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Ab initio prediction of the spectra of carbon cumulenes
Chemical Physics ELSEVIER
Chemical Physics 223 (1997) 149-158
Ab initio prediction of the spectra of carbon cumulenes Gad Fischer a,*, John P. Maier b a Department of Chemistry Australian National UniL,ersio, Canberra, ACT 0200, Australia b Institute for Physical Chemisto', University of Basel, Klingelbergstrasse 80, CH-4056 Basel, Switzerland Received 29 May 1997
Abstract
Ground state calculations of the optimized geometries and vibrational frequencies of the carbon cumulenes, H 2 C n (n = 3-15), have been variously carried out at HF/6-31G(d), MP2/6-31G(d), MP2/6-31G(d,p) and CASSCF/6-31G(d) levels, both with and without the constraint of linearity of the carbon chains. Substantial reductions in the A rotational constant are obtained when the constraint to linearity is removed, in agreement with the trend observed for the short cumulenes for which experimental results are available. Excited state calculations at the CIS/6-31G(d) level have focussed mainly on the intense JAt excited state (the third ~Aj for n > 5). Also, in these calculations the tendency for non-linearity of the carbon chain has been noted, although energy differences are smaller. CASSCF calculations have been restricted to the two smallest cumulenes. They are in good agreement with the CIS calculations for the first excited IA~ and IA 2 states, but differ for the higher ~Ai states. Empirical corrections to the CIS-calculated transition energies for the intense state have been introduced through comparisons with the linear carbon chain molecules, C,, for which both observed results and equivalent accuracy ab initio computations are available. Cumulenes with sizes H2C 13 to about H 2C 34 have their first strong electronic transition in the 400-860 nm DIB range. © 1997 Elsevier Science B.V.
I. I n t r o d u c t i o n
Cumulene carbene molecules have been shown by radioastronomy to be constituents in interstellar gas and circumstellar shells. In particular, the small cyclic c a r b e n e , C 3 H 2 , appears widespread in space, where two cumulene carbon chains, H2C n (n = 3, 4), have also been identified [1]. The electronic spectra of carbon chains, Cn (n = 4 - 1 5 ) , their anions, C~-, ( n = 2 - 1 0 ) , and the monohydrogenated derivatives, C 2 , H ( n = 3 - 7 ) , have been experimentally observed and assigned [2,3]. This has been achieved in neon matrices at 5 K
* Corresponding author. Tel.: +61 6 249 2935; fax: +61 6 249 0760; e-mail: [email protected]
using mass-selected ion beams. On the basis of the spectral trends observed for these homologous series, it is possible to make predictions which types and sizes of these carbon chains would have the right spectral characteristics to be possible carriers of the absorption features classified as diffuse interstellar bands (DIB) [4]. On the other hand, the electronic transitions of the cumulene carbenes, H2C ., have not been observed for n > 3, and ab initio calculations for excited states have not been reported. Because of the fairly regular trends apparent for the carbon chains, it is proposed that a theoretical prediction of the wavelength range of the strong electronic transitions of the cumulenes would answer the question which sizes may be of relevance to the diffuse absorption bands. Thus, in
0301-0104/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 0 1 - 0 1 0 4 ( 9 7 ) 0 0 2 2 0 - 6
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G. Fischer, J.P. Maier/ Chemical Physics 223 (1997) 149 158
this article the results of ab initio calculations on the intense transition of the species HzC n (n = 3-15) are presented and discussed. Optimized ground state geometries and rotational constants, determined from ab initio molecular orbital calculations, have been previously reported for the cumulene molecules with n = 5 - 1 0 [5]. These calculations assumed linearity of the carbon chains and retention of the Czv molecular symmetry. Empirical corrections to the rotational constants were introduced from comparisons with related molecules for which both observed geometries and equivalent accuracy ab initio computed structures were available. In this work new ab initio molecular orbital calculations are reported for the ground state of the carbon cumulenes for which the constraint of linearity of the carbon chains has been relaxed.
2. Molecular orbital calculations Standard ab initio molecular orbital calculations were carried out on HzC,, ( n = 3-15) using the Gaussian 94 suite of programs [6]. Uncorrelated wavefunctions were obtained by Hartree-Fock (HF) theory for the ground state. Electron correlation was treated with a second-order perturbation expansion (MP2-frozen core). As well, some complete active space multiconfiguration self-consistent field (CASSCF) calculations were carried out using 4 electrons in 5 orbitals (2 in-plane b 2, and 3 out-of-plane b l) for H2C3, and 6 electrons in 7 orbitals for H2C 4. Excited state wavefunctions were determined at the Configuration Interaction Singles (CIS) level, and for HzC 3 and H2C 4, also at the CASSCF level. Most calculations were performed with the 631G(d) basis set. This is of double zeta type with polarization functions on all heavy atoms. For purposes of comparison with reported ground state calculations where polarization functions on the hydrogens were included, calculations were also carried out using the 6-31G(d,p) basis for a number of the cumulenes. Force constants and harmonic vibrational frequencies were obtained by analytic second differentiation of the energy with respect to nuclear displacements. Vibrational frequencies were scaled by 0.895 for the HF and CIS calculations, and by 0.9434 for the MP2 ones [7].
2.1. Ground state geometries
As a first step the optimized ground state geometries of the cumulenes constrained to C2~ molecular symmetry were determined at the HF/6-31G(d) and MP2/6-31G(d) levels for molecules H2C 3 up to HzCI0, and for the larger ones up to H2C15 only at the HF level (Table 1). They were also obtained for HzC 3 and HzC 4 at the CASSCF/6-31G(d) level, and are in good agreement with similar calculations undertaken earlier [5]. In addition, MP2 calculations were carried out using the 6-31G(d,p) basis for H2C 6, H2C 7 and H2C 8, and the optimized geometries essentially reproduce those previously determined using the same level of theory [5]. They are also in excellent agreement with the geometries determined using the 6-31G(d) basis, and for this reason the majority of calculations reported in this work used this. The validity of constraining the molecules to C2v symmetry was checked through frequency calculations, which are particularly valuable in determining the nature of the located stationary point [8]. For both the MP2 and HF calculations imaginary frequencies were obtained for some molecules, suggesting that the constraint to C2v molecular symmetry is too restrictive. Imaginary frequencies were obtained for H2C 6 and larger molecules with the MP2 calculations, while for the HF ones only for molecules larger than HzCl0. NO imaginary frequencies were obtained for the CASSCF calculations on the two smallest cumulenes studied, Reduction to C~ symmetry in the MP2 calculations successfully removed the imaginary frequencies. However, for H2C 9 a very shallow potential was obtained for one a" (b~) mode even in C~ symmetry, so much so that unless 'tight' optimization conditions were specified a small negative frequency remained. For the cumulenes with odd numbers of carbon atoms the C~ symmetry is defined by retention of the molecular plane, with the chain adopting a non-linear conformation, in which bending is small and alternating (zig-zag). For the evennumbered cumulenes the small extent of alternate bending of the carbon chain occurs in a plane normal to the molecular plane. The optimized geometries at the global energy minima (no imaginary frequencies) of H2C n (n = 6-10), not constrained to C2v symmetry (MP2, C~), are included in Table 1.
G. Fischer, J.P. Maier / Chemical Physics 223 (1997) 149-158
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G. Fischer, J.P. Maier/ Chemical Physics 223 (1997) 149-158
152
Table 2 Energy differences (cm - I ) between the optimized ground state geometries determined at C~ and C2v molecular symmetries
H2C 6 H2C 7 H2C 8 H2C 9 H2CI0
MP2/ 6-31G(d)
MP2/ 6-31G(d,p)
32.7 38.6 94.4 90.6 137.2
29.3 91.0
H 2 C 4. Although for these two cumulenes the cause
may lie only partly in the non-linearity of the carbon chain, because for these two molecules no imaginary frequencies were obtained, indicating the constraint of C2v symmetry, linear carbon chain, to be justified. 2.2. Ground state vibrational frequencies
The energy differences between the optimized minima corresponding to the C2v and C~ symmetries are very small (Table 2). In order to confirm that the use of polarization functions for hydrogen had little effect on these energy differences, they were also calculated for H 2 C 6 and H z C 8 using the 6-31G(d,p) basis. Only small and insignificant changes were noted (Table 2). Such small differences between the molecule constrained to C2v and C~ symmetries are indicative of but slight geometry changes; The bond distances change by less than 0.0005 A, and the carbon chain angles by less than 5° from the linear form.
