J. Phys. Chem. 1995,99, 8572-8581
8572
Molecular Orbital Theory of the lL, and lLt, States of Indole. 2. An ab Initio Study Lee S. Slater and Patrik R. Callis" Department of Chemistry and Biochemistry, Montana State University, Bozeman, Montana 5971 7 Received: November 30, 1994; In Final Form: March 14, 1995@
The 'Lb and 'La excited state optimized energies and wave functions of indole are computed at the CIS/321G and CIS-MP2/3-21G levels. The 'Lb geometry is found to be compatible with experimental vibronic structure. The optimized ground state electronic wave function and energy were obtained using the HF and MP2 methods with several basis sets ranging in quality up to MP2/6-31G*. Vertical transition energies, oscillator strengths, and transition dipole vectors were obtained using the STO-3G, 3-21G, 6-31G, 6-3 1+G, 6-31G*, and 6-31+G* basis sets for the CIS and CIS-MP2 treatments of the excited states. Results are generally similar to those found from INDO/S studies, with higher excited configurations essential for approaching the experimental 'La-ILb energy gap. The two energy surfaces exhibit an avoided crossing as the geometry is varied between the 'Lb and 'La minima. The apparent coupling between the diabatic states is quite weak, only about 50 cm-I.
I. Introduction Indole (Figure l), the chromophore of the amino acid tryptophan, is responsible for most of the UV absorption and fluorescence in proteins.' The strong 280 nm absorption band is the result of overlapping mc*transitions to two excited states denoted, using Platt notation, as 'L, and 'Lb. The 'La and 'Lb states have distinctive properties. The 'La transition is moderately intense, having an oscillator strength of approximately 0.12, while the 'Lb transition is comparatively weak, with an oscillator strength of about 0.012.2-4 The transition dipoles are approximately perpendicular, with the 'La transition dipole about -45" from the long axis in indole (see Figure 1 for convention), while that of 'Lb is about +45".3,5-7 The 'L, and 'Lb states have differential solvent interactions resulting in a highly environment-dependent transition energy order. This effect is thought to be due primarily to the large difference in the 'La and 'Lb excited state dipole moments. In vapor phase ILb is the lower, and therefore the fluorescing, state, whereas in polar solvent the 'La state is usually slightly lower. In addition, the dipole quickly reorients the local solvent environment after excitation, causing a greater red-shift (stabilization) than the smaller 'Lb dipole, resulting in an 'La fluorescencetransition energy less than the 'Lb transition energy by about 4000 ~ m - ' . ~This , ~ has been exploited in tryptophan as a sensitive probe of protein environment and sol~ation.~ Methyl substitution in indole can also result in transition energy shifts and reordering.I0 Interpreting indole spectra is complicated by intermingling of the 'L, and ILL, vibronic transitions. 'La transitions have been observed using polarization-resolved two-photon fluorescence excitation spectroscopy on jet-cooled ( - 2 K) indole, 3-methylindole, and 5-methylindole.",'* It does not, therefore, seem likely that the 'L, is predissociative, as has been ~uggested.'~ The precise location of the 'L, origin has been an interesting question. The lowest two peaks in the jet fluorescence excitation spectrum having 'La optical character were thought initially to be the 'L, origin, split by coupling with 'Lb states. However, more recent evidence from our laboratoryI4 strongly suggests that these peaks are a Herzberg-Teller false origin, split by a Fermi r e s ~ n a n c e . ' ~ An , ' ~ argon matrix study places the true @
Abstract published in Advance ACS Abstracts, M a y 1, 1995.
Figure 1. Structure and numbering system for indole, including the convention for defining the direction of the dipoles. The arrow points in the positive direction for permanent electric dipole vectors.
