ocaÊ+e—
(3)
dt
where
α(ω)«
£
Σ —
.=x,y,z
%
J
ω
ex
β
χ ,
6
-
—
W
ω
(5) i=x,y,z
ex
^ e x , g
"""
®
The specific rotation angle is proportional to β.
[«L * ω β(ω) 2
(6)
There are numerous ways to interpret eq 5, in which the physical foundation of optical activity, and its structure dependence, are buried. One can view the electric-dipole matrix element as inducing linear polarization of charge in the molecule, while the angular momentum matrix element drives circulation of charge. Combined, the polarization rotation is nonzero if the light field induces helical motion of charge [11,15]. A n alternative perturbative view of electronic response can be developed in the regime far from electronic resonances, ω « C0 ex [16]. In this limit, eq 5 simplifies to the inner product of two firstorder wave function corrections.
In Chirality: Physical Chemistry; Hicks, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
108
i=x y z dEi {ÔBt
Downloaded by PENNSYLVANIA STATE UNIV on September 13, 2012 | http://pubs.acs.org Publication Date: March 1, 2002 | doi: 10.1021/bk-2002-0810.ch008
y
(7)
t
The computational challenge, in this regime at least, is to compute the first-order changes in the ground state induced by combined electric and magnetic dipole perturbations. Coupled Hartree-Fock (CHF) methods compute these changes analytically (rather than by performing multiple calculations at many different field strengths). The C H F calculation appears complicated compared to textbook first-order perturbation theory because the effective-potential that the electrons "feel" changes to first-order itself, which in turn leads directly to firstorder corrections to the ground state [17]. Determining the first-order wave function corrections (the derivatives in eq 7) amounts to solving a set of linear equations following the initial Hartree-Fock (HF) analysis in zero field [17]. One strategy for developing structure-function relations in optical activity is to map the wave function derivatives of eq 7 for occupied molecular orbitals. This C H F strategy is a type of "analytical derivative" method, and is very closely related to Hartree-Fock finite-field strategies. In finite-field calculations, wave function perturbations are determined by repeatedly solving the Schrôdinger equation (in the H F approximation) for varied applied-field strengths. The C H F calculation has the advantage of generating the response property from a single HF ground state calculation and some linear algebra. The wave function derivative of eq 7 can be used to identify the contributions of specific molecular orbitals to optical rotation. For example, Figure 1 maps the E- and B-field derivatives of the π-symmetry H O M O of ethylene. Note that the overlap of these two wave function derivatives is exactly zero, as ethylene is achiral. Summing these overlaps over all occupied molecular orbitals produces zero in the case of achiral structures. A frequency-dependent response-theory formulation of β, somewhat more complex than the C H F analysis can be carried out as well. This theory, often called coupled-perturbed HF (CPHF), leads to explicitly frequencydependent response properties [18,19,20]. Such calculations account for specific electronic resonances. As such, one makes additional approximations in the analysis that amount to building increasingly complex descriptions of the ground and excited states. The most common approach is to base the description on a single configurational reference state with a double configuration interaction (CI) ground state and single CI excited states. This level of C P H F computation, known as the random-phase approximation (RPA), includes the influence of some ground state correlation in the calculation [21]. Simpler CPHF strategies, such as the Tamm-Dancoff approximation, have no ground state dynamical correlation built in. CPHF computations are formulated to satisfy the hypervirial theorem. A significant consequence is that the oscillator
In Chirality: Physical Chemistry; Hicks, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
Downloaded by PENNSYLVANIA STATE UNIV on September 13, 2012 | http://pubs.acs.org Publication Date: March 1, 2002 | doi: 10.1021/bk-2002-0810.ch008
109
Fig. 1. Wave function derivative maps for ethylene. Top: the HOMO π orbital of ethylene. Middle: the electric-field derivative of the HOMO is the orthogonal LUMO π* molecular orbital. Bottom: the magnetic-field derivative of the HOMO is also an orthogonal /r* orbital (LUMO+1). As such, the overlap of derivatives in eq 7 for the HOMO is exactly zero, and makes no contribution to optical rotation.