Although the geometry changes are small when the constraint to C2v symmetry is relaxed, they are nevertheless important in so far as the rotational constants are concerned (Table 3). This bears on ab initio predictions of rotational constants to be used in laboratory and astronomical searches of these molecules. It is seen in the table that the A rotational constant is significantly reduced, and that the fractional reduction increases with chain length. The B and C constants are only very marginally increased. This is exactly what Maluendes and McLean observed in a comparison of their ab initio results [5] with the reported experimental values for H2C 3 and
The vibrational frequencies and infrared absorption intensities have been determined for the optimized geometries of H2C n - - no imaginary frequencies - - for n in the range 3-10 at the MP2/6-31G(d) level, and are listed for n = even in Table 4, and for n = odd in Table 5. For n in the range 3-5, C zv symmetry is applicable since no imaginary frequencies were obtained with this constraint on the molecular symmetry. The reduced C~ symmetry must be used for n in the range 6-10. The distinction between the even and odd series is made because the reflection plane is different for the two series; a' vibrations for the even series correlate with a~ and b~ modes in Czv, and for the odd series a' vibrations correlate with a~ and b 2 modes. It is seen in the tables that the correlations of the vibrations between the C2v and C~ symmetries are good. The outstanding feature in these frequency calculations is the large absorption intensity associated with the totally symmetric, high-frequency mode corresponding to the in-phase combination of the CC bond stretches. The same mode is also evident in SCF and CASSCF calculations on the linear carbon chains C n, and the intensity is considered to be anomalously large [9]. The frequency of this vibration lies in the range 1800-2100 cm -1, and even if its intensity is overestimated in the ab initio calculations, it should still be the most prominent in infrared
Table 3 Rotational constants (GHz), for the optimized ground state geometries calculated at the MP2/6-31G(d) level, with the cumulene molecules constrained to C2v, and unconstrained, C~ molecular symmetries
C2v H 2C6 H2C 7 H2C 8 H2C 9 H 2C i 0
Cs
A
B
C
A
B
C
287.90166 291.17098 288.57228 290.97720 289.09417
1.329413 0.840085 0.566921 0.399264 0.292586
1.323302 0.837920 0.565810 0.398717 0.292290
273.57861 274.74766 255.01152 253.40784 220.65018
1.330401 0.840647 0.567342 0.399549 0.292809
1.324610 0.838083 0.566376 0.398920 0.292604
G. Fischer, J.P. Maier/ Chemical Physics 223 (1997) 149-158 Table 4 Ground state vibrational frequencies (cm - t ) and intensities (in parentheses) calculated at the MP2/6-31G(d) level, for the evennumbered cumulenes, H2C . al
bl
b2
H2C 4 870 [0.00] 1339 [0.12] 1648 [0.00] 2067 [7.72] 3030 [0.45]
a'
182 [0.00] 449 [0.02] 718 [0.90]
159 [0.04] 406 [0.00] 926 [0.08] 3118 [0.09]
a"
1--I2C 6 620 [0.00] 1148 [0.00] 1367 [0.01] 1674 [1.09] 1993 [0.28] 2098 [26.2] 3018 [0.38]
H2C 8 474 [0.00] 908 [0.06] 1279 [0.12] 1391 [0.07] 1696 [0.01] 1930[15.5] 1988 [37.0] 2120116.3] 3012 [0.32]
H2Ct0 384 [0.00] 746 [0.14] 1077 [0.07] 1342 [0.03] 1423[0.00] 1708 [3.32] 1839111.2] 1905 [4.50] 2044 [0.54] 2134 [19.8] 3008 [0.26]
95 [0.01] 56 [0.01] 213 [0.05] 141 [0.05] 447 [0.00] 219[0.011 537 [0.09] 424 [0.00] 736 [0.81] 479 [(3.01] 605 [0.16] 750 [0.76]
36 [O.O1J 96 [0.04] 173 [0.011 223 [0.03] 417 [0.00] 458 [0.01] 548 [0.06] 588 [0.18] 762 [0.72]
94 [0.03] 215 [0.07] 376 [0.00] 551 [0.07] 934 [0.03] 3105 [0.04]
59 [0.02] 141 [0.07] 242 [0.01] 356 [0.01] 486 [0.01] 555 [0.10] 941 [0.02] 3098 [0.02]
42 [0.01] 92 [0.05] 178 [0.01} 248 [0.04] 345 [0.00] 396 [0.01] 468 [0.01] 545 [0.11] 946 [0.02] 3093 [0.01]
absorption studies of the c u m u l e n e s . With increasing chain length other totally symmetric vibrations, which also c o n t a i n a large c o m p o n e n t of the in-phase combination of the CC stretches, are characterized by large absorption intensities. So, the infrared spectra of these long-chain carbon c u m u l e n e s should be characterized by a n u m b e r of strong bands in the region of 2000 c m -~. A n o t h e r p r o m i n e n t , but substantially weaker b a n d in the spectrum relative to the strong a I bands discussed above, is predicted to be the highest frequency b 1 (a' for n = even; and a" for n = odd) mode. It is nevertheless also associated with large intensity, and is f o u n d in the range 7 0 0 1000 c m - l , c o n v e r g i n g on a value of ~ 800 c m -~ for chain lengths larger than 10.
153
2.3. Excited state geometries and energies With the exception of the C A S S C F calculations undertaken for H 2 C 3 and H2C4, the excited state optimized geometries and vibrational frequencies have been calculated using a relatively low level of ab initio theory, n a m e l y the CIS technique. The only excited state work reported on the c u m u l e n e s is an experimental and theoretical study of the first three excited states of H2C 3 [10]. However, the intense transition, the subject of this study is not included. Hence, c o m p a r i s o n with e x p e r i m e n t is not available, and for the lowest excited states only with the H 2 C 3 data. For a n u m b e r of reasons that will be elaborated upon below, we believe the CIS calculations are of value, and allow us to predict the electronic absorp-
Table 5 Ground state vibrational frequencies (cm - I ) and intensities (in parentheses) calculated at the MP2/6-31G(d) level, for the oddnumbered cumulenes, H2C,, al
b2
bl
H2C3 1082[0.00] 144810.06] 1947 [3.00] 2996 [0.06]
H2C5 721 [0.00] 130510.11] 1453 [0.09] 1878 [0.57] 2103 [11.4] 2996 [0.02]
a'
258 [0.02] 117 [0.00] 1021 [0.02] 220 [0.09] 3079 [0.01] 369 [0.02] 1000 [0.00] 3079 [0.00]
223 [0.02] 998 [0.19]
123 [0.021 271 [0.07] 480 [0.04] 923 [0.35]
a"
H2C7 530 [0.03] 101210.01] 1375 [0.04] 1482 [0.00] 1846 [1.74] 2020 [9.16] 2089 [21.9] 2997 [0.00]
H2C9 424 [0.00] 81710.01] 1172 [0.00] 1395 [0.00] 1520[0.41] 1830[0.00] 1918 [51.11 2005 [0.14] 2120 [20.5] 2998 [0.01]
72 [0.00] 177 [0.09] 237 [0.001] 421 [0.00] 555 [0.08] 986 [0.00] 3081 [0.00]
45 [0.00] 117 [0.06] 201 [0.00] 246 [0.04] 401 [0.00] 477 [0.00] 599 [0.15] 981 [0.00] 3082 [0.00]
70[0.01] <40 178 [0.09] 65 [0.05] 284 [0.01] 134 [0.03] 370 [0.05] 214 [0.01] 493 [0.001] 313 [0.04] 880 [0.42] 460 [0.04] 466 [0.00] 858 [0.46]
G. Fischer, J.P. Maier/ Chemical Physics 223 (19971 149-158
154
tion spectra. These calculations have been focused on the relatively low-lying singlet A l state associated with the large transition intensity. This would be expected to make the largest contribution to the absorption features classified as diffuse interstellar bands. For the even numbered cumulenes larger than H 2 C 4 it is the third excited ~A 1 state that carries the greatest oscillator strength. The second and third e x c i t e d IA 1 states for the odd-numbered cumulenes are in general sufficiently close in energy that both states are associated with large oscillator strength. Only for H z C 9 and H2Ct_s is the oscillator strength associated with the third A~ state much larger. The optimized geometries for the excited (2)1A l and (3)lA~ states of the cumulenes constrained to C2v
symmetry are listed in Table 6. In analogy with the ground state calculations a tendency exists for the molecules to adopt a non-linear carbon chain structure, with alternating bends (zig-zag). But the effect is small, and the bond lengths and the HCH angle are changed by less than 0.1%. This is exemplified by the case of H2C9; the energy difference between the C~ (molecular plane of symmetry retained), and C2v geometries is only 0.044 cm -~. CIS type calculations have been described [11] as an~
adequate zeroth-order treatment for many of the excited states of molecules. However, even with these so-called lower-level cal-
Table 6 Calculated (CIS/6-31G(d)) molecular structures (C 2v) of H 2C,, in the excited (2)'A I and (3)IA t states H-C
C1-C2
C2-C3
H~_C3
2A, 3A I
1.084 1.090
1.365 1.400
1.433 1.432
H2C 4
2A, 3A,
1 . 0 7 5 1.403 1 . 0 9 5 1.322
1.305 1.329
1.294 1.363
H2C 5
2A I 3A,
1 . 0 7 7 1.381 1.094 1.303
1.271 1.328
1.293 1.339
1.313 1.271
H_.C 6
2A, 3A,
1 . 0 7 5 1.354 1.086 1.29
1.246 1.334
1.302 1.268
1.326 1.304
1.277 1.302
