Field-induced coordinates for the determination of dynamic vibrational nonlinear optical properties

Transcription

1 JOURNA O CHEMICA PHYSICS VOUME 115, NUMBER 10 8 SEPTEMBER 2001 eld-nduced coordnates for the determnaton of dynamc vbratonal nonlnear optcal propertes Josep M. us and Mquel Duran Insttute of Computatonal Chemstry and Department of Chemstry, Unversty of Grona, Campus de Montlv, Grona, Catalona, Span Bernard Krtman Department of Chemstry and Bochemstry, Unversty of Calforna, Santa Barbara, Calforna Receved 27 Aprl 2001; accepted 18 June 2001 The most mportant contrbuton to noesonant vbratonal nonlnear optcal propertes arses from nuclear relaxaton NR. In prevous work a set of statc feld-nduced vbratonal coordnates ICs has been developed to smplfy calculaton of the NR contrbuton wthn the nfnte optcal frequency approxmaton. Although the number of ICs s small and ndependent of molecular sze, these coordnates form a complete set. However, the nfnte optcal frequency approxmaton does not take nto account the frequency dsperson, whch we evaluate for three conjugated organc molecules that span a range of polarty and valence-bond/charge transfer characterstcs. Our results show that dsperson can be sgnfcant and that, n such cases, frequency-dependent ICs D-ICs are necessary for an adequate treatment. A complete, though stll small, set of D-ICs s presented for ths purpose. Computatonal tests reveal that a reduced subset, together wth an harmonc approxmaton, can be used to acheve hgh accuracy outsde the nfrared IR regon. That subset s complete for the electro-optc and Pockels and Kerr effects though not for other common propertes Amercan Insttute of Physcs. DOI: / I. INTRODUCTION It s well-establshed that vbratonal motons often play a major role n nonlnear optcal NO propertes. 1 4 or noesonant processes there exsts a strong theoretcal edfce whereby that role may be quantfed through quantum chemcal computatons of the relevant molecular hyperpolarzabltes at varous levels of approxmaton. Thus, an approprate frequency-dependent perturbaton treatment 5 7 has been developed by Bshop and Krtman BK. It s based on a clamped nucleus approxmaton, 8 whch separates electronc from nuclear motons, and s based on a doubly harmonc mechancal and electrcal zeroth-order model. Compact vbratonal hyperpolarzablty formulas, complete through second-order n mechancal plus electrcal anharmoncty, have been derved from ths treatment. 9 If we set asde the contrbuton from the zero-pont vbratonal average ZPVA, for the moment, then each term n the perturbaton expanson contans products of dervatves of lower-order electronc propertes wth respect to vbratonal coordnates. Ths provdes a bass for classfyng the terms nto unque square bracket types. or example, the vbratonal second hyperpolarzablty v ( ; 1, 2, 3 ) contans terms of the 2 square bracket type, meanng a product of two factors wth dervatves of the electronc dpole moment e and one factor wth a dervatve of the electronc lnear polarzablty e s nvolved. Here 1, 2, and 3 are the frequences of the three appled felds; ; and the perturbaton terms also depend on these frequences as well as the vbratonal force constants. It turns out that the 2 contrbuton vanshes n zeroth-order as well as n all other even orders. Ths pattern vares n a systematc way for dfferent square bracket types, e.g., 2 terms contrbute n zerothorder, but vansh n all odd orders; on the other hand, the frst nonvanshng 4 terms are second-order whle all subsequent odd orders vansh. The lowest-order nonvanshng terms of each type, taken n toto, consttute the nuclear relaxaton NR approxmaton. Ths name arses from consderng the case where one or more of the appled felds s statc and the remanng optcal felds f any are formally allowed to become nfnte. 10 In that event the statc felds wll nduce a shft n the equlbrum geometry.e., a nuclear relaxaton whle the nucle wll be totally uesponsve to the optcal felds. If there were no NR, then an expanson of the statc electronc dpole moment, e (0), as a power seres n the statc felds would gve e (0;0) as the lnear term, e (0;0,0) as the quadratc term, and e (0;0,0,0) as the cubc term. When the nuclear relaxaton s taken nto account ths expanson gves, n addton, the NR contrbutons (0;0), (0;0,0), and (0;0,0,0). 11 Smlarly, from the correspondng expanson of e (0;0) we get (;,0) and (;,0,0) ; and from e (0;0,0) we obtan (2;,,0). 12 To complete the pcture t should be evdent that both (2;,) and (3;,,) are zero. The NR hyperpolarzabltes found n ths fashon turn out to be dentcal to the correspondng propertes calculated as the sum of lowest-order nonvanshng perturbaton terms of each square bracket type. 10,13 Clearly, the statc and nfnte optcal frequency NR propertes lsted above can be determned by fnte feld computatons. Included are all but one see below of the /2001/115(10)/4473/11/$ Amercan Insttute of Physcs

