Static and frequency dependent polarizabilities and hyperpolarizabilities of H2Sn

— —< <

Static and Frequency DependentPolarizabilities andHyperpolarizabilities of H S2 n

S. G. RAPTIS,1, 2 S. M. NASIOU,3 I. N. DEMETROPOULOS,3

M. G. PAPADOPOULOS1

1 National Hellenic Research Foundation, 48 Vas. Constantinou Avenue, Athens 11635, Greece2 Chemical Engineering Department, National Technical University of Athens, Zografou Campus,Athens 15773, Greece3 Chemistry Department, University of Ioannina, Ioannina 45110, Greece

Received 17 February 1998; accepted 22 June 1998

ABSTRACT: The structure]polarization relationship was investigated in aseries of polysulfanes, H S . The reported results demonstrate that the forms of2 nchange of the polarizability components, a , and the second hyperpolarizabilityiicomponents, g , as well as the average values a and g , respectively, of H Siiii 2 nwith n are similar. This shows that polarizability components can be easily usedto determine corresponding hyperpolarizability data. A remarkable change ofthe hyperpolarizabilities with the molecular geometry of H S was found. This2 nresult can be used for the design of nonlinear optical materials with optimumproperties. The present study uses the flexible s bonded H S and is2 ncomplementary to the works that considered the effect of conformational changesof p-conjugated systems on their hyperpolarizabities. The present computationswere performed using the semiempirical approaches MNDO and MNDOrd, aswell as ab initio methods with STO-3G, extended with polarization and diffuse

w xfunctions, and 3s2 pr7s5p2 d sets for H S . At the ab initio level, the electronic2 nand the vibrational contributions to polarizabilities and hyperpolarizabilitieswere both computed for several members of H S . The frequency dependence2 nof the above contributions and the static limit were discussed. Electroncorrelation was taken into account for several test cases using MP2 theory. Theselected methods and the variety of the approximations on which they rely

Correspondence to: M. G. Papadopoulos; e-mail: [email protected]

This article contains Supplementary Material available fromthe authors or via the Internet at ftp.wiley.comrpublicrjournalsrjccrsuppmatr19r1698 or http:rrjournals.wiley.comrjccr

( )Journal of Computational Chemistry, Vol. 19, No. 15, 1698]1715 1998Q 1998 John Wiley & Sons, Inc. CCC 0192-8651 / 98 / 151698-18

POLARIZABILITIES AND HYPERPOLARIZABILITIES OF H S2 n

allow the systematic consideration of the effect of changes of the geometry ofH S on their polarizabilities and second hyperpolarizabilities. Q 1998 John2 nWiley & Sons, Inc. J Comput Chem 19: 1698]1715, 1998

Keywords: MNDO; MNDOrd; polarizabilities; hyperpolarizabilities; H S2 n

Introduction

olarizabilities and hyperpolarizabilites1 haveP attracted the interest of an increasing num-ber of researchers because they provide importantinformation for molecular structure.2, 3 In addition,the hyperpolarizabilities are of great importancefor the design of materials that have many applica-

Žtions e.g., optical processing of information, opti-. 4cal computing etc. . Design or selection of materi-

als, which are likely to have the optimum proper-ties, requires an in depth understanding of thestructure]polarization relationship.

In this work we investigated the way the changeof molecular geometry affects the polarizabilitiesand second hyperpolarizabilities of H S , which2 nwas selected as the model compound. The S Hand S S bonds are of considerable importance inchemical and biological systems.5a, b Sulfur is alsoinvolved in derivatives with promising nonlinearelectric properties.5c, d One notes that the struc-ture]property relationships, which are used toguide one’s efforts in molecular engineering,5e

were investigated to a considerable extent in linearpolyenes or in general in conjugated systems.5f ] l

In particular, the effect of conformational changesor disorders on the nonlinear optical properties

Žwas studied extensively in conjugated systems see.brief literature survey in later section . Thus, we

systematically selected a set of conformationalchanges in the structure of H S , because the2 neffect of disorders in this class of flexible systemshas not been studied before, as far as we know. Inaddition, understanding of the implications of thegeometry variations on the hyperpolarizabilities inthe above nonplanar systems is complementary tothat resulting from studies in planar conjugatedsystems. The selected polysulfanes could not beconsidered as nonlinear optical materials but ratheras test models to analyze the effect of the abovestructural changes on the polarizabilities and hy-perpolarizabilities of such materials.

For the present study we used semiempiricalŽ 6 7.MNDO and MNDOrd and ab initio methods.

Correlation was taken into account at the MP2level.8 At the ab initio level the static and fre-quency dependent electronic and vibrational con-tributions of some members of the series H S2 n

were calculated. This allowed a comprehensiveunderstanding of the nonlinear optical propertiesin this class of compounds. The significant contri-bution of the static vibrational second hyperpolar-izability, compared with electronic contribution, isalso documented.

Computational Methods

Of primary importance for the present work arethe trends and differences in the properties, be-cause these are necessary for discussing the struc-ture]polarization relationship. To check the ade-quacy of the employed semiempirical methods inorder to approach the required goal, we performedextensive tests and comparisons using several the-

Ž .oretical techniques semiempirical and ab initioand the experimental results where they were

Ž .available Tables I, II .At the semiempirical level we used the MNDO6

and MNDOrd7 methods. The latter involves dorbitals on S, due to which gives results of supe-rior quality in comparison to those produced bythe MNDO as will be shown. A detailed descrip-tion of these methods is given in refs. 6a and 7.

At the ab initio level the properties of H S2 n

were computed using the GAMESS program9 withthe two following basis sets:

1. The STO-3G set, extended with polarizationŽ .p on H and d on S and diffuse functions on

Ž . Ž . 9both H s and S s, p , denoted by STO-3G**qq ; and

Ž . w x2. The set 6 s4 pr14s10 p4d r 3s2 pr7s5p2 dpublished by Sadlej.10a, b

The former was selected because it is a small set,which gives satisfactory g values for H S in2 n

Ž .comparison to the Sadlej orbitals Table II . The

JOURNAL OF COMPUTATIONAL CHEMISTRY 1699

RAPTIS ET AL.

TABLE I.( )Dipole Moment of H S n = 1 – 9 Computed by Various Methods.2 n

m / au

[ ]Compound MNDO MNDO / d STO-3G**++ MP2 / STO-3G**++ 3s2p / 7s5p2d Experimental

19aH S 0.582 0.407 0.677 0.671 0.445 0.383 " 0.0022a0.353a0.369a0.386

a0.433 " 0.020H S 0.639 0.465 0.716 0.703 0.4792 2H S 0.236 0.227 0.223 0.217 0.1832 3H S 0.307 0.130 0.390 0.369 0.2372 4H S 0.607 0.386 0.675 0.638 0.4682 5H S 0.408 0.326 0.398 0.3762 6H S 0.119 0.004 0.192 0.1802 7H S 0.548 0.320 0.6232 8H S 0.525 0.380 0.5382 9

( )The properties of H S were computed using bond length and angles defined by model 3 see text . The asterisks denote that2 n( ) ( ) ( ) ( )polarization functions were used for H p and S d . The two pluses mean that diffuse functions were employed for H s and S sp

of H S .2 na Gas phase values cited by McClellan.19b

functions reported by Sadlej are medium size po-larized basis sets designed for the satisfactory cal-

culation of dipole moments and polariz-abilities.10a, b

The polarizability and hyperpolarizability val-ues presented in this work were computed usingthe following methods:

1. A finite perturbation theory approach, whichwas used in connection with the MNDOtechnique6, 11 and the ab initio procedure atthe MP2 level8; and

2. Analytical methods implemented in theMNDOrd7 and GAMESS,9 which are usedfor the computation of the properties at the

Ž .self-consistent field SCF level.

