We use the sum-over-states formalism to compute the imaginary-frequency dipole polarizabilities  for H2, as a function of the H–H bond length, at the full configuration interaction level of theory  using atom-centered d-aug-cc-pVQZ basis sets. From these polarizabilities, we obtain isotropic and  anisotropic C6 dispersion coefficients for a pair of H2 molecules as functions of the two molecules’  bond lengths

Hinde, Robert

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University of Tennessee, Knoxville: Trace

University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Chemistry Publications and Other Works Chemistry January 2005 Vibrational dependence of the H2–H2 C6 coefficients Robert Hinde University of Tennessee, rhinde@utk.edu Follow this and additional works at: https://trace.tennessee.edu/utk_chempubs  Part of the Atomic, Molecular and Optical Physics Commons, Biological and Chemical Physics Commons, and the Physical Chemistry Commons Recommended Citation Hinde, Robert, "Vibrational dependence of the H2–H2 C6 coefficients" (2005). Chemistry Publications and Other Works. https://trace.tennessee.edu/utk_chempubs/7 This Article is brought to you for free and open access by the Chemistry at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Chemistry Publications and Other Works by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact trace@utk.edu. Vibrational dependence of the H 2–H2 C6 dispersion coefficientsRobert J. Hindea!Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600sReceived 27 December 2004; accepted 26 January 2005; published online 8 April 2005dWe use the sum-over-states formalism to compute the imaginary-frequency dipole polarizabilitiesfor H2, as a function of the H–H bond length, at the full configuration interaction level of theoryusing atom-centered d-aug-cc-pVQZ basis sets. From these polarizabilities, we obtain isotropic andanisotropicC6 dispersion coefficients for a pair of H2 molecules as functions of the two molecules’bond lengths. ©2005 American Institute of Physics. fDOI: 10.1063/1.1873512gThe parahydrogen dimerspH2d2 formed by twoj =0 H2molecules is the molecular analog of the He2 van der Waalsdimer. Because thepH2–pH2 interaction is relatively weakand the H2–H2 reduced mass is small, the dimer formedbetween twov=0 pH2 molecules has just a single boundstate,1 one which is characterized by an intermolecular vibra-tional wave function that spans a wide range of intermolecu-lar distancesR. In the ground state of the empirical H2–H2potential given in Ref. 2, for example, fully 30% of thespH2d2 dimer’s probability density is located in the classi-cally forbidden regionRù10 a0, where thepH2–pH2 poten-tial is dominated by dispersion interactions. In this paper, wecomputeR−6 dispersion coefficients for H2–H2, H2–D2, andD2–D2 dimers in which one of the monomers is in thev=1 level; these coefficients provide constraints on the shapeof the H2–H2 potential in the classically forbidden largeRregion, and hence on the binding energies of near-thresholdstates of these van der Waals dimers.We begin by computing the H2 dipole polarizabilitiesaxxand azz at imaginary frequencies using the sum-over-statesformalism,aqqsivd = 2onÞ0sEn − E0duk0um̂qunlu2sEn − E0d2 + "2v2, s1dwhere m̂ is the dipole moment operator andq represents aCartesian coordinatesin this case, eitherx or zd in themolecule-fixed frame of reference; the H2 bond defines themolecule-fixedz axis. The ground state electronic wavefunction u0l and excited state electronic wave functionsu l,their energiesE0 andEn, and the dipole moment matrix ele-mentsk0um̂qunl are obtained from full configuration interac-tion sCId