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Linear and nonlinear electric properties and their dependence onthe conformation and intramolecular H-bonding: A model study

Aggelos Avramopoulos a, Mirosław Jabłonski b,Manthos G. Papadopoulos a,*, Andrzej J. Sadlej b

a Institute of Organic and Pharmaceutical Chemistry, National Hellenic Research Foundation, 48, Vas. Constantinou, EL-116 35 Athens, Greeceb Department of Quantum Chemistry, Institute of Chemistry, Nicolaus Copernicus University, PL-87 100 Torun, Poland

Received 3 April 2006; accepted 7 June 2006Available online 21 June 2006

Abstract

The 3-aminoacroleine molecule is used as a model to study the dependence of linear and nonlinear molecular electric properties on themolecular conformation. Among four conformers of this molecule one of them has the intramolecular hydrogen bond, and thus, permitsthe study of its effect on electric properties. It has been established on the basis of MP2 calculations that the most elongated conformer of3-aminoacroleine has the largest values of the dipole moment, dipole polarizability, and first hyperpolarizability. They were found tosignificantly depend on the conformation of the molecule. Also the electron correlation effect on the first hyperpolarizability is foundto be very large. The results calculated for four conformers permit also to estimate the effect of the intramolecular hydrogen bond.The incremental lowering of the dipole polarizability is negligible whereas that for the first hyperpolarizability is of the order of�120 a.u. Considerable attention has been paid to the calculation of vibration corrections to electric properties. For the dipole polariz-ability and first hyperpolarizability the so-called pure vibrational corrections are found to be unusually large. These large values arisefrom the use of the harmonic zeroth-order approximation in the perturbation treatment of contributions due low-frequency modes.It is concluded that the perturbation method for the evaluation of pure vibrational corrections to electric properties should be used withgreat care and possibly preceded by the analysis of low frequency modes.� 2006 Elsevier B.V. All rights reserved.

1. Introduction

The elongated chains of multiple bonds with donor–acceptor end groups are known to exhibit significant elec-trooptical nonlinearities [1,2]. Over the past years aremarkable effort has been invested in theoretical studiesof these systems and particular attention has been givento calculations of their first hyperpolarizability (b). Itsvalue determines the outcome of the dc–Pockels effectand second-harmonic generation [3]. Molecules of large bare needed for the development of molecular electronics.Not surprisingly the design and synthesis of efficient pho-

ton manipulating materials became the subject of consider-able interest (see e.g., Refs. [4–6], and references therein).

Although the relation between molecular electronicstructure and large values of b in conjugated donor–accep-tor (push–pull) [7] systems is well understood and can beelucidated in terms of simple few-states models [7–11] muchless appears to be known about geometrical factors. Usu-ally only the most elongated molecular conformations orconfigurations are investigated. However, the elongatedacene-like chains [12] exhibit a high level of flexibility andmay exist in several structural forms. Moreover, the pres-ence of different polar substituents may lead to importantintramolecular interactions. The role of these structuralfactors in the enhancement or reduction of molecular elec-tric properties is investigated in this paper for a model sys-tem of 3-aminoacroleine (3-aminopropenal). This model isnot as artificial as it may appear. Different structural forms

0301-0104/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2006.06.005

* Corresponding author.E-mail addresses: aavram@eie.gr (A. Avramopoulos), teojab@chem.

uni.torun.pl (M. Jabłonski), mpapad@eie.gr (M.G. Papadopoulos),teoajs@chem.uni.torun.pl (A.J. Sadlej).

www.elsevier.com/locate/chemphys

Chemical Physics 328 (2006) 33–44

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of the substituted 3-aminoacroleine, in particular the so-called enaminoketones [13,14] are known and have beenwidely investigated [15–17].

The conformation dependence of electric properties hasbeen earlier investigated for long conjugated chains withdonor/acceptor end groups (see e.g., [12]). However, themain attention has been given to the dependence of theirelectric properties on the chain length rather than to theelectric properties and conformation of the fundamentalstructural units. The influence of geometric factors andconformation on nonlinear optical responses has also beenstudied from the point of view of the spatial arrangementof relatively isolated absorbing groups [18]. In this respectthe hydrogen bonded multimeters received particular atten-tion [19–22]. One should also mention the investigation ofthe dependence of nonlinear electric properties on theintramolecular hindered rotation [23]. However, most ofthese studies appear to have been focused on the geometryfactors which are rather different from those addressed inthis paper.

In the present study we shall consider a single molecule,3-aminoacroleine, whose conformational flexibility offersseveral spatially different structures. The 3-aminoacroleinemolecule has four (see Fig. 1) stable conformations (config-urations) which differ by the arrangement of the –CHO and–NH2 groups with respect to the CC double bond (cis andtrans forms) and by the conformation at the CC singlebond (s-cis and s-trans forms). Additionally, the so-calledZZ (cis-s-cis) form (I) features the intramolecular hydrogenbond. This diversity of different structural isomers makes3-aminoacroleine a very suitable candidate for the investi-gation of the dependence of its electric properties on the

spatial arrangement of the donor–acceptor groups and onthe effect of the intramolecular hydrogen bonding.

The necessary definitions and some details of ournumerical calculations are presented in Section 2. Theresults of calculations of the dipole moment (l), dipolepolarizability (a), and first hyperpolarizability are dis-cussed in Section 3. Particular attention is given to thedependence of these electric properties on the configura-tion/conformation of the unsaturated molecular skeleton.Also the effect of the intramolecular hydrogen bonding isinvestigated. The role of the conjugation and intramolec-ular charge transfer between donor and acceptor groups isdiscussed by comparing several systems in which thedonor and/or acceptor groups are either in the nearestneighbourhood or are separated by saturated or unsatu-rated linkers. In addition to the calculation of the so-called pure electronic contributions to polarizabilitiesand hyperpolarizabilities of the four isomers of 3-amino-acroleine we have also investigated the role of vibrationalcontributions. The results of our investigations are sum-marized in Section 4.

