Modeling hyperpolarizabilities of some TICT molecules and their

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7491

J. Phys. Chem. 1993,97, 7491-7498

Modeling Hyperpolarizabilities of Some TICT Molecules and Their Analogues R. Sen, D. Majumdar, and S. P. Bhattacharyya’ Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India

S. N. Bhattacharyya Department of Pure Chemistry, University of Calcutta, 92, A.P.C. Rd., Calcutta 700 009, India Received: January 28, 1993

Ground-state hyperpolarizabilities of a prototypical twisted intramolecular charge transfer (TICT) molecule (DMABN) and its analogues have been calculated by the finite-field SCF method at the CNDO/S level of approximation. The ground-state geometry of each molecule has been optimized by the MNDO method. Excited-state hyperpolarizabilities have been calculated by invoking the finite-field S C F single excitation C I method at the CNDO/S level of approximation. The study focuses attention on the possibly important role played by the TICT state in generating large quadratic polarizability in some of these molecules. The importance of proper stereochemical disposition of the donor group is analyzed. It is predicted that an optimal combination of donating and accepting abilities of the donor and acceptor moieties and a suitable stereochemical disposition of the donor group could maximize quadratic polarizability j3 in this particular class of molecules. A nonlinear scale is proposed to quantify the effects of donor and acceptor groups on first hyperpolarizability. The observed features are rationalized in terms of an exactly solvable 2 X 2 model.

1. Introduction

SCHEME I

Looking for molecules and materials possessing large nonlinear optical (NLO) coefficients and other fascinating properties is a very active area of contemporary research in materials science and te~hnology.l-~While the ultimate suitability of a material or molecule for a specific technological application can only be decided by actual experiment, theoretical calculationand analysis of the desired properties and optimal conditions for their emergence at the microscopic and macroscopic levels can be a powerful tool for handling the basic design problem and may help to narrow the area of search for developinga specificmaterial. This has led to the development of the so-called “molecular engineering”approach to materialsdesigning problems. The main goal in the design of nonlinear optical material is to conceive of and synthesize molecules with high hyperpolarizabilities (quadratic or cubic) with the ultimate hope of maximizing the bulk hyperpolarizabilities. It is no wonder then that a considerable body of theoretical research on nonlinear optical properties is available in the current literature.4~5 Naturally, several molecular criteria have been proposed from time to time for selecting potentially active nonlinear optical m0lecules.6.~ For the quadratic polarizability, the very first requirement is the absence of a center of inversion. It is wellknown that highly polarizabledissymetricconjugated r-electron molecules usually have much larger hyperpolarizabilities than their nonconjugated counterparts. It is also known that the r-electrons play a dominant role in shaping actual values of the hyperpolarizabilities,the u-core being rather passive in the whole process. It has often been speculated that one can possibly optimizequadratic polarizability j3 by designing a comparatively long conjugated r-electron system carrying substituent radicals with high mesomeric moments, in appropriate position^.^^^ Thus, the presence of a donor (D) and an acceptor (A) at the opposite ends of a conjugated system accentuates intramolecular (D A) charge transfer that results in an additionalcontribution (&T) to the quadratic polarizability (j3 = 80 + BCT),*.~with contributing almost 90% to 8.

-

To whom correspondence should be addressed.

,CHJ

Numerous electron donor-acceptor molecules (D-X-A) with the donor and the acceptor groups linked to each other through X (X is usually an aromatic ring system) by means of single bonds are known to respond to electronic excitation with the formation of a highly polar (charge-transfer) state in which the D+ and A- subunits are held in a mutually perpendicular conformationl0Jl (Scheme I). The existence of such a “twisted” intramolecular charge transfer (TICT) state appears to be well established by spectroscopic, quantum chemical, and thermodynamic evidence. It is generallybelieved that the TICT process is primarily determined by the presence of a charge-transfer (CT) state that is stabilized by twisting around the D-X bond (Scheme I), the stabilization increasingwith the increase in polarity of the surrounding medium. Now, the so-called “optical model” suggest^^.^ that the availability of a low-lying charge-transfer state with a large dipole moment pcx,with pexmuch larger than the ground-state dipole moment M,, is one of the favorable components that could trigger the emergence of large 8. This assertion of the optical model makes the TICT molecules naturally important candidates in the search for molecules with large NLO coefficients. Our preliminary investigation12revealed that this conjecture has a substantial element of truth and needs further exploration. We have therefore undertaken a systematic study of a prototypical TICT molecule (viz., DMABN, see Table I) and some closely related species with a view to probing the following questions: (i) Do the (T)ICT molecules generally possess large j3? (ii) What kind of donor-acceptor combination tends to maximize 8 in a series of structurally related (T)ICT molecules? (iii) How important is the stereochemicaldisposition of the donor group in the emergence of large NLO response of the molecules

0022-3654/93/2097-7491$04.00/00 1993 American Chemical Society

7492 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993

Sen et al.

