J. Phys. Chem. 1994,98, 7782-7789
7782
One- and Two-Photon Absorption Spectra of Short Conjugated Polyenes Yi Luo, Hans Agren,' and Sven Stafstriim Institute of Physics and Measurement Technology, University of Link6ping, S-58183 Linkaping, Sweden Received: February 10, 1994; In Final Form: May 23, 1994'
One- and two-photon absorption spectra (OPA and TPA) and polarizabilities of ethylene, trans-butadiene, trans-hexatriene, and trans-octatetraene are investigated by means of linear and quadratic response theory calculations. It is demonstrated that accurate results for the one-photon excitation spectra of these short polyenes can be obtained already a t the lowest level of response theory, namely, the random phase approximation (RPA). The excellent agreement with the experimental one-photon spectra, generally within one or two tenths of an electronvolt, indicates that the comparatively simple but still well-defined R P A method has important ramifications for the study of linear optical properties of the conjugated polyenes. The polarizabilities generated by R P A are within a few percent of the vapor-phase values a t the corresponding frequencies. Two-photon spectra of the short polyenes are investigated by the quadratic R P A method. Two-photon absorption maxima have been found for the even states of trans-butadiene, trans-hexatriene, and trans-octatetraene located a t 1.5-1.7 E,. The first one-photon forbidden *A, state is found to have a very small two-photon amplitude and is therefore likely not to contribute significantly to the nonlinear properties of the polyenes, although it might intrude the one-photon bandgap. The mechanism of optical nonlinearity in the short conjugated polyenes is explored in relation to the so-called three- and four-state models. I t is found that the validity of these models for calculations of hyperpolarizabilities of the short polyenes is rather limited.
1. Introduction
Short polyenes are frequently used as test compounds for a variety of methods addressing properties of oligomers. The electron absorption spectra of ethylene (ET), butadiene (BD), hexatriene (HT) and octatetraene (OT) have thus been extensively studied both theoretically and experimentally; see ref 1 and references therein. Themost systematicof the theoretical studies of the excited states of these molecules seems to be "state-specific applications" using multireference determinant configuration interaction (MRDCI) methods24 and, very recently, using the multiconfiguration perturbation theorylJ (CASPT2) method. A good general agreement with experimental data has been obtained using these methods, although excitation energies of some important states still are unsatisfactory. For instance, the excitation energy for the 11B, state of BD, which represents the major feature in the electronicspectrum, is afflicted by a difficulty to balance Rydberg and valence contributionsand is overestimated in most of the ab initio calculations. Although most applications of short polyenes have concerned one-photon spectra and other linear properties, theoretical efforts of today focus on nonlinear properties of these systems. This is because the well-known fact that conjugated polymers show conducting properties and large, ultrafast, nonlinear optical responses bearing some important technical consequences. Up to now the nonlinear properties, viz., hyperpolarizabilities and multiphoton spectra, have been analyzed by model calculations of semiempirical type. These studies have mostly concerned hyperpolarizabilitiesof oligomer sequences, in particular polyenes and polyynes, for which the length dependence and convergence to the polymer value is of interest. Out of several ab inito methods the random phase approximation (RPA, equivalent to timedependent HartreeFock (TDHF)) seems to have the greatest potential for large scale (large molecule) applications on linear as well as nonlinear properties. This has recently been demonstrated through thedevelopment of so-called direct atomicorbital techniques.6J One main purpose of the present work is therefore to establish a level of confidence for such calculations concerning two-photon absorption and hyperpolarizabilities for the shorter Abstract published in Advance ACS
Abstracts, July 1, 1994.
0022-3654/94/2098-1782%04.50~0
members of the polyene series, for which undisputable experimental results are available for comparison. A second main purpose of the present work is to resolve some ambiguities concerning the mechanism of the optical linearity and nonlinearity in the short polyenes. The electronic one- and two-photon spectra become progressively more compressed with the length of the chain. It is therefore crucial to obtain reliable information on the one- and two-photon transitions moments in order to assign spectra. For polyenes, the two-photon absorption of even states is of particular interest, since it is the key point for understanding optical nonlinearity. The excited states of polyene molecules can be divided into two categories according to the selection rules imposed by symmetry. The first kind of states are one-photon allowed only, while the second kind only can be reached by two-photon transitions. As mentioned above, the energetics of one-photon transitions have already been carried out by different theoretical methods. We examine the one-photon transition moments with RPA using large basis sets. Two-photon transition probabilities of BD have previously been obtained with small basis set RPA calculations,* using approximationsgfor the transition moments between excited states that excludes the so-called EL31 and S[31 contributions.lO In our previous calculations for the two-photon transition moments of small molecules it was, however, shown that such contributions cannot be neglected." No ab initio calculations have so far been reported for two-photon transition moments of the HT and OT molecules. Several semiempirical calculations are a ~ a i l a b l e , ' ~ J ~ but with rather disparate conclusions. For instance, Sooset al.l4J2 found a strong two-photon absorption (TPA) around 1.5 E,, where E, refers to the excitation energy of the 1IB, state of the polyenes (except for the ET molecule) and proposed a three-state model for the hyperpolarizability. Dixit et al.13915 found, however, such a strong TPA to be close to E,, and argued that a four-state model would be required to determine the hyperpolarizability. Such disparate conclusions have led to different explanations for the mechanism of optical nonlinearity of the polyenes. In an attempt to resolve these problems we present the first systematic ab initio calculations of two-photon transition moments of polyenes. We use the results to assign the TPA spectra of these 0 1994 American Chemical Society
Spectra of Short Conjugated Polyenes
The Journal of Physical Chemistry, Vol. 98, No. 32, 1994 7783
compounds and to evaluate the validity of the three- or four-state approximations.