2. Computational details

The so-called static electric properties are defined interms of the expansion,

EðFÞ ¼ Eð0Þ � laF a �1

2!aabF aF b �

1

3!babcF aF bF c � � � � ;

ð1Þ

where the repeated Greek indices imply the summationover all cartesian components. The E(F) term is theBorn–Oppenheimer (BO) energy of a molecule in the exter-nal electric field F with the zero-field energy E(0) as a ref-erence. The coefficients of the power series are the dipolemoment vector (la) and polarizability (aab) and first hyper-polarizability (babc) cartesian tensors. Actually, all of themdepend on molecular geometry parameters and should berather referred to as the corresponding molecular propertysurfaces. For a particular set of geometry parameters whichusually corresponds to the local minimum of the BO hyper-surface, the components of the property tensors are re-ferred to as their pure electronic values.

In the time-dependent harmonic electric field, Fa = Fa(t),of frequency x, the static polarizability and hyperpolariz-ability tensors in Eq. (1) are replaced by their so-calleddynamic (frequency-dependent) counterparts, e.g., aab(x).In the case of the frequency-dependent first hyperpolariz-ability the x-dependence involves two, in general differentfrequencies, x1 and x2, whose particular combinationdefines the optical process of interest [3]. For the second-harmonic generation process x1 = x2 = x and the corre-sponding x-dependent first hyperpolarizability is denotedby babc(�2x;x,x) [3]. For the purpose of presentationthe dipole polarizability is usually reported as the isotropicrotational average,

Fig. 1. 6-31 G**/MP2 optimized planar structures of the four conformersof 3-aminoacroleine. These conformers are usually referred to by thefollowing terms: conformer I – ZZ or cis-s-cis, conformer II – ZE or cis-s-trans, conformer III – trans-s-cis, and conformer IV – EE or trans-s-trans.All planar conformers lie in the xz plane with the z axis runninghorizontally to the right. For fully optimized structures conformers II–IVshow some small deviation from nonplanarity.

34 A. Avramopoulos et al. / Chemical Physics 328 (2006) 33–44

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a ¼ 1

3aaa: ð2Þ

The b tensor is conveniently expressed in terms of its vectorcomponent (bv),

bv ¼3

5bzaa; ð3Þ

where the direction of the vector axis (z) is collinear withthe dipole moment axis.

The geometry dependence of molecular properties leads(in the rotationless approximation) to vibrational correc-tions Pv to pure electronic values (Pel) of each propertyP. They arise from the electric field dependence of theBO hypersurface. The total value of P in the given elec-tronic state and its vibrational ground state can be in gen-eral represented as:

P ¼ P el þ P v ¼ P el þ P zpva þ P pv; ð4Þwhere Pel is the pure electronic, Pzpva is termed the zero-point vibrational average and Ppv is referred to as the purevibrational correction. This partition depends on the choiceof the reference geometry which determines the value ofPel. Usually Pel is evaluated at the minimum of the BOhypersurface whose harmonic approximation for thefield-independent problem is chosen to define the zeroth-order perturbation equation (see however Ref. [24]). Withthe BO energy and property hypersurfaces expandedthrough certain order in normal coordinates, the mechani-cal anharmonicities and electric harmonic and anharmonicterms are treated as a perturbation. This approach was pio-neered by Bishop and Kirtman [25–28] and is known as theBishop–Kirtman perturbation theory (BKPT).

In BKPT the lowest-order non-vanishing Pzpva correc-tion to the electric property P is given by [28]:

P zpva ¼ ½P �ð1;0Þ þ ½P �ð0;1Þ; ð5Þwhere the superscript (n, m) indicates the order of the elec-tric (n) anharmonicity in P and the order (m) of themechanical anharmonicity. Hence, Eq. (5) will use the sec-ond derivative of P with respect to normal coordinates andthe first anharmonic term in the BO energy expansion. ThePzpva contribution is usually small compared to pure elec-tronic values of P.

For systems with low frequency modes and significantelectric anharmonicities the dominant part of Pv followsfrom the pure vibrational term. Upon restricting the elec-tric and mechanical anharmonicities to the second-orderthe Ppv corrections to elements of the dipole polarizabilityand first hyperpolarizability tensors become [28]:

apv ¼ ½l2�ð0;0Þ þ ½l2�ð1;1Þ þ ½l2�ð0;2Þ þ ½l2�ð2;0Þ ð6Þand

bpv ¼ ½la�ð0;0Þ þ ½l3�ð0;1Þ þ ½l3�ð1;0Þ þ ½la�ð1;1Þ þ ½la�ð2;0Þ þ ½la�ð0;2Þ;ð7Þ

respectively. The notation used here follows that of Bishopand Kirtman with [A](n,m) meaning that the electric anhar-

monicities of the order n apply to all items in the string A

and that the mechanical anharmonicity is taken into ac-count in the mth order. For apv the strings A involve onlydipole moments whereas for bpv they contain either only di-poles or dipoles and dipole polarizabilities. The approxi-mation used in this study can be alternatively referred toby the symbol (pqrs) = (4320), where p, q, r, and s representthe orders of the normal coordinate expansion of energies,dipole moments, dipole polarizabilities, and first hyperpo-larizabilities, respectively.

The 3-aminoacroleine molecule has four conformerswhich differ by the s-cis/s-trans and cis/trans arrangementof the electron donor and acceptor groups at single anddouble CC bonds, respectively. They are displayed inFig. 1. In all calculations reported in this paper the z coor-dinate axis corresponds to the direction of the dipolemoment vector and for planar structures all conformers(isomers) are assumed to lie in the xz plane.

For the purpose of calculations of the pure electroniccontribution to l, a and b we have assumed that all fourconformers are planar. The geometry of planar conformershas been optimized at the level of the second-order Møller–Plesset approximation (MP2) with 6-31G** basis set andthen used in HF and MP2 calculations of electric proper-ties with the so-called polarized basis sets (PolX,X = H, C, N, O) [29–31]. The same approach has beenused for other molecules whose electric properties havebeen calculated in order to discuss the effect due to donorand acceptor end groups and the character of the linkerbetween them. According to the assumptions of the BKPTapproach, the calculation of vibrational contributionsrequires the full geometry optimization and refers to thetrue minima on the potential energy surface. These calcula-tions have been performed with PolX basis sets in the HFapproximation.

The choice of the SCF HF method for the calculationof vibrational corrections to electric properties of 3-aminoacroleine is obviously disputable, in particular inview of the recent findings by Torrent-Sucarrat et al.[32]. These authors have compared electronic and vibra-tional contributions to polarizabilities and hyperpolariz-abilities calculated in SCF HF and MP2 approximationswith a series of Gaussian basis sets of increasing flexibil-ity. Their results for hexatriene and its derivatives withdonor–acceptor end groups show a very strong depen-dence on the electron correlation. On passing from theSCF HF to MP2 level of approximation the so-callednuclear relaxation contribution to first hyperpolarizabilityhas been found to decrease almost as much as twice [32].However, in both approximations the nuclear relaxationcontribution remains large compared to the pure elec-tronic part of b. Hence, the qualitative conclusions ofthe present study of 3-aminoacroleine should not beaffected by using only the SCF HF data.

