Mathematics for Nerds¶

The previous sections demonstrated how to prepare an operator file, run an MCTDH propagation, and interpret the output. While those chapters focused on the practical workflow, this section explains the mathematical framework that makes those calculations possible.

Unlike electronic structure methods such as Hartree–Fock or CASSCF, which solve the electronic Schrödinger equation, MCTDH solves the time-dependent Schrödinger equation (TDSE) for coupled nuclear and electronic motion. The operator file generated in the previous chapter represents the molecular Hamiltonian, the propagation input specifies the initial quantum state and numerical parameters, and the propagation algorithm integrates the wavefunction in time to produce populations, autocorrelation functions, and absorption spectra.

The mathematical development presented here follows the same sequence as an actual MCTDH calculation.

Mathematical workflow¶

An MCTDH simulation proceeds through the following mathematical steps:

Electronic Structure Calculations
          │
          ▼
 Potential Energy Surfaces
 Coupling Surfaces
 Vibrational Frequencies
          │
          ▼
Vibronic Hamiltonian Construction
          │
          ▼
Operator File
(Hamiltonian Representation)
          │
          ▼
Time-Dependent Schrödinger Equation
          │
          ▼
MCTDH Wavefunction Expansion
          │
          ▼
Variational Equations of Motion
          │
          ▼
Wavepacket Propagation
          │
          ▼
Observables
(Populations, Autocorrelation,
Spectra, Expectation Values)

Each chapter in this section corresponds to one stage of this mathematical pipeline.

Topics Covered¶

1. Time-Dependent Schrödinger Equation¶

We begin with the fundamental equation governing quantum dynamics,

\[ i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{Q},t) = \hat H\Psi(\mathbf{Q},t), \]

which describes how the nuclear wavepacket evolves on one or more coupled potential energy surfaces.

2. Vibronic Hamiltonian¶

The Hamiltonian used in MCTDH is constructed from electronic energies, vibrational frequencies, and vibronic coupling constants obtained from quantum chemistry calculations.

This Hamiltonian is precisely what is written into the operator file generated in the previous chapter.

3. MCTDH Wavefunction¶

Instead of representing the wavefunction on a prohibitively large direct-product grid, MCTDH expands it using time-dependent basis functions (Single Particle Functions, SPFs).

This is the key approximation that makes simulations involving tens or even hundreds of vibrational modes computationally feasible.

4. Variational Principle¶

The time evolution of both the expansion coefficients and the SPFs is derived from the Dirac–Frenkel variational principle.

This produces a set of coupled equations of motion that are solved simultaneously throughout the propagation.

5. Multi-Layer MCTDH¶

For systems containing many vibrational modes, the coordinates are organized into hierarchical trees.

This mathematical framework explains why choosing 24, 28, or 32 strongly coupled modes and constructing a balanced ML-tree significantly improves computational efficiency.

6. Numerical Propagation¶

Once the Hamiltonian and wavefunction are defined, MCTDH numerically propagates the quantum state through time using predictor–corrector integration together with repeated Hamiltonian applications.

This stage generates the quantities observed in the output file.

7. Physical Observables¶

Finally, the propagated wavefunction is used to calculate measurable quantities such as

Electronic state populations
Vibrational expectation values
Natural orbital weights
Autocorrelation functions
Absorption spectra

These correspond directly to the population, auto, and spectrum files generated during the propagation tutorial.

Connection with Earlier Chapters¶

The mathematical concepts introduced here explain every file created during the practical workflow.

Practical Chapter	Mathematical Interpretation
Operator File	Hamiltonian operator \( \hat H \)
Propagation Input	Initial wavefunction and numerical parameters
Propagation Output	Numerical solution of the TDSE
Population Analysis	Projection of the wavefunction onto electronic states
Autocorrelation Function	Overlap between initial and propagated wavepacket
Absorption Spectrum	Fourier transform of the autocorrelation function

By the end of this section, you will understand not only how an MCTDH calculation is performed, but also why each equation appears, how the operator file represents the Hamiltonian, how the wavepacket evolves, and how experimentally observable spectra emerge from quantum dynamics.

Chapters in this Section¶

Time-Dependent Schrödinger Equation
Molecular Hamiltonian
Vibronic Hamiltonian
MCTDH Wavefunction Expansion
Single Particle Functions (SPFs)
Multi-Layer MCTDH
Dirac–Frenkel Variational Principle
Equations of Motion
Numerical Time Propagation
Population Calculation
Autocorrelation Function
Absorption Spectrum
Complete MCTDH Algorithm