Dipole polarizability, sum rules, mean excitation energies, and long-range dispersion coefficients for buckminsterfullerene C60

Chemical Physics Letters 516 (2011) 208–211

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Dipole polarizability, sum rules, mean excitation energies, and long-range dispersion coefficients for buckminsterfullerene C60 Ashok Kumar 1, Ajit J. Thakkar ⇑ Department of Chemistry, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3

a r t i c l e

i n f o

Article history: Received 26 August 2011 In final form 30 September 2011 Available online 6 October 2011

a b s t r a c t Experimental photoabsorption cross-sections combined with constraints provided by the Kuhn–Reiche– Thomas sum rule and the high-energy behavior of the dipole-oscillator-strength density are used to construct dipole oscillator strength distributions for buckminsterfullerene (C60). The distributions are used to predict dipole sum rules Sk, mean excitation energies Ik, the frequency dependent polarizability, and C6 coefficients for the long-range dipole–dipole interactions of C60 with a variety of atoms and molecules. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction The gas-phase photoabsorption and photoionization crosssections for buckminsterfullerene (C60) have attracted a great deal of interest; see Refs. [1–4] and earlier references cited therein. Theoretical calculations of high accuracy are difficult because of the large number of electrons in C60. Experimental measurements often yield relative cross-sections which must then be normalized. Moreover, different experiments lead to cross-sections in different energy ranges. Patching together relative cross-sections from a variety of sources and energy ranges is a difficult and uncertain but rewarding process. Since the photoabsorption cross-section is proportional [1] to the differential dipole oscillator strength (DOS), (df/dE), such a synthesis leads to a complete mapping of (df/dE) for the full range of photon energies, Ec 6 E < 1, beyond the excitation threshold Ec. Knowledge of the DOS distribution (DOSD) is desirable because it allows one to calculate many useful properties including the dynamic polarizability a(x) as a function of frequency (x), the stopping power and straggling coefficient for collisions with fast charged particles, and van der Waals coefficients for long-range interactions with other molecules. Berkowitz [1,5] has reported a synthesized DOSD for C60 using dipole sum rules Sk to guide the process. However, a full range of properties was not computed from that DOSD. Moreover, additional experimental cross-section measurements [2,3] have appeared since then. Several experimental determinations [6–9] and theoretical calculations [10–13] of a(x) for C60 have been made. However, the uncertainties in the experimental values are quite large (5–10%).

⇑ Corresponding author. Fax: +1 506 453 4981. E-mail address: [email protected] (A.J. Thakkar). 1 Permanent address: Department of Physics, Ch. Charan Singh University, Meerut 250004, India 0009-2614/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2011.09.080

This Letter reports the construction of dipole oscillator strength distributions for C60 from experimental photoabsorption cross-sections combined with constraints provided by the Kuhn–Reiche–Thomas (KRT) sum rule and the high-energy behavior of the DOS density. The method used [14,15] is summarized in Section 2. The distributions we obtain, the resulting dipole sums Sk, mean excitation energies Ik, dynamic polarizability, and van der Waals C6 coefficients are presented and discussed in Section 3. Atomic units are used throughout this work except where explicitly stated otherwise. 2. Method Our method has been described in detail elsewhere [14,15]. Here, we provide only a terse summary. The moments of the DOSD are defined by

Sk ¼

Z

1

Ec

dE



 df k E dE

ð1Þ

for k < 5/2. The moments with integer 2 P k P 1 are ground state expectation values [16]. The Kuhn–Reiche–Thomas (KRT) sum rule states that S0 is the number of electrons in the molecule. The S2 moment is proportional to the sum of the electron density values at the nuclei [17] and S2 is the static dipole polarizability (a). The DOS data is divided into energy intervals and a constrained least-squares method [18] is used to obtain scale factors for each interval. The constraints used are typically the KRT sum rule and molar refractivity values at one or two wavelengths. In this work, refractivities are not used because sufficiently accurate refractivities are not available for C60. We impose the known high-energy asymptotic behavior [19] of the DOS density given by

df =dE ¼ AE7=2 þ bE4 þ cE9=2 þ . . .

