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JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER 13 1 OCTOBER 2001 Statistical mechanics of quantum-classical systems Steve Nielsen and Raymond Kapral Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, ON M5S 3H6, Canada Giovanni Ciccotti INFM and Dipartimento di Fisica, Università ‘‘La Sapienza,’’ Piazzale Aldo Moro, 2, 00185 Roma, Italy 共Received 29 May 2001; accepted 13 July 2001兲 The statistical mechanics of systems whose evolution is governed by mixed quantum-classical dynamics is investigated. The algebraic properties of the quantum-classical time evolution of operators and of the density matrix are examined and compared to those of full quantum mechanics. The equilibrium density matrix that appears in this formulation is stationary under the dynamics and a method for its calculation is presented. The response of a quantum-classical system to an external force which is applied from the distant past when the system is in equilibrium is determined. The structure of the resulting equilibrium time correlation function is examined and the quantum-classical limits of equivalent quantum time correlation functions are derived. The results provide a framework for the computation of equilibrium time correlation functions for mixed quantum-classical systems. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1400129兴 I. INTRODUCTION equations for the quantum-classical density matrix where the evolution operator is expressed in terms of a quantumclassical bracket have also been studied20–26 and their solutions have been formulated in terms of surface-hopping trajectories.26 –28 In this paper we develop the statistical mechanics of mixed quantum-classical systems. We take as a starting point the evolution equation for the mixed quantum-classical density matrix.21,23–26,29 This equation may be derived by partial Wigner transforming the quantum Liouville equation over the bath degrees of freedom corresponding to massive particles and expanding the resulting evolution operator to linear order in the mass ratio (m/M ) 1/2, where m and M are the characteristic masses of the quantum subsystem and bath particles, respectively.26 The resulting evolution equation can be recast as an integral equation in which classical trajectory segments are interspersed with environment-induced quantum transitions and corresponding bath momentum changes.27 The canonical equilibrium density matrix for quantumclassical systems is constructed to be stationary under the quantum-classical evolution. Its form is derived and compared with its full quantum analog. A linear response derivation is carried out to determine the response function and the forms of the equilibrium time correlation functions appearing in quantum-classical systems. Mixed quantum-classical dynamics and the associated correlation functions present differences from their full quantum analogs: Identities among quantum correlation functions hold only approximately in the quantum-classical limit and properties such as time translation invariance are also only approximately valid. The paper is organized as follows: Section II presents the formal structure of mixed quantum-classical dynamics and contrasts it with that of full quantum mechanics. In Sec. III we discuss the canonical equilibrium density matrix and in Sec. IV a linear response derivation of the response function Often situations arise where it is appropriate to study composite dynamical systems with interacting quantum mechanical and classical degrees of freedom. In condensed matter physics such situations occur when one is interested in the dynamics of a light quantum particle or set of quantum degrees of freedom interacting with more massive particles.1–3 Specific examples include proton transfer,4 solvation dynamics of an excess electron,5 and nonradiative relaxation processes of molecules in a liquid-state environment.6 In these circumstances it is not feasible to attempt a full quantum solution of the Schrödinger equation for the entire system. Consequently one is led to consider the dynamics of a quantum subsystem coupled to a bath where the environmental degrees of freedom are treated classically. Such mixed quantum-classical systems arise in other contexts as well.7 Different formulations of the dynamics of such mixed quantum-classical systems have appeared in the literature. In these reduced descriptions of the quantum dynamics the environmental degrees of freedom are accounted for by the inclusion of dissipative and decoherence terms in the equations of motion,8 –10 through multistate Fokker–Planck dynamics11 or representations by quantum stochastic processes.12 Other approaches utilize a more detailed treatment of the classical environment. These include simple adiabatic dynamics where the classical system evolves on a potential energy surface determined from a single adiabatic eigenstate, or Ehrenfest mean field models where the classical evolution is governed by a mean force determined from the instantaneous value of the quantum wave function.13,14 These descriptions produce a definite classical evolution but do not always give physically correct results. If we relax the continuity of the trajectory of the classical variables,15 we are led to consider surface-hopping algorithms.16 –19 Evolution 0021-9606/2001/115(13)/5805/11/$18.00 5805 © 2001 American Institute of Physics Downloaded 06 Nov 2003 to 130.91.67.156. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp 5806 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Nielsen, Kapral, and Ciccotti ⳵␳ˆ W 共 R, P,t 兲 i ⫽⫺ 共 Ĥ W e ប⌳/2i ␳ˆ W 共 t 兲 ⫺ ␳ˆ W 共 t 兲 e ប⌳/2i Ĥ W 兲 ⳵t ប is carried out and a general discussion of the properties of mixed quantum-classical correlation functions is given. Concluding remarks are made in Sec. V and Appendixes A–C contain additional details. ⫽⫺ ⬅⫺iL̂ W ␳ˆ W 共 t 兲 II. QUANTUM-CLASSICAL DYNAMICS Consider a quantum system which may be partitioned into two interacting subsystems, a quantum subsystem with particles of mass m, and a quantum bath with particles of mass M, (M Ⰷm). In order to describe the distinctive features of quantum-classical dynamics, we begin with a brief overview of the algebraic structure of quantum and classical dynamics in the Wigner representation of the quantum bath. ⬅⫺ 共 H W , ␳ˆ W 共 t 兲兲 Q . ជ ⌳ ) and left In these equations we have defined the right (H ឈ ⌳ ) acting operators, (H ជ ⌳ ⫽Ĥ W 共 R, P 兲 e ប⌳/2i , H 共1兲 where Ĥ is the Hamiltonian of the system. Its formal solution is ˆ 共2兲 with iL̂⫽(i/ប) 关 Ĥ, 兴 the quantum Liouville operator. An alternative form of the evolution equation may be obtained by taking the partial Wigner transform over the bath degrees of freedom. This partial Wigner transform of the density matrix is defined by ␳ˆ W 共 R, P 兲 ⫽ 共 2 ␲ ប 兲 ⫺3N 冕 dze i P•z/ប 冓冏冏冔 z z R⫺ ␳ˆ R⫹ , 2 2 冕冓冏冏冔 dz e ⫺i P•z/ប R⫹ z z Â R⫺ . 2 2 共3兲共 ÂB̂ 兲 W 共 R, P 兲 ⫽Â W 共 R, P 兲 e B̂ W 共 R, P 兲 , 共9兲 Here p̂ and q̂ are the momentum and position operators, respectively, of the quantum subsystem and V̂ W (q̂,R) is the partially Wigner transformed total potential energy operator, which is the sum of the quantum subsystem, quantum bath, and subsystem-bath potential energies. The second equality in Eq. 共7兲 defines the quantum Liouville operator iL̂ W in the partial Wigner representation and the associated Lie bracket (H W , ) Q . More generally the Lie bracket of two partially Wigner transformed operators is defined as i ជ B̂ ⫺B̂ W Aឈ ⌳ 兲共 Â W ,B̂ W 兲 Q ⫽ 共 A ប ⌳ W i 共 Â e ប⌳/2i B̂ W ⫺B̂ W e ប⌳/2i Â W 兲 . ប W 共10兲 ឈ ⌳ operators are defined as in Eq. 共8兲 with Here the Aជ ⌳ and A the replacement Ĥ W →Â W . The formal solution of Eq. 共7兲 is ជ 共4兲 ឈ ⫽e ⫺iL̂ W t ␳ˆ W 共 R, P,0兲 . 共6兲 and the arrows indicate the directions in which the gradient operators act. Using this result and taking the partial Wigner transform of Eq. 共1兲 we find 共11兲 A similar set of equations may be written for the evolution of any quantum operator Â. In the Wigner representation these equations and their solutions, respectively, take the form dÂ W 共 t 兲 ⫽iL̂ W Â W 共 t 兲 ⫽ 共 Ĥ W ,Â W 共 t 兲兲 Q dt 共5兲 where the left and right acting differential operator ⌳ is the negative of the Poisson bracket operator, ឈ P •“ ជ R ⫺“ ឈ R •“ ជ P, ⌳⫽“ P2 p̂ 2 ⫹ ⫹V̂ W 共 q̂,R 兲 . 2M 2m ␳ˆ W 共 R, P,t 兲 ⫽e ⫺iH ⌳ t/ប ␳ˆ W 共 R, P,0兲 e iH ⌳ t/ប Both the density matrix and quantum operators retain their abstract operator character in the quantum subsystem degrees of freedom but are now functions in the 共R, P兲 phase space of the quantum bath. To proceed we use the fact that the Wigner transform of a product of operators is given by30 ប⌳/2i Ĥ W 共 R, P 兲 ⫽ ⫽ while the corresponding transform of an operator Â is given by Â W 共 R, P 兲 ⫽ 共8兲 ឈ ⌳ ⫽e ប⌳/2i Ĥ W 共 R, P 兲 H The von Neumann evolution equation for the quantum mechanical density matrix ␳ˆ is ␳ˆ 共 t 兲 ⫽e ⫺iL̂t ␳ˆ 共 0 兲 ⫽e ⫺iĤt/ប ␳ˆ 共 0 兲 e iH t/ប , 共7兲 where the partially Wigner transformed Hamiltonian Ĥ W is A. Quantum and classical dynamics ⳵␳ˆ 共 t 兲 i ⫽⫺ 关 Ĥ, ␳ˆ 共 t 兲兴 , ⳵t ប i ជ ␳ˆ 共 t 兲 ⫺ ␳ˆ W 共 t 兲 H ឈ ⌳兲共H ប ⌳ W 共12兲 and ˆ ជ ឈ Â W 共 t 兲 ⫽e iL W t Â W ⫽e iH ⌳ t/ប Â W e ⫺iH ⌳ t/ប . 共13兲关We shall drop the dependence of quantities like Â W (R, P) on the bath phase space coordinates when confusion is unlikely to arise.兴 The Wigner transform of a product of operators satisfies the associative product rule, Downloaded 06 Nov 2003 to 130.91.67.156. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Statistical mechanics of quantum-classical systems 共 ÂB̂Ĉ 兲 W ⫽ 共共 Â W e ប⌳/2i B̂ W 兲 e ប⌳/2i Ĉ W 兲 ⫽ 共 Â W e ប⌳/2i 共 B̂ W e ប⌳/2i Ĉ W 兲兲 , 共14兲 with an obvious generalization to products of n operators. Consider a quantum operator Ĉ⫽ÂB̂ which is the product of two operators. Since the time evolution of Ĉ may be written as Ĉ(t)⫽Â(t)B̂(t), its partial Wigner transform is Ĉ W 共 t 兲 ⫽Â W 共 t 兲 e ប⌳/2i B̂ W 共 t 兲 . 共15兲 We may also obtain this result by considering the action of the Liouville operator on the operator product in the partial Wigner representation: i iL̂ W Ĉ W ⫽ 共 Ĥ W e ប⌳/2i 共 Â W e ប⌳/2i B̂ W 兲兲 ប ⫺ i 共共 Â W e ប⌳/2i B̂ W 兲 e ប⌳/2i Ĥ W 兲 ប ⫽ 共 iL̂ W Â W 兲 e ប⌳/2i B̂ W ⫹Â W e ប⌳/2i 共 iL̂ W B̂ W 兲 . The Poisson bracket is a Lie bracket and satisfies the Jacobi identity. Of course, products of classical phase space functions satisfy the associative property. Consequently, both quantum and classical dynamics have a Lie algebraic structure and products of quantum operators or classical phase space functions satisfy an associative product rule. B. Quantum-classical dynamics The formulation of quantum dynamics in the partial Wigner representation allows the limit of a mixed quantumclassical system to be taken easily. In such quantum-classical dynamics, the full quantum dynamics of the subsystem is taken into account while the bath, in isolation, evolves according to the classical equations of motion. This limit is ជ ⌳ and H ឈ ⌳ by their expansions to first taken by replacing H order in ប: 冉 ប⌳ , 2i ឈ ⌳ →H ឈ ⌳ ⫽ 1⫹ H ប⌳ Ĥ W . 2i ជ ⌳ →H ជ ⌳ ⫽Ĥ W 1⫹ H 共16兲 In writing the last line of this equation we have used the associative property in Eq. 共14兲. From this result we may immediately compute the repeated action of iL̂ W on Ĉ W and, using the definition of the exponential evolution operator as a power series, find Eq. 