion energy is the lead or dominant dispersion energy, and the success of the potential-energy models depends (26, 61) on a reliable value of the coefficient [C.sub.6].

The pseudo-DOSDs employed in this work to calculate the dipole-dipole and the triple-double dispersion-energy coefficients form a concise representation of the original recommended DOSDs for the alkene molecules, which are continuous functions of the excitation energy from the UV-absorption threshold to many thousands of eV. They give reliable results for the dispersion-energy coefficients and a variety of other molecular properties through the use of discretized analogues of the usual expressions for these properties (10-12). We emphasize that more than ten pseudo-states are needed to reliably evaluate some of these properties, (3) for example, the logarithmic dipole sums [L.sub.k] for "larger" values of k, which have significant negative and positive contributions (41, 49).

Acknowledgements

The authors wish to thank C.E. Brion, G. Cooper, and T.N. Olney for making their dipole oscillator strengths for ethylene available to us. This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC).

Received 24 April 2007. Accepted 17 May 2007. Published on the NRC Research Press Web site at canjchem.nrc.ca on 27 July 2007.

This report is dedicated to Professor G. Michael Bancroft, a colleague and friend for a long time, on the occasion of his 65th birthday.

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A. Kumar. Department of Physics, Ch. Charan Singh University, Meerut 250004, India.

B.L. Jhanwar. Department of Computer Application, Mody Institute of Technology and Science, Lakshmangarth, Distt. Sikar, Rajasthan 332311, India.

W.J. Meath. (1,2) Department of Chemistry, The University of Western Ontario, London, ON N6A 5B7,Canada.

This article is part of a Special Issue dedicated to Professor G. Michael Bancroft.

(1) Corresponding author (e-mail: [e-mail]).

(2) Associated with the Center for Interdisciplinary Studies in Chemical Physics, The University of Western Ontario.

Table 1. Integrated dipole oscillator strengths for [C.sub.2][H.sub.4].

Energy region        Recommended            Unconstrained
(eV)                 DOSD1                  DOSD1

6.020-6.8            8.240(-4)              8.242(-4)
6.8-8.0              2.827(-1)              3.009(-1)
8.0-8.6              1.034(-1)              1.049(-1)
8.6-9.0              5.399(-2)              5.430(-2)
9.0-10.4             3.147(-1)              3.228(-1)
10.4-10.7            5.507(-2)              5.525(-2)
10.7-11.8            2.336(-1)              2.361(-1)
11.8-12.3            2.428(-1)              2.450(-1)
12.3-18.0            2.768                  2.898
18.0-19.1            5.570(-1)              5.579(-1)
19.1-20.7            7.541(-1)              7.541(-1)
20.7-26.7            2.18                   2.157
26.7-35.0            1.685                  1.656
35.0-50.0            1.442                  1.414
50.0-70.0            7.285(-1)              7.202(-1)
70.0-100.0           4.197(-1)              4.168(-1)
100.0-200.0          3.743(-1)              3.718(-1)
200.0-300.0          3.849(-1)              3.823(-1)
300.0-500.0          1.926                  1.864
500.0-1000.0         1.078                  1.058
1000.0-[infinity]    4.156(-1)              4.125(-1)

                     High-resolution        Low-resolution
Energy region        data of Cooper         data of Cooper
(eV)                 et al. (8)             et al. (8)

6.020-6.8            8.242(-4)              2.735(-2)
6.8-8.0              3.009(-1)              2.438(-1)
8.0-8.6              1.049(-1)              1.101(-1)
8.6-9.0              5.430(-2)              6.830(-2)
9.0-10.4             3.228(-1)              3.021(-1)
10.4-10.7            5.525(-2)              6.173(-2)
10.7-11.8            2.361(-1)              2.765(-1)
11.8-12.3            2.450(-1)              2.111(-1)
12.3-18.0            2.898                  2.889
18.0-19.1            5.579(-1)              5.685(-1)
19.1-20.7            7.541(-1)              7.577(-1)
20.7-26.7            2.157                  2.137
26.7-35.0            1.656                  1.572
35.0-50.0            1.414                  1.302
50.0-70.0            --                     7.202(-1)
70.0-100.0           --                     4.168(-1)
100.0-200.0          --                     3.718(-1)
200.0-300.0          --                     --
300.0-500.0          --                     --
500.0-1000.0         --                     --
1000.0-[infinity]    --                     --

Energy region
(eV)                 Jhanwar et al.(1)

6.020-6.8            8.151(-4)
6.8-8.0              2.891(-1)
8.0-8.6              9.029(-2)
8.6-9.0              3.845(-2)
9.0-10.4             2.863(-1)
10.4-10.7            5.254(-2)
10.7-11.8            2.363(-1)
11.8-12.3            2.339(-1)
12.3-18.0            3.011
18.0-19.1            5.382(-1)
19.1-20.7            7.173(-1)
20.7-26.7            1.909
26.7-35.0            1.772
35.0-50.0            1.441
50.0-70.0            7.274(-1)
70.0-100.0           4.837(-1)
100.0-200.0          4.558(-1)
200.0-300.0          3.823(-1)
300.0-500.0          1.864
500.0-1000.0         1.058
1000.0-[infinity]    4.125(-1)

Note: Numbers in parentheses indicate a power of ten.