119.6 116.4
H2C 7
2A I
1.076
1.353
1.243
1.312
1.275
1 . 2 8 5 1.302
118.5
H2C s
2A I 3A ~
1 . 0 7 5 1.337 1 . 0 8 2 1.303
1.245 1.303
1.312 1.254
1 . 2 6 9 1 . 2 6 4 1 . 3 2 2 1.275 1 . 3 0 4 1 . 2 7 4 1 . 2 8 5 1.295
119.0 117.2
H2C 9
2A 1 3A~
1.076 1.08
1.335 1.315
1.242 1.279
1.317 1.27
1 . 2 5 3 1.291 1.296 1.27
118.2 117.4
H2C1o
2A I ~A I
1 . 0 7 5 1.327 1.08 1.306
1.247 1.285
1.311 1.261
1.25 1.301
1 . 2 8 8 1 . 2 8 8 1.25 1.3161.273 1 . 2 6 2 1 . 2 8 6 1.2771.2831.287
118.7 117.5
H 2C I I
2A I 3A I
1.076 1.080
1.326 1.297
1.245 1.291
1.313 1.252
1.244 1.304
1 . 3 0 2 1 . 2 6 3 1.2791.2761.2821.288 1 . 2 5 4 1 . 2 8 4 1.291 1.2491.3171.273
118.2 116.6
H2Ct2
eAj 3A,
1 . 0 7 5 1.321 1 . 0 7 9 1.306
1.249 1.277
1.308 1.266
1.244 1.294
1.301 1 . 2 6 4 1.2691.2971.2471.311 1 . 2 5 9 1 . 2 9 0 1.2711.2761.2761.283
1.271 1.282
118.6 117.7
H2CI3
2A I 3A~
1.076 1.079
1.248 1.288
1.308 1.252
1.241 1.305
1 . 3 0 7 1 . 2 5 2 1.2881.2711.2721.276 1 . 2 4 5 1.30(I 1.2651.2691.2971.246
1.281 1.283 1.311 1.271
118.2 117.0
H2CI4
2A I 3A,
1 . 0 7 5 1.318 1 . 0 7 8 1.306
1.251 1.274
1.305 1.269
1.242 1.287
1 . 3 0 6 1 . 2 5 0 1.2871.2761.2581.300 1 . 2 6 0 1.290 1.2651.2791.2761.271
1.245 1.308 1.270 1.275 1.284 1.279
118.4 117.7
H2CI5
2A I
1 . 0 7 6 1.316
1.251
1.303
1.242
1 . 3 0 7 1 . 2 4 6 1.2951.2621.2751.278
1.266 1.276 1.28l
1.280 118.1
1.320 1.297
C3
C4
C5
C6
C7
C8
C9
CI0
Cll
C12
C13
C14
(HCH) 117.8 114.5 121.5 112.6 119.2 113.2
1 . 2 7 6 1.2831.294 1 . 2 7 7 1.299 1.28
Energy minima for the (3)IA I states of H2C v and H2C~5 could not be determined.
155
G. Fischer, J.P. Maier/ Chemical Physics 223 (1997) 149-158
Table 7 Differences between the calculated (CIS/6-31G(d)) and observed transition energies (eV) for the linear carbon chain molecules, C,, C7 C9 C II Ct3 CI5
Calc.
Obs.
Difference
6.61 5.66 4.98 4.33 3.84
4.90 4.20 3.70 3.27 2.95
1.71 1.46 1.28 1.06 0.89
culations, the largest cumulene that could be investigated using the 6-31G(d) basis was restricted by computation limitations to H2Cms. In general it is found that although the CIS calculations give excited state energies that are too high by ~ 1 - 2 eV, the order o f the first few excited states is in general correctly maintained. This also appears to be the case for the cumulenes, at least in so far as H z C 3 is concerned, since experimental results are available. The highest filled and lowest vacant orbitals correlate with the "rr orbitals of the carbon chain molecules C n. W h e n viewed relative to the reduced symmetry (C2,,) of the H z C n species, the 'rr orbitals split into b m and b e orbitals. The lowest lying electronic transitions correspond largely to transitions between these. Thus, for all the cumulenes investigated, the lowest excited state is electric-dipole forbidden, IA 2, and corresponds to the promotion of an electron from b m to b 2 orbital. A number of mA m symmetries occur among the lowest excited states. These are electric-dipole allowed, but the first always has a low oscillator strength because of destructive interference between the two electron promotions, b I ~- b m and b 2 ~-- b 2. In the linear carbon chains this destructive interference is exact because of the 'rr orbital degeneracies. As indicated above, for the longer cumulenes the third excited, (3)~A m, state corresponds largely to the constructive interference between the two electron promotions, and accordingly large oscillator strengths are associated with transitions to these states. The smaller chains have the second and third JA~ states substantially mixed; H z C 3 and H2C 4 have them reversed such that it is the second that has the largest oscillator strength. For the longer chains the second excited mAl state is obtained mainly from the promotion of electrons from the next lowest pair of b I and b z ( r r ) orbitals.
In order to obtain a more quantitative estimate of the errors associated with the CIS calculations for the excitation energies of the (3)JAr states of the cumulenes, a comparison was made between the transition energies experimentally measured for the carbon chain molecules [3], and the equivalent theoretical level, CIS-calculated transition energies. A similar strategy of introducing empirical corrections through comparisons with related molecules for which both observed results and equivalent accuracy ab initio computations are available, was used in the prediction of the rotational constants [5]. The optimized excitation energies for the intense i ~ + states of the linear carbon chains C2,+1 (n = 3 - 7 ) were calculated at the C I S / 6 - 3 1 G ( d ) level of theory. Transition energies were taken relative to the H F / 6 - 3 1 G ( d ) optimized ground states. From calculations of the vibrational frequencies corrections were made for the scaled zero-point energies in the transition energies. A comparison of the measured and calculated transition energies is presented in Table 7. Also included in the table is a column listing the differences between the two sets of values, that is the corrections that need to be made to bring the CIScalculated values into agreement with the experimental.
1.8
1.6
.~
1.4
1.2 (eV)
1.0
0.8
.
. 4
, 5
.
, 6
. 7
CIS (eV) Fig. 1. Corrections to the CIS-calculated, zero-point corrected, transition energies for the intense ~,+ state of the carbon chain molecules C~, against the CIS-calculated transition energies.
o/
156
G. Fischer. J.P. Maier/ Chemical Physics 223 (1997) 149-158
500
E
400
J
300
200 2
4
6
8
10
12
14
16
n
Fig. 2. The corrected CIS-calculatedtransition wavelengths for the intense state of the cumulenes H2C,, [(3)lA~, but for n=4, 5
(2) 'A,].
A plot of these energy differences (corrections), against the CIS-calculated values (Fig. 1), is well represented by a linear relation. The addition of a small quadratic term allows for an excellent fit, as illustrated in the figure. On the assumption that a similar plot would apply for the cumulenes, corrected transition energies have been determined. It should be recalled that the intense ~u state of the linear carbon chain molecules correlates with the intense (3) 1A~ state of the cumulenes. A plot of the corrected CIS transition wavelengths is given in Fig. 2. In accordance with simple molecular orbital models [12], and work on related molecules [2,3], a linear relation is obtained. The transition wavelengths for H2C 4 and H2C 5 are for the (2)1A, state, because this is associated with the largest oscillator strength. Energy minima could not be determined for H2C 7 and H2C,5. Attempts at optimization for the (3)IAI states for these two molecules led to the minima corresponding to the (2)'A~ state. Furthermore, for the odd-numbered chains H 2C ~~ and H 2C 13 the second and third excited ~A~ states are energetically close, and as a consequence are heavily mixed. For these reasons the results for the odd-numbered chains do not fit the linear plot of Fig. 2 as well. Consideration of Fig. 2 then suggests that when account is taken of the high values of the CIS 1
calculations, the cumulenes H 2 C n (n = 9-- 12), should have strong absorptions in the 3 0 0 - 4 0 0 nm region, and the even longer chains could be expected to absorb in the visible. The magnitudes of the oscillator strengths show the expected monotonic increase with the number of carbon atoms (Fig. 3). These predictions may be compared to the experimental measurements on substituted cumulenes in solution, R=(C=C),=R (n=2-5; R being an end substituent) [13]. The substituted cumulene with n = 2 absorbs near 271 nm, and the increment per double bond is ~ 65 nm to longer wavelengths, reflecting the linear relation between absorption wavelength and chain length. For the lowest excited tA l and tA 2 states good agreement was achieved between the CIS and CASSCF predicted energies, although accord with the experimental results for H2C 3 was only obtained for the latter state. Much larger differences exist between the CIS and CASSCF energies for the higher ~At states. Although clearly not very relevant to the longer cumulenes, the comparison does show that the CASSCF predictions for the higher excited states may also be in error, and require some corrections. This is seen in the multireference configuration interaction calculations carried out for the smaller linear carbon chains [14]. The value (5.54 eV) for the
+
10
[]
6' m
f
4.