All geometry optimization and calculations of electricproperties have been carried out with Gaussian98 [33].The evaluation of vibrational corrections has been

A. Avramopoulos et al. / Chemical Physics 328 (2006) 33–44 35

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performed by using the codes of the CADPAC package[34]. Some limited results for the frequency dependenceof the dipole polarizability and first hyperpolarizability of3-aminoacroleine, which are also reported in this paper,follow from calculations using the DALTON set of pro-grams [35].

3. Results and discussion

3.1. Pure electronic contributions: general features

The pure electronic contributions to dipole moments,static dipole polarizabilities and static first hyperpolariz-abilities for the four (planar) conformers of 3-aminoacro-leine are listed in Table 1. These values correspond toSCF HF and MP2 calculations with PolX basis sets at 6-31G**/MP2 molecular geometries optimized under the con-straint of planarity. Upon relaxing this constraint we havefound that conformers II–IV show some deviations fromplanarity, although the N–C@C–C@O molecular skeletonremains essentially planar. The non–planarity of II–IV ismostly due to the pyramidalization of the –NH2 group.The largest deviation from planarity of the –NH2 grouphas been found for the most elongated structure IV andthis conformer is used to illustrate the magnitude of thenonplanarity effect on the calculated properties.

Table 2 displays the SCF HF results for electric proper-ties of the conformer IV evaluated with PolX basis set forplanar and fully optimized 6-31G**/MP2 geometries. It canbe seen that the nonplanarity effect for the most distortedconformer IV is rather small and does not affect the generalfeatures of the calculated properties. For conformers II andIII the deviation from nonplanarity is smaller than in thecase of conformer IV. Hence, the results for planar struc-tures can be considered as sufficiently representative forthe analysis of general features of the pure electronic con-tributions to the electric properties for all conformers of3-aminoacroleine investigated in this paper.

The presentation of our results for a and b refers to thecoordinate system whose z axis is aligned along the dipoledirection. With the CC double bond positively orientedtowards the nitrogen direction the angle between this bondand the z axis fully specifies the orientation of the moleculein the xz plane (see Fig. 1). One should note that for thegiven optimized planar structure the value and directionof the dipole moment depends on the method used for itscalculation. For the two methods (SCF HF and MP2) usedin the present study to evaluate molecular electric proper-ties the direction of the calculated dipole moment is shownin Table 3. The differences between the SCF HF and MP2values are marginal. However, for the sake of consistencyall SCF HF data refer to the z axis aligned along theSCF HF dipole moment. Similarly, the MP2 results corre-spond to the orientation of the z coordinate axis parallel tothe MP2 dipole moment.

The electron correlation effect on the magnitude (Table1) of the calculated dipole moment is not particularly largeand decreases its SCF HF value by about 10%. This is theusual magnitude of this effect in moderately polar systems.As already mentioned the electron correlation effect on thedipole moment direction is essentially negligible (Table 3).

Table 1Pure electronic contributions to dipole moments, dipole polarizabilities, and first hyperpolarizabilities of the four (I–IV) planar conformers of3-aminoacroleine

Propertyb Ia IIa IIIa IVa

HF MP2 HF MP2 HF MP2 HF MP2

lz = l 1.441 1.289 2.498 2.203 2.094 1.959 2.673 2.423axx 58.93 63.14 44.97 49.49 54.29 57.47 43.84 47.61ayy 34.17 36.26 33.88 35.94 33.65 35.74 33.75 35.74azz 59.89 68.41 80.09 85.75 68.52 79.85 84.09 93.08a 50.99 55.94 52.98 57.03 52.15 57.69 53.89 58.81bzxx �45.2 26.3 59.5 7.9 �19.8 42.7 43.9 43.6bzyy �27.6 �23.3 �20.7 �25.6 �9.8 �18.4 �14.2 �23.4bzzz 100.2 245.3 �24.9 371.4 284.4 543.8 112.7 549.5bv 16.4 148.9 8.3 252.4 152.9 340.9 85.4 341.8

The SCF HF (HF) and MP2 results are evaluated with PolX basis sets for optimized planar molecular geometries obtained from 6-31G**/MP2calculations. All data in a.u.

a See Fig. 1 for the numbering of conformers.b The cartesian components defined according to Fig. 2. For all conformers the z axis is the dipole moment axis.

Table 2The nonplanarity effect on the electric properties (electronic contributions)of conformer IV

Property Planar Nonplanar

lz = l 2.673 2.514axx 43.84 43.64ayy 33.75 34.55azz 84.08 82.56a 53.89 53.58bzxx 43.9 47.0bzyy �14.2 �14.3bzzz 112.7 104.3bv 85.4 82.2

Results of SCF HF calculations with PolX basis set obtained at planar andnonplanar 6-31**/MP2 optimized geometries. All results in a.u.

36 A. Avramopoulos et al. / Chemical Physics 328 (2006) 33–44

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Similar conclusions can be drawn concerning the electroncorrelation contribution to in-plane components of thedipole polarizability tensor. Moreover, its out-of-planecomponent shows much smaller electron correlation effectwhich amounts to about 2–3 a.u. Independently of the con-formation of 3-aminoacroleine the electron correlationcontribution to its dipole polarizability is positive. Let usalso mention that for all conformers the dipole momentdirection determines the largest component of the dipolepolarizability tensor.

The most spectacular electron correlation effect is seenfor bzzz. In the conformer with the largest spatial separa-tion of the donor and acceptor groups (conformer IV)the electron correlation increases the bzzz value by about430 a.u. On the other hand, the electron correlation effecton bzzz in the most compact system (conformer I) is muchsmaller and amounts to about 145 a.u. The choice of thedipole axis as the principal axis of the b tensor makes itsbzxx (in-plane) component relatively small in both SCFHF and MP2 approximations. The same applies to the bzyy

component. In consequence the pattern of the correlationcontribution to bzzz is also reflected in the values of theMP2 correlation contribution to bv.

