Abstract
We describe in detail a general system–bath strategy for investigating the quantum behavior of small systems interacting with complex environments. In this approach, a simplified heat bath is used as a surrogate for realistic environments, and explicit, unitary quantum simulations of the “universe” (the system plus the bath) are performed by means of high-dimensional wave-packet techniques. In this chapter, we describe the underlying Hamiltonians and the related reduced dynamical descriptions, show how to recast real-world problems into this form, introduce some of the methods currently used to deal with high-dimensional quantum dynamics, and present the results of this strategy when applied to numerous problems of physicochemical interest.
Keywords
- System–bath dynamics
- Multi-configuration time-dependent methods
- Generalized Langevin equation
- Brownian motion
- Effective modes
1. Introduction
Recent years have witnessed an ever growing interest in dynamical processes that occur in complex environments, for example, ground- and excited-state molecular reaction dynamics in condensed phase, charge and excitation energy transfer in organic functional materials and biomaterials, and elementary processes at the gas–solid interface [1–6]. Their importance is hardly overemphasized, because of the key role they play in fields as diverse as catalysis, optoelectronics, nanotechnology, biochemistry, and astrophysics, just to mention a few.
The common structure of these problems—a relatively simple system that can be measured and manipulated and that is coupled to an environment only partially under control—has long been subjected to thorough theoretical investigation. Since Einstein’s [7] seminal work on Brownian motion, many important analytical results have been obtained regarding the statistical description of the effect that a large medium—being it a surface or a solvent—has on the dynamics of the small system of interest [8]. The environment, usually designated as the “bath,” is seen to exert two different kinds of force, a friction, and a stochastic force. They result, respectively, in dissipation and fluctuations in the system dynamics and represent two opposite but intimately related effects that ultimately lead to the establishment of equilibrium. This has been made apparent since Langevin formulated the first sound description of an open-system dynamics in 1908, with his equation of motion (the Langevin Equation, LE) and its Generalized version (GLE) [8–10].
These ideas have been extensively deepened in classical mechanics in the following years and thoroughly validated in both model and realistic systems by several numerical experiments, that is, explicit molecular dynamics (MD) simulations of a “universe” (the system
In quantum dynamics, the situation is considerably more complicated. A
An intermediate possibility between the impractical
In this chapter, we present the work done in the last few years in mapping a physical problem of interest into a system–bath model—the so-called independent oscillator (IO) model—and in solving such model with multi-configuration wave-packet approaches. The dissipative processes investigated range from small amplitude, damped vibrations in model anharmonic systems to “real-world” problems such as hydrogen atom sticking on graphene.
The chapter is organized as follows: Firstly, we describe the IO model in the framework of classical and quantum statistical mechanics, with a focus on its relationship to the generalized Langevin equation and on the role played by the so-called spectral density (SD) of the environmental coupling J0(ω) [13,14,17–19]. In addition to the well-established results, we include some recent developments that improved the numerical appeal of the model [20–28]. Secondly, we address the problem of mapping a complex (realistic) dynamics into an IO model and deriving the appropriate SD [29–31]. We focus in particular on
2. Independent oscillator models
The IO Hamiltonian is a popular and extremely powerful tool to study the dynamics of an open system in a quantum setting [13–15]. Here, we discuss its connection with the generalized Langevin equation, emphasizing the role played by the SD in the mapping between the two [18,19]. Later, we introduce an effective mode transformation that casts the IO bath into a linear-chain form which suits well to truncation schemes [20–22,25,26,28].
2.1. Generalized Langevin equation and SD
The generalized Langevin equation describes the dynamics of a Brownian particle in both the classical and the quantum (Heisenberg) setting. It is a stochastic equation for a system degree of freedom
Causality of the memory kernel (
and fully determines the memory kernel by virtue of the Kramers–Kronig relations, namely through
where Here and in the following … 〉denotes an average over the canonical equilibrium.
here written for a quantum environment (the classical result follows in the limit of high temperatures, β = (kBT)− 1 → 0 and takes the form 〈
Once J0(ω) is known, it can be used to construct an IO Hamiltonian This is also known as Caldeira–Leggett Hamiltonian, after the seminal work by Caldeira and Leggett on the effects of dissipation on quantum tunneling [17].
