J. Phys. Chem. 1993,97, 11291-1 1299
11291
Ammonium Cation Complexation by 18-Crown-6 under Realistic Conditions: Simulations Using Proper Potentials Yuk Lung Ha and Amp K. Chakraborty’ Department of Chemical Engineering, University of California, Berkeley, California 94720 Received: March 5. 1993’
Macrocyclic (crown) polyethers exhibit a high degree of specificity for cation (e.g. metal, ammonium) binding. The strength and specificity of this binding depend not only upon the intrinsic interaction between the crown and its substrate but also upon the moderating influences of the solvent. In this study, we investigate the complexation between an ammonium cation and 18-crown-6 in polar and apolar solvents (water and carbon tetrachloride, respectively). We use statistical perturbation theory in conjunction with N P T Monte Carlo simulations a t 25 OC and atmospheric pressure. The interaction between ammonium and 18-crown-6 is characterized by an intermolecular potential derived from electronic density functional theory calculations within the Kohn-Sham, local density approximation formalism. We discuss our results in terms of the potential of mean force between the crown and substrate as a function of center-of-mass separation R,, equilibrium binding constants which are compared to experiment, conformational changes of the crown upon complexation, and, lastly, solvent structure around the complex.
I. Introduction The discovery of crown ethers first reported by Pedersen1S2in 1967 has stimulated considerableinterest in the ensuing years.Sl0 Their ability to complex both c a t i ~ n i c ~ land -~~ substrate species has motivated the use of crown ethers in diverse fields of practical appli~ation.~~33 Specific applications include membrane separation of cationic species, enantiomer resolution, and catalysis. Such apparently diverse usages of crown ethers and their analogues rely upon the high selectivity exhibited by these molecules in binding with various substrates. This high level of discrimination that enables crown ethers to recognize and distinguish among rather similar molecules mimics, at the simplest level, the behavior of naturally occurring enzymes. Consequently, our study of the simplest crown ether, 18-crown-6 (1,4,7,10,13-hexaoxacyclooctadecane),and its interaction with the ammonium cation seeks to elucidate not only the potential use of 18-crown-6 as a complexing agent in a prototypical separation process of amino hormones but also the simplest proctssesoccurring in enzyme/substrate systems. Our approach relies upon performing molecular simulationswith intermolecular potential functionsderived from quantum mechanicalcalculations. Statistical perturbation theory in conjunction with molecular simulationshas facilitated the calculation of free energy changes for many systems in recent years and promises to be a valuable tool in the future design of “molecularlyengineered”molecules. For instance, solvation energy difference~,j”~acid-base reactions,40941 simple nucleophilic substitution r e a c t i o n ~ , 4ion ~ . ~pair ~ interactions,U~45and conformational isomersws have all been investigated with free energy perturbation calculations. Additionally, via the thermodynamic cycle perturbation method, the relative binding strength of a host molecule with various guest species may be determined.4e53 This scheme may also be used todetermine how a site-specificmutationof the complexing agent affects its ability to bind to a specific substrate. In the present study, we utilize statistical perturbation theory in combination with NFT Monte Carlo simulations in order to calculate the Gibbs potential of mean force between 18-crown-6 and an ammonium cation in solutions of water and carbon tetrachloride (at 25 OC and 1 atm). In carrying out these calculations, we use a crown-cation intermolecular potential To whom all correspondence should be addressed. *Abstract published in Advance ACS Abstracts. October 1, 1993.
