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Third-order nonlinear optical properties of finite bridged polymers
Volume205, number 1
CHEMICAL PHYSICS LETTERS
2 April 1993
Third-order nonlinear optical properties of finite bridged polymers Xu Monica Wu and Mary Jo Ondrechen Department of Chemistry, Northeastern University,Bosfon, MA 02115, USA Received 30 September 1992; in final form 30 December 1992
The thirdarder microscopic hyperpolarizability y is calculated for finite bridged polymers of the form A-B-A-B-... using simple model Hamiltonians. Generally, the even- and odd-numbered chains show markedly different behavior, with the addition or subtraction of one chain unit resulting in a predicted change in 1y 1by a factor of up to a few orders of magnitude. The corresponding contribution to the elcctrochromic effect was found to be a sharply increasing function of the chain length 1yI alsois predicted to be significantly enhanced when the coupled A and B electronic states are off-resonant and when the total number of electrons is odd. When the total number of electrons is even, 1y 1may be maximized by adjustment of the A-B energy gap.
1. Introduction
In the present work binuclear chains are studied [ 71. The effects of chemical substitution upon the
Molecules and materials with nonlinear optical properties are the focus of much current interest and activity [ 1,2]. The present work examines the tbirdorder nonlinear properties of finite, bridged polymers. The ultimate goal of this work is to provide guidance in the synthesis and development of materials with large third-order effects. These materials hold promise for novel, marketable optical technologies. Some examples of effects arising from the thirdorder nonlinear susceptibility xc3) are third harmonic generation (THG) and degenerate four-wave mixing (DFWM ) . The bridged binuclear ABAB... chains studied herein serve as models for two different types of systems: (1) conjugated organic polymers with an AB repeat unit [ 31, and (2) bridged mixed-valence compounds where a string of metal atoms are joined together by bridging ligands [ 4,5 1. Beratan et al. have reported on the third-order hyperpolarizability as a function of chain length for homonuclear conjugated organic polymers [ 6 1. The two-band, one-electron tight-binding model of ref. [ 6 ] correctly predicts that as chain length increases there is a rapid rise in the third-order microscopic hyperpolarizability y which then approaches an asymptotic value at about 20-30 atoms.
third-order hyperpolarizability y and upon the thirdorder component of the electrochromism are examined. For simplicity, a Hiickel Hamiltonian is used in the first part of this Letter. We recognize that eleo tron correlation is important in conjugated systems [ 8,9] and that one-electron Hamiltonians constitute an oversimplification, but the first simple model should show some useful general trends. In the fourth section, electron interaction terms are added to the Hamiltonian and some results pertaining to the effects of correlation are discussed. Conclusions are presented in section 5, where some suggestions are made for possible strategies for the synthesis of largexc3) materials.
2. The model A Hiickel-type Hamiltonian for the N-membered chain, written as
RI=
2 O!,UfUi+:
i=l
B(UjUj_1
+UJ_*Uj)
j=2
is adopted. Here at and ai are the creation and annihilation operators respectively for the ith electronic state. q represents the energy of the isolated basis function on the ith chain member. For the chain
0009-2614/93/$ 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
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ABAB, the energy of the parent A state is taken to be zero, so (Y~=(Yfor the B species and q=O for the A species. Thus Q represents the energy gap between the B and A basis states. The resonance interaction energy between nearest neighbors is given by B. Because delocalized, conjugated systems are considered here, it is assumed that the conjugated ICelectrons can be treated separately and hence only one electronic state per chain member is important. As is usually the case in a Hiickel system, the two electronic spin states possess the same set of spatial functions. The spatial orbitals are occupied by up to two electrons of opposite spin, starting with the lowestenergy orbital. The AB and BA resonance integrals are assumed to be the same for the sake of simplicity. Non-nearest-neighbor interactions are neglected. At this stage, the interactions between electrons are neglected. Electron-electron repulsion effects are incorporated into the model in section 4. An orthogonal basis set is assumed. The finite polymers are also presumed to be free of defects [ 10 1. The microscopic polarization for a molecule with permanent dipole moment $ in electric field d may be expanded as [ 111 ~j=~Ptaii~t~8iik4~~+dy~~S~~grt...,
2 April1993
where the x axis is the chain axis. The third-order hyperpolarizability y is related to the fourth-order correction to the total energy Ec4) and is given by [ 61 y=
-46,4C2E$f' m
>
(7)
where the index m represents occupied molecular states,
3. Results for ABAB finite polymers Fig. 1 shows log Iy I as a function of the number of members N in the chain for the cases: (a) a= 0.5 eV, fi= - 1 eV and the number of electrons n is twice the number of A atoms (n=2N,); and (b) cuz2.0 eV, /?= - 1 eV and n= 2h’,.,- 1. This corresponds to a bridged chain MLML... consisting of metal atoms M and bridging ligands L where each M atom contributes one or two ICelectrons and each L ligand contributes no n electrons. Curve (a) corresponds to the fully reduced case; curve (b ) corresponds to a mixed-valence system. Fig. 2 depicts log 1y 1 as a function of N for cy= 0.5
(2) 1ol'
and y represents the third-order microscopic hyperpolarizability. Similarly, the bulk polarization may be written as P=P,+x
“‘.Q+x’*‘.b.~+x’3”6.b.~+...