2
0
i
4
6
•
i
•
8
,
10
,
i
12
•
J
14
16
n Fig. 3. Plot of calculated oscillator strength, f, for the intense transition against carbon chain length of the cumulenes H2C,, [(3)lAi, but for n=4, 5 (2)JAI].
157
G. Fischer, J.P. Maier / Chemical Physics 223 (1997) 149-158
vertical transition energy of the strong 1~+ ,-- X t Z g transition of C 7, is in error by half an eV with respect to the experiment ( ~ 5 eV). In comparison the CIS calculation yields 6.7 eV, which is in much greater error. To proceed at the same theoretical level with the C A S S C F calculations for the longer-chain cumulenes much larger active spaces are required. It is the larger cumulenes that are expected to absorb in the visible and near-infrared regions of the spectrum. Without drastic reductions in the active space and the number of electrons, CASSCF calculations of this magnitude are not possible. Even for the ground state, it was found in CASSCF calculations on some C n molecules that the predicted infrared intensities depend markedly on the size of the active space [9].
Table 8 Vibrational frequencies (cm - I ) and intensities (in parentheses) calculated at the CIS/6-31G(d) level for the (3)IA I states al
H2C9
470 [0.01] 879 [0.00] 1242 [0.28] 1396 [0.25] 1645 [0.87] 1765 [0.66] 1993 [76.0] 2719 [6.31] 2918 [0.27]
423 802 1143 1401 1516 1731 1817 2159 2916
bl
49 [0.00] 147 [0.05] 214 [0.01] 354 [0.01] 372 [0.03] 410 [0.00] 875 [0.59]
b2
46 [0.00] 128 [0.05] 188 [0.01] 280 [0.01] 341 [0.02] 586 [0.191 959 [0.00] 2964 [0.01]
2.4. Excited state vibrational frequencies The CIS-calculated frequencies and absorption intensities for the (3)~A states of a number of longchain cumulenes are presented in Table 8. Because of the ambiguity concerning which of the excited ~A states is the most intense for H 2 C 3 , H 2 C 4 and H2C 5, and that these short-chain cumulenes are not expected to absorb above 300 nm, see Fig. 2, excited state vibrational frequencies have not been listed for them. No vibrational frequencies could be calculated for the cumulenes larger than H2C 5, other than those appearing in Table 8, either because of problems with state hopping, or computation limitations. In parallel with the ground state, the (3)~A~ state is characterized by one or two vibrations with enormous transition intensity. The responsible vibration is totally symmetric, and as for the ground state, corresponds to the in-phase combination of the CC bond stretches. There is no marked dependence on the electronic state, but some of the trends in the changes of the vibrational frequencies with carbon chain length are of interest. Thus the frequency of the a I mode associated with the largest infrared intensity is found to decrease with chain length in the ground state, while the reverse is true for the (3) ~Aj state. Simple arguments suggest that for the excited state, where a bonding electron has been promoted to an anti-bonding orbital, increased chain length leads to a reduction of the contribution of the anti-bonding orbital per CC bond, and hence to
H2C8
H2CI2 [0.00] [0.00] [0.06] [0.02] [0.98] [1.36] [6.97] [61.0] [0.16]
324 630 922 1189 1387 1461 1692 1813 1824
[0.01] [0.06] [0.05] [0.00] [1.31] [0.07] [2.45] [1.49] [8.09]
4180 [4.29]
1929 [7.79] 2064 [206] 2941 [0.04] 3582 [0.19]
43 [0.00] 118 [0.07] 213 [0.01] 282 [0.02] 447 [0.01] 460 [0.03] 622 [0.08] 895 [0.51]
25 [0.00] 72 [0.03] 133 [0.00] 197 [0.00] 211 [0.03] 250 [0.00] 365 [0.00] 387 [0.00] 452 [0.02] 461 [0.00] 897 [0.561
< 40 [0.00] [0.05] [0.001 [0.02] [0.00] [0.00] [0.00] [0.02]
24 [0.00] 68 [0.04] 124 [0.01] 184 [0.031 240 [0.01] 275 [0.03] 316 [0.00] 396 [0.00] 504 [0.01] 696 [0.22] 983 [0.00] 3016 [0.00]
43 121 203 265 316 392 987 2993
increased frequencies. The largest frequency bl mode (out-of-molecular plane), increases in frequency with chain length for both the ground and (3)~A 1 states for the even-numbered chains, and decreases for the odd-numbered chains. Furthermore, it is noted that one al vibration has suffered a large increase in frequency. The increase can be attributed to vibronic coupling with the near-lying, lower-energy (2)1A 1 state. Thus, for H 2 C 8 , H 2 C 9 and H2C12 the energy gaps between the two electronic states are 5418,
158
G. Fischer, J.P. Maier/ Chemical Physics 223 (1997) 149-158
2874 and 4149 cm -~, respectively, mirroring the magnitudes of the vibrational frequency increases.
3. Relation to astrophysical measurements Among the molecules considered to be responsible for the absorption features known as the DIB [15], are carbon chains. Though the proposal was made quite some time ago [16], only in the past few years have the strong characteristic electronic transitions of individual carbon chains been identified [2,3]. On the basis of these data, one could conclude which size the carbon chains must have so that the electronic transitions are in the 4 0 0 - 9 0 0 nm region [4], where the DIBs are observed, and that they are large enough to be photochemically stable in the stellar radiation field. For the homologous series considered, this is the case for the C2,,+ j (n = 8 - 1 6 ) range, whereby these may be limited to the midtwenties carbon atoms range because of known preference for ring structures for the larger sizes [17]. The aim of the present contribution is to predict this for the cumulenes. This is particularly relevant as a number of such cumulenes have been detected by radioastronomy in dense clouds and there is the need to understand their electronic transitions in connection with the DIB identification [1 ]. It is seen in Fig. 2 that H2C~3 should absorb at ~ 400 nm, and that the increment per additional carbon atom is ~ 22 nm. Thus, it is the cumulenes with sizes H z C I 3 to about H2C34 that would have their first strong electronic transition in the 4 0 0 - 8 6 0 nm DIB range. These species should be sufficiently large to be stable to photodissociation. The goal should now be to measure the electronic absorption spectra of a few such species in neon matrices using the established mass-selected approach, and ultimately a gas-phase measurement for direct comparison with astronomical observations. Furthermore, for such large cumulenes, ab initio predictions of ground state rotational constants that are to be used in radioastronomy will need to take into account the non-linearity of the carbon chains.
Acknowledgements GF wishes to thank the Institute for Physical Chemistry, University of Basel, for kind hospitality. This study is part of a project financed by the Swiss National Science Foundation (No. 20-49104.96). Some of the calculations were carried out using the Silicon Graphics Power Challenge of the ANU Supercomputer Facility. K. Bultitude is thanked for her help with the CASSCF calculations.
References [1] P. Thaddeus, C.A. Gottlieb, R. Mollaaghababa, J.M. Vrtilek, J. Chem. Soc. Faraday Trans. 89 (1993) 2125. [2] P. Freivogel, J. Fulara, M. Jakobi, D. Forney, J.P. Maier, J. Chem. Phys. 103 (1995) 54. [3] D. Forney, P. Freivogel, M. Grutter, J.P. Maier, J. Chem. Phys. 104 (1996) 4954. [4] J.P. Maier, Chem. Soc. Rev. 26 (1997) 21. [5] S.A. Maluendes, A.D. McLean, Chem. Phys. Lett. 200 (1992) 511. [6] M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G. Johnson, M,A. Robb, J.R. Cheeseman, T. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. AI-Laham, V.G. Zakrzewski, J.V. Ortiz, J.B. Foresman, J. Cioslowski, B.B. Stefanov, A. Nanayakkara, M. Challacobe, C.Y. Peng, P.Y. Ayla, W. Chen, M.W. Wong, J.L. Andres, E.S. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. Defrees, J. Baker, J.P. Stewart, M. Head-Gordon, C. Gonzales, J.A. Pople, Gaussian 94, Revision A.I, Gaussian Inc., Pittsburgh, PA, 1995. [7] A.P. Scott, L. Radom, J. Phys. Chem. 100 (1996) 16502. [8] J.B. Foresman, A. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian, Inc., Pittsburgh, PA, p.199. [9] J.M.L. Martin, P.R. Taylor, Chem. Phys. Lett. 240 (1995) 521. [10] J.F. Stanton, J.T. DePinto, R.A. Seburg, J.A. Hodges, R.J. McMahon, J. Am. Chem. Soc. 119 (1997) 429. [11] J.B. Foresman, M. Head-Gordon, J.A. Pople, M.J. Frisch, J. Phys. Chem. 96 (1992) 135. [12] H. Suzuki, Electronic Absorption Spectra and Geometry of Organic Molecules, Academic Press, New York, 1967. [13] H. Krauch, J. Chem. Phys. 102 (1958) 898. [14] M. Kolbuszewski, J. Chem. Phys. 102 (1995) 3679. [/5] G.H. Herbig, Annu. Rev. Astrophys. 33 (1995) 19. [16] A.E. Douglas, Nature (London) 269 (1977) 130. [17] H.W. Kroto, Int. J. Mass Spectrom. Ion Process. 138 (1994) 1.