TABLE I: Molecules Studied, Their Acronyms Used in the Text, and the Computed Dipole Moments of the Ground (h)and the Excited (T)ICI’ State (pa)’ molecule acronym Pg (D) k (D) AE (ev) AP (D) 4-(N,N-dimethylamino)benzonitrile DMABN 5.0001 17.8664 4.912 12.8663 MMABN 4.8774 17.7882 4.733 12.9108 4-(N-methylamino)benzonitrile ABN 4.6633 15.0115 4.481 10.3482 4-aminobenzonitrile 4-(NJV-dimethylamino)nitrobenzene DMANB 7.2949 17.6873 4.703 10.3924 MMANB 3.6987 6.1088 6.813 2.4101 4-(N-methylamino)nitrobenzene 6.9137 16.7160 4.698 9.8023 4-aminonitrobenzene(p-nitroaniline) PNA 4.6896 14.2476 6.154 9.5580 4-(N,N-dimethylamino)nitrosobenzene DMANSB MBN 4.7569 15.9213 4.715 11.1644 4-methoxybenzonitrile DMAS 1.4549 7.0727 6.302 5.6178 4-(N,N-dimethylamino)styrene 4-(N,N-dimethyfamino)-2’,2’-dicyanostyrene DMADCS 5.4072 17.0032 4.845 11.5960 3.4235 16.2015 4.788 12.7780 2-(N,N-dimethylamino)-5-cyanopyrimidine DMACP 2-(N,N-dimethylamino)-5-nitropyrimidine DMANP 5.6690 15.4563 4.945 9.7873 *Theincrease in dipole moment (&) following excitation into the (T)ICTstate and the corresponding excitation energy (AE)of these molecules are also included.

concerned? (iv) How important is the role or the TICT state itself in controlling the generation of large NLO response in these molecules? (v) Can the trends observed in the calculation of B be rationalized in terms of a simple model? The outlay of our paper is as follows. In section 2, we briefly discuss the salient features of the method used. The molecules studied by us are described in section 3. Section 4 summarizes our results, while section 5 is devoted to modeling the computed j3 values and rationalizing the observed trends within the limitations of an exactly solvable two-state model.

2. Method When a molecule of dipole moment p~ is placed in a homogeneous electric field E, a dipole moment p is generated which can be developed in increasing powers of the applied field E as follows:

It is a simple matter to calculatethe field-modifieddipole moment of the molecule (E) as (c() = Tr POI.),wkere D is the dipole matrix corresponding to the dipole operator ( p ) defined earlier; expand ( p ) as a function of the applied field strength (E); and obtain the desired tensorial coefficients by straightforward numerical differentiation or polynomial fitting, etc. Alternatively,one may adopt a perturbativeapproachand obtain thespecificcomponents, a,@, y, etc., by using appropriate sum-over-states expressions. One can extend the same variational technique to obtain the polarizability and hyperpolarizabilities in excited states. To do that, we invoke the singly excited configuration interaction (SECI) procedure to obtain the field-modulated excited-state wave functions

3f = C C I @ [

(6)

where @IS are single determinant configuration functions of appropriate spin symmetry. The Zth excited state wave function is then contracted to generate the one-electron density matrix in the MO (&) basis14

where p ~ isi the ith component ( i = x, y , z) of the permanent dipole moment h a n d the tensorial terms ag,&,and yijklrepresent linear and higher polarizabilities of the molecule. In the presence of a homogeneous dc field (E), the manyelectron free molecular electronic Hamiltonian (&) takes on an additional potential energy term:

f i 0 = H,-;E where; is the total dipole moment operator of the molecule given by

(3) i= I

where i = is and j = j,. P I is then diagonalized, generating the field-modified natural orbitals {$# and their occupation numbers (n:) for the Zth excited state. This leads to the naturaloneelectron density matrix [ p i ] for the given Zth excited state, which is then transformed to the A 0 basis This is used to compute the excited-state dipole moment ( pn*) as follows:14

(E).

A

In eq 3, the first sum runs over the (valence) electrons and the second over all the nuclei in the molecule, wiih r.4 denoting the positional coordinate of the atom A. Let P be a trial wave function, a single Slater determinant for a closed-shell molecule. One can then invoke the variational recipe to determine the fieldmodified Hartree-Fock orbitals which leads to the so-called finitefield SCF equation. When a finite fixed basis set expansion is used to represent the field-modified Hartree-Fock orbitals, we must solve the following matrix equation13

pc

(7)

sc;

4,

(4)

where = &?p:xpr OSC = 1, and P = FO- PE. Once the field-modified orbitals are determined by the traditional SCF procedure, we can easily construct a field-modified ground-state one-electron density matrix (PO),in x basis, as follows:

(4,)

Po = 2

3

(9)

e

= cpfict

The theoretical framework outlined above is perfectly general and can be applied in an ab initio as well as in a semiempirical framework. In fact, for the majority of molecules of practical interest the question of computational tractability would often force one to resort to semiempirical theories. In the context of polarizability calculations (both linear and nonlinear), semiempirical MO theories have often been invoked fairly successfully. Extensive calculations using the finite-field method within the INDO framework have been carried out by Zyss et allS Lalama and Garito,16 on the other hand, have invoked an excited-state perturbation theory within the framework of the CNDO method and computed the hyperpolarizabilitiesof substituted benzenes.17

