Accurate higher electric multipole moments for carbon monoxide

2 February 2001

Chemical Physics Letters 334 (2001) 214±219

www.elsevier.nl/locate/cplett

Accurate higher electric multipole moments for carbon monoxide George Maroulis * Department of Chemistry, Physical Chemistry Laboratory, University of Patras, GR-26500 Patras, Greece Received 16 October 2000; in ®nal form 1 December 2000

Abstract We have obtained accurate values for the higher electric multipole moments of CO. We rely on ®nite-®eld coupled cluster calculations with large gaussian-type basis sets. At the CCSD(T) level of theory the quadrupole moment varies as ‰H…R† ÿ H…Re †Š=ea20 ˆ 1:03…R ÿ Re † ‡ 0:40…R ÿ Re †2 around the experimental bond length of Re ˆ 2:132221 a0 . For the octopole and hexadecapole moment we ®nd ‰X…R† ÿ X…Re †Š=ea30 ˆ 1:49…R ÿ Re † ÿ 1:81…R ÿ Re †2 ÿ 0:45…R ÿ Re †3 ÿ 0:17…R ÿ Re †4 ; ‰U…R† ÿ U…Re †Š=ea40 ˆ ÿ4:61…R ÿ Re † ‡ 0:67…R ÿ Re †2 ‡ 2:29…R ÿ Re †3 ‡ 1:04…R ÿ Re †4 : Our best values X ˆ 3:46 ea30 and U ˆ ÿ9:07 ea40 are in reasonable agreement with recent experimental estimates deduced from farinfrared absorption in the high-density gas spectrum of CO±Ar. Agreement is less satisfactory with values obtained from the liquid spectrum. Ó 2001 Elsevier Science B.V. All rights reserved.

1. Introduction and theory Recently, Roco et al. [1,2] reported estimates of the magnitude of the quadrupole (H), octopole (X) and hexadecapole (U) moment of carbon monoxide extracted from measurements of far-infrared absorption in CO±Ar. The experimental determination of higher electric multipole moments represents a task of considerable diculty [3,4]. Very few measurements of the octopole and hexadecapole moments are available in the literature, most of them restricted to small tetrahedral molecules [4]. The occasional experimental reference for a polar molecule represents a memorable tour de force. A most interesting case is hydrogen chloride. Isnard et al. [5] attempted to determine the octopole mo-

*

Fax: +30-61-997118. E-mail address: [email protected] (G. Maroulis).

ment of HCl from an analysis of pressure broadening of spectral lines while Flygare [6] extracted a value of the hexadecapole moment from infrared measurements of HCl trapped in a Ar lattice. The multipole moments of carbon monoxide are currently of interest in view of the numerous investigations of the structure and properties its weakly bonded complexes [7±9] or problems involving signi®cant intermolecular studies of this molecule [10,11]. In this Letter we re®ne our previous theoretical predictions of the higher multipole moments of CO [12,13]. In addition, we report a study of the R-dependence of these properties around the experimental bond length. The importance of the multipole moment curves has been emphasized early enough by van Kranendonk [14]. We do not include the dipole moment in our present e€ort, as this property has been exhaustively studied [15,16]. We adopt the conventional de®nition of the electric multipole moments [17]. A detailed

0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 0 ) 0 1 4 1 3 - 5

G. Maroulis / Chemical Physics Letters 334 (2001) 214±219

presentation of our ®nite-®eld approach to the calculation of these properties for a polar diatomic molecule may be found elsewhere [18]. We rely on large, carefully optimized basis sets of gaussiantype functions (GTFs) for our self-consistent ®eld (SCF) and post-Hartree±Fock calculations. Electron correlation e€ects are accounted for via Mùller±Plesset perturbation theory (MP) and coupled cluster techniques (CC) [19]. The secondand fourth-order Mùller±Plesset methods, MP2 and MP4, used in this work are relatively inexpensive methods for electric property calculations on small molecules [20]. We lean heavily on the predictive capability of CC techniques [21]. We use the singles and doubles coupled cluster (CCSD) method and CCSD(T), which includes an estimate of connected triple excitations by a perturbational treatment. In addition to the above, we de®ne the electron correlation correction (ECC) as the difference between the presumably most accurate CCSD(T) level of theory and the SCF one, ECC ˆ CCSD…T† ÿ SCF:

