4581
J. Phys. Chem. 1991, 95,4581-4582
to the sampling of diatomic and triatomic moleculesC8 and to polyatomic m ~ l e c u l e s . ~ . ~ * ~ An efficient implementation of microcanonical, classical variational transition-state theory (TST)9 based on the use of the EMS procedure has been recently applied to simple bond fissions in SiH2 and S i 2 H t and in linear polyatomic chains.6 This EMS-TST method evaluates the microcanonicalrate constant as8
of probabilities like Q,and P,, the probabilities of larger sets of solid sites must be specified. Specifically, in one dimension, the occupation probabilities for three consecutive sites on the solid sublattice are needed. The requirement that the corresponding probabilities on the N sublattice be equal will, in turn, require the probabilitiesof still larger sets on the S sublattice, etc., leading to a hierarchy, requiring some truncation procedure in order to obtain even an approximate solution. An analogous problem arises in the two-dimensional calculation. There the number of squares (basic units) with various occupation on the N sublattice must be determined in terms of the Q s and Ps. In order to do this, probabilities giving the number of Occurrences of various 3 X 3 squares of sites on the solid lattice would be required, again leading to a hierarchy. Department of Chemistry, 0340 University of California. Sun Diego La Jolla, California 92093-0340
k ( E ) = j / 2 j d q w(q) 6(qRC - qC)(k?RCl)MC.K/jdq w(q) (2) where W(q) is the microcanonical weight function corresponding to the number of degrees of freedom excluding the center-of-mass motion, qRC = qRC(q) is the reaction coordinate, which may be a function of some or all of the coordinates q, and qc is the critical value required for reaction. The integrals over q in (1) are understood to be over reactant configuration space. The average absolute velocity through the critical surface is given by
John C . Wheeler
( I ~ R C I ) M C , K=
Received: November 20, I990
j d p ~ ( T ( P) K ) M R C I ~~ (~TP( P ) K)
where
K = E - U(q)
Sir: In a recent article’ Litniewski described the derivation of the microcanonical weight function
W(q) = (I,ZJc)-’/Z[E - E,,, - U(q)](3N-8)/Z
(1)
where E is the energy of the system, U(q) is the potential energy at a configuration q, and L is the total number of degrees of freedom in the momentum space. The weight function in (1) was used to derive expressions for partial derivatives of thermodynamic quantities. These expressions were employed in a molecular dynamics (MD) simulation of Lennard-Jones particles. The microcanonical weight function in (1) was previously derived by Severin et ala2in the context of the efficient microcanonical sampling (EMS) procedure.” They considered the case L = 3N, which is relevant to the microcanonical sampling of a 3D N-atom system with regard to the total energy. The trivial extension has also been made to the cases L = N - 1 3 s 6 for the sampling of a 1D N-atom chain with respect to the internal energy, and L = 3N - 3‘s6t8 for the sampling of a 3D N-atom system with respect to the internal energy. The microcanonical weight function has been found very useful in the microcanonical sampling of polyatomic systems via the EMS procedure.” Microcanonical initial states (q,p) are generated by performing a Markov walk in reactant configuration space q subject to the weight function in (1). The appropriate momenta p are chosen from Gaussian distributions that are scaled to reflect the correct kinetic energy, E - U(q). The EMS procedure and its derivation and efficiency are discussed at length elsewhere.6 The EMS procedure has been successfully applied in various forms
I
(5)
where I,, Ibrand I, are the three principal moments of inertia and E,, is the rotational energy. For a linear molecule W(q) = T ’ [ E- E, - U(q)](”-’)/2
(6)
where I is the principal moment of inertia. These weights have been tested in applications to simple models of OH and H20.’ The sampling of energy and angular momentum resolved states is of significance, for example, in the field of unimolecular reaction dynamics. Potential applications include the estimation of the angular momentum resolved microcanonical rate constant k(E,J)9 and collisional transition probabilities P(E’,JIE,J).’ The energy and angular momentum resolved weight functions in (5) and (6) can be used to evaluate thermodynamic quantities in an MD context in a way analogous to the approach employed by Litniewski’ for the microcanonical weight function in (1). In fact MD simulations of a completely isolated set of interacting molecules normally conserve both energy and the total angular momentumi0and thus an energy and angular momentum resolved (EJ)ensemble would therefore be more appropriate than the microcanonical ensemble characterized by (1). However, in the case of MD performed with periodic boundary conditions, the total angular momentum is not conserved.1° To explore the relationship of the EJ ensemble with the other ensembles, let us consider a system of N atoms with internal energy E and total angular momentum J enclosed in a volume V. Let us also restrict ourselves to consideration of nonlinear configurations as the contribution of exactly linear configurations is negligible. The statistical entropy for the EJ ensemble may be as defined by analogy with the microcanonical S = kB In [ p ( E , J ) dE dJl (7)
(1) Litniewski, M. J . Phys. Chem. 1990, 94,6472. (2) Severin, E. S.; Frcasier, E. C.; Hamer, N . D.; Jolly, D. L.; Nordholm, S. Chem. Phys. Lett. 1978, 57, 117. Severin, E. S. B.Sc. Honours Thesis, University of New South Wales, R. M. C. Duntroon, 1977. (3) Schranz, H. W.; Nordholm, S.; Frcasier, B. C. Chem. Phys. 1986,108, 69, 93, 105. Schranz, H. W. Ph.D. Thesis, University of Sydney, 1984. (4) Hippler, H.; Schranz. H. W.; Troe, J. J . Phys. Chem. 1986,90,6158. Schranz, H. W.; Troe, J. J . Phys. Chrm. 1986, 90,6168. ( 5 ) Nyman, G.; Rynefors. K.; Holmlid. L. J. Chem. Phys. 1988.88.3571. Nyman, G.; Rynefors, K.; Holmlid, L. Chem. Phys. 1988, 134, 355. (6) Schranz, H. W.; Nordholm. S.;Nyman, G. J . Chem. Phys. 1991,94, 1487. Schranz, H. W., manuscript in preparation. (7) Nyman, G.; Nordholm, S.; Schranz. H. W. J . Chem. Phys. 1990,93, 6767. (8) Schranz. H. W.; Raff, L. M.; Thompson. D. L. Chem. Phys. krr. 1990, 171.68. Schranz, H. W.; Raff, L. M.; Thompson, D. L. J. Chem. Phys. 1991, 94. 4219.