,
(3)
where the xci) are the ith order susceptibility coefficients [ 111. The third-order susceptibility is related to the microscopic hyperpolarizability y as X”‘(-w,;wl,WZ,W3)=li(W,)F(W2)~i(W3)F(W4)
x
;
9
(4)
where the F function is a local field correction and the summation is over orientationally averaged sites
f.121. Perturbation theory is used to obtain y with the Hamiltonian for the chain in an external electric field JXwritten as: A&1
+Y,
V= -e&X,
86
(5) (6)
2
7
12 Chain
17 Units
22
27
32
N
Fig. 1. Log 1yI as a function of number of chainunits N for AB finite polymersfor the cases: (a) (Y=0.5 eV, p= - 1eV and where the numberof electrons equals twice the number of A atoms; (b ) cY=2_0eV,b=-I eVandn=2N,,-1.
2 April 1993
CHEMICAL PHYSICS LETTERS
Volume 205, number 1
60 I
-2, 2
7
12
17 Chain
Units
22
27
32
0.0
0.4 Inl
N
Fig. 2. Log 1yI as a function of number of chain units N for AB finite poIymers for the case ar=O.5 eV, /?= - 1eV and where the
numberof electronsequalsthe numberof chainunits. eV, /I= - 1.OeV and the number of electrons n equals the number of chain members N, These parameters correspond to a polyene with alternate atoms substituted. One notes that in all cases shown, the odd and even chains behave differently, although the odd-only and even-only curves resemble those of Beratan et al. [ 6 ] . The dependence of I yI on the B-A energy gap a! for fixed chain length is also of interest. For even and odd finite chains with n=2N, and for even chains with n=N, the third-order hyperpolarizability y is maximized at finite CL Fig. 3 plots the third-order hyperpolarizability Iy] as a function of (Y for the 4-, 6- and &membered chains with /3= - 1.O eV and with one electron per chain member (II =N) . In cases wheren=NwithNoddorwheren=2N,-1, IyI increases as I a I increases. Fig. 4 shows log I y 1 as a function of (Yfor the 3-, 5- and 7-membered chains with fi= - 1.OeV and where the number of electrons is given by II= 2N, - 1. The case of fig. 4 corresponds to a bridged metallic chain which has been oxidized by one electron. For both even and odd chains, we find that the third-order hyperpolarizability for the very short
0.8 (in
eV)
1.2 A-B
1.6
Energy
2.0
Gap
Fig. 3. The third-order microscopic polarizability y as a function of the B-A energy gap (Yfor 4,6- and I-membered AB chains. ,% - 1 eV and the number of electrons equals the number of chain
-6
-4
0
-2 a(in
eV)
A-E!
2 Energy
4
6
Gap
Fig. 4. y as a function of the B-A energy gap(Yfor 3-, 5- and 7membered AB chains. p= - 1 eVand n=2N,- 1.
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polymers is less strongly dependent on the B-A energy gap than it is for the longer polymers. Since very large values for 1yI were calculated for some of the above examples, it was decided to look at the contributions of the fourth-order energy corrections Ej4) to the electrochromic effect. The HOMO-LUMO transition was studied for variable chain length, Hamiltonian parameters and number of electrons. It was found that for a wide range of cases, the Et4) contribution to the HOMOLUMO transition energy rises, often sharply, with increasing chain length. Of course the zero-order transition energy generally falls slowly as the number of chain members increases, so that the relative contribution of the fourth-order terms is even more strongly dependent on chain length. For instance, for odd chains for the case OI= 1 eV, /I= - 1 eV, n = ZN, and 8,~ 5 X lo6 V/cm, the fourth-order contribution to the HOMO-LUMO transition energy surpasses the zero-order contribution at N= 15. When a! is varied for the same case with fixed chain length, the magnitude of the E(‘) contribution to the HOMOLUMO transition energy is minimal at about a=0 and rises sharply as 1(Y1 is increased.