215

were designed for calculations of electric properties on CO. Their construction is presented in some detail elsewhere [18]. Both D and Q were used in calculations at the experimental bond length of Re ˆ 2:132221 a0 [22]. Calculations were also performed with basis D at bond lengths Re  0:2, Re  0:4 and Re  0:6. Very weak electric ®elds were used in the calculations. Typical magnitudes are jQ=R3 j ˆ 4 ÿ1 0:0000125 eÿ1 aÿ2 0 Eh for H, jQ=R j ˆ 0:00001 e ÿ3 5 ÿ1 ÿ4 a0 Eh for X and jQ=R j ˆ 0:000000032 e a0 Eh for U. All calculations were performed with GA U S S I A N 94 [23]. Atomic units are used throughout this Letter. Conversion factors to SI units are: Length, 1 a0 ˆ 0:529177249  10ÿ10 m, dipole moment, 1 ea0 ˆ 8:478358  10ÿ30 cm, quadrupole moment, 1 ea20 ˆ 4:486554  10ÿ40 cm2 , octopole moment, 1 ea30 ˆ 2:374182  10ÿ50 cm3 and hexadecapole moment, 1 ea40 ˆ 1:256363  10ÿ60 cm4 .

…1† 3. Results and discussion

2. Computational details Two ¯exible basis sets were used in our calculations, D  ‰6s4p4d2fŠ and Q  ‰9s6p4d3fŠ. They

SCF and post-Hartree±Fock values of H, X and U at Re are shown in Table 1. In Table 2 we display the method dependence of the ®rst derivative of the electric moments at Re . In Table 3 a selection of

Table 1 Electric multipole momentsa for carbon monoxide at the experimental bond length Re ˆ 2:132221 a0

a b

Methodb

H

X

U

D  ‰6s4p4d2fŠ SCF MP2 MP4 CCSD CCSD(T) ECC

)1.51 )1.49 )1.51 )1.45 )1.46 0.06

4.40 3.46 3.42 3.56 3.46 )0.94

)10.59 )8.98 )8.70 )9.07 )8.81 1.78

Q  ‰9s6p4d3fŠ SCF MP2 MP4 CCSD CCSD(T) ECC

)1.54 )1.50 )1.52 )1.47 )1.47 0.07

4.39 3.43 3.42 3.57 3.46 )0.93

)10.92 )9.18 )8.92 )9.36 )9.07 1.85

Relative to the centre of nuclear mass. The oxygen nucleus on the positive z axis. The two innermost MO were kept frozen in all post-Hartree±Fock calculations.

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G. Maroulis / Chemical Physics Letters 334 (2001) 214±219

Table 2 Electron correlation e€ects on the ®rst derivative of the multipole moments of carbon monoxide at Re a

a

Method

…dH=dR†e

…dX=dR†e

…dU=dR†e

SCF MP2 MP4 CCSD CCSD(T)

1.09 1.01 0.83 1.09 1.03

2.54 1.34 1.38 1.60 1.49

)6.84 )4.65 )4.08 )4.94 )4.61

Basis set D  ‰6s4p4d2fŠ.

Table 3 Comparison of theoretical and experimental values for the higher electric moments of CO Method

H

X

U

Theory NHFa SD-CIb SD-CIc ACCDd CCSD(T)e SDQ-MP4f CCD + ST(CCD)g CCSD(T)h CCSD(T)i

)1.53001 )1.450 )1.5164 )1.4902 )1.460 )1.48 )1.502 )1.46 )1.47

4.42239 3.771 3.903 3.8196

)10.6883 )9.42 )9.848

3.59

)9.01

3.46 3.46

)8.81 )9.07

Experiment MW asymmetric Zeeman shiftsj Far IR rotational spectrak Ion±molecule scattering cross-sectionsl MBER Stark±Zeeman spectram Far IR absorption in CO±Arn

)1.5  0.7 )1.4  0.1 )1.44  0.3 )1.5 )1.58 ()1.68)

4.15 (4.45)

