Quantum Dynamical Simulation of Electron-Transfer Reactions in an Anharmonic Environment † Haobin Wang Department of Chemistry and Biochemistry, MSC 3C, New Mexico State UniVersity, Las Cruces, New Mexico 88003 Michael Thoss* Department of Chemistry, Technical UniVersity of Munich, 85748 Garching, Germany ReceiVed: March 26, 2007; In Final Form: May 25, 2007 The multilayer multiconfiguration time-dependent Hartree theory is applied to study the quantum dynamics of ultrafast electron-transfer reactions in a condensed-phase environment with anharmonic potential functions. Effects of anharmonicity for both the nuclear degrees of freedom of the environment and the intramolecular vibrational degrees of freedom are investigated. Whereas the former can in principle be mapped to a fictitious harmonic bath, the latter cannot be represented in this way and, thus, go beyond the commonly employed linear response approximation. Numerical examples are presented to illustrate these findings. I. Introduction The accurate description of quantum effects for large mo- lecular systems is a challenging task in theoretical chemical dynamics. Due to the rapid development in time-resolved nonlinear spectroscopy techniques, more detailed information on the reaction dynamics of complex molecular systems has become available in recent years. As a consequence, it has been realized that in many complex processes, quantum tunneling and coherence effects may play important roles. Such effects cannot be described by purely classical methods, such as, for example, molecular dynamics simulation. This has stimulated the development of theoretical methods that are capable of describing the quantum dynamics in systems with many degrees of freedom. According to their different nature, these methods can be broadly divided into two major classes: rigorous quantum dynamical methods and semiclassical approaches. The multi- configuration time-dependent Hartree (MCTDH) theory 1-4 and, in particular, its multilayer (ML) generalization, the ML- MCTDH theory, 5 are promising examples of the former class. The feasibility of the MCTDH method has been demonstrated by many applications to gas-phase reactions of relatively large molecules. 6-12 For reactions in a condensed-phase environment, there is currently no universal rigorous method available that is capable of simulating the quantum dynamics for a general complex molecular system with arbitrary potential functions. However, the MCTDH method has been proven extremely useful for treating certain classes of quantum dynamical processes in large molecular systems in a numerically exact way, in which a moderate number of degrees of freedom has been explicitly included in the dynamical treatment. 13-17 An important example along this line is the system-bath Hamiltonian that models reactions in the condensed phase, for example, the spin- boson model 18,19 for donor-acceptor electron transfer (ET) processes. 20 The MCTDH method, together with the self- consistent hybrid approach, 14,21,22 has been shown to compare favorably with the alternative path integral approach 23-30 based on Feynman-Vernon influence-functional technique. 31 The original MCTDH method is limited to treating a few tens of degrees of freedom. This is adequate for describing the dynamics of the spin-boson model in a relatively limited physical regime. To simulate quantum dissipative dynamics in a broader parameter space, the more versatile multilayer (ML) generalization of the MCTDH method has been proven useful. 5,32-34 This is particularly important for treating donor- acceptor ET reactions in a complex condensed-phase environ- ment in which both the intramolecular vibrational degrees of freedom of the donor-acceptor complex (inner sphere) and the continuous distribution of solvent modes (outer sphere) con- tribute to the overall vibronic dynamics. In most previous studies of ET reactions in the condensed phase, the influence of the nuclear degrees of freedom is modeled by a bath of harmonic oscillators, 19,20,35 which corre- sponds to a linear response model 36,37 for the outer sphere solvent environment and a harmonic approximation for the inner sphere vibrational modes. This may be justified if the interaction of the donor-acceptor complex with the environment is evenly distributed over many nuclear degrees of freedom. There are, however, many situations in which this is not the case. Examples include strongly coupled low-frequency intramolecular modes (such as torsional motion) for which the harmonic approximation is not appropriate and ET in nonpolar liquids for which the linear response treatment may fail. Even in cases that the linear response approach is valid and, thus, a mapping to a fictitious harmonic bath is possible, it may be more convenient to simulate the quantum dynamics of the ET reaction in the original anharmonic environment. This is because such a mapping to a harmonic bath is temperature-dependent, which makes it difficult to systematically study the dependence of ET dynamics on various physical parameters (e.g., temperature, time scale, and coupling of the bath, etc.). From a theoretical perspective, it is also important to develop methods that can directly simulate ET reactions in an anharmonic environment. This not only † Part of the special issue “Robert E. Wyatt Festschrift”. 10369 J. Phys. Chem. A 2007, 111, 10369-10375 10.1021/jp072367x CCC: $37.00 © 2007 American Chemical Society Published on Web 07/19/2007