In Chirality: Physical Chemistry; Hicks, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
110
Downloaded by PENNSYLVANIA STATE UNIV on September 13, 2012 | http://pubs.acs.org Publication Date: March 1, 2002 | doi: 10.1021/bk-2002-0810.ch008
strength and the rotational strength sum rules are valid [12,22]. These are important constraints, leading to molecular response properties that one expects to be more "balanced" than might result from unconstrained variational calculations in a finite basis with an approximate description of correlation. The sum-on-states expression for β suggests the importance of maintaining balance among a very large set of transition element properties in order to make reliable property predictions. Alternative density functional approaches to computing response properties are becoming available for rotation angle computation. One method performs a Kramers-Kronig transformation of a simulated C D spectrum [23]. Linear-response DFT methods analogous to C H F and C P H F appear promising as well [24].
III.
Computation of optical rotations: Small Molecules
Before implementing computational methods to assign the absolute stereochemistry of natural products, it is important to point out that the C H F and CPHF methods are satisfactory in describing the optical rotations of small molecules. For a summary of these calculations the reader is referred to refs 16 and 25-35. These rotations were computed using the standard C A D P A C software [36] for off-resonance results and D A L T O N [37] for explicit frequency dependent analysis. The D A L T O N package has the added advantage of being able to employ gauge independent (London) atomic orbitals. We have found that the results for smaller molecules resulting from placing the gauge origin at the molecular center of mass produce satisfactory results in such systems [33]. However, this is probably not the case for larger molecules. We have also found that for small molecules (and for composite descriptions of larger structures) a 6-31G* basis set is adequate. Cheeseman and coworkers are performing important investigations of the limitations of modest basis set computation, as well as gauge origin dependence in small rigid structures [24].
IV. Computation of optical rotation: Natural Products Our computational approach to determining the optical rotation of natural products containing multiple stereocenters is: divide the structure of interest into fragments, each with stereogenic centers several bonds removed from one another; use Monte Carlo sampling (based on molecular mechanics force fields) to build a family of thermally accessible structures; compute the rotation angle for this family of structures; Boltzmann weight the resulting angles
In Chirality: Physical Chemistry; Hicks, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
Downloaded by PENNSYLVANIA STATE UNIV on September 13, 2012 | http://pubs.acs.org Publication Date: March 1, 2002 | doi: 10.1021/bk-2002-0810.ch008
Ill to predict a composite-molecule rotation angle. We have demonstrated the viability of this strategy for the marine natural product hennoxazole A, making a stereochemical prediction consistent with experiment. In this case, theory had to choose one of eight possible absolute configurations. As described in ref. 3, we computed theoretical molar rotation angles for three molecular fragments (shown in Table 1). This allowed us to add together contributions to the rotation angle and to predict the optical rotation of all 8 possible enantiomers and diasteriomers. This strategy is a kind of "van't Hoff summation" [3] of contributions from the distinct fragments. Summations of this kind are known to succeed when the motion of the individual units is essentially independent. In structures where the assumption of unconstrained relative motion is not adequate, larger fragments are required in the calculations (helicenes present an extreme example where the strategy of additivity is problematic). Table 2 shows the predictions of fragment computations of molar rotations based on C P H F R P A gauge independent atomic orbital (DALTON) analysis of hennoxazole A based on the fragments below [33]. Analysis of the predicted rotation angle for the 8 composite hennoxazole A stereoisomers appears in ref. 33. A similar analysis of pitiamide A - performed in advance of chemical synthesis - produced excellent agreement between computed and measured optical rotations for the synthesized diastereomers. Interestingly, the N M R analysis of the natural product and the synthetic diastereomers leads to an assignment of (7R IQR) or (7S,10S) and a stereochemistry that is inconsistent with the reported specific rotation of -10.3. It is possible that the natural product contained degradation products, a mixture of enantiomers or diasteromers, or chiral impurities. The fragmentation strategy is shown below [29]. 9
9
Table 1. Calculated molar rotations for the three fragments (l-III) in the specific enantiomeric configurations (6-31G basis), averaged from four calculations. From examining the 8 possible linear combinations of these three numbers we were able to assign the hennoxazole stereochemistry correctly as (2R,4R,6R,8R,22S) [33]. Fragment I was known to be 2R,4R,6R or 2S,4S,6S from N M R experiments [33]. Fragment I II III
[M] ±2a (Configuration, Sign) D
162 ± 11 (2R,4R,6R, -ve; 2S,4S,6S, +ve) 105 ±22 (8Λ, +ve; 8£ -ve) 146 ± 17 (22Λ, +ve ; 225, -ve)
In Chirality: Physical Chemistry; Hicks, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
Downloaded by PENNSYLVANIA STATE UNIV on September 13, 2012 | http://pubs.acs.org Publication Date: March 1, 2002 | doi: 10.1021/bk-2002-0810.ch008
112
In Chirality: Physical Chemistry; Hicks, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
113 Table 2. Computed pitiamide fragment molar rotation values. Results in the final row represent the calculation ^(Fragments I + II + III).