Chemical Physics 223 (1997) 149-158
Ab initio prediction of the spectra of carbon cumulenes Gad Fischer a,*, John P. Maier b a Department of Chemistry Australian National UniL,ersio, Canberra, ACT 0200, Australia b Institute for Physical Chemisto', University of Basel, Klingelbergstrasse 80, CH-4056 Basel, Switzerland Received 29 May 1997
Abstract
Ground state calculations of the optimized geometries and vibrational frequencies of the carbon cumulenes, H 2 C n (n = 3-15), have been variously carried out at HF/6-31G(d), MP2/6-31G(d), MP2/6-31G(d,p) and CASSCF/6-31G(d) levels, both with and without the constraint of linearity of the carbon chains. Substantial reductions in the A rotational constant are obtained when the constraint to linearity is removed, in agreement with the trend observed for the short cumulenes for which experimental results are available. Excited state calculations at the CIS/6-31G(d) level have focussed mainly on the intense JAt excited state (the third ~Aj for n > 5). Also, in these calculations the tendency for non-linearity of the carbon chain has been noted, although energy differences are smaller. CASSCF calculations have been restricted to the two smallest cumulenes. They are in good agreement with the CIS calculations for the first excited IA~ and IA 2 states, but differ for the higher ~Ai states. Empirical corrections to the CIS-calculated transition energies for the intense state have been introduced through comparisons with the linear carbon chain molecules, C,, for which both observed results and equivalent accuracy ab initio computations are available. Cumulenes with sizes H2C 13 to about H 2C 34 have their first strong electronic transition in the 400-860 nm DIB range. © 1997 Elsevier Science B.V.
I. I n t r o d u c t i o n
Cumulene carbene molecules have been shown by radioastronomy to be constituents in interstellar gas and circumstellar shells. In particular, the small cyclic c a r b e n e , C 3 H 2 , appears widespread in space, where two cumulene carbon chains, H2C n (n = 3, 4), have also been identified [1]. The electronic spectra of carbon chains, Cn (n = 4 - 1 5 ) , their anions, C~-, ( n = 2 - 1 0 ) , and the monohydrogenated derivatives, C 2 , H ( n = 3 - 7 ) , have been experimentally observed and assigned [2,3]. This has been achieved in neon matrices at 5 K
* Corresponding author. Tel.: +61 6 249 2935; fax: +61 6 249 0760; e-mail: [email protected]
using mass-selected ion beams. On the basis of the spectral trends observed for these homologous series, it is possible to make predictions which types and sizes of these carbon chains would have the right spectral characteristics to be possible carriers of the absorption features classified as diffuse interstellar bands (DIB) [4]. On the other hand, the electronic transitions of the cumulene carbenes, H2C ., have not been observed for n > 3, and ab initio calculations for excited states have not been reported. Because of the fairly regular trends apparent for the carbon chains, it is proposed that a theoretical prediction of the wavelength range of the strong electronic transitions of the cumulenes would answer the question which sizes may be of relevance to the diffuse absorption bands. Thus, in
0301-0104/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 0 1 - 0 1 0 4 ( 9 7 ) 0 0 2 2 0 - 6
150
G. Fischer, J.P. Maier/ Chemical Physics 223 (1997) 149 158
this article the results of ab initio calculations on the intense transition of the species HzC n (n = 3-15) are presented and discussed. Optimized ground state geometries and rotational constants, determined from ab initio molecular orbital calculations, have been previously reported for the cumulene molecules with n = 5 - 1 0 [5]. These calculations assumed linearity of the carbon chains and retention of the Czv molecular symmetry. Empirical corrections to the rotational constants were introduced from comparisons with related molecules for which both observed geometries and equivalent accuracy ab initio computed structures were available. In this work new ab initio molecular orbital calculations are reported for the ground state of the carbon cumulenes for which the constraint of linearity of the carbon chains has been relaxed.
2. Molecular orbital calculations Standard ab initio molecular orbital calculations were carried out on HzC,, ( n = 3-15) using the Gaussian 94 suite of programs [6]. Uncorrelated wavefunctions were obtained by Hartree-Fock (HF) theory for the ground state. Electron correlation was treated with a second-order perturbation expansion (MP2-frozen core). As well, some complete active space multiconfiguration self-consistent field (CASSCF) calculations were carried out using 4 electrons in 5 orbitals (2 in-plane b 2, and 3 out-of-plane b l) for H2C3, and 6 electrons in 7 orbitals for H2C 4. Excited state wavefunctions were determined at the Configuration Interaction Singles (CIS) level, and for HzC 3 and H2C 4, also at the CASSCF level. Most calculations were performed with the 631G(d) basis set. This is of double zeta type with polarization functions on all heavy atoms. For purposes of comparison with reported ground state calculations where polarization functions on the hydrogens were included, calculations were also carried out using the 6-31G(d,p) basis for a number of the cumulenes. Force constants and harmonic vibrational frequencies were obtained by analytic second differentiation of the energy with respect to nuclear displacements. Vibrational frequencies were scaled by 0.895 for the HF and CIS calculations, and by 0.9434 for the MP2 ones [7].
2.1. Ground state geometries
As a first step the optimized ground state geometries of the cumulenes constrained to C2~ molecular symmetry were determined at the HF/6-31G(d) and MP2/6-31G(d) levels for molecules H2C 3 up to HzCI0, and for the larger ones up to H2C15 only at the HF level (Table 1). They were also obtained for HzC 3 and HzC 4 at the CASSCF/6-31G(d) level, and are in good agreement with similar calculations undertaken earlier [5]. In addition, MP2 calculations were carried out using the 6-31G(d,p) basis for H2C 6, H2C 7 and H2C 8, and the optimized geometries essentially reproduce those previously determined using the same level of theory [5]. They are also in excellent agreement with the geometries determined using the 6-31G(d) basis, and for this reason the majority of calculations reported in this work used this. The validity of constraining the molecules to C2v symmetry was checked through frequency calculations, which are particularly valuable in determining the nature of the located stationary point [8]. For both the MP2 and HF calculations imaginary frequencies were obtained for some molecules, suggesting that the constraint to C2v molecular symmetry is too restrictive. Imaginary frequencies were obtained for H2C 6 and larger molecules with the MP2 calculations, while for the HF ones only for molecules larger than HzCl0. NO imaginary frequencies were obtained for the CASSCF calculations on the two smallest cumulenes studied, Reduction to C~ symmetry in the MP2 calculations successfully removed the imaginary frequencies. However, for H2C 9 a very shallow potential was obtained for one a" (b~) mode even in C~ symmetry, so much so that unless 'tight' optimization conditions were specified a small negative frequency remained. For the cumulenes with odd numbers of carbon atoms the C~ symmetry is defined by retention of the molecular plane, with the chain adopting a non-linear conformation, in which bending is small and alternating (zig-zag). For the evennumbered cumulenes the small extent of alternate bending of the carbon chain occurs in a plane normal to the molecular plane. The optimized geometries at the global energy minima (no imaginary frequencies) of H2C n (n = 6-10), not constrained to C2v symmetry (MP2, C~), are included in Table 1.
G. Fischer, J.P. Maier / Chemical Physics 223 (1997) 149-158
i 51 ©
I
o I
I.