共15兲 by identification of the two power series expansions. We shall make use of this lengthier route to Eq. 共15兲 in the sequel where the analogous demonstration for mixed quantum-classical systems is not so straightforward. The quantum mechanical Lie bracket, either in its original form as (i/ប) 关 Â,B̂ 兴 or in its partially Wigner transformed form (Â W ,B̂ W ) Q , satisfies the Jacobi identity, 冉共19兲 ⳵␳ˆ W 共 R, P,t 兲 i 1 ⫽⫺ 关 Ĥ W , ␳ˆ W 共 t 兲兴 ⫹ 共兵 Ĥ W , ␳ˆ W 共 t 兲其 ⳵t ប 2 ⫺ 兵 ␳ˆ W 共 t 兲 ,Ĥ W 其兲 ⬅⫺ 共 Ĥ W , ␳ˆ W 共 t 兲兲 ⬅⫺iL̂␳ˆ W 共 t 兲 . 共17兲 We may remark on several limiting situations of this general formulation of quantum dynamics. If the quantum bath is absent and the system comprises only quantum subsystem degrees of freedom, we simply have the usual quantum dynamical description in terms of the von Neumann equation 共1兲. If one considers the quantum bath dynamics alone without a quantum subsystem, one has the ordinary Wigner representation of quantum mechanics and all partially Wigner transformed operators become simple phase space functions: Â W (R, P)→A W (R, P). The classical limit of the quantum bath dynamics, which consists in keeping only terms of order ប 0 in the evolution operator, is obtained by the following truncation of the power series expression of the exponential operator: exp(ប⌳/2i)⫽1⫹ប⌳/2i. In this limit the bracket (H W , ) Q reduces to the Poisson bracket 兵 H W , 其 and the Wigner representation of the quantum Liouville equation becomes the classical Liouville equation, ⳵␳ C / ⳵ t ⫽⫺ 兵 H W , ␳ C 其 ⫽⫺iL C ␳ C (t), whose solution may be written as 共18兲共20兲 The last equalities in Eq. 共20兲 define the mixed quantumclassical Liouville operator L̂, and the bracket which takes the form23,24 共 Â W ,B̂ W 兲 ⫽ i ជ B̂ ⫺B̂ W A ឈ ⌳兲共A ប ⌳ W 冉冉冊冉冊冊 ⫽ i ប⌳ ប⌳ Â W 1⫹ B̂ W ⫺B̂ W 1⫹ Â W ប 2i 2i ⫽ i 1 关 Â W ,B̂ W 兴 ⫺ 共兵 Â W ,B̂ W 其 ⫺ 兵 B̂ W ,Â W 其兲 , ប 2 共21兲 ជ ⌳ is defined as H ជ ⌳ in Eq. 共19兲 with Ĥ W →Â W . The where A bracket reduces to the quantum Lie bracket (iប) ⫺1 关 Â W ,B̂ W 兴 when the classical bath is absent and to the classical Lie bracket 共Poisson bracket兲兵 A W ,B W 其 when the quantum subsystem is absent. Using this notation, the evolution equations for ␳ˆ W (t) and an operator Â W are given, respectively, by ⳵␳ˆ W 共 t 兲 ⫽⫺ 共 Ĥ W , ␳ˆ W 共 t 兲兲 ⳵t ␳ C 共 R, P,t 兲 ⫽e ⫺iL C t ␳ C 共 R, P,0兲 ⫽e ⫺1/2H W ⌳t ␳ C 共 R, P,0兲 e 1/2⌳H W t . 冊冊 The full system evolution, which includes the interaction between the quantum subsystem and classical bath, is then given by the mixed quantum-classical Liouville equation:20–26 共 Â W , 共 B̂ W ,Ĉ W 兲 Q 兲 Q ⫹ 共 Ĉ W , 共 Â W ,B̂ W 兲 Q 兲 Q ⫹ 共 B̂ W , 共 Ĉ W ,Â W 兲 Q 兲 Q ⫽0. 5807 共22兲 and Downloaded 06 Nov 2003 to 130.91.67.156. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp 5808 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 dÂ W 共 t 兲 ⫽ 共 Ĥ W ,Â W 共 t 兲兲 , dt Nielsen, Kapral, and Ciccotti 共23兲 in apparent complete analogy with pure quantum and classical dynamics. However, a number of important differences exist which we now examine. Consider the product of three operators in the mixed quantum-classical limit. From Eq. 共14兲 we have 冉冉冉冊冊冉冊冊冉冉冊冉冉冊冊冊共 ÂB̂Ĉ 兲 W → Â W 1⫹ ⫽ Â W 1⫹ ប⌳ B̂ W 2i ប⌳ 2i 1⫹ B̂ W 1⫹ ប⌳ Ĉ W 2i ប 共共共 Â W ⌳B̂ W 兲 ⌳Ĉ W 兲 4 ⫺ 共 Â W ⌳ 共 B̂ W ⌳Ĉ W 兲兲兲 , 共24兲 where the last term is O(ប 2 ). Thus, the associative product rule is no longer exact but is valid only to terms O(ប 2 ). This has important implications for the dynamics. In particular, in this limit, consider the action of the mixed quantum-classical Liouville operator on the product of two operators, Ĉ W ⫽Â W (1⫹ប⌳/2i)B̂ W : iL̂Ĉ W ⫽ 冉冉冉冊冊冊冉冉冉冊冊冊冉冊冉冊 i ប⌳ ជ ⌳ Â W 1⫹ H B̂ W ប 2i ⫺ i ប Â W 1⫹ ⫽ 共 iL̂Â W 兲 1⫹ ⫹Â W ប⌳ ឈ⌳ B̂ W H 2i ˆ ˆ 冉冉 ⫽Â W 共 t 兲 1⫹ 冊共25兲冊共26兲 Consequently, the evolution of a composite operator in quantum-classical dynamics cannot be determined exactly in terms of the quantum-classical evolution of its constituent operators, but only to terms O(ប), in contrast both to quantum and classical dynamics. This result has implications for the computation of time correlation functions discussed in Sec. V. The Jacobi identity involving the quantum-classical bracket is valid only to terms O(ប): 共 Â W , 共 B̂ W ,Ĉ W 兲兲 ⫹ 共 Ĉ W , 共 Â W ,B̂ W 兲兲 ⫹ 共 B̂ W , 共 Ĉ W ,Â W 兲兲 ⫽O共 ប 兲 ; Here the operator S specifies the rule that must be used to evaluate the exponential operators in the second equality in Eq. 共28兲. Appendix A gives the details of the rules that must be followed in order for the formal solution involving the ឈ ⌳ and right H ជ ⌳ operators to be exponentials of the left H equivalent to that obtained using the quantum-classical Liouville operator. Given this compact formulation of mixed quantumclassical dynamics that exploits the formal similarity with full quantum mechanics, we turn to a description of the statistical mechanics of such systems. In the next section we consider the determination of the equilibrium density matrix for a mixed quantum-classical system and then turn in the following section to a more general study of the properties of equilibrium time correlation functions. ˆ ប⌳ ˆ 共 e iLt B̂ W 兲 ⫹O共 ប 兲 2i ប⌳ B̂ W 共 t 兲 ⫹O共 ប 兲 . 2i 共28兲 The quantum mechanical equilibrium canonical density matrix has the familiar form Thus, while Eq. 共16兲 is exact, the corresponding quantumclassical equation is valid only to terms O(ប). From this result it follows that Ĉ W 共 t 兲 ⫽e iLt Ĉ W ⫽ 共 e iLt Â W 兲 1⫹ ឈ III. EQUILIBRIUM DENSITY MATRIX ប⌳ B̂ W 2i ប⌳ 1⫹ 共 iL̂B̂ W 兲 ⫹O共 ប 兲 . 