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Stochastic Quantum Molecular Dynamics for Finite and Extended Stochastic quantum molecular dynamics for finite and extended systems Heiko Appela,b,c,∗, Massimiliano Di Ventrab aFritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany bUniversity of California San Diego, La Jolla, California 92093, USA cEuropean Theoretical Spectroscopy Facility Abstract We present a detailed account of the technical aspects of stochastic quantum molecular dynamics, an ap- proach introduced recently by the authors [H. Appel and M. Di Ventra, Phys. Rev. B 80 212303 (2009)] to describe coupled electron-ion dynamics in open quantum systems. As example applications of the method we consider both finite systems with and without ionic motion, as well as describe its applicability to extended systems in the limit of classical ions. The latter formulation allows the study of important phenomena such as decoherence and energy relaxation in bulk systems and surfaces in the presence of time-dependent fields. 1. Introduction the measurement apparatus itself which necessarily projects non-unitarily the state of the system onto Time-dependent density-functional theory states of the observables. This is generally true for (TDDFT) calculations in the linear-response both electrons and ions, so that a first-principles limit are currently enjoying a large popularity description of their coupled dynamics in the presence due to their efficiency and success in describing of one or more environments is of fundamental low-lying excitation energies in molecular systems importance in order to describe phenomena and [1]. Beyond linear-response, many applications have compare with experiments. At this point, it is worth been investigated with TDDFT. Examples include noting that present quantum molecular dynamics electronic transport [2, 3, 4, 5], nonlinear optical (QMD) approaches, (e.g., the Born-Oppenheimer, response [6], or atoms and molecules in strong laser Ehrenfest or Car-Parrinello methods) either do not fields [7, 8]. In the latter cases, the time-dependent allow excited states dynamics (Born-Oppenheimer Kohn-Sham (TDKS) equations are usually evolved and Car-Parrinello methods) or, if they do (e.g., in real-time. However, the majority of these stud- Ehrenfest QMD), they do not permit electronic ies pertains to the description of closed quantum coupling to external environments. Indeed, in all systems, since the corresponding TDKS equations these approaches, energy dissipation and thermal describe a set of N particles evolving coherently coupling to the environment are usually described in time. 1 On the other hand, most experimental with additional thermostats coupled directly to the situations involve some level of energy dissipation classical nuclear degrees of freedom, which fall short and/or decoherence induced by either some envi- of describing the numerous physical phenomena ronments to which the given system is coupled, or associated with quantum decoherence and energy dissipation. In order to overcome these shortcomings, we have ∗Corresponding author: Email address: [email protected] (Heiko Appel) recently introduced a novel time-dependent den- 1Notable exceptions are the references [4, 9, 10, 11, 12]. sity functional approach based on stochastic time- Preprint submitted to Elsevier March 24, 2011 dependent Kohn-Sham equations [13], where we allow bulk systems and surfaces. We are in the process of the coupling of both electrons and (in principle quan- implementing SQMD for extended systems and we tum) ions with external baths. This approach - we will report these results in a forthcoming publication have named stochastic quantum molecular dynamics [14]. (SQMD) - extends the previously introduced stochas- The paper is organized as follows. In Section 2 we tic time-dependent-current density-functional theory give an introduction to the theory of stochastic quan- (STDCDFT) [9, 10] to the coupled dynamics of elec- tum molecular dynamics. For completeness, this in- trons and ions. The latter was formulated to account cludes general aspects of open quantum systems as for electrons interacting with external environments, well as the basic theorem of SQMD. In Section 3 we without however including atomic motion. There- discuss the aspects of a practical implementation of fore, SQMD combines and improves on the strengths SQMD. Finally, in Section 4 we illustrate with some of STDCDFT and present QMD methods by greatly examples the application of SQMD to finite systems expanding the physical range of applications of these with and without ionic motion, and outline its exten- methods. sion to periodic systems. Conclusions are reported in Clearly, from a practical point of view the present Section 5. approach suffers - like all density-functional theory (DFT) based methods - from our limited knowledge of the properties of the exact exchange-correlation 2. Theory functional. Furthermore, in the present case, the ex- act functional depends not only on the electronic de- 2.1. Stochastic Schr¨odingerequation grees of freedom, but also on the ionic and bath(s) de- grees of freedom [13]. Nevertheless, due to the weak In the following, we consider an electron-ion many- system-bath(s) coupling assumption of the present body system coupled to a bosonic bath. For simplic- theory, as well as the limited number of systems ity, we will consider only a single bath, but the for- where quantum nuclear effects are of disproportion- mulation is trivially extended to the case of several ate importance, we may start by considering the limit environments. The total Hamiltonian of the entire of SQMD to classical nuclei and adopt the available system is then functionals of standard closed-system TDDFT. Like in any other practical application of DFT, it is the ^ ^ ^ ^ ^ ^ predictions that we obtain and comparison with ex- H = HS ⊗ IB + IS ⊗ HB + λHSB: (1) periments that will be the ultimate judge of the range of validity of the approximate functionals used. Our system of interest is described by the many-body ^ In Ref. [13] we have outlined the details of the proof Hamiltonian HS and the environment degrees of free- ^ of the theorem at the core of SQMD, and provided a dom are given in terms of HB. The interaction of the simple example of the relaxation dynamics of a finite system with the environment is given by the Hamil- ^ system (a molecule) prepared in some excited state tonian HSB and is assumed to be weak in the sense and embedded in a thermal bath. However, there are that a perturbation expansion in terms of this cou- some technical details behind an actual implementa- pling can be performed. With λ we denote the cor- tion of this approach we have not reported yet, and responding coupling parameter for the system-bath which are nonetheless important if one is interested interaction. ^ in using this method for practical computations. In The total system described by the Hamiltonian H this work we then present all the technical aspects follows a unitary time-evolution, which can be for- for a practical implementation and use of SQMD. In mulated for pure states either in terms of the time- addition, we present the theory behind its applicabil- dependent Schr¨odingerequation (TDSE), withh ¯ = 1 ity to extended systems which is of great importance in the study of decoherence and energy relaxation in i@tΨ(t) = H^ (t)Ψ(t); (2) 2 or, alternatively for mixed states, in terms of the with xB the bath's coordinates (including possibly Liouville-von Neumann equation spin), and expand the total wavefunction of Eq. (2) d h i in the complete set of orthonormal states formed by ρ^(t) = −i H^ (t); ρ^(t) ; (3) fχ x g 2 dt n( B) , namely whereρ ^ is the statistical operator. X Ψ(x ; x ; t) = φ (x ; t)χ (x ); (5) Since H^ is a many-body Hamiltonian and we have S B n S n B n to deal, in principle, with infinitely many degrees of with φ (x ; t) some functions (not necessarily nor- freedom (due to the bath) it is not possible to solve n S malized) in the Hilbert space of the system S. Eq. (2) or (3) in practice, except for a few simple In order to see that in the presence of a bath the model cases. In addition, in most cases of interest the functions φ (x ; t) form a statistical ensemble de- microscopic knowledge about the bath is limited and n S scribing the properties of the subsystem S, let us only its macroscopic thermodynamic properties are proceed as follows. For a general observable O^ of known, e.g., one typically assumes that the bath is in S the system S we find after simple algebra (and using thermal equilibrium. However, we are only interested the orthonormality of the bath states χ (x )) in the dynamics of the system degrees of freedom. It n B is therefore desirable to find an effective description hΨ(xS; xB; t)jO^SjΨ(xS; xB; t)i = for the system only. X hφ x t jO^ jφ x t i: (6) To accomplish this we may trace out the bath de- n( S; ) S n( S; ) grees of freedom at the level of the statistical opera- n tor, namely we perform the operationρ ^S = TrBfρ^g, Let us now normalize the functions φn(xS; t) by writ- ρ ing where ^S is called the reduced statistical operator of p system S. It is worth pointing out here that this n(xS; t) ≡ φn(xS; t)= pn(t) (7) procedure does not generally lead to a closed equa- with tion of motion for the reduced statistical operator and pn(t) = hφn(xS; t)jφn(xS; t)i; (8) one needs further approximations. Depending on the which, according to Eq. (5), is nothing other than the approximations involved, one may arrive at an effec- probability for the bath to be in the state χ (x ).
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