Table 2.
Relative stabilities (kcal/mol) of [n]collarene-cation complexes in the gas phase and in aqueous solution
| [n]collarene |
n = 6
|
n = 8
|
n = 10
|
||||||
|---|---|---|---|---|---|---|---|---|---|
| Li+ | Na+ | Na+ | K+ | Rb+ | K+ | Rb+ | Cs+ | ||
| ΔEgas | HF/3–21G | −68.3 | −35.7 | −65.9 | −48.1 | −32.9 | −48.7 | −46.2 | −40.8 |
| B3LYP/6–31G* | −67.7 | −34.2 | −56.0 | −43.1 | — | — | — | — | |
| MC/OPLS | −44.8 | −20.5 | −33.6 | −29.3 | −24.1 | −25.6 | −24.5 | −22.1 | |
| MD/Polar. Pot. | −71.3 | −41.8 | −46.5 | −40.8 | −36.3 | −32.8 | −31.7 | −29.9 | |
| ΔGgas | HF/3–21G | −58.8 | −25.6 | −57.7 | −38.5 | −22.7 | — | — | — |
| ΔΔEgas | HF/3–21G | 0 | 32.6 | −17.8 | 0 | 15.2 | −7.9 | −5.4 | 0 |
| B3LYP/6–31G* | 0 | 33.5 | −12.9 | 0 | — | — | — | — | |
| MC/OPLS | 0 | 24.3 | −4.3 | 0 | 5.2 | −3.5 | −2.4 | 0 | |
| MD/Polar. Pot. | 0 | 29.5 | −5.8 | 0 | 4.4 | −2.9 | −1.8 | 0 | |
| ΔΔGgas | HF/3–21G | 0 | 25.7 | −19.3 | 0 | 16.7 | — | — | — |
| ΔΔEaq | SCRF/HF/3-21G | 7.9 | 0 | ||||||
| ΔΔGaqcomplex | MC/OPLS | 0 | 15.5 | 3.4 | 0 | 2.5 | 3.9 | 2.2 | 0 |
| ΔΔGaqsingle | MC/OPLS | 0 | 12.1 | 0.4 | 0 | (10.8) | 0.5 | (−0.3) | 0 |
| ΔΔGaq | MD/Polar. Pot. | 0 | 3.3 | 4.9 | 0 | 0.9 | 4.3 | 2.3 | 0 |
ΔE and ΔG are the zero-point uncorrected interaction energy and Gibbs free energy, respectively. See Fig. 2 for the definitions of ΔΔGaqcomplex and ΔΔGaqsingle. M1 ⇒ M2 denotes the substitution reaction in aqueous solution: [n]collarene-M1+ + M2+ → [n]collarene-M2+ + M1+. ΔΔGaq denotes the Gibbs free energy change of the substitution reaction at room temperature and at 1 atm pressure. Positive signs of ΔΔE and ΔΔG in the free energy perturbation calculations indicate that the corresponding cation-[n]collarene complex is less stable than the reference cation-collarene complex (ΔΔE = 0, ΔΔG = 0). To calculate the free energy change between two different cation-[n]collarene systems in the MC simulations, we followed the approach used by Kumpf and Dougherty (7); ΔΔGaqcomplex is used for the cases where the interactions between a rigid [n]collarene and a cation in the complexes were explicitly taken into account in free energy perturbation calculations; ΔΔGaqsingle is used when the OPLS interactions between a rigid [n]collarene and cations were neglected by treating the complex as one single rigid solute in the free energy perturbation calculations, but the ab initio interaction energy change was added to the resulting free energy change instead of the OPLS interaction energy change. The interaction energies of cations with [6]-collarene and [8]collarene were corrected with the B3LYP/6–31G* values, and those with [10]collarene with the HF/3–21G values (because the B3LYP/6–31G* values are not available). The ΔΔGaqsingle may not be reliable when the OPLS interaction energy difference between two cation-collarene complexes is significantly different from the corresponding ab initio interaction energy difference. However, the bounds can be estimated for the errors resulting from the deficiencies in the empirical potential. The MC simulations were carried out at constant temperature (298 K) and pressure (1 atm) with the complex at the center of a cubic box 24.8 Å containing 512 TIP4P water molecules under the periodic boundary condition. The cutoff distance was 11 Å. A series of 20 simulations with double wide sampling was performed for each system. Each perturbation step involved 106 configurations of equilibration followed by averaging over 5 × 106 configurations. The SCRF calculations were carried out at a single point HF/3–21G level in polarized dielectric medium (water: ɛ = 78.5). The interaction energies were obtained using the equation: ΔESCRF = ESCRF([n]collarene-M+ + (H2O)6) − ESCRF ([n]collarene) − ESCRF(M+ + (H2O)6); see Fig. 3. In case of MD simulations using the polarizable potential, we followed the approach used by Caldwell and Kollman (11); the potential was composed of charges obtained from a restrained electrostatic potential (RESP) fit of quantum mechanical electrostatic potential, the van der Waals parameters determined at the additive level, the atomic polarizabilities taken from Applequist except for the benzene sp2 carbon (0.36 Å3), and the POL3 water potential (11, 50, 53). The Gibbs free energy perturbation MD simulations were carried out as an NPT ensemble (298 K, 1 atm). The box sizes for the simulations of [6]-, [8]-, and [10]-collarenes were 27.0 Å × 26.6 Å × 25.3 Å, 28.9 Å × 28.9 Å × 25.3 Å, and 30.4 Å × 30.6 Å × 25.4 Å, with 487, 540, and 618 water molecules included. The cutoff distance was 10 Å. The number of simulation windows was 20. Each simulation consisted of 3 ps equilibration and 3 ps sampling. A 2-fs time step was used with all OH bonds and HOH bond angles of water molecules constrained by shake algorithm.