2i ជ ˆ Â W 共 t 兲 ⫽e iLt Â W ⫽S共 e iH⌳ t/ប Â W e ⫺iH⌳ t/ប 兲 . ប⌳ Ĉ W 2i 2 ⫺ thus, the quantum-classical bracket is not strictly a Lie bracket. As a consequence, Poisson’s theorem, which states that the Poisson bracket 共or the quantum Lie bracket兲 of any two constants of the motion is also a constant of the motion, fails. More generally, from this result we conclude that products or powers of the constants of the motion are constants of the motion only to O(ប). The formal solutions of the equations of motion that parallel those of full quantum mechanics also require some discussion. For example, the formal solution of Eq. 共23兲 can be written as 共27兲 ␳ˆ Qe ⫽Z ⫺1 e ⫺ ␤ H , 共29兲 where Z is the partition function, Z⫽Tr exp(⫺␤Ĥ). Its partial Wigner transform is given by Q ␳ˆ We 共 R, P 兲 ⫽ 冕冓冏冏冔 dz e i P•z/ប R⫺ z Q z ␳ˆ e R⫹ . 2 2 共30兲 We denote the corresponding canonical equilibrium density matrix in the quantum-classical limit by ␳ˆ We (R, P) Q (R, P) that is which is defined to be the approximation to ␳ˆ We stationary under quantum-classical dynamics: i ជ ⌳ ␳ˆ We ⫺ ␳ˆ We H ឈ ⌳ 兲 ⫽0. iL̂␳ˆ We ⫽ 共 H ប 共31兲 The solution of Eq. 共31兲 can be found in the following way: we first write ␳ˆ We as a power series in ប 关or in the mass ratio (m/M ) 1/2 if scaled variables are used兴 ⬁ ␳ˆ We ⫽ 兺 n⫽0 n兲 ប n ␳ˆ 共We . 共32兲 Substituting this expression in Eq. 共31兲 and grouping by powers of ប, we obtain the following recursion relations: for n⫽0: 0兲 i 关 Ĥ W , ␳ˆ 共We 兴 ⫽0 共33兲 Downloaded 06 Nov 2003 to 130.91.67.156. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Statistical mechanics of quantum-classical systems and for n⭓0, n⫹1 兲 n兲 n兲 ,Ĥ W 其 . i 关 Ĥ W , ␳ˆ 共We 兴 ⫽ 21 兵 Ĥ W , ␳ˆ 共We 其 ⫺ 21 兵 ␳ˆ 共We 共34兲 An analogous set of recursion relations may be written for the partial Wigner transform of the full quantum mechanical canonical equilibrium density matrix. These recursion relations are given in Appendix B where it is shown that Q ␳ˆ We (R, P) and ␳ˆ We (R, P) are identical to O(ប). We shall exploit this feature below where the solutions of the mixed quantum-classical recursion relations are presented. It is convenient to analyze the structure of the recursion relations and obtain their solutions in an adiabatic basis. The adiabatic states are defined to be the eigenstates of the quantum subsystem with fixed classical coordinates, ĥ W 共 R 兲兩 ␣ ;R 典 ⫽E ␣ 共 R 兲兩 ␣ ;R 典 , 共35兲 where 共36兲 Taking matrix elements of Eqs. 共33兲 and 共34兲, respectively, we find, 0 兲 ␣␣ ⬘ ⫽0, iE ␣␣ ⬘ ␳ 共We 共37兲 n⫹1 兲 ␣␣ ⬘ n 兲 ␣␣ ⬘ iE ␣␣ ⬘ ␳ 共We ⫽⫺iL ␣␣ ⬘ ␳ 共We ⫹ n 兲 ␯␯ ⬘ J ␣␣ ⬘ , ␯␯ ⬘ ␳ 共We . 兺 ␯␯ ⬘ 共38兲 ␯␯ ⬘ 兺 ␯␯ ⬘ iL␣␣ ⬘ , ␯␯ ⬘ ␳ We ⫽0 This set of equations is equivalent to ␯␯ ⬘ written as a power series in ប. In these equations, with ␳ We using the definitions in Ref. 26, we have the classical Liouville operator, iL ␣␣ ⬘ ⫽ P ⳵ ⳵ 1 ␣ ␣⬘ ⫹F W • ⫹ 共FW , 兲• M ⳵R 2 ⳵P 共39兲 ␣ ⫽⫺ 具 ␣ ;R 兩 ⳵ V̂ W (q̂,R)/ ⳵ R 兩 ␣ ;R 典 FW is the Hellmann– where Feynman force for state ␣, the energy difference E ␣␣ ⬘ (R) ⫽E ␣ (R)⫺E ␣ ⬘ (R), and the term responsible for nonadiabatic transitions, J ␣␣ ⬘ , ␯␯ ⬘ ⫽⫺ 冉冊 ⳵ P 1 •d ␣ ␯ 1⫹ S ␣ ␯ • ␦ M 2 ⳵ P ␣⬘␯⬘ 冉冊 ⳵ P 1 ⫺ •d ␣*⬘ ␯ ⬘ 1⫹ S ␣*⬘ ␯ ⬘ • ␦ . M 2 ⳵ P ␣␯ 共40兲 In this equation the nonadiabatic coupling matrix element is d ␣␣ ⬘ ⫽ 具 ␣ ;R 兩 ⳵ 兩 ␣ ⬘ ;R 典 , ⳵R 共41兲 while the quantity S ␣ ␯ is defined as 冉 P ␣␯ ␣ ⫺F W ␦ ␣␯ 兲 S ␣␯⫽共 F W •d M ␣␯ ⫽E ␣ ␯ d ␣ ␯ 冉 P •d M ␣␯ 冊 Equation 共37兲 provides no information about the diago(0) ␣␣ (0) ␣ nal elements ␳ We ⬅ ␳ We but requires that the off-diagonal (0) ␣␣ ⬘ be zero. Similarly, Eq. 共38兲 for n⫽0 proelements ␳ We (1) ␣␣ vides no information about the diagonal elements ␳ We (1) ␣ (1) ␣␣ ⬘ ⬅ ␳ We but relates the off-diagonal elements ␳ We to the (0) ␣ diagonal elements ␳ We . For n⭓1 Eq. 共38兲 determines the (n) ␣ in terms of the off-diagonal elements diagonal elements ␳ We (n⫹1) ␣␣ ⬘ in of the same order, and the off-diagonal elements ␳ We (n) ␣␣ ⬘ . terms of the diagonal and off-diagonal elements of ␳ We (0) ␣ Thus, provided we know the diagonal elements ␳ We there is sufficient information to determine the complete solution of ␣␣ ⬘ ␳ We . The method used to solve Eq. 共38兲 for the diagonal ele(n) ␣ requires some discussion. Writing this equation ments ␳ We explicitly for the diagonal elements we have n 兲␣ iL ␣ ␳ 共We ⫽ P̂ 2 ĥ W 共 R 兲 ⫽ ⫹V̂ W 共 q̂,R 兲 . 2m 冊兺 ␯⬎␯ ⬘ n 兲 ␯␯ ⬘ 2R共 J ␣␣ , ␯␯ ⬘ ␳ 共We 兲, 共43兲 ␣␣ ⬘ ␣⬘␣* * , ␯ ␯ and ⫽ ␳ We and J ␣␣ , ␯␯ ⬘ ⫽J ␣␣ where we have used ␳ We ⬘ the fact that J ␣␣ , ␯␯ ⫽0 when a real adiabatic basis is chosen. The left-hand side of Eq. 共43兲 consists of a linear self-adjoint differential operator iL ␣ acting on the diagonal elements of (n) ␣ (n) ␣ ␣ the density matrix, iL ␣ ␳ We ⫽兵HW ,␳W 其 and, due to the Poisson bracket form of this operator, one can determine (n) ␣ ␳W only up to a constant of the motion under the adiabatic ␣ . To insure that a solution can be found we Hamiltonian H W must invoke the theorem of the Fredholm alternative which requires that the right-hand side of Eq. 共43兲 be orthogonal to the null-space of iL ␣ . 31 This null-space consists of the con␣ —at stants of the motion under the adiabatic Hamiltonian H W ␣ least any function F(H W (R, P)) of this Hamiltonian. Hence we must verify that the following condition is satisfied: 冕 dR d P 兺 ␯⬎␯⬘ n 兲 ␯␯ ⬘ ␣ 2R共 J ␣␣ , ␯␯ ⬘ ␳ 共We 兲F共 HW 兲 ⫽0. 共44兲 (n) ␯␯ ⬘ ) is an odd We show in Appendix C that R(J ␣␣ , ␯␯ ⬘ ␳ We ␣ ) is an function of P. This, along with the fact that F(H W even function of P since it is a function of the Hamiltonian, guarantees the validity of Eq. 共44兲. Thus, we may write the formal solution of Eq. 共43兲 as n 兲␣ ␳ 共We ⫽ 共 iL ␣ 兲 ⫺1 兺 ␯⬎␯ ⬘ n 兲 ␯␯ ⬘ 2R共 J ␣␣ , ␯␯ ⬘ ␳ 共We 兲, 共45兲 and the formal solution of Eq. 