TABLE 1: Excitation Energies (in eV) and Oscillator Strengths of Lower-Lying States of Ethylene
2. Method
states
E
l'B3, llB1, l1BIS 11B2, 21Ag z1B3, llA, 3lB3,
7.15 7.38 7.77 7.93 8.59 8.94 9.02 9.37
BAS1"
The linear and quadratic response theories used in the present work are well d o c ~ m e n t e d , ~ ~ and J ~ J 7extensive applications of these theories have been reported. For instance, accurate values of polarizabilities and hyperpolarizabilities have been obtained for H20,"3 N2,19 and NH3z0molecules. Two-photon absorption spectra of H20," benzene,21 and pyrimidinezz have also been studied, where the vibronic induced two-photon spectrum for benzene21 and pyrimidineZZ have been calculated for the first time at the a b initio level. In the response theory, the dipole polarizability is determined by the linear response function. The single and double residues for the linear response function give the excitation energies and the transition moments. The quadratic response function refers to the hyperpolarizability when the dipole operators areemployed. The two-photon transition matrix elements and the transition moments between excited states aregiven by the singleand double residues of the quadratic response function. The formula of linear and quadratic response functions and their residues can be found in ref 10. 3. Calculations
We choose one set of optimized geometries, obtained with the MP2 meth0d,~3and two sets of basis functions, the polarizability consistent contracted basis set C(5~3p2d)/H(3~2p) of Sadlej24925 and the generally contracted basis sets of atomic natural orbitals (ANO) of Widmark et al.z6for the calculations of excited states. The latter was obtained from (14s,9p,4d) and (8s,4p) primitive sets for carbon and hydrogen, respectively, and used in a recent multiconfiguration perturbation theory calculations.' In the A N 0 calculations of ET, BD, and HT we use the same addition of diffuse functions and modifications as in ref 1. The A N 0 basis set for the OT molecule has been contracted from the primitive A N 0 set as C(4slpld)/H(2slp) together with ap(0.008) diffuse function on carbon. For the two-photon transition moments and polarizabilities we have, besides the A N 0 basis sets, used a 431G* basis set, which is a standard 431-G basis set plus diffuse p(0.05) and d(0.05) functions on carbon. All results have been obtained with the SIRIUS program for multiconfiguration selfconsistent field (MCSCF) wave functions and linear and quadratic response f~nctions.16J~~2~ The HERMIT program28is used for generation of electrostatic intergrals.
4. Results I. Excitation Energies for ET, BD, HT, and OT Molecules. Excited-state energies of ethylene have previously been studied with the random-phase approximation (RPA).29,30 The RPA results of Bouman and Hansen30 for ethylene using a moderate basis set were found to be in quite good agreement with experimental data. Molecular dipole polarizabilities of ET, BD, and HT calculated by RPA are also in good agreement with the experimental data.3' In this work, we have used extended basis sets and obtained results which we believe are close to the RPA basis set limit for one-photon spectra and polarizabilities. The excellent agreement with experiment sustains this contention. The results for excited states of the ET, BD, HT, and OT molecules are listed in Tables 1-4 together with experimental data and best previous theoretical results. Below we go into some details of the results for each molecule. I.A. Ethylene. The excitation energies for E T are shown in Table 1. For the lower-lying states, the results from both basis sets are of almost the same quality. However, the A N 0 basis set gives better results than Sadlej's basis set for the Rydberg states. The A N 0 basis set used here is the same as the one used
oscstr
BAS2b
CASFT2F BAS2C
oscstr ex@ error('
E
0.090 7.10 0.087 0.401 7.36 0.400 7.70 7.84 8.17 0.001 8.57 0.0001 8.76 0.200 8.81 0.025
7.11 7.60 7.80 7.90 8.28 8.62
-0.01 -0.24 -0.10 -0.06 -0.11 -0.05
8.90 -0.09
E
oscstr
7.17 8.40 7.85 7.95 8.40 8.66 8.94 9.03
0.0067 0.16
0.00094 0.00057
Sadlej's b a s i s ~ e t . A ~ ~N 0 basis set 1. Quoted in ref 1. Difference between the A N 0 results and experiment. Reference 1; oscillator strengths from CASSI.
TABLE 2 Excitation Energies (in eV) and Oscillator Strengths of Lower-Lying States of Butadiene BASl"
states
E
llB, llB, 1'A. 2IA, 2lB, 2IB, 2'A; 3IB. 3'A; 3'B,
5.91 6.19 6.53 6.69 7.49 7.42 7.25 7.44 8.28 9.56
oscstr
BAS2b
E
0.8111 5.93 6.26 0.005 6.52 0.057 6.63 0.050 6.96 7.30 7.31 7.35 7.48 0.005 7.81
CASPT2F BAS2f
oscstr expf errorC
E
oscstr
+0.01 -0.01 -0.14 -0.17 -0.11 +0.02 -0.09
6.23 6.29 6.56 6.59 6.70 7.30 6.27
0.686
0.00 7.48 0.002 8.00 -0.19
7.47 7.79
0.803 5.92 6.27 0.005 6.66 0.050 6.80 0.057 7.07 7.28 7.4d
0.002 0.037 0.080
0.036