'La origin near 1400 cm-' above the 'Lb origin.'5 In the case of jet-cooled 3-methylindole,many more 'L, lines are observed a few hundred cm-' above the ILb origin, which, however, gains significant 'L, character.I2 Interpretation of this spectrum has not been possible with the same simplicity as was possible for indole.I6 This could well be because of avoided crossing complications. To aid in understandingthese spectral complexities and other interesting questions involving indole-solvent complex formation, the electronic structure and potential energy surfaces of indole need to be determined accurately. The initial electronic structure calculations of the 'La and 'Lb excited states have been performed using semiempirical method^.^,^,]^,'^^'^ With the advent of new theoretical methods and advances in algorithm and hardware development, systematic ab initio excited state molecular orbital procedures are emerging with applications feasible for "large" molecule^.'^^^^ An exploratory ab initio study on indole has recently been reported by Chabalowski et a1.,2' which used first-order configuration interaction in the context of a complete active space self-consistent-field (CASSCF) reference and also the configuration interaction with single replacements only (CIS) approach. Their study was somewhat successful in reproducing some of the observed general properties of the 'L, and 'Lb states. Much of the emphasis of their study was to determine the effect of a dipole reaction field on the transition energies and properties of the 'La and 'Lb excited states of indole. A point of interest was the discarding of the CIS second root in favor of the CASSCF wave function as the preferred 'Lb wave function and optimized geometry. This is of interest because the geometry seems extremely unusual for
0022-3654/95/2099-8572$09.0010 0 1995 American Chemical Society
MO Theory of the 'Laand ILb States of Indole
J. Phys. Chem., Vol. 99,No. 21, 1995 8573
TABLE 1: Ground State Optimized Internuclear Distances (in Angstroms) ~~~
Nl-C2 C2-C3 c3-c9 C8-C9 c4-c9 c4-c5 C5-C6 C6-C7 C7-C8 Nl-C8 a
HF/ STO-3G
HF/ 3-21G
HF/ 6-31G
HF/ 6-31G*
MP21 6-31G*
exDtla
1.400 1.342 1.450 1.401 1.406 1.371 1.410 1.370 1.403 1.396
1.388 1.349 1.446 1.404 1.394 1.375 1.401 1.374 1.390 1.377
1.383 1.353 1.444 1.405 1.398 1.379 1.404 1.379 1.393 1.379
1.373 1.348 1.442 1.401 1.399 1.379 1.404 1.375 1.394 1.37 1
1.380 1.376 1.430 1.423 1.408 1.388 1.413 1.389 1.401 1.377
1.377 1.344 1.451 1.380 1.412 1.397 1.386 1.391 1.400 1.391
Crystal structure (ref 23).
the 'Lbstate of an aromatic molecule and is very different from that predicted by semiempirical methods.17~'* In this paper, we address not only the 'Lb geometry issue but also the important question of the effect of higher electron correlation as implemented by an "MP2" correction to the CIS energyI9 because it is known from semiempirical studies that the 'Lb state energy is much more sensitive to doubly excited configurations than is Ab initio studies on other aromatics show the same pattem.I9 The effect of improved basis sets upon vertical transition properties is examined, and the extent of interaction of the 'Lband 'La energy surfaces is explored along the coordinate connecting the 'Lband 'Laexcited state energy minima. Critical comparison with previous semiempirical results from Part 1'' is made to help establish a unified theoretical view of the 'Laand 'Lb excited states of indole. 11. Computational Methods
The calculations were done using GAUSSIAN 9222on a Cray Y-MP C90 computer. The CIS and CIS with second-order Moller-Plesset perturbation correction (CIS-MP2) calculations included all electrons. The ground state MP2 optimizations used the frozen-core approximation. Standard basis sets were used without modification. The Hartree-Fock (SCF), CIS generalized density matrix (GDM), MP2 GDM, and one-particle density matrix (1PDM) were kept in terms of the canonical HF MOs to calculate the total electron density planes. The MO and density xy planes parallel to the molecular plane were calculated at 0.2 8, intervals extending for 5 A in the k x and &y directions from the center of charge. These planes were plotted as contour illustrations. 111. Ground State
Optimized Geometries. Hartree-Fock (HF)-optimized ground state geometries were calculated for indole using the following series of basis sets: STO-3G, 3-21G, 6-31G, and 6-31G*. The effect of MP2 correlation was assessed using the 6-31G* basis. The indole starting structure for all optimizations was adapted from a tryptophan crystal structure23 for which the ring was made planar and all hydrogen bond lengths were taken as 1.08 8,. The constraint of C, symmetry was maintained throughout all optimizations, partly because of restricted computing time and partly because experimental evidence indicates planarity in the ground and 'Lb states. However, normal vibration calculations on the ground state with each of the above basis sets produced no negative eigenvalues, demonstrating that the minimum energy geometry is planar. The optimized intemuclear distances and crystal starting structure are listed in Table 1. Bond length alternation in the calculated structures is observed in the benzene ring bonds not common to the pyrrole ring. In many bond lengths the
TABLE 2: Ground State Dinole Moments geometry//density
Pa
Ob
HFISTO-3GIISCF HF/3-2 lG//SCF HF/6-3lG//SCF HF/6-3lG*//SCF MP2/6-3 lG*//SCF MP2/6-3 1G*//MP2 exptl
1.80 2.02 1.93 2.05 2.12 2.21 2.13c
-48.8 -45.3 -45.9 -45.5 -45.2 -47.6
a Dipole moment magnitude in debyes. Dipole moment direction in degrees. See Figure 1 for dipole orientation. Reference 25.