The results obtained in this paper for bzzz give yetanother example of the known sensitivity of the first hyper-polarizability to the treatment of the electron correlation[19,36]. The present MP2 electron correlation contributionsto bzzz of the 3-aminoacroleine conformers are much largerthan those usually found in other systems involving thesame or similar electron donor and acceptor groups[19,36]. One should note, however, that very large (MP2)electron correlation contributions have been found in sev-eral other systems of similar electronic structure [32,37].Thus, it is quite likely that the higher-order correlationeffects may considerably change the MP2 results. On theother hand, the pattern of convergence of the electron cor-relation contribution to molecular electric properties con-siderably depends on the studied system and large MP2contributions do not necessarily mean that the higher-order corrections will be also large [20,36,38]. In the caseof comparative studies of closely related conformers of

the same molecule the use of the high-level electronic the-ory methods does not appear to be necessary.

Including the electron correlation contribution to elec-tric properties of 3-aminoacroleine is particularly impor-tant in the case of the b tensor. For the I! IIisomerization the SCF HF approximation predicts thechange in bzzz opposite to that which follows fromthe MP2 data. This indicates significant deficiencies ofthe SCF HF approximation [32] which tends to give tooexaggerated separation of charges. This deficiency of theone-electron approximation is, at least to some extent, rem-edied by including the electron correlation effects and theMP2 results are considered to be more reliable than theSCF HF data [32]. Also the electron correlation effect onbzzz is particularly large for the most elongated structureIV, i.e. for the structure with the largest spatial separationbetween the donor and acceptor group.

As shown by the dipole moment and polarizability datathe four conformers of 3-aminoacroleine have similar elec-tronic structure and should not differ very much in the exci-tation spectrum. Thus, one can argue that the higher-ordercorrelation effects are not expected to bring significantchanges in the relative values of bzzz of the four conformersof 3-aminoacroleine. The value of bzzz in different conform-ers is mostly governed by the differences between theground and excited state dipole moments and by the vari-ation of the transition dipoles (see Section 3.2). Since thepresent investigations are primarily concerned with theconformation dependence of the calculated electric proper-ties we shall rather focus on their relative values for differ-ent conformers. Most of this discussion will be based onthe MP2 data of Table 1.

3.2. Pure electronic contributions: conformation and

intramolecular hydrogen bonding

The change of the conformation (configuration) of3-aminoacroleine modifies the geometry of the conjugationpath represented by the N–C@C–C@O skeleton. It alsochanges the mutual direct distance between the two endgroups. This change in the charge separation will affectthe permanent dipoles, transition dipoles, and in conse-quence, polarizabilities and hyperpolarizabilities. More-over, a unique situation occurs in the case of conformer Ibecause of the intramolecular hydrogen bond formedbetween the electron donor and acceptor groups and it willbe interesting to estimate the magnitude of its effect on thecalculated electric properties.

The estimation of the contribution due to intramolecu-lar hydrogen bonding must assume that there is certainadditivity of the conformation and hydrogen bondingeffects. Obviously such an assumption is highly approxi-mate and prone to criticism as regards the numerical valuesof the conformation/H-bonding increments. On the otherhand, this way of partitioning of certain number, whichrefers to the molecule as a whole, works reasonably wellfor estimates of the intramolecular hydrogen bond energy

Table 3The orientation of different conformers of 3-aminoacroleine in thecoordinate system of Fig. 1 as defined by the angle (in degrees) betweenthe CC(N) double bond direction and the z (dipole moment) axis

Conformer SCF HFa MP2b

I 128.7 132.0II 179.5 182.8III 113.3 116.5IV 158.4 161.7

The reported angles correspond to planar structures optimized at the levelof the MP2 approximation with 6-31G** basis set.

a Angles for the dipole moment calculated in the SCF HF approxima-tion with PolX basis set.

b Angles for the dipole moment calculated in the MP2 approximationwith PolX basis set.

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[39–43] and appears to be worth trying for other molecularproperties.

The conformers I and II differ by the rotation of theCHO group about the C–C bond which causes the s-cis! s-trans isomerization and increases the separationof the NH2 and CHO groups. This leads to the obviousincrease of the dipole moment. Also the I! II isomeriza-tion breaks the hydrogen bond in I and this should haveadditional consequences for the changes of electric proper-ties. In the case of the dipole moment its relatively smallvalue in I follows mostly from the shortest distancebetween the donor/acceptor end groups. Since the dipolemoment orientation with respect to the N–C@C–C@Oskeleton is different in different conformers, the separationof the conformation and hydrogen bonding effects in I doesnot seem to be possible. However, the mutual short–rangeinteraction through the hydrogen bond between the twoend groups in I must certainly influence the total value of l.

As regards conformers II and III they differ simulta-neously by the cis/trans configuration at the CC doublebond and the s-trans/s-cis arrangement at the formally sin-gle C–C bond. This leads to a considerable change in thevalue of the dipole moment and its orientation with respectto the CC double bond (see Tables 1 and 3), although thespatial separation between oxygen and nitrogen in the twoconformers remains approximately the same (see Fig. 1).On the other hand, there is certain, relatively small,increase of the dipole moment on passing from II to IV.Since structures II and IV differ only by the cis/trans con-figuration at the CC double bond one concludes that thelowering of the dipole moment on passing from II to IIIis mostly due to the change of conformation at the C–Csingle bond.

On inspecting the polarizability data of Table 1 onefinds that there is a large difference between the zz compo-nents of the dipole polarizability of I and all other con-formers. However, for the in-plane components of the atensor neither of them (axx and azz) is definitely dominant.Hence, for the purpose of comparisons between differentconformers one should rather use the in-plane rotationaverage apl ¼ 1

2ðaxx þ azzÞ. The MP2 values of apl for con-

formers I, II, III, and IV, calculated from the data of Table1, are equal to 65.8 a.u., 67.6 a.u., 68.7 a.u., and 70.3 a.u.,respectively. The large differences calculated for the longi-tudinal component azz become quenched to relatively smallvalues and indicate that in terms of the in-plane average ofthe a tensor the four conformers of 3-aminoacroleine arenot very different.

In spite of small differences in the calculated apl values,one can attempt to use them for the approximate separa-tion of the conformation and hydrogen bonding effectsaccording to the following scheme. The hypothetical struc-ture I without the intramolecular hydrogen bond woulddiffer from the conformer II only by the conformation atthe C–C single bond. The contribution due to this s-cis! s-trans isomerization in the cis configuration can beestimated from the data for conformers II, III, and IV.