0022-3654/93/2097-1129 1!§04.00/0
function that is derived from the results of KohnSham calculations reported by us in a recent publication.54 The Gibbs free energy of binding is determined in each case and compared to experiment where available. Furthermore, we characterize the conformational state of the crown molecule through its OC-CO torsion angle distribution and detail its evolution as a function of the center-of-mass separation, Rcr between the crown and ammoniumcation. These results shed light on the reorganization process that the crown undergoes when binding the cationic substrate under environments of differing solvent polarity. Furthermore, the radial distribution functionscharacterizing the solvent structure about the complex provide some insight into the restructuring of the solvating molecules during the complexation process as well. In section 11, we provide some background on previous work that is relevant to the issues addressed in this paper. In section 111,our method of investigatingthe problem is described. In section IV, we discuss the results in detail, and finally, some concluding comments are presented in section V. 11. Background
A considerable body of experimental data, both thermodynamic and structural, exists regarding the complexation of 18-crown-6 and its derivatives with a multitudeof alkylammonium substrates under various temperature and solvent conditions.1418 Unfortunately, thermodynamic information by its very nature is very coarse grained, providing little insight into the molecular events that lead to specific complexation. Information regarding these molecular events would aid in achieving a deeper understanding of the structure/function relationships that govern the complexation of ammonium cations with crown ether macrocycles. Similarly, although rather precise structural information is available for 18-crown-6/ammonium cations from X-ray crystallographic e~periments,55-5~ its direct correspondence to the conformations adopted in solution is not exactly clear. Even though experiments conducted in solution such as NMR,60v61 IR,62 Raman,63 neutron scattering,u.65 and dipole moment measurements66.67 probe the system on a microscopic level, it is often difficult to correlate what is observed experimentallyto specific molecular events since the data embody a combination of many complex phenomena. As a result, numerous theoretical studies have previously attempted to elucidate the nature of 18-crown6/ammonium cation complexation with detailed atomistic techniques such as ab initio54 and semiempirical quantum mechanical 0 1993 American Chemical Society
Ha and Chakraborty
11292 The Journal of Physical Chemistry, Vol. 97, No. 43, 1993
-
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molecular mechanics
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calculations,6* molecular mechanic^,^^-^^ as well as simulation schemes that include molecular d y n a m i c ~ ~ . ~and ~ , ~Monte O Carl0.71+~2 On the basis of past work, it is well established that 18-crown-6 is structurally a highly flexible molecule adopting many conformations in solution. In addition, 18-crown-6 assumes a conformation of Crlike symmetry in apolar conditions and as a crystalline solid; in contrast, it assumes a conformation akin to D3slike symmetry in polar conditions and as a crystalline complex with many cationic species (e.g. ammonium). Figure la,b illustrates the ideal C, structure of 18-crown-6 along with its ideal D3d structure as an ammonium complex. Consequently, under apolar situations, the complexation between the crown and its ammonium substrate necessitates a restructuring of the crown conformation from a C, to a D3~likesymmetry as the system maximizes the specific crown/cation interactions while overcoming the unfavorable free energy change associated with crown intraconversion. We note, however, that the restructuring has not been "observed" directly heretofore. In contrast, in polar environments, the crown is preorganized for complexation. Several molecular simulations have addressed the issue of 18crown-6 conformation in solution. Ranghino et performed Monte Carlo simulations of 18-crown-6 in water wherein the crown was held rigid at predefined symmetry conformations. Ha and Chakraborty72have examined the conformational statistics of 18-crown-6 in polar water and apolar carbon tetrachloride at two temperatures. In their NPT Monte Carlo simulations, the only constraint on the crown is one of constant bond lengths. They find that the crown samples many local conformations but adopts a D ~ l i k structure e in HzO and a Crlike structure in Cc4. They characterize the average conformation of the crown in solution via OC-CO and CO-CC torsion angle distributions. More extensivesimulations have since been conducted, and Figure 2a,b shows the OC-CO torsion distributions of 18-crown-6 in water and carbon tetrachloride, respectively, at 25 OC along with results for perfect C,and D3d symmetric structures as determined in a previous molecular mechanics ~ a p e r . ~In3 a similar fashion,
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Figure 2. (a, top) OC-CO torsion angle distribution in degrccs for 18crown-6 in CC4 at 25 OC, 1 atm7*(-) and for minimum energy Cr structure from molecular mechanics73 (b, bottom) OC-CO torsion angle distribution in degrccs for 18-crown-6 in HzO at 25 O C , 1 atm'z (-) and for minimum-energyD3d structurefrom molecular (-e).
(...)
.