The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7493

Modeling Hyperpolarizabilities of TICT Molecules

SCHEME I1

dispersion function of the two-level model defined as follows:

F(hE,w) =

(fw4 [(W2 - (h421[(W2 - (2h421

(14)

A reparametrized version of the CNDO/S in conjunction with the excited-state perturbation theoretical model of Lalama and Garit016 has been used by Docherty et a1.18to compute the first hyperpolarizability of a large number of organic ICT molecules. More recently, INDO/S19J0 and MNDO methodsz1have been used successfully for computing both linear and nonlinear polarizabilities within the framework of coupled Hartree-Fock theory and in the finite-field SCF method, respectively. In our calculations reported here, we have first invoked the semiempirical MNDO methodz2 to obtain the ground-state equilibrium geometry of the molecules concerned. This has been followed by finite-field SCF calculation on the ground state of the molecule at the CNDO/S level of approximation.23 In excitedstate calculations,a maximum of 100singly excited configurations have been included in the CI expansion.

Assuming therefore that large SHG coefficients, if generated in the TICT molecules, primarily originate from participation of the TICT state in the nonlinear optical resonance process, we could anticipate the following: (i) the lower the energy of the TICT state, the more enhanced will be the value of j3; (ii) the larger the charge transfer in the TICT state, the higher will be the value of A p and the larger will be the magnitude of j3; and (iii) twisting around the D-X bond will affect the j3 values. An examination of the assertion in (ii) suggests that the combination of a very strong donor and a powerful acceptor will cause a large charge separation even in the ground state. In that case, the excited CT state will not exhibit a dramatically larger dipole moment compared to the ground state, resulting in a smaller than expected ground-state first hyperpolarizability. One can speculate, therefore, that maximization of j3 in a series of related “push-pull” molecules like the ones we are dealing with would perhaps require an optimal combination of a donor and acceptor. However, if excited states other than the (T)ICT state significantly participate in the process of nonlinear optical resonanceresponsible for the generation of large SHG coefficients in these molecules, expectations based on the two-state model may be largely invalidated.

3. Molecules

4. Results and Discussion

All the molecules studied by us have the following feature in common: they have an electron-donorgroup (D) and an electronacceptor group (A) attached at the 1 and 4 positions of a benzene ring (Scheme 11). The D groups considered by us are NH2, NH(CH3), N(CH3)2, and OCH3, while the A groups include CN, NOz, NO, CH=CH2, and CH=C(CN)2. A D-A combination of this kind is expected to give rise to some contribution of the canonical quinonoid structure b (Scheme 111) in the ground state in addition to the normal aromatic structure a resulting in a small additional charge transfer from the donor to the acceptor moiety. This essentially means that the ground-state wave function can be expressed as a superposition of wave functions $a and $b representing structures a and b (Scheme 111), respectively:

Ground-State Structure and Charge Distribution in the TICT Molecules. Table I containsa compilationof the molecules studied by us, their acronyms, the predicted increase in dipole moment (Ap) in going from the ground to the excited charge-transfer state, and the correspondingexcitation energy (AE).Figure la-1 summarizes the important ground-state structural parameters predicted by MNDO geometry optimization of the molecules mentioned in Table I. Figure 2a-1 displays the net charges on the different atoms in the ground state and the so-called TICTlike excited state of all the molecules studied by us. A look at Figure 2a clearly shows that in going from the ground to the TICT state of DMABN a very significant amount of charge has been transferred from the N(CH3)z group (D) to the CN moiety (A). The net increase in dipole moment accompanyingexcitation into the TICT state is rather large for DMABN, about 13 D (see Table 11). The picture is almost the same when we consider MMABN or ABN. In fact, except for MMANB and DMAS the increase in dipole moment following excitation into the (T)ICT state is -10 D. The case of DMADCS also deserves special mention sincethis molecule has an extended conjugateds-electron system. Although the charge transfer in the (T)ICT state of DMADCS is less pronounced than that in DMABN, the change in dipole moment (Ap) is fairly large (about 11 D). For DMAS, the small change in dipole moment is not quite unexpected as the CH=CH2 moiety is not likely to possess a very large electron accepting power. The behavior of MMANB, however, appears to be rather unusual. A look at Figure 2e reveals that excitation into what we call the “TICT like” state is not accompanied by any significant amount of charge transfer from the NHCH, to the NO2 group, although a significant amount of electron density is seen to have flown to the ring carbon atoms. One explanation could be that the lone pair of electrons on the nitrogen atom of the NHCH3 group of MMANB is not properly oriented for the conjugative electron transfer to take place. This may also be a consequence of an improper balance between the donating and accepting abilities of the D and A moieties (see section 5 ) . Introduction of heteroatoms in the ring introduces inhomogeneity in the electron-transfer pathway and may affect the charge transfer. In the pyrimidine analogue of DMABN (DMANP),

SCHEME III

a

b

= ca$a + Cb$b;

>> c~

(11) For the excited state, $-, the picture would be just the reverse: $g

-

$ex

ca

= cb$a - Ca$& ca >> c b

(12)

