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J. Phys. Chem. 1983, 87,5304-5314
FEATURE ARTICLE Theoretical Studies of Medium Effects on Conformational Equilibria Wllllam L. Jorgensen Department of Chemistty, Purdue Unkersity, West Lafayette, Indiana 47907 (Received: June IO, 1983)
A summary of recent theoretical studies of condensed-phase effects on conformational equilibria is presented Emphasis is placed on Monte Carlo statistical mechanics and molecular dynamics calculations as general means for probing the origin of solvent effects. Technical aspects of the theoretical procedures are discussed particularly with regard to obtaining reliable results for conformational equilibria. Comparisons with experimental conformational data on pure liquids including high-pressure studies are very favorable. The work on dilute solutions is also promising and should become an important source of detailed information on solvent effects in chemistry and biochemistry.
Introduction The influence of solvent effects on reaction rates and chemical equilibria is well appreciated.’ Indeed, even the most basic characteristic of molecules, their structure, may be strongly influenced by the medium. This is apparent in the sensitivity of proteins to denaturation2 and in studies of tautomeric3 and conformational e q ~ i l i b r i a . ~It~ ~is clearly important to obtain a detailed understanding of the origin of solvent effects so that observations can be properly interpreted and that accurate predictions can be made in the course of designing and investigating new chemistry. For example, trans-1,2-dichloro- or trans-172-dibromocyclohexane prefer to be diequatorial in polar solvents.6 This result could be interpreted as reflecting the disfavoring of axial substituents; however, since the diaxial forms are more stable in nonpolar solvents and in the gas phase, a more considered explanation is r e q ~ i r e d An.~~~ other interesting example is provided by the recent observation of the strong influence of solvent on the conformation of phencyclidine (eq l).7 The equilibrium is
n
shifted substantially to the right upon transfer from a (1) (a) Amis, E. S. “Solvent Effects on Reaction Rates and Mechanisms”;Academic Press: New York, 1966. (b) Amis, E. S.; Hinton, J. F. “Solvent Effects on Chemical Phenomena”; Academic Press: New York, 1973. (c) Parker, A. J. Chem. Reu. 1969,69,1. (d) Reichardt, C. “Solvent Effects in Organic Chemistry”; Verlag Chemie: Weinheim, 1979. (2) Nemethy, G.; Pear, W. J.; Sheraga, H. A. Annu. Rev. Biophys. Bioeng. 1981, 10, 459. (3) Beak, P. Acc. Chem. Res. 1977,10, 186. (4) (a) Lemieux, R. U. Pure Appl. Chem. 1971,27, 527. (b) Eliel, E. L. Angew. Chem., Int. Ed. (Engl.) 1972, 11, 739. (5) Abraham, R. J.; Bretschneider, E. In “Internal Rotation in Molecules”; Orville-Thomas, W. J., Ed.; Wiley: London, 1974; Chapter 13. (6) Klaboe, P.; Lothe, J. J.; Lunds, K. Acta Chem. Scand. 1957, 11,
nonpolar (CD2C12) to a hydrogen bonding (CD30D/ CD2C12)solvent. Consequently, the form of the drug may be anticipated to change upon traversing a cell membrane. So that such phenomena can be understood it is important to develop theoretical procedures or models with demonstrated predictive ability. A traditional approach for solvent effects on conformational equilibria is based on reaction field t h e ~ r y . ~The ? ~ stabilization of a solute with dipole moment p and radius r in a solvent modeled as a uniform dielectric with dielectric constant E can be expressed as in eq 2. The difference in solvation energy
for two species in equilibrium (A s B) is then given by eq 3 by assuming they have the same size. Qualitatively, Amso= l
PA^ - P B ’ ) x / ~ ~
this expression provides the maxim that the more polar isomer will be preferentially stabilized in more polar media. However, the formula, even after adjustment for backpolarization of the solute by its own reaction field, significantly overestimates the solvent effects quantitatively. It was found that this problem could be remedied in many cases by also including the effect of the solute’s quadrupole on the reaction field.5 Nevertheless, further modifications are required to treat highly polar solvents, and cases where there is well-defined solvent structure or specific solutesolvent interactions, as with hydrogen-bonding solvents, cannot be treated with reaction field theory. Thus, although polarity arguments rationalize the solvent effects they do not explain the for trans-1,2-dihalocyclohexanes, observations for phencyclidine. It appears that hydrogen bonding to the nitrogen is less encumbered when the piperidyl group is e q ~ a t o r i a l . ~ A general theoretical treatment of fluids and solvent effects has been evolving through the computer simulation techniques of molecular dynamics and Monte Carlo statistical mechanics. These procedures model systems at the molecular level and include the specific solute-solvent and
1677. ~~
(7) Manoharan, M.; Eliel, E. L.; Carroll, F. I. Tetrahedron Lett. 1983, 24, 1855.
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(8) Onsager, L. J. Am. Chem. SOC. 19h6,58, 1486.
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The Journal of Physical Chemistty, Vol. 87, No. 26, 1983
Feature Article
solvent-solvent interactions needed to provide a detailed picture of liquid structure and the origin of solvent effects. A variety of pure liquids and dilute solutions have now been studied including conformational equilibria. The present paper provides a summary of the theoretical techniques used in such investigations and the results that have been obtained. Although accord with experimental findings for pure liquids is excellent, the limitations of the theoretical methods are discussed particularly for dilute solutions.