2 4474 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 us, Duran, and Krtman most commonly studed NO processes. Although care must be taken to avod molecular rotatons wth respect to the feld, 14 the method has proven to be an attractve way to calculate nfnte optcal frequency propertes to be further dscussed later on not only for ndvdual molecules, 15 but also for molecules n soluton 16 and for the lnear polarzablty of nfnte polymers. 17 It s evdent, however, that ths procedure cannot be employed for NR propertes at arbtrary frequences. Thus, t cannot be appled to obtan frequency dsperson nor even to obtan (;,,). The value for the latter quantty found from the perturbaton treatment s smply the statc zeroth-order 2 multpled by 2/3. An analogous treatment has been developed for the next hgher-order nonvanshng terms of each type n the BK perturbaton expanson. or ths purpose the statc electronc propertes are replaced by ther statc frst-order ZPVA correctons. 18 In evaluatng these correctons one must take nto account the fact that the vbratonal wave functons are feld-dependent. Relaxaton.e., change due to the feld of the vbratonal wave functons s caused only n part by relaxaton of the nuclear geometry. or ths, and for hstorcal reasons, the terms that replace,, are denoted by c-zpva, c-zpva, c-zpva, where C refers to the curvature of the feld-dependent vbratonal potental. The key pont s that the C-ZPVA terms consttute the next hgher-order nonvanshng perturbaton contrbutons of each type to the statc and nfnte optcal frequency vbratonal hyperpolarzabltes. kewse, usng the second-order ZPVA leads to the next succeedng hgher-order nonvanshng perturbaton terms of each type, and so forth, untl the entre perturbaton seres s accounted for. 19 As already noted the approach does not take nto account frequency dsperson. There have been some studes of NR.e.,,, n small molecules 20,21 showng that dsperson s relatvely small whenever ths contrbuton s mportant. Smlar nvestgatons have not been conducted, however, for conjugated organc molecules of the sort that are of nterest as NO materals. That s one purpose of ths paper. The other s to present a convenent analytc method for computng the frequency-dependence usng so-called feld-nduced coordnates ICs. We note here that an alternatve scheme for generalzng the procedure to accomplsh the same purpose has prevously been proposed 22 but t s more dffcult to mplement and does not apply to all NO propertes. It turns out that the 3N6 normal coordnates employed n the perturbaton treatment of vbratonal NO propertes are more than needed to determne the NR terms. Wthn the nfnte optcal frequency approxmaton t has been demonstrated 14,23 analytcally and numercally that a small set of statc ICs, whch reman fxed n number regardless of molecular sze, s able to duplcate exactly the result obtaned wth all 3N6 normal modes. 24 These coordnates can be used drectly to smplfy and evaluate the perturbaton theory formulas for the statc and nfnte optcal frequency NR hyperpolarzabltes. Other statc ICs can be used to advantage to determne the correspondng ZPVA and C-ZPVA hyperpolarzabltes 25 whle stll others are readly developed for effcent calculaton of vbratonal effects n a wde varety of electrcal, magnetc and spectroscopc propertes. 26 In the next secton a small set of dynamc or frequencydependent ICs.e., D-ICs s defned whch proved to be suffcent to gve the exact frequency-dependent NR hyperpolarzablty for any process. General formulas for the hyperpolarzabltes n terms of the D-ICs are presented. or the most common NO propertes the statc ICs used prevously wll occur as a subset of the dynamc ICs. However, there are other nterestng cases where that s not so. An example s sum-frequency generaton SG nvolvng the second-order NO response.e., frst hyperpolarzablty as appled ether to chral molecules 27,28 or to surface vbratonal spectroscopy wth one fxed frequency n the UV doman and one tunable nfrared frequency. 