Theoretical study of the structures and racemization barriers of [n]helicenes (n = 3–6, 8)

Chemical Physics ELSEVIER

Chemical Physics 204 (1996) 41 t-417

Theoretical study of the structures and racemization barriers of [n] helicenes (n = 3-6, 8) * S. Grimme, S.D. Peyerimhoff Institutfir Physikalische und Theoretische Chemie, Universitiit Bonn. Wegelerstrasse 12, D-531 15 Bonn. Germany Received 20 December

1994

Abstract The minima and saddlepoint structures of [ nlhelicenes (n = 3-6, 8) have been optimized by semiempirical AM1 and ab initio SCF methods. The racemization barriers have been calculated as the energy difference between the ground state

minima possessing CZ symmetry and a twisted transition state having a C, structure. A qualitatively correct dependence of the barrier heights with n in comparison with experimental data is obtained with both methods showing the reliability of the assumed reaction path. The relative large overestimations in the ab initio SCF barriers (lo-14 kcal/mol for n > 5) are attributed to neglect of correlation energy which is larger in the more strained transition state than in the ground state structures. Single-point density-functional calculations yield barrier heights for the racemization of all helicenes accurate to + 1 kcal/mol.

1. Introduction Helicenes are benzologues of phenantbrene ( [ 31 helicene, 3H) in which a regular cylindrical helix is formed through an all-ortho annelation of the aromatic rings (see Scheme 1) . The helical structure is a consequence of the repulsive steric interactions between terminal aromatic rings. Since helicenes posses a number of unusual and interesting physical properties (large optical rotations, spontanous formation of enantiomorphic crystals, shielding effects in NMR spectroscopy, racemization barriers) excellent reviews on the substances exist [ l-31. There has been a continued interest in relationships between the structure and physical and chemical properties of strained or sterically hindered aromatic hydrocarbons [ 451. The helicenes

*Dedicated to Professor Bernard Pullman 75th birthday.

on occasion

of his

030 I-O 104/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved SSD10301-0104(95)00275-8

Dimethyl[J]helicene

[n]Helicene

(n = 4-6, 8)

Scheme 1

represent a special class of such compounds due to their obvious structural relationship with biological macromolecules (DNWDNA helix) and macroscopic mechanic springs. In this paper we want to focus on two basic aspects of helicenes: the structure of the molecules in the ground state and the pathway for interconversion of the two enantiomers (racemization) . Perhaps the most intriguing observation is the unexpected ease

412

S. Grimme, S.D. Peyerimhoff/Chemical

with which the helicenes racemize thermically. A direct inversion process has been proposed (instead of bond-breaking or Diels-Alder processes) and was confirmed by force-field calculations of Lindner [ 61 and Armstrong et al. [ 71 (for 4,5-dimethyl3H, DM3H). DM3H has also been the subject of semiempirical configuration interaction studies to explore its circular dichroism spectrum [ 81 and the red-shift in the UV spectrum induced by twisting of the phenanthrene chromophore [ 8,9]. However, to our knowledge no detailed study on these subjects with modern molecular orbital (MO) methods like semiempirical NDDO (AM1 parameterization [ lo] ) or ab initio Hat-tree-Fock self-consistentfield (SCF) approaches exists for a wide range of helicenes. It is therefore the goal of this paper to study the structures and racemization energies of a number of helicenes and elucidate their energetic and structural properties in dependency of the number of aromatic units.

2. Theoretical

methods

All semiempirical AM1 calculations were performed with a modified version of the MOPAC 6.0 program system [ 111. Ab initio SCF and density functional calculations were performed with the TURBOMOLE program package [ 12,131. The Gaussian basis sets are of split-valence quality (SV, [ 3s2p] / [ 2s] for C and H) [ 141. This basis is subsequently augmented with polarization d-functions at the carbon atoms (&!d = 0.8, SV+d basis) and polarization p-functions at the hydrogen atoms ( ap = 0.8, SV+d+p basis). Density-functional theory (DFT) calculations were performed with the SV+d+p basis set and the Becke/Lee-Yang-Parr (BLYP) [ 15,161 exchange-correlation functional. The helicene minima/saddlepoints on the potential energy surface were optimized with the constrain of CJC, symmetry. A normale coordinate analysis performed at the AM1 level for all helicenes results in one imaginary vibrational frequency (displacement along this normal coordinate gives a distorted Ct structure with resemblance to the corresponding minimum, see Fig. 2) in the case of the C, structures showing that they are genuine transition states for the racemization reaction. For a comparison of experimental and theorectical