calculated geometries differ significantly from the crystal structure. The HF bond lengths are relatively consistent, but the HF/STO-3G geometry is in most discord with the HF/split valence basis geometries in the bonds involving the nitrogen (Nl-C2 and Nl-C8). This is probably due to the enhanced ability of the split valence basis to produce carbon orbitals in the anisotropic environment which overlap the contracted nitrogen orbitals, thereby better producing shorter bond lengths. The effect of polarization functions is also most pronounced for these bonds, enabling a shift of electron density into the carbon-nitrogen intemuclear region, resulting in shorter bond lengths. The polarization functions have little effect on the other bonds. Increasing the number of Gaussian primitives in going from the 3-21G to the 6-31G basis has only a small effect on the HF geometries. Compared to the HF/6-3 lG* structure, the MP2 correlation tends to lengthen all bonds by about 0.01 8, except for the C3C9 and Nl-C8 bonds, which are involved in the linkage of the enamine moiety to the benzene ring. The C3-C9 bond length is contracted by about 0.01 A, and the Nl-C8 bond is invariant. The C2-C3 and C8-C9 MP2/6-31G* bond lengths elongate significantly from both the HF and crystal values. This is interesting in light of the observation that MP2/6-3 lG* structures of many other molecules are found to closely agree with their experimental structure. This is thought to be due to a cancellation of errors induced by anharmonicity and MP2 bond length extension.24 Interestingly, it is the C2-C3 and C8-C9 bonds which are in nearly perfect agreement with those of the SCF ground state structure of ref 21, calculated using a double93%). This is probably due to the restricted CIS space in the INDO/S calculations (196 singly excited configurations), whereas complete CIS (within the 3-21G basis) is performed in this study (2816 SEC). With diffuse functions in the basis, CSFs involving extremely diffuse virtual n MOs became more important in the 'La and 'Lb wave functions. Excited State Energies. The CIS/3-21G- and CIS-MP2/321G-optimized 'La and 'Lb state energies are shown in Table 4. At the CIS/3-21G level of theory, the 'La state is lower in energy than the ILb by about 0.0043 au (940 cm-I). This contradicts the experimental observation that the vapor phase 'La 0-0 transition has higher energy than the 'Lb 0-0 transition.''.'2.'4.'5 With the CIS-MP2 method, the state energy order is correct, with the 'Lb state lower than the 'La state by about 0.0251 au (5510 cm-I). While vibronic peaks having 'La character are known to start 455 cm-' to the blue of the ILL, origin,''.l2 it is most likely that the true origin lies about
MO Theory of the 'Laand 'Lb States of Indole
J. Phys. Chem., Vol. 99, No. 21, 1995 8575
[to. 0 2 6
( I.436)
(1.382) El.3621
(-0.03 I ) [-0.0511
I1 . 4 3 6 1
II .3701
1
(-0.035)
(to. 0 1 3 ) LtO.0 131
[+0.0721
(1.430) (1.359) I .3571
(-0.002)
[+O.O I 2 1 (1.4 II ) I .4251
r
r I .4
I91
Figure 4. 'La (a) and 'Lb (b) optimized geometry differences (excited state-ground state) using the MP2/6-3 1G* ground state geometry (in
Figure 3. 'L, (a) and 'Lb (b) optimized internuclear distances (in A) (CIS/3-2 1G) [CIS-MP2/3-2lG].
A) (CIS/3-21G) [CIS-MP2/3-21G].
1400 cm-' above the 'Lb rigi in.'^*'^ Thus, the CIS values are in error by about 2300 cm-' too low, and the CIS-MP2 values are about 4100 cm-' too high. It is evident that the ILb energy has a larger dependence on higher order correlation than the IL, state. We interject the observation that "correctness" of the 'Laand 'Lbstate ordering is not a particularly meaningful issue at the 3-21G level. The separation is only &0.01 au, whereas we demonstrate in section V that single-point computations using better basis sets lower the ground and excited states by more than 2 au. The estimated contributions of the doubly and triply excited configurations to the CIS-MP2 correction, also shown in Table 4, provide interesting insight into the character of the 'Laand 'Lb excited states. Both excited state energies have greater dependence on triples than doubles, with the 'La state being stabilized more by 0.01 au. It is the larger contribution of the doubles to the ILb state energy that produces the correct state energy order. This pattern is seen in the CIS and CIS-MP2 'La and 'Lbenergies of pyridineI9and is also noted in semiempirical computations. Optimized Geometries. The 'Laand 'Lb optimized excited state geometries of indole are calculated at the CIS/3-21G and CIS-MP2/3-21G levels. Again, the C, symmetry constraint was applied. For the 'La state, this constraint was relaxed after the planar minimum was located. Starting with the N pyramidal still resulted in a planar minimum. The optimized internuclear distances, provided in Figure 3, are substantially different from the ground state MP2/6-31G* geometry. Our optimized CIS 'Lageometry agrees perfectly with the CIS structure obtained by Chabalowski et al.21 However, they did not report a CIS 'Lb optimized geometry, but instead gave a 'Lb geometry obtained from a CASSCFI3-21G wave function having a fourelectron, four-n-orbital active space. Their CASSCF structure