The vibrational polarizabilities and hyperpolar-Ž .izabilities of H S n s 1]3 were computed using2 n

the perturbation theoretic approach proposed byBishop and Kirtman12a ] c and implemented by Co-hen et al.12d The required derivatives of the energyŽ . Žsecond and third order , dipole moment first and

. Žsecond order , polarizability first and second or-. Ž .12dder , and first hyperpolarizability first order

were computed at the SCF level using CADPAC12e

and the vibrational properties were calculated em-ploying SPECTRO.12

The property values presented in this work areŽ . 12ggiven in atomic units au .

Results and Discussion

The material presented in this section is orga-nized in the following way: first a literature surveyis presented; then the employed geometric modelsare discussed; followed by the analysis of the elec-tronic dipole moment, polarizabilities, and secondhyperpolarizabilities; and finally comments aremade about the vibrational polarizabilities andhyperpolarizabilities.

LITERATURE SURVEY

The structure]polarization relationship has beeninvestigated in considerable detail in conjugatedsystems.13 ] 15 One may distinguish three themes onwhich the attention of the various workers is fo-cused:

1. The derivation of scaling rules that con-Ž .nect the size number of double bonds etc.

Žwith the properties polarizability or hyper-polarizability components or average val-

. 13a, b, c, h, 14ues .2. The use of molecular geometry as a tool to

achieve the required nonlinearities.13f, g

Marder et al.13f showed that by reducing theŽbond length alternation which is the average

VOL. 19, NO. 151700

POLARIZABILITIES AND HYPERPOLARIZABILITIES OF H S2 nTA

BLE

II.

Co

mp

aris

on

of

bfo

rH

SD

eter

min

edb

yM

ND

O,M

ND

O/// //d

,STO

-3G

**+

+,a

ndM

P2

/// //STO

-3G

**+

+.

2n

()

()

b0;

0,0

/au

by

2v

;v,v

/au

aa

ab

aa

,ca

,c[

][

]C

ompo

und

MN

DO

MN

DO

/dS

TO-3

G**

++

MP

2/S

TO-3

G**

++

3s2

p/7

s5p

2d

STO

-3G

**+

+3

s2p

/7s5

p2

dE

xper

imen

tal d

HS

y72

.389

y29

.843

y12

.488

y15

.153

3.31

7y

19.2

953.

904

y9.

95"

2.08

2H

Sy

102.

871

y61

.822

y4.

815

y10

.451

2.20

9y

14.8

64.6

282

2H

Sy

51.3

56y

14.5

01y

2.27

5y

7.37

1y

13.0

13y

15.4

72y

21.4

482

3H

Sy

77.8

70y

100.

263

y5.

018

y8.

195

18.8

344.

699

27.4

892

4H

Sy

152.

300

y14

0.00

61.

484

y5.

376

10.4

392.

635

25

HS

y94

.456

y52

.670

8.32

75.

410

26

HS

y53

.587

92.4

251.

012

2.64

616

.005

27

HS

y16

6.91

1y

173.

392

13.3

3912

.924

24.0

722

8H

Sy

141.

466

y10

6.21

915

.143

14.4

1812

.948

29

()

The

prop

erty

valu

esof

this

tabl

ew

ere

dete

rmin

edus

ing

for

HS

,bo

ndle

ngth

san

dan

gles

optim

ized

byM

ND

Om

odel

3.

2n

aTh

ese

resu

ltsw

ere

dete

rmin

edby

ape

rtur

batio

nth

eory

met

hod

asim

plem

ente

din

MN

DO

/d7

and

GA

ME

SS

.9

bTh

ese

data

wer

eca

lcul

ated

bya

finite

pert

urba

tion

theo

ryap

proa

ch.

cTh

ere

sults

ofth

isco

lum

nw

ere

com

pute

dat

694.

3nm

unle

ssot

herw

ise

spec

ified

.d

Val

uede

term

ined

usin

gdc

-indu

ced

seco

ndha

rmon

icge

nera

tion

inth

ega

sph

ase.

19a

difference in length between the single and13g .double bonds , one gets large hyperpolar-

izabilities. It is thus demonstrated that geom-etry can be employed as a key factor toproduce large nonlinearities.

3. The study of the effect of conformationalchanges or disorders on various properties ofthe electronic structure and in particular onthe polarizabilities and hyperpolarizabili-ties.13d, e, i, 15a

It should be noted that the effect of conforma-tional changes on the nonlinear optical propertieshas been studied using mainly conjugated sys-tems. The present study, which employs four dif-ferent models or conformations, allows us to traceand analyze the effect of regular and irregularchanges in the geometry of H S on their m, a and2 ng as well as their development with n.

Two important review articles, which are re-lated to some aspects of the present work, recentlyappeared. These were written by Bishop15b andKirtman and Champagne.15c

GEOMETRIC MODELS

We used several models for H S in order to2 n

study the effect of changes in the molecular geom-Ž .etry on the properties of interest m, a and g . The

first of those, model 1, assumes that the moleculesŽ .are planar Fig. 1 and initially all dihedral angles

are set at 1808. The optimization is performedŽhaving as variables the bond lengths H S and

. Ž .S S and angles S S S and H S S ,constraining the dihedrals to the set value. Thebond lengths S S and angles S S S of agiven H S were not constrained to have the same2 n

value. In model 2 the initial geometry for eachmolecule H S is set as in model 1. The geometry2 n

optimization is performed for all bonds, angles,Ž .and dihedral angles unconstrained minimization ,

yielding conformers that can be described as he-lices and segments of helices with reverse turns.Helices are secondary structures built from a se-quence of consecutive dihedral angles with ap-

Žproximately equal values, say g in our case.aproximately 908 , then gggggg is the string de-

scribing a nine atom helix. Reverse turns are theintervention to this consequence at some point bya yg dihedral angle producing a string ggg-ggg.

Ž .Model 3 assumes a helical structure Fig. 1 . Thisconfiguration was adopted because it was thought


RAPTIS ET AL.