I
A
I
E J
- o&
~6
~< b I
II
~= I
Y
~s
o I
E
~E _-_=
G
d
C ~
G
~
d
~
G G G ~
o
G. Fischer, J.P. Maier/ Chemical Physics 223 (1997) 149-158
152
Table 2 Energy differences (cm - I ) between the optimized ground state geometries determined at C~ and C2v molecular symmetries
H2C 6 H2C 7 H2C 8 H2C 9 H2CI0
MP2/ 6-31G(d)
MP2/ 6-31G(d,p)
32.7 38.6 94.4 90.6 137.2
29.3 91.0
H 2 C 4. Although for these two cumulenes the cause
may lie only partly in the non-linearity of the carbon chain, because for these two molecules no imaginary frequencies were obtained, indicating the constraint of C2v symmetry, linear carbon chain, to be justified. 2.2. Ground state vibrational frequencies
The energy differences between the optimized minima corresponding to the C2v and C~ symmetries are very small (Table 2). In order to confirm that the use of polarization functions for hydrogen had little effect on these energy differences, they were also calculated for H 2 C 6 and H z C 8 using the 6-31G(d,p) basis. Only small and insignificant changes were noted (Table 2). Such small differences between the molecule constrained to C2v and C~ symmetries are indicative of but slight geometry changes; The bond distances change by less than 0.0005 A, and the carbon chain angles by less than 5° from the linear form.
Although the geometry changes are small when the constraint to C2v symmetry is relaxed, they are nevertheless important in so far as the rotational constants are concerned (Table 3). This bears on ab initio predictions of rotational constants to be used in laboratory and astronomical searches of these molecules. It is seen in the table that the A rotational constant is significantly reduced, and that the fractional reduction increases with chain length. The B and C constants are only very marginally increased. This is exactly what Maluendes and McLean observed in a comparison of their ab initio results [5] with the reported experimental values for H2C 3 and
The vibrational frequencies and infrared absorption intensities have been determined for the optimized geometries of H2C n - - no imaginary frequencies - - for n in the range 3-10 at the MP2/6-31G(d) level, and are listed for n = even in Table 4, and for n = odd in Table 5. For n in the range 3-5, C zv symmetry is applicable since no imaginary frequencies were obtained with this constraint on the molecular symmetry. The reduced C~ symmetry must be used for n in the range 6-10. The distinction between the even and odd series is made because the reflection plane is different for the two series; a' vibrations for the even series correlate with a~ and b~ modes in Czv, and for the odd series a' vibrations correlate with a~ and b 2 modes. It is seen in the tables that the correlations of the vibrations between the C2v and C~ symmetries are good. The outstanding feature in these frequency calculations is the large absorption intensity associated with the totally symmetric, high-frequency mode corresponding to the in-phase combination of the CC bond stretches. The same mode is also evident in SCF and CASSCF calculations on the linear carbon chains C n, and the intensity is considered to be anomalously large [9]. The frequency of this vibration lies in the range 1800-2100 cm -1, and even if its intensity is overestimated in the ab initio calculations, it should still be the most prominent in infrared
Table 3 Rotational constants (GHz), for the optimized ground state geometries calculated at the MP2/6-31G(d) level, with the cumulene molecules constrained to C2v, and unconstrained, C~ molecular symmetries
C2v H 2C6 H2C 7 H2C 8 H2C 9 H 2C i 0
Cs
A
B
C
A
B
C
287.90166 291.17098 288.57228 290.97720 289.09417
1.329413 0.840085 0.566921 0.399264 0.292586
1.323302 0.837920 0.565810 0.398717 0.292290
273.57861 274.74766 255.01152 253.40784 220.65018
1.330401 0.840647 0.567342 0.399549 0.292809
1.324610 0.838083 0.566376 0.398920 0.292604
G. Fischer, J.P. Maier/ Chemical Physics 223 (1997) 149-158 Table 4 Ground state vibrational frequencies (cm - t ) and intensities (in parentheses) calculated at the MP2/6-31G(d) level, for the evennumbered cumulenes, H2C . al
bl
b2
H2C 4 870 [0.00] 1339 [0.12] 1648 [0.00] 2067 [7.72] 3030 [0.45]
a'
182 [0.00] 449 [0.02] 718 [0.90]
159 [0.04] 406 [0.00] 926 [0.08] 3118 [0.09]
a"
1--I2C 6 620 [0.00] 1148 [0.00] 1367 [0.01] 1674 [1.09] 1993 [0.28] 2098 [26.2] 3018 [0.38]
H2C 8 474 [0.00] 908 [0.06] 1279 [0.12] 1391 [0.07] 1696 [0.01] 1930[15.5] 1988 [37.0] 2120116.3] 3012 [0.32]
H2Ct0 384 [0.00] 746 [0.14] 1077 [0.07] 1342 [0.03] 1423[0.00] 1708 [3.32] 1839111.2] 1905 [4.50] 2044 [0.54] 2134 [19.8] 3008 [0.26]
95 [0.01] 56 [0.01] 213 [0.05] 141 [0.05] 447 [0.00] 219[0.011 537 [0.09] 424 [0.00] 736 [0.81] 479 [(3.01] 605 [0.16] 750 [0.76]
36 [O.O1J 96 [0.04] 173 [0.011 223 [0.03] 417 [0.00] 458 [0.01] 548 [0.06] 588 [0.18] 762 [0.72]
94 [0.03] 215 [0.07] 376 [0.00] 551 [0.07] 934 [0.03] 3105 [0.04]
59 [0.02] 141 [0.07] 242 [0.01] 356 [0.01] 486 [0.01] 555 [0.10] 941 [0.02] 3098 [0.02]
42 [0.01] 92 [0.05] 178 [0.01} 248 [0.04] 345 [0.00] 396 [0.01] 468 [0.01] 545 [0.11] 946 [0.02] 3093 [0.01]
absorption studies of the c u m u l e n e s . With increasing chain length other totally symmetric vibrations, which also c o n t a i n a large c o m p o n e n t of the in-phase combination of the CC stretches, are characterized by large absorption intensities. So, the infrared spectra of these long-chain carbon c u m u l e n e s should be characterized by a n u m b e r of strong bands in the region of 2000 c m -~. A n o t h e r p r o m i n e n t , but substantially weaker b a n d in the spectrum relative to the strong a I bands discussed above, is predicted to be the highest frequency b 1 (a' for n = even; and a" for n = odd) mode. It is nevertheless also associated with large intensity, and is f o u n d in the range 7 0 0 1000 c m - l , c o n v e r g i n g on a value of ~ 800 c m -~ for chain lengths larger than 10.
