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Inelastic neutron scattering spectra of pagodane: experiment and DFT calculations
Chemical Physics Letters 386 (2004) 356–363 www.elsevier.com/locate/cplett
Inelastic neutron scattering spectra of pagodane: experiment and DFT calculations Damian G. Allis a, Horst Prinzbach b, Bruce S. Hudson a
a,*
Department of Chemistry, 1-014 Center for Science and Technology, Syracuse University, Syracuse, NY 13244-4100, USA b Chemisches Laboratorium der Universit€at Freiburg im Breisgau, Institut f€ur Organische Chemie und Biochemie, Albertstrasse 21, 0-7800 Freiburg, Germany Received 8 December 2003; in final form 21 January 2004
Published online:
Abstract The inelastic neutron scattering (INS) spectrum of [1.1.1.1]-pagodane is presented through 1600 cm1 . The INS spectrum is divisible into a region of well-resolved vibrational features (200–750 cm1 ) and a second region of overlapping transitions (750–1600 cm1 ) that contains the majority of all molecular modes. Comparison of the INS spectrum with a B3LYP/6-31G** calculation for the isolated molecule reveals notable differences in several low frequency modes while generally agreeing at higher frequency. Periodic density functional theory (DFT) calculations are employed to determine whether intermolecular interactions are the origin of these differences between the B3LYP/6-31G** and INS spectra. Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction Pagodane ([1.1.1.1]-pagodane) is the trivial name assigned to the D2h -symmetry undecacyclic polyquinane undecacyclo-[9.9.0.01;5 .02;12 .02;18 .03;7 .06;10 .08;12 .011;15 .013;17 .316;20 ]-eicosane (Fig. 1). The high symmetry and rigidity of the three-dimensional pagodane carbon framework is apparent in its crystal structure from the slight changes in molecular geometry that occur in the Ci -site group symmetry of its P 1 unit cell (Fig. 1) [1]. The high symmetry of the solution-phase pagodane framework is readily apparent from the simplicity of its 1 H and 13 C NMR spectra [2]. While unequivocally confirming the shape and symmetry of pagodane, the crystal structure and NMR studies provide the only reported characterization for this molecule. While initially envisioned as a structural precursor to its higher symmetry (Ih ) isomer dodecahedrane, efforts to convert pagodane through superacid chemistry yielded an interesting bishomoaromatic dication of significant theo-
*
Corresponding author. Fax: +3154434070. E-mail address: [email protected] (B.S. Hudson).
0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.01.076
retical interest and, owing to its stability, spectroscopic characterization [1]. The interest in the INS vibrational spectrum of pagodane stems in part from previous work on the solidstate INS spectrum of dodecahedrane [3]. The DFT (B3LYP/6-31G**) normal mode frequencies for the isolated dodecahedrane are found to differ from the INS spectrum of its 15 K crystal in a manner consistent with the breakdown of H and G vibrational mode degeneracies as the molecule deforms from Ih - to Th -symmetry in the cubic unit cell lattice. While the loss of degeneracy in these modes is expected from the packing of the icosahedral molecule into a cubic (Fm3m) space group, the size of the normal mode frequency splittings attributed to these 4- and 5-fold degenerate modes are, at up to 5 cm1 , large for the small observed changes in dodecahedrane geometry. Pagodane provides another test of a rigid hydrocarbon framework with a known crystal structure in which the molecular symmetry is reduced in the lowered (Ci ) lattice site symmetry. Pagodane is also of interest because it contains C–C bond lengths that span the range from 1.533 to 1.589 A. The restrictions of selection rules for Raman and IR absorptions as imposed by the dipole approximation are
D.G. Allis et al. / Chemical Physics Letters 386 (2004) 356–363
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Fig. 1. [1.1.1.1]-pagodane, showing the IUPAC numbering scheme (left) and the unit cell of pagodane (right), showing the side-on packing interaction (top) and top-down structure alignment (bottom). Figures were generated with VMD [18].
absent in INS spectroscopy, where lattice and molecular vibrations are observed in proportion to the contribution of the hydrogen-atom motions that constitute a particular normal mode. The accessibility of all vibrational normal modes in the INS spectrum is ideal for theoretical comparisons of the INS spectrum and solidstate DFT normal mode calculations. This accessibility is also useful when comparing the isolated molecule and solid-state normal mode calculations for the examination of crystal effects on the structure and vibrations of molecules. In the case of dodecahedrane, the INS spectrum made available 19 of 30 vibrational modes inaccessible by optical spectroscopic studies. In the D2h point group to which the isolated pagodane belongs, all but the (10 of 114) Au symmetry normal modes are either IR or Raman active. Accordingly, the INS spectrum will account for a small fraction of the normal modes inaccessible to optical studies while containing significant vibrational structure consistent in frequency with the combined IR and Raman spectra obtained from subsequent studies of the molecule and its crystal.
2. Methods The INS experiment was carried out at the ISIS facility of the Rutherford Appleton Laboratory using the time-focusing, crystal analyzer spectrometer TOSCA [4–8]. A polycrystalline sample of ca. 0.6 g of [1.1.1.1]pagodane was held at 15 K for this experiment. A more detailed description of neutron scattering in general, including results using this spectrometer, is given in [9]. The synthesis of pagodane is discussed in [2]. The pa-
godane crystal structure used for the solid-state DFT calculations was taken from [1] and is shown in Fig. 1. The crystallographic data for the room temperature unit cell is as follows: Space group P1, a ¼ 7:304ð5Þ A, c ¼ 6:336ð3Þ A, a ¼ 105:50ð4Þ°, b ¼ b ¼ 8:182ð4Þ A, 112:07ð5Þ°, c ¼ 64:97ð5Þ°, Z ¼ 1. An isolated molecule geometry optimization and normal mode vibrational analysis was performed with Gaussian 98 [10] at the DFT level of theory using the B3LYP generalized gradient approximation (GGA) density functional [11], 6-31G** basis set (B3LYP/ 6-31G**) [12], ÔtightÕ convergence criteria (density matrix RMS < 1.00D-08), and ÔultrafineÕ grid spacing (integration grid of 99 590). Both isolated molecule and zone-centered ðk ¼ 0Þ solid-state DFT calculations were performed using the program DM O L 3 [13,14] on the SGI Origin Array at the National Center for Supercomputing Applications (NCSA). Adequate reproduction of the pagodane INS vibrational spectrum was achieved in the pagodane solid-state calculation with the dn numerical basis set BLYP CGA density functional [15], and ÔfineÕ program option (corresponding to grid spacing (BLYP/ a k-point separation of 0.04 1/A) dn). The dn numerical basis set is reported to compare in accuracy with the 6-31G Gaussian basis set [13,14]. The BLYP density functional was selected after trial calculations on the isolated molecule gave results consistent with the B3LYP/6-31G** calculations performed with Gaussian 98 (the B3LYP density functional is not available in DM O L 3 ). All B3LYP/ 6-31G** vibrational modes were scaled by 0.987 to improve agreement with the INS results. BLYP/dn vibrational modes were left unscaled.
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C4–C19 methylene carbon atoms about the central fourmember ring.