we shall characterize the average conformation of the crown in solution during the complexation process by its OC-CO torsion angle distribution. Figure 2a,b provides reference points against which we compare when discussing how the average conformation of the crown molecule evolves during the complexation process with ammonium. Several previous theoretical indicate that structural reorganization must occur during complexation. Straatsma and McCammon,& applying free energy perturbation with molecular dynamics of 18-crown-6 in water, find that "the free energy of an ensemble of crown ether molecules with D3d symmetry is 2.59 kJ mol-' higher than an ensemble of unconstrained crown ether molecules". As noted earlier, and as found by Straatsma and McCammon,& 18-crown-6 in water adopts many conformations which are not exclusively D3dr and by so forcing a Dgd conformation, the entropy of the system is greatly reduced. Such an entropic penalty may be paid when a cation complexes with the crown and locks it into the Dsdconformation. Sun and Kollman4' explicitly determined that the free energy difference in water between 18-crown-6 D3d and C, structures via free energy perturbation wherein the crown conformation was slowly intraconverted between the two conformations. They find the D3d structure to be 5.0 kcal mol-' lower in free energy in aqueous surroundings than the Ci structure. Although both these studies consider the free energy effects of conformational reorganization of the crown in water, neither explicitly incorporates the presence
Ammonium Cation Complexation by 18-Crown-6
The Journal of Physical Chemistry, Vol. 97, No. 43, 1993 11293
of a complexingcation that would induce such structural changes. TABLE I: Potential Parameters Used for Ammonium' In this paper, we address this issue directly. i Av Cil 41 Previous molecular mechanics studies have also facilitated our N 4.752 X lo8 4.032 X lo5 -1.635 X 10' understanding on the nature of alkylammonium/crown complexes, H 1.431 X lo2 especially the insightful work of Gehin et wherein the Units are KA~Zfor Aft, K As for C,!, and (K A)l/z for qi. Ammonium energetics of different structures were minimized with respect to treated as rigid tetrahedral with bond lengths fixed at 1.01 A. various degrees of freedom. Semiempirical as well as ab initio molecular orbital calculations have also addressed the most TABLE II: Ammonium/l8-Crown-6 Model Potential rudimentary ammonium/ 18-crown-6system. Our past ~ o r k ~ ~ Parameters. v ~ ~ has included density functional theory calculations (within the i=N, i = N, i=H, i = H, KohnSham, local density approximation framework) to charj=C j=O j=C j=O acterize the interaction between 18-crown-6 and the ammonium "ii 6.064 5.614 10.50 10.05 ion for many degrees of freedomes4 We find that the substrate Bo = "ill2 5.25 5.026 has a relatively high degree of mobility for certain types of motion Sij 3.595 x 107 2.171 x 105 1.022 x ioio 2.469 x 1 ~ within the crown cavity. Additionally,significant charge transfer 2.738 X 1Olo 1.945 X lo8 and polarization effects are present which are typically unac-9.346 X lo9 1.422 X lo9 counted for in molecular mechanics calculations. Because the -7.512 x 109 -1.201 x 109 nature of these strong and specific interactions extend beyond 4.711 X lo9 -6.204 X lo8 simple two-body effects, it is necessary to generate a nonpairwise 4.040 X lo8 1.104 X lo8 additive potential that adequately captures such substantial -1.028 X lo9 3.459 X 1@ binding. For our present simulations, we develop such a model a Units are A-1 for all and &; K for Si, and q;K A-1 for and potential between ammonium and 18-crown-6 from more than ql;and K A-2 for l$ and i$ 150 quantum mechanically derived data points. These values cover a greater range of system degrees of freedom than those in the system. It assumes a form akin to the Morse potential with reported in our previously published study.54 The exact functional gij = aij/2. Since Tij depends upon the local geometry in a form of this interaction potential will be discussed in detail in the nonlinear manner as exemplified by eq 2b, it captures aspects of following section. We note, however, that the molecular meboth the angular orientation and the many-body dependence of chanics and electronic structure calculations represent a zero the interaction. In other words, in order to fit the quantum temperature and pressure environment that is unrepresentative mechanical results,54 it is necessary to go beyond simple twoof experimental situations. Statistical mechanical simulations body potentials for this system. performed with these proper crown/cation interaction potentials, The bases for our potential parameters are quite varied. A however, offer a satisfying approach toward studyingtemperature, detailed description of the crown, carbon tetrachloride, and water pressure, and solvation effects. parameters that we utilize can be found in our previous study of the conformational statistics of 1 8 - c r o ~ n - 6 .We ~ ~ merely note 111. Method that the bond lengths of the crown are fixed at C-O = 1.430 A The total energy of our system may be expressed in the following and C-C = 1.525A and that both the methyleneand CCLgroups manner are treated in the united atom approximation. For ammonium, Jorgensen's OPLS parameter^'^ define the cation as a rigid assemblage (tetrahedral bond angles and bond lengths of 1.01 A) of partial charges centered on each constituent atom along with a Lennard-Jones site about the nitrogen. Table I lists the parameters used for ammonium in our computations. Standard combining rules provide the values necessary to characterize Lennard-Jones interactions between the remaining unlike sites; where the first and second terms represent the bond bending and Aij = (AiiAjj)l/z and Cij = (CiiCjj)1/2. torsion potentials of the crown molecule, respectively; 0 is the As stated earlier, due to the strong intermolecular binding bond angle, 8, its equilibrium value, and q5 the torsion angle. The between 18-crown-6 and ammonium, we have performed and nonbonded energetics embodied by the third term consist of reported54 quantum mechanical calculations that elucidate the Lennard-Jones 6-1 2 and coulombic potentials between all intrinsic interactions between the two species. More than 150 interaction site pairs (except those between solutes) separated by data points from our KohnSham calculations were utilized in at least three bonds. Lastly, EEomplcx characterizes the interaction developing our model potential of the form indicated in eqs 2a energetics between the ammonium cation and the 18-crown-6 and 2b. Specifically, the parameters found in Table I1 were molecule. Specifically, obtained via a least-squares fit to