Sadlej's basis A N 0 basis set.' Quoted in ref 1. Reference 34. Difference between the A N 0 results and experiment. f Reference 1; oscillator strengths from CASSI. by Serrano-Andres et al.' in their CASPT2 calculations. Our RPA results show the same accuracy with respect to experimental data as those obtained by CASPT2. For the 11B3,, llBIg, and 1'Bzg states, however, Sadlej's basis set gives the best results. The energy of the second excited state of the ethylene molecule, the llB1, state, is associated with some uncertainty. In ref 32 it was claimed that the vertical energy of the 1lBlu state, defined as the band maximum, occurs at 7.60 eV. However, from theoretical studies it was suggested that the vertical energy might be about 8.0 eV; see ref 1 and references therein. In any case, it seems that this state may be underestimated by our RPA calculation and overestimated by the CASPT2 calculation. There are no experimental data available for the l'A, state, which is both dipole and quadrupole forbidden. Our RPA results for this state are in good agreement with the CASPT2 result, 9.02 eV (BAS1) or 8.76 eV (BAS2) compared with 8.94 eV. I.B. Butadiene. An accuratedescription of the first 11B, state of the butadiene molecule has been considered a challenge for theoretical calculation^.^^^ A large number of theoretical studies have been reported for this state, but with disparate and rather unsatisfactory result^.^ For instance, the excitation energy for the l'B,state, which represents themajor feature in theelectronic spectrum of BD, is overestimated in most a b initio calculations presented so far. Only two very recent results have given an excitation energy for this state in reasonable agreement with the experiment, 6.1433and 6.12 eVI compared to the experimental value of 5.92 eV (quoted in ref 1). It has been claimed that the theoretical treatment of this state is extremely difficult due to the sizable Rydberg-valence mixing. From this point of view, the excellent agreement obtained by RPA is quite remarkable, the deviation being only 0.01 eV for both basis sets employed (see Table 2). Actually, in addition to this state, our RPA results from A N 0 basis set calculations give the excitation energies for 10 singlet states with an error less than 0.19 eV compared with available experimental data. The 21Ag state has been experimentally assigned to be at 7.4 eV34whichis in very good agreement
7784 The Journal of Physical Chemistry, Vol. 98, No. 32, 1994
TABLE 3: Excitation Energies (in eV) and Oscillator Strengths of Lower-Lying States of Hexatriene BAS 1* BASZb CASPT2D BAS2C states E oscstr E oscstr expc errord E oscstr llB, lIA, lIB, ZIB, 2IA, 2'A, 3IA, 2IB, 3IB, 4IA, 3'B, 41B,
5.06 5.68 5.87 6.11 6.69 6.61 6.67 6.69 7.73 7.16 7.30 7.41
5.08 5.73 5.83 6.02 6.14 0.024 6.51 0.012 6.52 0.003 6.63 0.019 6.73 0.0009 6.79 6.91 6.95 1.324 0.012
1.333 0.010
4.93 +0.15 5.67 +0.06 6.06 -0.04 6.24 -0.10 6.53 -0.02
0.025 0.003 0.001 0.013 6.90 -0.17 0.00006
5.01 5.84 6.12
0.85 0.0015
5.19 6.11
0.071
Sadlej's basis setZ4 but with the diffuse p function of H removed. A N 0 basis sets1 Reference 40. Difference between the A N 0 results and experiment. e Reference 1; oscillator strengths from CASSI. with our calculated value of 7.31 eV. Extensive searches by multiphoton ionization,35 thermal lensing,36 and fluorescence spectroscopy37 have provided no assistance on the location of a lower 'A, state as predicted from CASPT2 ca1culations.l Both of RPA and CASPT2 calculations have placed the 'A,stateabove the IB, state. The resonance Raman scattering studies of Hudson and co-workers found an 'A, state lower than the llB, state by 0.25 eV;3*,39however, there is no indication in their work that this difference refers to the vertical energies. It should be mentioned that the two states have quite different geometries and that relaxation effects could place the 0-0 transition of the 1'A, state below that of the llB, state. As for the ET molecule, Sadlej's basis set is of somewhat poorer quality for the Rydberg states. For the two 'A, states, however, Sadlej's basis set is better than the A N 0 basis set, which probably is due to the fact that it gives a better description for the out-of-plane component for the polarizability. Z.C. Hexatriene. In Table 3 we show the excitation energies for HT. Much of the observations accounted for in the previous paragraphs for the E T and BD molecules are repeated in the case of hexatriene. An evident difference is that the spectrum of hexatriene is more compressed. Our RPA results from the A N 0 basis set are in very good agreement with the experimental observations.40 Contrary to E T and BD there are rather few theoretical studies of HT. Our assignment for the experimental spectra is, however, on several points different from the one given by Serrano-Andres et a1.l As for E T and BD our BAS2 basis set is the same as the A N 0 basis set used in ref 1. The 11B, and lIA, states are found at 5.08 and 5.73 eV, in good agreement with experimental and CASPT2 results. There should be two possible B, 3p Rydberg states in HT (trans). One arises from the 3p Rydberg orbital oriented predominantly along the carbon atoms of the molecule, Le., B,(3pX), and the other from the 3p Rydberg orbital lying in the molecular plane but oriented perpendicular to the line connecting the carbon atoms, Be(3pJ. These two states are not distinguishable experimentally (by symmetry), and only a weak band at 6.06 eV has been observed.40 By comparing with the spectrum of cis-hexatriene, the B,(3py) state of trans-hexatriene has been assigned to 6.06 eV.40 We have located those two lB, states at 5.83 and 6.02 eV, respectively. A band at 6.53 eV has been found by Sabljic and McDiarmid,40 who suggested it would be an 'A, state, and by Gavin and Rice,41who considered it might be a 'B,, state. From the analysis of oscillator strengths, we find that it should correspond to the 2IA, state. The 2IB, state is harder to detect experimentally due to the weakness of this transition. The band occurring at 6.9 eV was tentatively assigned by Sabljic and McDiarmid40 to be an A,, state. However, our results indicate that this band should correspond to the 31B, state.