If $a is orthogonal to $br ICalzcan be thought of as a measure of the D A transfer of charge. Naturally, the amount of charge transfer in the excited state is expected to be much larger than that in the ground state-a feature that should lead to a very large increase in the dipole moment (AN) upon excitation. In the TICT molecules,it is believed that the magnitude of Cais sensitive to twisting about the D-X bond, and Ca 1 in the excited state as the twisting brings the two subunits (D and X) to mutually orthogonal planes (Scheme I). If we now invoke the two-state model for the second-harmonic generation (SHG) coefficients with $g and qCxas basis vectors, we have9

-

where AE is the energy of the charge-transfer transition, w is the fundamental wave frequency, f is the oscillator strength of the qS transition, and A p is the difference of the dipole moment between the ground and the excited (CT) states. F(AE,o) is the

-

Sen et al.

7494 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 ..

io.oo4l

'1221 0.104

:1.

0.110

0.003

DMABN

A0NlC)

la)

7(7-2-l-l0)=

I C 717-2-1'

300

II )=l50.0

MMCBNlbl

OMANBldl

- 0 390 1-0.5431

0029

0124

0 000

PNA

MMANB

If)

(el 0.050

10.0631

-0.014

1-0.0551

1-0.006) IO

C l 3 - 2-l-l01=158.l3 ( g i ? l 3 - 2 - I - l 1 I = 38.13 T W - 5 - 8 - 9 12-178.14

l h ) T 13- 2-1-10)=

0.84

CH3 10.0411 0.001

,

1-0.017 I 0.017

o"62

0.005

-0.262

OMAN58

MBN

191

ih l

cj)

I1 L l T 1 3 - 2 - l - I I )r210.0 i t 1 4 - 5 - 8 - 9 1 . -1.15 f'(3-2-1- 12 1'3300 i +14-5-8-101=17B.78

Figure 1. Computed equilibriumstructural parameters of the molecules studied (a-1). For identification of the molecules, see Figure 2a-1.

however, the extent of charge transfer in the (T)ICT state is seen to be large although the ring heteroatoms (nitrogen) introduce inhomogeneityin the mesomeric path of charge transfer. It would beinteresting to examinethe possibleeffect of this inhomogeneity on the first hyperpolarizabilitysince it is supposed to be intimately related to the ease with which charge transfer can take place (see later) when an electric field is applied. First Hyperpolarizability of (T)ICT Molecules in the Ground State: TrendsExhibitedby the Largest Componentand /3 Projected on Dipole. Table I1 summarizes computed values of groundstate 8,,, @xy,,, and,,3j components (magnitudewise the largest among the components) of all the molecules studied by us (for the calculationof /3 we have followed thedefinition used by Zyss13. In general, we have not constrained the disposition of the NRlR2 or OCH3 groups to be planar, as our previous experience12 has shown that pyramidalization is essential for maximization of 8. Pyramidal DMABN is found to possess a fairly large value of 8- in the ground state. Thecorrespondingvalue is approximately 15% less for planar DMABN (Table 11). However, the computed ground-state value of 8, is seen to decrease as we have pass on to ABN through MMABN. DMANSB is predicted to have a much larger BX,,value compared to DMABN, although the nitroso

10.0341

1-0.431)

I2

11

I-0.1971

(0.i241

(-0.541 I -0.399

0.008 10014)

DMANP

(I)

F w2. Computed chargedensity on different molecules studied (a-I). The quantities in parentheses refer to charge densities in the T(ICT) state. moiety is a weaker electron acceptor than the CN group. MBN, on the other hand, is predicted to have a #Ixxx value nearly equal to that of ABN, apparently suggesting that the OCH3 and NH2 groups have almost identical electron-releasingpower as reflected in the magnitudeof the first nonlinear polarizability (8,). Again, even though the CH=CH2 moiety is expected to have a much lower electron pull than either the C N or NO moieties, the pxXx value for DMAS is, however, found to be larger than that of DMABN. Some factor other than the simple electron-donating (-accepting) ability of thedonor (acceptor) moiety must therefore be operative. Incidentally, the conjugative path for electron transfer is more homogeneous in DMAS as the conjugated side chain contains only carbon atoms. Replacement of the two H atoms of the CH=CH2 unit by C N groups is expected to increase its electron pull and also the length of the conjugated .rr-network.

Modeling Hyperpolarizabilities of TICT Molecules

The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7495

TABLE II: Computed 8,

8 and,0 Values (in 10-30 mu) of the Ground State of thexolecules Studied molecule Bxxx DMABN 11.087 -0.869 -0.260 (10.452)"

MMABN 8.063 ABN 6.378 DMANB 32.022 MMANB 2.604 PNA 22.116 DMANSB 18.581 MBN 6.625 DMAS 11.986 DMADCS 74.395 DMACP 5.106 DMANP 21.392 Data for planar DMABN (& = 0').