FIGURE 1. The structure of H S , models 1]3, using2 nH S as an example.2 12

that it is likely to be the most favorable, using theenergy as a criterion.16a This hypothesis was con-firmed after detailed computations that were per-formed in the following steps using H S as a test2 12

Ž . 16b, ecase Fig. 2 . First, the suite of programs GMMXwas used for the optimization. This employs the

Ž .MM2 Molecular Mechanics, program 2 , forcefield.16c, d It searches the conformational space us-ing the stochastic procedure described by Saun-ders et al.16e From these computations resulted thesix, lower lying in energy, conformers. Subse-quently, the GMMX generated conformers weregeometry optimized with the MNDO6 Hamilto-nian. These calculations showed that the helicalstructure is associated with the global energy min-imum and the conformers depicted as helices seg-mented by reverse turns are local energy minima.In addition, an MNDO systematic conformationalsearch was performed for the H S molecule; the2 12outcome verified the helical structure as the moststable conformation for H S . All model 3 station-2 12ary points ended up with a force matrix calcula-tion; no imaginary frequencies were found, hence,H S helical structures are true minima. Green-2 n

wood and Earnshaw17 noted that ‘‘ . . . it is nowestablished that fibrous S consists of infinite chainsof S atoms arranged in parallel helices.’’ Similar

FIGURE 2. The six conformers of lower energy forH S .2 12

information is also provided by Cotton andWilkinson.18a

Summarizing the main features of models 2 and3, we note that calculations for the structure ofH S , which were performed using model 2, pro-2 n

duced helices and helices with reverse turns whileonly helical geometries were computed usingmodel 3.

Ž .Model 4 Fig. 3 involves structures in the formof closed chains. We consider S , S , S , and S .8 12 16 20The geometries of models 1]4 were optimizedusing the MNDO6 approach.

The semiempirical computations using models1]4 at the MNDO level were performed usingMOPAC 6.0.6b The keyword POLAR, which wasemployed for the calculations of the polarizabili-ties and hyperpolarizabilities, leads to the orienta-tion of the molecule so that the tensor of inertia isdiagonal. The coordinates of the first ninemolecules, which were used for the ab initio calcu-lations, are given as supplementary material.

ŽHowever, all the above structural data models.1]4 can also be provided on request.

A natural bond order analysis on H S and2 2H S confirmed the s nature of the S S bond.2 3The 3d atomic orbitals of the sulfur atoms, whichplay a significant role on hypervalent sulfur com-

VOL. 19, NO. 151702


FIGURE 3. The structure of S , where n = 8, 12, 16,nand 20.

18b Ž .pounds, are of a minor importance 1.3% in theconsidered H S . The stability of the helical con-2 nformation can be explained by the advantageousorientation of the sulfur lone pairs that, in thisposition, have minimum repulsion energy.18c, d Thelack of p bonds is the main reason for the smallersecond hyperpolarizabilities of polysulfanes incomparison with those of polyenes.5g The p bondthat serves as a bridge for the flow of electronsdoes not exist in H S and thus the p electron2 ndelocalization along the chain direction18e that ac-counts for the large second hyperpolarizabilities ofthe conjugated systems is absent.

In the following paragraphs a comparative studyof the dipole moment, polarizability, and hyperpo-larizabilities of H S computed with MNDO,2 n

ŽMNDOrd, and ab initio methods is presented Ta-.bles I]III . For the geometries of H S , the above-2 n

defined models 1]3 will be used. Thus, the effectof the molecular geometries on the properties ofinterest will be analyzed.

DIPOLE MOMENTS

It is likely that the best dipole moment valuesare those given by the Sadlej basis set.10 Thishypothesis is supported by comparing the com-puted dipole moment value of H S, using the2methods employed in the present work, with the

19a, b Ž .experimentally determined one Table I .However, the variation of m of H S as a function2 nof n is correctly described by the MNDO methodŽ .Table I, Fig. 4 .

POLARIZABILITIES

( )Frequency Dependent Values, a Iv; v

The frequency dependent polarizabilities ofŽH S were computed using the STO-3G**qq for2 n

. w x Ž .n F 9 and 3s2 pr7s5p2 d for n F 5 basis sets.The results are shown in Table II. The dynamicpolarizability of H S was calculated at four fre-2quencies. One observes that there is a reasonableagreement between the computed and the experi-

Ž .mental results Table II . Figure 5 presents theŽ .variation of a yv ; v for H S as a function of2 4

the frequency. From the results of Table II weŽ . Ž .observe that the ratio a yv ; v ra 0; 0 , at l s

Ž .694.3 nm, for increasing n H S , is approxi-2 nmately constant for both the employed basis setsŽ w x.STO-3G**qq and 3s2 pr7s5p2 d . This con-stant is approximately 1.03 for STO-3G**qq .

( )Static Values, a 0; 0

Ž .We observe that Table II

w x w xa 3s2 pr7s5p2 d ) a MP2rSTO-3G**qqw x) a STO-3G**qqw x w x) a MNDOrd ) a MNDO .

The difference

Ž . Ž .Da s a MNDOrd -a MNDO

is small. For n s 1, 4, and 9, Da s 1.22, 2.4 and0.0 au, respectively. Thus, for n G 9, MNDO andMNDOrd most likely give the same polarizabilityvalues for H S . From the STO-3G**qq results2 nwe observe that the correlation contribution in-creases with n. For H S this is 9.4%. The ratio2 7

w x w xa 3s2 pr7s5p2 d ra MNDO or MNDOrd

for H S decreases with n.2 nThe polarizability values of H S computed2 n

with the MNDO are now discussed. In general weŽ . Ž .observe that P n q 1 ) P n , where P s a , ax x y y

1a Ž .a and a Fig. 6 . However, there are severalz zexceptions to the above trend, most of which areobserved for model 2. We found that for model 1

Ž .we have Fig. 6

a 4 a ) a .x x y y z z

For models 2 and 3

a 4 ax x y y


RAPTIS ET AL.

( )FIGURE 4. The dipole moment of H S determined using the MNDO method as a function of n for models 1]3.2 n

VOL. 19, NO. 151704


( ) ( )FIGURE 5. Plot of a yv; v and y2v; 0, v , v of( )H S STO-3G**++ versus the frequency.2 4

and in most cases the values of a and a differy y z z

very little. For models 1 and 3 a , a , a , andx x y y z z

a change linearly with n for n G 40. However, itwould be a reasonable approximation to consider

Žthat this holds for even smaller values of n e.g.,.for n G 15 . The change of the polarizability prop-

Ž .erties e.g., a , a with n for model 2 presentsx xŽ .several irregularities Fig. 6 , which are induced by

the irregularities in the structure. The dependenceof the longitudinal polarizability and second hy-perpolarizability on the chain length in conjugatedhydrocarbons has been discussed by various au-

Ž .thors e.g., ref. 5g . In a more recent study Tretiaket al.14 connected the jth order off-resonant polar-izabilities of conjugated polymers with the numberof carbon atoms, the bond length alternation, andthe exciton coherence size.

Ž .Employing the three a values models 1]3 ofH S , we calculated the averaged values using a2 n

Boltzmann distribution19c

0 Ž . 0 Ž .P s P exp yH i rkT exp yH i rkT ,Ž . Ž .Ý Ýav i f fi i

Ž .1

where P denotes the a or g values; i defines theŽ . omodel 1]3 ; H , k, and T are the heat of forma-f

Žtion, the Boltzmann constant, and temperature 300.K , respectively. The plot of average values, a , ofav

H S versus n is shown in Figure 7. It has been2 nfound that a of H S are close to the polarizabil-av 2 nity values computed using model 3. This is due tothe fact that model 3 gives structures of lowerenergy in comparison to those produced by mod-

Žels 1 and 2. However, some exceptions have beenobserved, e.g., for H S , where n s 13, 19, and 20,2 nmodels 2 and 3 have approximately, the same

.energy.We found that the molecular shape greatly af-

fects the magnitude of the polarizability and, inparticular, the second hyperpolarizability. This isconfirmed by comparing the a values of model 4with those of models 1]3. It is found that model 4,which involves closed chains, has a smaller a than

Ž .models 1]3 Table III .