)8.15 ()17.52)

a

Fully numerical Hartree±Fock values at Re ˆ 2:132 a0 . Laaksonen et al. [24]. Basis set [5s4p2d] at Re ˆ 2:132 a0 . Amos [25]. c Basis set [8s5p3d1f] at Re ˆ 2:132 a0 . Obtained as average values of the respective operators. Diercksen and Sadlej [26]. d ELP basis set. Dykstra et al. [27]. e Best estimate at Re ˆ 2:1322 a0 of Rizzo et al. [28]. f Maroulis [12]. g Maroulis and Thakkar [13]. Basis set [6s4p3d2f]. h Present investigation. Basis set [6s4p4d2f]. i Present investigation. Basis set [9s6p4d3f]. j Gustafson and Gordy [30]. k Buontempo et al. [31]. l Budenholzer et al. [32]. m Meerts et al. [33]. n Far-infrared absorption in the high-density gas spectrum. The values in parentheses are from the liquid spectrum. The quantities measured are jHj, jXj and jUj. Roco et al. [2]. b

theoretical values are compared to the available experimental data. Our SCF values of H, X and U calculated with basis sets D and Q are quite close to the accurate numerical Hartree±Fock (NHF) results [24]. It should be noted that the NHF data pertain to a bond length slightly shorter than Re . Electron

correlation reduces the magnitude of H. The change is rather small, the SCF value is reduced by 4.54% for basis Q. The predicted ECC is 0.06 (D) and 0.07 (Q) ea20 . The performance of the two basis sets is virtually identical for the octopole moment. The ECC is )0.94 (D) and )0.93 (Q) ea30 . The total correction corresponds to 21.2% of the SCF value

G. Maroulis / Chemical Physics Letters 334 (2001) 214±219

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for Q. All post-Hartree±Fock methods predict reliable values for X. Electron correlation reduces the magnitude of U by 16.8% for D and 16.9% for Q. The ECC is 1.78 for D and 1:85 ea40 for Q. We have also obtained the R-dependence of all properties using basis set D. We ®nd that at the CCSD(T) level of theory the quadrupole moment is very well represented by the parabolic form ‰H…R† ÿ H…Re †Š=ea20 2

ˆ 1:03…R ÿ Re † ‡ 0:40…R ÿ Re † :

…2†

For the octopole we have

Fig. 2. R-dependence of the octopole moment of carbon monoxide.

‰X…R† ÿ X…Re †Š=ea30 ˆ 1:49…R ÿ Re † ÿ 1:81…R ÿ Re †

2

3

ÿ 0:45…R ÿ Re † ÿ 0:17…R ÿ Re †

4

…3†

4

…4†

and for the hexadecapole moment ‰U…R† ÿ U…Re †Š=ea40 ˆ ÿ4:61…R ÿ Re † ‡ 0:67…R ÿ Re † 3

2

‡ 2:29…R ÿ Re † ‡ 1:04…R ÿ Re † :

The quantities …dP =dR†e , P  H, X and U, in Table 2 show that all correlated methods predict rather stable values for the ®rst derivative. For the quadrupole moment the stability includes the SCF value as well. We have traced in Figs. 1±3 the R-dependence of H, X and U. Only the SCF, MP4 and CCSD(T) curves are shown for H. For the other two properties we show all methods used in

Fig. 1. R-dependence of the quadrupole moment of carbon monoxide.

Fig. 3. R-dependence of the hexadecapole moment of carbon monoxide.

this work. The curves show that for X and U the performance of the theoretical methods diverges signi®cantly for bond lengths R > Re . In Table 3 we have included previous theoretical values for H, X and U. The singles and doubles con®guration interaction (SD-CI) results of Amos [25] and Diercksen and Sadlej [26] for the quadrupole moment are in good agreement with our CCSD(T) values. Our X and U values are smaller in magnitude than theirs. The same trend is observed for the approximate doubles coupled cluster (ACCD) H and X of Dykstra et al. [27]. We note also a recent study for the quadrupole moment that produced an equilibrium value of )1.460 ea20 [28]. We have included in Table 3 previous H, X and U values from a partial MP4 treatment [12]

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G. Maroulis / Chemical Physics Letters 334 (2001) 214±219

and an approximate coupled cluster result for H [13]. To render comparison with experiment meaningful we have estimated the zero-point vibrational correction (ZPVC) as [29]   1 Be P0 ÿ Pe ˆ 2 xe     ae xe 1 2 2 D P ‡ R D P ; …5† R  3 1‡ e e 6B2e where Dk P  …dk P =dRk †e ; k ˆ 1; 2. We rely on experimental spectroscopic constants [22] and the derivatives obtained in this work. The ZPVC is found to be small for all properties: 0.01 ea20 for H, 0.004 ea30 for X and ÿ0:03 ea40 for U. The old experimental values for the quadrupole moment of Gustafson and Gordy [30], Buontempo et al. [31], Budenholzer et al. [32] and Meerts et al. [33] are in fair agreement with the best theoretical predictions. One should also mention here the available induced birefringence measurements of the quadrupole moment of CO, the most recent being that of Graham et al. [34]. The induced birefringence values are not relative to the centre of mass and their comparison to the other results in Table 3 is a rather non-trivial matter. It is readily seen that the high-density gas spectrum values of Roco et al. [2] should be closer to the free molecule values than those obtained from the liquid spectrum. We judge the agreement of our values with the experimental estimates of Roco et al. [2] to be reasonable enough. The experimental estimates reproduce rather well the relative size of the higher moments of carbon monoxide. 4. Conclusions We have calculated accurate higher electric multipole moments for carbon monoxide. We have also obtained the R-dependence of these properties around the experimental equilibrium bond length. For the quadrupole and octopole moments we recommend values of H ˆ ÿ1:46  0:03 ea20 and X ˆ 3:46  0:07 ea30 . For the hexadecapole moment we combine the NHF result, our CCSD(T) value at Re obtained with basis Q  ‰9s6p4d3fŠ and