153
2.3. Excited state geometries and energies With the exception of the C A S S C F calculations undertaken for H 2 C 3 and H2C4, the excited state optimized geometries and vibrational frequencies have been calculated using a relatively low level of ab initio theory, n a m e l y the CIS technique. The only excited state work reported on the c u m u l e n e s is an experimental and theoretical study of the first three excited states of H2C 3 [10]. However, the intense transition, the subject of this study is not included. Hence, c o m p a r i s o n with e x p e r i m e n t is not available, and for the lowest excited states only with the H 2 C 3 data. For a n u m b e r of reasons that will be elaborated upon below, we believe the CIS calculations are of value, and allow us to predict the electronic absorp-
Table 5 Ground state vibrational frequencies (cm - I ) and intensities (in parentheses) calculated at the MP2/6-31G(d) level, for the oddnumbered cumulenes, H2C,, al
b2
bl
H2C3 1082[0.00] 144810.06] 1947 [3.00] 2996 [0.06]
H2C5 721 [0.00] 130510.11] 1453 [0.09] 1878 [0.57] 2103 [11.4] 2996 [0.02]
a'
258 [0.02] 117 [0.00] 1021 [0.02] 220 [0.09] 3079 [0.01] 369 [0.02] 1000 [0.00] 3079 [0.00]
223 [0.02] 998 [0.19]
123 [0.021 271 [0.07] 480 [0.04] 923 [0.35]
a"
H2C7 530 [0.03] 101210.01] 1375 [0.04] 1482 [0.00] 1846 [1.74] 2020 [9.16] 2089 [21.9] 2997 [0.00]
H2C9 424 [0.00] 81710.01] 1172 [0.00] 1395 [0.00] 1520[0.41] 1830[0.00] 1918 [51.11 2005 [0.14] 2120 [20.5] 2998 [0.01]
72 [0.00] 177 [0.09] 237 [0.001] 421 [0.00] 555 [0.08] 986 [0.00] 3081 [0.00]
45 [0.00] 117 [0.06] 201 [0.00] 246 [0.04] 401 [0.00] 477 [0.00] 599 [0.15] 981 [0.00] 3082 [0.00]
70[0.01] <40 178 [0.09] 65 [0.05] 284 [0.01] 134 [0.03] 370 [0.05] 214 [0.01] 493 [0.001] 313 [0.04] 880 [0.42] 460 [0.04] 466 [0.00] 858 [0.46]
G. Fischer, J.P. Maier/ Chemical Physics 223 (19971 149-158
154
tion spectra. These calculations have been focused on the relatively low-lying singlet A l state associated with the large transition intensity. This would be expected to make the largest contribution to the absorption features classified as diffuse interstellar bands. For the even numbered cumulenes larger than H 2 C 4 it is the third excited ~A 1 state that carries the greatest oscillator strength. The second and third e x c i t e d IA 1 states for the odd-numbered cumulenes are in general sufficiently close in energy that both states are associated with large oscillator strength. Only for H z C 9 and H2Ct_s is the oscillator strength associated with the third A~ state much larger. The optimized geometries for the excited (2)1A l and (3)lA~ states of the cumulenes constrained to C2v
symmetry are listed in Table 6. In analogy with the ground state calculations a tendency exists for the molecules to adopt a non-linear carbon chain structure, with alternating bends (zig-zag). But the effect is small, and the bond lengths and the HCH angle are changed by less than 0.1%. This is exemplified by the case of H2C9; the energy difference between the C~ (molecular plane of symmetry retained), and C2v geometries is only 0.044 cm -~. CIS type calculations have been described [11] as an~
adequate zeroth-order treatment for many of the excited states of molecules. However, even with these so-called lower-level cal-
Table 6 Calculated (CIS/6-31G(d)) molecular structures (C 2v) of H 2C,, in the excited (2)'A I and (3)IA t states H-C
C1-C2
C2-C3
H~_C3
2A, 3A I
1.084 1.090
1.365 1.400
1.433 1.432
H2C 4
2A, 3A,
1 . 0 7 5 1.403 1 . 0 9 5 1.322
1.305 1.329
1.294 1.363
H2C 5
2A I 3A,
1 . 0 7 7 1.381 1.094 1.303
1.271 1.328
1.293 1.339
1.313 1.271
H_.C 6
2A, 3A,
1 . 0 7 5 1.354 1.086 1.29
1.246 1.334
1.302 1.268
1.326 1.304
1.277 1.302
119.6 116.4
H2C 7
2A I
1.076
1.353
1.243
1.312
1.275
1 . 2 8 5 1.302
118.5
H2C s
2A I 3A ~
1 . 0 7 5 1.337 1 . 0 8 2 1.303
1.245 1.303
1.312 1.254
1 . 2 6 9 1 . 2 6 4 1 . 3 2 2 1.275 1 . 3 0 4 1 . 2 7 4 1 . 2 8 5 1.295
119.0 117.2
H2C 9
2A 1 3A~
1.076 1.08
1.335 1.315
1.242 1.279
1.317 1.27
1 . 2 5 3 1.291 1.296 1.27
118.2 117.4
H2C1o
2A I ~A I
1 . 0 7 5 1.327 1.08 1.306
1.247 1.285
1.311 1.261
1.25 1.301
1 . 2 8 8 1 . 2 8 8 1.25 1.3161.273 1 . 2 6 2 1 . 2 8 6 1.2771.2831.287
118.7 117.5
H 2C I I
2A I 3A I
1.076 1.080
1.326 1.297
1.245 1.291
1.313 1.252
1.244 1.304
1 . 3 0 2 1 . 2 6 3 1.2791.2761.2821.288 1 . 2 5 4 1 . 2 8 4 1.291 1.2491.3171.273
118.2 116.6
H2Ct2
eAj 3A,
1 . 0 7 5 1.321 1 . 0 7 9 1.306
1.249 1.277
1.308 1.266
1.244 1.294
1.301 1 . 2 6 4 1.2691.2971.2471.311 1 . 2 5 9 1 . 2 9 0 1.2711.2761.2761.283
1.271 1.282
118.6 117.7
H2CI3
2A I 3A~
1.076 1.079
1.248 1.288
1.308 1.252
1.241 1.305
1 . 3 0 7 1 . 2 5 2 1.2881.2711.2721.276 1 . 2 4 5 1.30(I 1.2651.2691.2971.246
1.281 1.283 1.311 1.271
118.2 117.0
H2CI4
2A I 3A,
1 . 0 7 5 1.318 1 . 0 7 8 1.306
1.251 1.274
1.305 1.269
1.242 1.287
1 . 3 0 6 1 . 2 5 0 1.2871.2761.2581.300 1 . 2 6 0 1.290 1.2651.2791.2761.271
1.245 1.308 1.270 1.275 1.284 1.279
118.4 117.7
H2CI5
2A I
1 . 0 7 6 1.316
1.251
1.303
1.242
1 . 3 0 7 1 . 2 4 6 1.2951.2621.2751.278
1.266 1.276 1.28l
1.280 118.1
1.320 1.297
C3
C4
C5
C6
C7
C8
C9
CI0
Cll
C12
C13
C14
(HCH) 117.8 114.5 121.5 112.6 119.2 113.2
1 . 2 7 6 1.2831.294 1 . 2 7 7 1.299 1.28
Energy minima for the (3)IA I states of H2C v and H2C~5 could not be determined.
155
G. Fischer, J.P. Maier/ Chemical Physics 223 (1997) 149-158
Table 7 Differences between the calculated (CIS/6-31G(d)) and observed transition energies (eV) for the linear carbon chain molecules, C,, C7 C9 C II Ct3 CI5
Calc.
Obs.
Difference
6.61 5.66 4.98 4.33 3.84
4.90 4.20 3.70 3.27 2.95
1.71 1.46 1.28 1.06 0.89
culations, the largest cumulene that could be investigated using the 6-31G(d) basis was restricted by computation limitations to H2Cms. In general it is found that although the CIS calculations give excited state energies that are too high by ~ 1 - 2 eV, the order o f the first few excited states is in general correctly maintained. This also appears to be the case for the cumulenes, at least in so far as H z C 3 is concerned, since experimental results are available. The highest filled and lowest vacant orbitals correlate with the "rr orbitals of the carbon chain molecules C n. W h e n viewed relative to the reduced symmetry (C2,,) of the H z C n species, the 'rr orbitals split into b m and b e orbitals. The lowest lying electronic transitions correspond largely to transitions between these. Thus, for all the cumulenes investigated, the lowest excited state is electric-dipole forbidden, IA 2, and corresponds to the promotion of an electron from b m to b 2 orbital. A number of mA m symmetries occur among the lowest excited states. These are electric-dipole allowed, but the first always has a low oscillator strength because of destructive interference between the two electron promotions, b I ~- b m and b 2 ~-- b 2. In the linear carbon chains this destructive interference is exact because of the 'rr orbital degeneracies. As indicated above, for the longer cumulenes the third excited, (3)~A m, state corresponds largely to the constructive interference between the two electron promotions, and accordingly large oscillator strengths are associated with transitions to these states. The smaller chains have the second and third JA~ states substantially mixed; H z C 3 and H2C 4 have them reversed such that it is the second that has the largest oscillator strength. For the longer chains the second excited mAl state is obtained mainly from the promotion of electrons from the next lowest pair of b I and b z ( r r ) orbitals.
In order to obtain a more quantitative estimate of the errors associated with the CIS calculations for the excitation energies of the (3)JAr states of the cumulenes, a comparison was made between the transition energies experimentally measured for the carbon chain molecules [3], and the equivalent theoretical level, CIS-calculated transition energies. A similar strategy of introducing empirical corrections through comparisons with related molecules for which both observed results and equivalent accuracy ab initio computations are available, was used in the prediction of the rotational constants [5]. The optimized excitation energies for the intense i ~ + states of the linear carbon chains C2,+1 (n = 3 - 7 ) were calculated at the C I S / 6 - 3 1 G ( d ) level of theory. Transition energies were taken relative to the H F / 6 - 3 1 G ( d ) optimized ground states. From calculations of the vibrational frequencies corrections were made for the scaled zero-point energies in the transition energies. A comparison of the measured and calculated transition energies is presented in Table 7. Also included in the table is a column listing the differences between the two sets of values, that is the corrections that need to be made to bring the CIScalculated values into agreement with the experimental.
1.8
1.6
.~
1.4
1.2 (eV)
1.0
0.8
.
. 4
, 5
.
, 6
. 7
CIS (eV) Fig. 1. Corrections to the CIS-calculated, zero-point corrected, transition energies for the intense ~,+ state of the carbon chain molecules C~, against the CIS-calculated transition energies.
o/
156
G. Fischer. J.P. Maier/ Chemical Physics 223 (1997) 149-158
500
E
400
J
300
200 2
4
6
8
10
12
14
16
n
Fig. 2. The corrected CIS-calculatedtransition wavelengths for the intense state of the cumulenes H2C,, [(3)lA~, but for n=4, 5
(2) 'A,].