FIRST HYPERPOLARIZABILITIES

We now discuss in some detail the first hyper-polarizability of H S for which there are reliable2experimental and ab initio results. Sekino andBartlett19d used the polarizability consistent basisset proposed by Sadlej10a, b and two extended ver-

wsions of it at various levels of theory SCFrTDHF,Ž . xCCSD T , etc. and found for H S that at the2

Ž .SCFrTDHF b ) 0, while at the CCSD T level theircomputed value is negative and agrees satisfacto-rily with the dc-SHG measurement of Ward andMiller.19a Thus, the great importance of the corre-lation contribution for b of H S is clear. This2observation explains why our SCF results, staticand dynamic, computed with Sadlej’s basis set arenot in satisfactory agreement with the experimen-

Ž .tal value Table II . It may be fortuitous, but stillŽ .pleasant, to observe that the STO-3G**qq static

b is in good agreement with the experimentalvalue. The MP2rSTO-3G**qq and the frequencydependent STO-3G**qq b values are less satis-factory. Using the STO-3G**qq results as refer-ence, we observe that the MNDO and MNDOrdgive b values that do not even have the right sign.Shelton and Rice3i noted that some widely used

Ž .semiempirical methods AM1, PM3, MNDO, etc.


RAPTIS ET AL.

( )FIGURE 6. The polarizabilities a , a , a and a of models 1, 2, and 3. The results were calculated using thex x yy zzMNDO method.

VOL. 19, NO. 151706


[ ( )]FIGURE 7. Plot of a and g eq. 1 versus nav av( )H S . The results were computed employing the2 nMNDO method.

‘‘ . . . are not reliable for either the quantitative orthe qualitative determination . . . ’’ of the first hy-perpolarizabilities. At least part of this deficiencyis due to the overestimation of b at the expensez z z

of b and b .3i Taking into account the abovez x x z y y

observations, we shall not proceed to report andanalyze the b values of H S for n ) 11, using2 n

the MNDO or MNDOrd methods, because it hasbeen shown that their reliability for the first hyper-polarizability is questionable. For completeness itis added that Malagoli and Thiel7c developed anextended version of MNDO and MNDOrd where2 p functions for H are also used. They concludedthat ‘‘The resulting specialized MNDO andMNDOrd treatments . . . are not useful for b.’’ŽThese methods, however, give reasonable polariz-ability values and reproduce trends in the second

.hyperpolarizabilities.

SECOND HYPERPOLARIZABILITIES

dc-Electric Field Induced Second Harmonic( )Generation: g y2v; 0, v, v

ŽIt is observed that the computed g y2v ;.0, v, v for H S is in fair agreement with the2

available experimental value measured by Ward19a Ž .and Miller Table II . In Figure 5 we present the

Ž .variation of g y2v ; 0; v, v of H S as a function2 4Žof the frequency the STO-3G**qq basis set was

.employed . It is observed that for a given changeŽin frequency, the resulting effect in g y2v ;

. Ž .0, v, v is larger than that in a yv ; v . The ratioŽ . Ž .g y2v ; 0, v, v rg y0; 0, 0, 0 initially decreases

Žand then stabilizes at 1.4 for n G 6 STO-3G**qq ;.Table II .

( )Static Values: g 0; 0, 0, 0 .

From the results of Table III one observes thatŽ . Ž .g H S yg H S , at both the STO-3G**qq2 nq1 2 n

and the MP2rSTO-3G**qq , pass through a min-imum. It would be perhaps appropriate to addthat this basis may not be particularly good fortreating correlation effects.

We observe that MNDO gives second hyperpo-larizabilities for H S and H S having the wrong2 2 2

Ž .sign they should be positive; Table II . As n in-creases the discrepancy of the semiempiricalŽ . Ž .MNDO and MNDOrd results g with thosedetermined by ab initio methods decreases. For

Ž .certain value of n there is a crossing Table II , forexample, for n s 5

Ž . Ž .g MNDOrd - g Sadlej ,

but for n s 7

Ž . Ž .g MNDOrd ) g Sadlej .

The crossing between the MNDO g values andthose computed with the Sadlej basis set occurs for10 - n - 11. However, the major point of interestin this work is to semiquantitatively show theeffect of geometry variations on the polarizabilitiesand hyperpolarizabilities of H S and this task can2 nbe accomplished satisfactorily by the MNDO

Ž .method Table III .The second hyperpolarizabilities of H S calcu-2 n

lated with the MNDO method will now be dis-cussed. They take values that greatly depend onthe adopted geometry. Thus, for H S , where n s2 n


RAPTIS ET AL.TA

BLE

III.

()

Co

mp

aris

on

of

aan

dg

for

HS

n=

1–

7D

eter

min

edb

yM

ND

O,M

ND

O/// //d

,STO

-3G

**+

+,a

ndM

P2

/// //STO

-3G

**+

+.

2n

()

()

a0;

0/a

ua

yv

;v/a

ua

aa

ba

a,c

a,c

d[

][

]C

ompo

und

MN

DO

MN

DO

/dS

TO-3

G**

++

MP

2/S

TO-3

G**

++

3s2

p/7

s5p

2d

STO

-3G

**+

+3

s2p

/7s5

p2

dE

xp.

ee

HS

9.98

11.2

14.1

14.6

23.2

14.5

24.3

26.0

2f

f24

.225

.9g

g23

.825

.5h

23.7

HS

22.4

24.4

28.2

29.7

40.5

29.0

41.4

22

HS

37.2

39.5

45.8

49.5

59.8

47.0

61.2

23

HS

53.0

55.4

62.6

68.2

79.6

64.2

81.6

24

HS

69.8

71.9

79.7

87.4

100

81.8

103

25

HS

87.4

89.2

97.3

107

99.8

26

HS

106

107

115

127

144

118

27

HS

124

125

133

147

137

28

HS

144

144

151

168

190

156

29

y2

y2

c(

)(

)g

0;0,

0,0

=10

/au

gy

2v

;0,v

,v=

10/a

u

eH

Sy

1.98

0.04

35.0

43.8

53.8

53.1

133

2f

117

g85

.5h

h,i

78.2

103.

4"

0.69

HS

y5.

0215

.972

.195

.783

.210

511

92

2H

S9.

0758

.713

920

213

619

919

92

3H

S31

.710

819

028

418

126

726

62

4H

S65

.717

224

437

723

034

333

32

5H

S11

725

630

147

742

12

6H

S18

835

835

857

834

250

12

7H

S27

547

441

968

858

72

8H

S37

860

548

384

847

967

72

9

()

The

prop

erty

valu

esof

this

tabl

ew

ere

dete

rmin

edus

ing

for

HS

,bo

ndle

ngth

san

dan

gles

optim

ized

byM

ND

Om

odel

3.