4. Electron-electron repulsion effects To get a general idea of the effects of electron-electron repulsion on the trends observed above, the Hiickel Hamiltonian of eq. (1) was augmented to include same-site electron-electron repulsion. The resulting Hubbard Hamiltonian is given by
where p is the spin index (t or 1) and Vi is the electron-electron repulsion on the ith site. Eq. (8) is solved approximately by the HartreeFock (HF) method, which has been shown [ 13 ] to give results virtually identical to the exact [ 141 results for UG I II _ When U= 2 IJ], the HF energy is too high by about 10% [ 131. Solutions to eq. (8) for a variety of parameter values show that for I a I> I/31 and I /.?I> U, y changes somewhat as V is increased from zero, but the general form of the dependence on chain length is maintained. Thus when the A and B basis states are off88
2 April 1993
CHEMICALPHYSICSLETTERS
8=-l n=N
eV
Odd Chains
6
i 6 !
6 P
4-
2-
0
I_ 1
5
9
13
17
21
Chain Units N
Fig. 5. Log IyI as a function of chain units for odd chains. (a) a=2 eV, U=O; (b) a=2 eV, U~0.5 eV, (c) a~0.5 eV, U=O; (d)(Y=OSeV,CJ=0.5eV./I=-leVandn=Ninallcases.
resonant (nonzero a), Iy 1 is reduced slightly or may even increase with increasing U, as long as U is not too large (U- ]J])_ For systems where the number of electrons equals the number of chain units ( n = N), the even and odd chains respond differently to an increase in U from zero. For the odd chains when cr is nonzero, ]r] becomes less sensitive to changes in U, also, an increase in U diminishes Iyl for the shorter chains but increases I y 1 for the longer chains. The corresponding threshold chain length is reduced as the A-B energy gap a! increases. For even chains with cyset close to its optimum values, an increase in U from zero causes I y I to be reduced for the shorter chains but increased for the longer chains. Fig. 5 plots log I y I as a function of chain units for the odd chains with I= - 1 eV and n=N for four cases: (a) a=2 eV, U=O; (b) a=2 eV, U~0.5 eV, (c) cu~O.5 eV, U=O; (d) (~~0.5 eV, U~0.5 eV. 5. Discussion and conclusions This work suggests some possible means to control and to improve the third-order nonlinear susceptibility xf3) in finite bridged (or alternating) polymers.
Volume 205, number 1
CHEMICAL PHYSICS LETTERS
1yl is found to be enhanced significantly when the A and B electronic basis states which participate in the conjugation are different in energy (off-resonant) wherever the total number of electrons is odd. The sensitivity of 1yI to the B-A energy gap (Yis greater for the longer chains. While the present results suggest that, for a mixed-valence system with an odd number of electrons, one should try to choose A and B so that the energy difference (Ybetween the coupled states is as large as possible, there of course will be a practical limit on the magnitude of o: if 1a I is too large, the resonance interaction will be diminished. However, loI=2 1/II is reasonable for Kbonded metallic systems [ 15 1. Some effect also may be created through substitution of alternate atoms on a conjugated carbon backbone without forfeiture of conjugation. When the total number of electrons is even, 1yl may be maximized by adjustment of cyfor finite chains (for n even and N very large, the optimum (Yapproaches zero [ 7] ). For the cases studied, the optimum (Yis in the range O-21/31to O-71/31,which should be achievable by chemical substitution. For systems with one electron per chain unit (n =N) and N odd, electron correlation resulting from single-site electron-electron repulsion generally causes I y I to be diminished for smaller chain lengths and for smaller values of the B-A energy gap. When a is large the functional dependence of I yI upon chain length is not sensitive to the on-site Coulomb repulsion U: for smaller values of the chain length, I y I is diminished somewhat by an increasing U, but the substantial even-odd effects and the general functional form are preserved. When cy is nonzero (i.e. A and B chemically different), it was found that Iy I is less sensitive to electron correlation than in the a=0 case. This further suggests that, at least in some cases, a difference in energy (of the B and A basis states) is a promising path to.follow when an increase in 171is desired. The different behavior of the odd and even ABAB chains is of interest. In the n=2N, (e.g. fully reduced bridged metallic) case, the even-numbered chains show somewhat higher third-order microscopic polatizability than the corresponding oddmembered chains, but only when (Yis close to the optimum value. The situation is reversed for the n =iV (e.g. conjugated polyene) case, where the odd-