First, let us note that upon the transition IV! II, whichinvolves solely the trans! cis isomerization with the s-trans arrangement at the C–C single bond, the apl valueis changed by �2.7 a.u. To obtain I from II requires thes-trans! s-cis isomerization. The corresponding confor-mational contribution can be estimated from the s-trans! s-cis isomerization IV! III as equal to �1.6 a.u.Hence, without the hydrogen bond formation, the changeIV! I would contribute �4.3 a.u., whereas the calculateddifference is equal to �4.5 a.u. This leads to the estimate ofthe intramolecular hydrogen bond contribution as equal to�0.2 a.u. The intramolecular hydrogen bond contributionto the in-plane average apl is found to be negative and verysmall. Its value is much smaller than the interaction-induced polarizabilities found in the case of intermolecularhydrogen bonds [20,38].

Much larger changes are computed for the longitudinalcomponent of the b tensor and it is worthwhile to followthe same estimation scheme to obtain the correspondingcontribution due to the hydrogen bond formation in I.At the level of the MP2 approximation the total increaseof bzzz upon the I! II isomerization amounts to about126 a.u. The transition from IV to II leads to the changein bzzz by about �178 a.u. and within our estimationscheme corresponds to the contribution due to the trans–cis isomerization. The contribution of the s-trans! s-cis

isomerization at the trans configuration at the CC doublebond, estimated from the IV! III process amounts toabout �5 a.u. On combining these data one finds that thehypothetical structure I without the hydrogen bond wouldhave the bzzz value equal to about 366 a.u., whereas thecomputed value is equal to ca. 245 ca. Within the presentestimation scheme the difference of about �120 is inter-preted as the contribution due to the intramolecular hydro-gen bond. Let us add that in the case of b using the in-planeaverage would not make any significant difference becauseof the small value of bzxx.

Let us also mention that the lowering of bzzz (per mono-mer) was also calculated for linear chains of the hydrogenbonded hydrogen halides [44]. Related studies of the so-called interaction hyperpolarizabilities in hydrogen bondeddimers have been published earlier [20,38,45,46] and showthat intermolecular hydrogen bonds may lead to either neg-ative or positive interaction hyperpolarizabilities. Theintramolecular hydrogen bond, whose contribution to bzzz

of 3-aminoacroleine is estimated to be of the order of �102,appears to behave differently than intermolecular hydrogenbonds. On the other hand, the estimated small value of theH-bond contribution to apl of I is consistent with nearadditivity of polarizabilities calculated for systems withintermolecular hydrogen bonds [20].

On analysing the data of Table 1 one also finds that hetransition from II to either III or IV leads to a largeincrease of bzzz. Since the change of bzzz on passing fromII to IV is approximately the same as that calculated forthe transition from II to III, one concludes that the mostimportant factor is the trans arrangement of the end groups

38 A. Avramopoulos et al. / Chemical Physics 328 (2006) 33–44

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at the CC double bond. The conformation of the endgroup, as long as it does not lead to the intramolecularhydrogen bonding, appears to be less important. Hence,one may anticipate that the trans-s-cis elongated systemsmay have larger values of b than those with the cis-s-trans

configuration of the donor–acceptor groups. This observa-tion can be useful in syntheses of the high b materials [12].

The origin of large differences between the bzzz valuesfor I and IV can be elucidated in terms of the two-statemodel [7,10]. According to this model the main contribu-tion to b follows from the lowest symmetry-allowed elec-tronic excitation of energy Dge and is given by:

b �3ðle � lgÞl2

ge

D2ge

; ð8Þ

where lg, le, and lge denote the dipole moments in theground and excited states, and the transition dipole mo-ment, respectively. For the two most distinct conformers,I and IV, the dipole moment values in the ground statecan be found in Table 1. The estimates of the excited statedipole moments in I and IV obtained from CIS calculationsare equal to 1.763 a.u. and 3.099 a.u., respectively. Thesame method gives for the z component of lge the corre-sponding values of 1.428 a.u. and 2.278 a.u. The respectiveCIS excitation energies are 0.205 a.u. and 0.216 a.u. Similarvalues (ca. 0.183 a.u.) of the excitation energy are obtainedin the TDHF approximation and are almost the same forthe two conformers.

Upon combining the dipole moment and transitionenergy data one finds that the ratio of bzzz for conformersIV and I is about 3. This estimate gives the correct order ofmagnitude of the corresponding ratio (2.2) evaluated fromthe MP2 data of Table 1. However, one should note thatthe SCF HF results of Table 1 give the corresponding ratioas equal to about 1.1. The estimate (8) is based on the CISdata and neither SCF HF nor MP2 results are the appro-priate reference. Hence, one can only conclude that a sim-ple two-state model correctly predicts the increase of bzzz

on passing from I to IV. Simultaneously one finds thatthe major cause of the increase of bzzz upon the conforma-tion change I ! IV is the change in the length of the tran-

sition dipole and the difference between the excited andground state dipole moments. For systems with donorand acceptor end groups both the transition dipole andthe dipole moments strongly depend on molecular geome-try, i.e., on the conformation and configuration. The mostelongated structures are expected to give the highest valuesof the longitudinal component of the b tensor. In the caseof I additional lowering of bzzz arises from the presence ofthe intramolecular hydrogen bond.

3.3. Pure electronic contributions: conjugation and

intramolecular charge transfer

The MP2 data of Table 1 show that the mostextended quasilinear structure IV leads to the largestMP2 values of l, azz, and bzzz. The lowest values ofthese properties are obtained for the most compactH-bonded structure I. According to Wu et al. [47] theelectronic structure of I, as compared to that of IV, cor-responds to the ’reduction of the effective length overwhich p electrons can respond to an applied optical elec-tric field’. However, the idea of the ’effective length’ ofthe p electron path is highly qualitative and does notseem to explain the similarity of bzzz for III and IVand its difference for II and III.

