Gaussian wavepacket methods are an attractive way to solve the time-dependent Schrödinger equation (TDSE). They have an underlying trajectory picture that has a natural connection to semi-classical mechanics, allowing a simple pictorial interpretation of an evolving wavepacket. They also have better scaling with system size compared to conventional grid-based techniques. Here we review the variational multi-configurational Gaussian (vMCG) method. This is a variational solution to the TDSE, with explicit coupling between the Gaussian basis functions, resulting in a favourable convergence on the exact solution. The implementation of the method and its performance will be discussed with examples from non-adiabatic photo-excited dynamics and tunneling to show that it can correctly describe both of these strongly quantum mechanical processes. Particular emphasis is given to the implementation of the direct dynamics variant, DD-vMCG, where the potential surfaces are calculated on-the-fly via an interface to quantum chemistry programs.

The MCTDH method has been used successfully to treat the non-adiabatic dynamics of a number of systems. These are hard problems due to the number of modes that need to be included in a calculation, and the strong coupling between the nuclear and electronic motion at conical intersections connecting electronic states in these systems. In this review, an overview of the basic theory of the method is given highlighting how it is able to treat larger systems than other quantum dynamics methods. The vibronic coupling model Hamiltonian is also described, which provides a good starting point for the description of these systems. Examples of calculations made and systems treated are given. Finally, a development of the basic MCTDH method in which some of the usual time-dependent basis functions are replaced by Gaussian wavepackets is outlined. This method promises not only to treat larger systems, but to provide a consistent quantum–semiclassical framework.

The Gaussian-based multiconfiguration time-dependent Hartree (G-MCTDH) method is applied to calculate the S2(ππ∗) absorption spectrum of the pyrazine molecule, whose diffuse structure results from the ultrafast nonadiabatic dynamics at the S2-S1 conical intersection. The 24-mode second-order vibronic-coupling model of Raab et al. [J. Chem. Phys. 110, 936 (1999)] is employed, along with 4-mode and 10-mode reduced-dimensional variants of this model. G-MCTDH can be used either as an all-Gaussian approach or else as a hybrid method using a partitioning into primary modes, treated by conventional MCTDH basis functions, and secondary modes described by Gaussian particles. Comparison with standard MCTDH calculations shows that the method converges to the exact result. The variational, nonclassical evolution of the moving Gaussian basis is a key element in obtaining convergence. For high-dimensional systems, convergence is significantly accelerated if the method is employed as a hybrid scheme.

A review of direct dynamics methods is given, focusing on their application to non-adiabatic photochemistry–i.e. systems in which a conical intersection plays an important role. Direct dynamics simulations use electronic structure calculations to obtain the potential energy surface only as it is required ‘on-the-fly’. This is in contrast to traditional methods that require the surface to be globally known as an analytic function before a simulation can be performed. The properties and abilities, with descriptions of calculations made, of the three main methods are compared: trajectory surface hopping (TSH), ab initio multiple spawning (AIMS), and variational multi-configuration Gaussian wavepackets (vMCG). TSH is the closest to classical dynamics, is the simplest to implement, but is hard to converge, and even then not always accurate. AIMS solves the time-dependent Schrödinger more rigorously, but as its basis functions follow classical trajectories again suffers from poor convergence. vMCG is harder to implement, but its basis functions do not follow classical trajectories and it converges much faster.

We present here a direct quantum dynamics method using variational multi-configuration Gaussian wavepackets. Based on the efficient multi-configuration time-dependent Hartree wavepacket propagation algorithm, it uses on-the-fly quantum chemical calculation of the potential energy and its derivatives rather than fitted surfaces. Intermediate results are stored in a database so that expensive quantum chemical computations can be recycled. This method is intended to treat quantum effects in the photochemistry of large molecules and the use of Cartesian coordinates to perform direct dynamics is discussed with a comparison between Cartesian coordinates of Jacobi vectors and Cartesian coordinates of the nuclei, using various free and constrained approaches depending on the way the rotation is treated. As a test calculation to be compared to full quantum dynamics it is applied here to the computation of the photodissociation spectrum of nitrosyl chloride (NOCl).