that can be made (quasi) equivalent to the GLE above by appropriately choosing the harmonic oscillator (HO) frequencies and the coupling coefficients. In Eq. (5), the system degree of freedom In the following we will adopt, without loosing generality, the same mass for all the oscillators,
(but yet such that
Notice that in Eq. (5), a system potential counter-term
The equivalence between the two dynamical formulations [Eqs. (1) vs. (5)] is established when the coupling coefficients sample the SD J0(ω) of the problem, for example, for evenly spaced bath frequencies ωk = kΔω, when the coefficients are set according to
It rigorously holds for a finite time only, determined by the size of the bath in Eq. (5). For longer times, Eq. (1) keeps on describing a dissipative dynamics, whereas the Hamiltonian dynamics of Eq. (5) displays the consequences of discretely sampling the bath frequencies. More precisely, the equivalence is guaranteed up to the Poincaré recurrence time tP = 2π ∕ Δω of the finite system, which thus needs to be set larger than the any time scale of interest. This has to be done by choosing the appropriate discretization
Unlike the starting GLE, the IO Hamiltonian of Eq. (5) can be quantized by applying standard quantization rules. Furthermore, the relatively trivial dynamics followed by the harmonic oscillators of the bath makes the use of standard time-dependent wave-packet approaches to the dynamics possible. As long as a system–bath Hamiltonian can be effectively mapped into a GLE, this represents a powerful and general methodology to tackle an open-system quantum dynamical problem.
2.2. Chain representation of the bath
Recent work has shown that the IO Hamiltonian of Eq. (5) can be expressed in an alternative representation which is particularly suited to truncation schemes or hierarchical descriptions of the dynamics [20,23,24,27,42]. The idea is incredibly simple and reads as follows. Eq. (5) naturally introduces a collective or “effective” mode
The definition of this effective mode fixes the first “column” of an orthogonal transformation of the original bath coordinates into a new set of coordinates, otherwise arbitrary. The rest of the transformation matrix can be fixed by requiring that the “residual bath” is in normal form. In this way, Eq. (5) becomes an equivalent IO Hamiltonian for the
Here,
and
determines the coupling coefficients between the
and reads, for mass-scaled coordinates, as
Here,
And {(
by introducing new coordinates,
One interesting issue concerning Eq. (13) is whether a limiting residual SD exists and, in that case, which forms takes
Strictly speaking such “Markovian reduction” rigorously holds in classical mechanics only; in a quantum setting the very definition of Markovian dynamics is still debated. Thus, one should better refer here to an “Ohmic embedding”.
One further issue on the effective mode construction is of much practical interest and concerns its role in defining approximate representations of the bath [24]. In fact, the construction of the linear chain amounts to “unroll” the memory kernel in time—any excitation initially localized in the system necessarily moves sequentially along the chain starting from its end attached to the system. This is contrast to what happens with Eq. (5), where the coupling pattern is appropriate for a frequency resolution of the kernel. As a consequence, truncated or Markov-closed chains with
3. Mapping of a complex system to an IO model
The possibility of using the IO Hamiltonian for simulations of “real-world” systems relies on the existence of a general strategy to derive a GLE from a given microscopic model. In the past, the fundamental problem of mapping in an exact way a reduced dynamics into a GLE was addressed by many authors, The problem is essentially classical in nature, since the statistical properties of the bath (when subsumed in the spectral density J0(ω) are the same for both the classical and quantum GLE.
3.1. Inversion of classical dynamics
When the Brownian motion along
namely through
that follows from Eq. (1) by harmonic analysis. This equation relates the reduced dynamics of the system to its coupling with the environment and can be “inverted” to give an analytic expression for J0(ω) in terms of
Where
that is, the Cauchy transform of the function
3.2. Numerical tests
In Ref. [29], we tested the inversion procedure of Eq. (18) on some model systems to elucidate how its effectiveness is affected by the presence of an harmonicities and/or by a “Debye” cutoff frequency in the environment. A number of IO Hamiltonians of the kind of Eq. (5) and its variants were used to generate the dynamics, with a reasonable choice of the parameters that mimicked typical molecular problems. Some key alternatives were considered (
The main conclusions of such numerical analysis can be summarized as follows. When the system is a harmonic oscillator, the transformation perfectly recovers the original SD up to the bath Debye frequency Numerical evaluation of Eq. (19) requires the introduction of a high-frequency cutoff. The problem arises when using an “unbiased” cut-off frequency well above the spectral range of interest ( A fictitious coupling to the bath appears here because numerically the autocorrelation function needs to be damped.