Physics 204 (1996) 411-417

a)

4

d)

d

Fig. I. Optimized minimum (left, Cz) and transition state structures (right, C,) of the [n] helicenes. (a) 4,5-dimethyl[ 31helicene (SCFLW). (b) [4]helicene (SCF/SV). (c) [5]helicene (SCF/SV). (d) [6]helicene (AMI). (e) [SJhelicene (AMI).

racemization barriers we assume that the activation enthalpies (Ai?Zt(exp)) can be approximated by the energy difference of the corresponding stationary points, i.e AHS (exp) x AE = E(C,) - E(C2) since zero-point and temperature-dependent vibrational contributions to AE have been estimated at the Ah41 level to be less than 1 kcal/mol.

3. Geometries The calculated local minima on the helicene potential energy surfaces of all molecules investigated in the present work are found to correspond to structures with C2 symmetry which is confirmed by Xray analysis for the compounds with n = 3, 5 and 6 [ 7,17,183. Previous force-field calculations and an ab initio study of 4H [ 191 also found these structures as

S. Grimme, S.D. Peyerimhoff/Chemical Physics 204 (I 996) 41 I-41 7

413

Table 1 Selected geometrical data of the Ca minima of the [n] helicenes. The C-C-C-C dihedral angles (in degrees) correspond to the atoms of the inner helix; the first angle refers to the first four atoms beginning with C( 1) and following values refer to angles where the the first atom is replaced by the second one. The shortest nonbonding distances am given in A. Values in parentheses refer to experimental X-ray data Compound

C-C-C-C

angles

Shortest nonbonding

distancesa

AM1

SCFI sv

AM1

SCFISV

DM3H h

31.1

33.1 (31.5)

4H

19.9

i8.8

5HC

18.0 28.7 14.0 28.5 15.4 27.2 27.7

18.0 30.0 14.6 (13.3) 28.6 (30.2)

2.89 2.28 2.91 2.01 2.82 2.59 3.10 2.39 3.34 2.53

2.99 (2.98) 2.40 3.06 1.98 2.96 2.64 3.20 (3.22) 2.51 (2.63)

6H 8H

il The conventinal atomic numbering scheme of helicenes (see Ref. [ 21) is used. h X-ray data from Ref. [ 71. c The errors of the X-ray data ( [ 171, reported in 1954) are too large to allow comparison minima. The quality of our geometrical data is shown in an exemplary manner for the C-C-C-C dihedral angles of the inner helix and for a selected number of nonbonded distances between the aromatic units (Table 1) . The C-C bond lengths do not show any extraordinary behaviour and fall for the inner bonds in the range 1.44 f 0.02 A while the outer bonds alternate between 1.36 and 1.43 8, (AM1 data of 8H). At the ab initio level the calculated bond alternation is slightly larger and the inner bonds become longer by 0.01-0.02 A. Compared to the X-ray data of 6H the AM1 bond lengths seem to be of slightly better quality than the ab initio data. Due to the size and the structure of the molecules the dihedral angles and nonbonded distances are much more sensitive to determine the quality of the theoretical geometries. Here we notice a very good agreement between AM1 and ab initio structures. The pitch of the helicenes is usually described within a helical model for the inner, middle and outer carbon atoms (triple helical model). The pitch p of the helix (in A/degree) is obtained by fitting the Cartesian coordinates of the corresponding atoms to a regular screw-t e curve of the form z = p(p with (p = arcsin(y/ + x + y ). The standard deviation u of this linear least-squares fit gives a measure of which the atoms deviate from a regular helix (i.e. 9 being independent of z ) . The data obtained at the

C(methyl)...C(methyl) H(methyl)...H(methyl) C( l)...C( 12) H( l)...H( 12) C( l)...C( 14) H( l)...H( 14) C(l)...C(16) H( l)...C( 19) C( l)...C(22) H( l)...C(23)

with the theoretical

values.