Table 2. Recommended values of the molar refractivity ([cm.sup.3]
[mol.sup.-1]) of [C.sub.2][H.sub.4] (ideal gas, STP) evaluated
using the recommended DOSD1 and a comparison with literature values.

[lambda]      Recommended    Watson and
[Angstrom]    DOSD1          Ramaswamy (33)

6709          10.6132        --
6440          10.6362        10.6362 (a)
5792          10.7065        --
5462          10.7531        10.7538
5087          10.8186        10.8202
4917          10.8539        --
4801          10.8804        10.8825
4603          10.9309        --
4359          11.0043        11.0061
4109          11.0958        --
3985          11.1488        --
3342          11.5515        --
2968          11.9624        --
2926          12.0227        --
2894          12.0713        --
2857          12.1304        --
2760          12.3024        --
2753          12.3159        --
2675          12.4776        --
2577          12.7152        --
2464          13.0522        --
2447          13.1101        --
2302          13.7115        --

[lambda]
[Angstrom]    Friberg (34)   Lowery (56)

6709          --             10.6069
6440          --             10.6306
5792          10.7094        --
5462          10.7538        10.7522
5087          --             10.8185
4917          10.8565        --
4801          --             10.8819
4603          --             --
4359          --             11.0079
4109          11.0991        --
3985          11.1523        --
3342          11.5540        --
2968          11.9363        --
2926          12.0270        --
2894          12.0752        --
2857          12.1349        --
2760          12.3079        --
2753          12.3193        --
2675          12.4823        --
2577          12.7220        --
2464          13.0564        --
2447          13.1140        --
2302          13.7115 (a)    --

(a) Used as constraints in constructing the DOSD.

Table 3. The dipole sums [S.sub.k], logarithmic dipole sums [L.sub.k]
and mean excitation energies [I.sub.k] for [C.sub.2][H.sub.4] evaluated
using the recommended and unconstrained DOSD1.

                   Unconstrained    Recommended
Property           DOSD1            DOSD1

[S.sub.2]          6.553(3)         6.619(3)
[S.sub.1]          9.956(1)         1.012(2)
[S.sub.0]          1.598(1)         1.600(1)
[S.sub.-0.5]       1.450(1)         1.438(1)
[S.sub.-1]         1.656(1)         1.630(1)
[S.sub.-1.5]       2.100(1)         2.056(1)
[S.sub.-2]         2.841(1)         2.770(1)
[S.sub.-2.5]       4.048(1)         3.933(1)
[S.sub.-3]         6.029(1)         5.840(1)
[S.sub.-4]         1.496(2)         1.442(2)
[S.sub.-6]         1.261(3)         1.207(3)
[S.sub.-8]         1.341(4)         1.277(4)
[S.sub.-10]        1.588(5)         1.508(5)
[L.sub.2]          3.906(4)         3.940(4)
[L.sub.1]          3.024(2)         3.075(2)
[L.sub.0]          9.394            9.781
[L.sub.-1]        -6.480           -6.189
[L.sub.-2]        -1.873(1)        -1.805(1)
[I.sub.2] (eV)     1.0556(4)        1.0471(4)
[I.sub.1] (eV)     5.672(2)         5.674(2)
[I.sub.0] (eV)     4.898(1)         5.015(1)
[I.sub.-1] (eV)    1.840(1)         1.862(1)
[I.sub.-2] (eV)    1.407(1)         1.419(1)

Property           Jhanwar et       Olney et al. (57)
                   al. (1)

[S.sub.2]          6.556(3)         --
[S.sub.1]          1.001(2)         --
[S.sub.0]          1.600(1)         --
[S.sub.-0.5]       1.438(1)         --
[S.sub.-1]         1.631(1)         1.590(1)
[S.sub.-1.5]       2.057(1)         --
[S.sub.-2]         2.770(1)         2.779(1)
[S.sub.-2.5]       3.930(1)         --
[S.sub.-3]         5.828(1)         5.918(1)
[S.sub.-4]         1.435(2)         1.470(2)
[S.sub.-6]         1.202(3)         1.240(3)
[S.sub.-8]         1.286(4)         1.318(4)
[S.sub.-10]        1.541(5)         1.562(5)
[L.sub.2]          3.906(4)         --
[L.sub.1]          3.034(2)         --
[L.sub.0]          9.712            --
[L.sub.-1]        -6.189           -7.011
[L.sub.-2]        -1.802(1)        -1.848(1)
[I.sub.2] (eV)     1.0535(4)        --
[I.sub.1] (eV)     5.628(2)         --
[I.sub.0] (eV)     4.993(1)         --
[I.sub.-1] (eV)    1.862(1)         --
[I.sub.-2] (eV)    1.420(1)         --

Note: Numbers in parentheses indicate a power of ten.