29 Secton III contans test calculatons of the most common NO propertes for three -conjugated molecules spannng a range of polarty and valence bond-charge transfer characterstcs. Our frequency-dependent results allow us to make an ntal assessment of the condtons lkely to yeld sgnfcant errors when the nfnte optcal frequency approxmaton no dsperson s appled. requency dsperson s ntroduced approxmately even f only statc ICs are employed n the frequency-dependent BK perturbaton theory expressons. Numercal tests reveal that ths procedure s not satsfactory and, therefore, that D-ICs are requred. In addton to the complete set of D-ICs we have found a reduced set that leads to only a small loss n accuracy outsde the IR regon and, further, that neglectng anharmoncty contrbutons s a good approxmaton. Ths reduced set s exact for the electro-optc Pockels and Kerr effects though not for other NO propertes. nally, n Sec. IV we revew our conclusons and present some plans for future work. II. REQUENCY DEPENDENT IED INDUCED COORDINATES Under the nfluence of a unform statc electrc feld,, the equlbrum geometry of a molecule wll relax to a new feld-dependent equlbrum poston. We call ths change n geometry nuclear relaxaton NR. The value of the th feldfree normal coordnate nduced by NR may be wrtten as a power seres n the feld, x,y,z Q x, y, z. Q x,y,z 2 Q 1 2, Expanson of the potental energy as a double power seres n terms of the normal coordnates and, followed by energy mnmzaton, 1,11 leads to expressons for the dervatves of Q that appear n Eq. 1, Q q 1,, 1 2

3 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Dynamc vbratonal nonlnear optcal propertes where 2 Q q, 2 2 j1, j...,,,... a nm 3N6 j, a 21 a 20 q 1, 3N6 jk 3a j,k1 30 q j, 2a 1 q k, 1, 3 20 n!m! 1 nm VQ 1,...,Q 3N6, x, y, z Q Q j q, 1 a, 11,, Q0,0 q, 2 a, 12, 6 and the sums over 3N6 normal coordnates reduce to 3N 5 for lnear molecules. The coeffcent a 10 s zero, a 20 s equal to 1 2 2, wth beng the harmonc vbratonal frequency, and the a nm wth n1 and m0 are harmonc electrcal property frst dervatves. or n2 and m0 the a nm are mechancal anharmoncty parameters; for n1 and m0 these coeffcents characterze the electrcal anharmoncty. rom Eqs. 2 and 3 the statc lnear and quadratc ICs are defned by 23 3N N N6 1 Q 3N6 Q 1 2 Q Q q 2, j1 q 1, Q, 3N6 j, a 21 j, q a N6 jk 3a j,k1 30 q j, 2a 1 q 1 k,q Whereas the frst-order ICs ( 1 ) depend only on harmonc a nm, the second-order ICs ( 2 ) depend also on the anharmoncty parameters a 21 and a 30. The latter are far more dffcult to compute than the correspondng harmonc parameters a 11 and a 20 snce they nvolve hgher-order dfferentaton and occur n greater number. Removng the anharmonc terms from Eq. 8 one can defne the harmonc second-order statc IC, 3N6 2,har N6 Q q harq, 2 Q. 1 As mentoned n the Introducton the statc ICs are advantageous over normal modes because they reduce the number of coordnates needed to calculate the NR contrbuton to electrc, magnetc and spectroscopc propertes 19 n the nfnte optcal frequency approxmaton 12 and n the 9 statc case. or nstance, the approxmate NR contrbuton to the dagonal elements of the IDRI tensor contans terms nvolvng all 3N6 normal coordnates, 10,12 but when statc ICs are used nstead only one vbratonal coordnate ( 2,har ) for each Cartesan drecton s needed, 23,24.e., 3N6 ;,, 2 1 Q 2 2,har 2 2 2,har 2 Q 2 har har. 10 Note that Eq. 10 s exact. In addton, the statc ICs gven by Eqs. 7 9 permt a smlar smplfcaton for all NO processes that nvolve at least one statc feld such as (;,0,0). The statc ICs yeld exact results only n the statc case and n the nfnte optcal frequency approxmaton. It s desrable to have frequency-dependent ICs D-ICs that can be used when ths approxmaton s not suffcent to determne ( ; 1 ), ( ; 1, 2 ), and ( ; 1, 2, 3 ) for arbtrary frequences. To that end we start wth the expressons for the exact frequencydependent NR contrbuton to the dynamc hyperpolarzabltes as derved by BK. 6 Thus, for the lnear polarzablty, 3N6 ; 1 P 2 1 where, a, 11 2, q 1, 3N6 1 2 P 1 a,, 11 a 11 2a 20 2 a, 11 q, 1,, and P ndcates a sum over both permutatons of the, ndces and. Snce q 1, reduces to q, 1 when 0 ths suggests that Eq. 7 for the frst-order statc IC can be generalzed to the frequency-dependent form, 3N6 1, 1 q 1, Q. 13 Ths defnes the frst-order D-IC. Smlarly, one can replace q, 1 n Eq. 2 wth q, 1, and, smultaneously, wth to obtan the frequency-dependent lnear nuclear relaxaton per unt feld, Q q, 1,. 14 It s easy to demonstrate that (;) can be wrtten exactly n terms of 1, and 1, wthout nvokng any other coordnates. The frst step s to construct two separate sets of vbratonal coordnates, whch are lnear combnatons of the feld-free normal coordnates, such that 1