AM1 and SCF/SV levels corresponding to the n + 1 inner atoms are given in Table 2 for 4H, 6H and 8H. The standard deviation for the fit is small (< 0.04 A) which shows that the molecules form nearly regular helices. The largest deviations are seen for the atoms near the CZ axes. As expected the pitch increases strongly from n = 4 to n = 6 while there is only little change in going to n = 8. The mean radius decreases in the same direction since the aromatic units avoid short distances to each other by the twisting motion which allows relaxation of the bending angle deformation. Compared to the SCF/SV data the pitch is smaller at the AM1 level (smaller dihedral angles) which results in shorter nonbonded distances (see also Table 1) . The only exception is 4H which is predicted to be more helical distorted at the Ah41 level. This comes from the fact that the dominant steric interaction is a very short H( l)...H( 12) distance which is known to be too repulsive in the AM1 Hamiltonian [ lo]. However, both methods predict all geometrical data to be within the error limits of the X-ray analysis (i.e. the deviation of the experimental structures from C2 symmetry due to packing and disordering effects) . The transition state (TS) structures for the helicene racemization are also shown in Fig. 1. The TS structures found for DM3H and 4H are nearly planar while all other geometries have C, symmetry with

S. Grimme, S.D. Peyerimhoff/Chemical Physics 204 (1996) 411-417

414 Table 2 Helix parameters

according

Compound/method

to the triple helical model. The data given correspond

to the helix formed by the n + 1 inner atoms

Pitch ( x 1O-2 A/deg)

Height

Mean radius

Standard

(A)

(A)

(A)

4H/AMI

0.54

1.40

SCFISV 6H/AMl SCFISV X-ray a RH/AM

0.50 0.88 0.90 0.9 I 0.95

1.29

1.38 1.42 1.34 1.37 1.33 1.24

0.03 0.03 0.02 0.02 0.03 0.04

I

’ Data from Ref.

3.09 3.14 3.16 4.70

deviation

[ 181.

Fig. 2. Transition state structure for the racemization of [ 51 helicene (AM I ) The arrows show the direction of atomic displacements along the imaginary vibrational mode (vi = 62 cm-‘).

face-to-face orientated terminal aromatic rings. In the course of the racemization of DM3H the rotation of the methyl groups play an important role; the transition state shown in Fig. 1 with the methyl groups faceto-face is a true stationary point with one imaginary frequency. In contrast to results of the force-field study by Lindner [ 61 we have also found a nonplanar (carbon skeleton) TS for 5H at both levels of theory. Inspection of the TS geometries reveals the existence of two types of ‘strain’ components. The dihedral angles describing the twisting of the benzene units towards each other are increased by M 20 degrees relative to the C:! minima. Furthermore, the nonbonded distances between the terminal rings are decreased. Shortest H...H contacts of 1.84, 1.53, 1.63, 1.81 and 2.30 A are found in DM3H, 4H, 5H, 6H and 8H, respectively (AM1 level). The C...C distances are not significantly shortened relative to those of the C2 minima. The directions of atomic displacements along the imaginary mode are displaced for 5H in Fig. 2. This plot is typical for all helicenes with a nonplanar transition state. At the ini-

tial stage of the reaction starting from the TS the right terminal ring moves up while the other moves slightly in the opposite direction. Additionally, the left terminal ring shows a stretching component. This picture is only qualitative since the harmonic approximation for such a soft mode (vi = 62 cm-t ) is only valid for very small displacements. However, this movement clearly leads to the C:! minimum structure so that a reaction path with intermediate Ct symmetry 5H(C2, left-handed) -+ 5H(C,) + MSH(C;?, right-handed) seem to represent the chemical reality. For the larger helicenes with n > 8 we expect that the C, structure is also a stationary point on the hypersurface. However, the movement of the face-to-face orientated terminal rings in these systems as described above seems not to be a low-energy pathway for racemization. The imaginary frequencies of the TS modes are 111, 243, 62, 24 and 7 cm-’ for DM3H, 4H, 5H, 6H and 8H, respectively. Beginning with n = 4 we notice an inverse proportionality between the wavenumber of the TS mode and n (the best nonlinar least-squares fit is ~i = 70 cm-’ /(n - 3.7) ) . The increasing softness of this mode with n clearly demonstrates the increasing flexibility of the larger helicenes. This is mainly attributed to the numerous degrees of freedom in the larger compounds which is also responsible for the almost constant racemization barriers for n > 6 (see below).

4. Racemization barriers The results of the computations for the racemization barriers are given in Table 3 and are graphically shown as a function of n in Fig. 3. All data show the

S. Grimme,

Table 3 Comparison

of calculated

and experimental

SD.

Peyerimhoff/Chemical

activation

enthalpies

Physics 204 (1996)

AH t a for the racemization

411-417

415

process of the helicenes

Method

4,5-DM3H

4H

5H

6H

8H

AM1 h ab initio SCF (SP) ’ ab initio SCF (OPT) d DFI-BLYP (SP) ’ exp.