Table 4. The values of the pseudo-DOSD excitation
energies (in units of [E.sub.H]) and oscillator strengths for
ground state [C.sub.2][H.sub.4].

[E.sub.i]          [f.sub.i]

2.49559(-1)        7.70604(-3)
2.72445(-1)        2.28700(-1)
3.13376(-1)        2.99037(-1)
4.02588(-1)        8.03816(-1)
5.44545(-1)        2.47268
7.92270(-1)        3.92625
1.33778            3.12855
3.21851            1.43078
1.77416(1)         3.60023
2.31130(2)         1.02252(-1)

Note: Numbers in parentheses indicate a power of ten.

Table 5. The values of the pseudo-DOSD excitation
energies (in units of [E.sub.H]) and oscillator strengths for
ground state [C.sub.3][H.sub.6].

[E.sub.i]               [f.sub.i]

2.45374(-1)             4.44189(-2)
2.65480(-1)             2.19447(-1)
3.17393(-1)             3.49069(-1)
4.02328(-1)             1.35368
5.45075(-1)             3.94058
7.88288(-1)             5.57534
1.33880                 4.71831
3.24561                 2.34981
1.77062(1)              5.29692
2.31219(2)              1.52409(-1)

Note: Numbers in parentheses indicate a power of ten.

Table 6. The values of the pseudo-DOSD excitation
energies (in units of [E.sub.H]) and oscillator strengths for
ground state [C.sub.4][H.sub.8].

[E.sub.i]         [f.sub.i]

2.44401(-1)       5.47815(-2)
2.63912(-1)       2.41008(-1)
3.16143(-1)       4.28545(-1)
4.02901(-1)       1.77394
5.43816(-1)       5.14232
7.90723(-1)       7.48640
1.33569           6.41832
3.22456           3.17907
1.76814(1)        7.07170
2.30866(2)        2.03906(-1)

Note: Numbers in parentheses indicate a power of ten.

Table 7. Recommended values for the dipole- dipole dispersion-energy
coefficients [C.sub.6]([C.sub.2][H.sub.4], B) for the interaction of
[C.sub.2][H.sub.4] with various species B (in units of [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII]).

B                              [C.sub.6]([C.sub.2][H.sub.4], B)

[C.sub.2][H.sub.4]                     300.2
H                                       43.80
Li                                     418.1
He                                      20.07
Ne                                      40.66
Ar                                     138.0
Kr                                     196.8
Xe                                     292.8
[H.sub.2]                               60.23
[N.sub.2]                              147.0
[O.sub.2]                              133.9
[Cl.sub.2]                             341.7
HF                                      73.73
HCl                                    197.7
HBr                                    254.9
CO                                     155.6
C[O.sub.2]                             216.4
NO                                     143.1
[N.sub.2]O                             234.4
[H.sub.2]O                             115.9
S[O.sub.2]                             296.2
CS.sub.2]                              507.0
COS                                    347.2
[H.sub.2]S                             254.6
S[F.sub.6]                             403.9
Si[H.sub.4]                            319.8
Si[F.sub.4]                            306.1
C[Cl.sub.4]                            779.2
[C.sub.2][H.sub.2]                     247.5
[C.sub.6][H.sub.6]                     719.1
N[H.sub.3]                             163.5
C[H.sub.3]N[H.sub.2]                   301.7
[(C[H.sub.3]).sub.2]NH                 440.6
[(C[H.sub.3]).sub.3]N                  564.5
C[H.sub.2]O                            222.3
C[H.sub.3]CHO                          346.8
[(C[H.sub.3]).sub.2]CO                 487.6
C[H.sub.3]OH                           257.5
[C.sub.2][H.sub.5]OH                   400.2
[C.sub.3][H.sub.7]OH                   540.0
C[H.sub.4]                             197.2
[C.sub.2][H.sub.6]                     338.3
[C.sub.3][H.sub.8]                     479.9
[C.sub.4][H.sub.10]                    616.6
[C.sub.5][H.sub.12]                    755.6
[C.sub.6][H.sub.14]                    891.2
[C.sub.7][H.sub.16]                   1028
[C.sub.8][H.sub.18]                   1164
--                                       --
--                                       --
--                                       --

Table 8. Recommended values for the dipole-dipole dispersion-energy
coefficients [C.sub.6]([C.sub.3][H.sub.6], B) for the interaction of
[C.sub.3][H.sub.6] with various species B (in units of [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII]).