deviates markedly from our CIS and CIS-MP2 structures and appears very inconsistent with spectroscopic data on the ILb Whereas the CIS structure is seen from Figure 4 to predict changes which tend to alternate between increase and decrease about the perimeter, with the largest changes being increases of 0.04 A at the 1-2 bond and the bridging bond, the MCSCF structure predicts very large increases (0.08, 0.04, and 0.08 A) in three adjacent bonds from C4 to C7, while all bonds in the pyrrole ring decrease by about 0.02 A. We have estimated the Franck-Condon factor pattern anticipated from the latter structure and found it to bear no resemblance to the experimentally observed spectrum.I6 In contrast, the CIS and CISMP2 geometry differences used in simulated spectra reproduce critical features of the experimental 'Laand 'Lb vibronic spectra,26as outlined below. The simulated spectra were produced with the FranckCondon factor program used in ref 18 for computing spectra from the geometries and normal modes computed by the semiempirical program QCFFPI. In this case we used the difference between the 'Lb CIS/3-21G geometry and the HF/ 3-21G geometry combined with normal coordinates from the W 6 - 3 lG* results, for which the unscaled frequencies matched experiment within 5%. This combination provided excellent agreement with the experimental fluorescence spectrum for indole in jet, a stringent test because the low symmetry allows all 29 of the in-plane vibrations to be Franck-Condon active. Experimentally, in-plane vibrations 11-29 are observed,I6with the exception of 25, 19, and 18. These three are correctly predicted to be exceptionally weak. The rest exhibit a wide range of intensities which are qualitatively well matched by the calculation, except that 15 and 16 are reversed. An important point is that the origin is the most intense line by about a factor of 2, causing the overall width of the band to be narrow, a result also found in the calculation. This is a direct consequence of rather small changes in bond length accompanying the transition.
Slater and Callis
8576 J. Phys. Chem., Vol. 99, No. 21, 1995
TABLE 6: CIS//MP2/6-31G* Energies
TABLE 5: Excited State Dipole Moments state
geometrylldensity
Pa
Ob
'La
CW3-2 1GI11PDM CISl3-21GllCIS exptl CIS/3-2 1Gl/lPDM CISl3-2 1GIICIS exptl
3.42 3.22 5.4' 2.42 2.15 2.3d
-20.1 -27.3
'Lb
-44.0 -41.3
Dipole moment direction a Dipole moment magnitude in debyes. in degrees. See Figure 1 for dipole orientation. Reference 29. Reference 28.
As noted in Table 4, changes of 0.02 8, are typical, with the maximum being 0.044 A. In contrast, the 'Lb geometry found in ref 21 has two bonds in the benzene ring increasing by 0.08 A, while most other bonds decrease by 0.02 A. The size and pattern of the changes have the consequence that the origin line is weaker than several other lines, which leads to a much larger width (factor of 2 or 3) and extremely poor fit with experiment. The most telling comparison is the absolute Franck-Condon factor for the origin. It is about 0.19 experimentally and calculated to be 0.22 using the CIS 'Lb geometry. The MCSCF geometry gives 0.015, Le., an order of magnitude lower. The optimized geometry differences from the ground state primarily reflect changes in nearest neighbor off-diagonal density matrix elements incurred upon excitation, vide infra. The CIS and CIS-MP2 excited state geometry differences from the ground state MP2/6-31G* reference structure are provided in Figure 4. A striking feature is the alternation of bond length contractions and expansions around the ring perimeter starting from (24 in 'La and c 7 in ILb. In the CIS 'La geometry difference bond contractions occur in the C5-C6, C7-C8, C3C9, and N1 -C2 bonds, while in the CIS ILb geometry difference bond contractions occur, to a lesser degree, in the C5-C6, C4C9, C8-N1, and C2-C3 bonds. The effect of the CIS-MP2 correlation is different for the IL, and 'Lb geometries and shows the very different character of these two states. The CIS-MP2 correction to the CIS 'La structure involves bond length contractions and elongations of large magnitude with many changes greater than 0.02 A, while the correction to the CIS 'Lb structure involves bond length elongations less than 0.01 8,. The origin intensity in simulated spectra suggests that the CIS-MP2 method overcorrects the 'La CIS-optimized geometries while undercorrecting the ILb CIS geometry.'6 Excited State Dipole Moments. The CIS/3-21G 'La and ILb permanent dipole moments calculated with the lPDM and CIS GDM'9.27are shown in Table 5. It seems that the dipole magnitude for the 'La state is grossly underestimated compared to experiment, while the 'Lb magnitude is in close agreement. However, the reported experimental 'Lb-ground dipole difference magnitude was measured in vapor phase,28while the 'L,-ground dipole difference magnitude was a benzene solution measureme~~tA . * ~1.1 D difference between the vapor and benzene solution indole I L b dipole m a g n i t ~ d e salong , ~ ~ with the larger polarizability of the 'La state (transitions with larger oscillator strengths are more p ~ l a r i z a b l e and ~ ~ )benzene solvent, suggests that the calculated 'La magnitude may not be far from the unknown vapor phase experimental value. The effect of a polar solvent model (water) has been calculated to increase the 'La dipole magnitude substantially in explicit solvent molecular dynamics simulations.* The standard INDOIS-CIS 'La and I L b state dipoles are overestimated compared to the ab initio and experimental magnitudes but match the solvent measurements
]La
'Lb
basis
transition (IO3cm-I)
state (au)
transition (lo3cm-I)
state (au)
STO-3G 3-21G 6-3 1G 6-31+G 6-3 1G* 6-31+G* exptl"
58.2 48.6 47.7 45.8 46.5 44.6 38.9
-356.764 -359.230 -361.117 -361.137 -361.253 -361.273
57.1 49.0 48.3 46.9 47.2 45.1 35.2
-356.169 -359.228 -361.115 -36 1.132 -361.250 -361.268
Reference 9.