Downloaded by PENNSYLVANIA STATE UNIV on September 13, 2012 | http://pubs.acs.org Publication Date: March 1, 2002 | doi: 10.1021/bk-2002-0810.ch008
[MJ^for
indicated stereoisomer
Fragment I
+111(75)
Fragment II
+66 (105)
Fragment III
+123 (75,105)
+17(75,10/?)
Pitiamide A
+150(75,105)
+31 (75,10Λ)
V. Optical rotation computation as a probe of reaction mechanism With confidence in our ability to compute the sign and magnitude of rotation angles in organic molecules, we turned to unanswered questions of reaction mechanism that can be approached from optical rotation. In Curran's group at Pittsburgh, new methods are under study to use radical cyclization to generate chiral products. The reaction of achrial atropisomer to form an enantioenriched indolinone was examined (Fig. 2), with the hypothesis that the P- stereochemistry of the
starting material would be preserved (assuming rotation around the C N bond is slow compared to the rate of ring closure) in the product. Calculations of rotation angles for the product specific rotation at the sodium D line of -17 were fully consistent with optical rotation of -16 measured for the dominant enantiomer [30]. Experimental attempts to test this prediction via crystallization or derivatization have, so far, not met with success.
VI. Toward structure-function relationships for optical rotation The non-resonant rotation angles in eq 5 can be written in terms of individual occupied molecular orbital contributions:
In Chirality: Physical Chemistry; Hicks, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
In Chirality: Physical Chemistry; Hicks, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
Fig. 2. The optical rotation of the indolinone formed by cyclization of the atropisomer indicates that the stereochemistry is retained in the course of this reaction.
Downloaded by PENNSYLVANIA STATE UNIV on September 13, 2012 | http://pubs.acs.org Publication Date: March 1, 2002 | doi: 10.1021/bk-2002-0810.ch008
115
β(ω)^Σ
5 - ) Σ'""' dE,
α
Σ Σ ^
() 8
n PA occ
where C is a product of atomic orbital coefficients associated with the perturbation of each polarized occupied molecular orbital and S is the overlap between atomic orbitals [32]. This is a Mulliken-like analysis of optical rotation. We have made atomic maps of these contributions, combining the diagonal (p=q) contributions with equally divided off-diagonal terms (p*q) [31,32]. This mapping reveals, for example, the relative contributions to optical rotation that arise from stereogenic centers as opposed to twisted chains that may be remote from these centers [32]. Atomic contribution map analysis for an indoline [31] indicate that - for a specific conformer - the contribution of a side-chain twist to the rotation angle is comparable to the contribution from the stereogenic center itself. We have used this analysis to probe the decay of optical rotation contributions as a function of distance of groups from a stereocenter [32] and to understand the origin of sign differences in the rotation angles for (+)-calyculin A vs. (-)calyculin B, which differ only in the Ζ vs. Ε configuration of a C N substituent that is nine bonds removed from the nearest stereogenic center [28]. Atomic contribution maps for achiral oxirane and chiral chiral chloro-oxirane appear in Fig. 3.