ð2Þ

pffiffiffi in which, for atoms and homonuclear molecules, A ¼ 2 2ZS2 =p and 2 b = 2Z S2 where Z is the atomic number common to all the atoms

A. Kumar, A.J. Thakkar / Chemical Physics Letters 516 (2011) 208–211

in the system. Extrapolation of the DOS data to high energies is carried out using a Padé approximation given by

" # pffiffiffi df 1 A þ b= E pffiffiffi : ¼ 7=2 dE E 1 þ ðb  bÞ=ðA EÞ

ð3Þ

An iterative procedure [14,15] is used to obtain self-consistent values of S2 and the parameters in Eq. (3). This method is applied to a representative selection of the many distributions that can be constructed using different combinations of the available photoabsorption data. The distribution that leads to the smallest standard deviation of the scale factors is chosen as the best one. 3. Dipole properties for C60 We examined 45 initial distributions with emphasis placed on using recently recommended photoabsorption data [1–3,20–24] based upon the equilibrium vapor pressures [25] of C60 that are considered [1,3,5,22,26] to be the most reliable. For energies above 118 eV we used, when necessary, an atomic-additivity model (AM) in which the value of the photoabsorption cross-section rabs for C60 is 60 times the pertinent rabs for atomic carbon [27,28]. The sources of the initial data used to construct our recommended DOSD are summarized in Table 1. The constrained least-squares procedure led to scale factors for this DOSD that were within the interval (1,1.0011). The striking closeness to unity of these scale factors is testimony to the consistency of the input data selected from different sources and to the care with which the normalization of relative cross-sections was carried out by Berkowitz [1,5] and Kafle et al. [3]. We begin with a comparison of our Sk listed in Table 2 with the predictions of the atomic-additivity model. The polarizability of a molecule is normally smaller than the sum of the polarizabilities of its constituent atoms; see, for example, the parameters in an additive model [29,30] for the polarizabilities of more than 100

Table 1 Sources of initial data used to construct the recommended DOSD for C60. Energy range

Source of initial data

3.1–5.1 eV 5.1–11 eV 10–25 eV 25–118 eV 118–280 eV 280–340 eV 340 eV–10 keV 10–100 keV 100 keV and above

Ref. [22] Ref. [23] scaled as in Ref. [3] Ref. [24] scaled as in Ref. [3] Ref. [3] Atomic-additivity model with data from Ref. [27] Composite of Refs. [20,21]from Ref. [1] Atomic-additivity model with data from Ref. [27] Atomic-additivity model with data from Ref. [28] Eq. (3) with b = 2.23730  107

Table 2 Dipole sum rules Sk and Lk in atomic units and mean excitation energies Ik in eV for C60. A(n) is short for A  10n. k

Sk

Lk

Ik

2 1 0 1 2 3 4 5 6 7 8 9 10

1.961(5) 3013 360.0 301.7 558.6 1674 7222 3.829(4) 2.257(5) 1.415(6) 9.271(6) 6.286(7) 4.386(8)

1.154(6) 9430 351.1 102.7 485.0

9806 622.2 72.15 19.36 11.42

209

heteroaromatic molecules. Indeed, all the recent measurements [6,8] and calculations [11–13] of a for C60 are smaller than the additive estimate of SAM 2 ¼ a  60  11:67 ¼ 700:2 obtained from the best available ab initio value of a for atomic carbon [31]. Since S2 < SAM 2 , an argument based on the positive definite nature and normalization of the DOSD [15] leads us to expect

Sk < SAM k

for k < 0

ð4Þ

for k > 0

ð5Þ

and

Sk > SAM k

where the AM superscript indicates a value from the atomic-additivity model. Analytical [32] and numerical [33] Hartree–Fock (HF) calculations of the electron density at the nucleus in the carbon atom both lead to SAM ¼ 1:923  105 , and atomic HF values [34] of 2 AM Sk lead to S1 ¼ 2903 and SAM 1 ¼ 410:8. Comparison with our values shows that Eqs. (4) and (5) are satisfied as expected. The SAM and SAM values are 1.9% and 3.7% less than the corre2 1 sponding Sk, and SAM 1 is 36% larger than S1. The decrease in the difference between SAM and Sk as k increases is due to two factors. As k k increases, the Sk are increasingly determined by the high energy portion of the DOSD. For example, 92.8% of S2 for C60 comes from energies larger than 500 eV whereas 92.3% of S1 comes from energies less than 50 eV. Atomic-additivity of rabs and (df/dE) should work best for high energies. Hence, the SAM should increase in k accuracy as k increases. Moreover, the neglect of electron correlation in the atomic HF values of Sk used to construct SAM is much less k important for the one-electron expectation value S2 than for the two-electron expectation values S1 and S1 which are related to statistical electron correlation coefficients [35]. We estimate the uncertainties in our Sk to be ±3% for 2 P k P 2, ±4% for 3 P k P 6, and ± 5% for 7 P k P 10. The only previous values of Sk with k > 2 are those of Berkowitz [5]. His sum rules [5] differ from ours by only 0.41%, 0.46%, and 0.56% for S2, S1, and S1, respectively. We are unaware of any previous values of Sk for k < 2. We discuss S2 in the paragraph after the next one. Table 2 also lists the logarithmic moments of the DOSD defined by