2n

aTh

ese

resu

ltsw

ere

dete

rmin

edby

ape

rtur

batio

nth

eory

met

hod

asim

plem

ente

din

MN

DO

/d7

and

GA

ME

SS

.9

bTh

ese

data

wer

eca

lcul

ated

bya

finite

pert

urba

tion

theo

ryap

proa

ch.

cTh

ere

sults

ofth

isco

lum

nw

ere

com

pute

dat

694.

3nm

unle

ssot

herw

ise

spec

ified

.d

The

cite

dm

ean

mol

ecul

arpo

lariz

abili

tyva

lues

26

wer

ede

rived

bya

criti

cale

valu

atio

nof

liter

atur

eda

ta.

eA

t48

8nm

.26

fA

t51

4.5

nm.2

6

gA

t63

2.8

nm.2

6

hA

t69

4.3

nm.

iM

etho

d:dc

-ele

ctric

field

indu

ced

seco

ndha

rmon

icge

nera

tion

inth

ega

sph

ase.

19a

VOL. 19, NO. 151708


1]51, the computed values are in atomic unitsŽ .Fig. 8 :

y0.198 = 103 F g F 0.73 = 105 model 1,

y0.198 = 103 F g F 9.05 = 105 model 2,

y0.198 = 103 F g F 9.05 = 105 model 3.

Figure 8 clearly shows that the dominant com-ponent in all three models is g . In several casesx x x xthe absolute values of g and g are largerx x y y x x z zthan those of g and g . In particular, thisy y y y z z z zwas found for n G 14 of model 1. The variation ofthe components g , g , and g as a func-x x y y x x z z y y z z

Ž .tion of n H S is given as supplementary infor-2 nŽ .mation Fig. A .

The geometry adopted in model 1 shows thatby varying g versus n we observe one local mini-

Ž .mum for n s 7 Fig. 9 . Model 2 shows that varia-tion of g versus n presents several irregularities,while model 3 shows a smooth increase of g withŽ .n Fig. 9 . In models 2 and 3, g varies within the

same bounds, although the change of g as a func-tion of n presents many dissimilarities in the twomodels. Models 1 and 3 differ widely in the abso-lute values of g ; for example, for H S2 51

Ž . Ž .g model 3 rg model 1 s 12.4.

The second hyperpolarizabilities of moleculeswith structures described by model 4 are smallerthan those of molecules with geometries given bymodels 2 and 3, but larger than the hyperpolariz-

Žabilities of molecules that belong to model 1 Table.IV . In Figure 7 the g of H S versus n is pre-av 2 n

sented. The averaged values were calculated usingŽ .eq. 1 . We found that g of H S is close to the gav 2 n

of molecules, having structures that belong tomodel 3. The explanation of this observation wasdiscussed in connection with the plot of a ver-avsus n.

From the results in Figures 6 and 8 one deducesthat there is a similarity in the change of a andi ig of H S with n, which may be explained byi i i i 2 ntaking into account that there is no charge transferin the considered compounds. A similar trend isfound for the pair arg . This observation may be

Ž . Ž .used to predict g or g from a or a . Fori i i i i iexample, using the a values of H S , n s 13]202 nand 30, we predicted the g values of H S , n s2 n1]12 and 40]51 for models 1]3. The computedŽ .MNDO and the predicted values are shown inFigure 10.

We found that the a and g of H S vary with n2 nfollowing a completely different pattern than that

Ž .of m Figs. 4, 6, 8 .

VIBRATIONAL POLARIZABILITIES ANDHYPERPOLARIZABILITIES

The properties discussed in the previous sec-tions were the electronic contributions. The follow-ing analysis refers to the vibrational contributionsto the polarizabilities and second hyperpolarizabil-ities. The superscripts e and v will be used todenote the electronic and vibrational contributions

Ž e v .respectively e.g., g and g .There is a growing body of evidence that sug-

gests that static g v is not negligible. Some well-known cases taken from the excellent review arti-cle of Bishop20 are SF12a, where the vibrational6contribution is 20 times larger than the electronicand Hq for which the vibrational hyperpolariz-2ability is 10 times larger then the electronic one.21, 22

Some other examples, where the vibrational hyper-polarizability is significant but less spectacular,

Ž 23 23 24have already been reported e.g., NH , CH , Li ,3 4 225. 23aand LiH . Bishop and Dalskov recently pre-

sented a survey of methods by which one cancalculate the vibrational contributions to polariz-abilities and hyperpolarizabilities.

Bishop and Kirtman,12a ] c whose approach weemploy in this work, define the vibrational contri-butions to the polarizability and second hyperpo-larizability, av and g v, respectively, by

2, 0 1, 10, 0v 2 2w x w x w xa s m q m q m ,2

0, 0 2, 0 1, 1 0, 0v 2 2w x w x w x w xg s a q a q a q mb

1, 02, 0 1, 1 2w x w x w xq mb q mb q m a

0, 1 2, 0 1, 12 4 4w x w x w xq m a q m q m .

w x i, jExpressions for A , where i and j denoteorders of electrical and mechanical anharmonicity,respectively, were given by Bishop and Kirtman.12c

These authors considered i F 2 and j F 1. In thepresent work we took into account quadratic and

Ž .cubic energy derivatives m s 0, 1 , first- and sec-Ž .ond-order dipole moment derivatives n s 1, 2 ,

first- and second-order polarizability derivativesŽ .o s 1, 2 , and first-order hyperpolarizability

Ž .derivatives p s 1 . The various levels of anhar-monicity considered in this work are denoted byŽ .mnop .


RAPTIS ET AL.

( )FIGURE 8. The hyperpolarizabilities g , g , g and g of models 1]3. The properties were calculated usingx x x x yyyy zzzzthe MNDO technique.

VOL. 19, NO. 151710


( )FIGURE 9. A comparative study of g MNDO of H S for models 1]3.2 n


RAPTIS ET AL.

TABLE IV.Comparison of Polarizabilities of Models 1 – 4.

a / au

H S S2 n nNo.Sulfur Atoms Model 1 Model 2 Model 3 Model 4

8 119 115 126 10812 190 189 206 17416 262 267 287 24120 335 370 370 308

These computations were performed by using the MNDO6

method.

( )FIGURE 10. g 9 = g / au values of H S , n = 1 ]122 nand 40 ]51, estimated by using the formula g 9 = c a92

1+ c a9 + c where a9 = a / au The constants are2 3determined from least squares fitting using the a9 andg 9 values of H S , n = 13 ]20 and 30.2 n

TABLE V.Comparison of Second Hyperpolarizabilities ofModels 1 – 4.

y2g = 10 / au

H S S2 n nNo.Sulfur Atoms Model 1 Model 2 Model 3 Model 4

8 y51.7 139 275 13.112 y15.2 455 773 19116 44.3 970 1440 40020 114 2210 2210 676

The MNDO6 method was used for these computations.