The role of the ‘effective length’ of the p conjugationpath can be also addressed in terms of the intramolecularcharge transfer in the excited state. For this purpose wehave calculated the pertinent electric properties of relatedcompounds which differ from 3-aminoacroleine either bythe end groups or by the linker between them. The calcu-lated values of l = lz, azz, and bzzz, with the z axis coincid-ing with the dipole direction, are shown in Table 4. Theresults correspond to SCF HF and MP2 calculations withPolX basis sets at the 6-31G**/MP2 optimized geometries.The SCF HF and MP2 electric properties of five structur-ally and electronically related molecules are compared withthose for the most elongated structure IV of 3-aminoacro-leine. These molecules are displayed in Fig. 2 which alsoshows their orientation with respect to the dipole moment(z) axis (horizontal, oriented to the right).

Table 4Comparative study of the conjugation and the end group effect on electric properties (electronic contributions) of a series of quasilinear systems

Moleculea PolX/HF PolX/MP2

lzb azz bzzz lz

b azz bzzz

H2N–CHO 1.705 31.96 �30.0 1.532 36.94 �17.6H2C@CH–CHO 1.546 62.51 �47.2 1.273 60.97 91.9H2N–CH@CH2 0.677 50.24 104.7 0.649 52.80 200.8H2N–CH2–CH2–CHO 1.670 53.69 �55.6 1.434 57.43 �39.7H2N–CH2–CH2–CH2OH 1.013 43.88 �42.9 0.993 51.66 � 8.4H2N–CH@CH–CHOc 2.673 84.09 112.7 2.423 93.08 549.5

All values in a.u.a The optimized 6-31G**/MP2 geometries are shown in Fig. 2. The calculated local energy minima correspond to the most elongated structures.b The z axis is the direction of the dipole moment and the molecule orientation with respect to this axis is shown in Fig. 2.c Conformer IV of 3-aminoacroleine, see Table 1.

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pyThe conjugation path and intramolecular charge trans-

fer between donor and acceptor end groups are the mainfactors which determine molecular electric properties andin particular the first hyperpolarizability. The effect of theextension of the conjugation path can be seen from thecomparison of the electric property data for formamide(H2N–CHO) and 3-aminoacroleine (conformer IV). The –CH@CH– linker causes the increase of the formamidedipole moment by almost 1 a.u. One should note, however,that this may not necessarily be connected with the conju-gation-assisted increase of the charge transfer; the separa-tion of charges on the end groups also increases onpassing from formamide to 3-aminoacroleine (IV) andcould be equally responsible for the increase of the dipolemoment. The choice between these two interpretations,which simultaneously avoids the highly unreliable toolsof the population analysis, can be made by comparing IVwith the H2N–CH2–CH2–CHO molecule (the most elon-gated structure, see Table 4). The saturated –CH2–CH2–linker between the amino and formyl end groups does onlyslightly affect the dipole moment calculated for formamide.Since the distances between the end groups in the mostelongated structure of H2N–CH2–CH2–CHO (see Fig. 2)and 3-aminoacroleine (IV) are very similar, this compari-

son rules out the essential importance of the distance fac-tor. One concludes that the –CH@CH– linker must alsoconsiderably enhance the intramolecular charge transfer.The same role of the unsaturated linker can be concludedfor longitudinal components of a and b.

It is interesting to note that already the one-sided substi-tution of the ethylene moiety by the amino or formylgroups leads to considerably large MP2 values of bzzz.Although the ground state dipole moment of H2N–CH@CH2 is small compared to that of 3-aminoacroleine(IV), its longitudinal value of b is quite large. On the otherhand, the fully unsaturated molecule H2N–CH2–CH2–CH2OH (see Table 4 and Fig. 1) has a very small of bzzz.This shows that in this molecule the low-energy electronicexcitations contributing to bzzz have purely local character.Moreover, this also indicates that the polarization of thecharge distribution along the unsaturated –C–C– chainhas only a small contribution to the change of locally deter-mined value of bzzz.

One more interesting feature of the molecules listed inTable 4 is the magnitude of the electron correlation effectupon bzzz. In the case of the direct connection betweendonor and acceptor groups as well as in the case of the sat-urated linker between them, the electron correlation effectis of the order of about 12–16 a.u. (H2N–CHO andH2N–CH2–CH2–CHO) and very small compared to thevalues calculated for H2C@CH–CHO, H2N–CH@CH2,and in particular for 3-aminoacroleine (IV). Hence, fordonor–acceptor systems with unsaturated linkers betweenthe end groups the SCF HF approximation is most likelyinsufficient, though it may work reasonably well for analo-gous systems with saturated linkers [18].

3.4. Vibrational corrections: problems of the perturbation

approach

With the recognition of the importance of pure vibra-tional contribution to dipole polarizabilities and hyperpo-larizabilities it became quite common to supplement thepure electronic calculations with the corresponding vibra-tional terms. The same is followed in the present paper.The results for the pure vibrational contribution to a andb tensors for the two most distinct conformers I and IVare presented in Tables 5 and 6, respectively. The corre-

Fig. 2. Fully optimized 6-31G**/MP2 structures of molecules presented inTable 4. The z coordinate (dipole moment) axis is horizontal and directedto the right.

Table 5Pure vibrational contributions to the dipole polarizability tensor forconformers I and IV of 3-aminoacroleine

Conformer Component [l2](0,0) [l2](1,1) [l2](0,2) [l2](2,0)

I xx 7.11 3.55 2.01 2.62yy 83.31 6.21 �25.53 �12.32zz 14.44 1.43 1.14 1.11

IV xx 11.02 �0.80 4.98 �0.65yy 35.87 7.85 46.79 �4.21zz 47.39 �12.96 26.09 �5.10

Vibrational corrections follow from PolX/HF calculations at fully opti-mized minimum energy geometries. All values in a.u.

40 A. Avramopoulos et al. / Chemical Physics 328 (2006) 33–44

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pysponding data for conformers II and III are essentially sim-ilar to those calculated for I and IV. Including them inTables 5 and 6 would only increase the amount of numer-ical data of similar character without adding any significantnovelty.

The data presented in Tables 5 and 6 correspond to SCFHF calculations with PolX basis sets at fully optimizedmolecular geometries. It is worthwhile to stress that theuse of the minimum energy geometry is mandatory forthe BKPT approach [24,27]. All energy and electric prop-erty derivatives which enter Eqs. (5)–(7) must be calculatedwith the same basis set and at the same fully optimizedmolecular geometry. This could be one of the major prob-lems of BKPT in the case of several different minima sepa-rated by low barriers.