2 April 1993
numbered chains have significantly larger values for I y I. For the n = N and n = 2N,- 1 cases, the difference between the even- and odd-membered chains is generally greater for larger values of the B-A energy gap; the even-odd difference is minimized when a is small (but not necessarily zero). Upon examination of the effects of the B-A energy gap cy on the energy level diagrams, some understanding of the even-odd effects emerges. When (Y= 0 and U= 0, the energy level diagrams are symmetrical in energy space for both even and odd chains. As (Y is increased from zero, the energy levels lying below the median energy level are bunched more closely together while the energy levels lying above the median level are spread further apart. Therefore when there is one electron per chain unit (n =N), the odd chains have a number of low-lying, closely-spaced excited configurations. The corresponding even chains have higher-lying excited configurations which are spread further apart. However, when the number of electrons is 2N,, and (Yis close to its optimum value, the excited configurations for the odd chains are spread still further apart and thus the values for I yI are lowest of all in this latter case. Hopefully the strategies suggested here for the synthesis of higher-X(3) materials will be tested, as materials of the above type appear to show promise.
Acknowledgement The support of National Science Foundation grant No. CHE-8820340 is gratefully acknowledged. We are indebted to Dr. Steven Risser for many useful criticisms and comments. We thank Dr. James Shirk for valuable discussions about nonlinear optics. We thank Messrs. Ihsan Shehadi and Leone1 Murga for many good conversations.
References [ 11 S.R. Marder, J.E. Sohn and G.D. Stucky, eds., Materials for nonlinear optics: chemical perspectives, ACS Symp. Ser. No. 455 (American Chemical Society, Washington, 199 1). [ 21H. Kuhn and J. Robilkud, eds., Nonlinear optical materials (CRC Press, Boca Raton, 1992).
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[ 31 H. Kuzmany, M. Mehring and S. Roth, eds., Electronic properties of conjugated polymers III, Springer series in solid state sciences Vol. 9 1 (Springer, Berlin, 1989) . [ 41 M.J. Ondreehen, S. Gozashti, LT. Zhang and F. Zhou, in: Electron transfer in biology and the solid state, M.K. Johnson, R.B. King, D.M. Kurtz, C. Kutal, M.L. Norton and R.A. Scott, eds., Advan. Chem. Ser. No. 226 (American Chemical Society, Washington, 1990) pp. 225-235. [ 51LB. Bersuker and S.A. Borahch, Advan. Chem. Phys. 8 I (1992) 703. [ 61 D.N. Beratan, J.N. Gnuchic and J.W. Perry, J. Phys. Chem. 91 (1987) 2696. [ 71 SM. Risser and K.F. Ferris, Chem. Phys. Letters 170 ( 1990) 349. [ 81 K. Schulten, U. Dinur and B. Honig, J. Chem. Phys. 73 (1980) 3927.
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[9] Z.G. Soos and S. Ramasesha, Phys. Rev. B 29 (1984) 5410. [ 101 D.N. Beratan, J. Phys. Chem. 93 (1989) 3915. [ 111G.D. Stucky, S.R. Marder and J.E. Sohn, in: Materials for nonlinear optics: chemical perspectives, ACS Symp. Ser. No. 455,eds. S.R. Ma&r, J.E. Sohnaod G.D. Stucky (American Chemical Society, Washington, 1991) pp. 2-30. [ 121 P.N. Prasad, in: Materials for nonlinear optics: chemical perspectives, ACS Syrup. Ser. No. 455, eds. S.R. Marder, J.E. Sohn and G.D. Stucky (American Chemical Society, Washington, 199 1) pp. 50-66. [ 131 J. Linderberg, Physica Scripta 21 (1980) 373. [ 141 E.H. Lieband F.Y. Wu, Phys. Rev. Letters 20 (1968) 1445. [ 151 M.J. Ondrechen, J. Ko and L.-T. Zhang, J. Am. Chem. Sot. 109 (1987) 1672.