共38兲 for ␣ ⫽ ␣ ⬘ as n⫹1 兲 ␣␣ ⬘ ␳ 共We ⫽ 冉冊 i n 兲 ␣␣ ⬘ n 兲 ␯␯ ⬘ iL ␣␣ ⬘ ␳ 共We ⫺ J ␣␣ ⬘ , ␯␯ ⬘ ␳ 共We . E ␣␣ ⬘ ␯␯ ⬘ 共46兲兺 Using the formal results outlined above, we may construct the form of the mixed quantum-classical canonical equilibrium density matrix. In Appendix B we show that ⫺1 ⫺1 , 5809 共42兲 and specifies the momentum transfer to and from the classical subsystem arising from quantum transitions. ␣ 0 兲␣ ⫺␤HW ␳ 共We ⫽Z ⫺1 , 0 e Z 0⫽ 兺␣ 冕 ␣ dR d Pe ⫺ ␤ H W , 共47兲 (1) ␣ and that ␳ We ⫽0. Then, taking n⫽0 in Eq. 共46兲 we obtain, after some algebra, Downloaded 06 Nov 2003 to 130.91.67.156. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp 5810 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 1 兲 ␣␣ ⬘ ␳ 共We ⫽⫺i ⫹ Nielsen, Kapral, and Ciccotti 冉 ␤ P •d ␳ 共 0 兲 ␣ 共 1⫹e ⫺ ␤ E ␣ ⬘ ␣ 兲 M ␣␣ ⬘ We 2 冊 1⫺e ⫺ ␤ E ␣ ⬘ ␣ 共 1⫺ ␦ ␣␣ ⬘ 兲 . E ␣␣ ⬘ now assume that the system with Hamiltonian Ĥ W is in thermal equilibrium from t⫽⫺⬁ up to t 0 . With this boundary condition, to first order in the external force Eq. 共51兲 yields 共48兲 We may then use these results in the recursion relations to obtain all higher order terms in the expansion. The higher order terms are more difficult to evaluate and, as we shall show in the following section, the explicit solutions to O(ប) will suffice since all results we can obtain are reliable only to this order. ␳ˆ W 共 t 兲 ⫽ ␳ˆ We ⫹ IV. TIME CORRELATION FUNCTIONS A. Response function Standard descriptions of linear response theory32 assume that a system which is in thermal equilibrium in the distant past is subjected to a time dependent external force. The response of the system to this external force is determined by computing the average value of some property at time t using the density matrix determined to linear order in the force. We adopt a similar point of view here except that we suppose the system follows quantum-classical dynamics instead of full quantum mechanics. The mixed quantum-classical system is subjected to a time dependent external force F(t) which couples to the ob† 共† is the adjoint兲 and is applied from the distant servable Â W past. The partially Wigner transformed Hamiltonian of the system in the presence of the external force is † F共 t 兲. ĤW 共 t 兲 ⫽Ĥ W ⫺Â W 共49兲 The evolution equation for the density matrix is obtained ᠬ ⌳ (t) and H ᠭ ⌳ (t), ជ ⌳ and H ឈ ⌳ by H from Eq. 共22兲 by replacing H respectively, to yield ⳵␳ˆ W 共 t 兲 ᠬ ⌳ 共 t 兲 ␳ˆ W 共 t 兲 ⫺ ␳ˆ W 共 t 兲 H ᠭ ⌳ 共 t 兲兲 ⫽ 共 iប 兲 ⫺1 共 H ⳵t ⫽⫺ 共 iL̂⫺iL̂A F 共 t 兲兲 ␳ˆ W 共 t 兲 , 共50兲 ᠬ ⌳ (t)⫽H ជ ⌳ ⫺A ជ ⌳ F(t) and iL̂A has a form analogous where H † † , ). The formal to iL̂ with Â W replacing Ĥ W , iL̂A ⫽(Â W solution of this equation is found by integrating from t 0 to t, ␳ˆ W 共 t 兲 ⫽e ⫺iL̂共 t⫺t 0 兲 ␳ˆ W 共 t 0 兲 ⫹ 冕 t t0 dt ⬘ e ⫺iL̂共 t⫺t ⬘ 兲 iL̂A ␳ˆ W 共 t ⬘ 兲 F 共 t ⬘ 兲 . 冕 ⫺⬁ ⫽⫺ ⬅ t Choosing ␳ˆ W (t 0 ) to be the equilibrium density matrix, ␳ˆ We , defined to be invariant under quantum-classical dynamics, iL̂␳ˆ We ⫽0, the first term on the right-hand side of Eq. 共51兲 coincides with ␳ˆ We and is independent of t 0 . We can ⫺⬁ 冕冕 dt ⬘ Tr⬘ t ⫺⬁ dt ⬘ e ⫺iL̂共 t⫺t ⬘ 兲 iL̂A ␳ˆ We F 共 t ⬘ 兲 . 共52兲 t ⫺⬁ 冕 dR d P B̂ W e ⫺iL̂共 t⫺t ⬘ 兲 iL̂A ␳ˆ We F 共 t ⬘ 兲 † dt ⬘ 具共 Â W ,B̂ W 共 t⫺t ⬘ 兲兲典 F 共 t ⬘ 兲 dt ⬘ ␾ BA 共 t⫺t ⬘ 兲 F 共 t ⬘ 兲 , 共53兲 where ⌬B W (t)⫽B W (t)⫺ 具 B̂ W 典 . In this equation we made use of integration by parts and cyclic permutations under the trace to move the evolution operator onto B̂ W and iL̂A onto B̂ W (t⫺t ⬘ ). Then using the definition of iL̂A as the quantumclassical bracket and the definition of the average 具 f̂ W 典 ⫽Tr⬘ 兰 dR d P f̂ W ␳ˆ We for any operator f̂ W , we arrive at the form in the second line. The prime on the trace denotes the fact that the trace is over the quantum subsystem states. The last line defines the response function ␾ BA which is given by † ␾ BA 共 t 兲 ⫽⫺ 具共 Â W ,B̂ W 共 t 兲兲典 . 共54兲 The response function in Eq. 共54兲 was derived by considering a system following quantum-classical dynamics, in equilibrium in the past, subject to a time dependent external force. This quantum-classical response function should now be compared with the reduction of the response function for a fully quantum mechanical system to its quantum-classical form. The quantum expression for the response function is32 Q ␾ BA 共 t 兲 ⫽⫺ i i † ˆQ Tr关 Â † , ␳ˆ Q e 兴 B̂ 共 t 兲 ⫽ Tr关 Â ,B̂ 共 t 兲兴 ␳ e , 共55兲 ប ប which, taking the partial Wigner transform of the second equality in Eq. 共55兲, may be written as Q ␾ BA 共 t 兲 ⫽⫺Tr⬘ 冕 dR d P i 共 Â † e ប⌳/2i B̂ W 共 t 兲 ប W † Q ⫺B̂ W 共 t 兲 e ប⌳/2i Â W 兲 ␳ˆ We ⫽⫺Tr⬘ 共51兲 t The nonequilibrium average value of any operator B̂ W over the density matrix ␳ˆ W (t), B(t)⫽Tr⬘ 兰 dR d P B̂ W ␳ˆ W (t) may be computed to determine the response of the system to the external force. Using Eq. 共52兲 we may write this as ⌬B W 共 t 兲 ⫽ To derive expressions for transport properties in terms of equilibrium time correlation functions, one may consider the response of a system to an external force using linear response theory or monitor the equilibrium fluctuations in a system. In this section we examine linear response theory for mixed quantum-classical systems. 冕冕 dR d P i Q ជ B̂ 共 t 兲 ⫺B̂ W 共 t 兲 Aឈ ⌳ 兲 ␳ˆ We 共A ប ⌳ W † ⫽⫺ 具共 Â W ,B̂ W 共 t 兲兲 Q 典 . 共56兲 To obtain the first equality we have used the fact that the partial Wigner transform of Tr ÂB̂⫽Tr⬘ 兰 dR d P Â W B̂ W . We wish to take the mixed quantum-classical limit of this expression but this cannot be done by a simple expan- Downloaded 06 Nov 2003 to 130.91.67.156. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Statistical mechanics of quantum-classical systems sion in ប. Mixed quantum-classical dynamics is defined by the evolution equation 共20兲 where the evolution operator is obtained by an expansion to first order in ប 共or the mass ratio兲; however, the time-evolved density matrix contains all orders in ប since it involves the exponential of the evolution operator. The same arguments apply to operators at time t and the equilibrium density matrix. As an ansatz one may replace the time dependent operaQ tor B̂ W (t) and the quantum equilibrium density matrix ␳ˆ We by their quantum-classical forms. In addition, we may replace the quantum Lie bracket ( , ) Q by its quantum classical analog, ( , ) Q →( , ) which is equivalent to replacing the exponential operator exp(ប⌳/2i) by its expansion to linear order in ប, exp(ប⌳/2i)→1⫹ប⌳/2i. After these replacements the response function takes the form 冕 ␾ BA 共 t 兲 ⫽⫺Tr⬘ i 共 Â † 共 1⫹ប⌳/2i 兲 B̂ W 共 t 兲 ប W dR d P † ⫺B̂ W 共 t 兲共 1⫹ប⌳/2i 兲 Â W 兲 ␳ˆ We † ⫽ 具共 Â W ,B̂ W 共 t 兲兲典 , 共57兲 which is exactly the result in Eq. 共54兲 obtained by performing a linear response derivation directly on the mixed quantum-classical equations of motion. This entitles one to use the above intuitive correspondence rule to convert a quantum mechanical correlation function to its mixed quantum-classical analog. In the linear response approach to dynamical phenomena there are various useful equivalent ways to represent the response function. Moreover, since the response function is essentially an equilibrium correlation function, its computation is generally simplified by using the symmetry properties of the equilibrium system. As we shall see in the next sections the situation is much less favorable in the quantumclassical case since many of the rigorous equivalences established in quantum or classical response theory are only approximately true in the quantum-classical limit. The first and most important equivalence is obtained through the Kubo identity; therefore, we begin our discussion with this case. B. Kubo transformed correlation functions In quantum mechanics, correlation functions often appear in Kubo transformed form. Making use of the quantum mechanical identity, i 关 Â † , ␳ˆ Q e 兴⫽ ប 冕 ␤ 0 ˙† d␭ ␳ˆ Q e Â 共 ⫺iប␭ 兲 , 共58兲 in the first equality of Eq. 共55兲 we may write the response function as Q ␾ BA 共 t 兲 ⫽⫺ ⫽⫺ 冕冕 ␤ 0 ␤ 0 d␭ Tr Â˙ † 共 ⫺iប␭ 兲 B̂ 共 t 兲 ␳ˆ Q e d␭ Tr⬘ 冕 Q ⫻e ប⌳/2i B̂ W 共 t 兲兲 ␳ˆ We , where we have again written the second line in partially Wigner transformed form. Using the correspondence rule to take the quantum-classical limit of this expression, we obtain ␾ BA 共 t 兲 ⫽⫺ 冕 ␤ 0 d␭ Tr⬘ 冕共60兲 We shall show now that Eqs. 共57兲 and 共60兲 are not equal and agree only to O共ប兲. The quantum mechanical identity 关Eq. 共58兲兴 used to obtain the Kubo transformed correlation function uses the explicit form for the thermal density matrix ˆ ⫺1 ⫺ ␤ H of ␳ˆ Q e . In the quantum-classical case the structure e ⫽Z of ␳ˆ We is complicated 共see Sec. III兲 and we do not have a closed form expression for it. Consequently, it is necessary to obtain an expression for ␳ˆ We that is analogous to ␳ˆ Q e so that manipulations parallel to those in the quantum derivation may be performed. We shall see that this is possible only to terms O(ប 2 ). The unnormalized equilibrium quantum density matrix, , satisfies the Bloch equation, ⍀̂ Q e ⳵ ⍀̂ Qe Q ⫽⫺Ĥ⍀̂ Q e ⫽⫺⍀̂ e Ĥ. ⳵␤ 共61兲 Taking its partial Wigner transform we may write it in the form Q ⳵ ⍀̂ We Q Q ឈ ជ ⌳ ⍀̂ We ⫽⫺H ⫽⫺⍀̂ We H⌳ , ⳵␤ 共62兲 Q ( ␤ ⫽0)⫽1. with the boundary condition ⍀̂ We The normalized formal solution 共taking into account the boundary condition兲 is ជ ឈ Q ␳ˆ We ⫽Z ⫺1 共 e ⫺ ␤ H ⌳ 1 兲 ⫽Z ⫺1 共 1e ⫺ ␤ H ⌳ 兲 . 共63兲 By the usual rule the quantum-classical limit should be obtained by the replacement H ⌳ →H⌳ . However, by direct ជ ⌳ )1) and calculation one may show that (exp(⫺␤H 33 ឈ ⌳ )) agree only to first order in ប. Moreover, to (1 exp(⫺␤H first order in ប they also agree with ␳ˆ We whose explicit form to this order is given in Eqs. 共47兲 and 共48兲. Thus, to this order these expressions may be used interchangeably. The Kubo identity 共58兲 in the partial Wigner representation is Q † ,Â W 兲Q⫽ 共 ␳ˆ We 冕 ␤ 0 Q ប⌳/2i ˙ † d␭ ␳ˆ We e Â W 共 ⫺iប␭ 兲 , 共64兲 and its quantum-classical analog, which holds only approximately, is 冕 ␤ 0 冉 d␭ ␳ˆ We 1⫹ ⬅K⫹O共 ប 兲 . 共59兲 † dR d P 共 Â˙ W 共 ⫺iប␭ 兲 ⫻ 共 1⫹ប⌳/2i 兲 B̂ W 共 t 兲兲 ␳ˆ We . † 兲⫽ 共 ␳ˆ We ,Â W dR d P 共 Â˙ † W 共 ⫺iប␭ 兲 5811 冊 ប⌳ ˙ † Â W 共 ⫺iប␭ 兲 ⫹O共 ប 兲 2i 共65兲 This result is derived as follows: performing the integral over ␭ we obtain Downloaded 06 Nov 2003 to 130.91.67.156. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp 5812 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 冉冉 iបK⫽⫺ ␳ˆ We 1⫹ ⫹ ␳ˆ We 1⫹ 冊冊 ប⌳ † Â W 2i ឈ 冉 The last equality has been obtained by substituting Eq. 共65兲 and then using cyclic permutation of the trace and integration by parts. Following Kubo, we may define a new correlation function, also in the mixed quantum-classical limit, by ប⌳ ជ ឈ † ⫺␤H ⌳兲 S共 e ␤ H⌳ Â W e 2i 冉 ⫽⫺Z ⫺1 共 1e ⫺ ␤ H⌳ 兲 1⫹ ⫹ ␳ˆ We 1⫹ Nielsen, Kapral, and Ciccotti 冊冊 ប⌳ ជ ឈ † ⫺␤H ⌳兲 S共 e ␤ H ⌳ Â W e 2i ប⌳ † Â W ⫹O共 ប 2 兲 , 2i 具 Â W ;B̂ W 共 t 兲典 ⫽ 冉冉 1⫹ 冊 ប⌳ ជ ⫽e ⫺ ␤ H⌳ ⫹O共 ប 2 兲 , 2i 共67兲 ប⌳ ជ ឈ 共 e ⫺ ␤ H⌳ 1 兲 ⫽e ⫺ ␤ H⌳ ⫹O共 ប 2 兲 , 2i 共68兲 ឈ 共 1e ⫺ ␤ H⌳ 兲 1⫹ 冊 and use these results to manipulate Eq. 共66兲. We have iបK⫽⫺Z ជ ⫺1 ⫺ ␤ H ⌳ e 冉 ⫹ ␳ˆ We 1⫹ S共 e ជ ␤H ⌳ 冊共69兲 To further simplify the right-hand side of this equation we use the relation ជ ជ ឈ ឈ † ⫺ ␤ H⌳ † ⫺ ␤ H⌳ e e ⫹O共 ប 2 兲 , e ⫺ ␤ H⌳ S共 e ␤ H⌳ Â W 兲 ⫽Â W 共70兲 which may be proved by direct expansion of the exponential operators. Then using Eq. 共68兲 we have 冉 † iបK⫽⫺Z ⫺1 Â W 1⫹ 冊冉冉 ␤ 0 冊冉冕 ⬘冕 ⫽⫺Tr Tr⬘ 冕 ⫽⫺ 冕冉 ␤ 0 ⫻ 1⫹ 冕冊 ⫽Tr dR d PÂ W 共 e ⫺iLt ␳ˆ We 兲 dR d PÂ W ␳ˆ We , 共75兲共76兲 Indeed, for the exact equality to hold one needs 冉 † Â W 共 ␶ 兲 1⫹ 冊冉冉冊冊 ប⌳ ប⌳ † B̂ W 共 t⫹ ␶ 兲 ⫽ Â W 1⫹ B̂ W 共 t 兲共 ␶ 兲 . 