3. Results and discussion 3.1. Molecular geometry
3.2. Molecular normal mode analysis The symmetry-unique framework bond lengths for the isolated molecule and unit cell calculations are provided with the Ci -symmetry crystallographic lengths in Table 1. All DFT framework bond lengths are longer than their corresponding crystallographic bonds. The B3LYP/6-31G** bond lengths differ from the crystal This rms averages with an rms deviation of 0.004 A. deviation in the isolated molecule BLYP/dn bond lengths In the Ci -symmetry unit cell structure, the rms is 0.025 A. from a uniform comdeviation decreases to 0.022 A pression of all framework bonds. A previous RHF/ 6-31G* calculation yielded consistently shorter frame [16]. work bond lengths with an rms deviation of 0.009 A A B3LYP/tzv* calculation [17] for isolated pagodane which is less than the yields an rms deviation of 0.0014 A estimated error of 0.002 A for the crystallographic determination reported in [1]. The corresponding value for the larger basis set DM O L 3 /dnp calculation is, at 0.013 A, large relative to the Gaussian basis set calculations but an improvement over the BLYP/dn isolated molecule calculations. The pagodane geometry obtained from the BLYP/dn unit cell optimization is only slightly deformed from its D2h structure to Ci -symmetry, with (D2h ) symmetry-related C–C bonds in the (Ci ) isolated molecule The only changes to the differing by less than 104 A. BLYP/dn structure in the solid-state calculation are a uniform compression of the framework bonds by ca. and a slight twisting of the C9–C14 and 0.0033 A
The 15 K INS and B3LYP/6-31G** simulated spectra of pagodane from 200 to 1600 cm1 are shown in Fig. 2. The phonon region of the INS spectrum (to 200 cm1 ) will be discussed in conjunction with the solid-state calculations (see below). The INS spectrum above 1600 cm1 is virtually featureless, consistent with the gap between the bending (to 1504.0 cm1 ) and stretching (from 3007.5 cm1 ) vibrational modes in the B3LYP/
Fig. 2. INS spectrum (black) and B3LYP/6-31G** simulated spectrum (gray) from 200 to 1600 cm1 . The B3LYP/6-31G** spectrum has been scaled for clarity.
Table 1 Framework bond lengths for the crystallographic (Ci ), isolated molecule (D2h ), and solid-state DFT (Ci ) pagodane geometries Bond
From [1]
Solid-state Ci
Isolated molecule D2h
Crystal
Crystal average
B3LYP/6-31G**
RHF 6-31G*a
BLYP/dn
BLYP/dn
C1–C2, C11–C12
1.573(2)
1.573
1.5752
1.556
1.6016
1.5996
C1–C5, C12–C13 C1–C20, C8–C12 C2–C3, C11–C15 C2–C18, C10–C11
1.532(2) 1.536(2) 1.530(2) 1.534(2)
1.533
1.5368
1.532
1.5527
1.5500
C1–C11, C2–C12
1.549(2)
1.549
1.5545
1.541
1.5741
1.5715
C3–C4, C14–C15 C4–C5, C13–C14 C8–C9, C19–C20 C9–C10, C18–C19
1.546(2) 1.540(2) 1.543(2) 1.547(2)
1.544
1.5466
1.539
1.5675
1.5630
C3–C7, C15–C16 C5–C6, C13–C17 C6–C10, C17–C18 C7–C8, C16–C20
1.561(2) 1.557(2) 1.557(2) 1.559(2)
1.559
1.5621
1.553
1.5825
1.5783
C6–C7, C16–C17
1.589(2)
1.589
1.5952
1.583
1.6170
1.6132
C9–C14, C4–C19
3.530(2)
3.530
3.5392
3.528
3.5659
3.5753
a
RHF/6-31G* values are taken from [16].
D.G. Allis et al. / Chemical Physics Letters 386 (2004) 356–363
6-31G** calculation. The limited resolution of the TOSCA spectrometer in the pagodane stretching region prohibits the identification or assignment of any spectral features that coincide with the DFT vibrational modes. The INS spectrum and isolated molecule B3LYP/ 6-31G** and BLYP/dn calculations are considered together prior to the solid-state DFT analysis in order to discuss their many similarities and to identify where discrepancies exist that require the solid-state DFT treatment to remedy. The INS spectrum derived from the B3LYP/tzv* calculation noted above is indistinguishable from the one derived from the B3LYP/ 6-31G** calculation. The bending modes of the pagodane INS spectrum are divided into two regions based on INS peak resolution. The region of the INS spectrum between 200 and 750 cm1 (Region I) is highlighted by clear separations between 6 groupings of 18 vibrational modes. It is within this smaller region that the most complete comparisons between the isolated molecule and unit cell DFT calculations and the INS results can be made. The vast majority of pagodane bending modes (76) lie between 750 and 1600 cm1 (Region II), the result of which is a complicated summation of smaller, individual INS intensities into many indistinguishable features and overlapping peaks. A series of four INS bands of some structure occur between 750 and 1600 cm1 that, by inspection, are duplicated in the simulated B3LYP/631G** spectrum, including the occurrence of normal
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modes with prominent INS intensities and the shoulders of these more prominent modes. The resolvable INS and D2h -symmetry (isolated molecule) BLYP/dn and B3LYP/6-31G** normal modes in Region I (200–750 cm1 ) are listed in Table 2. An expanded view of this region, showing the INS and calculated normal modes, is provided in Fig. 3. In a number of instances the normal mode frequencies differ between the B3LYP/6-31G** and BLYP/dn results by less than 10 cm1 . This result is noteworthy in light of the differences in their respective molecular geometries (Table 1). The INS and B3LYP/6-31G** spectra are in excellent agreement with one another in the three regions at ca. 500, 600 and 700 cm1 , reproducing both the peak spacings and relative intensities. The BLYP/dn calculation provides the correct overall shape and position of the 500 cm1 region while underestimating the frequencies of the normal mode groupings to higher frequency. The INS peak at 268.8 cm1 includes at least one shoulder to lower frequency. Both the B3LYP/ 6-31G** and BLYP/dn calculations predict two distinct normal modes in this region of the spectrum. The BLYP/dn calculation places one mode at 263.1 cm1 , in good agreement with the observed INS peak, and a second (18.5 cm1 ) lower frequency mode at 244.6 cm1 . These two modes in the B3LYP/6-31G** calculation occur at 259.9 and 246.0 cm1 , corresponding to a 15 cm1 separation. The atomic motions of these two modes are shown in Table 3. The lowest frequency DFT
Table 2 INS and DFT molecular vibrational modes for the 200–750 cm1 region of the pagodane INS spectrum INS
Solid-state, Ci
Isolated molecule, D2h B3LYP/6-31G**
BLYP/dn
Symmetry D2h
BLYP/dn (unit cell)
BLYP/dn (Ci )
Symmetry Ci (D2h )
268.8
246.0 259.9
244.6 263.1
Au B2u , IR
261.0 266.8
240.9 256.9
Au (Au ) Au (B2u )
336.4 353.6
338.5 343.8 355.7
336.3 348.6 361.2
B1u , IR Ag , Raman B3u , IR
334.8 354.4 366.4
330.8 340.3 346.4
Au (B1u ) Ag (Ag ) Au (B3u )
402.5 425.2 436.0
402.8 415.4 427.4
408.7 409.4 420.4
B1g , Raman B2g , Raman B1u , IR
412.0 415.5 428.7
396.9 407.8 418.8
Ag (B1g ) Ag (B2g ) Au (B1u )
486.5 508.9 516.5 527.0
488.9 505.6 515.9 524.8 524.9 530.9
489.8 501.9 512.9 515.1 520.5 530.3
B2g , Raman Au B1g , Raman B3g , Raman Ag , Raman B2u , IR
496.7 508.4 514.1 520.5 524.0 531.7
488.0 499.6 511.1 514.3 517.2 520.1
Ag Au Ag Ag Ag Au
602.9 609.0 627.5
600.5 607.7 626.0
588.7 599.7 619.0
Ag , Raman B3u , IR B3g , Raman
595.8 607.1 625.5
593.0 603.3 619.8
Ag (Ag ) Au (B3u ) Ag (B3g )
696.7
693.2
684.3
B1u , IR
689.3
684.1
Au (B1u )
(B2g ) (Au ) (B1g ) (B3g ) (Ag ) (B2u )
Optical activity is provided with the symmetry assignments for the B3LYP/6-31G** calculations. Groupings are provided based on the INS analysis. See text.