3
$! d G
c,
j=c,o
The first term of eq 2a represents the long-range interactions while the second and third embody the strong short-range complexation behavior. Specifically, the second term accounts for exchange repulsion between all ammonium/crown sites while the third term accounts for charge transfer and thus the bonding
6
where Uwmplcx, U N ~ U1scmn-6 +, are the quantum mechanically derived energies of the complex and the isolated components (at the same geometry as in the complex). The long-rangecoulombic and dispersive interactions of Eoomplsx are assigned the literature values mentioned above. Furthermore, no special weighting of the low-energy points was necessary since the form of the model potential was able to capture accurately the many features of the interaction energy hypersurface. The quality of our fit as seen in Figure 3 is quite good with a standard deviation of 3.39 kcal mol-'. More important is that the fit captures the topology of the interaction surface without introducing anomalous peaks and valleys. Because of the number of parameters which required fitting, simulatedannealingprovided an efficient and rapid method
~
7
Ha and Chakraborty
11294 The Journal of Physical Chemistry, Vol. 97, No.43, 1993 -200
,
0,
-300 t
2
-400
1
I
.loo!OOO
rotation of the entire molecule about a randomly chosen Cartesian coordinate passing through its center of mass and of internal motions where a single crown site is rotated about a circle defined by its two adjacent bounds. Concomitantly,in order to maintain a constant center-of-mass separation, the ammonium cation undergoes a simultaneous translation with the internal crown motion in addition to its own Eulerian rotation. Finally, volume fluctuations were accomplished by rescaling the simulation box. The ranges of all the moves were periodically adjusted during the simulations to maintain acceptance rates in the range 30-50%%. Furthermore, we applied umbrella and preferential sampling to improve our simulationstatistics. Umbrella sampling ensured that the internal conformationsof 18-crown-6were well sampled and not locked within a single local structure. This was accomplished by mitigating the high angular and torsional barriers with a weighting function
P -900
-800
-700 -600 -50 0 E(mmp1ex) [kcalimole]
-400
-300
-200
Figure 3. Comparison of 18-crown-6/ammonium interaction energies from ab initio KohnSham calculations, U - w - UNK+- UIE.cmm-6, and model potential, EmpiA line of unit slop indicating perfect agreement is shown.
for determining a very good, if not the best, fit of parameters for our chosen functional form. We utilize statistical perturbation theory in order to calculate the potential of mean force as a function of center-of-mass separation between the 18-crown-6 macromolecule and the ammonium substrate in solution. The free energy differencefor a small perturbation fX along the center-of-massseparation R, is given by
with H-A and G ~being A the system Hamiltonian and free energy, respectively, of the perturbed state at & f X; HR,and GR,represent the same quantities for the reference state at &. The difference of the Hamiltonians reduces to the difference in solvation energies of the solutes and their direct interaction with each other. We evaluate the ensemble average via NPT Monte Carlo simulations at a pressure of 1 atm and a temperature of 25 OC. In order to map out the complete potential of mean force, a series of Monte Carlo simulations must be performed over a range of &‘s. Subsequently, from the potential of mean force, we are able to obtain an association constant,K,,and its conjugate free energy of binding, AGB, as
- e,)*
W = exp
2 kdn + -E-[ 1 + cos(n4)l
5 ,
2
kT Configurational averages for a property M were then obtained by
(W = W / W / ( l / W
(7) In addition, preferential sampling was achieved following the sphere scheme of Owicki and Scheraga.’5 We define all in molecules as those lying within a distance k of any crown or ammonium nitrogen site, the remainder being out molecules. k was chosen to be 7.5 A, which is approximately the extent of the first solvation shell in CC4 and the second in water. fi, an arbitrarily adjustable probability, defines how often to move the out molecules relative to the in ones. The out molecules are moved as frequently as the in molecules as fi approaches 1 and less so as fi approaches 0. This parameter was adjusted during the simulations to ensure that the in molecules attempted moves twice as frequently as the remaining out molecules. Volume moves were attempted every six cycles, a cycle being completed when the number of attempted moves equal the number of particles in the system. This corresponds to 762 and 2706 attempted configuration generations between each volume move for CCh and HzO, respectively. The new configurations were accepted with a slightly modified Monte Carlo criterion
K, = NAc47r? exp( F -w)dr AGB = -kT In K,
(5b)
with w being the potential of mean force, N AAvogadro’s number, and r, the geometric association limit. NPT Monte Carlo simulations, the details of which can be found in our previous paperJ2were conducted with one 18-crown-6 molecule and one ammonium cation in solution wherein 432 water and 108 carbon tetrachloride molecules represented polar and apolar environments,respectively. TheNPT ensemblereproduces experimental conditions and thus generates Gibbs free energies. Our simulationswere conducted in a periodic cubic box of length -25 A wherein the minimum image criterion was imposed by truncating interactions beyond the cutoff limit of half the box length. New configurations were generated with standard Monte Carlo moves. The water molecules undergo rotation in Eulerian coordinates and translation in Cartesian coordinates, whereas the carbon tetrachloride molecules undergo only translation since they are treated in the united atom approximation. New configurations of the crown consist of a bulk motion with the