Downloaded by PENNSYLVANIA STATE UNIV on September 13, 2012 | http://pubs.acs.org Publication Date: March 1, 2002 | doi: 10.1021/bk-2002-0810.ch008
p q
n
VII. Prospects Having access to reliable computational tools for determining optical rotation angles has established new strategies for absolute stereochemical assignment. Moreover, the new tools are leading toward structure-function relations for a rather poorly understood macroscopic property. We have shown that molecular fragmentation, geometry sampling, linear-response computation, and Boltzmann weighting of contributions leads to reliable rotation-angle predictions. Future directions in our research program will include extension to molecules of increasing complexity. For example, reducing the number of plausible stereoisomers in a natural product with 10 stereocenters from 2 (1024) to a few dozen will place the laboratory synthesis of the structure within the realm of possibility. Complementary studies of the "chiral imprint" imposed on solvent by a dissolved chiral solute must be understood. Also, the competing influence of tetrahedral stereogenic centers vs. twisted chain contributions to optical rotation angles, predicted by theory, must now be tested in carefully 1 0
In Chirality: Physical Chemistry; Hicks, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
Downloaded by PENNSYLVANIA STATE UNIV on September 13, 2012 | http://pubs.acs.org Publication Date: March 1, 2002 | doi: 10.1021/bk-2002-0810.ch008
116
Fig 3. Maps of atomic contributions to [a] ο for achiral oxirane (top) and chiral chloro-oxirane (bottom). Note the canceling contributions to the angle in oxirane and the non-zero sum in the chiral species. A 6-31 G** basis was used in CADPAC calculations of optical rotation with 6-31G** based geometry optimization.
In Chirality: Physical Chemistry; Hicks, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
117
Downloaded by PENNSYLVANIA STATE UNIV on September 13, 2012 | http://pubs.acs.org Publication Date: March 1, 2002 | doi: 10.1021/bk-2002-0810.ch008
designed experiments. In the near future, increasingly reliable electronic structure methods will undoubtedly emerge that will prove valuable for making predictions of increasing reliability on structures of ever increasing size and complexity. The methods discussed in this chapter complement other chirooptical methods [38,39] - and were developed specifically to address stereochemical assignments in complex natural products containing multiple stereogenic centers, conformational flexibility, and novel chemical bonding.
Acknowledgments We thank ACS-PRF (33532-AC), NSF (GHE-9727657,CHE-9453461) and N I H (GM-55433) for support of this research project. D N B also thanks A l l Souls College, the Burroughs-Wellcome Foundation, and the J.S. Guggenheim Foundation for support while some of this work was conducted. We thank D.P. Curran, D.W. Pratt, A . B . Smith III, and G.K Friestad for their important contributions to our research.
References
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
Cragg, G. M . ; Newman, D. J . ; Snader, K. M . , J. Nat. Prod. 1997, 60, 52-60. Shu, Y.-Z., J. Nat. Prod. 1998, 61, 1053-1071. Eliel, E.L.; Wilen, S.H.; Mander, L.N. Stereochemistry of Organic Compounds; Wiley; New York, 1994. Nakanishi, K.; Berova, N.; Woody, R. W. Circular Dichroism. Principles and Applications; V C H Publishers Inc.; New York, 1994. Crews, P.; Rodriguez, J.; Jaspars, M . Organic structure analysis, Oxford University Press; New York, 1998. Caldwell, D.J; Eyring, H. The Theory of Optical Activity, WileyInterscience, New York, 1971. Brewster, J.H. J. Am. Chem. Soc. 1959, 81, 5475-. Applequest, J. J. Chem. Phys. 1973, 10, 4251-. Rosenfeld, L.Z. Physik 1928, 52, 161-. Selected Papers on Natural Optical Activity, SPIE Milestone Series, Vol. MS 15, Lakhtakia, A . Ed., SPIE Press, Bellingham, W A 1990. Kauzmann, W. Quantum Chemistry, Academic Press, New York, 1957. Atkins, P. Molecular Quantum Mechanics, 2 ed., Oxford, 1983. Besley, N.A.; Hirst, J.D. J. Am. Chem. Soc. 1999, 121, 9636-9644. Atkins, P. Physical Chemistry, 6 ed., Freeman, New York, 1998. nd
th
In Chirality: Physical Chemistry; Hicks, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.