The presence of these features warns against blind application of the transformation of Eq. (18), particularly when the system frequency is larger than
3.3. Nonlinear extensions of the IO Hamiltonian
As mentioned in the previous section, system anharmonicity poses no real problem to modeling, and only generates a spurious high-frequency coupling that can be minimized by working at low enough temperature. In realistic situations, however, structured features in the spectral region
In Eq. (5), our definition of IO Hamiltonian, we have already included a coupling which is not linear with respect to the system coordinate. It involves a shape function
As for the bath, the consideration of a nonlinear coupling poses more problems. In general, an exponential interaction model seems to be appropriate in typical physicochemical situations, where relaxation occurs as a consequence of close encounters between the molecular system of interest and the atoms of the environment. One simple
Such coupling can be justified in the context of the linear-chain representation of the bath (see Section 2.2) and follows from Eq. (15) upon replacing the first harmonic term of the series with a Morse potential with the same frequency. Here,
As a last possibility, there may be realistic situations in which the dynamics of the bath
where, as before, (
The bath dynamics entailed by Eq. (21) can be highly nonlinear, especially for those low-frequency oscillators which undergo large amplitude motion. In principle, such model also describes energetic processes that irreversibly modify the environment, a phenomenon that can be mimicked by the dissociation of one or more oscillators.
4. Techniques for high-dimensional wave-packet dynamics
As is well known, the brute-force numerical solution of the Schrodinger equation rapidly runs into troubles when increasing the system dimensions. This is clearly seen by a simple scaling argument: With
Clearly, one central point to address when devising a method that could fulfill the above requirements is how to define approximate equations of motion. According to our experience, the most general strategy relies on the use of a variational principle that, apart from being physically transparent and mathematically sound, endows the resulting scheme with nontrivial properties. These features have important consequences in practice (
4.1. Variational principle and Hamiltonian flows
The time-dependent variational principle is usually stated in terms of the Dirac–Frenkel condition
where In fact, this is the
The above analysis shows that the time-dependent variational principle provides a In fact, the best approximation would just be the point on the manifold that lies closest to the exact solution at time
Energy and norm conservation follow immediately from Eq. (22) under quite mild conditions. For the energy, just notice that ∂ This is the
We expand somehow on this issue in the following since it does not seem to be widely appreciated, despite its deep significance and its potential practical utility. To this end, let us first briefly introduce the concept of Hamiltonian flows and symplectic manifolds [46]. A symplectic manifold is a differentiable manifold equipped with a This condition restricts the analysis to even dimensional manifolds.
With this premise in mind, we (smoothly) introduce a set of real variational parameters That is, we set
Where
or, equivalently, for a generic tangent vector
Hence, if
which are defined by {
(where
The importance of the symplectic structure thus described is hardly overemphasized. Apart from its possible consequences on fundamental issues such as the emergence of classicality, or the coexistence of quantum and classical worlds, it offers in practice the possibility of introducing
4.2. MCTDH, G-MCDTH, LCSA, and related methods
In this section, we give a brief account of (wave packet) quantum dynamical methods that have been applied in the past to system–bath problems of the kind discussed in this Chapter. The presentation is necessarily limited and, following authors’ personal experience, focuses on the so-called multi-configurational methods only. In these methods, the wave function is written as a superposition of “Hartree products” of single-particle states (or single-particle functions,
Among the various possibilities, the conceptually simplest method is the
where the sum runs over the possible configurations labeled by the multi-index
The equations of motion follow from the application of the Dirac–Frenkel condition to the above MCDTH
only corrected for the “ This expression is generally written as an explicit equation for the orbitals, by adding the above mentioned projection onto the occupied space of the spf velocity.
and involve the projection Strictly speaking,
Some general considerations are in order. The MCDTH method does
A second issue concerns the kind of problems MCTDH may handle. The method is “general purpose” and can tackle arbitrary problems, provided the interaction terms between modes can be reasonably described as (sum of) products of terms involving one mode at a time. This is due to the appearance of the mean-field operators above, whose evaluation requires “tracing” over potentially many degrees of freedom. Apart from this, there exists no limitation in the form of the system Hamiltonian and indeed, MCTDH has been applied with success to a very large number of different problems. The application to system–bath problems to be described below represents just one possible problem where the method applies; further applications can be found in the original research papers and in the extensive review literature [16]. Here, we just mention that a user-friendly, highly efficient, general MCDTH code which takes arbitrary Hamiltonians as input is freely available upon request to the author [47].