Lk ¼

Z



1

dE

Ec

 df k E ln E dE

ð6Þ

and the mean excitation energies, Ik = exp(Lk/Sk). The average energy associated with the total inelastic scattering cross-section for grazing collisions of fast charged particles with the target species is I1. Radiation damage theory requires I0 and I1 which are related to the average energy loss (stopping power) and its mean fluctuation (straggling), respectively, in these collisions [16]. I2 is related to the Lamb shift [16]. Kamakura et al.’s value [36] of I0 = 72.0 eV is only 0.21% lower than ours. No other values of the Ik for C60 have been published earlier. We estimate that our Ik are accurate to ±4%. The frequency-dependent polarizability a(x) is obtained from the DOSD via

aðxÞ ¼

Z

1

Ec

dE

ðdf =dEÞ E 2  x2

:

ð7Þ

Our values of the electronic a(x) at some selected wavelengths k are shown in Table 3. First, consider the infinite wavelength or static electronic polarizability S2 = a. Berkowitz’s [5] value of 566.0 is 1.3% higher than our value of 558.6. Table 3 shows that our S2 lies toward the high end of Antoine et al.’s value [6] measured by a molecular beam deflection technique, and marginally below the low end of Berninger et al.’s range [8] obtained by near-field matter-wave interferometry. Next, we compare our S2 with values obtained by linear-response (LR) theory at the HF, coupled-cluster

210

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Table 3 Frequency dependent polarizability a(x), in atomic units, for C60. This work

k/nm

558.6 572.7 575.9 577.7 582.9 591.4 596.0 603.8 611.6 629.5

1 1064. 966.0 922.7 826.7 724.7 685.0 633.0 594.1 532.0 a b c d e

LR-CCSDa

Table 4 C6 coefficients, in atomic units, for C60–X interactions. A(n) is short for A  10n.

Expt.

X b

555.27 564.85

516 ± 52 , 600 ± 40 533 ± 27d

598.64

607 ± 74e

c

Linear-response coupled-cluster singles and doubles (LR-CCSD), Ref. [13]. Molecular beam Stark deflection, Ref. [6]. Near-field matter-wave interferometry, Ref. [8]. Light-force technique, Ref. [7]. Near-field matter-wave interferometry, Ref. [9].

with single and double substitutions (CCSD), and density functional theory (DFT) levels. The LR–HF (or time-dependent (TD)–HF) value [12] of the static polarizability is 539.5 which is 3.4% below our value. It is satisfying to see in Table 3 that the best electron-correlated static polarizability, the LR–CCSD value [13], is only 0.60% below our value. Remarkably, the LR–DFT (or TDDFT) value [12] of 557.2 is only 0.25% below ours. Table 3 shows that our polarizability at k = 1064 nm is marginally above the high end of Ballard et al.’s range obtained by a light force technique [7]. Our polarizability at k = 532 nm is comfortably within Hackermüller et al.’s range obtained by near-field matterwave interferometry [9]. The ab initio LR–CCSD values [13] at k = 1064 nm and k = 532 nm are 1.4% and 4.9%, respectively, lower than ours. The very low energy portion of the recommended DOSD may be somewhat inaccurate because no refractivity constraints were used in its construction. Perhaps this is why our a(x) deviates more from its LR–CCSD counterpart as the wavelength is shortened. The spherically averaged induced-dipole-induced-dipole C6 term dominates the dispersion interaction –C6R6 – C8R8 – C10R10 – . . . between freely rotating closed-shell molecules separated by a large distance R. The C6 coefficients are vital ingredients in the construction of model intermolecular potentials that are valid for all distances [37–41]. If both C6 and C8 dispersion coefficients are available, then C10 and higher-order coefficients can be calculated to good accuracy from simple models [42,43]. Good approximations to C6 can be obtained by using I2 as the average energy [44] in the London formula. An exact way to compute C6 is to substitute the frequency-dependent polarizabilities for the two species into the Casimir–Polder expression [45]