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D.G. Allis et al. / Chemical Physics Letters 386 (2004) 356–363 Table 3 B3LYP/6-31G** and isolated molecule BLYP/dn mode frequencies and atomic displacements for the first eight pagodane vibrations. Mass-weighted atomic displacements (motions along the normal mode coordinate) have been scaled by 200% for clarity. Correspondence of the B3LYP/6-31G** and BLYP/dn symmetry assignments was confirmed through eigenvector analysis
Ground State Pagodane Geometry B3LYP/6-31G**
246.0 cm-1 BLYP/dn
Fig. 3. INS spectrum (gray) and simulated BLYP/dn and B3LYP/ 6-31G** spectra (black) from 200 to 750 cm1 . Calculated spectra have been scaled for clarity.
vibrational modes are of Au symmetry in the D2h point group, making them both IR and Raman forbidden by optical studies. The only other optically forbidden mode in this region occurs as a resolved peak at 508.9 cm1 (INS) as part of a larger collection of modes with a maximum intensity at 527.0 cm1 (INS). The calculated vibrational mode frequencies in Table 2 include four more molecular modes than are resolved in the INS spectrum. The eight lowest frequency internal modes, which together account for the regions of greatest discrepancy between the molecular and solidstate results, are shown in Table 3. First among these are the B3LYP/6-31G** and BLYP/dn calculated peaks at 246.0 and 244.6 cm1 , respectively. While their occurrence in the INS spectrum is likely as the shoulder feature at lower frequency on the 268.8 cm1 INS peak, the required shift in frequency to account for this INS feature is significant at 15 cm1 (B3LYP/6-31G**) and 20 cm1 (BLYP/dn). The two sharp INS peaks at 336.4 and 353.6 cm1 are the result of three molecular vibrational modes. The relative intensities of the INS peaks would indicate the grouping to be 1(336.4 cm1 ):2(353.6 cm1 ). The relative spacing of the B3LYP/6-31G** normal modes is more consistent with a 2:1 grouping and the BLYP/dn normal modes are spaced enough to be easily resolved as three peaks at the same scale. The B3LYP/6-31G** modes at 400 cm1 agree in number but disagree in relative intensities with the INS results in this region. The BLYP/dn modes agree in intensity between the larger peak and either of the two smaller peaks by the summation of two closely-spaced modes, but disagree in number. This region will be examined in greater detail in the solid-state DFT analysis. Finally, the INS peaks at 516.5 and 527.0 cm1 combine to form one larger vibrational region including five molecular vibrational modes. A 1:4 grouping of these five modes is consistent with the spacing of the calculated peaks for
244.6 cm-1 B3LYP/6-31G**
259.9 cm-1 BLYP/dn
263.1 cm-1 B3LYP/6-31G**
338.5 cm-1 BLYP/dn
336.3 cm-1 B3LYP/6-31G**
343.8 cm-1 BLYP/dn
348.6 cm-1 B3LYP/6-31G**
355.7 cm-1 BLYP/dn
361.2 cm-1 B3LYP/6-31G**
402.8 cm-1 BLYP/dn
408.7 cm-1 B3LYP/6-31G**
415.4 cm-1 BLYP/dn
409.4 cm-1 B3LYP/6-31G**
427.4 cm-1 BLYP/dn
420.4 cm-1
the various DFT methods, but higher resolution is required for the accurate assignment of the group of four modes to specific INS peak features.
D.G. Allis et al. / Chemical Physics Letters 386 (2004) 356–363 Table 4 Selected INS peaks and B3LYP/6-31G** modes and assignments above 750 cm1 INS
B3LYP/6-31G**
Symmetry
Region 1 769.8 785.3 817.3 854.5
763.9 788.1 818.0 851.0
B3u B3u B1g B2g
Region 2 885.2 935.1 968.3 987.0 1017.8
882.2 931.4 967.8 981.2 1022.9
B1u B2u B1u B2u B3g
Region 3 1054.0 1091.5 1107.9 1141.58
1057.5 1093.7 1112.3 1141.0
B3g Au Ag B1u
Region 4 1188.1 1206.0 1242.6 1255.0 1274.0 1286.7 1319.2 1332.0
1184.4 1204.5 1250.8 1267.3 1277.0 1284.0 1307.0 1322.9
Au B1u B1g B1u B1u B2g B2u B2g
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INS spectrum. A selection of the most prominent INS peaks is correlated with their respective B3LYP/6-31G** peaks in Table 4. While the assignments are neither rigorously accurate nor complete, they provide a reasonable picture of this very complex region. Given the wealth of spectral features versus the limited resolution in this region of the INS spectrum, it is expected, in contrast to many previous INS experiments, that vibrational studies by optical methods will provide far more detail and certainty in assignments than is available with TOSCA. 3.3. Solid-state normal mode analysis
Region II of the INS spectrum (750–1600 cm1 ) is rich with 76 densely-packed INS and calculated vibrational modes. The overall shape of this region is reproduced remarkably well in both isolated molecule calculations, with strong calculated peaks in good agreement with maxima in the INS spectrum. This region of the INS spectrum is further divided into smaller sections in Fig. 2 for the purpose of highlighting those calculated groupings that correspond to features in the
The phonon region of the pagodane INS spectrum and the solid-state BLYP/dn simulated phonon spectrum are shown in Fig. 4. The phonon region between 25 and 90 cm1 is highlighted by one strong INS peak containing two narrow maxima centered at 66 cm1 . A broad peak is observed to higher frequency with a maximum intensity at 92.0 cm1 . With a single molecule per unit cell, the calculated phonon spectrum consists of three torsional modes at 54.3, 70.7 and 96.9 cm1 that split along the three nonequivalent crystal axes. The two lower frequency peaks fall within a broad region of INS intensity, with the 70.7 cm1 calculated peak closest in frequency to the strong INS absorption at 66 cm1 . The third calculated peak at 96.9 cm1 is clearly isolated from the remaining calculated peaks, as is the INS peak at 92.0 cm1 . The normal modes of the solid-state BLYP/dn calculations and the Ci -symmetry isolated molecule BLYP/ dn normal modes from the unit cell molecular geometry are provided in Table 2. Fig. 5 is provided to show the relative changes in positions and intensities of the BLYP/dn isolated molecule normal modes (BLYP/dn D2h ), the normal modes of the pagodane in the solid state (BLYP/dn unit cell), and the molecular normal
Fig. 4. INS (black) and BLYP/dn simulated (gray) phonon region.
Fig. 5. INS spectrum (gray) and BLYP/dn simulated spectra (black) from 200 to 750 cm1 . Calculated spectra are scaled for clarity.