electronic structure calculations employing a cc-pVQZ atom-centered Gaussian basis set for hydrogen3 thathas been doubly augmented4,5 sdenoted d-aug-cc-pVQZdwith diffuse functions that facilitate the accurateab initiocomputation of dipole polarizabilities.6These electronic structure calculations are performed atH2 bond lengthsr ranging from 0.9a0 to 2.1a0 in steps of0.1 a0, and also atravg=1.449a0, which is close to thev=0 vibrationally averaged H2 bond length. Calculations atr =0.9 a0, r =1.449a0, and r =2.1 a0 are also performed us-ing the smaller d-aug-cc-pVTZ atom-centered Gaussian basisset for hydrogen. All electronic structure calculations are per-formed using the 22 November 2004 R1 version ofGAMESS.7At each value ofr, the sum in Eq.s1d is evaluated over theentire set of excited electronic states obtained from the fullCI calculations. The relevant electronic state energies anddipole moment matrix elements may be downloaded fromthe EPAPS depository.8As a check on the completeness of our electronic basisset, we use our d-aug-cc-pVQZ dipole moment matrix ele-ments to evaluate the sumsonÞ0uk0um̂xunlu2, onÞ0uk0um̂zunlu2, s2dwhich should give the ground state expectation valuesk0um̂x2u0l andk0um̂z2u0l in the limit of a complete basis set. Wethen compare these quantities with the corresponding onescomputed directly from the electronic second moments forH2 given in Table V of Ref. 9. At the bond lengths where thiscomparison can be made, our results differ from those ob-tained from Ref. 9 by 0.2% or less. The static dipole polar-izabilities given by Eq.s1d at v=0 are in similarly goodagreement with the H2 polarizabilities computed in Ref. 10.Equipped with the polarizabilities defined by Eq.s1d, wenext compute the isotropic polarizabilitya=s2axx+azzd /3and the polarizability anisotropyDa=azz−axx. Using thesequantities and one-dimensional quadrature, we then obtainthe C6 dispersion coefficient for a pair ofj =0 H2 moleculeswith bond lengthsr1 and r2,C6sr1,r2d =3"ps4pe0d2E0`asiv,r1dasiv,r2ddv, s3dand the anisotropic dispersion coefficientsC6,02, C6,20, andC6,22 defined by11C6,absr1,r2d ="ps4pe0d2E0`fasiv,r1dfbsiv,r2ddv. s4dIn Eq. s4d, f0sivd=asivd while f2sivd=Dasivd.For each pair of H2 bond lengthssr1,r2d, we evaluateEqs. s3d and s4d using trapezoid rule quadrature with a stepsize ofDv=0.1 a.u. over the range 0øvø50 a.u.sWe findthat the values of the integrals do not change significantlywhen we use smaller step sizes or when we extend the inte-adElectronic mail: rhinde@utk.eduTHE JOURNAL OF CHEMICAL PHYSICS122, 144304s2005d0021-9606/2005/122~14!/144304/2/$22.50 © 2005 American Institute of Physics122, 144304-1Downloaded 11 Feb 2010 to 160.36.192.127. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jspgrals to include larger values ofv.d At r1=r2=ravg we obtainC6=12.0571 a.u., C6,20=C6,02=1.2256 a.u., and C6,22=0.3941 a.u. These values are in good agreement with thoseobtained by Bishop and Pipin12 using a similar computa-tional approachsbut a different basis set for describing theelectronic wave functionsd: C6=12.058 168 a.u., C6,20=C6,02=1.2194 a.u., andC6,22=0.3898 a.u. Dispersion coef-ficients computed using the d-aug-cc-pVTZ basis set differfrom the d-aug-cc-pVQZ values by 0.7% or less, indicatingthat the d-aug-cc-pVTZ dispersion coefficients are alreadynearly converged with respect to basis set. We therefore ex-pect that further expansion of the atomic basis set beyond thed-aug-cc-pVQZ level will provide only minor improvementsin the accuracy of the dispersion coefficients presented here.We fit our computed dispersion coefficients to polyno-mials of the formPsr1,r2d = ob=02oc=02Abcsr1 − r0dbsr2 − r0dc, s5dwhere r0=1.4 a0. We then use the H2 and D2 vibrationalmatrix elements given in Table I to integrate these polyno-mial fits over specific H2 and D2 vibrational wave functions,thereby obtaining vibrationally averaged dispersion coeffi-cients with the general formkv1,v2uPsr1,r2duv18 ,v28l for a pairof j =0 molecules; these are listed in Table II. We find that, asexpected, dispersion interactions are slightly stronger inv=0+v=1 molecular pairs than inv=0+v=0 pairs. We alsofind that off-diagonal terms of the formkv1=0,v2=1uPsr1,r2duv18=1,v28=0l are small but non-negligible; theC6 coefficient for H2sv=0d+H2sv=1d pairs, for example,changes by about 2% depending on whether the vibrationalexcitation is delocalized across the pair of molecules in asymmetric or antisymmetric fashion.ACKNOWLEDGMENTThis work was supported by the National Science Foun-dation through Grant No. CHE-0414705.1A. R. W. McKellar, J. Chem. Phys.92, 3261s1990d.2M. J. Norman, R. O. Watts, and U. Buck, J. Chem. Phys.81, 3500s1984d.3T. H. Dunning, Jr., J. Chem. Phys.90, 1007s1989d.4R. A. Kendall, T. H. Dunning, Jr., and R. J. Harrison, J. Chem. Phys.96,6796 s1992d.5D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys.100, 2975s1994d.6The basis sets used in this work were obtained from the 02/25/04 versionof the Extensible Computational Chemistry Environment Basis Set Data-base, located atkhttp://www.emsl.pnl.gov/forms/basisform.htmll.7M. W. Schmidt, K. K. Baldridge, J. A. Boatzet al., J. Comput. Chem.14,1347 s1993d.8See EPAPS Document No. E-JCPSA6-122-307515 for text files contain-ing these energies and dipole moment matrix elements. A direct link tothis document may be found in the online article’s HTML reference sec-tion. The document may also be reached through the EPAPS homepageshttp://www.aip.org/pubservs/epaps.htmld or from ftp.aip.org in the direc-tory /epaps/. See the EPAPS homepage for more information.9W. Kołos and L. Wolniewicz, J. Chem. Phys.43, 2429s1965d.10L. Wolniewicz, J. Chem. Phys.99, 1851s1993d.11A. J. Stone and R. J. A. Tough, Chem. Phys. Lett.110, 123 s1984d.12D. M. Bishop and J. Pipin, Int. J. Quantum Chem.45, 349 s1993d.13J. W. Cooley, Math. Comput.15, 363 s1961d.TABLE I. Vibrational matrix elementskvusr −r0dnuv8l for j =0 H2 and D2molecules. These integrals are based on vibrational wave functions com-puted from the potential curve in Ref. 9 using the Numerov–CooleysRef. 13d method.Molecule v v8 n=1 sa0d n=2 sa02dH2 0 0 0.048 34 0.030 53H2 0 1 0.166 39 0.026 74H2 1 1 0.144 95 0.106 87D2 0 0 0.034 35 0.020 90D2 0 1 0.139 52 0.015 84D2 1 1 0.101 93 0.070 15TABLE II. Vibrationally averagedC6 dispersion coefficientssin atomicunitsd for various molecular pairs.Pair C6 C6,20 C6,02 C6,22H2 sv1,v18d H2 sv2,v28ds0, 0d s0, 0d 12.077 1.246 1.246 0.408s0, 0d s1, 1d 12.799 1.327 1.450 0.479s0, 1d s1, 0d 0.137 0.035 0.035 0.027H2 sv1,v18d D2 sv2,v28ds0, 0d s0, 0d 11.972 1.234 1.217 0.398s0, 0d s1, 1d 12.477 1.291 1.358 0.447s1, 1d s0, 0d 12.687 1.437 1.296 0.467D2 sv1,v18d D2 sv2,v28ds0, 0d s0, 0d 11.869 1.205 1.205 0.388s0, 0d s1, 1d 12.368 1.260 1.345 0.435s0, 1d s1, 0d 0.096 0.024 0.024 0.018144304-2 Robert J. Hinde J. Chem. Phys. 122, 144304 ~2005!Downloaded 11 Feb 2010 to 160.36.192.127. 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Vibrational dependence of the H\u3csub\u3e2\u3c/sub\u3e–H\u3csub\u3e2\u3c/sub\u3e \u3ci\u3eC\u3c/i\u3e\u3csub\u3e6\u3c/sub\u3e coefficients

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Vibrational dependence of the H\u3csub\u3e2\u3c/sub\u3e–H\u3csub\u3e2\u3c/sub\u3e \u3ci\u3eC\u3c/i\u3e\u3csub\u3e6\u3c/sub\u3e coefficients

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