The multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method—also known in applied mathematics as the (time-dependent) hierarchical Tucker or tree tensor network decomposition—is a powerful variational technique for simulating quantum dynamics in high-dimensional systems. It constructs efficient, recursive, time-adaptive representations of wavefunctions using dynamically contracted tensor networks. This short course introduces the theoretical foundations, numerical formulations, and computational implementations of ML-MCTDH, with emphasis on its application to condensed phase chemical physics. Relevant connections to tensor methods in numerical analysis and data science will also be briefly noted.

Course outline:
1. Standard variational approach to solve time-dependent Schrödinger equations.
(a) Conventional eigenfunction (basis) expansion method to the Schrödinger equations - linear homogeneous PDEs to ODEs.
(b) Variational principle and Dirac-Frenkel framework.
(c) Continuous and grid basis functions, finite basis representation and discrete variable representation, contraction of the basis functions.
(d) Initial conditions and integration schemes.
(e) Scaling issues and motivation for advanced methods.
2. The multiconfiguration time-dependent Hartree (MCTDH) method.
(a) The Heidelberg formulation of the MCTDH method: Tucker decomposition of the full tensor.
(b) Nonlinear equations of motion and projection
(c) Initial condition preparation and numerical propagation; regularization of the equations of motion.
(d) Applications and scaling analysis.
3. The ML-MCTDH method.
(a) Tree tensor networks and multilayer structure.
(b) Recursive equations of motion.
(c) Technical details of the simulations: basis functions, integration schemes, and tree structures.
(d) Models of condensed phase chemistry/physics and specific ML-MCTDH implementations.
(e) Discussion of scaling and numerical challenges.
4. Treating identical particles.
(a) The second quantized representation: dummy encoding and the permutation symmetry/anti-symmetry.
(b) Treating identical fermions: Jordan-Wigner transformation.
(c) Technical considerations and sample applications.
5. Calculating energy eigenstates.
(a) Tree tensor representation for time-independent problems.
(b) Multilayer Improved Relaxation method: Krylov subspace iteration and alternating optimization.
(c) Example applications.
(d) Current limitations and possible extensions.

The school spreads over five days with theory sessions in the morning and hands-on exercises in the afternoon. The list of topics covered includes:

- Numerical methods for quantum dynamics
- MCTDH theory
- Introduction to polyspherical coordinates
- Mode combination and multilayer trees
- Vibronic Hamiltonians for photophysics and photochemistry
- Scattering and molecular dissociation
- High-dimensional problems
- Hands-on exercises with the Quantics / Heidelberg MCTDH package, including
- Spectroscopy & quantum control
- Reactive scattering
- Direct dynamics / GWP
- Practical use of the TANA program for deriving analytical kinetic operators
- Potential energy operators in sum-of-products form

Conference Themes
• Advances in Discretisation Schemes
• Multiscale and Multiphysics Problems
• Hardware-Aware Scientific Computing
• Inverse Problems
• Uncertainty Quantification 
• Data-Driven Modelling and Simulation
• Scientific Machine Learning
• Reduced Order Models and Surrogates
• Randomised Numerical Algorithms
• Numerical Optimisation and Optimal Control

For more information, please visit the event website.