A second class of multi-configurational methods is represented by the
Though the method has several variants depending on the number of Gaussians introduced, it was originally formulated for system–bath-like problems, where one easily identifies primary modes (to be described at the high, fully flexible level) and secondary, less important modes that can be managed with moving Gaussians. In that case, the equations of motion for the amplitude coefficients and for the fully flexible
Along this line of thought, LCSA was
where These are highly localized objects in configuration space which underlie the use of any “grid”. See Refs. [49,50] for a formal introduction.
where
and, as a result, the bath dynamics is described by a set of coupled, Rigorously speaking, the system reduced density matrix
Equations of motion can be derived from the Dirac–Frenkel condition, using
in which the elements of the system DVR Hamiltonian are
(here,
which can be explicitly written down with the bath equations below. The bath equations are
and contain both a classical (local) force (−
This concludes the description of the original LCSA method. Several variants are possible (
In fact, among the features of LCSA, one key strength of the method is that it reduces the bath dynamics to classical-like evolution, with a number of trajectories that scales linearly with the bath dimensions. This means that the method itself has a power-low scaling with such dimensions, the exponent of this scaling depending (eventually) on the interaction between bath modes. For bath modes coupled to the system only [as in Eq. (5)],
Coupled trajectories also arise in a number of closely related approaches, namely the coupled coherent-state method of Shalashilin and Child [50,51] and the G-MCTDH method mentioned above. The latter, in fact, is strongly connected with LCSA and reduces to it as a limiting case (see Appendix B of Ref. [34]). The main difference between the two is that in LCSA
Finally, one interesting property about the pseudo-classical description of the bath degrees of freedom is that it suits well to induce The same applies to finite temperature cases where, as expected, both a friction
5. Applications
Here, we present some numerical applications of the IO model, starting from simple simulations of a Brownian anharmonic oscillator—used as a testing ground for new dynamical methodologies [34,36–38] and/or for different representation of the Hamiltonian [27]. The last part of the contribution will be devoted to a “real-world” application, namely the hydrogen atom dynamics on the graphene surface [40,41].
5.1. Model systems
We consider here a model Hamiltonian describing an anharmonic (Morse) oscillator coupled to a heat bath. A typical problem considered in this context is the small amplitude, damped motion of the oscillator. The initial state is taken in product form, with the bath in its ground state (to mimic relaxation at
This type of simulations was used in Ref. [34,36] to test the performances of different quantum dynamical approximations (the LCSA and its energy-local version, eLCSA). In Figure 2, the results of these techniques are shown along with benchmark MCTDH results, for different Ohmic spectral densities sampled with a bath of 80 oscillators. A Markovian exponential decay of the energy was found for all but the strongest coupling case, where some energy oscillations are clearly evident. The graph illustrates the main problems of standard LCSA, an inherent numerical instability related to saturation of the bath. These problems are solved in either its “damped” version [34] or with eLCSA [36]. The good agreement between LCSA and MCTDH is impressive, especially in light of the timing of the calculations (for LCSA only 2–3 min on a standard PC).
Similarly, in Ref. [27], the small amplitude relaxation of the Morse oscillator was used to illustrate the advantages of the chain transformation (Section 2.2). Here, MCTDH was used and different degrees of correlation were introduced along the chain, namely a small number of oscillators were described by a full, many particle expansion, whereas the rest of the chain was treated with one
Some results for small amplitude relaxation with the bath in linear-chain form are reported in Figure 3, for different structured SDs. The agreement with the benchmark is rather satisfactory and, as expected, the minor discrepancies were removed by increasing the correlation level. This is true both for the average system energy and for more detailed quantities like the position correlation functions.