TABLE 7: CIS-MP2//MP2/6-31G* Energies 'L.3
'Lb
basis
transition (lo3cm-I)
state (au)
transition (lo3cm-I)
state (au)
STO-3G 3-21G 6-31G 6-3 1+G 6-31G* 6-3 1+G* exptl"
69.7 61.4 61.4 58.3 63.2 60.3 38.9
-357.229 -359.970 -361.851 -361.890 -362.354 -362.390
55.3 54.3 54.1 52.7 51.4 56.1 35.2
-357.295 -360.002 -361.885 -361.916 -362.380 -362.409
a
Reference 9.
more c10sely.l~ Some parametrizations produce good agreement with the ab initio magnitudes.17 The difference between the CIS GDM and lPDM dipole angles is slightly larger for the 'La state compared to the ILb state. A similar pattern is observed in the semiempirical calculations between the INDO/S-CIS and CISD dipole angles," suggesting that the 'La one-electron properties may have a larger dependence on correlated electron distributions. The 'La dipole directions agree slightly better between the ab initio and semiempirical results than the 'Lb dipole directions do. V. Vertical Transitions State and Transition Energies. CIS and CIS-MP2 vertical excitation energies are calculated at the MP2/6-3 1G*-optimized ground state geometry using a common progression of basis sets: STO-3G, 3-21G, 6-31G, 6-31+G, 6-31G*, 6-31+G*. All CIS and CIS-MP2 ground and excited state energies decrease with this basis set progression. The CIS excited state and transition energies are shown in Table 6, and those for CISMP2 are in Table 7 . The largest incremental effects are going from STO-3G to 3-21G (2.5 au) and from 3-21G to 6-31G (1.9 au). Polarization functions stabilized the 6-31G energies by 0.14-0.50 au and diffuse functions by 0.02-0.04 au. The MP2 correction has increasing impact on the state energies as the basis is improved, increasing from about 0.5 to 1.1 au from the worst to the best basis. It also consistently stabilizes the 'Lb state slightly more than the 'La state by about 0.03 au (6500 cm-I). As noted earlier in the paper, it is the greater sensitivity of the ILb state energy to doubly excited configurations which is at the root of this ubiquitous effect evident in semiempirical and ab initio calculations on the 'Lb and 'La states of many aromatic molecules.'7,'9 Underlying the large state energy changes are subtle differences between the stabilization of the ground and excited states. The general trend is a gradual lowering of the transition energies as the basis set quality increases because the excited states are lowered slightly more than the ground state. However, even with the best basis, the transition energies are several thousand cm-' (0.5- 1 eV) from converging to the experimental transition
J. Phys. Chem., Vol. 99, No. 21, 1995 8577
MO Theory of the 'La and 'Lb States of Indole TABLE 8: CIS//MP2/6-31G* Transition Properties
'L, basis STO-3G 3-21G 6-31G 6-31+G 6-3 lG* 6-31+G* exptlc.d
0" -37 -30 -30 -30 -27 -26 -46
ILb
f 0.210 0.198 0.200 0.223 0.180 0.207 0.12
0 34 57 59 48 56 48 42
f 0.056 0.05 1 0.047 0.050 0.044 0.047 0.012
a 0 : transition dipole direction in degrees. See Figure 1 for dipole orientation. bf: oscillator strength. Experimental transition dipole directions, refs 3, 5-7. Oscillator strengths, refs 2-4.