4 4476 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 us, Duran, and Krtman 1, ( 1 1, ) and 2, 3,, 3N6 ( 2, 3,, 3N6 ) s the orthogonal complement of 1 ( 1 ). Hence, 3N6 j1 M, j Q j, 15 where M, s an orthogonal matrx and M, 1 j q j, 1,. There s, of course, an exactly analogous transformaton matrx M, that determnes the coordnate set 1. Note that the same transformaton as n Eq. 15 connects the lnear relaxaton of the coordnates wth the lnear relaxaton of the normal coordnates,.e., 3N6 j1 M, j Q j. Thus, by applyng the chan rule to determne 1 /, 3N6 3N6 M, 1 M, 1 M Q, , 17 and, from the propertes of an orthogonal matrx, 3N6 M j M 1 0 for j1. 18 nally, usng Eqs and the chan rule n the expresson for (;).e., Eq. 11 we obtan 3N6 ; 1 P 2 3N6 1 2 P 3N6 1 2 P j 1 2 P 1, a,, 11 q 1, Q j 1 Q 3N6. M,, j M 1 19 The above relaton demonstrates that for (;) a sum over two approprately chosen D-ICs s completely equvalent to the usual sum over all 3N6 normal coordnates. It should be clear that for the statc case ths s exactly the same result as found n Ref. 23. ollowng a smlar procedure t s easy to show that for ( ; 1, 2 ) three ICs are suffcent to reproduce the 3N6 normal coordnate result. As n the prevous demonstraton we begn wth the BK 3N6 normal coordnate equaton, ; 1, 2 P 3N6 a,, 12 q 1,2 3N6 a j,, j, 21 q 1,1 q 1,2 j 3N6 a jk, j, k, 30 q 1, q 1,1 q 1,2, jk 20 where P ndcates a sum over all sx permutatons of the pars of ndces /, 1/, and 2/. But, n ths nstance, ntroduce three sets of vbratonal coordnates defned ether by 1 1, or 1 1,1 or 1 1,2 and ther orthogonal complements. or nstance, the second term on the rhs of Eq. 20 leads to P j 3N6 a j,, j, 21 q 1,1 q 1,2 3N6 1 P 2 j 3N6 1 P 2 kl 3N6 j 2 Q Q j, M 2, lj M 2 1 j Q 1 Q j 2 2 3N6, k l M 1, k M P 2 1,1 1, Applyng the same procedure to the other two terms of ( ; 1, 2 ) one obtans 24 ; 1, 2 P ,2 2 1,1 1, V , 1,1 1, Here. Snce 1 / vanshes n the lmt t s easy to verfy that Eq. 22 reduces to the expresson gven n an earler work 23 for zzz (;,0) and for zzz (0;0,0). or ( ; 1, 2, 3 ) we requre second-order D-ICs. These are obtaned, as n the frst-order case, by generalzng the statc IC expresson. Thus, from Eq. 8, where 2, 1 1 3N6 2 Q Q, 23

5 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Dynamc vbratonal nonlnear optcal propertes Q, 2 1 q 2, 1,a q 2, 1,a a N6 j1 3N6 3 j,k1 j, q 1, j, 2a 2l j, 2a q 1, 1 jk a 30 2a k, q 1,1, 24 Proceedng as before we now need to ntroduce ten sets of D-ICs assumng all frequences are dfferent based on 1,, 1,1, 1,2, 1,3, 2, 1, 2, 2, 2, 3, 2,1 2, 2,1 3, and 2,2 3. Substtutng nto the formula for ( ; 1, 2, 3 ) whch can be wrtten from the square bracket BK terms gven n Ref. 20 yelds 24 ; 1, 2, 3 P 3N6 a 13 2, j,, q 1,3 3N6 a j,, j, 21 q 1,1 q 2,2 3 3a 30 jk, j, k, q 1, q 1,1 q 2,2 3 3N6, a 40 j,k,l 9 2 3N6, j,k,l,m P 1 6, a 12 2 a, 2, 2 3, j,k 3N6 a 22, j 3N6 a jk,, j, k, 31 q 1,1 q 1,2 q 1,3 3N6, j,k jkl, j, k, l, q 1, q 1,1 q 1,2 q 1,3 3N6 a jk klm 30 a 30 kk 1 2 q, 1, 1, ,2 3 2a j, jk, 21 a 21 jj 2 2 q, 1, 2, j,k,l ,1 2, , 3 V 1 j,, j, q 1,2 q 1,3 k, q 1,3 6a jk kl, 30 a 21 kk 3 2 q, 1, 1 j, l, m, q 1,1 q 1,2 q 1,3 j, l, q 1,2 q 1, ,2 1, , ,1 2, ,2 1, V , 1,1 1,2 1, Snce 1 / vanshes n the lmt ; the harmonc and anharmonc parts of 2 2 / j vansh n the lmt j ; and the anharmonc part of 2 2 / j vanshes f and/or j ; ts easy to verfy that ths general expresson reduces to what has been found for the specal cases consdered prevously. 23 Equatons 19, 22, and 26 are the key results of our treatment. In the study of NO materals one s often nterested n only one or, perhaps, a few components of the hyperpolarzablty tensor. In that event the number of unque D-ICs s reduced. or example, the longtudnal component of ( ; 1, 2, 3 ), for whch, s completely determned by seven D-ICs: 1,, 1,1, 1,2, 1,3, 2,1 2, 2,1 3, and 2,2 3 rather than ten. If, n addton, some of the optcal frequences are duplcated, then a further reducton occurs. In the case of the Kerr effect.e., 1 ; 2 3 0, for nstance, there reman only two unque frst-order D-ICs ) and two unque second-order ICs ( 1,0, 1, ( 2,0, 2, ) for a total of four. In the statc lmt only one frst-order and one second-order IC ( 1,0, 2,0 ) endure and they are the same as those obtaned n our earler treatment. 23 In fact, as we have already noted, t s straghtforward to verfy that, n the statc lmt, our key formulas for the hyperpolarzabltes are dentcal to those found prevously.