BY""

BXZZ

(-1.329) -0.865 -0.940 -2.041 -0.201 -2.066 -1.087 -0.901 -0.684 -0.725 -2.570 -3.185

(-0.373) -0.194 -0.266 0.056 0.035 0.021 0.094 -0.179 0.118 -2.007 -0.049 0.364

These two favorable factors apparently add to produce a very high &,,value for DMADCS. The nitro analogue of DMABN (DMANB) also is seen to have a much higher 0,value compared to DMABN, a feature that possibly reflects higher electron pull of the NO2 compared to the CN moiety. A closer look at the entries of Table 11, however, fails to reveal any firm correlation between the computed ground-state PxXxvalues and the A p or A&T values as would have been expected from the optical model. This, in turn, suggests that apart from the (T)ICT state itself someother state(s) possibly participates significantly in theNLO resonance process responsible for the generation of large /? coefficients. The ground state 8, and bxZzcomponents computed by us for all these molecules are seen to be approximately 2-10% of the, 0 values. These components, as well as the others not reported here, are presumably underestimated since the basis set is too restricted and no polarization functions are included in the basis. Therefore, for modeling of the 0 values computed by us we may perhaps base our analysis solely on the computed bxxx components. For comparison with experimental 6, the components of the 0 tensor need to be converted into a quantity that can be measured. The quantity of interest is b,,, the scalar component of computed i3 projected along p:

where

TABLE IIk Comparison of Computed Values of B (B Projected on Dipole) and 8 (Experimental) for the &round State of the Molecules Studied.

DMABN

49.7 (18.0)' 42.0 43.1 40.6 47.6 42.8 40.0 14.9 39.7 40.3 39.9 21.9

5.826 (5.250) 4.037 2.952 17.717 1.260 11.823 10.020 3.279 2.012 29.893 1.277 10.794

5.0

MMABN ABN 3.1 DMANB 12.0 MMANB PNA 9.2 DMANSB 12.0 MBN 1.9 DMAS DMADCS 32.0 DMACP DMANP "The orientation of the lone pair on the central atom of the donor moiety, ,e is also included for comparison. Cheng, L.-T.; Tam, W.; Stevenson, S.H.; Meredith, G. R.;Rikken, G.; Marder, S.R.J . Phys. Chem. 1991,95,10631. Cheng,L.-T.;Tam, W.;Marder,S.R.;Stiegman, A. E.; Rikken, G.; Spangler, C. W. J. Phys. Chem. 1991, 95, 10643. Data for planar DMABN.

NRlR2 moiety generates a larger /3 in DMABN than planar NR1R2. One may try to rationalize the observation made earlier by arguing that the nitrogen lone pair in NRlR2 is properly oriented for taking part in the excited-state electron-transfer process (from the lone pair on nitrogen to a C-N antibonding u* system) if the NR1R2 unit is pyramidal. To test this argument, we have transformed the SCF orbitals of the pyramidal TICT molecules into a set of localized MOs using the localization method of Perkins and S t e ~ a r t . 2Table ~ I11 displays the orientation of the axis of the lone pair on the donor moiety in DMABN and other molecules with respect to the molecular coordinate axes (Figure 3). If the central atom of the donor moiety is planar (sp* hybridized), the lone-pair orbital is an almost pure p-type orbital perpendicular to the molecular plane (small 0, in Figure 3). This kind of lone-pair orbital can enhance the extent of D A charge transfer in the ground state through the ring ?r system. But the extent of D A transfer of charge in the excited state decreases, because the lone pair cannot have significant overlap with the u* orbital which is opposite in phase. The charge transfer in the excited state can be facilitated if the donor lone pair is reoriented, say, by a change in the state of hybridization of the central atom of the donor. Pyramidalization at the central atom of the donor moiety opens up this possibility. If the state of hybridization changes as sp2 sp3,0, would increase, and this in turn would decrease the extent of D A charge transfer in the ground state. The picture for the excited state would be just the opposite. In addition to the effect of overlap, energetics also seems to play a role. Thus, mixing with s orbital stabilizes the lone pair in a pyramidal donor group much more relative to the planar donor moiety, making the D A transfer of charge energetically less favorable in the ground state but more favorable in the excited state. A close look at Table I11shows that molecules with smaller values of 0, usually have smaller /3 values as well. However, quite a few exceptions to the rule can be noted. It would therefore appear that, apart from the stereochemical disposition of the lone pair on the central atom of the donor moiety, other competing factors must be simultaneously operative.

-

-

- -

Values for B,, along with 0, (orientation angle of the lone pair on the central atom of the donor moiety) are reported in Table 111. Available experimental data are also included for comparison. Just as was the case with the &, component, the (3, values fail to exhibit any meaningful correlation with A p or ACT. Hyperpolarizability data for the excited states are really scarce. Therefore, we refrain from discussing excited-state hyperpolarizability of these molecules in a general way. But to give the reader some idea about the hyperpolarizability of the excited states, including the (T)ICT state, we report computed values of pxXxand P,, for six representative molecules in the first five excited states in Table IV. The magnitudes are quite large in a number of excited states. Stereochemical Disposition of the Electron-Donor Group and Its Influence on 8. We have previously12 shown that twisting about the N-C bond of DMABN or PNA has no important consequence on the magnitude of 8. On the other hand, the stereochemical disposition of the NRlR2 group is an important factor in the emergence of large 0 coefficients. A pyramidal

-

5. Modeling of Hyperpolarizabfities

From a chemist's point of view, the chemical properties of the molecules are the properties of the functional groups they contain. The properties of specific functional groups in a molecule in turn may be somewhat modified by substitution in other parts of the molecule. A response property such as B is a property of the entire system and is not localized at the functional groups.