energies. The CIS-MP2 result is worse from this viewpoint because the MP2 correction stabilizes the ground state more than the excited states. On a more optimistic note, we observe that the 'La-ILb difference appears to have converged to near the observed vertical energy difference. Because this gap is on the order of the error in the transition energies, it is reasonable that this is a meaningful result. At the finest level of detail, we note that polarization functions actually increase the transition energies in the presence of MP2. Addition of diffuse functions in the basis set caused a Rydberg state to drop between the 'La and 'Lb states, intermediate in CIS energy. With higher electron correlation, the 'Lb state should drop below the 'La state, making this Rydberg state the third or higher excited state depending on the behavior of the B states. Transition Properties. The CIS transition dipole directions and oscillator strengths calculated at the MP2/6-3 lG* ground state geometry using the CIS GDM are provided in Table 8. The calculated oscillator strengths are overestimated compared to experimental a result also common to semiempirical CIS calculations. The value of about 0.2 found for 'La is very close to that found from INDO/S-CIS,I7 while that for 'Lb is several times larger. This is probably because of the nearness of the 'La state in the CIS calculations reported here. For the 'La state the transition dipole angle tends to smaller angles from the long axis as the basis improves and appears to converge to about -25" at the CIS level. The deviation from experiment by -20" is again similar to what is found in semiempirical INDOIS-CIS calculation^.'^ Given the lack of symmetry and that the CIS transition moment length is considerably overestimated, some error in the direction could be expected. For perspective it should be noted that the 'La transition dipole directions are from condensed phase experiments, and older reported values range from -33 to -54°.3s5 Recent condensed phase experiments report a value of -46" at band maximum.' It should be noted that in the INDOIS-CISD calculations the 'La transition dipole angle is in very good agreement with the recent measurement, suggesting the possible importance of doubles in the 'La transition properties. The ILb transition dipole direction, while varying somewhat with basis, is in good agreement with experiment, where excellent agreement exists between the condensed phase LD measurement in stretched polyethylene5 and the supersonic jet, vapor phase measurement.6 Electron Density Differences. Electron density difference plots for the 'La and ILb states calculated at the MP2/6-31G* ground state geometry are provided in Figures 5 and 6. The total electron densities were calculated at 0.6 %, above the molecular plane using the CIS GDM/6-3 1+G* density/basis in the excited state and subtracting the MP2 GDM/6-31G* density/ basis for the ground state. The density differences are consistent
--7-
I
Figure 5. 'La(CIS GDW6-3 l+G*)-ground state (MP2 GDM/6-31G*) total electron density difference at 0.6 A above molecular plane. Solid contours represent positive difference; dashed contours represent negative difference. The contour separation is 0.001 A3,and the range is [-0.01,0.01] A3.
Figure 6. ILb (CIS GDW6-31+G*) - ground state (MP2 GDM/631G*) total electron density difference at 0.6 A above molecular plane. Sign convention and contour levels are as specified in Figure 5.
with the CIS/3-21G geometry differences discussed above. In both the 'La and 'Lb plots, the outer contour is positive, indicating that the excited state electron density is greater than the ground state density in regions distant from the molecule. This is due to excitation of electrons in compact, occupied MOs into more diffuse virtual MOs. This is also the reason for the conspicuous high degree of density loss at each of the carbon centers, a consequence of the somewhat arbitrary selection of the 0.6 %, contour, which emphasizes loss of density from the innermost of the split valence AOs. As noted earlier, the lowest virtual MOs are weighted much more strongly in the outer AO. The nitrogen appears very different because its AOs are smaller. Density differences using the lPDM/6-31+G* as the excited state total electron density and the same ground state density and geometry (not shown) have structure at the 0.6 8, level nearly identical to those in Figures 5 and 6. Comparison with corresponding semiempirical x density matrix difference plots is awkward because of the split valence basis related effects just noted. However, some consistencies can be noted. For the 'La transition, the large loss of n bond
8578 J. Phys. Chem., Vol. 99, No. 21, 1995
Slater and Callis CIS Difference Vector Fraction
w -0.4
-0.2
I
I
I
I
I
I
I
0.2
0.4
0.6
0.8
1
1.2
&4
-359.215
3
I-
-359.22
Y u -359.23
-359.235
-359.24
character in the C2-C3, C4-C9, and C6-C7 bonds and the gain in the C3-C9 and Nl-C2 bonds of Figure 5 are reflected by parallel effects in the INDO/S x density matrix difference p10ts.I~ The INDO/S results indicate major density gains at C4, C7, and C9. This is manifest in Figure 5 by smaller losses at C4, C7, and C9 compared to other carbons in Figure 5. For 'Lb, alternation of the strength of the loss of x bond order is in qualitative accord with the density difference map of Figure 6.