the ZPVC obtained from the basis D  ‰6s4p4d2fŠ results to recommend a value of U ˆ ÿ9:0 ea40 . We also recommend ®rst derivative values …dH=dR†e  1:0 ea0 , …dX=dR†e  1:5 ea20 †, and …dU=dR†e  ÿ4:6 ea30 : References [1] J.M.M. Roco, A. Calvo-Hernandez, S. Velasco, J. Chem. Phys. 103 (1995) 9161. [2] J.M.M. Roco, A. Medina, A. Calvo Hernandez, S. Velasco, J. Chem. Phys. 103 (1995) 9175. [3] A.D. Buckingham, in: H. Eyring, Henderson, W. Jost (Eds.), Physical Chemistry an Advanced Treatise, Academic, New York, 1970, p. 349 (Chapter 8). [4] C.G. Gray, K.E. Gubbins, Theory of Molecular Fluids, vol. 1, Oxford University Press, Oxford, 1984. [5] P. Isnard, C. Boulet, A. Levy, J. Quant. Spectrosc. Radiat. Transfer 13 (1973) 1433. [6] W.H. Flygare, J. Chem. Phys. 39 (1963) 2263. [7] M.O. Bulanin, V.P. Bulychev, K.G. Tokhadze, Opt. Spectrosc. 67 (1989) 338. [8] A. van der Pol, A. van der Avoird, P.E.S. Wormer, J. Chem. Phys. 92 (1990) 7498. [9] C.A. Parish, J.D. Augspurger, C.E. Dykstra, J. Phys. Chem. 96 (1992) 2069. [10] D. Priem, F. Rohart, J.-M. Colmont, G. Wlodarczak, J.-P. Bouanich, J. Mol. Struct. 517±518 (2000) 435. [11] A. Predoi-Cross, J.-P. Bouanich, D.C. Benner, A.D. May, J.R. Drummond, J. Chem. Phys. 113 (2000) 158. [12] G. Maroulis, Z. Naturforshung A 47 (1992) 480. [13] G. Maroulis, A.J. Thakkar, J. Chem. Phys. 92 (1990) 812. [14] J. van Kranendonk, Physica 24 (1957) 347. [15] G.E. Scuseria, M.D. Miller, F. Jensen, J. Geertsen, J. Chem. Phys. 94 (1991) 6660. [16] G. Maroulis, J. Phys. Chem. 100 (1996) 13466. [17] A.D. Buckingham, in: B. Pullman (Ed.), Intermolecular Interactions: From Diatomic to Biopolymers, Wiley, Chichester, 1978, p. 1. [18] G. Maroulis, J. Chem. Phys. 108 (1998) 5432. [19] A. Szabo, N.S. Ostlund, Modern Quantum Chemistry, Macmillan, New York, 1982. [20] M. Urban, I. Cernusak, V. Kell o, J. Noga, in: S. Wilson (Ed.), Methods in Computational Chemistry, vol. 1, Plenum, New York, 1987, p. 117. [21] J. Paldus, X. Li, Adv. Chem. Phys. 110 (1999) 1. [22] K.P. Huber, G. Herzberg, Molecular Spectra and Molecular Structure IV, Van Nostrand-Reinhold, New York, 1979. [23] M.J. Frisch et al., GA U S S I A N 94, Revision E.1, Gaussian, Inc., Pittsburgh, PA, 1995. [24] L. Laaksonen, P. Pyykk o, D. Sundholm, Comput. Phys. Rep. 4 (1986) 313. [25] R.D. Amos, Chem. Phys. Lett. 68 (1979) 536.

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