CHEMICAL PHYSICS LETTERS
2 April 1993
Third-order nonlinear optical properties of finite bridged polymers Xu Monica Wu and Mary Jo Ondrechen Department of Chemistry, Northeastern University,Bosfon, MA 02115, USA Received 30 September 1992; in final form 30 December 1992
The thirdarder microscopic hyperpolarizability y is calculated for finite bridged polymers of the form A-B-A-B-... using simple model Hamiltonians. Generally, the even- and odd-numbered chains show markedly different behavior, with the addition or subtraction of one chain unit resulting in a predicted change in 1y 1by a factor of up to a few orders of magnitude. The corresponding contribution to the elcctrochromic effect was found to be a sharply increasing function of the chain length 1yI alsois predicted to be significantly enhanced when the coupled A and B electronic states are off-resonant and when the total number of electrons is odd. When the total number of electrons is even, 1y 1may be maximized by adjustment of the A-B energy gap.
1. Introduction
In the present work binuclear chains are studied [ 71. The effects of chemical substitution upon the
Molecules and materials with nonlinear optical properties are the focus of much current interest and activity [ 1,2]. The present work examines the tbirdorder nonlinear properties of finite, bridged polymers. The ultimate goal of this work is to provide guidance in the synthesis and development of materials with large third-order effects. These materials hold promise for novel, marketable optical technologies. Some examples of effects arising from the thirdorder nonlinear susceptibility xc3) are third harmonic generation (THG) and degenerate four-wave mixing (DFWM ) . The bridged binuclear ABAB... chains studied herein serve as models for two different types of systems: (1) conjugated organic polymers with an AB repeat unit [ 31, and (2) bridged mixed-valence compounds where a string of metal atoms are joined together by bridging ligands [ 4,5 1. Beratan et al. have reported on the third-order hyperpolarizability as a function of chain length for homonuclear conjugated organic polymers [ 6 1. The two-band, one-electron tight-binding model of ref. [ 6 ] correctly predicts that as chain length increases there is a rapid rise in the third-order microscopic hyperpolarizability y which then approaches an asymptotic value at about 20-30 atoms.
third-order hyperpolarizability y and upon the thirdorder component of the electrochromism are examined. For simplicity, a Hiickel Hamiltonian is used in the first part of this Letter. We recognize that eleo tron correlation is important in conjugated systems [ 8,9] and that one-electron Hamiltonians constitute an oversimplification, but the first simple model should show some useful general trends. In the fourth section, electron interaction terms are added to the Hamiltonian and some results pertaining to the effects of correlation are discussed. Conclusions are presented in section 5, where some suggestions are made for possible strategies for the synthesis of largexc3) materials.
2. The model A Hiickel-type Hamiltonian for the N-membered chain, written as
RI=
2 O!,UfUi+:
i=l
B(UjUj_1
+UJ_*Uj)
j=2
is adopted. Here at and ai are the creation and annihilation operators respectively for the ith electronic state. q represents the energy of the isolated basis function on the ith chain member. For the chain
0009-2614/93/$ 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.
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CHEMICAL PHYSICSLETTERS
ABAB, the energy of the parent A state is taken to be zero, so (Y~=(Yfor the B species and q=O for the A species. Thus Q represents the energy gap between the B and A basis states. The resonance interaction energy between nearest neighbors is given by B. Because delocalized, conjugated systems are considered here, it is assumed that the conjugated ICelectrons can be treated separately and hence only one electronic state per chain member is important. As is usually the case in a Hiickel system, the two electronic spin states possess the same set of spatial functions. The spatial orbitals are occupied by up to two electrons of opposite spin, starting with the lowestenergy orbital. The AB and BA resonance integrals are assumed to be the same for the sake of simplicity. Non-nearest-neighbor interactions are neglected. At this stage, the interactions between electrons are neglected. Electron-electron repulsion effects are incorporated into the model in section 4. An orthogonal basis set is assumed. The finite polymers are also presumed to be free of defects [ 10 1. The microscopic polarization for a molecule with permanent dipole moment $ in electric field d may be expanded as [ 111 ~j=~Ptaii~t~8iik4~~+dy~~S~~grt...,
2 April1993
where the x axis is the chain axis. The third-order hyperpolarizability y is related to the fourth-order correction to the total energy Ec4) and is given by [ 61 y=
-46,4C2E$f' m
>
(7)
where the index m represents occupied molecular states,
3. Results for ABAB finite polymers Fig. 1 shows log Iy I as a function of the number of members N in the chain for the cases: (a) a= 0.5 eV, fi= - 1 eV and the number of electrons n is twice the number of A atoms (n=2N,); and (b) cuz2.0 eV, /?= - 1 eV and n= 2h’,.,- 1. This corresponds to a bridged chain MLML... consisting of metal atoms M and bridging ligands L where each M atom contributes one or two ICelectrons and each L ligand contributes no n electrons. Curve (a) corresponds to the fully reduced case; curve (b ) corresponds to a mixed-valence system. Fig. 2 depicts log 1y 1 as a function of N for cy= 0.5
(2) 1ol'
and y represents the third-order microscopic hyperpolarizability. Similarly, the bulk polarization may be written as P=P,+x
“‘.Q+x’*‘.b.~+x’3”6.b.~+...