Region divisions are shown in Fig. 2.
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modes of a single Ci -symmetry pagodane taken from the unit cell optimization and treated as an isolated molecule (BLYP/dn Ci ). These calculated normal modes share a number of features between each other, the B3LYP/6-31G** modes, and the INS spectrum. The vibrational features at 600 and 700 cm1 are nearly identical in all cases, although all three BLYP/dn calculations underestimate the positions of these features relative to the B3LYP/6-31G** peaks and the INS features. The structure of the normal mode grouping at 500 cm1 is largely consistent between all three BLYP/ dn calculations. The Ci -symmetry grouping of these modes is slightly modified by the 10 cm1 shift of the highest intensity (and highest frequency) peak to lower frequency and the resulting grouping of four of the normal modes in this region to a single broad feature. A clear improvement is made in the unit cell calculations to the positions of the molecular modes assigned to the 268.8 cm1 INS peak. The unit cell optimization reduces the relative splitting of the lowest frequency modes from 15 and 20 cm1 (in all of the isolated molecule calculations) to 6 cm1 and shifts this pair 7 cm1 to higher frequency, providing excellent agreement between the solid-state DFT normal mode pair and the experimental peak (with shoulder at lower frequency) at 268.8 cm1 . The Ci -symmetry isolated molecule normal mode pair assigned to the 268.8 cm1 INS peak is nearly identical to the D2h mode pair in both position and relative intensity, an indication that this shift in calculated frequencies towards correspondence with the INS peak is a result of solid-state features and not the change in symmetry of the pagodane in the unit cell. The motion of these modes, shown in Table 3, reveals that the largest shift occurs to the mode whose motion is a twisting of the methylene groups about the central four-member ring in the direction of the c-crystal axis shown in Fig. 1. While the Ci -site group symmetry in the pagodane crystal would make this lowest frequency (Au ) mode IR active, it would only be accessible by optical methods in the solid state, as this mode symmetry is both IR and Raman-forbidden in the D2h point group. The INS peak pair at 350 cm1 is composed of three molecular normal modes. The two highest frequency modes in this group are the in- (lower frequency) and out-of-phase (higher frequency) wagging motions of the methylene carbons (Table 3). The large hydrogen displacements that occur with the methylene motions in each of these modes, combined with the near-equivalence of their frequencies, accounts for the intensity of this single peak. The lowest-lying peak in this region corresponds to a sliding motion of the four-member ring and limited methylene motion (Table 3). The BLYP/dn normal mode analysis of the isolated Ci structure is the only calculation in which these two in- and out-of-phase methylene peaks are not separated by approximately
12 cm1 . In light of the otherwise close mode grouping and the shared molecular origin of this mode pair in the INS spectrum, it is assumed that the other DFT methods are slightly overestimating this mode pair difference relative to the INS spectrum. As a summation of in- (Ag ) and out-of-phase (B3u ) methylene motions, this single peak in the INS spectrum is both IR- and Ramanallowed. The origin of the differences between the various DFT methods and the experimental features of the 400 cm1 region of the INS spectrum cannot be resolved from the available data alone, but a case can be made that the error lies, for this specific region, in the differences between the room temperature and 15 K unit cell lattice constants. The B3LYP/6-31G** and BLYP/dn Ci -symmetry normal mode calculations produce three resolvable simulated peaks as observed in the INS spectrum. The relative intensities in the INS spectrum, however, are not reproduced by either method. The two lower frequency peaks in the BLYP/dn unit cell and isolated (D2h ) molecule calculations are spaced closely enough that their intensities sum to form one strong peak in the simulated spectrum consistent in position with the strong INS peak. This summation leaves only one peak to higher frequency and therefore conflicts with the INS results. The low-frequency mode is dominated by methylene motions perpendicular to the c-crystal axis (Table 3 and Fig. 1) and is always the lowest-frequency mode in this region by all DFT calculations. The two higher-frequency modes are dominated by methylene motions in the direction of the c-crystal axis (Table 3). Our ability to accurately reproduce the 400 cm1 region is presumed to be limited by the use of the room temperature unit cell lattice parameters in the DM O L 3 solid-state calculation to analyze the 15 K pagodane INS spectrum (DM O L 3 does not optimize lattice constants). The lowest-frequency molecular vibrational mode at 246 cm1 (B3LYP/6-31G**) corresponds to methylene motions in the direction of the c-crystal axis and is clearly altered in the unit cell calculation, where this peak shifts to higher frequency by almost 17 cm1 in the BLYP/dn calculations. Compression of the c-crystal axis upon cooling would be expected to affect motions in this direction, such as the two high frequency modes in the 400 cm1 region, the most, just as the presence of other pagodane molecules along this axis affects the position of the lowest-frequency vibrational mode in the BLYP/dn calculations. Thus a proposed compression along the c-axis is expected to provide frequencies in agreement with experiment. The relative intensities of the observed features, with the lowest mode being more intense than the two higher frequency c-axis modes, must derive from a shift in the normal mode eigenvectors for these higher modes away from methylene carbons to methane and quaternary carbons. This realignment of the normal mode eigenvectors may re-
D.G. Allis et al. / Chemical Physics Letters 386 (2004) 356–363
quire an enhanced basis set as indicated by examination of the C–C bond lengths. The B3LYP/6-31G** peaks at 402.8 and 415.4 cm1 are Raman-allowed in the D2h point group, while the 427.4 cm1 peak is IR-allowed. This ordering by symmetry is consistent in the three BLYP/dn calculations as determined by eigenvector analysis. Optical spectroscopic studies are expected to provide the required information to complete the assignment of peaks in this region.
Acknowledgements The Rutherford Appleton Laboratory is thanked for neutron beam access at the ISIS Facility where the TOSCA spectrometer was used. This work was supported by US National Science Foundation grant CHE 0240104 and by the US Department of Energy grant DE-FG02-01ER14245 and utilized the SGI Origin Array at the National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign. We thank Dr. Dale Braden of Schr€ odinger, Inc., Portland, Oregon for performing the B3LYP/tzv* calculation using the Schr€ odinger program Jaguar. References [1] G.K.S. Prakash, V.V. Krishnamurthy, R. Herges, R. Bau, H. Yuan, G.A. Olah, W.-D. Fessner, H. Prinzbach, J. Am. Chem. Soc. 110 (1988) 7764.