A plot of these energy differences (corrections), against the CIS-calculated values (Fig. 1), is well represented by a linear relation. The addition of a small quadratic term allows for an excellent fit, as illustrated in the figure. On the assumption that a similar plot would apply for the cumulenes, corrected transition energies have been determined. It should be recalled that the intense ~u state of the linear carbon chain molecules correlates with the intense (3) 1A~ state of the cumulenes. A plot of the corrected CIS transition wavelengths is given in Fig. 2. In accordance with simple molecular orbital models [12], and work on related molecules [2,3], a linear relation is obtained. The transition wavelengths for H2C 4 and H2C 5 are for the (2)1A, state, because this is associated with the largest oscillator strength. Energy minima could not be determined for H2C 7 and H2C,5. Attempts at optimization for the (3)IAI states for these two molecules led to the minima corresponding to the (2)'A~ state. Furthermore, for the odd-numbered chains H 2C ~~ and H 2C 13 the second and third excited ~A~ states are energetically close, and as a consequence are heavily mixed. For these reasons the results for the odd-numbered chains do not fit the linear plot of Fig. 2 as well. Consideration of Fig. 2 then suggests that when account is taken of the high values of the CIS 1
calculations, the cumulenes H 2 C n (n = 9-- 12), should have strong absorptions in the 3 0 0 - 4 0 0 nm region, and the even longer chains could be expected to absorb in the visible. The magnitudes of the oscillator strengths show the expected monotonic increase with the number of carbon atoms (Fig. 3). These predictions may be compared to the experimental measurements on substituted cumulenes in solution, R=(C=C),=R (n=2-5; R being an end substituent) [13]. The substituted cumulene with n = 2 absorbs near 271 nm, and the increment per double bond is ~ 65 nm to longer wavelengths, reflecting the linear relation between absorption wavelength and chain length. For the lowest excited tA l and tA 2 states good agreement was achieved between the CIS and CASSCF predicted energies, although accord with the experimental results for H2C 3 was only obtained for the latter state. Much larger differences exist between the CIS and CASSCF energies for the higher ~At states. Although clearly not very relevant to the longer cumulenes, the comparison does show that the CASSCF predictions for the higher excited states may also be in error, and require some corrections. This is seen in the multireference configuration interaction calculations carried out for the smaller linear carbon chains [14]. The value (5.54 eV) for the
+
10
[]
6' m
f
4.
2
0
i
4
6
•
i
•
8
,
10
,
i
12
•
J
14
16
n Fig. 3. Plot of calculated oscillator strength, f, for the intense transition against carbon chain length of the cumulenes H2C,, [(3)lAi, but for n=4, 5 (2)JAI].
157
G. Fischer, J.P. Maier / Chemical Physics 223 (1997) 149-158
vertical transition energy of the strong 1~+ ,-- X t Z g transition of C 7, is in error by half an eV with respect to the experiment ( ~ 5 eV). In comparison the CIS calculation yields 6.7 eV, which is in much greater error. To proceed at the same theoretical level with the C A S S C F calculations for the longer-chain cumulenes much larger active spaces are required. It is the larger cumulenes that are expected to absorb in the visible and near-infrared regions of the spectrum. Without drastic reductions in the active space and the number of electrons, CASSCF calculations of this magnitude are not possible. Even for the ground state, it was found in CASSCF calculations on some C n molecules that the predicted infrared intensities depend markedly on the size of the active space [9].
Table 8 Vibrational frequencies (cm - I ) and intensities (in parentheses) calculated at the CIS/6-31G(d) level for the (3)IA I states al
H2C9
470 [0.01] 879 [0.00] 1242 [0.28] 1396 [0.25] 1645 [0.87] 1765 [0.66] 1993 [76.0] 2719 [6.31] 2918 [0.27]
423 802 1143 1401 1516 1731 1817 2159 2916
bl
49 [0.00] 147 [0.05] 214 [0.01] 354 [0.01] 372 [0.03] 410 [0.00] 875 [0.59]
b2
46 [0.00] 128 [0.05] 188 [0.01] 280 [0.01] 341 [0.02] 586 [0.191 959 [0.00] 2964 [0.01]
2.4. Excited state vibrational frequencies The CIS-calculated frequencies and absorption intensities for the (3)~A states of a number of longchain cumulenes are presented in Table 8. Because of the ambiguity concerning which of the excited ~A states is the most intense for H 2 C 3 , H 2 C 4 and H2C 5, and that these short-chain cumulenes are not expected to absorb above 300 nm, see Fig. 2, excited state vibrational frequencies have not been listed for them. No vibrational frequencies could be calculated for the cumulenes larger than H2C 5, other than those appearing in Table 8, either because of problems with state hopping, or computation limitations. In parallel with the ground state, the (3)~A~ state is characterized by one or two vibrations with enormous transition intensity. The responsible vibration is totally symmetric, and as for the ground state, corresponds to the in-phase combination of the CC bond stretches. There is no marked dependence on the electronic state, but some of the trends in the changes of the vibrational frequencies with carbon chain length are of interest. Thus the frequency of the a I mode associated with the largest infrared intensity is found to decrease with chain length in the ground state, while the reverse is true for the (3) ~Aj state. Simple arguments suggest that for the excited state, where a bonding electron has been promoted to an anti-bonding orbital, increased chain length leads to a reduction of the contribution of the anti-bonding orbital per CC bond, and hence to
H2C8
H2CI2 [0.00] [0.00] [0.06] [0.02] [0.98] [1.36] [6.97] [61.0] [0.16]
324 630 922 1189 1387 1461 1692 1813 1824
[0.01] [0.06] [0.05] [0.00] [1.31] [0.07] [2.45] [1.49] [8.09]
4180 [4.29]
1929 [7.79] 2064 [206] 2941 [0.04] 3582 [0.19]
43 [0.00] 118 [0.07] 213 [0.01] 282 [0.02] 447 [0.01] 460 [0.03] 622 [0.08] 895 [0.51]
25 [0.00] 72 [0.03] 133 [0.00] 197 [0.00] 211 [0.03] 250 [0.00] 365 [0.00] 387 [0.00] 452 [0.02] 461 [0.00] 897 [0.561
< 40 [0.00] [0.05] [0.001 [0.02] [0.00] [0.00] [0.00] [0.02]
24 [0.00] 68 [0.04] 124 [0.01] 184 [0.031 240 [0.01] 275 [0.03] 316 [0.00] 396 [0.00] 504 [0.01] 696 [0.22] 983 [0.00] 3016 [0.00]
43 121 203 265 316 392 987 2993
increased frequencies. The largest frequency bl mode (out-of-molecular plane), increases in frequency with chain length for both the ground and (3)~A 1 states for the even-numbered chains, and decreases for the odd-numbered chains. Furthermore, it is noted that one al vibration has suffered a large increase in frequency. The increase can be attributed to vibronic coupling with the near-lying, lower-energy (2)1A 1 state. Thus, for H 2 C 8 , H 2 C 9 and H2C12 the energy gaps between the two electronic states are 5418,
158
G. Fischer, J.P. Maier/ Chemical Physics 223 (1997) 149-158
2874 and 4149 cm -~, respectively, mirroring the magnitudes of the vibrational frequency increases.
3. Relation to astrophysical measurements Among the molecules considered to be responsible for the absorption features known as the DIB [15], are carbon chains. Though the proposal was made quite some time ago [16], only in the past few years have the strong characteristic electronic transitions of individual carbon chains been identified [2,3]. On the basis of these data, one could conclude which size the carbon chains must have so that the electronic transitions are in the 4 0 0 - 9 0 0 nm region [4], where the DIBs are observed, and that they are large enough to be photochemically stable in the stellar radiation field. For the homologous series considered, this is the case for the C2,,+ j (n = 8 - 1 6 ) range, whereby these may be limited to the midtwenties carbon atoms range because of known preference for ring structures for the larger sizes [17]. The aim of the present contribution is to predict this for the cumulenes. This is particularly relevant as a number of such cumulenes have been detected by radioastronomy in dense clouds and there is the need to understand their electronic transitions in connection with the DIB identification [1 ]. It is seen in Fig. 2 that H2C~3 should absorb at ~ 400 nm, and that the increment per additional carbon atom is ~ 22 nm. Thus, it is the cumulenes with sizes H z C I 3 to about H2C34 that would have their first strong electronic transition in the 4 0 0 - 8 6 0 nm DIB range. These species should be sufficiently large to be stable to photodissociation. The goal should now be to measure the electronic absorption spectra of a few such species in neon matrices using the established mass-selected approach, and ultimately a gas-phase measurement for direct comparison with astronomical observations. Furthermore, for such large cumulenes, ab initio predictions of ground state rotational constants that are to be used in radioastronomy will need to take into account the non-linearity of the carbon chains.
Acknowledgements GF wishes to thank the Institute for Physical Chemistry, University of Basel, for kind hospitality. This study is part of a project financed by the Swiss National Science Foundation (No. 20-49104.96). Some of the calculations were carried out using the Silicon Graphics Power Challenge of the ANU Supercomputer Facility. K. Bultitude is thanked for her help with the CASSCF calculations.
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