6 4478 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 us, Duran, and Krtman TABE I. Structural formula of molecules studed n ths paper. establsh the crcumstances under whch statc ICs are nsuffcent and the frequency-dependent verson s requred. III. TEST CACUATIONS The second-order D-IC defned by Eq. 24 contans two anharmonc terms whch are tedous to calculate. In the statc case these terms can be obtaned by fnte feld geometry optmzaton but that s not possble when frequency dependence s taken nto account. On the other hand, the harmonc approxmaton, 2, 1,har 1 3N N6 1 2 Q 1, q 2, Q 1 Q har 27 s suffcent to obtan most propertes ether n the statc or nfnte optcal frequency lmt wth the notable excepton of the statc. or the latter property usng the harmonc approxmaton causes errors of second-order n anharmoncty. Snce these second-order anharmoncty contrbutons are expected to fall off rapdly wth ncreasng frequency the harmonc approxmaton s an attractve possblty for frequences beyond the nfrared regon. In the next secton we wll explore that possblty along wth other test calculatons to Ths secton has several goals wth respect to the calculaton of NR propertes; to examne the lmts of the nfnte frequency approxmaton; to determne when substtuton of statc ICs nto the frequency-dependent BK perturbaton formulas s adequate to descrbe frequency dsperson; n nstances where s nadequate, to quantfy the performance of varous subsets of the complete set of D-ICs; and, n conjuncton wth that study; v to nvestgate the valdty of the harmonc approxmaton for secondorder ICs. or ntal testng purposes ab nto RH/6-31G calculatons were carred out on see Table I p-methylpyrdone I, 1-amno-4-ntrobuta-1,3-dene II, and hexatrene III. These three organc molecules are each representatve of a dfferent class of compound n terms of polarty and valence bond-charge transfer VB-CT characterstcs; 30 I s polar wth a domnant CT ground state; II s polar wth a domnant VB ground state; and III s nonpolar. The nfnte optcal frequency approxmaton has been found to perform satsfactorly for small molecules by Bshop and Dalskov 20 as well as by Qunet and Champagne. 21 However the behavor of ths approxmaton for medum sze organc molecules has not prevously been tested. In Tables II IV we present results for the frequencydependent NR contrbuton to zz (;), zzz (;,0), zzz (2;,), zzzz (;,0,0), zzzz (2;,,0), zzzz (3;,,), and zzzz (;,,) n molecules I III, where z s the longtudnal drecton assocated wth the largest component of the hyperpolarzablty tensor. In our calculatons we took z to be along the prncpal axs assocated wth the largest rotatonal constant. or comparson purposes we also gve the statc electronc property value. The propertes n Tables II IV were evaluated at the Hartree ock 6-31G H/6-31G level usng the dervatve expressons of Eqs. 19, 20, and 26. Although a hgher level treatment s needed for accurate results, the calculatons TABE II. Exact frequency-dependent NR hyperpolarzabltes zz (;), zzz (;,0), and zzz (2;,) calculated at the H/6-31G level usng BK perturbaton theory. Molecule I s p-methylpyrdone; II s 1-amno-4-ntrobuta-1,3-dene; and III s hexatrene. The percentage error of the nfnte optcal frequency NR values wth respect to the sum of the electronc and nfnte optcal frequency NR values are gven n parentheses. Those cases where the error exceeds 5% are blocked off wth dashed lnes. All quanttes are n a.u.