VI. Adiabatic Potential Energy Surfaces To obtain some measure of the coupling between the two excited state adiabatic surfaces, i.e., the extent to which the surfaces avoid each other in the 'L,-'Lb singlet manifold, a section (plane) intersecting the 29-dimensional (in-plane vibrational motion) adiabatic potential energy surface was examined. After the determination of the CIS and CIS-MP2 'La and 'Lb optimized geometries, a 29-D vector of internal coordinates (from the Z matrix) connecting the corresponding 'La and ILb minima ('Lb-'La difference vector) was constructed and extended on both sides of the minima. Each component of the 'Lb-'La difference vector represents a difference in a Z-matrix intemal coordinate corresponding to a degree of freedom in indole constrained to C, symmetry. The CIS/3-21G or CIS-
MP2/3-21G total state energy was computed at each nuclear configuration incremented along the corresponding extended 'Lb-'La difference vector. Using a linear geometry increment, uniform in each internal coordinate, energies were calculated at 19 geometry points (including the optimized geometries). The CIS and CIS-MP2 state energies are plotted as a function of the fraction of the corresponding extended difference vector. The 0.0 and 1.0 fractions correspond to the 'Lb and 'La optimized geometries, respectively. CIS Potential Energy Surface Sections. The CIS/3-21G surface section is shown in Figure 7. The lower energies form a smooth curve with two minima, with energies corresponding to optimized 'Lb and 'La energies. Near these minima the structure of the surface is nearly harmonic. The higher energies also form a smooth curve with a single minimum just above the maximum in the lower curve, the signature of an avoided crossing. That the curves do not come arbitrarily close, Le., that there is indeed an avoided crossing, is further demonstrated by a plot of the electronic transition dipole angles in Figure 8. Figure 8 shows that the transition dipole angles associated with the first and second excited states change character well before the projected crossing and that the states near the crossing region
MO Theory of the 'La and 'Lb States of Indole
J. Phys. Chem., Vol. 99, No. 21, 1995 8579
CIS Difference Vector Fraction
root2
-140 Figure 8. Indole CIS/3-21G transition dipole angle as a function of nuclear configuration along the CIS IL,-'Lb difference vector.
are well mixed. For increasing ILb-ILa difference vector fraction, this corresponds to a change in a wave function having more ILb character to a wave function having more 'La character in the first excited state (root 1) and from more 'La character to more ILb character in the second excited state (root 2). The point of equal transition dipole length corresponds to equal mixing of the 'La and 'Lb states expected at a point in the 2-D avoided crossing region. Here the lower transition dipole angle will have swung toward that of the higher, while the higher root's direction will have swung toward that of minus the lower. The angle between them at the point of equal mixing will be 2 tan-'(A/B), where A and B are the lengths of the dipoles of the unmixed states (assuming they are perpendicular). In the limiting case of an infinitesimal avoided crossing, the transition dipole directions would change discontinuously. The slight increase in the 'Lb transition dipole angle just before the avoided crossing (0.1 difference vector fraction of root 1) has been noted. Further refinement of the transition dipole angle plot in this region is warranted. From the nearest approach of the two curves, about 100 cm-I, one deduces that the coupling between the diabatic states is about 50 cm-I. This is about half that found from INDO/S calculations on 3-methylind01e.I~ In any event, the smallness
of the coupling suggests that a diabatic limit for 'La and 'Lb surfaces is nearly obtained. This predicts that mixing of the state properties will not be extensive, a fact that has been made evident in spectroscopic studies, which find that most 'Lb and 'La vibronic states have distinctive properties which are invariant across the absorption band.2.3.5.7.9.1 CIS-MP2 Potential Energy Surface Sections. The CISMP2 surface section given in Figure 9 has a much different structure than the CIS section. The presence of higher correlation produces the correct ordering of the potential minima in the CIS-MP2 'La-ILb manifold as a result of the ILb surface being lowered more than the 'La surface. A slight bump is evident at the 0.2 fraction of the difference vector. This is due to the avoided crossing in the CIS surface since the CIS-MP2optimized geometries are similar to the CIS-optimized geometries and the CIS-MP2 state energy functional is a correction to the CIS energy. In monitoring the CIS energy during the CIS-MP2 surface calculation a CIS avoided crossing is also observed along the CIS-MP2 difference vector. We believe that the CIS-MP2 surfaces actually cross near the 1.2 fraction of the difference vector. The lack of an avoided crossing results because the CIS-MP2 energy functional is constructed from the addition of perturbation correction terms
Slater and Callis
8580 J. Phys. Chem., Vol. 99, No. 21, 1995 CIS-MP2 Difference Vector Fraction
w -0.4
-0.2
I
I
I I
I I
I
I I
I I
0.2
0.4
0.6
0.8
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\\ -359.95
-359.