,
(3)
where the xci) are the ith order susceptibility coefficients [ 111. The third-order susceptibility is related to the microscopic hyperpolarizability y as X”‘(-w,;wl,WZ,W3)=li(W,)F(W2)~i(W3)F(W4)
x
;
9
(4)
where the F function is a local field correction and the summation is over orientationally averaged sites
f.121. Perturbation theory is used to obtain y with the Hamiltonian for the chain in an external electric field JXwritten as: A&1
+Y,
V= -e&X,
86
(5) (6)
2
7
12 Chain
17 Units
22
27
32
N
Fig. 1. Log 1yI as a function of number of chainunits N for AB finite polymersfor the cases: (a) (Y=0.5 eV, p= - 1eV and where the numberof electrons equals twice the number of A atoms; (b ) cY=2_0eV,b=-I eVandn=2N,,-1.
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CHEMICAL PHYSICS LETTERS
Volume 205, number 1
60 I
-2, 2
7
12
17 Chain
Units
22
27
32
0.0
0.4 Inl
N
Fig. 2. Log 1yI as a function of number of chain units N for AB finite poIymers for the case ar=O.5 eV, /?= - 1eV and where the
numberof electronsequalsthe numberof chainunits. eV, /I= - 1.OeV and the number of electrons n equals the number of chain members N, These parameters correspond to a polyene with alternate atoms substituted. One notes that in all cases shown, the odd and even chains behave differently, although the odd-only and even-only curves resemble those of Beratan et al. [ 6 ] . The dependence of I yI on the B-A energy gap a! for fixed chain length is also of interest. For even and odd finite chains with n=2N, and for even chains with n=N, the third-order hyperpolarizability y is maximized at finite CL Fig. 3 plots the third-order hyperpolarizability Iy] as a function of (Y for the 4-, 6- and &membered chains with /3= - 1.O eV and with one electron per chain member (II =N) . In cases wheren=NwithNoddorwheren=2N,-1, IyI increases as I a I increases. Fig. 4 shows log I y 1 as a function of (Yfor the 3-, 5- and 7-membered chains with fi= - 1.OeV and where the number of electrons is given by II= 2N, - 1. The case of fig. 4 corresponds to a bridged metallic chain which has been oxidized by one electron. For both even and odd chains, we find that the third-order hyperpolarizability for the very short
0.8 (in
eV)
1.2 A-B
1.6
Energy
2.0
Gap
Fig. 3. The third-order microscopic polarizability y as a function of the B-A energy gap (Yfor 4,6- and I-membered AB chains. ,% - 1 eV and the number of electrons equals the number of chain
-6
-4
0
-2 a(in
eV)
A-E!
2 Energy
4
6
Gap
Fig. 4. y as a function of the B-A energy gap(Yfor 3-, 5- and 7membered AB chains. p= - 1 eVand n=2N,- 1.
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polymers is less strongly dependent on the B-A energy gap than it is for the longer polymers. Since very large values for 1yI were calculated for some of the above examples, it was decided to look at the contributions of the fourth-order energy corrections Ej4) to the electrochromic effect. The HOMO-LUMO transition was studied for variable chain length, Hamiltonian parameters and number of electrons. It was found that for a wide range of cases, the Et4) contribution to the HOMOLUMO transition energy rises, often sharply, with increasing chain length. Of course the zero-order transition energy generally falls slowly as the number of chain members increases, so that the relative contribution of the fourth-order terms is even more strongly dependent on chain length. For instance, for odd chains for the case OI= 1 eV, /I= - 1 eV, n = ZN, and 8,~ 5 X lo6 V/cm, the fourth-order contribution to the HOMO-LUMO transition energy surpasses the zero-order contribution at N= 15. When a! is varied for the same case with fixed chain length, the magnitude of the E(‘) contribution to the HOMOLUMO transition energy is minimal at about a=0 and rises sharply as 1(Y1 is increased.