To complete the data for vibrational corrections let usadd that the zpva contribution to the dipole moment, i.e.the sum of the terms in Eq. (5) with P = l, is equal to�0.053 a.u. for the hydrogen bonded structure I and+0.032 for the structure IV. In the case of the dipole polar-izability (P = a) the zpva corrections to the xx, yy, and zz

components of a for I are equal to 1.64 a.u., 0.43 a.u., and1.57 a.u., respectively. The corresponding numbers for thestructure IV are 1.86 a.u., 0.68 a.u., and 1.46 a.u. In thecase of the b tensor only the [P](0,1) is taken into accountwithin the (4320) approximation of BKPT. The zpva cor-rections to bzxx, bzyy, and bzzz for I are equal to 3.47 a.u.,0.13 a.u., and 1.02 a.u, respectively. For the structure IVthe corresponding numbers are 2.45 a.u., �1.31 a.u., and4.73 a.u. One concludes that, compared to the magnitudeof pure electronic contributions, the zpva corrections forall investigated electric properties are essentially negligible.It is worthwhile to note that the zpva contributions wouldvanish in the double harmonic approximation. Hence,within the approximation (5), the zpva results indicatethe magnitude of the first-order electric anharmonicity(for l and a) as well as the cubic anharmonicity of thefield-independent BO potential.

The zpva corrections are small for all investigated prop-erties and their magnitude does not indicate any particu-larly large vibrational effects which can be seen for pvcorrections. As one can see from the data of Table 5already for the dipole polarizability some of the pv termsare unusually large. This applies mostly to the resultsobtained in the double harmonic approximation (the[l2](0,0) term) which involves the first derivatives of the

dipole moment function evaluated at the equilibriumgeometry [25,26,28]. Similar pattern is also observed forthe double harmonic term for bzzz. The very large [l3](0,1)

and [l3](1,0) contributions to bzyy in I appear unlikely tobe physically justified. Already for moderately strong exter-nal electric fields (e.g., 0.1 a.u.) the large negative values ofthese contributions in conformer I will lead to a very largepositive shift of the ground state BO surface, which in turnmay come close to the excited state surfaces. This wouldmean the violation of the BO approximation which under-lies the BKPT scheme. One should also note that the[la](n,m) contributions to bzyy of the conformer I are quitesmall, although they correspond to the second-order per-turbation expansion [28]. The [l3](n,m) follow from thethird-order terms in BKPT and one would expect themto be smaller than the second-order contributions.

The very large (negative or positive) vibrational contri-butions most likely arise from the failure of the perturba-tion approach in the case of highly flexible molecules.The same can be concluded for the corresponding contri-butions to bzyy and bzzz in the elongated structure IV. Sim-ilar features of pv corrections to a and b have beenobserved earlier for hydrogen bonded dimers [48] andattributed to the failure of the harmonic approximationwhich underlies the BKPT approach. More recently thisproblem was analysed by Torrent-Sucarrat et al. [37],who found that the higher-order BKPT terms do not exhi-bit divergence pattern. The unusually large pv correctionsseem to appear mostly in low-orders of BKPT.

The easiest to analyse is the double harmonic approxi-mation for pv contributions to a and b. This approxima-tion is equivalent to the parallel field-dependent shift ofthe BO surface; the force constants (harmonic vibrationfrequencies) are assumed to be left unaffected. Hence, theonly contribution of the electric field is to change the equi-librium configuration of the system embedded in the exter-nal electric field whereas the shape of the BO surfaceremains field-independent. This lowest-order perturbationtreatment of pv terms, which involve the dipole momentdependence on nuclear coordinates, is also referred to asthe nuclear relaxation approximation (NR) [49–51]. Inthe sense of the convergent perturbation expansion theNR terms should be the leading pv contributions. How-ever, the higher-order terms are known to be sometimesmuch larger than the NR contributions [32,37,48]. In thepresent case all terms which follow from the third-order

Table 6Pure vibrational contributions to the first hyperpolarizability tensor for conformers I and IV of 3-aminoacroleine

Conformer Component [la](0,0) [l3](0,1) [l3](1,0) [la](1,1) [la](2,0) [la](0,2)

I zxx 40.0 18.6 15.3 12.0 �42.4 32.3zyy �10.9 �4668.2 �6440.3 �88.4 �0.4 �5.9zzz 349.9 �39.5 210.7 21.5 �33.9 36.8

IV zxx 86.1 807.3 �300.4 �1.7 �9.9 61.9zyy 60.0 7954.5 �168.5 �133.4 �8.9 96.6zzz 1010.5 3554.9 �2638.7 109.8 �64.0 138.4

Vibrational corrections follow from PolX/HF calculations at fully optimized minimum energy geometries. All values in a.u.

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perturbation theory (the [l3] terms) are definitely domi-nant. This can be seen from the present data for pv correc-tions to first hyperpolarizability (Table 6).

Since already the double harmonic approximation forpv contributions to a and b leads to unusually large vibra-tional corrections (see Tables 5 and 6) it is easy to identifythe responsible modes. The analysis of the dipole momentderivatives and frequencies shows that for I the very largepv contribution to ayy is principally due to two low-fre-quency modes. Both of them (243 cm�1 and 270 cm�1

in the SCF HF approximation) correspond to out-of-plane vibrations of hydrogen atoms with the largestamplitudes for the (N)H atom which does not participatein the hydrogen bond. They contribute 17.34 a.u. and60.87 a.u., respectively, to the total [l2](0,0) vibrationalcorrection to ayy. The low-frequency modes are in generalhighly anharmonic and correspond to very flat regions ofthe BO surface. Approximating them by a harmonicpotential may result in a serious underestimation of thefirst transition frequency and will result in artificially large[l2](0,0) correction. This is partly remedied by the anhar-monic contributions [l2](0,2) and [l2](2,0), i.e., by takinginto account the second-order electric and mechanicalanharmonicities.

The pv vibrational corrections to ayy and azz for con-former IV are also considerably large. In the double har-monic approximation the yy component has the largestcontribution (30.62 a.u.) from the 394 cm�1 mode whichcorresponds to the (out-of-plane) wagging vibration ofthe –NH2 group. The BO potential for this vibration isactually a double minimum and cannot be represented inthe harmonic approximation. Using this approximationmeans that the minimum is arbitrarily positioned in oneof the two minima of the double well potential. The barrierbetween them, though small, is simultaneously largeenough to give numerically a single minimum. This willstrongly affect the corresponding vibration mode and itsfrequency. Hence, the large value of [l2](0,0) is not surpris-ing. This interpretation is further supported by the largevalue of the [l2](0,2) contribution to ayy. It is worthwhileto note that any attempt to represent a double minimumpotential in terms of anharmonic corrections to a singleminimum harmonic potential must lead to divergent per-turbation expansion [52,53].