Luo et al. Thedipole forbidden 2'A, states turn out to be the problematic ones. From their multiphoton experiments, Parker et al.42assigned the first 'A, peak in gas-phase hexatriene to 6.2 eV. This assignment was later confirmed (at 6.24 eV), by Sabljic and M ~ D i a r m i d .Fujii ~ ~ et al.43 assigned a weak structure in their CARS two-photon experiment on liquid phase hexatriene to 5.21 eV. We could not find an IA, state close to this value, instead, the first RPA 'A, pole appears a t 6.14 eV which is very close to the experimentally observed gas-phase value around 6.2 eV.40.42 One should, however, note the rather large basis set dependence for this particular state. The possible Rydberg transitions, i.e., three optically forbidden 3p, one optically allowed 3s, and five optically allowed 3d Rydberg transitions, are assigned quite accurately from our RPA response calculations. It is the first time that this complicated spectrum can be assigned theoretically. Z.D. Octatetraene. The RPA response theory results for excitation energies and oscillator strenghts of the octatetraene molecule are listed in Table 4. As for the smaller members of the studied polyene series the ANO-based RPA results are in very good agreement with experimental observation^.^-^^ In contrast to the smaller members the number of theoretical studies is quite small for octatetraene. Expect for the work of Cave and Davidson& the multiconfigurational second-order perturbation theory (CASPT2) calculations of Serrano-Andres et al.5 is the only a b initio study of the excited states of octatetraene (OT) reported to date. However, these calculations could not assign the one-photon spectrum. The llB, state is found a t 4.41 and 4.55 eV with the 431-G* and the A N 0 basis sets, respectively, which is in very good agreement with experimental value 4.41 eV, and also in good agreement with the CASPT2 result, 4.42 eV (the A N 0 type basis set used here for the OT molecule is somewhat smaller than the one used by Serrano-Andres et a1 in their CASPT2 calcuationss). The CI calculation of Cave and Davidson6 gives a value of 4.76 eV. Except for this 1'B, state the RPA results are, however, significantly different from those obtained by the CASPT2 method. Three bands a t 5.69,5.88, and 6.04 eV have been found by optical fluorescence spectroscopy." Using EELS, Allan et al.45 found two bands a t 5.85 and 6.04 eV and gave a tentative assignment of the 6.04 eV peak as a 2lB, state. This state was, however, assigned as an optically forbidden state by CASPT2.5 By contrast, the ANO-based RPA results assign all experimentally observed bands unambiguously. In the relevant energy region (5.60-6.1 eV), we have found three one-photon allowed states at 5.63, 5.88, and 6.03 eV, which are assigned as the llA,,, 2IA,, and 2IB, states, respectively. The llB, state is found at 5.79 eV and is thus also in this energy region but should, being one-photon forbidden, remain unobserved. Semiempirical calculations on all-trans OT have favored the 21A,statelowerthan the l'B,statewhile thereversewas predicted in large scale CI calculations.& These disparate results can undoubtedly be referred to both geometrical distortion and to the unusually correlated nature of the 2'A, state with strong mixing of singly and doubly excited configurations.47 Because the 21A, state is long-lived and because it has a strong nonadiabatic mixing and strong nonradiative decay on a short time scale48the vertical approximation is called into question. Three very recent studies on gas-phase trans,truns-octatetraeneall favor a low 2IA, state assignment; the fluorescence spectrum by Bouwman et al.;49 the fluorescence one- and two-photon spectra in free jet expansions by Peteket al.;48and the CASPT2 calculations of Serrano-Andres et al.5 The one- and two-photon fluorescence excitation and emission spectra of OT in free jet expansions measured the 0-0 transition of the 2IA, state below that of the lIB, state$* which basically was confirmed by the CASPT2 calculations of SerranoAndres et al.5 However, no experimental data for the vertical transition to the 2IA, state have been reported.
Spectra of Short Conjugated Polyenes
TABLE 4
The Journal of Physical Chemistry, Vol. 98, No. 32, I994 7785
Excitation Energies (in eV) and Oscillator Strengths of Lower-Lying States of Octatetraene BAS 10
states 1 'B, 1 'A, llB, 2'A, 2lB, 2lB, 31B, 2lA, 3lA, 3lA, 4lB, 4lA,
E 4.41 5.59 5.53 5.96 6.75 6.35 6.38 6.43 6.48 6.96 6.94 7.20
CASPT2D BAS2e
BAS2b
osc str 1.9070 0.0000001 0.0240 0.0022
0.0179 0.0066
E 4.55 5.63 5.79 5.88 6.03 6.22 6.34 6.41 6.47 6.63 6.64 6.70
osc str
expe
error('
1.8411 0.00001
4.41 5.69
+0.14 4.06
0.0174 0.0006
5.88 6.04
4.00 -0.01
E 4.42 5.48 5.32 5.68 5.70
osc str 1.8316 0.000003 0.0054 0.0213
4.38 6.10 0.0022 0.0002
43 1-G*,see text. A N 0 basis set, see text. References 44 and 45. d Difference between the A N 0 results and experiment. * Reference 5 ; oscillator strengths from CASSI.