C6 ¼

3

p

Z

1

aA ðiyÞaB ðiyÞ dy

ð8Þ

0

pffiffiffiffiffiffiffi in which i ¼ 1. As in previous work [46], we evaluate the Casimir–Polder integral in Eq. (8) using pseudospectral representations of a(ix). A representation which reproduces 20 DOSD moments, (Sk, 2 P k P 17), was constructed for C60 and is available as Supplementary material. Pseudospectra were taken from ab initio calculations for H [43] and He [47], from recent DOSD constructions for the ethers [48], Ne, Ar, Kr, and Xe [14], O2 [15,49], O3 [15], and pseudospectra for the remaining species were from older DOSD constructions cited in Ref. [50]. The resulting C6 coefficients are listed in Table 4 for interactions of C60 with itself and 54 other species. Our C6 for interactions between a pair of C60 molecules differs by only 0.70% and 1.5% from the TDHF and TDDFT values [12] of 1.010  105 and 1.018  105, respectively. Given the limitations of the TDHF and TDDFT

H He Ne Ar Kr Xe Li H2 N2 O2 Cl2 HF HCl HBr CO CO2 NO N2O C2H2

C6 801.7 364.7 737.3 2511 3592 5362 8066 1098 2674 2434 6230 1341 3604 4654 2834 3938 2605 4269 4519

X a

O3(B1) O3(C)a SO2 CS2 SCO H2S H2O NH3 CH3OH C2H5OH 1-Propanol H2CO CH3CHO (CH3)2CO SF6 SiH4 SiF4 NH2CH3 NH (CH3)2

C6

X

C6

4214 4008 5399 9300 6347 4652 2110 2982 4690 7290 9841 4052 6321 8885 7343 5850 5561 5499 8031

N (CH3)3 C2H4 Propene 1-Butene CCl4 CH4 C2H6 C3H8 n-C4H10 n-C5H12 n-C6H14 n-C7H16 n-C8H18 O (CH3)2 CH3OC3H7 O (C2H5)2 C6H6 C60

1.029(4) 5479 8135 1.063(4) 1.421(4) 3593 6165 8745 1.124(4) 1.377(4) 1.624(4) 1.873(4) 2.122(4) 7284 1.248(4) 1.247(4) 1.313(4) 1.003(5)

a The two values for O3 labeled B1 and C correspond to the B1 and C DOSDs that bracket the true one [15].

methods, this extraordinary agreement must be fortuitous at least in part. In summary, a DOSD for C60 was constructed from photoabsorption cross-sections by a least squares procedure [14] that imposes the constraints provided by the Kuhn–Reiche–Thomas sum rule and the high-energy behavior of the dipole-oscillator-strength density. The predicted dipole sum rules Sk, mean excitation energies Ik, dynamic polarizability a(x), and C6 coefficients for the long-range dipole–dipole interactions of C60 with a variety of atoms and molecules should prove useful. Acknowledgements We thank Koichiro Mitsuke for sharing his data with us, Cara Nordstrom for technical assistance, and Uwe Hohm for useful discussions. The Natural Sciences and Engineering Research Council of Canada supported this work. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.cplett.2011.09.080. References [1] J. Berkowitz, Atomic and Molecular Photoabsorption: Absolute Total Cross Sections, Academic, San Diego, 2002. [2] A. Reinköster et al., J. Phys. B: At., Mol. Opt. Phys. 37 (2004) 2135. [3] B.P. Kafle, H. Katayanagi, M.S.I. Prodhan, H. Yagi, C. Huang, K. Mitsuke, J. Phys. Soc. Jpn. 77 (2008) 014302. [4] Y. Kawashita, K. Yabana, M. Noda, K. Nobusada, T. Nakatsukasa, J. Mol. Struct. (Theochem) 914 (2009) 130. [5] J. Berkowitz, J. Chem. Phys. 111 (1999) 1446. [6] R. Antoine, P. Dugourd, D. Rayane, E. Benichou, M. Broyer, F. Chandezon, C. Guet, J. Chem. Phys. 110 (1999) 9771. [7] A. Ballard, K. Bonin, J. Louderback, J. Chem. Phys. 113 (2000) 5732. [8] M. Berninger, A. Stefanov, S. Deachapunya, M. Arndt, Phys. Rev. A 76 (2007) 013607. [9] L. Hackermüller, K. Hornberger, S. Gerlich, M. Gring, H. Ulbricht, M. Arndt, Appl. Phys. B 89 (2007) 469. [10] K. Ruud, D. Jonsson, P.R. Taylor, J. Chem. Phys. 114 (2001) 4331. [11] T.B. Pedersen, A.M.J.S. de Merás, H. Koch, J. Chem. Phys. 120 (2004) 8887. [12] A. Jiemchooroj, P. Norman, B.E. Sernelius, J. Chem. Phys. 123 (2005) 124312. [13] K. Kowalski, J.R. Hammond, W.A. de Jong, A.J. Sadlej, J. Chem. Phys. 129 (2008) 226101. [14] A. Kumar, A.J. Thakkar, J. Chem. Phys. 132 (2010) 074301. [15] A. Kumar, A.J. Thakkar, J. Chem. Phys. 135 (2011) 074303. [16] M. Inokuti, Rev. Mod. Phys. 43 (1971) 297.