B

[C.sub.3][H.sub.6]              662.1
H                                65.00
Li                              617.3
He                               29.84
Ne                               60.48
Ar                              205.0
Kr                              292.3
Xe                              434.9
[H.sub.2]                        89.45
[N.sub.2]                       218.5
[O.sub.2]                       198.9
[Cl.sub.2]                      507.5
HF                              109.6
HCl                             293.7
HBr                             378.5
CO                              231.1
C[O.sub.2]                      321.5
NO                              212.7
[N.sub.2]O                      348.2
[H.sub.2]O                      172.2
S[O.sub.2]                      440.1
C[S.sub.2]                      752.3
COS                             515.4
[H.sub.2]S                      378.0
S[F.sub.6]                      600.5
Si[H.sub.4]                     474.7
Si[F.sub.4]                     455.1
C[Cl.sub.4]                    1157
[C.sub.2][H.sub.2]              367.6
[C.sub.6][H.sub.6]             1068
N[H.sub.3]                      242.9
C[H.sub.3]N[H.sub.2]            448.2
[([CH.sub.3]).sub.2]NH          654.5
[([CH.sub.3]).sub.3]N           838.5
C[H.sub.2]O                     330.3
C[H.sub.3]CHO                   515.2
[(C[H.sub.3]).sub.2]CO          724.3
C[H.sub.3]OH                    382.5
[C.sub.2][H.sub.5]OH            594.5
[C.sub.3][H.sub.7]OH            802.2
C[H.sub.4]                      292.9
[C.sub.2][H.sub.6]              502.6
[C.sub.3][H.sub.8]              712.8
[C.sub.4][H.sub.10]             915.9
[C.sub.5][H.sub.12]            1122
[C.sub.6][H.sub.14]            1324
[C.sub.7][H.sub.16]            1527
[C.sub.8][H.sub.18]            1729
[C.sub.2][H.sub.4]              445.8
--                           --
--                           --

Table 9. Recommended values for the dipole-dipole dispersion-energy
coefficients [C.sub.6]([C.sub.4][H.sub.8], B) for the interaction of
[C.sub.4][H.sub.8] with various species B (in units of [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII]).

B                             [C.sub.6]([C.sub.4][H.sub.8], B)

[C.sub.4][H.sub.8]             1130
H                                84.85
Li                               802.0
He                               39.07
Ne                               79.22
Ar                               268.1
Kr                               382.1
Xe                               568.1
[H.sub.2]                        116.9
[N.sub.2]                        285.7
[O.sub.2]                        260.2
[Cl.sub.2]                       663.2
HF                               143.4
HCl                              383.8
HBr                              494.5
CO                               302.2
C[O.sub.2]                       420.4
NO                               278.2
[N.sub.2]O                       455.2
[H.sub.2]O                       225.1
S[O.sub.2]                       575.3
C[S.sub.2]                       982.0
COS                              673.2
[H.sub.2]S                       493.6
S[F.sub.6]                       785.8
Si[H.sub.4]                      619.8
Si[F.sub.4]                      595.5
C[Cl.sub.4]                     1512
[C.sub.2][H.sub.2]               480.2
[C.sub.6][H.sub.6]              1395
N[H.sub.3]                       317.4
C[H.sub.3]N[H.sub.2]             585.7
[(C[H.sub.3]).sub.2]NH           855.3
[(C[H.sub.3]).sub.3]N           1096
C[H.sub.2]O                      431.6
C[H.sub.3]CHO                    673.3
[(C[H.sub.3]).sub.2]CO           946.6
C[H.sub.3]OH                     500.0
[C.sub.2][H.sub.5]OH             776.9
[C.sub.3][H.sub.7]OH            1048
C[H.sub.4]                       382.7
[C.sub.2][H.sub.6]               656.7
[C.sub.3][H.sub.8]               931.5
[C.sub.4][H.sub.10]             1197
[C.sub.5][H.sub.12]             1467
[C.sub.6][H.sub.14]             1730
[C.sub.7][H.sub.16]             1995
[C.sub.8][H.sub.18]             2260
[C.sub.2][H.sub.4]               582.5
[C.sub.3][H.sub.6]               865.0
--                                 --

Table 10. Recommended values for the triple-dipole
dispersion-energy coefficients [C.sub.9] for all the
three-body interactions involving the A = [C.sub.2][H.sub.4],
B = [C.sub.3][H.sub.6], and C = [C.sub.4][H.sub.8] molecules,
in units of ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).

Interaction     [C.sub.9]

A-A-A               5944.8
A-A-B               8810.1
A-A-C              11481
A-B-C              17016
B-B-B              19351
B-B-A              13057
B-B-C              25220
C-C-C              42837
C-C-A              22176
C-C-B              32868