We failed to find a vertical transition to a 'A, state below the llB, state. Instead, our first 'A, state is located as high as 6.41 eV by the ANO-type basis set calculation. However, beside this state, the ANO-based RPA results are in good agreement with the CASPT2 results for theother 'A,states; we find the excitation energies of 3lA,, 41A,, and S'A, to be 6.47, 6.83, and 7.15 eV, respectively, compared to 6.10,6.56, and 7.14 eV, respectively, obtained from the CASPT2 calculations. The two basis sets employed give different number of intensive 'A, TP states in the appropriate energy region; however, the (probably unresolved) TP peak maximum, remains the same, around 8 eV. The mismatch for the sole first excited 'A, state is remarkable consideringthe experimental agreement for all other states in the polyene series here investigated but also considering the agreements obtained for other 'A, states in the series of octatetraene (note, however, the assignment of the 2lA, states in BD and HT). Further investigations on expansion of correlated reference spaces (multiconfigurationallinear and quadratic response calculations) and on the role of geometric relaxation is called for, though both effects are beyond the scope of the present investigation. The important conclusion drawn from the present study is that due to the very small TP amplitude the 21A, state is not importnat for the nonlinear behavior of polyenes as previously expected. Larger TPA intensities are instead found for transitions to higher lying 'A, states. These states must be included in a proper treatment of the nonlinear susceptibility. The discussion in subsection 5 on three- and four-state models for hyperpolarizabilities further illuminates this point. 11. OscillatorStrength. The data from the RPA and CASSCF state interaction (CASSI)' methods give rather different oscillator strengths for ET and HT, but similar for the BD molecule; see Tables 1-4. The RPA oscillator strengths for the ET molecule are generally larger than those obtained from CASSI' but are closer to the experimental values and are also in good agreement with other similar c a l ~ u l a t i o n s . ~For ~ ~ instance, 5~ the experimental oscillator strength for the 11B1, state is 0.34, in fair agreement with the RPA value of 0.4. The oscillator strengths for the 1'B, stateofOTis 1.84for theANObasissetand 1.91 forthe431-G* basis, this time in reasonable agreement with the CASSI result of 1.83.5 The RPA oscillator strength for the first state, the 11B, state, of the HT molecule is 1.32 and 1.33 for Sadlej's and A N 0 basis sets, respectively, and close to the result obtained by Cave and Davidson,s1 which is 1.24, but different from the CASSI result of 0.85.' For the next state, the llA, state, the RPA and CASSI results of the oscillator strength differ by a factor of 10. The successful assignment of the experimental spectrum of the HT molecule gives, however, confidence to the present results for the oscillator strengths, despite that experimentallyderived oscillator strengths are lacking. The general experience from aromatic compounds,where experimental oscillator strengths occasionally
are available, is that they indeed are well recapitulated by linear response theory, in a relative sense better than excitation energies.52 III. Two-Photon Absorption of Even States. The hyperpolarizability of polyenes is dominated by the component along the chain. Except for the ethylene molecule, the polyenes belong to the common C2h point group. The one-photon allowed 'B, (odd) states and two-photon allowed 'A, (even) states are thus very important for understanding the nonlinear optical response of polyenes. The odd states (lB,) and even states ('A,) are antisymmetric and symmetric with the respect to the mirror plane passing through the center of the chain, respectively. In this section, we discuss the results of response theory calculations of the two-photon transitionsof even states for butadiene, hexatriene, and octatetraene molecules. The even states with the excitation energies lower than 1.8 E, have been calculated, where E, is the excitation energy of the first odd state (llB,,), Le., the optical gap. The resonance effect is thus not discussed. The calculated two-photon transition matrix elements of even states for BD, HT and OT molecuels are shown in Tables 4-6,respectively. Data for both the A N 0 type and the 43 1-G*basis setsareincluded. Calculations of the TPA intensity, &, assuming two linearly polarized light beams with parallel polarization vectors are also carried out (the definition of the TPA intensity can be found in refs 21 and 22). This intensity is expressed as
The polarization ratio for two-photon transitions for a circularly versus a linearly polarized one-color beam is obtained as
The basis set dependence is quite evident from the data shown in Tables 4-6. With the ANO-type basis set more states show up in the energy region below 1.8 E,. To be able to describe the excited states of polyenes accurately, an extended basis set has to be employed, and we regard, therefore, two-photon absorption results obtained from the ANO-type basis sets as more reliable. However, as will be discussed below, some common features of two-photon absorptioncan still be found for the results irrespective of the choice of basis sets. The TPA maximum for the BD molecule using the A N 0 type basis set is found at 9.03 eV, corresponding to the seventh 'A, state (see Table 5). With the 431-G* basis set the maximum is instead located at 8.93 eV, which corresponds to the fourth 'A, state. Although the number of states is not the same, the positions of the maxima are very close, at about 1.5 E, in both cases. The
Luo et al.
7786 The Journal of Physical Chemistry, Vol. 98, No. 32, 1994
TABLE 5: Two-Photon Transition Moment Matrix Elements (Sb i, j = x, y, z), Intensities (St), and Polarization Ratios (A) of Even States (lAc) for Butadiene Molecule (in Atomic Units) energy TP matrix elements S", 6, ( ~ 1 0 3 ) A basis state eV E. sxx sxv SZZ
BAS2 2'A, 31A, 41A8 5lA, 6IAg 7IAg 8IA,
7.31 7.48 7.90 8.05 8.22 9.03 10.08
1.23 1.26 1.33 1.36 1.39 1.52 1.70
18.10 22.51 3.24 12.64 3.26 0.01 41.20
52.30 23.25 17.85 11.61 3.53 123.81 20.72
11.77 11.95 5.60 6.50 2.43 50.59 -2.00
1.02 6.79 1.17 3.82 1.86 19.23 -1.65
11.81 5.52 1.28 1.58 0.15 62.10 7.91
0.42 0.25 0.53 0.25 0.27 0.68 0.35
2lA, 3IA, 4IA, 5IA,
7.50 8.71 8.93 10.63
1.24 1.44 1.48 1.76
11.49 42.62 15.72 14.66
57.02 67.47 116.79 7.81
9.37 29.36 40.55 -80.36
1.53 25.32 20.09 -33.13
12.03 35.80 58.45 28.69
0.48 0.22 0.50 1.49
431-G*
TABLE 6 Two-Photon Transition Moment Matrix Elements (Sh i, j = x, y, z), Intensities (St) and Polarization Ratios (A) of Even States ('A#) for Hexatriene Molecule (in Atomic Units) energy TP matrix elements basis energy eV E, SZ, sxx SX" S", 8, (xi031 A BAS2 21A8 3'A, 4lA, 5IA, 6lA, 71A,