7 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Dynamc vbratonal nonlnear optcal propertes 4479 TABE III. Exact frequency-dependent NR hyperpolarzabltes zzzz (;,0,0) and zzzz (2;,,0) calculated at the H/6-31G level usng BK perturbaton theory. Molecule I s p-methylpyrdone; II s 1-amno-4-ntrobuta-1,3-dene; and III s hexatrene. The percentage error of the nfnte optcal frequency NR values wth respect to the sum of the electronc and nfnte optcal frequency NR values are gven n parentheses. Those cases where the error exceeds 5% are blocked off wth dashed lnes. All quanttes are n a.u. reported are suffcent for the exploratory purposes to whch they have been put here. or our three molecules t s feasble to use the GAUSSIAN 98 sute of programs 31 to obtan analytcal H/6-31G results for a 20, a 01, a 11, a 02, a 12, and a 03 where the vbratonal dervatves are wth respect to atomc Cartesan coordnates. These dervatves may be combned to obtan dervatves wth respect to normal coordnates or D-ICs. Then, numercal dfferentaton of a 20, a 11, a 12, and a 03 wth respect to the normal coordnates or D-ICs yelds a 30, a 21, a 22, and a 13, respectvely. a 40 and a 31 were computed by double numercal dfferentaton of a 20 and a 11. The NR contrbutons that le nsde the dashed lne area of the tables represent those cases where there s a sgnfcant dfference between the frequency-dependent NR contrbuton and the nfnte frequency approxmaton. As a measure of sgnfcance we have chosen to compare ths dfference to the sum of the statc electronc property value and the nfnte frequency NR contrbuton. The crteron for sgnfcance was arbtrarly set at 5%. We note that the pecular behavor n the tables between 0.00 and 0.02 a.u. arses because these frequences le n the vbratonal absorpton regon. or those propertes that have a vanshng NR contrbuton under the nfnte frequency approxmaton,.e., zz (;), zzz (2;,), and zzzz (3;,,), the frequency-dependent NR term always tends rapdly to zero as ncreases. Tables II IV show that for these propertes the error s nsgnfcant when 0.04 a.u. The only excepton s zzz (3;,,) of molecule I, whch has a small statc electronc value and, n that case, the error becomes nsgnfcant when 0.08 a.u. On the other hand, our results for zzz (;,0), zzzz (;,0,0), zzzz (2;;;0), and zzzz (;,,) ndcate, that for these propertes, the nfnte frequency approxmaton can not be systematcally appled wthout sgnfcant error when the optcal frequency s lower than 0.06 a.u. ( cm 1 ). Ths s consstent wth the theoretcal assumpton underpnnng the nfnte frequency approxmaton that ( v /) 2, wth v a fundamental vbratonal frequency, s neglgble compared to unty. There s a way to ntroduce frequency-dependence usng only the statc ICs and that s to substtute these ICs for the normal coordnates drectly nto the usual frequencydependent perturbaton formulas. or the polarzablty and frst hyperpolarzablty only the frst-order statc IC s utlzed. or the second hyperpolarzablty the second-order IC s requred as well. In that case there are two possbl- TABE IV. Exact frequency-dependent NR hyperpolarzabltes zzzz (3;,,) and zzzz (;,,) calculated at the H/6-31G level usng BK perturbaton theory. Molecule I s p-methylpyrdone; II s 1-amno-4-ntrobuta-1,3-dene; and III s hexatrene. The percentage error of the nfnte optcal frequency NR values wth respect to the sum of the electronc and nfnte optcal frequency NR values are gven n parentheses. Those cases where the error exceeds 5% are blocked off wth dashed lnes. All quanttes are n a.u.

8 4480 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 us, Duran, and Krtman TABE V. requency-dependent NR hyperpolarzabltes zz (;), zzz (,,0), and zzz (2;,) calculated at the H/6-31G level usng theory and statc ICs. Molecule I s p-methylpyrdone; II s 1-amno-4-ntrobuta-1,3-dene; and III s hexatrene. The percentage error wth respect to the sum of the electronc and exact dynamc NR values are gven n parentheses. Those cases where the error exceeds 5% are blocked off wth dashed lnes. All quanttes are n a.u. tes. The complete second-order statc IC yelds the exact statc NR second hyperpolarzablty but, for optcal processes, the harmonc IC s necessary to obtan the exact soluton n the nfnte frequency lmt. Thus, the frequencydependent NR propertes presented n Tables V VII were calculated utlzng harmonc statc ICs. Of course, these results are approxmate for all optcal processes at fnte, nonzero frequences. As n Tables II IV, those results contanng a sgnfcant 5% error are marked off by dashed lnes. In ths case the percent error was calculated by comparng to the sum of the statc electronc property and the exact NR contrbuton. Although substtuton of statc ICs nto the perturbaton expressons often mproves upon the nfnte frequency approxmaton, the extent of that mprovement depends consderably on the property and the molecule. Errors are reduced between 28% for zzzz (2;,,0) at 0.02 a.u..e., from 29.3% to 21.2% and 68% for zz (;) at 0.12 a.u. 0.10% to 0.03% n the case of molecule II, whereas the reducton s between 2.3% for zz (;) at 0.02 a.u % to 0.168% and 35.2% for zzzz (3;,,) at 0.12 a.u. 0.17% to 0.11% n the case of molecule III. or molecule I the error n zzzz (;,0,0) and zzzz (2;,,0) actually ncreases, although reductons up to 54% are acheved for the remanng propertes. Snce the mprovement s lmted, the above method s not that attractve over the nfnte frequency approxmaton, especally when one consders the necessty of determnng hgher-order dervatves. In order to calculate the dynamc NR contrbuton to the second hyperpolarzablty t s necessary to determne fourth dervatves of the energy wth respect to vbratonal coordnates (a 40 ) and thrd dervatves of the dpole moment (a 31 ), nether of whch occur n the nfnte optcal frequency approxmaton for the same quanttes. If we use D-ICs one stll has to determne the hgher-order dervatves but, then, the calculated results are exact as long as a complete set s employed cf. Sec. II. In connecton wth D-ICs an nterestng queston s whether suffcent accuracy can be acheved wth less than a complete set. In order to address that pont we focus on the most problematc case, namely on the hyperpolarzabltes of molecule I at 0.02 and In addton, we also carred out calculatons at a.u. (1097 cm 1 ) n order to analyze the performance of the D-ICs wthn the IR regon. Apart from usng just a subset of the second-order D-ICs we also consdered only the harmonc approxmaton to the second-order D-ICs for the second hyperpolarzabltes. Ths consderably smplfes the calculatons because, then, all numercal dfferentatons are done wth respect to the D-ICs. Otherwse, a 20, a 11, and a 12 must be dfferentated wth respect to the 3N6 normal coordnates. Our results for several subsets are gven n the Tables TABE VI. requency-dependent NR hyperpolarzabltes, zzzz (;,0,0) and zzzz (2;,,0) calculated at the H/6-31G level usng BK perturbaton theory and statc ICs. Molecule I s p-methylpyrdone; II s 1-amno-4-ntrobuta-1,3-dene; and III s hexatrene. The percentage error wth respect to the sum of the electronc and exact dynamc NR values are gven n parentheses. Those cases where the error exceeds 5% are blocked off wth dashed lnes. All quanttes are n a.u.