with p ~ being m the NPT ensembleprobability density and a! the underlying stochastic transition matrix. The subscriptsdesignate the stateof the system before (old) and after (new) the attempted move. A sequence of simulations spanning a range of Re's were conducted for each distinct solvent environment. For the CC4 case, a random initial configuration was generated at & = 10 A, and then the system was equilibrated for 3400 volume moves corresponding to -2.6 X 106 configurations. Following this, simulations of 3400 volume moves in length were conducted sequentially at & = 10, 10 - X, ..., X (A = 0.125 A). In each instance, the first 400 volume moves ( 3.0 X 1Os configurations) were utilized for equilibration and the remaining 3000 (-2.3 X 106configurations)for data collection. Analogously for the H20 case, a random initial configuration was generated at & = 8 A, and then the system was equilibrated for 1100 volume moves (-3.0 X 106 configurations). A sequence of simulations were then performed at & = 8, 8 - A, ..., X (A = 0.125 A) for 1100
-
Ammonium Cation Complexation by 18-Crown-6 volume moves each, 100 (-2.7 X los configurations) for equilibration and 1000 (-2.7 X lo6 configurations) for data analysis. For both CC4 and HzO, the final configuration of each preceding run, with the & distance accordingly decreased by A, served as the starting point for the next one. Physically, we could imagine the ammonium cation being brought closer to the crown incrementallywith step size X. In our simulations,we also consider the effects of the perturbation step size, A, upon the calculated potentials of mean force and related quantities. We will discuss this X dependence in detail in the following section. In order to maintain computational tractability, a decrease of X by one-half or one-fourth for each sequence of simulations is accompanied by a corresponding decrease in the total simulation length for each individual run.
The Journal of Physical Chemistry, Vol. 97, No. 43, 1993 11295 0.0
T -
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1 a 0
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-40.0
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IV. Results and Discussion We first examine the potential of mean force as well as AGB in the two solvents for different perturbation step sizes, A. We then examine the reorganization of the crown molecule during the complexation prccess, and finally, we examine the restructuring of solvent molecules about the ammonium cation. The perturbation method permits the determination of free energy changes in both directions from a singlesimulation. Hence, for each series of simulations along the & coordinate, we map out two free energy curves corresponding to perturbations from the reference state in either the +A or -A direction. Ideally, the two curves should be identical since the free energy is a state function. This lack of correspondence between the two curves may be attributed to some irreversibility in the simulations which can be systematically reduced by using a smaller step size, A, given that the configurational space is being adequately sampled. Nevertheless, the two free energy curves should bound the true curve and provide an estimate of the statistical error of the simulations. In all figures of the potential of mean force, we show only the average of the two curves for each value of A, omitting the other two for clarity. However, we shall report the bounds explicitly when discussing AGB. The choice of the step size X is crucial from the point of view of computational tractability. This is so because the system under consideration is rather complicated with many internal and orientational degrees of freedom. Prior free energy perturbation ~tudies35*3~ have suggested that A should be sufficiently small so that the free energy difference at each interval not exceed twice the thermal energy. For each of our systems, the x's chosen satisfied this general recommendation except for X = 0.125 A. In fact, as X decreases, the gap between the two curves decreases as expected. In the case of carbon tetrachloride, three series of runs were conducted with perturbation step sizes of X = 0.125,0.0625, and 0.031 25 A. We anchor the potential of mean force (PMF) to zero at a center-of-massseparation of & = 10A. Figure 4 depicts the PMF curves. We note that the PMF remains relatively constant, thus indicating that the solutes are well solvated, until it reaches & 8 A where it begins to drop rapidly. We see that the general features of the three curves are the same with each exhibiting a deep minimum at approximately & = 1 %I centerof-mass separation. Note that for X = 0.125 A we find a nearly monotonic decrease in the PMF with decreasing separation whereas the X = 0.0625 A and 0.031 25 A cases appear to exhibit two distinct regions between ranges 1-4 and 4-8 A. In the latter region, the PMF undergoes a sharp drop followed by a gentle decline. We can attribute this to the squeezing out of the solvent between the 18-crown-6 and ammonium molecules. This effect has been observed previously in similar system^.^^.^^ The second region corresponds to the movement of the cation into the crown cavity. Lastly, the slight upturn may be explained by looking at the quantum mechanical calculations or, equivalently, the model potential. Basically, the cation prefers not to sit in the mean
-20.0
Y
4.0
2.0
6.0
100
8.0
CENTER OF MASS SEPARATION [angstrom]
Figure 4. Potential of mean force for 18-crown-6/ammoniumcomplex as a function of center-of-massseparation. Monte Carlo simulationsin carbon tetrachloride at 25 OC and 1 atm for various perturbation step sizes, A.