16.7 23.5 17.6 14.0

7.6 4.2 4.0 3.5

23.9 32.0 29.0 22.7 22.8 s

31.4 48.6 45.3 35.2 34.9 h

34.9 55.2 41.5 40.9

i

d We assume AHI -A_@

=E(C) -::::. 2 h AMI Heat of Formations of theSCa minima are 56.3, 81.2, 103.5, 126.8 and 175.9 kcallmol. c Single point calculation with the AM1 optimized geometries. Total energies of the Ca minima are -613.33196, -687.85496, -840.31999, -902.78821 and -1297.81873 au. d Fully optimized geometries. Total energies of the C2 minima are -613.34127, -687.86021, -840.32707 and -992.79652 au. ’ Smgle point calculation with the SCF/SV optimized geometries. Total energies (SV+d+p basis) of the CT minima are -617.39715, -692.36099, -845.81943, -999.27973 and -1306.19719 au. ’ Ref. 171. s Ref. [201. h Ref. [23]. i Ref. [24].

60 ;

50

E f ii . 7. 0

40

WT ^. 0

20

z

10

--e-

experiment AM1

-

SCF (OPT) SCF (SP) scaled

d-

SCF (SP)

30 d

0

I

3

I

4

5

6

7

8

[nlhel icene Fig. 3. Dependence of the barrier height AE on the number of aromatic units (n) with different theoretical treatments. The scaling procedure (dashed line with open triangles) is described in the text.

same trend, i.e. a strong increase in the barrier heights A E in going from n = 4 to n = 5,6 (the value of unsubstituted 3H is zero; the minimum structure of this molecule is planar, &). It is clearly seen that for n > 6 a plateau is reached where the AE values remain nearly constant. As was noted earlier by Lindner [ 61, this behaviour is surprising from the older common viewpoint of chemists who assume a strong tendency of aromatic hydrocarbons to form planar structures. If such assumptions were true the increasing number of aromatic units should raise the barriers monotonically.

Generally, the dependence of the AE values with respect to n is qualitatively reproduced by the semiempirical AM1 and ab initio SCF data. However, the theoretical approaches differ somewhat in detail. The AM1 values are very good for small n but become significantly too low for n > 5 (errors are 5-7 kcal/mol) . The ab initio SCF data show exactly the reverse dependence, i.e. the barriers become too large for n > 5. The single point (SP) ab initio data using the AM 1 geometries are only slightly shifted (by x 3 kcal/mol) relative to the values obtained from the very expensive ab initio SCF optimizations. This demonstrates the reliability of the AM1 geometries and their usefulness in single-point ab initio calculations; the combination of both approaches allows the proper calculation of systems as large as 8H. Detailed analysis of the errors of the SCF(SP) values shows an increase with n. A linear scaling of the form A E( corrected) = 0.83 x AE(SCF/SP) - 20.0 (in kcal/mol) may be deduced from comparison of the experimental data for n > 4. The errors of these corrected values (the dashed line in Fig. 3) are below 1 kcal/mol and may be of use in future studies of larger compounds and/or heterohelicenes (a further discussion on this topic is given at the end of this section). The results and conclusions given above are also valid for the smallest member of the series DM3H. The nonbonding interactions between the two methyl groups introduce a barrier of 13.9 kcal/mol 171. The calculated data fit nicely into the curves of the larger helicenes if a value for n of 4.5 is assumed. How-

416 Table 4 Dependence racemization Method/basis

S. Grimme, SD. Peyerimhoff/Chemical

of the calculated barrier height (in kcal/mol) of [ 5 1helicene on the theoretical method a set

SCFISV h SCF/SV+d’ SCF/SV+d+pd DFI-BLYP/SV+d+p’ exp.

E(G)

for the

- E(G)

29.0 28.1 28.2 22.7 22.8

n The ab initio SCF/SV optimized geometries are used. h E(C2) = -840.31999 au. c E(C2) = -840.62699 au. d E(C2) = -840.66880 au. o Density-functional theory with the gradient corrected Becke/Lee-Yang-Parr exchange-correlation functional [ 15,161. The error of the integrated number of electrons is 2 x 10e4. The total energy of the Ca minimum is -845.81943 au.