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-359.99 -360
-360.02 Figure 9. Indole CIS-MP2/3-21G 'L,-'Lb energy surface section. to the CIS energies in an approximate manner analogous to ground state MP2 perturbation theory. The problem lies in that the CIS-MP2 correction is a single-state correction, applied to only one excited state at a time, which does not allow coupling between molecular electronic states. Such corrections do not sense the degeneracy in the way that a variational calculation or multistate perturbative treatment would, since these methods incorporate electron correlation into the solution of many states simultaneously. VII. Conclusions and Future Directions This study has examined the ab initio electronic structure, transition energies and properties, and partial potential energy surfaces of the 'La and 'Lb excited states of indole at the CIS and CIS-MP2 relative to the HF/3-21G reference. The 'Lb geometry differs radically from the CASSCF geometry favored by Chabalowsky et al.,*I and we believe the CIS and CIS-MP2 geometries to be much more in accord with spectroscopic measurements. Reasonable agreement with experimental results is found for calculated properties, but large errors in the transition energies remain, although the vertical 'La- ILb energy gap is well represented using the better basis sets. A weakly avoided crossing is observed in the CIS 'La-ILb singlet
manifold. In addition, good agreement with previous INDO/S results leads to improved confidence in the theoretical understanding of indole. The large effect of the CIS-MP2 correction, especially on the 'Lb state energy suggests that it will be important to evaluate transition properties of the more highly correlated wave functions. Such studies are underway, as well as analysis of vibrational properties obtained with the energy surfaces computed in this work. Acknowledgment. This work was supported by NIH Grant GM3 1824 and by Grant CHE92005P of computer time from Pittsburg Supercomputing Center. References and Notes (1) Demchenko, A. P. Ultraviolet Spectroscopy of Proteins; SpringerVerlag: Berlin, 1986. (2) StricMand, E. H.; Billups, C. Biopolymers 1973, 12, 1989. (3) Yamamoto, Y . ;Tanaka, J. Bull. Chem. SOC.Jpn. 1972, 45, 1362. (4) Lami, H. J . Chem. Phys. 1977, 67, 3274. ( 5 ) Albinsson, B.; Dubista, M.; Norden, B.; Thulstrup, E. J . Phys. Chem. 1989, 93, 6646. (6) Phillips, L.; Levy, D. H. J . Chem. Phys. 1986, 85, 1327. (7) Albinsonn, B.; Norden, B. J . Phys. Chem. 1992, 96, 6204. (8) Muifio, P.; Callis, P. R. J . Chem. Phys. 1994, ZOO, 4093.
MO Theory of the 'La and 'Lb States of Indole (9) Strickland, E. H.; Honvitz, J.; Billups, C. Biochemistry 1970, 9, 4914. (10) Rehms, A. A,; Callis, P. R. Chem. Phys. Lett. 1987, 140, 83. (11) Sammeth, D. M.; Yan, S.: Spangler, L.; Callis, P. R. J. Phys. Chem. 1990, 94, 7340. (12) Sammeth. D. M.: Siewert. S.: Suangler, L.: Callis. P. R. Chem. Phys. Lett. 1992, 193, 532. (13) Glasser, N.; Lami, H. J . Chem. Phys. 1981, 74, 6526. (14) Muifio, P. M.; Callis, P. R. Proc. SPZE 1994, 2137, 278. (15) Fender, B.; Callis, P. R. Chem. Phys. Lett., in press. (16) Bickel, G. A,; Demmer, D. R.; Outhouse, E. A.; Wallace, S . C. J . Chem. Phys. 1989, 91, 6013. (17) Callis, P. R. J . Chem. Phys. 1991, 95, 4230. (18) Vivian, J. T.; Callis, P. R. Chem. Phys. Lett. 1994, 229, 153. (19) Foresman, J. B.; Head-Gordon, M.; Pople, J. A,; Frisch, M. J. J . Phys. Chem. 1992, 96, 135. (20) Fiilscher, M. P.; Andersson, K.; Roos, B. 0. J . Phys. Chem. 1992, 96, 9204. (21) Chabalowski. C. F.: Garmer. D. R.: Jensen. J. 0.: Gauss, M. J . Phys. Chem. 1993, 97, 4609. (22) Fnsch, M. J.: Trucks, G. W.; Head-Gordon, M.; Gill, P. M. W.; Wong, M. W.; Foresman, J. B.; Johnson, B. G.; Schlegel, H. B.; Robb, M. ~I
J. Phys. Chem., Vol. 99, No. 21, 1995 8581 A,; Replogle, E. S.; Gomperts, R.; Andres, J. L.; Raghavachari, K.; Binkley, J. S.; Gonzalez, C.; Martin, R. L.; Fox, D. J.; Defrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople, J. A. Gaussian 92, Revision A; Gaussian, Inc.: Pittsburgh, PA, 1992. (23) Takigawa, T.; Ashida, T.; Sasada, Y.; Kakudo, M. Bull. Chem. SOC. Jpn. 1966, 39, 2369. (24) 'Wiberg, K. B.; Hadad, C. M.; LePage, T. J.; Breneman, C. M.; Frisch, M. J. J . Phys. Chem. 1992, 96, 671. (25) McClellan, A. L. Tables of Experimental Dipole Moments; Freeman: London, 1963. (26) Slater, L. S.; Vivian, J. T.; Callis, P. R. Manuscript in preparation. (27) Wiberg, K. B.; Hadad, C. M.; Foresman, J. B.; Chupka, W. A. J . Phys. Chem. 1992, 96, 10756. (28) Chang, C.-T.; Wu, C.-Y.; Muirhead, A. R.; Lombardi, J. R. Photochem. Photobiol. 1974, 19, 347. (29) Lami, H.; Glasser, N. J . Chem. Phys. 1986, 84, 597. (30) Atkins, P. W. Molecular Quantum Mechanics, 2nd ed.; Oxford University Press: Oxford, 1983. JP943 1696