0022-3654191 1S02.50IO , ,12095-458 -
(4)
A significant increase in computational efficiency in the evaluation of the microcanonical rate constant k(E) was achieved by performing Markov sampling in configuration space q as well as by employing analytical averages of ( 1qRcl)MC,K over momentum space P . ~ A recent extension of the microcanonical weight function has been made to the case of an angular momentum resolved ensemble.’ In the case of a nonlinear 3D molecule with N atoms
On the Mkrocanonical Weight Function
W(q) = (E - U(q))(L-2)/2
(3)
(9) Miller, W. H. J . Chem. Phys. 1974,61,1622; 1976,65,2216. Gamtt, B. C.; Truhlar, D. G. J . Phys. Chem. 1979,83, 1052. Truhlar, D. G.; Garrett, B. C. Acc. Chem. Res. 1980,13,440. Pechukas, P. Annu. Rev. Phys. Chem. 1981,32,159. Truhlar, D. G.; Hasc, W. L.; Hynes, J. T. J. Phys. Chem. 1983. 87, 2664, 5523. (10) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon: Oxford, 1987. (1 1) McQuarrie, D. A. Srcrrisrical Mechanics; Harper and Row: New York. 1976.
0 1991 American Chemical Societv -
4582 The Journal of Physical Chemistry, Vol. 95, No. 11, 1991
where kB is Boltzmann's constant and the EJ density of states p(E,J) can be expressed by using (5) as7
Comments ensemb1e.l From (7) the pressure is given by
p(E,J) = C j d q (ZJd,)-1/2[E - E,,, - U(q)](3N-8)/2 (8) where C is a constant independent of the configuration q. The corresponding temperature and pressure for the EJ ensemble can be obtained by differentiating the equation for the entropy:",'*
where (8) can be written as
(10) where (X)denotes the ensemble average of X over the EJ weight function of (5). The derivation of (9) for the temperature T i s straightforward. However, the derivation of (10) for the pressure P is somewhat more involved and is given in more detail in the Appendix. The derivation of further partial derivatives of thermodynamic quantities is possible using standard thermodynamic relations."-'2 For the particular case where the total potential energy of the N-body system is pairwise additive'-"
where rl1is the distance between atoms i and j , the pressure can be written as
Each of the moments of inertia depend quadratically on the distance so that from (A3) aAq)/av
2:
-Aq)/V
(A81
Differentiation of (A4) with respect to volume yields where1."
aww -av _. 3N - aE;ld+
[
2
!%!I
Equations 9 and 12 are clearly analogous to eqs 7 and 35 and eqs 8 and 36, respectively, in ref 1. Because of the conservation of both energy and angular momentum, the EJ ensemble and (9), (lo), and (12) could be considered as more appropriate for use with or comparison with results of the MD method. However, if as is common in MD simulations, the number of atoms N is large and the angular momentum is low such that the rotational energy Em is negligible, then there is negligible difference of the definitions of T and P for the EJ ensemble from the corresponding definitions for the microcanonical and canonical ensembles.' Acknowledgment. Thanks are due to Prof. Lionel Raff and Prof. Donald Thompson for helpful discussions and encouragement. Financial support from the Air Force Office of Scientific Research under Grant AFOSR-89-0085 is gratefully acknowledged. Appendix
The derivation of the equation for the pressure P in the EJ ensemble is sketched here in more detail. The approach is similar to that of McQuarrie (ref 11, p 261) for the canonical ensemble and leads to results analogous to those for the microcanonical (12) Berry, R. S.;Rice, S.A.; Ross, J. Physical Chemistry; Wiley: New York, 1980. Mohling, F. Statistical Mechanics: Methods and Applicationr; W h y : New York, 1982. Chandler, D. Introduction to Modern Statistical Mechanics; Oxford University Reas: Oxford, 1987. Hecht, C. E.Statistical Thermodynamics and Kinetic Theory; Freeman: New York, 1990.
[E - E,(q)
av
- ~(~)](3N-I0)/2 (A9)
The rotational energy is7 E,,,
(A 10)
Xw-J
where the angular velocity is
= I-'J (A1 1) and the total angular momentum vector relative to the center of mass position r, and momentum p, is w
N
J
X(ri - rem)
I= 1
X (PI
- ~cm)
By substituting (AS) into (AlO)-(A12), we find that J Thus, E,, a and and I a p/3, so that w 0:
(Ala 0:
v'f3
~&,(q)/av= 0
(A1 3) Substitution of (A4)-(A9) and (A13) into (Al) yields the desired result in (10). ~
~~
~~~
Current address: Research School of Chemistry, Australian National University, G.P.O. Box 4, Canberra, ACT 2601, Australia.
Department of Chemistry Oklahoma State University Stillwater, Oklahoma 74078
Harold W. ScbrPazt
Received: October 1 1 , 1990