Ž . v vThe static and dynamic at 694.3 nm a and gof H S, H S , and H S are reported in Tables V2 2 2 2 3and VI. Specifically, the contribution related to theelectric field induced second harmonic generationŽ . v Ž .ESHG , g y2v ; 0, v, v , is presented. Thesecomputations were performed at the SCF levelusing the contracted Gaussian type orbitalsw x 103s2 pr7s5p2 d suggested by Sadlej.

v Ž .We observe from Table VI that a 0; 0 in-Ž .creases with n H S . Indeed the static vibrational2 n

polarizability is zero for H S and almost negligible2for H S but becomes a more substantial contribu-2 2tion for H S . For all three molecules, the dynamic2 3

v Ž .property a yv ; v is zero at the selected fre-quency.

v n ŽAs expected, the results for g of H S n s2.1]3 are more interesting. We observe from Table

v Ž .VII that g 0; 0, 0, 0 is small but not negligible forH S and increases dramatically with n; it is almost2half the g e for H S , while for H S it exceeds the2 2 2 3electronic contribution . W e found that

v Ž . Ž .g y2v ; 0, v, v is very small for H S n s 1]3 .2 nA similar observation was made by Bishop andDalskov 23a for CH , NH , H O, and HF, who4 3 2

TABLE VI.Static and Frequency Dependenta Electronic andVibrational Contribution to Polarizabilities of H S,2

bH S , and H S Computed at SCF Level.2 2 2 3

e v 3 v( ) ( ) ( ) ( )a 0; 0 a 0; 0 a yv; v a yv; v/au /au /au /au

H S 23.2 0.0 23.7 0.02H S 40.5 0.9 41.4 0.02 2H S 59.8 4.3 61.2 0.02 3

a At 694.3 nm.b [ ] 10Basis set: 3s2p / 7s5p2d .

VOL. 19, NO. 151712


TABLE VII.Static and Frequency Dependenta Electronic and Vibrational Contribution to Second Hyperpolarizabilities of

bH S, H S , and H S , Computed at SCF Level.2 2 2 2 3

v e v( ) ( ) ( ) ( )g 0; 0, 0, 0 g 0; 0, 0, 0 g y2v ; 0, v , v g y2v; 0, v , vy2 y 2 y 2 y 2=10 / au =10 / au =10 / au =10 / au

H S 53.8 6.8 78.2 y0.22H S 83.2 38.2 119 y0.12 2H S 136 171 199 2.82 3

a At 694.3 nm.b [ ] 10Basis set: 3s2p / 7s5p2d .

v Ž .found that g y2 v ; 0, v , v for the abovemolecules was less than 5% of the correspondingg e. It is useful to note that the ratio of staticcontributions to the hyperpolarizability g erg v are9.4 and 7.9, for H O23 and H S, respectively. Ten-2 2tative computations of the static g v of H S con-2 4firm the above findings and show that it is 256%of g e.

v Ž .To gain some insight into a static and inv Ž .particular g static , it is useful to analyze these

properties at various anharmonicity levels. Fromthe results of Table VIII we observe that av shows

Ž .small variation with mnop . Harmonic potentialapproximation appears to adequately describe av

for H S and H S . For H S , adding cubic terms2 2 2 2 3leads to a small increase in av. For both m s 0 andm s 1, electrical anharmonicity has no effect onav. In contrast, g v shows considerable dependenceon the level of electrical anharmonicity, particu-larly for H S . However, although derivatives of a2 3appear to have a significant effect on g v, the first-order derivatives of b appear to be less important;

Ž .for example, the difference between levels 1210

Ž . Ž .and 1220 far exceeds that between 1220 andŽ .1221 . This observation is interesting because thefirst hyperpolarizability of H S is not satisfactorily2described at the Hartree]Fock level using the em-ployed basis set. However, the reduced effect offirst-order derivatives of b may be associated with

w x i, jthe fact that out of the 10 terms A that de-v Ž .scribe g in eq. 3 , only two terms involve the

Ž .first hyperpolarizability derivatives first orderw x0, 0 w x1, 1 w x2, 0mb and mb . The term mb involvessecond-order derivatives of the first hyperpolariz-ability, which are not considered in this work.

Summarizing the results of this section, we notev Ž .that g static of H S makes a significant contri-2 n

bution to the second hyperpolarizability, exceptfor H S, where it is small but not negligible.2

Synopsis and Conclusions

We used several theoretical ab initio andsemiempirical methods to systematically study theway in which the variation of the geometry of

TABLE VIII.v v ( )Static a and g of H S n = 1 – 3 Calculated at Various Levels of Anharmonicity.2 n

H S H S H S2 2 2 s 3

a y 2 y 2 y 2( )mnop a / au g = 10 / au a / au g = 10 / au a / au g = 10 / au

( )0200 0.01 0.00 0.85 1.68 4.11 25.1( )0210 0.01 6.58 0.85 27.7 4.11 68.6( )0220 0.01 6.68 0.85 42.4 4.11 163( )0221 0.01 6.62 0.85 43.0 4.11 176

( )1200 0.01 0.00 0.86 2.67 4.26 59.2( )1210 0.01 6.58 0.86 23.3 4.26 69.5( )1220 0.01 6.83 0.86 37.7 4.26 159( )1221 0.01 6.77 0.86 38.2 4.26 171

a ( )mnop defines the order of anharmonicity. Specifically, m, n, o, and p defines the order of energy, the dipole moment, the( )polarizability, and the hyperpolarizability derivatives, respectively. The maximum anharmonicity employed in this work is 1221 .


RAPTIS ET AL.

Ž .H S affects their properties a , a, g , etc. and2 n i i i i i ito comment on the implications of our observa-tions. The ab initio study involved the computationof the static and frequency dependent electronicand vibrational contributions to the polarizabilitiesand second hyperpolarizabilities of several mem-bers of the series H S . Two basis sets were em-2 n

Ž w x.ployed STO-3G**qq and 3s2 pr7s5p2 d . Cor-relation was taken into account for some of thederivatives by using the MP2 theory. At thesemiempirical level the MNDO6 and MNDOrd7

methods were used. Of primary importance for thepresent work are trends and differences, the valid-ity of which is safeguarded to a great extent by theconsidered variety of methods. The main findingsof the present work follow.

There is a similarity in the variation of the pairsa rg and arg of H S with n, which may bei i i i i i 2 nexplained by taking into account that there is nocharge transfer in the considered molecules. Thisobservation may be used to predict the second

Žhyperpolarizabilities if the polarizabilities which.require much lower computational cost are

known. We demonstrated that the longitudinalcomponent has the dominant contribution to thepolarizability and second hyperpolarizability ofH S . A similar conclusion was reached for the2 ncontribution of this component to the average sec-ond hyperpolarizability of polyenic chains.13e

Ž .In the considered set of test molecules H S a2 nremarkable variation of the hyperpolarizabilitieswith the geometries was found. This result is sig-nificant because it shows that molecular geometryis an important tool for designing nonlinear opticalmaterials.

v Ž .The vibrational hyperpolarizabilities g staticof H S increase with n, and for H S it was2 n 2 3demonstrated that g v ) g e. The present results, inconnection with those found in the literature, sug-

v Ž .gest that g static should be computed wheneverone aims at reasonably accurate molecular hyper-

v Ž .polarizabilities. Our values for g y2v ; 0, v, vare only a small percentage of g e. Studies on othermolecules also found that the g v associated withESHG is small.