2i 2i 共77兲 A special case of this is, for any operators Ĉ W and D̂ W , Ĉ W (t) 关 1⫹(ប⌳/2i) 兴 D̂ W (t)⫽(Ĉ W 关 1⫹(ប⌳/2i) 兴 D̂ W )(t). A necessary and sufficient condition for this to be true is 冉冉冉 † dR d P Â˙ W 共 ⫺iប␭ 兲冊冊 dR d P 共 e iLt Â W 兲 ␳ˆ We 具共 Â W ,B̂ W 共 t 兲兲典 ⫽ 具共 Â W 共 ␶ 兲 ,B̂ W 共 t⫹ ␶ 兲兲典 ⫹O共 ប 兲 . 共71兲 ប⌳ B̂ W 共 t 兲 ␳ˆ We ⫹O共 ប 兲 . 2i 共74兲 where we have used the stationarity of ␳ˆ We , while for the time correlation function in Eq. 共57兲 we only have † d P 共 ␳ˆ We ,Â W 兲 B̂ W 共 t 兲 d␭ Tr⬘ 冕 ⬘冕 ⬘冕 dR d P Â W 共 t 兲 ␳ˆ We ⫽Tr⬘ † dR d P 共 Â W ,B̂ W 共 t 兲兲 ␳ˆ We dR 共73兲 For both quantum and classical systems in equilibrium the time evolution of an observable generates a stationary random process. In particular, ensemble averages of observables are independent of time and time correlation functions do not depend on the time origin but only on time differences. For quantum-classical dynamics the ensemble average of an observable is independent of time, ⫽Tr Starting with Eq. 共54兲 and using Eq. 共65兲 gives ␾ BA 共 t 兲 ⫽⫺Tr⬘ 冊冊 ប⌳ B̂ W 共 t 兲 ␳ˆ We . 2i Equation 共74兲 may be useful in practical calculations. However, since the equivalence between the two forms of the response function 关Eqs. 共57兲 and 共74兲兴 cannot be established to all orders in ប, the magnitude of their numerical difference must be evaluated in specific applications in order to determine the applicability of Eq. 共74兲. 冊 ប⌳ ប⌳ † 1⫹ ␳ˆ We ⫹ ␳ˆ We 1⫹ Â W ⫹O共 ប 2 兲 2i 2i † ⫽iប 共 ␳ˆ We ,Â W 兲 ⫹O共 ប 2 兲 . 冉 † dR d P Â W 共 ⫺iប␭ 兲 ␾ BA 共 t 兲 ⫽⫺ ␤ 具 Â˙ W ;B̂ W 共 t 兲典 ⫹O共 ប 兲 . ប⌳ ប⌳ † ជ Â W 共 e ⫺ ␤ H⌳ 1 兲 ⫹ ␳ˆ We 1⫹ 2i 2i ⫹O共 ប 兲冉冕 Using this notation, the response function may be written simply as 2 † ⫽⫺Â W d␭ Tr⬘ C. Time translation invariance ឈ † ⫺␤H ⌳兲 Â W e ប⌳ † Â W ⫹O共 ប 2 兲 . 2i 冕 ⫻ 1⫹ 共66兲 where we have used one of the ‘‘equivalent’’ expressions for ␳ˆ We , and we remind the reader that the interpretation of S( ) is given in Appendix A. Next, expanding the exponential operator and using the quantum-classical associative product rule 关valid to terms O(ប 2 )兴 we may show that 1 ␤ iL̂ Ĉ W 1⫹ 共72兲冊冊 ប⌳ D̂ W 2i 冉 ⫽ 共 iL̂Ĉ W 兲 1⫹ 冊冉冊 ប⌳ ប⌳ D̂ W ⫹Ĉ W 1⫹ 共 iL̂D̂ W 兲 . 2i 2i 共78兲 Downloaded 06 Nov 2003 to 130.91.67.156. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Statistical mechanics of quantum-classical systems However, we have shown in Eq. 共25兲 that this is true only to O共ប兲. Thus, time translation invariance is valid only approximately in quantum-classical dynamics. The first few terms 共thought of as a power series expansion in it/ប兲 are, from Eq. 共A2兲, ˆ e iLt Â W ⫽Â W ⫹ V. CONCLUSION The results obtained in this paper provide the basis for the computation of equilibrium time correlation functions in many-body quantum-classical systems. The numerical computation of time correlation functions entails both the simulation of the evolution of dynamical variables or operators that enter in the correlation function of interest and sampling over a convenient weight function (exp(⫺␤ĤW)) which is part of the quantum-classical canonical equilibrium density matrix. Algorithms have been developed for the simulation of quantum-classical evolution and their further development is a topic of continuing research. Sampling methods for the calculation of quantum-classical time correlations require additional considerations. Typically, in simulations of classical equilibrium time correlation functions the ensemble average is replaced by a time average and the information needed to sample from the equilibrium distribution is obtained from a long molecular dynamics trajectory. The validity of such a procedure hinges on the assumed ergodicity and stationarity of the system and replaces direct sampling from the equilibrium distribution using Monte Carlo methods by a time average. This ensemble average method has been used occasionally but since it is not necessary and is algorithmically more complex, requiring both Monte Carlo and molecular dynamics programs, its practical importance has been minor. For quantum-classical systems neither the assumption of ergodicity nor stationarity can be made and sampling from the equilibrium distribution or another suitable weight function must be carried out to evaluate the correlation functions. The analysis presented in this paper gives all of the information needed to compute equilibrium time correlations in the quantum-classical limit in a consistent fashion and, thus, provides a useful way to treat a large class of manybody systems where the environmental degrees of freedom have a classical character. ACKNOWLEDGMENTS This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. Acknowledgment is made to the donors of The Petroleum Research Fund, administered by the ACS, for partial support of this research. One of the authors 共S.N.兲 would like to acknowledge a Walter C. Sumner Memorial Fellowship. APPENDIX A The meaning of the formal solution in the second equality in Eq. 共28兲 will be established here by comparing the two forms in this equation ជ ឈ Â W 共 t 兲 ⫽S共 e iH⌳ t/ប Â W e ⫺iH⌳ t/ប 兲 , 共A1兲 and Â W 共 t 兲 ⫽e ˆt iL Â W . 共A2兲 5813 it t2 ជ Â ⫺Â H ឈ ជ H ជ Â 共H 共H ⌳ W W ⌳兲⫺ ប 2ប 2 ⌳ ⌳ W ជ ⌳ 共 Â W H ឈ ⌳兲⫺共 H ជ ⌳ Â W 兲 H ឈ ⌳ ⫹Â W H ឈ ⌳H ឈ ⌳兲 ⫺H it 3 ជ ⌳H ជ ⌳H ជ ⌳ Â W ⫺H ជ ⌳H ជ ⌳ 共 Â W H ឈ ⌳兲 ⫺ 3 共H 6ប ជ ⌳ 共共 H ជ ⌳ Â W 兲 H ឈ ⌳ 兲 ⫹H ជ ⌳ 共 Â W H ឈ ⌳H ឈ ⌳兲 ⫺H ជ ⌳H ជ ⌳ Â W 兲 H ឈ ⌳⫹共 H ជ ⌳ 共 Â W H ឈ ⌳ 兲兲 H ឈ⌳ ⫺共 H ជ ⌳ Â W 兲 H ឈ ⌳H ឈ ⌳ ⫺Â W H ឈ ⌳H ឈ ⌳H ឈ ⌳ 兲 ⫹¯ , 共A3兲 ⫹共 H and, from Eq. 