7496 The Journal of Physical Chemistry, Vo1.97, No. 29, I993

and Br Values (lkMesu) for the First Five Excited States of Six Selected Molecules' state

TABLE IV: Computed 8, molecule DMABN

1

6.566 (6.941) ii.316. (7.841) 101.087 (40.025) -106.306 (-100.377) -38.757 (-1 1.273) 66.866 (43.508)

ABN DMANB PNA DMANSB DMANP 0

Sen et al.

2 -3 1.293 (-12.464) '252.482 (148.926) -2.240 (-42.813) 215.861 (-1 56.103) 34.067 (17.531) 269.409 (157.626)

3

4 -18.703) (-11.197.) 21.!74 (-25.001) -293.584* (-18 1.888*) -297.994, (-181.892*) -45.801 (-28.595) 53.260 (2 1.094)

-1 59.120

(-92.570) -501.652' (-299.951)) -127.145 (-77.831) -1 10.459 (-68.328) 76.471 (45.580) 115.283 (80.529)

5 53.575 (30.490) . 7.598. (7.110) 84.135 (44.837) 242.769 (139.282) 93.305* (40.831*) -87.218* (-65.682,)

fi values for the (T)ICT states are indicated by an asterisk. fi, values are given in parentheses.

dY p \ = /

"y7-J

- - - - +x

'/

Figure 3. Orientation angle (e),

of the lone pair defined.

However, substituents can modify or distort the overall electronic charge distribution in a molecule and can conceivably affect the response properties as well. Is it possible to model the variation in 0observed in the molecules studied in terms of specificproperties of the important substituents, namely, the donor and the acceptor units (D and A) present in the molecule? In what follows, we explore this question from different points of view. For previous attempts in this direction, we refer readers to an excellent review by Nicoud and Twieg.' L e ~ i n eobtained ~~ good linear correlation between j3 and a composite substituent parameter) : :Y( expressed in terms of substituent constants (YD and YA) for the donor and the acceptor assigned by Petruska26 from spectral shift data. Dulcic and Sauteretzs observed that in a series of disubstituted benzenes the increase in j3 values is followed in general by a red shift in the first electronic absorption band, and they emphasized the need to choose a proper acceptor for a particular donor. The 3~" parameters of Levine24 for many of the groups of our interset are not available. Moreover, our calculated values of hardly reveal any linear trend when examined qualitatively against the pushpull strengthsof the groups. We therefore search for an alternative model. The amount of D -c A electron transfer will obviously depend upon the electron-releasing power (push) of the donor, a property that can be modeled well in terms of the first ionization potential of the donor group. The lower the IP, the greater is the ease with which an electron may be released to a given acceptor provided that there is no hindrance to the conjugative electron-transfer process caused by stereochemical or other factors. Similarly, an acceptor with a high electron affinity (pull) will induce a higher degree of electron transfer from a given donor unit. Our intention is to devise a composite parameter, hopefully a function of the ionization potential (IP) of the donor (IPD) and the electron affinity (EA) of the acceptor (EAA)units, that can appropriately model changes in the extent of electron transfer in response to changes in substituents. This parameter may perhaps also help to model substituent effect, if any, on j3. In terms of the donor IP and the EA of the acceptor groups, we may define a composite push-pull parameter 6, the absolute magnitude of which increases with the increase in EA of the acceptor and the decrease in IP of the donor. Figure 4 exhibits a 8-6 profile where 6 = (IPD - EA&' and IPD and EAA are theoretical quantities calculated at the A SCFlevel by the MNDO

Figure4. Profile of computed fiagainstthe effectivepush-pull parameter (6) of the donor groups, 6 = (IPD- EA*)-'.

method. 0 is not observed to be a monotonic function of 6. Maximization of 0as a function of 6 is seen to occur over a very narrow range of 6 values. We may observe at this point that the IP of the free donor group (D) alone cannot possibly model its electron-releasing power or push in these molecules. This is because the strong electron demand of the acceptor moiety at the para position would surely modify the extent of conjugative electron release by the donor. We need to have, therefore, a continuum of "push-measuring scales" for modeling the substituent effect on the D A electron transfer properly. One possible way to derive such a parameter is to introduce an electron demand factor (e, say) and define a modified IP for the donor. Let us call it 6,, where

-

6, = IP,(~ -

EA*)-'

Clearly an acceptor group (A) with high electron affinity in the para position can substantially affect the electron-releasingpower of the donor group. Moreover, the magnitude of the effect will be a nonlinear function of the electron demand of the acceptor.30 Once the scale is fixed," we can model the 0values in terms of a single parameter (6J. Figure 5 shows how the computed pxXx values behave as a function of 6,. There appears to be a definite range of 6, values where maximization of j3xxx occurs. This essentially means that for the present series of molecules maximization of 0demands a judicious choice of a donor and an acceptor that generates an effective push-pull parameter (6,) in the appropriate range. Mere increase in electron push or pull or

Modeling Hyperpolarizabilities of TICT Molecules

The Journal of Physical Chemistry, Vol. 97, No. 29, 1993 7497

80.00

60.00

5

?