6.15 7.03 7.15 7.31 8.14 8.61
1.21 1.39 1.41 1.44 1.60 1.69
18.01 12.30 8.46 -1.78 5.18 -13.19
46.90 88.51 13.71 21.50 408.01 38.09
17.12 27.66 18.98 1.87 173.99 18.78
9.93 6.88 14.74 -1.02 70.83 9.19
12.02 30.72 3.76 1.30 698.4 5.99
0.33 0.56 0.59 0.83 0.66 1.02
2IA, 3IA, 4IA, 5IA,
7 -00 7.50 7.97 8.71
1.35 1.45 1.54 1.69
21.31 -7.78 7.51 32.80
110.76 29.39 333.69 216.08
35.29 4.07 130.82 125.25
16.41 -5.54 52.53 69.61
53.01 2.23 451.56 269.41
0.46 1.21 0.64 0.56
431-G*
TABLE 7: Two-Photon Transition Moment Matrix Elements (S, i, j = x, y, z), Intensities (St) and Polarization Ratios (A) of Even States (1Ad for Octatetraene Molecule (in Atomic Units) energy TP matrix elements basis state eV E. SZ, S., S," S"" 8,1x103) A ~~
BAS2(O) 2IA, 3IA, 4lA, 5IA, 6IA, 7IA,
6.41 6.47 6.83 7.15 7.25 7.82
1.40 1.42 1.50 1.57 1.59 1.72
27.12 8.44 7.38 0.90 -3.05 -3.74
106.87 3.31 190.85 79.13 52.04 905.92
38.88 18.00 8.67 29.37 -19.13 405.70
18.73 16.72 41.16 10.20 -23.96 174.31
54.38 2.83 163.75 24.32 8.67 3535.55
0.43 0.78 0.43 0.66 1.32 0.68
2IA8 3IA, 4lA, 5IA,
6.43 6.48 7.17 7.66
1.46 1.47 1.63 1.74
10.67 10.96 12.50 21.69
122.58 92.97 840.30 92.01
39.59 36.81 358.39 32.62
14.41 17.10 152.83 2.96
58.76 38.18 2984.27 35.76
0.57 0.54 0.65 0.55
431-G*
See text. 2lA, state shows a considerable TPA intensity, it is the second strongest TPA band in the BD molecule as calculated with the A N 0 basis set. It should, therefore, be observable in future TPA experiments. The multiphoton ionization experiment reported in ref 35, shows no sign of an 'A, state below 6.05 eV. From the electron energy loss investigation, a diffuse structure near 7.4 eV has been assigned as a lAg+ 'A, transition.34~53Our result seems to confirm this assignment. The polarization ratio A for the 3lA, state has been found to be 0.3 by multiphoton ionization experiment,$4 which is in good agreement with our calculated value of 0.25 with the A N 0 basis set. Gas phase multiphoton spectra of HT show a relatively strong TPA at 6.24 eV.40 It corresponds well with the calculated 'A, state at 6.15 eV obtained with the A N 0 basis set; see Table 6. The differences between the results from the A N 0 and 43 1-G* basis sets for HT are very similar to those for BD. The TPA maximum is located around 8.1 eV (1.6 E,), corresponding to the 6IA, state for the A N 0 basis set and to the 4IA, state for the
431-G* basis set. A quite strong TPA band is also predicted around 7.0 eV. The TPA maximum for the OT molecule is located at 7.82 eV for the A N 0 basis set, and a t 7.17 eV for the 4-31G* basis set (see Table 7), thus approximately at 1.7 E,. The TPA spectra of the even states for BD, HT, and OT are also shown in Figure 1. Irrespective of basis set, the TPA spectrum of those three molecules show the common feature of a band maximum around 1.5-1.7 E,. Such a'feature has also been found by McWilliams et al.,l* who used the Parise-Parr-Pople (PPP) model to carry out the TPA spectrum for 6 C N C 12 polyenes, where N is the number of carbon atoms. By using the Hubbard and extended Hubbard models up to N = 8, Dixit et al. found that the TPA maximum is close to the lB, state.13 Since our study is limited to the even states of the lowest three members of the polyene series, the generality of these features must be verified by calculations of longer polyene chains. Further investigation along this line is in progress in our laboratory.
Spectra of Short Conjugated Polyenes
The Journal of Physical Chemistry, Vol. 98, No. 32, 1994 I181
1
1
.-5
Q
0.8
2
i
0.8
.-Y
c E
$.-I 2
0.6
>
o.6 0.4
0.2 0
0
I
1
.-n
A
0.8
-
0.8
: -
0.6
-
I
OT-431G.
u E
0.6 0.4 0.2
0.2 0'4
0
1 1
0 : I .2
.
4
1
1.3
I .4
'
1
I .5 b I E l
'
1
I\
I .6
1.7
-
1 .E
Figure 1. Relative TPA intensities of even states for the BD,HT, and OT molecules. A Gaussian line shape with a line width of 0.01 Eg for each individual transition is used: (a, left) from AN0 basis sets; (b, right) from 431-G*basis set.
TABLE 8 Frequency-DependentPolarizabilities of Ethylene (ET), Butadiene (BD), Hexatriene (HT),and Octatetraene (OT, All Values in Atomic Units)from AN0 and 4314' Basis Sets ~w = 0.00 w = 0.0656 ET BD HT OT ET BD HT OT AN@ ~
a, a, azz a
36.94 24.15 23.03 28.24
86.92 42.84 33.69 54.48
159.64 67.64 47.02 91.43
247.32 91.94 60.20 133.15
38.23 25.12 23.62 28.99
92.37 43.58 34.36 56.77
175.24 69.70 47.9 1 97.62
280.47 96.45 61.10 145.98
axx avv azz
36.03 22.26 18.68 25.66
87.57 43.04 32.07 54.23
160.87 67.22 44.97 91.02
268.03 96.35 57.55 140.64
37.29 22.60 19.22 26.37
92.98 43.88 32.85 56.57
176.19 69.27 45.99 97.15
307.61 101.50 58.80 155.97
431-G'
a
exP
28.7Ob 58.3lC * The same basis sets as the onw used for two-photon transition element matrix elements. * Reference 55. C Reference 56. acrp
IV. Polarizabilities. Table 8 shows results from RPA calculations on static and dynamic dipole polarizabilities in comparison with experimental data and with previous theoretical results. ANO-Sadlej's polarizability consistent-and 431-G* basis sets have been used. The basis set effect is quite small in general. A noticable difference is obtained only for OT, where A N 0 and Sadlej basis sets predict a smaller value than the 43 1G* basis set. It reflects the fact that the A N 0 basis set gives higher energy for the first 1B, state of OT than that calculated with the 43 1-G* basis set. Experimental values of polarizabilities have been reported for the ET and BD molecules in the gas pha~e.5~~56 Our results for dynamic polarizabilities of ET and BD are in excellent agreement with these experimental data (see Table 8). For the HT and OT molecules, the polarizabilities at o = 0.0656 au are 97.15 and 155.97 au, respectively, which are close to previous theoretical results of 98.67 and 145.84 au given
in ref 3 1. The static polarizabilities of the three molecules are also listed in Table 8. It can be seen that the dispersion of polarizability is increased when the chain is longer. V. Mechanism of Optical Nonlinearity. The hyperpolarizabilities of short polyenes have been intensively studied recently. Those investigations fall into two categories. One is using ab initio methods attempting to accurately reproduce experimental hyperpolarizabilitie~.31.5~*58 The other category of investigations attempt to find out the mechanism behind optical nonlinearitylzlJJs in conjugated polymers. The latter has involved semiempirical methods. Using ab initio coupled Hartree-Fock theory (equivalent to the RPA), Karna et al. computed the frequency-dependent hyperpolarizabilities of the ET, BD, HT, and OT molecules and abtained good agreement with available experimental gas-phase res~lts.3~The basis set dependence has been discussed by Hurst et aI.,s7who concluded that the basis set
Luo et al.