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[2] W.-D. Fessner, H. Prinzbach, Tetrahedron Lett. 24 (1983) 5857. [3] B.S. Hudson, D.A. Braden, S.F. Parker, H. Prinzbach, Angew. Chem. Int. Ed. 39 (2000) 514. [4] S.F. Parker, C.J. Carlile, T. Pike, J. Tomkinson, R.J. Newport, C. Andreani, F.P. Ricci, F. Sachetti, M. Zoppi, Physica B 241–243 (1998) 154. [5] Z.A. Bowden, M. Celli, F. Cilloco, D. Colognesi, R.J. Newport, S.F. Parker, F.P. Ricci, V. Rossi-Albertini, F. Sacchetti, J. Tomkinson, M. Zoppi, Physica B 276–278 (2000) 98. [6] M. Celli, F. Cilloco, D. Colognesi, R.J. Newport, S.F. Parker, V. Rossi-Albertini, F. Sacchetti, J. Tomkinson, M. Zoppi, Notiziario Neutrone e Luce di Synchrotrone 6 (2001) 33. [7] S.F. Parker, J. Neutron Res. 10 (2002) 173. [8] D. Colognesi, M. Celli, F. Cilloco, R.J. Newport, S.F. Parker, V. Rossi-Albertini, F. Sacchetti, J. Tomkinson, M. Zoppi, Appl. Phys. A 74 (Suppl.) (2002) S64. [9] B.S. Hudson, J. Phys. Chem. A 105 (2001) 3949. [10] M.J. Frisch et al., Gaussian 98 (Revision A.7), Gaussian, Inc., Pittsburgh PA, 2001. [11] The B3LYP functional is BeckeÕs three parameter hybrid method with LYP correlation functional, A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [12] O.C. Hariharan, J.A. Pople, Theoret. Chim. Acta 28 (1973) 213. [13] B. Delley, J. Chem. Phys. 113 (2003) 7756. [14] B. Delley, J. Chem. Phys. 92 (1982) 508. [15] The BLYP functional is BeckeÕs one parameter hybrid method with LYP correlation functional, A.D. Becke, J. Chem. Phys. 88 (1988) 2547; C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [16] J.M. Schulman, R.L. Disch, J. Mol. Struct. (Theochem.) 358 (1995) 51. [17] D.A. Braden, Schr€ odinger, Inc., Portland, Oregon, USA, private communication. [18] Graphics were generated with VMD. W. Humphrey, A. Dalke, K. Schulten, J. Mol. Graphics 14 (1996) 33.
Inelastic neutron scattering spectra of pagodane: experiment and DFT calculations Damian G. Allis a, Horst Prinzbach b, Bruce S. Hudson a
a,*
Department of Chemistry, 1-014 Center for Science and Technology, Syracuse University, Syracuse, NY 13244-4100, USA b Chemisches Laboratorium der Universit€at Freiburg im Breisgau, Institut f€ur Organische Chemie und Biochemie, Albertstrasse 21, 0-7800 Freiburg, Germany Received 8 December 2003; in final form 21 January 2004
Published online:
Abstract The inelastic neutron scattering (INS) spectrum of [1.1.1.1]-pagodane is presented through 1600 cm1 . The INS spectrum is divisible into a region of well-resolved vibrational features (200–750 cm1 ) and a second region of overlapping transitions (750–1600 cm1 ) that contains the majority of all molecular modes. Comparison of the INS spectrum with a B3LYP/6-31G** calculation for the isolated molecule reveals notable differences in several low frequency modes while generally agreeing at higher frequency. Periodic density functional theory (DFT) calculations are employed to determine whether intermolecular interactions are the origin of these differences between the B3LYP/6-31G** and INS spectra. Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction Pagodane ([1.1.1.1]-pagodane) is the trivial name assigned to the D2h -symmetry undecacyclic polyquinane undecacyclo-[9.9.0.01;5 .02;12 .02;18 .03;7 .06;10 .08;12 .011;15 .013;17 .316;20 ]-eicosane (Fig. 1). The high symmetry and rigidity of the three-dimensional pagodane carbon framework is apparent in its crystal structure from the slight changes in molecular geometry that occur in the Ci -site group symmetry of its P 1 unit cell (Fig. 1) [1]. The high symmetry of the solution-phase pagodane framework is readily apparent from the simplicity of its 1 H and 13 C NMR spectra [2]. While unequivocally confirming the shape and symmetry of pagodane, the crystal structure and NMR studies provide the only reported characterization for this molecule. While initially envisioned as a structural precursor to its higher symmetry (Ih ) isomer dodecahedrane, efforts to convert pagodane through superacid chemistry yielded an interesting bishomoaromatic dication of significant theo-
*
Corresponding author. Fax: +3154434070. E-mail address: [email protected] (B.S. Hudson).
0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.01.076
retical interest and, owing to its stability, spectroscopic characterization [1]. The interest in the INS vibrational spectrum of pagodane stems in part from previous work on the solidstate INS spectrum of dodecahedrane [3]. The DFT (B3LYP/6-31G**) normal mode frequencies for the isolated dodecahedrane are found to differ from the INS spectrum of its 15 K crystal in a manner consistent with the breakdown of H and G vibrational mode degeneracies as the molecule deforms from Ih - to Th -symmetry in the cubic unit cell lattice. While the loss of degeneracy in these modes is expected from the packing of the icosahedral molecule into a cubic (Fm3m) space group, the size of the normal mode frequency splittings attributed to these 4- and 5-fold degenerate modes are, at up to 5 cm1 , large for the small observed changes in dodecahedrane geometry. Pagodane provides another test of a rigid hydrocarbon framework with a known crystal structure in which the molecular symmetry is reduced in the lowered (Ci ) lattice site symmetry. Pagodane is also of interest because it contains C–C bond lengths that span the range from 1.533 to 1.589 A. The restrictions of selection rules for Raman and IR absorptions as imposed by the dipole approximation are
D.G. Allis et al. / Chemical Physics Letters 386 (2004) 356–363
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Fig. 1. [1.1.1.1]-pagodane, showing the IUPAC numbering scheme (left) and the unit cell of pagodane (right), showing the side-on packing interaction (top) and top-down structure alignment (bottom). Figures were generated with VMD [18].
absent in INS spectroscopy, where lattice and molecular vibrations are observed in proportion to the contribution of the hydrogen-atom motions that constitute a particular normal mode. The accessibility of all vibrational normal modes in the INS spectrum is ideal for theoretical comparisons of the INS spectrum and solidstate DFT normal mode calculations. This accessibility is also useful when comparing the isolated molecule and solid-state normal mode calculations for the examination of crystal effects on the structure and vibrations of molecules. In the case of dodecahedrane, the INS spectrum made available 19 of 30 vibrational modes inaccessible by optical spectroscopic studies. In the D2h point group to which the isolated pagodane belongs, all but the (10 of 114) Au symmetry normal modes are either IR or Raman active. Accordingly, the INS spectrum will account for a small fraction of the normal modes inaccessible to optical studies while containing significant vibrational structure consistent in frequency with the combined IR and Raman spectra obtained from subsequent studies of the molecule and its crystal.
2. Methods The INS experiment was carried out at the ISIS facility of the Rutherford Appleton Laboratory using the time-focusing, crystal analyzer spectrometer TOSCA [4–8]. A polycrystalline sample of ca. 0.6 g of [1.1.1.1]pagodane was held at 15 K for this experiment. A more detailed description of neutron scattering in general, including results using this spectrometer, is given in [9]. The synthesis of pagodane is discussed in [2]. The pa-
godane crystal structure used for the solid-state DFT calculations was taken from [1] and is shown in Fig. 1. The crystallographic data for the room temperature unit cell is as follows: Space group P1, a ¼ 7:304ð5Þ A, c ¼ 6:336ð3Þ A, a ¼ 105:50ð4Þ°, b ¼ b ¼ 8:182ð4Þ A, 112:07ð5Þ°, c ¼ 64:97ð5Þ°, Z ¼ 1. An isolated molecule geometry optimization and normal mode vibrational analysis was performed with Gaussian 98 [10] at the DFT level of theory using the B3LYP generalized gradient approximation (GGA) density functional [11], 6-31G** basis set (B3LYP/ 6-31G**) [12], ÔtightÕ convergence criteria (density matrix RMS < 1.00D-08), and ÔultrafineÕ grid spacing (integration grid of 99 590). Both isolated molecule and zone-centered ðk ¼ 0Þ solid-state DFT calculations were performed using the program DM O L 3 [13,14] on the SGI Origin Array at the National Center for Supercomputing Applications (NCSA). Adequate reproduction of the pagodane INS vibrational spectrum was achieved in the pagodane solid-state calculation with the dn numerical basis set BLYP CGA density functional [15], and ÔfineÕ program option (corresponding to grid spacing (BLYP/ a k-point separation of 0.04 1/A) dn). The dn numerical basis set is reported to compare in accuracy with the 6-31G Gaussian basis set [13,14]. The BLYP density functional was selected after trial calculations on the isolated molecule gave results consistent with the B3LYP/6-31G** calculations performed with Gaussian 98 (the B3LYP density functional is not available in DM O L 3 ). All B3LYP/ 6-31G** vibrational modes were scaled by 0.987 to improve agreement with the INS results. BLYP/dn vibrational modes were left unscaled.