A. Kumar, A.J. Thakkar / Chemical Physics Letters 516 (2011) 208–211 [17] J.P. Vinti, Phys. Rev. 41 (1932) 432. [18] G.D. Zeiss, W.J. Meath, J.C.F. MacDonald, D.J. Dawson, Can. J. Phys. 55 (1977) 2080. [19] M.C. Struensee, Phys. Rev. A 30 (1984) 2339. [20] S. Krummacher, M. Biermann, M. Neeb, A. Liebsch, W. Eberhardt, Phys. Rev. B 48 (1993) 8424. [21] B.S. Itchkawitz, J.P. Long, T. Schedel-Niedrig, M.N. Kabler, A.M. Bradshaw, R. Schlögl, W.R. Hunter, Chem. Phys. Lett. 243 (1995) 211. [22] A.L. Smith, J. Phys. B: At., Mol. Opt. Phys. 29 (1996) 4975. [23] H. Yasumatsu, T. Kondow, H. Kitagawa, K. Tabayashi, K. Shobatake, J. Chem. Phys. 104 (1996) 899. [24] R. Jaensch, W. Kamke, Mol. Mater. 13 (2000) 143. [25] V. Piacente, G. Gigli, P. Scardala, A. Giustini, D. Ferro, J. Phys. Chem. 99 (1995) 14052. [26] P.F. Coheur, M. Carleer, R. Colin, J. Phys. B: At., Mol. Opt. Phys. 29 (1996) 4987. [27] B.L. Henke, E.M. Gullikson, J.C. Davis, At. Data Nucl. Data Tables 54 (1993) 181. [28] C.T. Chantler, J. Phys. Chem. Ref. Data 24 (1995) 71. [29] N.E.-B. Kassimi, R.J. Doerksen, A.J. Thakkar, J. Phys. Chem. 99 (1995) 12790. [30] R.J. Doerksen, A.J. Thakkar, J. Phys. Chem. A 103 (1999) 2141. [31] A.K. Das, A.J. Thakkar, J. Phys. B: At., Mol. Opt. Phys. 31 (1998) 2215. [32] T. Koga, S. Watanabe, K. Kanayama, R. Yasuda, A.J. Thakkar, J. Chem. Phys. 103 (1995) 3000.

[33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

211

T. Koga, A.J. Thakkar, J. Phys. B: At., Mol. Opt. Phys. 29 (1996) 2973. F.E. Cummings, J. Chem. Phys. 63 (1975) 4960. A.J. Thakkar, V.H. Smith Jr., Phys. Rev. A 23 (1981) 473. S. Kamakura, N. Sakamoto, H. Ogawa, H. Tsuchida, M. Inokuti, J. App. Phys. 100 (2006) 064905. C. Douketis, G. Scoles, S. Marchetti, M. Zen, A.J. Thakkar, J. Chem. Phys. 76 (1982) 3057. R.R. Fuchs, F.R.W. McCourt, A.J. Thakkar, F. Grein, J. Phys. Chem. 88 (1984) 2036. K.T. Tang, J.P. Toennies, J. Chem. Phys. 80 (1984) 3726. B. Jeziorski, R. Moszynski, K. Szalewicz, Chem. Rev. 94 (1994) 1887. R.J. Wheatley, A.S. Tulegenov, E. Bichoutskala, Int. Rev. Phys. Chem. 23 (2004) 151. A.J. Thakkar, V.H. Smith Jr., J. Phys. B: At., Mol. Phys. 7 (1974) L321. A.J. Thakkar, J. Chem. Phys. 89 (1988) 2092. A.J. Thakkar, J. Chem. Phys. 81 (1984) 1919. H.B.G. Casimir, D. Polder, Phys. Rev. 73 (1948) 360. D.J. Margoliash, W.J. Meath, J. Chem. Phys. 68 (1978) 1426. A.J. Thakkar, J. Chem. Phys. 75 (1981) 4496. A. Kumar, W.J. Meath, Mol. Phys. 106 (2008) 1531. A. Kumar, W.J. Meath, P. Bündgen, A.J. Thakkar, J. Chem. Phys. 105 (1996) 4927. A. Kumar, B.L. Jhanwar, W. Meath, Can. J. Chem. 85 (2007) 724.