fs 2
60.0
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Figure 5. (a, top) Potential of mean force for 18-crown-6/ammonium complex as a function of center-of-mass separation. Monte Carlo simulations in water at 25 OC and 1 atm for various perturbation step sizes, X. (b, bottom) Potential of mean force for 18-crown-6/ammonium complex as a function of center-of-mass separation. Monte Carlo simulationsin water at 25 OC and 1 atm for X =,0.0625 A. Gap between bounding curves at three points & = 0.0, 1.0, and 2.0 A.
plane of the crown. In the gas phase, ammonium opts to sit 0.415
A displaced from the crown center.s4 Solvent effects lead to a greater displacement from the mean plane. The PMF for complexation in water is shown in Figure Sa. It differs significantly from that of carbon tetrachloride. Since greater electrostatic screening of NH4+ can occur in water than in carbon tetrachloride, we anchor the PMF to zero at 8 A rather than at 10 A. We see that instead of a deep and sharp minimum, the PMF has a relatively broad and shallow minimum about & = 1A. Again, the ammonium is further removed from the crown
Ha and Chakraborty
11296 The Journal of Physical Chemistry, Vol. 97, No. 43, 1993 cavity than gas-phase calculations indicate. However, the 1-A distance determined here compares quite well with the crystallographic value of 1.07 A.57 The shallowness of the region is attributable to unfavorabledehydration effects of the cation and its relative broadness to the cation attempting to maximize its interactions with the surrounding solvent along with the crown molecule. In fact, the interaction between ammonium and one water molecule74can become as large as -20 kcal mol-', which is 25% of the maximum crown/ammonium interaction. Thus, the features of the PMF are indicative of the strong competitive interactions that the cation and water molecules have with the 18-crown-6 and with each other. While the A = 0.125 A case indicates binding over a wide range of center-of-massseparations with the potential of mean force being less than zero, the X = 0.0625 A case seems to imply that most of the binding occurs at large & rather then near & = 1 A. Figure 5b shows the PMF for X = 0.0625 A along with the gap between the two bounding f X curves at three illustrative points. We note that, within statisticaluncertainty, the results suggest thepresenceofa binding minimum about Rc = 1 A. Computational resource limitations have precluded us from examiningthe effects of further reducing A. Nonetheless, we expect that the PMF would exhibit a relatively weak binding minimum in such a case. Structural aspects which we discuss below lend support to this conclusion. The lack of a deeper minimum at 1 A may also be attributable to the lack of a counterion in our simulations. The presence of the anion about ammonium would disrupt the water network surrounding thecation. Thus, the short-rangebinding interactions between the crown and ammonium would not necessitate breaking of the ordered structure of water about the ammonium solute as much as when there is no counterion present, consequently giving rise to a deeper contact minimum. We will discuss the order of water about ammonium shortly below. Weapply eqs Sa and 5b (r, = 10 A for CCl,, r, = 8 A for H2O) with the average potential of mean force in determining AGE. Likewise, the bounds on AGE are derived from eqs Sa and 5b in conjunction with the two curves corresponding to f X that bound the average potential of mean force. For the average potential of mean force with the smallest error bounds, the most reliable A G Bbinding ~ ~ that weobtain for thecarbon tetrachloride system is-39.78kcalmol-1(withboundsof-34.40,45.17;X =0.031 25 A). Unfortunately, no experimentaldata are availablefor direct comparison with this system. For water, we obtain a value of -1.50 kcal mol-' (with bounds of 0.83, -9.02; X = 0.0625 A), which compares quite well with the experimental value of -1.68 f 0.08 kcal mol-1.12 We note that by assuming only a radial dependence of the potential of mean force, w(r), eqs Sa and 5b would tend to overestimate the extent of binding since some trajectories between the cation and the crown would not experience any attraction. For instance, on a given sphere defined by &, regions where the two solutes may overlap are treated the same as when the cation interacts favorably with the crown. Thus, the extent of complexation is being underestimated more than the AGEvalue indicates. This would seem to suggest that the PMF for complexation in water (with X = 0.0625 A) is too large. Nevertheless, although the good correspondence may itself be overly fortuitous, it does augment our confidence in our model potential characterizing the complexing components. Most important are the general features that are observed in the PMF and the observed structural reorganization during complexation which we discuss next. The remaining discussion will concentrate on the X = 0.125 A simulations as these cases were run for the longest number of configurationsat each &, providing the most reliable ensemble averages. However, we observe similar trends for the other values of A. We first discuss the restructuring of the crown molecule during the complexation process. We characterize the conformation of the crown by its OC-CO torsion angle distribution.
o'20
2 - Rc ......