Acknowledgments

M. G. P. thanks Professor W. Thiel for providinghim with a copy of his MNDOrd program, Profes-sor N. C. Handy and Drs. A. Willetts and R. D.Amos for allowing him to use their programs

Ž .CADPAC and SPECTRO , and Professor D. M.Bishop for sending him a preprint of his reviewarticle in Advances in Chemical Physics. We alsothank Dr. V. E. Ingamells for reading themanuscript and making useful suggestions.

References

1Ž . Ž1. a The mean polarizability a is defined by a s a qx x3

.a q a , where the suffixes x, y, and z denote Carte-y y z zsian components. The average second hyperpolarizability is

1 Ždetermined by: g s g q g q g q 2g qx x x x y y y y z z z z x x y y5

. Ž .2g q 2g ; b M. P. Bogaard and B. J. Orr, In Physi-x x z z y y z zcal Chemistry, Series Two, vol. 2, A. D. Buckingham, Ed.,MTP International Review of Science, Butterworths, Lon-

Ž .don, 1975, p. 149; c A. D. Buckingham and B. J. Orr,Ž .Quantum Rev. Chem. Soc., 21, 195 1967 .

Ž .2. a S. M. Le Cours, H.-W. Guan, S. G. Di Magno, C. H.Wang, and M. J. Thierien, J. Am. Chem. Soc., 118, 1497Ž . Ž .1996 ; b S. Priyadarshy, M. J. Thierien, and D. N. Beratan,

Ž . Ž .J. Am. Chem. Soc., 118, 1504 1996 ; c P. Norman, D.˚Jonsson, O. Vahtras, and H. Agren, Chem. Phys., 203, 23

Ž .1996 .

Ž .3. a M. Nakano, S. Yamada, I. Shigemoto, and K. Yam-Ž . Ž .aguchi, Chem. Phys. Lett., 250, 247 1996 ; b Z. Shuai, S.

Ramasesha, and J. L. Bredas, Chem. Phys. Lett., 250, 14´Ž . Ž .1996 ; c F. Aiga and R. Itoh, Chem. Phys. Lett., 251, 372Ž . Ž .1996 ; d M. Nakano, S. Yamada, I. Shigemoto, and K.

Ž . Ž .Yamaguchi, Chem. Phys. Lett., 251, 381 1996 ; e M. L.Dekhtyar and V. M. Rozenbaum, J. Phys. Chem., 101, 809

˚Ž . Ž .1997 ; f P. Norman, Y. Luo, D. Jonsson, and H. Agren, J.Ž . Ž .Chem. Phys., 106, 1827 1997 ; g J. O. Morley, Int. J.Ž . Ž .Quantum Chem., 61, 991 1997 ; h B. Kirtman, B. Cham-

Ž . Ž .pagne, and J.-M. Andre, J. Chem. Phys., 104, 4125 1996 ; i´Ž . Ž .D. P. Shelton and J. E. Rice, Chem. Rev., 94, 3 1994 ; j D. R.

Kanis, P. G. Lacroix, M. A. Ratner, and T. J. Marks, J. Am.Ž . Ž .Chem. Soc., 116, 10089 1994 ; k D. Mukhopadhyay and Z.

Ž . Ž .G. Soos, J. Chem. Phys., 104, 1600 1996 ; l S. R. Marder, W.E. Torruellas, M. Blanchard]Desce, V. Ricci, G. I. Stegeman,S. Gilmour, J.-L. Bredas, J. Li, G. U. Bublitz, and S. G. Boxer,´

Ž . Ž .Science, 276, 1233 1997 ; m J. L. Toto, T. T. Toto, C. P. deMelo, B. Kirtman, and K. Robins, J. Chem. Phys., 104, 8586Ž .1996 .

Ž .4. a D. J. Williams, Ed., Nonlinear Optical Properties of Organicand Polymeric Materials, Americal Chemical Society, Wash-

Ž .ington, D.C., 1983; b S. R. Marder and J. W. Perry, Science,Ž . Ž .263, 1706 1994 ; c P. N. Prasad and D. J. Williams,

Introduction to Nonlinear Optical Effects in Molecules and Poly-mers, Wiley, New York, 1991.

Ž .5. a C. J. Marsden and B. S. Smith, J. Phys. Chem., 92, 347Ž . Ž . Ž . Ž .1988 ; b A. Rauk, J. Am. Chem. Soc., 106, 6517 1984 ; cD. Beljonne, Z. Shuai, and J. L. Bredas, J. Chem. Phys., 98,´

Ž . Ž .8819 1993 ; d M. G. Papadopoulos and J. Waite, NonlinearŽ . Ž .Opt., 7, 65 1994 ; e S. R. Marder, J. W. Perry, G. Bourhill,

C. B. Gorman, B. G. Tiemann, and K. Mansour, Science, 261,Ž . Ž . Ž .186 1993 ; f B. Kirtman, Chem. Phys. Lett., 143, 81 1988 ;

Ž .g G. J. B. Hurst, M. Dupuis, and E. Clementi, J. Chem.Ž . Ž .Phys., 89, 385 1988 ; h C. P. De Melo and R. Silbey, Chem.

Ž . Ž .Phys. Lett., 140, 537 1987 ; i B. Kirtman and M. Hasan,

VOL. 19, NO. 151714


Ž . Ž .Chem. Phys. Lett., 157, 123 1989 ; j D. Li, M. A. Ratner,Ž . Ž .and T. J. Marks, J. Am. Chem. Soc., 110, 1707 1988 ; k Y.-J.

Ž . Ž .Lu and S. L. Lee, Int. J. Quantum Chem., 44, 773 1992 ; l Z.Huang, D. A. Jelski, R. Wang, D. Xie, C. Zhao, X. Xia, and

Ž .T. F. George, Can. J. Chem., 70, 372 1992 .Ž .6. a M. J. S. Dewar and W. Thiel, J. Am. Chem. Soc., 99, 4899Ž . Ž .1977 ; b J. J. P. Stewart and F. J. Seiler, MOPAC 6.00,Research Laboratory, United States Air Force Academy,1990.Ž .7. a W. Thiel and A. A. Voityuk, Theor. Chim. Acta, 81, 391Ž . Ž .1992 ; b W. Thiel and A. A. Voityuk, J. Phys. Chem., 100,

Ž . Ž .616 1996 ; c M. Malagoli and W. Thiel, Chem. Phys., 206,Ž .73 1996 .

Ž .8. C. Møller and M. S. Plesset, Phys. Rev., 46, 618 1934 .9. M. W. Schmidt, K. K. Baldrige, J. A. Boatz, S. T. Elbert, M. S.

Gordon, J. H. Jensen, S. Kozeki, N. Matsunaga, K. A.Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A.

Ž .Montgomery, J. Comput. Chem., 14, 1347 1993 .Ž . Ž . Ž .10. a A. J. Sadlej, Col. Cz. Chem. Com., 53, 1995 1988 ; b A. J.

Ž .Sadlej, Theor. Chim. Acta, 79, 123 1991 .11. H. A. Kurtz, J. J. P. Stewart, and K. M. Dieter, J. Comput.