4. Electron-electron repulsion effects To get a general idea of the effects of electron-electron repulsion on the trends observed above, the Hiickel Hamiltonian of eq. (1) was augmented to include same-site electron-electron repulsion. The resulting Hubbard Hamiltonian is given by
where p is the spin index (t or 1) and Vi is the electron-electron repulsion on the ith site. Eq. (8) is solved approximately by the HartreeFock (HF) method, which has been shown [ 13 ] to give results virtually identical to the exact [ 141 results for UG I II _ When U= 2 IJ], the HF energy is too high by about 10% [ 131. Solutions to eq. (8) for a variety of parameter values show that for I a I> I/31 and I /.?I> U, y changes somewhat as V is increased from zero, but the general form of the dependence on chain length is maintained. Thus when the A and B basis states are off88
2 April 1993
CHEMICALPHYSICSLETTERS
8=-l n=N
eV
Odd Chains
6
i 6 !
6 P
4-
2-
0
I_ 1
5
9
13
17
21
Chain Units N
Fig. 5. Log IyI as a function of chain units for odd chains. (a) a=2 eV, U=O; (b) a=2 eV, U~0.5 eV, (c) a~0.5 eV, U=O; (d)(Y=OSeV,CJ=0.5eV./I=-leVandn=Ninallcases.
resonant (nonzero a), Iy 1 is reduced slightly or may even increase with increasing U, as long as U is not too large (U- ]J])_ For systems where the number of electrons equals the number of chain units ( n = N), the even and odd chains respond differently to an increase in U from zero. For the odd chains when cr is nonzero, ]r] becomes less sensitive to changes in U, also, an increase in U diminishes Iyl for the shorter chains but increases I y 1 for the longer chains. The corresponding threshold chain length is reduced as the A-B energy gap a! increases. For even chains with cyset close to its optimum values, an increase in U from zero causes I y I to be reduced for the shorter chains but increased for the longer chains. Fig. 5 plots log I y I as a function of chain units for the odd chains with I= - 1 eV and n=N for four cases: (a) a=2 eV, U=O; (b) a=2 eV, U~0.5 eV, (c) cu~O.5 eV, U=O; (d) (~~0.5 eV, U~0.5 eV. 5. Discussion and conclusions This work suggests some possible means to control and to improve the third-order nonlinear susceptibility xf3) in finite bridged (or alternating) polymers.
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1yl is found to be enhanced significantly when the A and B electronic basis states which participate in the conjugation are different in energy (off-resonant) wherever the total number of electrons is odd. The sensitivity of 1yI to the B-A energy gap (Yis greater for the longer chains. While the present results suggest that, for a mixed-valence system with an odd number of electrons, one should try to choose A and B so that the energy difference (Ybetween the coupled states is as large as possible, there of course will be a practical limit on the magnitude of o: if 1a I is too large, the resonance interaction will be diminished. However, loI=2 1/II is reasonable for Kbonded metallic systems [ 15 1. Some effect also may be created through substitution of alternate atoms on a conjugated carbon backbone without forfeiture of conjugation. When the total number of electrons is even, 1yl may be maximized by adjustment of cyfor finite chains (for n even and N very large, the optimum (Yapproaches zero [ 7] ). For the cases studied, the optimum (Yis in the range O-21/31to O-71/31,which should be achievable by chemical substitution. For systems with one electron per chain unit (n =N) and N odd, electron correlation resulting from single-site electron-electron repulsion generally causes I y I to be diminished for smaller chain lengths and for smaller values of the B-A energy gap. When a is large the functional dependence of I yI upon chain length is not sensitive to the on-site Coulomb repulsion U: for smaller values of the chain length, I y I is diminished somewhat by an increasing U, but the substantial even-odd effects and the general functional form are preserved. When cy is nonzero (i.e. A and B chemically different), it was found that Iy I is less sensitive to electron correlation than in the a=0 case. This further suggests that, at least in some cases, a difference in energy (of the B and A basis states) is a promising path to.follow when an increase in 171is desired. The different behavior of the odd and even ABAB chains is of interest. In the n=2N, (e.g. fully reduced bridged metallic) case, the even-numbered chains show somewhat higher third-order microscopic polatizability than the corresponding oddmembered chains, but only when (Yis close to the optimum value. The situation is reversed for the n =iV (e.g. conjugated polyene) case, where the odd-
