How Wavepacket Propagation Works¶

Wavepacket propagation is the heart of every MCTDH calculation. Unlike electronic structure methods such as Hartree–Fock or CASSCF, which determine stationary electronic states, MCTDH follows the time evolution of a molecular wavefunction after it has been prepared in an initial electronic state.

The propagation starts from an initial wavepacket, evolves it under the influence of the vibronic Hamiltonian defined in the operator file, and computes physical observables such as electronic populations, autocorrelation functions, and absorption spectra. Rather than solving the full time-dependent Schrödinger equation on an enormous multidimensional grid, MCTDH represents the wavefunction in a compact time-dependent basis that adapts continuously during propagation.

Overall Workflow¶

Initial Wavepacket
        │
        ▼
Read Operator File
        │
        ▼
Construct ML-MCTDH Tree
        │
        ▼
Expand the Wavefunction
        │
        ▼
Propagate in Time
        │
        ▼
Compute Observables
        │
        ▼
Generate Spectra

Each stage of the calculation is discussed below.

Step 1 — Prepare the Initial Wavepacket¶

Every MCTDH propagation begins with an initial quantum state. This wavepacket represents the state of the molecule immediately after photoexcitation or another external perturbation.

The initial state typically consists of

the occupied electronic state,
the vibrational ground-state wavefunction,
the initial position of the nuclear wavepacket.

For example, in a photoexcitation simulation the wavepacket is usually initialized on one excited electronic state while all vibrational modes start from their ground-state harmonic oscillator functions.

Ground-State Geometry
        │
Photoexcitation
        │
        ▼
Initial Nuclear Wavepacket

At the beginning of the propagation,

all expansion coefficients are initialized,
all single-particle functions (SPFs) are generated,
the wavefunction norm equals one.

Step 2 — Read the Vibronic Hamiltonian¶

The propagation program next reads the operator file generated during the previous stage.

The operator file contains

harmonic frequencies,
electronic energies,
linear vibronic couplings,
quadratic coupling terms,
interstate coupling elements,
kinetic energy operators.

Instead of calculating the potential energy surface during propagation, MCTDH simply evaluates the operator defined in this file.

Operator File
      │
      ▼
Hamiltonian Construction

This Hamiltonian governs the complete quantum dynamics throughout the simulation.

Step 3 — Construct the ML-MCTDH Tree¶

The vibrational modes are grouped into a multilayer tree according to the ML tree defined in the input file.

Rather than treating all vibrational coordinates simultaneously, ML-MCTDH organizes them hierarchically.

For example,

Root
├── Node 1
│   ├── Modes 1–8
│   └── Modes 9–16
└── Node 2
    ├── Modes 17–24
    └── Modes 25–32

Each node possesses its own time-dependent basis functions.

This hierarchical decomposition dramatically reduces computational cost while retaining the essential quantum correlations.

Step 4 — Expand the Wavefunction¶

Once the tree is constructed, the molecular wavefunction is expanded in terms of time-dependent basis functions.

Instead of using a fixed multidimensional basis, MCTDH continuously updates the basis functions during propagation.

Conceptually,

Wavefunction
      │
      ▼
Expansion Coefficients
      +
Time-Dependent SPFs

The wavefunction therefore evolves through

changing expansion coefficients,
changing single-particle functions,
changing multilayer basis functions.

This adaptive representation is the primary reason MCTDH can treat systems containing dozens or even hundreds of vibrational degrees of freedom.

Step 5 — Propagate the Wavepacket¶

After initialization, the wavepacket is propagated through time.

For each propagation step,

the Hamiltonian acts on the wavefunction,
expansion coefficients are updated,
SPFs are optimized,
normalization is maintained,
observables are evaluated.

Current Wavefunction
        │
Hamiltonian Action
        │
        ▼
Update Coefficients
        │
        ▼
Update SPFs
        │
        ▼
Next Time Step

This procedure is repeated until the final propagation time specified in the input file is reached.

Unlike classical molecular dynamics, every time step retains the complete quantum-mechanical description of both nuclear and electronic motion.

Step 6 — Compute Physical Observables¶

During propagation, MCTDH evaluates several physical quantities at every output time step.

Typical observables include

total energy,
wavefunction norm,
natural populations,
expectation values,
variances,
vibrational occupations,
electronic populations.

These quantities are written directly to the output file and provide detailed information about the evolution of the system.

Wavefunction
      │
      ▼
Expectation Values
      │
      ├── Energy
      ├── Population
      ├── Norm
      └── Mode Occupation

These observables are later used to analyse ultrafast molecular dynamics.

Step 7 — Generate Spectroscopic Information¶

Once propagation is complete, the saved wavefunction information can be transformed into experimentally observable spectra.

Common post-processing calculations include

electronic populations,
autocorrelation functions,
absorption spectra,
emission spectra,
Fourier transforms.

Propagation Output
        │
        ├── Population Analysis
        ├── Autocorrelation
        └── Spectrum Generation

These analyses connect the quantum dynamics simulation with measurable spectroscopic experiments.

Why ML-MCTDH is Efficient¶

The largest computational challenge in quantum dynamics is the exponential growth of the multidimensional wavefunction with increasing vibrational degrees of freedom.

ML-MCTDH avoids this problem by

using time-dependent basis functions,
optimizing only the important parts of the wavefunction,
grouping modes into a hierarchical tree,
reducing the number of required basis functions.

As a result, systems containing tens or even hundreds of coupled vibrational modes become computationally tractable.

Advantages¶

Efficient treatment of high-dimensional quantum dynamics.
Time-dependent basis adapts automatically during propagation.
Simultaneous treatment of electronic and vibrational motion.
Suitable for vibronic coupling and nonadiabatic dynamics.
Direct calculation of time-dependent observables.
Straightforward generation of spectra through post-processing.

Limitations¶

Requires a reliable vibronic Hamiltonian.
Construction of the operator file can be computationally demanding.
Choice of ML tree strongly influences efficiency.
Long propagation times may require significant computational resources.
Accuracy depends on the number of SPFs and primitive basis functions.

Key Takeaways¶

Wavepacket Propagation Workflow

Prepare an initial nuclear wavepacket.
Read the vibronic Hamiltonian from the operator file.
Build the multilayer tree representation.
Expand the wavefunction using adaptive time-dependent basis functions.
Propagate the wavepacket through time.
Compute observables such as energies and electronic populations.
Generate autocorrelation functions and spectra for comparison with experiments.

The propagation stage transforms a static vibronic Hamiltonian into a complete description of molecular quantum dynamics, revealing how electronic populations, nuclear motion, and spectroscopic properties evolve on the femtosecond timescale.