The Morse oscillator was also used to model a dissipative scattering event, one in which the system is initially asymptotically free and moves toward the potential well where energy exchange with the bath occurs. Typically, in the interaction region, the wave-packet splits into two parts: One gets trapped in the well and fully relaxes on the long run, while the other returns to the asymptotic region. The first fraction, possibly resolved over the collision energy of the incoming wave packet, defines the “sticking” probability (having in minds problems where the bath represents a surface and a projectile sticks to it).
The sticking problem was considered as a test case for both LCSA and for the linear-chain representation of the bath with MCTDH (Figure 3). Simulations with standard LCSA showed that the numerical instabilities were too severe to extract meaningful sticking coefficients, even if the energy dissipation was described quite accurately [35]. On the contrary, the energy-local variant eLCSA gave stable results but only in semi-quantitative agreement with the benchmark. A detailed analysis showed that this is due to an inadequate system–bath correlation in the adopted
5.2. Hydrogen atom dynamics on graphene
In the last decade, the
Recently, we have devised a rather elaborate system–bath model to describe hydrogen chemisorption of graphene and used it in a fully quantum study of the sticking dynamics with the MCDTH method. The model consists of an accurate description of the hydrogen atom and its bonding carbon atom (a 4D system), which were then coupled to the graphene sheet described by a phonon bath. It rests on the following, reasonable assumptions: (i) The energy exchange that occurs between the system and the lattice for near equilibrium configurations is representative of energy dissipation; (ii) relaxation proceeds through sequential energy transfer from the hydrogen atom to the carbon atom; (iii) a mapping holds, at least approximately, which relates the classical Hamiltonian dynamics of the interesting C and H atoms to a GLE description. On this basis, the following form was adopted for the Hamiltonian
Here,
The thus-obtained SD The “surface” stretching is one of the normal mode of the 4D system potential
Once the coupling of the C atom with its environment was introduced, the Hamiltonian model of Eq. (39) could be used to investigate the hydrogen atom dynamics. We start here by considering the relaxation problem of the C–H bond. In this case, the system was prepared in an eigenstate of the
As is shown in Figure 5, relaxation from the surface stretching mode proceeds over a very short time scale and is complete in a few tens of fs. Despite the fast relaxation dynamics indicates a strong coupling between this coordinate and the bath, the energy decay shows essentially a Markovian behavior, except for the slippage at short time which extends for a considerable fraction of the relaxation window. This feature is related to the prepared initial states and to the switching on of the coupling term, which actually causes a slight
Next, we consider the quantum simulations of the collinear sticking dynamics [41], that is, the process in which a gas-phase hydrogen atom colliding at normal incidence above a carbon atom exchanges energy with the surface and gets trapped in the chemisorption well. We used the MCTDH method once again and, in addition, classical and quasi-classical methods to single out quantum effects in the results (Figure 7).
Before discussing the quantum dynamical results, it is instructive to first focus on some classical aspects of the sticking process. In Figure 7, the classical results are reported for two different surface temperatures,
The quantum results differ from the classical ones in the whole energy range considered. While quantum effects (tunneling) can be invoked in the low-energy regime, above saturation the discrepancy is necessarily due to the quantum nature of the low-temperature surface which, in this
Acknowledgments
This work has arisen over the years through an intense and fruitful collaboration with several people. Among them, we are particularly in debt to Mathias Nest, Peter Saalfrank, Keith Hughes, and Irene Burghardt, who are sincerely acknowledged for their contribution. Support for the numerics from the Regione Lombardia and the CINECA High Performance Computing Center through the LISA initiative is also gratefully acknowledged.
References
- 1.
Nitzan A. Chemical dynamics in condensed phases—relaxation, transfer and reactions in condensed molecular systems. Oxford University Press; 2006. - 2.
Lee H, Cheng Y-C, Fleming GR. Coherence dynamics in photosynthesis: protein protection of excitonic coherence. Science 2007;316:1462–5. doi:10.1126/science.1142188. - 3.
Collini E, Scholes GD. Coherent intrachain energy migration in a conjugated polymer at room temperature. Science 2009;323:369–73. doi:10.1126/science.1164016. - 4.
Hwang I, Scholes GD. Electronic energy transfer and quantum-coherence in π-conjugated polymers. Chem Mater 2011;23:610–20. doi:10.1021/cm102360x. - 5.