7.00 anpslmms Rc -2.00 RC I 0.75
Figure 6. OC40 torsion angle distributions of 18-crown-6 at various center-of-separations, &, from ammonium cation. Simulations conducted in carbon tetrachloride at 25 O C and 1 atm with X = 0.125
A.
- Rc ......
-
I
7.00 angstroms
RC I 2.00 RC
0.75
Flgwe 7. OC-CO torsion angle distributions of 18-crown-6at various
center-of-mass separations, &, from ammonium cation. Simulations conducted in water at 25 O C and 1 atm with X = 0.125 A.
- Rc = 7.00 a n g s t " _ _ _ RC = 2.00
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10.0
r [mngmml
Figure 8. Radial distribution function of H2O oxygen (0) about ammonium hydrogen (H)at various centersf-mass separations, &. Simulations conducted in water at 25 OC and 1 atm with X = 0.125 A.
For CCL, Figure 6 shows threccurves representing thedistribution at center-of-massseparations of = 7.2, and 0.75 A. Initially, the torsion angledistribution indicatesthat at & = 7 A thecrown is in a Crlike structure where we have an approximately2: 1 ratio of gauche-to-trans angles. As the cation approaches & = 2 A, the distribution becomes distorted, and finally at & = 0.75 A, the distribution is exclusively gauche-like,indicative of a D ~ ~ l i k e structure. Thus, we are observing reorganization of the crown during complexation from first principle considerationsonly. The
The Journal of Physical Chemistry, Vol. 97, No. 43, 1993 11297
Ammonium Cation Complexation by 18-Crown-6 2.0
- Rc = 7.00 angstroms - - - RC= 2.00 1.5
_ _ _ _Rc _ =. 0.75
s
2 1.0 CD
0.5
0.0
0.0
2.0
6.0
4.0
1 3
8.0
r [angstrom]
Figure 9. Radial distribution function of CC4 (LJ) about ammonium hydrogen (H) at various center-of-mass separations, &. Simulations conducted in carbon tetrachloride at 25 OC and 1 atm with X = 0.125
A.
conformationalrestructuringis induced by the ammonium cation as it approachesthe crown. This results purely from our quantum mechanically derived potential and the mediating effects of the solvent since no constraints are placed on the macromolecule beyond one of constant bond lengths. In contrast, in Figure 7 with water, the distribution is initially gauche-like, signifying a &alike structure. Again, as the crown moves closer at Rc = 2 A,the distribution becomes distorted, and finally, at Rc = 0.75 A the distribution remains &alike, albeit rather distorted as in the carbon tetrachloride case. In the apolar solvent, on close approach, the structure of the complex is governed primarily by the direct interactions between ammonium and the crown due to the relative weak Lennard-Jones interactions of the CC4 molecules. Thus, the crown adopts a &alike conformationwhich optimizes the host-substrate interactions. In contrast, for the
polar solvent, the interactionsbetween all species arequitestrong. Consequently, when the cation is at & = 0.75 from the macrocycle, the conformation of the crown is governed not only by the ammonium cation but also by the mediating influences of the water molecules. Nevertheless, the crown retains a &alike structure. One important issue should be noted at this point. From previous simulations and experience, we expect the crown to be preorganized for cation complexation in polar water since it assumes the preferable &alike structure. For the case of CCld, we find a definite shift in the crown conformation from Cr to &like structures. If we were to consider only these factors, circumstances would seem to imply that cation binding should be stronger in water since the restructuring of the crown in CCb requires a definite free energy penalty?’ Given that the binding is stronger in carbon tetrachloride rather than in water clearly implies that solvation effects of the cations are significant enough to affect the total strength of binding in solution. Specifically, the dehydration energy of ammonium from water eliminates whatever advantage preorganization of the crown offers for more favorable binding. In contrast, the carbon tetrachloride solvent creates no such solvation penalty, and the crown provides a favorable microenvironment with a polar interior for cation inclusion and an apolar exterior for complex solubilization. Examining the radial distribution function (RDF) of the solvent molecules about the ammonium cation provides some insight into solvation effects upon the extent of binding. Specifically, for the case of water about ammonium (Figure 8) two large peaks characteristicof the hydrogen-bonded network occur when & = 7 A. However, as the separation distancediminishes,the network is completely destroyed. In contrast, because NH4+ and CC4 interact only through Lennard-Jones sites on N and C C 4 (CC4 has no charge), the RDF is typical of particles governed by Lennard-Jones interactions. Figure 9 shows the RDF of carbon tetrachloride molecules about the hydrogen of ammonium. As the separation decreases, we see that the peaks merely shift. Thus,
C
0
0
0 0 0
0
0 0
0 0
e
0
0
0
0
0
0
0 0
0
0
0 Figure 10. (a, left) Snapshots of configuration of 18-crown-6 in carbon tetrachloride at 25 OC and 1 atm with & = 1.25 A and A = 0.125 A. Ring structure is the crown with carbon darkly shaded and oxygen white. Ammonium nitrogen is lightly shaded. Top view is along perpendicular to crown oxygens’ mean plane with only solvent molecules within 4.5 A of 18-crown-6 or NH4+ shown. Corresponding side view immediately below. (b, right) Same as (a) but with & = 7 A.