9

-% 40.00 cp

20.00

0.00 i

3.20

d 5.20

7.20

9.20

11.20

0

6, (ev)

Figure 5. Profile of computed fl against an effective nonlinear push-pull parameter 6, = IPD( 1 - cEAa)-’.

Figure 7. Computed exact &6 profiles of the model diatomic for two

Y

different values of hopping matrix element (e). The upper curve corresponds to 0 = 0.2 au and the lower curve to 6 = 0.4 au.

T

Using eq 16, we obtain 1

0 = -qAR3(A2/4

Figure 6. Definition of the parameters of the model diatomic.

both will not always be effective. This basic observation has important bearing on the design problem of nonlinear optical materials and therefore needs to be rationalized. 6. A Two-State Model

Let us consider a diatomic species with one basis function (x, a s-orbital) on each atom, the atoms being separated from each other by a distance of 2R (Figure 6). Let c be the strength of the electrostatic field applied along the internuclear axis (A B). Employing a Huckel-like model, weobtain the field-modified energy eigenvalues and eigenfunctions by solving the following secular equation:

-

Here, a0 = ( x A ~ ~ x A ) ,a0 + A = (xB~&), and fi is the Hamiltonian operator for the system. The field-modified energy eigenvalues are

E,(€) = (a+ A/2)

* [A2/4 - AeR + tZR2+ 02]1/2

(16) The energy E can be expanded in a power series in the field strength (c):

From the expression E+(€)we can calculate p by taking the third limitingvaluederivativeofenergywith respect to the field strength

+

+ k ( A R ) ’ ( A 2 / 4 + ez)-s/z (19)

In this model, A essentially measures the difference between the electron-releasing powers of the two atoms (A and B), while 0 is a measure of the strength of a through-bond “conjugative” type of interaction. A typical plot of 0 values computed from eq 19 against A is shown in Figure 7. It is interesting to note the emergence of a well-defined maximum in the p-A plot which nicely mirrors what has already been observed with thecomputed 3/ values of real molecules. Recently, Marder et al.27 showed that a two-state four-orbital independent electron analysis of the first optical molecular hyperpolarizability, 8, leads to the prediction that Ij3I is maximized only at a particular combination of donor and acceptor strengths, a feature also noted by us in an earlier publication.12 It is heartening to note that our simpler two-state two-orbital model leads to similar predictions. In our model we can, however, go a step further and look for the conditions under which B of eq 18 reaches its maximum value as a function of A. The analytic condition is, of course SP/SA = 0 which leads to the following condition for extremization of

8:

A = *e That the condition A = f0 indeed maximizes B can be seen by calculating ti2/3/6A2at A = f0, which comes out to be negative. Physically, therefore, one can expect a maximization of 0 to take place in a series of related molecules if the difference between the IP of the donor and the EA of the acceptor (Le.,A parameter of our model) exactly matches the strength of the conjugative interaction or the so-called “hopping” matrix element (e parameter in eq 15). This is assuming, of course, a completely homogeneous path of electron transfer from D to A, which essentially means that the entire path of electron transfer can be characterized by a single value of the ‘hopping” matrix element. To put this conclusion to test, we first consider a substituted benzene molecule with a particular donor-acceptor combination that generates a fairly large value of 8. One may suppose that the condition A = 0 for the maximization of p has been approximately achieved for this molecule. If we now introduce heteroatoms in the aromatic ring which participate actively in

Sen et al.

7498 The Journal of Physical Chemistry, Vol. 97, No. 29, 1993

the conjugativecharge transfer from D to A, the A = 6 condition for maximization of 0 will be disturbed, and this in turn will lower thegvalue. Asacaseinpoint,wecancomparethecomputed B-values for DMABN and DMANBvis-&vis those for DMACP and DMANP. Our calculations show that BDMACPis smaller than ODMA~Nby about 55%, and ~ D M A N Pis smaller than BDMANB by about 32%. The ring nitrogen atoms introduce inhomogeneity in the path of electron transfer and disturb the A = 6 condition for maximization of fl that is approximatelyattained in DMABN and more completely in DMANB. This kind of ‘inhomogeneity” effect has also been observed by Meyers et a1.28 7. Conclusions Maximization of @ in a series of internal-charge-transfer molecules potentially capable of displaying the so-called TICT phenomenon requires proper stereochemical disposition of the donor moiety and an appropriate choice of the acceptor for a given donor. There appears to be a certain optimal range of combination of push-pull strengths which maximize 8. This behavior of /3 also follows from an analytically solvable 2 X 2 model problem which asserts that /3 is maximized if the hopping matrix element (i.e., the interaction term responsible for throughbond transfer of charge) is equal in magnitude to the energy difference between the HOMO of the donor and the LUMO of the acceptor moieties. Inhomogeneity introduced in the path of charge transfer generally has the effect of decreasing the magnitude of 0 for the same pair of donor and acceptor groups and an identical path length for charge transfer. These findings could be of considerable importancein designing nonlinear optical materials for efficient second-harmonic generation. We are extending these studies to stilbene derivatives with a view to analyzing the kind and extent of dependence of /3 values on the path length of charge transfer. We hope to communicate these results in due course.