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C4–C19 methylene carbon atoms about the central fourmember ring.
3. Results and discussion 3.1. Molecular geometry
3.2. Molecular normal mode analysis The symmetry-unique framework bond lengths for the isolated molecule and unit cell calculations are provided with the Ci -symmetry crystallographic lengths in Table 1. All DFT framework bond lengths are longer than their corresponding crystallographic bonds. The B3LYP/6-31G** bond lengths differ from the crystal This rms averages with an rms deviation of 0.004 A. deviation in the isolated molecule BLYP/dn bond lengths In the Ci -symmetry unit cell structure, the rms is 0.025 A. from a uniform comdeviation decreases to 0.022 A pression of all framework bonds. A previous RHF/ 6-31G* calculation yielded consistently shorter frame [16]. work bond lengths with an rms deviation of 0.009 A A B3LYP/tzv* calculation [17] for isolated pagodane which is less than the yields an rms deviation of 0.0014 A estimated error of 0.002 A for the crystallographic determination reported in [1]. The corresponding value for the larger basis set DM O L 3 /dnp calculation is, at 0.013 A, large relative to the Gaussian basis set calculations but an improvement over the BLYP/dn isolated molecule calculations. The pagodane geometry obtained from the BLYP/dn unit cell optimization is only slightly deformed from its D2h structure to Ci -symmetry, with (D2h ) symmetry-related C–C bonds in the (Ci ) isolated molecule The only changes to the differing by less than 104 A. BLYP/dn structure in the solid-state calculation are a uniform compression of the framework bonds by ca. and a slight twisting of the C9–C14 and 0.0033 A
The 15 K INS and B3LYP/6-31G** simulated spectra of pagodane from 200 to 1600 cm1 are shown in Fig. 2. The phonon region of the INS spectrum (to 200 cm1 ) will be discussed in conjunction with the solid-state calculations (see below). The INS spectrum above 1600 cm1 is virtually featureless, consistent with the gap between the bending (to 1504.0 cm1 ) and stretching (from 3007.5 cm1 ) vibrational modes in the B3LYP/
Fig. 2. INS spectrum (black) and B3LYP/6-31G** simulated spectrum (gray) from 200 to 1600 cm1 . The B3LYP/6-31G** spectrum has been scaled for clarity.
Table 1 Framework bond lengths for the crystallographic (Ci ), isolated molecule (D2h ), and solid-state DFT (Ci ) pagodane geometries Bond
From [1]
Solid-state Ci
Isolated molecule D2h
Crystal
Crystal average
B3LYP/6-31G**
RHF 6-31G*a
BLYP/dn
BLYP/dn
C1–C2, C11–C12
1.573(2)
1.573
1.5752
1.556
1.6016
1.5996
C1–C5, C12–C13 C1–C20, C8–C12 C2–C3, C11–C15 C2–C18, C10–C11
1.532(2) 1.536(2) 1.530(2) 1.534(2)
1.533
1.5368
1.532
1.5527
1.5500
C1–C11, C2–C12
1.549(2)
1.549
1.5545
1.541
1.5741
1.5715
C3–C4, C14–C15 C4–C5, C13–C14 C8–C9, C19–C20 C9–C10, C18–C19
1.546(2) 1.540(2) 1.543(2) 1.547(2)
1.544
1.5466
1.539
1.5675
1.5630
C3–C7, C15–C16 C5–C6, C13–C17 C6–C10, C17–C18 C7–C8, C16–C20
1.561(2) 1.557(2) 1.557(2) 1.559(2)
1.559
1.5621
1.553
1.5825
1.5783
C6–C7, C16–C17
1.589(2)
1.589
1.5952
1.583
1.6170
1.6132
C9–C14, C4–C19
3.530(2)
3.530
3.5392
3.528
3.5659
3.5753
a
RHF/6-31G* values are taken from [16].
D.G. Allis et al. / Chemical Physics Letters 386 (2004) 356–363
6-31G** calculation. The limited resolution of the TOSCA spectrometer in the pagodane stretching region prohibits the identification or assignment of any spectral features that coincide with the DFT vibrational modes. The INS spectrum and isolated molecule B3LYP/ 6-31G** and BLYP/dn calculations are considered together prior to the solid-state DFT analysis in order to discuss their many similarities and to identify where discrepancies exist that require the solid-state DFT treatment to remedy. The INS spectrum derived from the B3LYP/tzv* calculation noted above is indistinguishable from the one derived from the B3LYP/ 6-31G** calculation. The bending modes of the pagodane INS spectrum are divided into two regions based on INS peak resolution. The region of the INS spectrum between 200 and 750 cm1 (Region I) is highlighted by clear separations between 6 groupings of 18 vibrational modes. It is within this smaller region that the most complete comparisons between the isolated molecule and unit cell DFT calculations and the INS results can be made. The vast majority of pagodane bending modes (76) lie between 750 and 1600 cm1 (Region II), the result of which is a complicated summation of smaller, individual INS intensities into many indistinguishable features and overlapping peaks. A series of four INS bands of some structure occur between 750 and 1600 cm1 that, by inspection, are duplicated in the simulated B3LYP/631G** spectrum, including the occurrence of normal
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modes with prominent INS intensities and the shoulders of these more prominent modes. The resolvable INS and D2h -symmetry (isolated molecule) BLYP/dn and B3LYP/6-31G** normal modes in Region I (200–750 cm1 ) are listed in Table 2. An expanded view of this region, showing the INS and calculated normal modes, is provided in Fig. 3. In a number of instances the normal mode frequencies differ between the B3LYP/6-31G** and BLYP/dn results by less than 10 cm1 . This result is noteworthy in light of the differences in their respective molecular geometries (Table 1). The INS and B3LYP/6-31G** spectra are in excellent agreement with one another in the three regions at ca. 500, 600 and 700 cm1 , reproducing both the peak spacings and relative intensities. The BLYP/dn calculation provides the correct overall shape and position of the 500 cm1 region while underestimating the frequencies of the normal mode groupings to higher frequency. The INS peak at 268.8 cm1 includes at least one shoulder to lower frequency. Both the B3LYP/ 6-31G** and BLYP/dn calculations predict two distinct normal modes in this region of the spectrum. The BLYP/dn calculation places one mode at 263.1 cm1 , in good agreement with the observed INS peak, and a second (18.5 cm1 ) lower frequency mode at 244.6 cm1 . These two modes in the B3LYP/6-31G** calculation occur at 259.9 and 246.0 cm1 , corresponding to a 15 cm1 separation. The atomic motions of these two modes are shown in Table 3. The lowest frequency DFT
Table 2 INS and DFT molecular vibrational modes for the 200–750 cm1 region of the pagodane INS spectrum INS
Solid-state, Ci
Isolated molecule, D2h B3LYP/6-31G**
BLYP/dn
Symmetry D2h
BLYP/dn (unit cell)
BLYP/dn (Ci )
Symmetry Ci (D2h )
268.8
246.0 259.9
244.6 263.1
Au B2u , IR
261.0 266.8
240.9 256.9
Au (Au ) Au (B2u )
336.4 353.6
338.5 343.8 355.7
336.3 348.6 361.2
B1u , IR Ag , Raman B3u , IR
334.8 354.4 366.4
330.8 340.3 346.4
Au (B1u ) Ag (Ag ) Au (B3u )
402.5 425.2 436.0
402.8 415.4 427.4
408.7 409.4 420.4
B1g , Raman B2g , Raman B1u , IR
412.0 415.5 428.7
396.9 407.8 418.8
Ag (B1g ) Ag (B2g ) Au (B1u )
486.5 508.9 516.5 527.0
488.9 505.6 515.9 524.8 524.9 530.9
489.8 501.9 512.9 515.1 520.5 530.3
B2g , Raman Au B1g , Raman B3g , Raman Ag , Raman B2u , IR
496.7 508.4 514.1 520.5 524.0 531.7
488.0 499.6 511.1 514.3 517.2 520.1
Ag Au Ag Ag Ag Au
602.9 609.0 627.5
600.5 607.7 626.0
588.7 599.7 619.0
Ag , Raman B3u , IR B3g , Raman
595.8 607.1 625.5
593.0 603.3 619.8
Ag (Ag ) Au (B3u ) Ag (B3g )
696.7
693.2
684.3
B1u , IR
689.3
684.1
Au (B1u )
(B2g ) (Au ) (B1g ) (B3g ) (Ag ) (B2u )
Optical activity is provided with the symmetry assignments for the B3LYP/6-31G** calculations. Groupings are provided based on the INS analysis. See text.