Ž .Chem., 11, 82 1990 .Ž .12. a B. Kirtman and D. M. Bishop, Chem. Phys. Lett., 175, 601Ž . Ž .1990 ; b D. M. Bishop and B. Kirtman, J. Chem. Phys., 95,

Ž . Ž .2646 1991 ; c D. M. Bishop and B. Kirtman, J. Chem.Ž . Ž .Phys., 97, 5255 1992 ; d M. J. Cohen, A. Willetts, R. D.

Ž .Amos, and N. C. Handy, J. Chem. Phys., 100, 4467 1994 ;Ž .e R. D. Amos, I. L. Alberts, J. S. Andrews, S. M. Colwell,N. C. Handy, D. Jayatilaka, P. J. Knowles, R. Kobayashi, N.Koga, K. E. Laidig, P. E. Maslen, C. W. Murray, J. E. Rice, J.Sanz, E. D. Simandiras, A. J. Stone, and M.-D. Su, CADPAC,The Cambridge Analytic Derivatives Package Issue 5, Cam-

Ž .bridge, U.K., 1992; f A. Willetts, J. F. Gaw, W. H. Green,and N. C. Handy, In Advances in Molecular Vibrations andCollision Dynamics, vol. 1B, J. M. Bowman and M. A. Ratner,

Ž .Eds., JAI Press, Greenwich, CT, 1991; g Conversion fac-tors:

1 au dipole moment ; 2.54174 D; 8.47831 = 10y3 0 C m

1 au of polarizability ; 0.148176 = 10y24 esu; 0.164867 = 10y40 C2 m2 Jy1

1 au of second ; 0.503717 = 10y39 esuhyperpolarizability ; 0.623597 = 10y64 C4 m4 Jy3

Ž .13. a A. Dulcic, C. Flytzanis, C. L. Tang, D. Pepin, M. Fetizon,´Ž . Ž .and Y. Hoppilliard, J. Chem. Phys., 74, 1559 1981 ; b C.

Flytzanis, In Nonlinear Behaviour of Molecules, Atoms andIons in Electric, Magnetic or Electromagnetic Fields, L. Neel,

Ž .Ed., Elsevier, Amsterdam, 1979; c J. P. Hermann and J.Ž . Ž .Ducuing, J. Appl. Phys., 45, 5100 1974 ; d V. Dobrosavlje-

Ž . Ž .vic and R. M. Stratt, Phys. Rev. B, 35, 2781 1987 ; e C. P.´Ž . Ž .de Melo and R. Silbey, J. Chem. Phys., 88, 2567 1988 ; f S.

R. Marder, L.-T. Cheng, B. G. Tiemann, A. C. Friedli, M.Blanchard]Desce, J. W. Perry, and J. Skindhøj, Science, 263,

Ž . Ž .511 1994 ; g C. B. Gorman and S. R. Marder, Proc. Natl.Ž . Ž .Acad. Sci., 90, 11297 1993 ; h D. R. Kanis, M. A. Ratner,

Ž . Ž .and T. J. Marks, Chem. Rev., 94, 195 1994 ; i I. D. W.Samuel, I. Ledoux, C. Dhenaut, J. Zyss, H. H. Fox, R. R.

Ž .Schrock, and R. J. Silbey, Science, 265, 1070 1994 .

14. S. Tretiak, V. Chernyak, and S. Mukamel, Phys. Rev. Lett.,Ž .77, 4656 1996 .

Ž .15. a S. N. Yaliraki and R. J. Silbey, J. Chem. Phys., 104, 1245Ž . Ž . Ž .1996 ; b D. M. Bishop, Adv. Chem. Phys., to appear; c B.Kirtman and B. Champagne, Int. Rev. Phys. Chem., 16, 389Ž .1997 .Ž .16. a S. M. Nasiou, I. N. Demetropoulos, M. G. Papadopoulos,and S. Raptis, 16th Panhellenic Conference of Chemistry, 1995,

Ž . Ž .p. 858; b Serena Software, Bloomington, IN; c N. L.Ž . Ž .Allinger, J. Am. Chem. Soc., 99, 8127 1977 ; d U. Burkert

and N. L. Allinger, Molecular Mechanics, ACS Monograph177, American Chemical Society, Washington, D.C., 1982;Ž .e M. Saunders, K. N. Houk, Y. D. Wu, W. C. Still, M.Lipton, G. Chang, and W. C. Guida, J. Am. Chem. Soc., 112,

Ž .1419 1990 .

17. N. N. Greenwood and A. Earnshaw, Chemistry of the Ele-ments, Pergamon, Oxford, U.K., 1985.Ž .18. a F. A. Cotton and G. Wilkinson, Advanced InorganicChemistry, 5th ed., Wiley, New York, 1988, p. 495: ‘‘Whenmolten sulfur is poured into ice water, the so-called plasticsulfur is obtained; although normally this has S8 inclusions,it can be obtained as long fibers by heating S in nitrogen ata3008C for 5 min and quenching a thin stream in ice water.These fibers can be stretched under water and appear tocontain helical chains of sulfur atoms with ; 3.5 atoms per

Ž . Ž .turn.’’ b I. Mayer, J. Mol. Struct. Theochem. , 149, 81Ž . Ž . Ž . Ž .1987 ; c B. Meyer, Chem. Rev., 76, 367 1976 ; d K.

Ž . Ž .Steliou, Acc. Chem. Res., 24, 341 1991 ; e G. P. Agrawal,Ž .C. Cojan, and C. Flytzanis, Phys. Rev. B, 17, 776 1978 .

Ž .19. a J. W. Ward and C. K. Miller, Phys. Rev. A, 19, 826Ž . Ž .1979 ; b A. L. McClellan, Tables of Experimental Dipole

Ž .Moments, W. H. Freeman, London, 1963; c N. MatzuzawaŽ . Ž .and D. A. Dixon, Int. J. Quantum Chem., 44, 497 1992 ; d

Ž .H. Sekino and R. J. Bartlett, J. Chem. Phys., 98, 3022 1993 .Ž .20. D. M. Bishop, Rev. Modern Phys., 62, 343 1990 .

Ž .21. D. P. Shelton and L. Ulivi, J. Chem. Phys., 89, 149 1988 .

22. D. M. Bishop, J. Pipin, and J. N. Silverman, Mol. Phys., 59,Ž .165 1986 .

Ž .23. a D. M. Bishop and E. K. Dalskov, J. Chem. Phys., 104,Ž . Ž .1004 1996 ; b D. M. Bishop, B. Kirtman, H. A. Kurtz, and

Ž .J. E. Rice, J. Chem. Phys., 98, 8024 1993 .

24. M. G. Papadopoulos, A. Willetts, N. C. Handy, and A. D.Ž .Buckingham, Mol. Phys., 85, 1193 1995 .

25. M. G. Papadopoulos, A. Willetts, N. C. Handy, and A. E.Ž .Underhill, Mol. Phys., 88, 1063 1996 .

26. M. P. Bogaard, A. D. Buckingham, R. K. Pierens, A. W.Ž .White, Trans. Faraday Soc. I, 74, 3008 1978 .


Documents

Static and frequency dependent polarizabilities and hyperpolarizabilities of H2Sn