Ab initio investigation of the sum-frequencyhyperpolarizability of small chiral moleculesBeno^õt Champagnea,*, Peer Fischerb, A. David BuckinghambaLaboratoire de Chimie Theorique Appliquee, Facultes Universitaires Notre-Dame de la Paix, rue de Bruxelles, 61, Namur B-5000,BelgiumbDepartment of Chemistry, University of Cambridge, Lens®eld Road, Cambridge CB2 1EW, UKReceived 21 July 2000; in ®nal form 2 October 2000AbstractUsing a sum-over-states procedure based on con®guration interaction singles /6-311++G , we have computed thesum-frequency hyperpolarizability bijkÿ3x; 2x; x of two small chiral molecules, R-mono¯uoro-oxirane and R-(+)-propylene oxide. Excitation energies were scaled to ®t experimental UV-absorption data and checked with ab initiovalues from time-dependent density functional theory. The isotropic part of the computed hyperpolarizabilities,bÿ3x; 2x; x, is much smaller than that reported previously from sum-frequency generation experiments on aqueoussolutions of arabinose. Comparison is made with a single-centre chiral model. Ó 2000 Elsevier Science B.V. All rightsreserved.1. IntroductionIn the electric dipole approximation, coherentthree-wave mixing (TWM) processes in isotropicmedia are only symmetry-allowed for sum- anddierence-frequency generation in chiral media [1].Indeed, coherent second-order nonlinear optical(NLO) processes appear either in non-centrosym-metric media such as crystals and interfaces, or inisotropic media provided they contain chiral mol-ecules and are not racemic.Chirality-allowed sum-frequency generation(SFG) in isotropic media was ®rst reported over 30years ago for d- and l-arabinose solutions in waterwith a sum-frequency susceptibility about one-tenth that of quartz [2,3]. In a recent joint theo-retical±experimental study [4], it has been shownthat the SFG phenomenon in chiral liquids ismuch weaker than previously reported [2,3,5].Firstly, a single-centre chiral model and ab initiocalculations on small chiral molecules indicate thatthe isotropic part of the SFG ®rst hyperpolariz-ability,b, is at least two orders of magnitudesmaller than that reported experimentally forarabinose. Secondly, three- and four-wave mixingexperiments support the absence of a strong SFGresponse in simple chiral liquids.In this Letter, we describe the theoretical pro-cedure we have adopted for computing the dy-namic sum-frequency hyperpolarizability and itsisotropic part. Two simple chiral molecules areconsidered: propylene oxide, C3H6O, and mono-¯uoro-oxirane, C2H3OF.24 November 2000Chemical Physics Letters 331 (2000) 83±88www.elsevier.nl/locate/cplett*Corresponding author. Fax: +32-81-724567.E-mail address: [email protected] (B. Cham-pagne).0009-2614/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.PII: S 0 0 0 9 - 2 6 1 4 ( 0 0 ) 0 1 149-02. Methodology and computational procedureThe TWM in chiral liquids is related to thecompletely antisymmetric isotropic component ofthe second-order susceptibility v2or to its mi-croscopic analog [1,6]b  bxyzÿÿ bxzy byzxÿ byxz bzxyÿ bzyx=6 1with each tensor component written according toBoyd [7]bijkÿxqÿ xp; xq; xp hÿ2PIXnXmli0nljnmlkm0~xn0ÿ xpÿ xq~xm0ÿ xp8<:lj0nlinmlkm0~x n0 xq~xm0ÿ xplj0nlknmlim0~x n0 xq~x m0 xp xq9=;;2where the operator PIpermutes simultaneously theincident frequencies and the associated Cartesianindices, xq; j and xp; k. The summations runover all the excited states m and n, li0nh0 j^lij niis the ith component of the transition momentbetween the ground state (0) and the nth excitedstate,ljnm ljnmÿ lj00dnm. The validity of Eq. (2)near resonance is ensured by the complex nature ofthe transition frequency~xm0 xm0ÿi=2Cm0,where xm0is the real transition frequency and Cm0the width at half the maximum height of thetransition between states 0 and m. When we in-clude damping in the SOS expressions we choose alinewidth of 1000 cmÿ1for all excited states withxqand xpin the visible.The excitation energies and transition dipolemoments were computed using the GAUSSIANAUSSIAN94program [8] at the con®guration interaction singles(CIS) level of approximation using the 6-311++G [9,10] atomic basis set. Thenb wasevaluated by using a modi®ed version of the ab-SOS code [11,12]. Fig. 1 shows that for propyleneoxide the variation in thebÿ3x; 2x; xhx 1:165 eV  0:0428 a:u:; k  1064 nm due to thecontributions from higher-energy excited statestends to decrease after the inclusion of the ®rst80 excited states in the sum-over-states (SOS)expression. For propylene oxidebÿ3x; 2x; xconverges towards a small positive value. With theexception of the sign, similar behavior is obtainedfor mono¯uoro-oxirane (Fig. 1). For the comput-ed hyperpolarizabilities reported in this Letter weconsider the ®rst 120 excited states.The frequency dispersion ofbÿ3x; 2x; x issubstantial and much larger than for the vectorcomponent of the b-tensor in the direction of thedipole moment l, generally denoted bk3=5libijj=lklk1=2with summation assumedover repeated indices. For instance, in the case ofR-(+)-propylene oxide,bÿ3x; 2x; x0:00001,Fig. 1. CIS/6-311++G bÿ3x; 2x; x as a function of thenumber of excited states included in the SOS expression. (a)R-(+)-propylene oxide, k  1064 nm, scissor operator  2:675eV. (b) R-mono¯uoro-oxirane, k  694:3 nm, scissor operator 2:27 eV.84 B. Champagne et al. / Chemical Physics Letters 331 (2000) 83±880.00056, and )0.0202 a.u.1, whereas bkÿ3x;2x; x is )135, )160, and )251 a.u., respectively,for k  1970, 1064 and 694.3 nm k  2pc=x. Forthe same order of wavelengths,bÿ3x; 2x; x ofR-mono¯uoro-oxirane is )0.000004, )0.00031,and )0.0160 a.u. whereas, bkÿ3x; 2x; x attains)129, )147, and ) 206 a.u., respectively. Accurateexcitation energies are therefore mandatory to es-timate the order of magnitude ofbÿ3x; 2x; x.Toaccount for electron correlation as well as for sol-vatochromic eects, we have evaluated improvedexcitation energies by using a scissor operator toreduce all excitation energies by a ®xed amount. Theamplitude of the shift (2.675 eV) for propylene ox-ide has been determined to ®t the experimental UVabsorption spectra. Since