9 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Dynamc vbratonal nonlnear optcal propertes 4481 TABE VII. requency-dependent NR hyperpolarzabltes, zzzz (3;,,) and zzzz (;,,) calculated at the H/6-31G level usng BK perturbaton theory and statc ICs. Molecule I s p-methylpyrdone; II s 1-amno-4-ntrobuta-1,3-dene; and III s hexatrene. The percentage error wth respect to the sum of the electronc and exact dynamc NR values are gven n parentheses. Those cases where the error exceeds 5% are blocked off wth dashed lnes. All quanttes are n a.u. VIII and IX. We lst the D-ICs employed n the form ( 1, 1,.../ 2, 2,...) where the frst-order ICs are on the left of the dagonal lne whle the second-order ICs are on the rght. or zzz (;,0) and zzzz (;,0,0) the sets, 0/, 0 and 2,, 0/2,, 0 are both complete. Indeed, for ether set our value of the frst hyperpolarzablty agrees wth the exact result at all three frequences as t should. The second hyperpolarzablty, however, does not agree; the dfference s due to the harmonc approxmaton for the second-order ICs. A smlar analyss pertans to zzz (2;,), zzzz (2;,,0), and zzzz (;,,). In ths nstance, although the set 2,, 0/2,, 0 s complete, 0/, 0 s not. However, for all three propertes there are smaller subsets not lsted here that are complete. The errors n the second hyperpolarzabltes that occur for the complete set agan arse from the harmonc approxmaton for the second-order ICs. or zz (;) all of the D-IC sets chosen produce the exact value because they are all complete and we have, therefore, refraned from ncludng ths property n the table. The most mportant result that emerges from Tables VIII and IX s that outsde the IR regon the set, 0/, 0 s suffcent for all processes. Ths s true even n cases where ths set s not complete and regardless of the fact that the harmonc approxmaton s appled for the second-order ICs. The frst hyperpolarzabltes cannot be affected by the harmonc approxmaton. That s not true of the second hyperpolarzabltes but, n practce, zzzz (;,0,0) s not sgnfcantly altered even wthn the IR regon. IV. CONCUSIONS AND UTURE WORK We have demonstrated that ( ; 1 ), ( ; 1, 2 ), and ( ; 1, 2, 3 ) can be expressed exactly n terms of frequency-dependent feldnduced vbratonal coordnates called D-ICs. The man advantage of the D-ICs, as opposed to normal coordnates, s that they are few n number and the sze of the set remans the same regardless of the number of atoms n the molecule. In prevous work the hyperpolarzabltes lsted above TABE VIII. requency-dependent NR hyperpolarzabltes of zzz (;,0), zzz (2;;) and zzzz (;,0,0) of p-methylpyrdone calculated at the H/6-31G level usng BK perturbaton theory and varous sets of harmonc TD-ICs. The exact result s ncluded for sake of comparson. The percentage error wth respect to the sum of the electronc and exact dynamc NR values are gven n parentheses. Those cases where the error exceeds 5% are blocked off wth dashed lnes. All quanttes are n a.u.