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numbered chains have significantly larger values for I y I. For the n = N and n = 2N,- 1 cases, the difference between the even- and odd-membered chains is generally greater for larger values of the B-A energy gap; the even-odd difference is minimized when a is small (but not necessarily zero). Upon examination of the effects of the B-A energy gap cy on the energy level diagrams, some understanding of the even-odd effects emerges. When (Y= 0 and U= 0, the energy level diagrams are symmetrical in energy space for both even and odd chains. As (Y is increased from zero, the energy levels lying below the median energy level are bunched more closely together while the energy levels lying above the median level are spread further apart. Therefore when there is one electron per chain unit (n =N), the odd chains have a number of low-lying, closely-spaced excited configurations. The corresponding even chains have higher-lying excited configurations which are spread further apart. However, when the number of electrons is 2N,, and (Yis close to its optimum value, the excited configurations for the odd chains are spread still further apart and thus the values for I yI are lowest of all in this latter case. Hopefully the strategies suggested here for the synthesis of higher-X(3) materials will be tested, as materials of the above type appear to show promise.
Acknowledgement The support of National Science Foundation grant No. CHE-8820340 is gratefully acknowledged. We are indebted to Dr. Steven Risser for many useful criticisms and comments. We thank Dr. James Shirk for valuable discussions about nonlinear optics. We thank Messrs. Ihsan Shehadi and Leone1 Murga for many good conversations.
References [ 11 S.R. Marder, J.E. Sohn and G.D. Stucky, eds., Materials for nonlinear optics: chemical perspectives, ACS Symp. Ser. No. 455 (American Chemical Society, Washington, 199 1). [ 21H. Kuhn and J. Robilkud, eds., Nonlinear optical materials (CRC Press, Boca Raton, 1992).
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[ 31 H. Kuzmany, M. Mehring and S. Roth, eds., Electronic properties of conjugated polymers III, Springer series in solid state sciences Vol. 9 1 (Springer, Berlin, 1989) . [ 41 M.J. Ondreehen, S. Gozashti, LT. Zhang and F. Zhou, in: Electron transfer in biology and the solid state, M.K. Johnson, R.B. King, D.M. Kurtz, C. Kutal, M.L. Norton and R.A. Scott, eds., Advan. Chem. Ser. No. 226 (American Chemical Society, Washington, 1990) pp. 225-235. [ 51LB. Bersuker and S.A. Borahch, Advan. Chem. Phys. 8 I (1992) 703. [ 61 D.N. Beratan, J.N. Gnuchic and J.W. Perry, J. Phys. Chem. 91 (1987) 2696. [ 71 SM. Risser and K.F. Ferris, Chem. Phys. Letters 170 ( 1990) 349. [ 81 K. Schulten, U. Dinur and B. Honig, J. Chem. Phys. 73 (1980) 3927.
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[9] Z.G. Soos and S. Ramasesha, Phys. Rev. B 29 (1984) 5410. [ 101 D.N. Beratan, J. Phys. Chem. 93 (1989) 3915. [ 111G.D. Stucky, S.R. Marder and J.E. Sohn, in: Materials for nonlinear optics: chemical perspectives, ACS Symp. Ser. No. 455,eds. S.R. Ma&r, J.E. Sohnaod G.D. Stucky (American Chemical Society, Washington, 1991) pp. 2-30. [ 121 P.N. Prasad, in: Materials for nonlinear optics: chemical perspectives, ACS Syrup. Ser. No. 455, eds. S.R. Marder, J.E. Sohn and G.D. Stucky (American Chemical Society, Washington, 199 1) pp. 50-66. [ 131 J. Linderberg, Physica Scripta 21 (1980) 373. [ 141 E.H. Lieband F.Y. Wu, Phys. Rev. Letters 20 (1968) 1445. [ 151 M.J. Ondrechen, J. Ko and L.-T. Zhang, J. Am. Chem. Sot. 109 (1987) 1672.