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D.G. Allis et al. / Chemical Physics Letters 386 (2004) 356–363 Table 3 B3LYP/6-31G** and isolated molecule BLYP/dn mode frequencies and atomic displacements for the first eight pagodane vibrations. Mass-weighted atomic displacements (motions along the normal mode coordinate) have been scaled by 200% for clarity. Correspondence of the B3LYP/6-31G** and BLYP/dn symmetry assignments was confirmed through eigenvector analysis
Ground State Pagodane Geometry B3LYP/6-31G**
246.0 cm-1 BLYP/dn
Fig. 3. INS spectrum (gray) and simulated BLYP/dn and B3LYP/ 6-31G** spectra (black) from 200 to 750 cm1 . Calculated spectra have been scaled for clarity.
vibrational modes are of Au symmetry in the D2h point group, making them both IR and Raman forbidden by optical studies. The only other optically forbidden mode in this region occurs as a resolved peak at 508.9 cm1 (INS) as part of a larger collection of modes with a maximum intensity at 527.0 cm1 (INS). The calculated vibrational mode frequencies in Table 2 include four more molecular modes than are resolved in the INS spectrum. The eight lowest frequency internal modes, which together account for the regions of greatest discrepancy between the molecular and solidstate results, are shown in Table 3. First among these are the B3LYP/6-31G** and BLYP/dn calculated peaks at 246.0 and 244.6 cm1 , respectively. While their occurrence in the INS spectrum is likely as the shoulder feature at lower frequency on the 268.8 cm1 INS peak, the required shift in frequency to account for this INS feature is significant at 15 cm1 (B3LYP/6-31G**) and 20 cm1 (BLYP/dn). The two sharp INS peaks at 336.4 and 353.6 cm1 are the result of three molecular vibrational modes. The relative intensities of the INS peaks would indicate the grouping to be 1(336.4 cm1 ):2(353.6 cm1 ). The relative spacing of the B3LYP/6-31G** normal modes is more consistent with a 2:1 grouping and the BLYP/dn normal modes are spaced enough to be easily resolved as three peaks at the same scale. The B3LYP/6-31G** modes at 400 cm1 agree in number but disagree in relative intensities with the INS results in this region. The BLYP/dn modes agree in intensity between the larger peak and either of the two smaller peaks by the summation of two closely-spaced modes, but disagree in number. This region will be examined in greater detail in the solid-state DFT analysis. Finally, the INS peaks at 516.5 and 527.0 cm1 combine to form one larger vibrational region including five molecular vibrational modes. A 1:4 grouping of these five modes is consistent with the spacing of the calculated peaks for
244.6 cm-1 B3LYP/6-31G**
259.9 cm-1 BLYP/dn
263.1 cm-1 B3LYP/6-31G**
338.5 cm-1 BLYP/dn
336.3 cm-1 B3LYP/6-31G**
343.8 cm-1 BLYP/dn
348.6 cm-1 B3LYP/6-31G**
355.7 cm-1 BLYP/dn
361.2 cm-1 B3LYP/6-31G**
402.8 cm-1 BLYP/dn
408.7 cm-1 B3LYP/6-31G**
415.4 cm-1 BLYP/dn
409.4 cm-1 B3LYP/6-31G**
427.4 cm-1 BLYP/dn
420.4 cm-1
the various DFT methods, but higher resolution is required for the accurate assignment of the group of four modes to specific INS peak features.
D.G. Allis et al. / Chemical Physics Letters 386 (2004) 356–363 Table 4 Selected INS peaks and B3LYP/6-31G** modes and assignments above 750 cm1 INS
B3LYP/6-31G**
Symmetry
Region 1 769.8 785.3 817.3 854.5
763.9 788.1 818.0 851.0
B3u B3u B1g B2g
Region 2 885.2 935.1 968.3 987.0 1017.8
882.2 931.4 967.8 981.2 1022.9
B1u B2u B1u B2u B3g
Region 3 1054.0 1091.5 1107.9 1141.58
1057.5 1093.7 1112.3 1141.0
B3g Au Ag B1u
Region 4 1188.1 1206.0 1242.6 1255.0 1274.0 1286.7 1319.2 1332.0
1184.4 1204.5 1250.8 1267.3 1277.0 1284.0 1307.0 1322.9
Au B1u B1g B1u B1u B2g B2u B2g
361
INS spectrum. A selection of the most prominent INS peaks is correlated with their respective B3LYP/6-31G** peaks in Table 4. While the assignments are neither rigorously accurate nor complete, they provide a reasonable picture of this very complex region. Given the wealth of spectral features versus the limited resolution in this region of the INS spectrum, it is expected, in contrast to many previous INS experiments, that vibrational studies by optical methods will provide far more detail and certainty in assignments than is available with TOSCA. 3.3. Solid-state normal mode analysis
Region II of the INS spectrum (750–1600 cm1 ) is rich with 76 densely-packed INS and calculated vibrational modes. The overall shape of this region is reproduced remarkably well in both isolated molecule calculations, with strong calculated peaks in good agreement with maxima in the INS spectrum. This region of the INS spectrum is further divided into smaller sections in Fig. 2 for the purpose of highlighting those calculated groupings that correspond to features in the
The phonon region of the pagodane INS spectrum and the solid-state BLYP/dn simulated phonon spectrum are shown in Fig. 4. The phonon region between 25 and 90 cm1 is highlighted by one strong INS peak containing two narrow maxima centered at 66 cm1 . A broad peak is observed to higher frequency with a maximum intensity at 92.0 cm1 . With a single molecule per unit cell, the calculated phonon spectrum consists of three torsional modes at 54.3, 70.7 and 96.9 cm1 that split along the three nonequivalent crystal axes. The two lower frequency peaks fall within a broad region of INS intensity, with the 70.7 cm1 calculated peak closest in frequency to the strong INS absorption at 66 cm1 . The third calculated peak at 96.9 cm1 is clearly isolated from the remaining calculated peaks, as is the INS peak at 92.0 cm1 . The normal modes of the solid-state BLYP/dn calculations and the Ci -symmetry isolated molecule BLYP/ dn normal modes from the unit cell molecular geometry are provided in Table 2. Fig. 5 is provided to show the relative changes in positions and intensities of the BLYP/dn isolated molecule normal modes (BLYP/dn D2h ), the normal modes of the pagodane in the solid state (BLYP/dn unit cell), and the molecular normal
Fig. 4. INS (black) and BLYP/dn simulated (gray) phonon region.