10 4482 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 us, Duran, and Krtman TABE IX. requency-dependent NR hyperpolarzabltes of zzzz (2;,,0), zzzz (3;;,), and zzzz (;,,) of p-methylpyrdone calculated at the H/6-31G level usng BK perturbaton theory and varous sets of harmonc TD-ICs. The exact result s ncluded for sake of comparson. The percentage error wth respect to the sum of the electronc and exact dynamc NR values are gven n parentheses. Those cases where the error exceeds 5% are blocked off wth dashed lnes. All quanttes are n a.u. were evaluated for the most common cases usng statc ICs and the nfnte optcal frequency approxmaton. Tests of that approxmaton for fnte frequences are carred out here at the H/6-31G level for three medum sze conjugated organc molecules that are ether nonpolar, or polar wth a covalent ground state, or polar wth a charge transfer ground state. At frequences smaller than 0.06 a.u cm 1 or 1.63 ev errors n the nfnte frequency approxmaton may become sgnfcant for the longtudnal hyperpolarzabltes. A possble alternatve to the nfnte-frequency approxmaton s to combne statc ICs wth the BK dynamc perturbaton formulas. Although ths procedure leads to some overall mprovement, the extent of that mprovement f any depends upon the molecule and the property. On the contrary, when approprately chosen D-ICs are combned wth perturbaton theory the result s exact at all frequences. urthermore, for the -conjugated organc molecules studed z z n ths paper the reduced set consstng of 1,0, 1, plus zz zz the harmonc 2,0 and 2, yelds accurate values outsde the IR regon. Ths reduced set s complete for zzz (;,0) and zzzz (;,0,0). The harmonc approxmaton s exact for frst hyperpolarzabltes and yelds an nsgnfcant error for zzzz (;,0,0) even wthn the IR regon. Of course, more molecules need to be examned before our conclusons based on the computatons done here can be consdered general. In addton, the effect of the bass set and the electron correlaton must be nvestgated. We also plan to study other propertes of nterest such as sum-frequency generaton. ACKNOWEDGMENT Support for ths work under Grant No. PB C02-01 from the Dreccón General de Enseñanza Superor e Investgacón Centífca y Técnca MEC-Span s acknowledged. 1 B. Champagne and B. Krtman, n Handbook of Advanced Electronc and Photonc Materals, edted by H. S. Nalwa Academc, San Dego, D. M. Bshop and P. Norman, n Handbook of Advanced Electronc and Photonc Materals, edted by H. S. Nalwa Academc, San Dego, M. C. Magnon, P. Mondn, M. Del Zoppo, C. Castglon, and G. Zerb, J. Chem. Soc., Perkn Trans. 2 2, P. Macak, Y. uo, P. Norman, and H. Agren, J. Chem. Phys. 113, B. Krtman and D. M. Bshop, Chem. Phys. ett. 175, D. M. Bshop and B. Krtman, J. Chem. Phys. 95, D. M. Bshop and B. Krtman, J. Chem. Phys. 97, D. M. Bshop, B. Krtman, and B. Champagne, J. Chem. Phys. 107, D. M. Bshop, J. M. us, and B. Krtman, J. Chem. Phys. 108, J. M. us, J. Martí, M. Duran, J.. Andrés, and B. Krtman, J. Chem. Phys. 108, J. M. us, M. Duran, and J.. Andrés, J. Chem. Phys. 107, D. M. Bshop, M. Hasan, and B. Krtman, J. Chem. Phys. 103, J. Martí and D. M. Bshop, J. Chem. Phys. 99, J. M. us, M. Duran, J.. Andrés, B. Champagne, and B. Krtman, J. Chem. Phys. 111, B. Champagne, J. M. us, M. Duran, J.. Andrés, and B. Krtman, J. Chem. Phys. 112, E. Camm, B. Mennucc, and J. Tomas, J. Am. Chem. Soc. 120, P. Otto, A. Martnez, and J. adk, J. Chem. Phys. 111, B. Krtman, J. M. us, and D. M. Bshop, J. Chem. Phys. 108, B. Krtman, B. Champagne, and J. M. us, J. Comput. Chem. 21, D. M. Bshop and E. K. Dalskov, J. Chem. Phys. 104, O. Qunet and B. Champagne, J. Chem. Phys. 109, D. M. Bshop, and B. Krtman, J. Chem. Phys. 109, J. M. us, M. Duran, B. Champagne, and B. Krtman, J. Chem. Phys. 113, A sgn error n Eqs. 12, 13, 18, 19, 22, 23, and 25 n Ref. 23 has been corrected. 25 J. M. us, B. Champagne, and B. Krtman, Int. J. Quantum Chem. 80, B. Krtman, B. Champagne, and J. M. us, J. Comput. Chem. 21, 1572

11 J. Chem. Phys., Vol. 115, No. 10, 8 September 2001 Dynamc vbratonal nonlnear optcal propertes P. scher, D. S. Wersma, R. Rghn, B. Champagne, and A. D. Buckngham, Phys. Rev. ett. 85, B. Champagne, P. scher, and A. D. Buckngham, Chem. Phys. ett. 331, O. Qunet and B. Champagne, Int. J. Quantum. Chem. to be publshed. 30 D. M. Bshop, B. Champagne, and B. Krtman, J. Chem. Phys. 109, M. J. rsch, G. W. Trucks, H. B. Schlegel et al., GAUSSIAN 98, Revson A.6 Gaussan, Inc., Pttsburgh, 1998.