Engel GS, Calhoun TR, Read EL, Ahn T-K, Mancal T, Cheng Y-C, et al. Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature 2007;446:782–6. doi:10.1038/nature05678. - 6.
Collini E, Wong CY, Wilk KE, Curmi PMG, Brumer P, Scholes GD. Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature. Nature 2010;463:644–7. doi:10.1038/nature08811. - 7.
Einstein A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann Phys 1905;322:549–60. doi:10.1002/andp.19053220806. - 8.
Kubo R, Toda M, Hashitsume N. Statistical physics II—nonequilibrium statistical mechanics. Springer Verlag; 1991. - 9.
Lemons DS, Gythiel A. Paul Langevin’s 1908 paper “On the Theory of Brownian Motion” [“Sur la théorie du mouvement brownien,” C. R. Acad. Sci. (Paris) 146, 530–533 (1908)]. Am J Phys 1997;65:1079–81. doi:10.1119/1.18725. - 10.
Zwanzig R. Nonequilibrium statistical mechanics. USA: Oxford University Press; 2001. - 11.
Tuckerman M. Statistical mechanics: theory and molecular simulation. OUP Oxford; 2010. - 12.
Breuer HP, Petruccione F. The theory of open quantum systems. Oxford University Press; 2007. - 13.
Caldeira AO, Leggett AJ. Influence of damping on quantum interference: an exactly soluble model. Phys Rev A 1985;31:1059–66. doi:10.1103/PhysRevA.31.1059. - 14.
Ford GW, Lewis JT, O’Connell RF. Quantum Langevin equation. Phys Rev A 1988;37:4419–28. doi:10.1103/PhysRevA.37.4419. - 15.
Zwanzig R. Nonlinear generalized Langevin equations. J Stat Phys 1973;9:215–20. - 16.
Meyer H-D, Gatti F, Worth GA, editors. Multidimensional quantum dynamics: MCTDH theory and applications. Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA; 2009. - 17.
Caldeira AO, Leggett AJ. Influence of dissipation on quantum tunneling in macroscopic systems. Phys Rev Lett 1981;46:211–4. doi:10.1103/PhysRevLett.46.211. - 18.
Weiss U. Quantum dissipative systems. 3rd ed. Singapore: World Scientific; 2008. - 19.
Pottier N. Oxford University Press ;2010 . - 20.
Hughes KH, Christ CD, Burghardt I. Effective-mode representation of non-Markovian dynamics: a hierarchical approximation of the spectral density. I. Application to single surface dynamics. J Chem Phys 2009;131:024109. doi:10.1063/1.3159671. - 21.
Chin AW, Rivas Á, Huelga SF, Plenio MB. Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials. J Math Phys 2010;51:092109. doi:10.1063/1.3490188. - 22.
Prior J, Chin AW, Huelga SF, Plenio MB. Efficient simulation of strong system-environment interactions. Phys Rev Lett 2010;105:050404. doi:10.1103/PhysRevLett.105.050404. - 23.
Martinazzo R, Vacchini B, Hughes KH, Burghardt I. Communication: Universal Markovian reduction of Brownian particle dynamics. J Chem Phys 2011;134:011101. doi:10.1063/1.3532408. - 24.
Martinazzo R, Hughes KH, Burghardt I. Unraveling a Brownian particle’s memory with effective mode chains. Phys Rev E 2011;84:030102(R). doi:10.1103/PhysRevE.84.030102. - 25.
Burghardt I, Hughes KH, Martinazzo R, Tamura H, Gindensperger E, Köppel H, et al. Conical intersections coupled to an environment. In: Domcke W, Yarkony DR, Köppel H, editors. Conical Intersect., vol. 17, World Scientific; 2011, pp. 301–46. - 26.
Martinazzo R, Hughes KH, Burghardt I. Hierarchical effective-mode approach for extended molecular systems. In: Hoggan EP, Brändas JE, Maruani J, Piecuch P, Delgado-Barrio G, editors. Adv. Theory Quantum Syst. Chem. Phys., Dordrecht, Netherlands: Springer; 2012, p. 269–83. - 27.
Bonfanti M, Tantardini GF, Hughes KH, Martinazzo R, Burghardt I. Compact MCTDH wave functions for high-dimensional system–bath quantum dynamics. J Phys Chem A 2012;116:11406–13. doi:10.1021/jp3064504. - 28.