共A1兲, ជ ឈ Se iH⌳ t/ប Â W e ⫺iH⌳ t/ប ⫽Â W ⫹ it t2 ជ ⌳ Â W ⫺Â W H ឈ ⌳ 兲 ⫺ 2 S共 H ជ ⌳H ជ ⌳ Â W S共 H ប 2ប ជ ⌳ Â W H ឈ ⌳ ⫹Â W H ឈ ⌳H ឈ ⌳兲⫺ ⫺2H it 3 ជ ⌳H ជ ⌳H ជ ⌳ Â W S共 H 6ប 3 ជ ⌳H ជ ⌳ Â W H ឈ ⌳ ⫹3H ជ ⌳ Â W H ឈ ⌳H ឈ⌳ ⫺3H ឈ ⌳H ឈ ⌳H ឈ ⌳ 兲 ⫹¯ . ⫺Â W H 共A4兲 For these expressions to agree we must make assignments like ជ ⌳ Â W H ឈ ⌳H ឈ ⌳兲 S共 H ជ ⌳ Â W 兲 H ឈ ⌳H ឈ ⌳⫹共 H ជ ⌳ 共 Â W H ឈ ⌳ 兲兲 H ឈ⌳ ⬅ 13 共共 H ជ ⌳ 共 Â W H ឈ ⌳H ឈ ⌳ 兲兲 . ⫹H 共A5兲 ជ ⌳ ) j Â W (H ឈ ⌳ ) k ) is composed of ( j In general the term S((H ⫹k)!/ j!k! separate terms each with a prefactor of j!k!/( j ⫹k)!. Each of these separate terms corresponds to a unique association of terms indicating the order in which the H⌳ operators act. APPENDIX B This appendix concerns the relationship between the Q (R, P) and quantum-classical ␳ˆ We (R, P) equiquantum ␳ˆ We librium density matrices in the canonical ensemble. We begin by deriving the recursion relations for the quantum mechanical equilibrium density matrix. Q is stationary under the The thermal density matrix ␳ˆ We quantum dynamics in Eq. 共7兲, Q 0⫽ 共 Ĥ W , ␳ˆ We 兲Q i Q Q ប⌳/2i ⫽ 共 Ĥ W e ប⌳/2i ␳ˆ We ⫺ ␳ˆ We e Ĥ W 兲 . ប 共B1兲 Downloaded 06 Nov 2003 to 130.91.67.156. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp 5814 J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 Nielsen, Kapral, and Ciccotti If we substitute the expansion, ⬁ Q ␳ˆ We ⫽ 兺 n⫽0 Q共 n 兲 n ␳ˆ We ប , 共B2兲 into Eq. 共B1兲 and group by powers of ប we obtain 共using the fact that the P dependence of H is of the form P 2 /2M 兲 Q共 0 兲 0⫽ 共 iប 兲 ⫺1 关 Ĥ W , ␳ˆ We 兴, 共B3兲 Q共 1 兲 Q共 0 兲 Q共 0 兲 ⌳Ĥ W ⫹ 21 Ĥ W ⌳ ␳ˆ We 0⫽ប 0 共 i 关 Ĥ W , ␳ˆ We 兲 , 共B4兲兴 ⫺ 21 ␳ˆ We 冉 1 Q共 1 兲 1 Q共 2 兲 Q共 1 兲 0⫽ប 1 ⫺i 关 Ĥ W , ␳ˆ We ⌳Ĥ W ⫺ Ĥ W ⌳ ␳ˆ We 兴 ⫹ ˆ␳ We 2 2 ⫹ 冊 1 Q共 0 兲 2 1 Q共 0 兲 ˆ␳ ⌳ Ĥ W ⫺ Ĥ W ⌳ 2 ␳ˆ We , 8i We 8i 冉共B5兲 1 Q共 2 兲 1 Q共 3 兲 Q共 2 兲 ⌳Ĥ W ⫺ Ĥ W ⌳ ␳ˆ We 0⫽ប 2 ⫺i 关 Ĥ W , ␳ˆ We 兴 ⫹ ˆ␳ We 2 2 ⫹ 1 Q共 1 兲 2 1 1 Q共 0 兲 3 Q共 1 兲 ˆ␳ ⌳ Ĥ W ⫺ Ĥ W ⌳ 2 ␳ˆ We ⫺ ˆ␳ We ⌳ Ĥ W 8i We 8i 48 ⫹ 1 Q共 0 兲 Ĥ ⌳ 3 ␳ˆ We , 48 W 冊共B6兲 and so on. These results should be compared with the mixed quantum-classical recursion relations given in Eqs. 共33兲 and 共34兲. Notice that the first two recursion relations are identical in the quantum-classical and fully quantum mechanical systems 关see Eqs. 共33兲 and 共34兲 for n⫽0, 共B3兲, and 共B4兲兴. Hence, we can calculate the first two terms for the quantum mechanical system 共in the partially Wigner transformed representation兲 and the results will also be valid for the quantum-classical system. In quantum mechanics we have for the unnormalized Q canonical equilibrium density matrix, ⍀̂ We Q ⍀̂ We ⫽共 e ˆ ⫺␤H It is possible to show that in the adiabatic basis representation 共properly normalized兲 these are the results quoted in Eqs. 共47兲 and 共48兲. We have established the equivalence between the second term on the right-hand side of Eq. 共B8兲 and Eq. 共48兲 by a direct calculation which shows that the matrix representation of this term in Eq. 共B8兲 is the same as the Taylor expansion of Eq. 共48兲 in powers of ␤. APPENDIX C (n) ␯␯ ⬘ Here we establish that R(J ␣␣ , ␯␯ ⬘ ␳ We ) is an odd function of P. Using the definitions in Sec. III, this is * equivalent to the condition that R((⫺( P/M )•d ␣␣ ⬘ 1 (n) ␣␣ ⬘ * ⫺ 2 E ␣␣ ⬘ d ␣␣ ⬘ ( ⳵ / ⳵ P)) ␳ We ) be an odd function of P or, choosing to work with a real adiabatic basis so that d ␣␣ ⬘ is (n) ␣␣ ⬘ ) and real, this simplifies to the conditions that PR( ␳ We (n) ␣␣ ⬘ ⳵ / ⳵ PR( ␳ We ) be odd functions of P. The effect of multiplication by P or differentiation with respect to P is to (n) ␣␣ ⬘ ), if indeed this change the parity of the factor R( ␳ We factor has a definite parity; thus, we must show that (n) ␣␣ ⬘ ) is an even function of P. For technical reasons it R( ␳ We (n) ␣␣ ⬘ ) is an odd function will also be useful to show that J( ␳ We of P. We use an inductive proof. The statements are true for (1) ␣␣ (1) ␣␣ ⬘ ⫽0 and that ␳ We n⫽1. From Eq. 共48兲 we see that ␳ We is pure imaginary and odd in P for ␣ ⫽ ␣ ⬘ . Assuming n 兲 ␣␣ ⬘ R共 ␳ 共We 兲 n 兲 ␣␣ ⬘ I共 ␳ 共We 兲 ␤ Ĥ e ប⌳/2i Ĥ W ⫺¯ . 2 W n⫹1 兲 ␣␣ ⬘ R共 ␳ 共We 兲⫽ 共B7兲 ˆ n ⫻ 兺 j⫽0 n⫺ j 共 n⫺2 j 兲 Ĥ W n⫹2 ⳵ Ĥ W j Ĥ W ⫹O共 ប 2 兲 , ⳵R 兺 ␣ ⬙ 冉冉冉 d ␣␣ ⬙ 冊 1 ⳵ P n 兲␣⬙␣⬘ E ␣␣ ⬙ ⫹ J共 ␳ 共We 兲 2 ⳵P M 冊 1 ⳵ P n 兲 ␣␣ ⬙ E ␣⬘␣⬙ ⫹ J共 ␳ 共We 兲 2 ⳵P M 冊冎 . 共C3兲共B8兲 after isolating the zeroth and first order terms and proving by induction to first order in ប that n 共 Ĥ n⫹2 兲共W1st 兲 ⫽ 再 1 P ⳵ n 兲 ␣␣ ⬘ ⫺ • J共 ␳ 共We 兲 E ␣␣ ⬘ M ⳵R ⫹d ␣ ⬘ ␣ ⬙ ប ⳵ Ĥ W 共⫺␤兲 2i ⳵ P n⫽0 共 n⫹2 兲 ! 兺共C2兲 ⫺ This power series in ␤ must be converted into a power series with respect to ប. Extracting the first two terms in this new power series gives Q ⫽e ⫺ ␤ H W ⫹ ⍀̂ We odd function of P᭙ ␣ , ␣ ⬘ ,n, ⳵ 1 n 兲 ␣␣ ⬘ ⫺ 共 F ␣ ⫹F ␣ ⬘ 兲 J共 ␳ 共We 兲 2 ⳵P 2 ⬁ 共C1兲 we show that these conditions are true for n⫹1. From Eq. 共46兲 we have 兲W ⫽1⫺ ␤ Ĥ W ⫹ even function of P᭙ ␣ , ␣ ⬘ ,n, ⳵ Ĥ W j ប ⳵ Ĥ W n⫺ j Ĥ W . 共B9兲共 n⫺2 j 兲 Ĥ W 2i ⳵ P j⫽0 ⳵R 兺 The operators Ĥ W and ⳵ Ĥ W / ⳵ R do not commute and so the order must be preserved as shown. Each of the terms on the right-hand side is even with respect (n⫹1) ␣␣ ⬘ ) is even in P. The analysis of to P so R( ␳ We (n⫹1) ␣␣ ⬘ ) is similar and will be omitted. Thus, the stateJ( ␳ We ment is true for n⫹1 completing the inductive proof. 1 J. C. Tully, in Modern Methods for Multidimensional Dynamics Computations in Chemistry, edited by D. L. Thompson 共World Scientific, New York, 1998兲, p. 34. 2 M. F. Herman, Annu. Rev. Phys. Chem. 45, 83 共1994兲; J. Chem. Phys. 87, 4779 共1987兲. 3 Classical and Quantum Dynamics in Condensed Phase Simulations, ed- Downloaded 06 Nov 2003 to 130.91.67.156. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp J. Chem. Phys., Vol. 115, No. 13, 1 October 2001 ited by B. J. Berne, G. 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