11298 The Journal of Physical Chemistry, Vol. 97, No. 43, 1993
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Ha and Chakraborty
99
"$ Figure 11. (a, left) Snapshots of configuration of 18-crown-6 in water at 25 OC and 1 atm with & = 1.25 A, X = 0.125 A. Ring structure is the crown with carbon darkly shaded and oxygen white. Ammonium nitrogen is lightly shaded. Top view is along perpendicular to crown oxygens, mean plane with only solvent molecules within 3.5 Aof 18-crown-6 or NHJ+shown. Corresponding side view immediately below. (b, right) Same as (a) but with Rc = 7 A.
as the crown begins to surround the ammonium ion, the CC14 molecules, due to excluded-volume interactions, are forced to repack about the entire complex. To illustrate these observations, we show snapshots from our simulations for various cases. In each instance, we show the complex along with all surrounding CC14/H20 molecules within 4.5/3.5 A from a top and side perspective. Figure 1Oa,b shows the carbon tetrachloridesystem at R, = 1.25 and 7 A, respectively. At 1.25 A, we see the crown completely envelop the substrate, with the ammonium favorably oriented with the hydrogens directed toward the ether sites. In contrast, at 7 A, the crown is flat and ellipsoidal, indicative of a Cf like structure. The cation approaches the edge rather than the plane of the crown. In this manner, it can at least closely approach one of the ether sites as it attempts to maximize advantageouselectrostaticinteractionsgiven the constant centerof-massseparation constraint. Figure 1la,b depictssnapshots of simulationsinvolving water at 1.25 and 7 A, respectively. Again, at 1.25 A, the cation is situated favorably well into the center of the crown cavity. However, when far away at & = 7 A, we note two points of interest. Firstly, the ammonium is definitely well solvated with its adjoining water molecules oriented in such a fashion as to direct their oxygen atoms toward it. This is in contrast to the equivalent 1.25 A snapshot where although some of the water molecules are favorably oriented with respect to the ammonium, the high degree of organization seen at 7 A is definitely missing. Furthermore, at 7 A, the crown itself is no longer flat as in the equivalent case with carbon tetrachloride but is engaged in hydrogen bonding with the water molecules. These snapshots reinforce theconclusiondrawn from thechange in the RDF which demonstrate that complexation in water is accompanied by a destruction of the H-bonded network.
Although we do not show the radial distribution functions characterizing the structure of the solvent molecules about the crown, we note that, in the case of CCl,, the RDF's are only slightly changed for various center-of-mass separations. The curves maintain their general form with only a slight decrease of the peak heights upon complexation. The ammonium is merely displacing some of the first solvation shell CCb molecules about the crown. In contrast, the RDF's with respect to water demonstrate much more intricate behavior as the crown forms a complex with ammonium. In this situation, the cation significantly alters the structure of solvent molecules about the crown.
V. Conclusion We have reported results of NPT Monte Carlo simulations that clarifycertain issues concerningthe energetics (via statistical perturbation) and structural aspects of 18-crown-6/ammonium cation complexation under realistic conditions. Specifically, the free energy calculationsbetween the complex in solution utilized a potential derived from electronic structure calculations. We discover that 18-crown-6 binds NH4+ with a deeper and sharper minimum in C C 4than in water. Furthermore, in terms of crown conformation, 18-crown-6 reorganizes from a Crlike to a D3a like conformation when complexing NH4+ in CCt. We would also like to emphasize that the only constraint placed on the crown macromolecule is the one of constant bond lengths and that we never diminish the flexibility of the crown by imposing a predetermined conformation. The transition from a Cr to 036 like symmetry occurs naturally as a consequence purely of the intermolecular interactions delineated by the model potential.
Ammonium Cation Complexation by 18-Crown-6 Lastly, the structure of CCl, about 18-crown-6 and NH4+ is governed primarily by packing considerations in contrast with the complex features seen in water arising from the specific and detailed interactions therein. Specifically,we observe a H-bonded network of water about the ammonium cation which is destroyed during complexation with 18-crown-6. This mitigates the advantage of preorganization in the D3plike symmetry for complexation in aqueous environments.
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