ever, we notice a large (and for this small molecule unexpected) difference between the theoretical SCF (AMI: 16.7 kcal/mol, SCF/SV: 17.6 kcal/mol) and experimental AE values so that electron correlation might not be negligible in this small compound. The relative large deviations of the ab initio SCF barriers for IE >4 has prompted us to investigate the performance of the theoretical treatments in more detail. Since [5]helicene is the smallest helicene which has a significant racemization barrier (22.8 kcal/mol [ 201) we have used this molecule as a model compound. The results of various ab initio calculations of the barrier height are collected in Table 4. Starting with the full SCF optimization SCF/SV (no polarization functions) value of 29 kcal/mol we can notice only a marginal improvement to 28.1 kcal /mol by adding d-functions to the basis set of the carbon atoms. This is surprising since prior studies on strained [n] paracyclophane hydrocarbons [ 21,221 have shown that these polarization functions are necessary for reliable energetic predictions. Adding pfunctions to the hydrogen atoms has also a negligible effect on the energy difference (28.2 kcal/mol) . The stability of these results with respect to the geometries used is demonstrated by the small difference between SCF( SP) and SCF( OPT) values (3 kcal/mol). Thus we expect that even with much larger basis sets there remains an error of approximately +5 kcal/mol relative to the experimental barrier of 23 kcal/mol. Electron correlation effects neglected so far will cer-

Physics 204 (I 996) 41 l-4 I7

tainly be important here because the transition state of the reaction has a significant different structure than the ground state. The stronger twisting of the benzene units present in the C, structures favors electron correlation compared to the ground state structure. Furthermore, there may be stronger attractive dispersion forces between the face-to-face orientated benzene rings which are neglected in the SCF treatment. The nearly perfect agreement between the densityfunctional calculation (which includes to some extend electron correlation effects) and the experimental value (22.7 vs. 22.8 kcal/mol [ 201) seems to support this viewpoint. Thus we have carried out DFT-BLYP calculations (SV+d+p basis) for all other compounds also (see Table 3). The discrepancy relative to the measured values is less than one kcal/mol in all cases. Even the value for the small DM3H is now in perfect agreement with experimental data. It appears as if the DFI approach with a gradient corrected correlation functional and a A0 basis of moderate size is very feasible for calculating racemization barriers in helicenes. MP2 calculations may give results of similar quality but become prohibitively costly (with respect to the amount of disc space required) in the case of the larger compounds with more than 400-500 basis functions. If one takes electron correlation into account the linear scaling of the ab initio SCF( SP) barriers with AE(SP) as the independent variable seems not entirely appropriate. A physical meaningful1 scaling procedure should depend on the number of aromatic units since the correlation energies (and hence the error of the SCF data) also depend on n. Inspection of the SCF(SP) errors (6E) with n shows an S-type curvature (the errors are small for small II and become constant for n > 6). A nonlinear least-squares fit with a gaussian-type function of the form SE = kl(l exp[kz(n-3)*]) gives kt = 14.57kcal/mol and k2 = -0.263. For large n the correlation energy contributions of additional benzene units are expected to be equal in the ground- and the TS state structures (i.e. most of the rings are not twisted toward each other) and thus the SCF error in the barrier height is expected to approach a constant value of 14.6 kcal/mol.

S. Grimme, S.D. Peyerimhof/Chemical

5. Summary

and conclusions

Our investigation of the structures and racemization barriers has given a detailed understanding of these features based on well-founded theoretical methods. The theoretical approaches clearly show that fused benzene rings are much more flexible than previously thought. Theoretical studies on [n] paracyclophanes also demonstrate that the benzene ring itself is not a very rigid structure (see Ref. [ 2 1] and references therein). The increasing racemization barrier of the helicenes with n is attributed to an increasing number of nonbonded interactions between the terminal rings in the TS structures. These nonbonded interactions are dominated by short H...H contacts for small n and by longer C...C contacts for larger n. Due to the flexibility of the benzene rings for twisting towards each other these distances become essentially constant for n > 6 so that the racemization barriers reach a plateau around 40-45 kcal/mol. A similar conclusion can be drawn from inspection of the twisting angles and the pitch of the helicenes which is almost constant for n > 5-6. Due to the presence of the two methyl groups the 4,ZLdimethyl 3H shows some structural resemblance with 5H (nonbonded C...C and H...H contacts are equal to within 0.1 A). The lower barrier of DM3H compared to 5H (the difference is 9-12 kcal/mol) is explained with a nearly free arrangement of the H atoms of the methyl groups and a lower out-of-plane bending potential (C( aromatic) C (methyl vs. benzene ring twisting). The inclusion of correlation effects for the prediction of racemization barriers is important and can be accounted for quite easily with a gradient-corrected density-functional.

Acknowledgement The services and computer time made available by the Sonderforschungsbereich 334 ( ‘Wechselwirkungen in Molekiilen’) have been essential to this study which was financially supported by the Deutsche Forschungsgemeinschaft.

Physics 204 (1996) 411-417

417

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