The same wagging mode of the –NH2 group modemakes very large the [l2](0,0) contribution to azz of IV.One should recall the fully optimized structure IV is non-planar and the wagging vibration of the –NH2 group willalso contribute to the [l2](0,0) term in the vibration correc-tion to azz. In general one concludes that in the presence ofdouble minima with low barriers, the BKPT approachbecomes invalid and other more sophisticated techniquesneed to be used [54]. Actually the same conclusion has beenrecently drawn by Torrent-Sucarrat et al. [55]. Theseauthors have solved the problem of high anharmonicitiesby using different vibrational mean-field, perturbation,and configuration interaction methods.

For both I and IV the large double harmonic contribu-tions ([la](0,0)) to b arise only for bzzz. For the nonplanarstructure IV the dominant contribution (424.6 a.u.) comesagain from the 394 cm�1 wagging vibration of the –NH2

group. However, for the [la](0,0) pv contribution to bzzz

of the structure IV also some high–frequency bendingmodes are important, e.g., the 1833 cm�1 mode contributes188.4 a.u. and the 1948 cm�1 mode adds 179.4 a.u. Thesetwo modes are reasonably harmonic and the large contri-butions should appear primarily due to electric anharmo-nicities. This is confirmed by the very large (and negative)value of the [l3](1,0) term. Also the [l3](0,1) contribution tobzzz of the nonplanar structure IV is unacceptably large.Its large values involve the highly anharmonic low-fre-quency mode of 394 cm�1, which has been discussed ear-lier. Hence, it becomes evident that the perturbationseries will diverge.

For the planar structure I only the zyy component of bsuffers from large anharmonic contributions. In this casethey are again primarily due to the two low frequencymodes already discussed for azz. Both the electric andmechanical (cubic) anharmonicity arising from these vibra-tions cannot be efficiently treated by the perturbationexpansion.

In general one can conclude that the BKPT approachneeds to be used with particular care. Using the harmonicpotential as the zeroth-order approximation is valid onlyfor systems whose vibrations are almost harmonic andwhose electric properties are near-linear in vibration coor-dinates. Moreover, the use of BKPT of much higher-ordersis not a remedy since the perturbation series for the anhar-monic oscillator is known to diverge [52,53].

Finally, it is interesting to note that the large pv correc-tions to the static dipole polarizabilities and first hyperpo-larizabilities of I and IV are significantly reduced at non-zero frequencies. At the oscillatory field frequency of0.072 a.u. the total pv correction to azz is reduced to�0.11 a.u. and �0.19 a.u., for I and IV respectively. Thesevalues are to be compared with the static (total) results ofTable 5, i.e., 18.12 a.u. and 55.42 a.u., respectively. In thecase of bzzz the corresponding numbers for bzzz(�x;x, 0)at the same frequency are 122.1 a.u. and 389.3 a.u. The(total) static corrections from Table 6 for conformers Iand IV are equal to 545.6 a.u. and 2110.8 a.u., respec-tively. These data show that the importance of pv termswill be anyway strongly reduced by the oscillatory electro-magnetic field. The present conclusions are fully compat-ible with those presented earlier by Torrent-Sucarrat et al.[32].

4. Summary and conclusions

We have investigated different structural and electronicfactors which affect the magnitude of linear and nonlin-ear molecular electric properties. The 3-aminoacroleinemolecule has been chosen as a suitable model for thestudy of the conformation dependence of dipole

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moments, dipole polarizabilities, and first hyperpolariz-abilities. Additionally, one of the conformers (I) has beenused to study the effect of the intramolecular hydrogenbond on these properties.

For all four conformers of 3-aminoacroleine the electroncorrelation contribution to dipole moments and dipolepolarizabilities, as estimated at the level of the MP2approximation is of the usual magnitude. However, inthe case of b its value has been found to be significantlychanged by including the electron correlation contribution.The largest electron correlation effect is observed for themost elongated conformer IV. This shows that for conju-gated donor–acceptor systems the results obtained for thefirst hyperpolarizability tensor in the SCF HF approxima-tion can be quite unrealistic and may strongly depend onthe molecular conformation.

The data obtained for four different conformers permitthe approximate estimation of the hydrogen bond effecton the calculated electric properties. It has been found thatintramolecular hydrogen bonding leads to small reductionof dipole polarizabilities. In the case of the zzz componentof b the H bond effect is much larger and in the present caseis estimated at the level of �120 a.u.

The pure electronic contributions to l, a, and b havebeen also analysed in terms of the conjugation and intra-molecular charge transfer effects. Most of the enhancementof these properties follows from the presence of the unsat-urated linker. In the case of the first hyperpolarizability bzzz

it leads to the increase of the ground and excited statedipole moments and also increases the length of the transi-tion dipole moment. This is accompanied by the loweringof excitation energy. All these factors lead to a largeincrease of bzzz as compared to the value calculated forthe same pair of the donor and acceptor groups joinedby the unsaturated linker.

To follow similar studies of electric properties of con-jugated donor–acceptor systems we have also calculatedvibration corrections to the considered electric properties.The zpva have been found to be small and essentiallynegligible. However, the pv contributions to a, and inparticular to b, are found to be unphysically large. Theorigin of these large pv contributions has been analysed.The unusually large terms are due to highly anharmonicmodes of low frequency. Particularly large are those con-tributions which involve nuclear motions in a double wellpotential. In BKPT approach they originate from theassumption of the zeroth-order harmonic approximationat one of the two minima. The corresponding perturba-tion series cannot recover the pattern of energy levelsof e.g. the double minimum potential. This shows thatfor floppy molecules BKPT should be used with greatcare and the validity of its assumptions needs to be a pri-

ori established by the analysis of normal modes in thegiven system. It is, however, quite consoling that thelarge pv contributions to a and b are significantlyreduced already at relatively low frequencies of the elec-tromagnetic field.

Acknowledgments

Two of us (M.J and A.J.S) acknowledge the use of thecomputer cluster at the Information & CommunicationTechnology Centre of the Nicolaus Copernicus Universityand the very helpful attitude of its staff.

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