7788 The Journal of Physical Chemistry, Vol. 98, No. 32, 1994 and dispersion effects play the dominant roles in determining the hyperpolarizabilities of polyenes. The correlation effect might, however, be rather small.31 Although, the hyperpolarizabilities of short polyenes calculated with the RPA method have reached high accuracy, the mechanism of the optical nonlinearity has not been discussed in previous ab initio calculations. Using a semiempirical method, Heflins9 proposed that the hyperpolarizability of polyenes can be well reproduced by a three-state approximation. Later, this proposal was supported by Soos et al,,12J4 while Dixit et al.1596’Jfound that it is necessary to use at least a four-state approximation to determine the hyperpolarizability. The hyperpolarizability of polyenes is dominated by the component along the chain, in our case, the xxxx component. Such a component is controlled by the transition route 1‘A,
+
ulBu
+
b’A,
- c’Bu
>’+--
-4 M l A l h ( MIAl,’ WIAl:
WIAIB
MIAn:)
(3)
OlAnE
The corresponding expression for the three-state model is y3=-4M1A1.’[ WIAl,’
MmAl: WmAIB
AN@ 1IA, 2IAg 3IAg 4IAg 5IA, 6IA, 7IA, 8 IA, 9IAg
2.295 0.514 3.136 0.912 3.848 1.568 -0.133 0.764 0.303 -0.794 0.731 0.930 1.043 0.250 -0.037 0.204 0.082 0.216 0.509 1.079 0.277 0.415 -0,313 0.087 0.405 2.455 1.278 0.224 0.346 -0.271 1.366 0.790 0.191 0.103 2.808 0.220 0.027 -0.135 0.015 -0.117 0.108 0.170 -0.223 0.106 -0.084
1.305 -0.304 -0.266 0.592 0.154 -0,268 1.506 0.074 -0.041
431-G*
l‘A,
where u, b, and c denote state numbers. We have seen that the first odd state, 1IB,, presents the major feature of the one-photon absorption spectrum of polyenes (see Tables 1-3). In the threestate approximation, only the llA,, lIB,,, and “A, states are considered.12 The “A, state, with the strongest TPA intensity, is found at 1.7 E,. For the four-state approximation, an additional oddstate (nlB,) has been included15 and thestrongest TPA (“A,) state has been claimed to be very close to the llB, state. From the sum-over-states expression of the cubic response function (see ref lo), it is fairly straightforward to derive the four-state model expression for the static hyperpolarizability as
+ MIAnB‘mAnB @lAnE
TABLE 9 Transition Moments between llB, and the Even States for Butadine, Hexatriene, and Octatetraene Molecule (in Atomic Units) even butadine hexatriene octatetraene basis state Mx My Mx My Mx My
MIAl,’] WIAIB
(4)
Here y4and 73 are the hyperpolarizabilities given by the fourstate and three-state models, respectively. Mi/ and q j represent the transition moment and the energy difference between states i andj. The expression for the three-statemodel is straightforward to analyze. When the transition moment between the odd (lB,) state and the excited even state (mA,) is larger than that between the ground state and the odd (lB,) state, the hyperpolarizability is positive, otherwise it is negative. For polyenes it is well-known experimentally that thexxxx component of the hyperpolarizability has large positive values. Thus, for the three-state approximation to be valid, it is required that the magnitude of M m ~ l eshould 2 be much larger than that of Such a requirement should also be sufficient for the four-state model, since the transition moment between the ground state and the nB, state is always very small. To examine the validity of the simplified models discussed above, the calculations for the transition moments between excited states, especially between the lIB, and the even states, have to be carried out. The transition moments between the 11B, and the even states for BD, HT, and OT molecuels are shown in Table 9. Although the results from both basis sets are quantitatively different, they show qualitatively the same features, namely, relatively strong transition moments for the even states around 1.4 E, and 1.6 E,. As a result, both even states show strong two-photon transition intensities as we reported in the previous section. This is most pronounced for HT, but similar features are also found for the BD and O T molecules. However, none of the transition moments between the even states to the lIB, state
1IA, 2.397 0.503 3.202 0.913 3.995 1.292 1.209 0.037 1.162 0.296 0.682 0.221 2IA, 31Ag -1.018 -0.652 0.248 -0.032 -0.535 -0.240 1.560 0.675 2.379 4IA, 1.105 3.275 1.663 5lA, -0.064 0.820 1.056 0.860 0.253 0.075 3.242 1.453 6‘Ag 0.107 0.182 0.562 -0.472 7IAg 0.533 0.363 0.264 0.040 -0.542 -0.346 0.145 -0.214 8IA, 0.049 0.005 1.368 0.925 0.099 0.066 9IA, -0,273 0.225 -0.630 -0.221
The same basis sets as theones used for two-photontransitionelement matrix elements. is larger than the transition moment between the ground state and the lIB, state. It is thus evident that the three-state approximation does not hold for the short polyenes. A similar analysis of the four-state model shows that this approximation is not valid for short polyenes either. Furthermore, the results obtained using a ?r-electron Hamiltonian (Hubbard, extended Hubbard, or PPP) indicate that only one even state (mA,) is important for yxxXx.s9In our all-electron calculations this is not the case. Even though there is always one transition which is dominating, there are several other transitions that contribute considerably to the hyperpolarizability. By taking full account of all possible transitions, we obtain a large positive value for the xxxx component of the hyperpolarizability. Using the 431-G* basis set, we have calculated the xxxx component of the static and dynamic hyperpolarizability for HT. In the static case, yxxxx = 9677 au, and yxxxx= 30584 au for wavelength 694 nm, using the perturbation convention. We thus find from the present RPA calculations that although there is an even state which dominates the TPA spectrum, it cannot be used as a candidate to form three- or four-state approximations for determining the hyperpolarizability. Calculations of hyperpolarizabilities of polyenes including also long-chain members are in progress at our laboratory using direct SCF and direct RPA methods.’
