J. Phys. Chem. 1990, 94, 431-434
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Calculation of the Molar Volume of Electron Solvation in Liquid Ammonia Massimo Marchi, Chemistry Department, McMaster University, Hamilton, Ontario L8S 4 M l . Canada
Michiel Sprik, and Michael L. Klein* Chemistry Department, University of Pennsylvania, Philadelphia, Pennsylvania 191 04-4323 (Received: January 23, 1989; In Final Form: July 13, 1989)
Path integral Monte Carlo simulations in the (NpT) ensemble have been used to estimate the volume change that accompanies electron solvation in liquid ammonia. By comparison with existing results in the (NvT) ensemble, we find only minor structural differences between the solvated electron at constant pressure and at constant volume. At constant pressure, the electron wavefunction is slightly expanded. However, the calculated volume change (about 75 cm3 mol-I) is in fair agreement with the experimental value (about 100 cm3 mol-').
1. Introduction The increasing power and speed of modern computers have led to an enormous number of studies on classical molecular systems by means of molecular dynamics (MD) and Monte Carlo (MC) calculations.' Recently, these simulation techniques have been extended to investigate quantum system^.^-^^ By far the most popular approach is derived from the Feynman path integral formulation of quantum equilibrium statistical mechanic^.^^*^^ Path integral MC6*7*'2*20 or MD293,8*13 simulations have now been performed on a variety of systems. Other approaches to solvation problems involving a quantum particle (electron) interacting with solvent molecules are also being e ~ p l o r e d . ~ ~ J ~ , * ~ - * ~ Our initial path integral Monte Carlo (PIMC) simulation of an electron in a molecular liquid was carried out in ammonia4 ( I ) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon: Oxford, 1987. (2) Parrinello, M.; Rahman, A. J . Cfiem. Phys. 1984, 80, 860. (3) DeRaedt B.; Sprik, M.; Klein, M. L. J . Chem. Pfiys. 1984,80, 5719. (4) Pollock, E. L.; Ceperley, D. M . Phys. Reu. E 1984, 30, 2555. (5) Thirumalai, D.; Hall, R. W.; Berne, B. J. J . Cfiem. Phys. 1984, 81, 2523. (6) Kuharski, R. A.; Rossky, P. J . J . Cfiem. Pfiys. 1985, 82, 5164. (7) Sprik, M.; Impey, R. W.; Klein, M. L. J . Cfiem.Phys. 1985,83, 5802. (8) Landman, U.; Scharf, D.; Jortner, J. Phys. Reu. Lett. 1986, 54, 1860. (9) Parrinello, M.; Car, R. Pfiys. Reu. Lett. 1985, 55, 2471. (IO) Sprik, M.; Impey, R. W.; Klein, M. L. Pfiys. Rev. Lett. 1986, 55, 2326. ( I I ) Jonah, C. D.; Romero, C.; Rahman, A . Cfiem.Pfiys. Lett. 1986,123, 209. (12) Ceperley, D. M.; Pollock, E. L. Phys. Rev. Lett. 1986, 56, 351. (13) Schnitker, J.; Rossky, P. J . J . Cfiem. Phys. 1986, 86, 3471. (14) Berne, B. J.; Thirumalai, D . Annu. Rev. Pfiys. Chem. 1987, 37, 401. ( I 5 ) Coker, D. F.; Berne, B. J.; Thirumalai, D. J . Cfiem.Pfiys. 1987,86, 5689. (16) Wallqvist, A.; Thirumalai, D.; Berne, B. J. J . Chem. Pfiys. 1987,86, 6404. (17) Selloni, A.; Carnevali, P.; Car, R.; Parrinello, M . Pfiys. Rev. Lett. 1987, 59, 823. (18) Car. R.: Parrinello. M. Pfivs. Reu. Left. 1988. 60. 204. (19) Schnitker, J.; Motakabbir,'K.; Rossky, P. J.; Friesner, R. A . Pfiys. Rev. Lett. 1988, 60, 456. (20) Wallqvist, A,; Martyna, G.J.; Berne, B. J. J . Pfiys. Chem. 1988, 92,
by using a novel sampling technique called staging that efficiently sampled the phase space of the s y ~ t e m . ~ ~The - ~ 'main focus of that work was on structural aspects of the solvent molecules in the vicinity of the solvated electron and the nature of the solvated electron state. The electron was found to produce a hole and become self-trapped in the solvent in a rather compact (localized) state. Analogous results have since been found for the hydrated electron.13J6 In a more recent PIMC simulation we have provided some new insights into the thermodynamics of the electron-ammonia syst e ~ n .In~ particular, ~ it was shown that the volume expansion of the solvent produced by the electron was responsible for the positive sign of the experimental entropy change on solvation. In the present paper we tackle the problem of calculating directly the volume change on electron solvation. To do so, we have extended the PIMC simulation technique to the (NpT) ensemble. The direct calculation of the volume change provides an additional test for the electron-ammonia model potential used in our previous Anticipating our results, we find that the large experimental volume change for the solvated electron (ca. 100 cm3 mol-]) is satisfactorily reproduced even though the pseudopotential we employ is rather simple. The paper is arranged as follows: in Section 2 a brief description of the underlying theory and the algorithm used in our simulation is presented. The details and results of the simulation are given in Section 3.