Pure Liquids-Theoretical Methods Monte Carlo simulation^.^ Statistical mechanics simulations are typically performed on systems with 100-300 solvent molecules. The monomers are contained in a cube that is surrounded by images of itself to remove edge effects, thereby providing a reasonable model for an infinite liquid. The calculations are usually carried out in the NVT (canonical) or N P T (isothermal-isobaric) ensemble in which the number of molecules, the temperature, and either the volume or pressure are fixed. From classical statistical mechanics the average value for a property Q is then given by eq 4 and 5 for the NVT and N P T en-
(4) P(X) = exp(-PV(X))/ $exp(-PVW) (Q) =
Qk
+
$$Q(X,V)
P(X,V) = exp(-PH(X,V))/$
dX
P(X,V) dX d V $exp(-PH(X,V))
(5)
dX d V
sembles, respectively, where P is the appropriate Boltzmann factor and the integrals are taken over all possible geometric configurations for the system (X) and for eq 5 all possible volumes. Furthermore, Qk represents the contribution from the kinetic energy which is assumed to be separable from the configurational integral; P = (kT)-l and the enthalpy, H(X,V) = U(X) + PV(X). The key problem of how to evaluate the configurational integrals was solved by Metropolis et al. using a sampling algorithm that enables configurations to be picked such that they occur with a probability proportional to their Boltzmann factors.12 With this procedure and converting the integral to a sum over discrete configurations, eq 4 and 5 are simplified to eq 6 where M is the number of con(Q) =
1 M
Qk
+ -XQi(x’) M i
(6)
figurations considered and X‘ indicates a Metropolis sampled configuration. The generation of a new configuration is performed by selecting a solvent molecule and randomly altering the values for its external and internal degrees of freedom. Typically, the entire molecule is translated in all three Cartesian directions and rotated about a randomly chosen axis. Bond lengths and bond angles are usually fixed, though to study conformational problems variation of appropriate dihedral angles must be included. For NPT simulations, the volume of the system is also changed periodically and the coordinates of the molecules are scaled accordingly. (9) References 10 and 11 can be consulted for additional details and references. (10)Jorgensen, W. L.; Binning, R. C.; Bigot, B. J. Am. Chem. SOC. 1981,103,4393. Jorgensen, W.L. Ibid. 1981,103,677. (11) Jorgensen, W.L.; Ibrahim, M. J.Am. Chem. SOC.1981,103,3976. (12) Metropolis, N.;Rosenbluth, M. N.; Teller, A. H.; Teller, E. J. Chem. Phys. 1953,21,1087.
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Briefly, the Metropolis algorithm involves computing a probability that a move should be made from configuration i to j . This probability can be expressed as p = T ~ / Twhere , for the NVT ensemble T, = exp(-PU,) and for the N P T ensemble T ] = exp(-PH,)VjN = exp(-P(U, + PV, - N k T l n V,)). It may be noted that the VJNterm is introduced by decoupling the volume and coordinate integrations in eq 5. Now, if p 1 1 which for the NVT ensemble means U, IU,, the move is accepted. If p < 1,p is compared to a random number x between 0 and 1,and if p 1 x , the move is still accepted, otherwise configuration j is rejected and i repeats. The chain of configurations generated in this way allows the configurational integrals to be evaluated by eq 6. Furthermore, it is clear that the Metropolis algorithm concentrates the sampling on configurations with low energy. This speeds convergence of the averages by avoiding excessive sampling of high-energy configurations which contribute little to the integrals. Nevertheless, convergence of the calculations must be carefully monitored and different averages converge at different rates. In Monte Carlo simulations of water or organic liquids with N = 100-200, the energy, enthalpy, volume, and radial distribution functions are normally converged in 0.5-1 M (lo6) configurations after equilibrium has been established.1° Other properties such as the heat capacity, isothermal compressibility, and coefficient of thermal expansion, which are calculated from fluctuations in E , H, and V, converge more s10wly.l~ Convergence for conformer populations is considered further below. In view of the sampling procedure it is essential to know the potential energy of a configuration. This may be represented by the sum over all pairwise interactions between monomers within a fixed cutoff radius and with an intramolecular term for the internal rotational energy of each monomer, V(@), where Q is the set of dihedral angles (eq 7). Again intramolecular vibrations have been ignored; N
u ( X ) = C cmn m 1) enhancement processes which were generated by adsorption of pyridine,
cyanide, etc. on a clean silver surface.2 Silver-metal generated enhancement theories can be simplistically analyzed in terms of surface morphology. They involve flat surfaces, microscoPicallY roughened surfaces, Or (chemisorption On) atomically roughened surfaces.’ In situ Cathodic cleaning procedures have been developed for silver electrode surfaces3 and such cleaned silver surfaces
(1) R. P. Cooney, M. R. Mahoney, and A. J. McQuillan in “Advances in Infrared and Raman Spectroscopy”, Vol. 9, R. J. H. Clark and R. E. Hester, Ed., Heyden, 1982, Chapter 4, and references therein.
(2) T. E. Furtak and J. Reyes, Surf.Sci., 93, 351 (1980). (3) M. W. Howard, R. P. Cooney, and A. J. McQuillan, J. Raman Spectrosc., 9, 273 (1980).
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