Woods MP, Groux R, Chin AW, Huelga SF, Plenio MB. Mappings of open quantum systems onto chain representations and Markovian embeddings. J Math Phys 2014;55:032101. doi:10.1063/1.4866769. - 29.
Bonfanti M, Hughes KH, Burghardt I, Martinazzo R. Vibrational relaxation and decoherence in structured environments: a numerical investigation. Ann Phys 2015;527:556–69. doi:10.1002/andp.201500144. - 30.
Gottwald F, Ivanov SD, Kühn O. Applicability of the Caldeira–Leggett model to vibrational spectroscopy in solution. J Phys Chem Lett 2015;6:2722–7. doi:10.1021/acs.jpclett.5b00718. - 31.
Gottwald F, Ivanov SD, Kühn O. Vibrational spectroscopy via the Caldeira–Leggett model with anharmonic system potentials. ArXiv E-Prints 2016. - 32.
Meyer H-D, Manthe U, Cederbaum LS. The multi-configurational time-dependent Hartree approach. Chem Phys Lett 1990;165:73–8. doi:10.1016/0009-2614(90)87014-I. - 33.
Beck MH, Jäckle A, Worth GA, Meyer H-D. The multiconfiguration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets. Phys Rep 2000;324:1–105. doi:10.1016/S0370-1573(99)00047-2. - 34.
Martinazzo R, Nest M, Saalfrank P, Tantardini GF. A local coherent-state approximation to system-bath quantum dynamics. J Chem Phys 2006;125:194102. doi:10.1063/1.2362821. - 35.
Martinazzo R, Burghardt I, Martelli F, Nest M. Local coherent-state approximation to system-bath quantum dynamics. In: Shalashilin DV, De Miranda MP, editors. vol. Multidimensional quantum mechanics with trajectories, CCP6; 2009. - 36.
López-López S, Nest M, Martinazzo R. Generalized CC-TDSCF and LCSA: The system-energy representation. J Chem Phys 2011;134:014102. doi:10.1063/1.3518418. - 37.
Nest M, Meyer H-D. Dissipative quantum dynamics of anharmonic oscillators with the multiconfiguration time-dependent Hartree method. J Chem Phys 2003;119:24–33. doi:10.1063/1.1576384. - 38.
Burghardt I, Nest M, Worth GA. Multiconfigurational system-bath dynamics using Gaussian wave packets: energy relaxation and decoherence induced by a finite-dimensional bath. J Chem Phys 2003;119:5364–78. doi:10.1063/1.1599275. - 39.
López-López S, Martinazzo R, Nest M. Benchmark calculations for dissipative dynamics of a system coupled to an anharmonic bath with the multiconfiguration time-dependent Hartree method. J Chem Phys 2011;134. doi:10.1063/1.3556940. - 40.
Bonfanti M, Jackson B, Hughes KH, Burghardt I, Martinazzo R. Quantum dynamics of hydrogen atoms on graphene. I. System-bath modeling. J Chem Phys 2015;143. doi:10.1063/1.4931116. - 41.
Bonfanti M, Jackson B, Hughes KH, Burghardt I, Martinazzo R. Quantum dynamics of hydrogen atoms on graphene. II. Sticking. J Chem Phys 2015;143. doi:10.1063/1.4931117. - 42.
Hughes KH, Christ CD, Burghardt I. Effective-mode representation of non-Markovian dynamics: a hierarchical approximation of the spectral density. II. Application to environment-induced nonadiabatic dynamics. J Chem Phys 2009;131:124108. doi:10.1063/1.3226343. - 43.
Mori H. Transport, collective motion, and Brownian motion. Prog Theor Phys 1965;33:423. doi:10.1143/PTP.33.423. - 44.
Zwanzig R. Memory effects in irreversible thermodynamics. Phys Rev 1961;124:983–92. doi:10.1103/PhysRev.124.983. - 45.
Kramer P, Saraceno M, editors. Geometry of the time-dependent variational principle in quantum mechanics. Lecture notes in physics. vol. 140. Berlin: Springer Verlag; 1981. - 46.
Flanders H. Differential forms with applications to the physical sciences. Dover Publications; 1963. - 47.