Fig. 5. INS spectrum (gray) and BLYP/dn simulated spectra (black) from 200 to 750 cm1 . Calculated spectra are scaled for clarity.
Region divisions are shown in Fig. 2.
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modes of a single Ci -symmetry pagodane taken from the unit cell optimization and treated as an isolated molecule (BLYP/dn Ci ). These calculated normal modes share a number of features between each other, the B3LYP/6-31G** modes, and the INS spectrum. The vibrational features at 600 and 700 cm1 are nearly identical in all cases, although all three BLYP/dn calculations underestimate the positions of these features relative to the B3LYP/6-31G** peaks and the INS features. The structure of the normal mode grouping at 500 cm1 is largely consistent between all three BLYP/ dn calculations. The Ci -symmetry grouping of these modes is slightly modified by the 10 cm1 shift of the highest intensity (and highest frequency) peak to lower frequency and the resulting grouping of four of the normal modes in this region to a single broad feature. A clear improvement is made in the unit cell calculations to the positions of the molecular modes assigned to the 268.8 cm1 INS peak. The unit cell optimization reduces the relative splitting of the lowest frequency modes from 15 and 20 cm1 (in all of the isolated molecule calculations) to 6 cm1 and shifts this pair 7 cm1 to higher frequency, providing excellent agreement between the solid-state DFT normal mode pair and the experimental peak (with shoulder at lower frequency) at 268.8 cm1 . The Ci -symmetry isolated molecule normal mode pair assigned to the 268.8 cm1 INS peak is nearly identical to the D2h mode pair in both position and relative intensity, an indication that this shift in calculated frequencies towards correspondence with the INS peak is a result of solid-state features and not the change in symmetry of the pagodane in the unit cell. The motion of these modes, shown in Table 3, reveals that the largest shift occurs to the mode whose motion is a twisting of the methylene groups about the central four-member ring in the direction of the c-crystal axis shown in Fig. 1. While the Ci -site group symmetry in the pagodane crystal would make this lowest frequency (Au ) mode IR active, it would only be accessible by optical methods in the solid state, as this mode symmetry is both IR and Raman-forbidden in the D2h point group. The INS peak pair at 350 cm1 is composed of three molecular normal modes. The two highest frequency modes in this group are the in- (lower frequency) and out-of-phase (higher frequency) wagging motions of the methylene carbons (Table 3). The large hydrogen displacements that occur with the methylene motions in each of these modes, combined with the near-equivalence of their frequencies, accounts for the intensity of this single peak. The lowest-lying peak in this region corresponds to a sliding motion of the four-member ring and limited methylene motion (Table 3). The BLYP/dn normal mode analysis of the isolated Ci structure is the only calculation in which these two in- and out-of-phase methylene peaks are not separated by approximately
12 cm1 . In light of the otherwise close mode grouping and the shared molecular origin of this mode pair in the INS spectrum, it is assumed that the other DFT methods are slightly overestimating this mode pair difference relative to the INS spectrum. As a summation of in- (Ag ) and out-of-phase (B3u ) methylene motions, this single peak in the INS spectrum is both IR- and Ramanallowed. The origin of the differences between the various DFT methods and the experimental features of the 400 cm1 region of the INS spectrum cannot be resolved from the available data alone, but a case can be made that the error lies, for this specific region, in the differences between the room temperature and 15 K unit cell lattice constants. The B3LYP/6-31G** and BLYP/dn Ci -symmetry normal mode calculations produce three resolvable simulated peaks as observed in the INS spectrum. The relative intensities in the INS spectrum, however, are not reproduced by either method. The two lower frequency peaks in the BLYP/dn unit cell and isolated (D2h ) molecule calculations are spaced closely enough that their intensities sum to form one strong peak in the simulated spectrum consistent in position with the strong INS peak. This summation leaves only one peak to higher frequency and therefore conflicts with the INS results. The low-frequency mode is dominated by methylene motions perpendicular to the c-crystal axis (Table 3 and Fig. 1) and is always the lowest-frequency mode in this region by all DFT calculations. The two higher-frequency modes are dominated by methylene motions in the direction of the c-crystal axis (Table 3). Our ability to accurately reproduce the 400 cm1 region is presumed to be limited by the use of the room temperature unit cell lattice parameters in the DM O L 3 solid-state calculation to analyze the 15 K pagodane INS spectrum (DM O L 3 does not optimize lattice constants). The lowest-frequency molecular vibrational mode at 246 cm1 (B3LYP/6-31G**) corresponds to methylene motions in the direction of the c-crystal axis and is clearly altered in the unit cell calculation, where this peak shifts to higher frequency by almost 17 cm1 in the BLYP/dn calculations. Compression of the c-crystal axis upon cooling would be expected to affect motions in this direction, such as the two high frequency modes in the 400 cm1 region, the most, just as the presence of other pagodane molecules along this axis affects the position of the lowest-frequency vibrational mode in the BLYP/dn calculations. Thus a proposed compression along the c-axis is expected to provide frequencies in agreement with experiment. The relative intensities of the observed features, with the lowest mode being more intense than the two higher frequency c-axis modes, must derive from a shift in the normal mode eigenvectors for these higher modes away from methylene carbons to methane and quaternary carbons. This realignment of the normal mode eigenvectors may re-
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quire an enhanced basis set as indicated by examination of the C–C bond lengths. The B3LYP/6-31G** peaks at 402.8 and 415.4 cm1 are Raman-allowed in the D2h point group, while the 427.4 cm1 peak is IR-allowed. This ordering by symmetry is consistent in the three BLYP/dn calculations as determined by eigenvector analysis. Optical spectroscopic studies are expected to provide the required information to complete the assignment of peaks in this region.
Acknowledgements The Rutherford Appleton Laboratory is thanked for neutron beam access at the ISIS Facility where the TOSCA spectrometer was used. This work was supported by US National Science Foundation grant CHE 0240104 and by the US Department of Energy grant DE-FG02-01ER14245 and utilized the SGI Origin Array at the National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign. We thank Dr. Dale Braden of Schr€ odinger, Inc., Portland, Oregon for performing the B3LYP/tzv* calculation using the Schr€ odinger program Jaguar. References [1] G.K.S. Prakash, V.V. Krishnamurthy, R. Herges, R. Bau, H. Yuan, G.A. Olah, W.-D. Fessner, H. Prinzbach, J. Am. Chem. Soc. 110 (1988) 7764.
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