118 15. Cantor, C.R. and Schimmel, P.R. Biophysical Chemistry, Part II, Chapter 8, W H Freeman and Co, San Francisco, 1980. 16. Polavarapu, P.L. Molec. Phys. 1997, 91, 551-554. 17. Stevens, R.M.; Pitzer, R.M.; Lipscomb, W.N. J. Chem. Phys. 1963, 38, 550. 18. Jensen, F. Introduction to Computational Chemistry, Wiley; New York, Chapter 10, 1999. 19. McWeeny, R. Methods of Molecular Quantum Mechanics, 2 ed., Academic Press, New York, 1992. 20. Cook, D.B. Handbook of Quantum Chemistry, Oxford Science, Oxford, 1998. 21. Hansen, A.E.; Bouman, T.D., Adv. Chem. Phys. 1980, 44, 545-644. 22. E. Merzbacher, Quantum Mechanics, 3rd ed.,Wiley, New York, 1998. 23. De Meijere, Α.; Khlebnikov, A.F.; Kostikov, R.R.; Kozhushkov, S.I.; Schreiner, P.R.; Wittkopp, Α.; Yufit, S.S. Angew. Chem. 1999, 38, 3473-. 24. Cheeseman, J.R.; Frisch, M.J.; Devlin, F.J.; Stephens, P.J. J. Phys. Chem. A 2000, 104, 1039-1046. 25. Polavarapu, P.L. Tetrahedron Asymmetry 1997, 8, 3397-3401. 26. Polavarapu, P.L.; Zhao, C.X. Chem. Phys. Lett. 1998, 296, 105-110. 27. Polavarapu, P.L.; Chakraborty, D. K. J. Am. Chem. Soc. 1998, 120, 61606164. 28. Kondru, R.K.; Beratan, D.N.; Friestad, G.K.; Smith III, A . B . ; Wipf, P. Org. Lett., 2000, 2, 1509-1512. 29. Ribe, S.; Kondru, R.K.; Beratan, D.N.; Wipf, P. J. Am. Chem. Soc. 2000, 122, 4608-4617. 30. Kondru, R.K.; Chen, C.H.-T.; Curran, D.P.; Wipf, P.; Beratan, D.N. Tetrahedron; Asymmetry 1999, 10, 4143-4150. 31. Kondru, R.K.; Wipf, P.; Beratan, D.N. J. Phys. Chem. A 1999, 103, 66036611. 32. Kondru, R.K.; Wipf, P.; Beratan, D.N. Science 1998, 282, 2247-2250. 33. Kondru, R.K.; Wipf, P.; Beratan, D.N. J. Am. Chem. Soc. 1998, 120, 22042205. 34. Kondru, R.K.; Lim, S.; Wipf, P.; Beratan, D.N. Chirality 1997, 9, 469-477. 35. Kondru, R.K.; Wipf, P.; Beratan, D.N. in preparation. 36. Amos, R.D.; Rice, J.E. The Cambridge Analytic Derivative Package, issue 4.0, 1987. 37. Helgaker, T. et al., DALTON, an ab initio electronic structure program, Release 1.0, 1997. 38. F. Furche, R. Ahlrichs, C. Wachsmann, E. Weber, A. Sobanski, F.Vögtle, and S. Grimme, J. Am. Chem. Soc. 2000, 122, 1717-1724. 39. F.J. Devlin and P.J. Stephens, J. Am. Chem. Soc. 1999, 121, 7413-7414.
Downloaded by PENNSYLVANIA STATE UNIV on September 13, 2012 | http://pubs.acs.org Publication Date: March 1, 2002 | doi: 10.1021/bk-2002-0810.ch008
nd
In Chirality: Physical Chemistry; Hicks, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.