Acknowledgment. We thank the CSIR, Government of India, New Delhi, for financial assistance (Grant No. [5( 136)/88-EMRZq) and one of the reviewers for valuable comments. References and Notes (1) Twieg, R. J.; Jain, K. In Nonlinear Optical Properties of Organic and Polymeric Materials; Williams, D. J., Ed.; ACS Symposium Series 233; American Chemical Society: Washington, DC, 1983.

(2) Chemla, D. S.;Zyss, J., Eds. Nonlinear OpticalPropertiesof Organic Molecules and Crystals; Academic Press: New York, 1987; Vols. 1 and 2. (3) Gordon, P. F.; Gregory, P. Organic Chemistry in Colour; Springer: Berlin, 1983. (4) Pugh,D.; Morley, J. 0. In Nonlinear Optical Properties of Organic Moleculesand Crystals; Chemla, D. S.,Zyss, J., Eds.;Academic Press: New York, 1987; Vol. 1, p 193 and references cited therein. (5) Andre, J.; Barbier, C.; Bodart, V.; Delhalle, J. In Nonlinear Optical Properties of Organic Molecules and Crystals; Academic Press: New York, 1987; Vol. 2 and references cited therein. (6) Zyss, J.; Chemla, D. S. In ref 2, pp 23-187 and references cited therein. Nicond, J. F.; Twieg, R. J. In ref 2, pp 227-291 and references cited therein. (8) Oudar, J. L. J. Chem. Phys. 1983,67,446. (9) Oudar, J. L.; Chemla, D. S . J. Chem. Phys. 1977,66,2644. (10) Rotkiewcz, K.; Grellman, K. H.; Grabowski, Z. R. Chem. Phys. k i t . 1973, 19, 315. (1 1) Lippert, E.; Rettig, W.; Konacci-Koutccky, V.;Heisel, F.; Mihe, J. A. Adv. Chem. Phys. 1987,68, 1. (12) Sen, R.; Majumdar, D.; Bhattacharyya, S.P.; Bhattacharyya, S.N. Chem. Phys. Lett. 1991,181,288. (13) Hurley, A. C. Proc. R. Soc. 1954, A226, 170, 193. (14) Davidson, E. R. In ReducedDensity Matrices in Quantum Chemistry; Academic Press: New York, 1976. (15) Zvss. J. J. Chem. Phvs. 1979. 70. 3333. 3341: 1979. 71.909. Zvss. J.; irthie;, G.J . Chem. Phy;. 1982,77,3635. Following Zyss, we have ;sed &jk sa ’ 1 2 8EjEkb-0. (16) Lalama, S.J.; Garito, A. F. Phys. Rev. 1979, A208, 1179. (17) Garito,A. F.;Teng,C.C.; Wong,K.Y.; Zamman’Khamiri,O.Mol. Cryst. Liq. Cryst. 1984, 106, 219. (18) Docherty, V. J.; Pugh, D.; Morley, J. 0. J. Chem. Soc., Faraday Trans. 2 1985, 81, 1179. (19) See, for example: Parkinson, W. A.; Zerner, M. C. J . Chem. Phys. 1991, 94, 476; 1989, 90, 5606 and references cited therein. (20) Parkinson, W. A.; Zerner, M. C. Chem. Phys. Lett. 1989,139,563. (21) Kurtz,H.A.;Stewart, J. J.P.;Dieter,K.M. J. Comput. Chem. 1990, 11, 82. (22) Dewar, M. J. S.;Thiel, W. J. Am. Chem. Soc. 1977,99,4899,4907. (23) DelBene, J.; Jaffe, H. H. J . Chem. Phys. 1968, 48, 1807. (24) Levine, B. F. J. Chem. Phys. 1975,63, 115. (25) Dulcic, A.; Sauteret, C. J. Chem. Phys. 1978, 69, 3453. (26) Petruska, J. J. Chem. Phys. 1961,34, 1120. (27) Marder, S.R.; Beratan, D. M.; Cheng, L. T. Science 1991,252,103. (28) Meyers, F.; Adant, C.; Bredas, J. L. J. Am. Chem. Soc. 1991,113, 3715. (29) Perkins, P. G.; Stewart, J. P. J. Chem. SOC.,Faraday Trans. 2 1982, 78, 285. (30) Bromilow, J.; Brownlee, R. T. C.; Craik, D. J.; Sadek, M.; Taft, R. W. J. Org. Chem. 1980,45,2429. (31) For the NO1 and CN groups, the c values are available in literature from an analysis of substituent chemical shift in NMR spe-ctra-m A linear relationshipbetween tand theelectronaffnityoftheacceptorhasbeenassumed for estimating the c parameter for the other groups ( t ~=a-0.72, = -0.6, €NO -0.14, CCH-CH~ = -0.097, and ~ c H ~ ( c N=) -0.46). ~

(7)