Worth GA, Beck MH, Jäckle A, Meyer H-D. The MCTDH Package, Version 8.2, (2000). H.-D. Meyer, Version 8.3 (2002), Version 8.4 (2007), Version 8.5 (2011). Current version: 8.5.3 (2013). See http://mctdh.uni-hd.de. University of Heidelberg, Germany; n.d. - 48.
Burghardt I, Meyer H-D, Cederbaum LS. Approaches to the approximate treatment of complex molecular systems by the multiconfiguration time-dependent Hartree method. J Chem Phys 1999;111:2927–39. doi:http://dx.doi.org/10.1063/1.479574. - 49.
Zhang DH, Bao W, Yang M. Continuous configuration time-dependent self-consistent field method for polyatomic quantum dynamical problems. J Chem Phys 2005;122:091101. doi:10.1063/1.1869496. - 50.
Shalashilin DV, Child MS. Time dependent quantum propagation in phase space. J Chem Phys 2000;113:10028. - 51.
Shalashilin DV, Child MS. The phase space CCS approach to quantum and semiclassical molecular dynamics for high-dimensional systems. Chem Phys 2004;304:103–20. doi:10.1016/j.chemphys.2004.06.013. - 52.
Sakong S, Kratzer P. Hydrogen vibrational modes on graphene and relaxation of the C–H stretch exmixed-citation from first-principles calculations. J Chem Phys 2010;133:054505. doi:10.1063/1.3474806.
Notes
- Here and in the following … 〉denotes an average over the canonical equilibrium.
- This is also known as Caldeira–Leggett Hamiltonian, after the seminal work by Caldeira and Leggett on the effects of dissipation on quantum tunneling [17].
- In the following we will adopt, without loosing generality, the same mass for all the oscillators, i.e. mk≡μ for all k, where μ is a numerically convenient value.
- Strictly speaking such “Markovian reduction” rigorously holds in classical mechanics only; in a quantum setting the very definition of Markovian dynamics is still debated. Thus, one should better refer here to an “Ohmic embedding”.
- The problem is essentially classical in nature, since the statistical properties of the bath (when subsumed in the spectral density J0(ω) are the same for both the classical and quantum GLE.
- Numerical evaluation of Eq. (19) requires the introduction of a high-frequency cutoff. The problem arises when using an “unbiased” cut-off frequency well above the spectral range of interest (ωc = 4000cm− 1 in the simulations) and can be easily amended by setting ωc equal to the bath Debye frequency (if known).
- A fictitious coupling to the bath appears here because numerically the autocorrelation function needs to be damped.
- In principle, such model also describes energetic processes that irreversibly modify the environment, a phenomenon that can be mimicked by the dissociation of one or more oscillators.
- In fact, this is the condicio sine qua non for the existence of a variational principle.
- In fact, the best approximation would just be the point on the manifold that lies closest to the exact solution at time t (whose identity may further depend on the adopted metrics).
- This is the true variational principle, i.e. a stationarity condition of some cleverly designed functional (something which Eq. (22) is not). However, the two formulations can be shown to be equivalent under quite mild conditions that are usually satisfied in practice.
- This condition restricts the analysis to even dimensional manifolds.
- That is, we set Ψ〉 = Ψ(x)〉, where x are, e.g., the real and the imaginary parts of a set of complex parameters specifying the wave function. Though not necessarily finite in number or numerable, it is conceptually easier to think of a large but finite number of parameters.
- This expression is generally written as an explicit equation for the orbitals, by adding the above mentioned projection onto the occupied space of the spf velocity.
- Strictly speaking, ρk is the transpose of the 1-particle density for the kth mode, in the spf’s basis.
- These are highly localized objects in configuration space which underlie the use of any “grid”. See Refs. [49,50] for a formal introduction.
- Rigorously speaking, the system reduced density matrix ρ also requires the overlap between bath state, ραβ = CαCβ〈Zβ ∨ Zα〉 in the underlying DVR.
- The same applies to finite temperature cases where, as expected, both a friction and a fluctuating term appear in the LCSA equations of each realization.
- The “surface” stretching is one of the normal mode of the 4D system potential Vs(xH, zC). This eigen-mode lying at ~460 cm− 1 approximately corresponds to block oscillations of the C-H unit above the surface plane.