5. Discussion The excellent agreement found in the present work between SCF linear response theory calculations and the experimental data for the excitation energies and polarizabilities has many ramifications. The results provide strong evidence that the electron correlation contributions are small for the ET, BD, HT, and OT molecules. RPA is generally considered appropriate for states described by wave functions with dominating single excitations. It is also often claimed (from a state-specific point of view) that the static (valence) and dynamic (nonvalence) electron correlations in general tend to have opposite effects for hydrocarbons of the kinds investigated here, which may provide one rationalization of the presented results. This is probably also the reason that only the CASPTZ method can give results compatible with RPA since it provides a balance between static and dynamic correlations and is therefore better than, e.g., methods based solely on welectron or valence electron correlation. Actually, it has been found that the state-specific CASSCF
Spectra of Short Conjugated Polyenes calculations even give a reversed order for the IB, states of BD and HT with the valence state above the Rydberg states.’ Naturally, calculations performed with Hamiltonians including ?r-electrons only also fail to account for the dynamic correlation. It is therefore not surprising that the results obtianed using methods like PPP or extended Hubbard are quite different from those reported here. The conclusions referring to the role of correlation obviously rest on how we define this quantity for an excitation spectrum. Response theory calculations cannot be compared directly with state-specific methods in this respect, something which becomes quite obvious from the results presented in the tables of this work. Already RPA includes the correlation contribution of the fluctuation potential to first order, it is a single-particle-hole approximation to excited states which allows for the presence of two-particle and two-hole pairs in the ground state.29~~~ Its multiconfigurational analogueI6 (MCLR) “adds correlation” to the excited states more than that covered by the ground-state correlating space. However, if we define correlation as anything we cannot get out of a single SCF determinant, then, indeed, correlation is not important for excitation spectra of small polyenes. Another obvious conclusion to be drawn is that for larger polyene members with a high density of states in comparison with the number of experimental structures, it is necessary to include reliable property calculations in order either to sustain assignments or to test the type of calculation one is performing. In this respect linear response theory, providing a set of excitation energies and oscillator strengths in one batch of calculations, becomes very powerful. In summary, we have presented the first systematic ab initio study of the two-photon transition moments for the even states of BD, HT, and OT molecules. A TPA band maximum has been located at 1.5-1.7 Eg for all these molecules. The calculations for the transition moments between excited states have shown that previously suggested three- or four-state approximations can not give correct descriptions for the hyperpolarizabilities of the short polyenes. Further investigations along this line for larger polyenes are warranted.
Acknowledgment. This work was supported by CRAY Research Inc. and the Swedish Research Council for Engineering Sciences (TFR). References and Notes (1) Serrano-Andres, L.; Merchan, M.; Nebot-Gil, I.; Lindh, R.; Roos, B. 0. J. Chem. Phys. 1993, 98, 3151. (2) Petrongolo, C.; Buenker, R. J.; Peyerimhoff, S. 0. J. Chem. Phys. 1982, 76, 3655. (3) Malrieu, J. P.; Nebot-Gil, I.; Sanchez-Marin, J. Pure Appl. Chem. 1984,56, 1241. (4) Serrano-Andres,L.; Merchan, M.;Nebot-Gil, I. J . Chem.Phys. 1992, 97, 7499. (5) Serrano-Andres, L.; Lindh, R.; Roos, B. 0.;Merchan, M. J . Phys. Chem. 1993. 97. 9360. (6)-Kkh, H.;Agren, H.; Jsrgensen, P.; Helgaker, T.; Jensen, H. J. Aa. Chem. Phys. 1993, 172, 13. (7) Agren, H.; Vahtras, 0.;Koch, H.; Jsrgensen, P.; Helgaker, T. J. Chem. Phys. 1993, 98,6417. (8) Galasso, V. J. Chem. Phys. 1988,89,4529. (9) Yeager, D. L.; Jsrgensen, P. Chem. Phys. Lett. 1976, 65, 77. (10) Olsen, J.; Jsrgensen, P. J. Chem. Phys. 1985,82, 3235. (11) Luo, Y.; Vahtras, 0.;Agren, H.; Jsrgensen, P. Chem. Phys. Lett. 1993, 204, 587. (12) McWilliams, P. C. M.; Hayden, G. W.; Soos, Z . G . Phys. Rev. E 199.1, 43, 9777. (13) Dixit, S.N.; Guo, D.; Mazumdar, S.Phys. Reu. E 1991, 43, 6781.
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