2. Theory 2.1. Path Integral and Staging. According to the Feynman path integral formulation of statistical mechanics, the partition function of a quantum particle in a potential field is isomorphic with that of a closed chain polymer (necklace) composed of P classical particles (beads). Each bead interacts with its adjacent neighbors in the necklace through a harmonic potential, but the interaction with the external field is reduced by a factor Pi.zs The isomorphism becomes exact in the limit P- m . In the (NvT) ensemble, the partition function for a quantum solute can be written schematically as
1711
1 I L 1 .
(21) Barnett, R. N . ; Landman, U.; Cleveland, C . L.; Jortner, J. J . Cfiem. Pfiys. 1988, 88, 4421. (22) Martyna, G. J.; Berne, B. J. J . Chem. Phys. 1988, 88, 4516. (23) Sprik, M.; Klein, M. L. J . Cfiem. Pfiys. 1988, 89, 1592; 1989, 90,
where ri is the position of the ith bead, r l = rP+'
7614. (24) Coker, D. F.; Berne, B. J. J . Chem. Pfiys. 1988, 89, 2128. (25) Marchi, M.; Sprik, M.; Klein, M . L. J . Pfiys. Cfiem. 1988, 92, 3625. (26) Rossky, P. J.; Schnitker, J . J . Pfiys. Chem. 1988, 92, 4277. (27) Sprik, M.; Klein, M. L. Comput. Phys. Rep. 1988, 7, 147. (28) Feynman, R. P.; Hibbs, A . R. Quantum Mechanics and Path Integrals; McGraw-Hill: New York, 1965. (29) Feynman, R. P. Statistical Mechanics; Benjamin: Reading, MA, 1972.
and
0022-3654/90/2094-043 1$02.50/0
(30) Sprik, M.; Klein, M . L.; Chandler, D. Pfiys. Rev. E 1985, 31,4234. (31) Sprik, M.; Klein, M. L.; Chandler, D. J . Chem. Pfiys. 1985,83, 3042.
0 1990 American Chemical Society
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Marchi et al.
The Journal of Physical Chemistry, Vol. 94, No. 1 , 1990
ensemble has to be sampled by a M C simulation according to the probability (6) exp(-@pc - P U N N In L.)
+
The first term in eq 3 represents the harmonic potential acting Although the Metropolis importance sampling algorithm can between adjacent beads. It should be noted that the force constant be readily adapted to sample the phase space of a classical system is proportional to P. In the second term, q5(ri) is the potential field in the (NpT) ensemble, the extension to the coupled quantum acting on the ith electron bead with coordinates ri. The significance electron-classical solvent system is not conceptually straightforof this formulation is that through eq 1 the quantum solute ward. problem is reduced to a classical many-body problem whose Intuitively, one can argue that the fluctuations in the volume Hamiltonian is HefPIn principle, the canonical averages of such of the simulation box should not couple directly to the electron a system can be calculated by using either the M C or MD simwavefunction if the electron is localized in a region small compared ulation technique.32 to the size of the box. Volume fluctuation will then affect an Due to the increase in strength of the harmonic spring that electron in a localized state only indirectly, i.e., via changes in connects two adjacent beads, standard Monte Carlo importance the coordinates of the solvent. Indeed, only in this case does the sampling33of the isomorphic polymer chain becomes increasingly system satisfy the quantum mechanical virial t h e ~ r e m ~for *
