Multi-Layer MCTDH (ML-MCTDH)¶

After expressing the wavefunction in terms of Single Particle Functions (SPFs), a practical problem still remains. For systems containing tens or hundreds of vibrational modes, the number of SPFs required becomes extremely large, making even standard MCTDH computationally demanding.

ML-MCTDH overcomes this limitation by representing the wavefunction as a hierarchy (tree) of wavefunctions instead of a single-layer expansion. This recursive representation drastically reduces the computational cost and enables simulations of systems containing dozens or even hundreds of coupled degrees of freedom.

Why Standard MCTDH Eventually Becomes Expensive¶

In standard MCTDH the wavefunction is written as

\[ \Psi(Q,t)= \sum_{j_1}^{n_1} \sum_{j_2}^{n_2} \cdots \sum_{j_f}^{n_f} A_{j_1j_2\cdots j_f}(t) \prod_{\kappa=1}^{f} \varphi^{(\kappa)}_{j_\kappa}(q_\kappa,t) \]

where

\(A\) is the coefficient tensor,
\(f\) is the number of logical modes,
each logical mode contains several physical vibrational coordinates.

Although this is already much smaller than a full grid representation, the tensor

\[ A_{j_1j_2\cdots j_f} \]

still grows exponentially with the number of logical modes.

For a molecule containing 32 vibrational modes, even grouping them into logical modes may still require an extremely large coefficient tensor.

The ML-MCTDH Idea¶

Instead of treating every SPF as a fixed function, ML-MCTDH expands each SPF itself into another MCTDH wavefunction.

Thus the wavefunction becomes hierarchical.

Instead of

\[ \Psi \rightarrow A \times \varphi_1 \times \varphi_2 \times \cdots \]

we obtain

\[ \Psi \rightarrow A \rightarrow \text{SPFs} \rightarrow \text{SPFs of SPFs} \rightarrow \text{SPFs of SPFs of SPFs} \]

until the bottom of the tree reaches primitive basis functions.

This recursive decomposition gives ML-MCTDH its remarkable efficiency.

Recursive Expansion¶

A parent SPF can itself be expanded as

\[ \varphi^{(\kappa)}_j = \sum_{i_1,i_2,\ldots} B^{(\kappa)}_{j,i_1,i_2,\ldots} \prod_r \varphi^{(\kappa,r)}_{i_r} \]

where

\(B^{(\kappa)}\) is another coefficient tensor,
the child SPFs are again time dependent,
each child node can be expanded once more.

This recursive procedure continues until primitive basis functions are reached.

Tree Structure¶

Instead of one enormous tensor, the wavefunction becomes a collection of many smaller tensors arranged in a tree.

                    Ψ(t)
                      │
                Coefficient Tensor
                      │
        ┌─────────────┴─────────────┐
        │                           │
      Node 1                     Node 2
        │                           │
    ┌───┴────┐                 ┌────┴────┐
    │        │                 │         │
  Node A   Node B           Node C    Node D
    │        │                 │         │
   Modes    Modes            Modes     Modes

Each node represents another MCTDH expansion.

Only the bottom layer contains primitive harmonic oscillator basis functions.

Relation to the Input File¶

The ML-MCTDH hierarchy is completely determined by the ML tree specified in the propagation input.

For example,

ML-basis-section

0> 2 2
 1> [el]
 1> 3 3
    ...

defines

the number of layers,
how vibrational modes are grouped,
how many SPFs belong to every node.

Changing the ML tree changes the mathematical representation of the wavefunction but does not change the Hamiltonian.

It only changes how efficiently the Schrödinger equation is solved.

Relation to the Output File¶

During propagation the output repeatedly prints

Natural weights

node: 1
node: 2
node: 3
...

Each node listed corresponds to one node of the ML tree.

For every node MCTDH prints the natural populations of its SPFs.

For example,

node: 7

m1: 999.325 0.555 0.085 0.035

indicates that the first SPF carries almost all of the probability, while the remaining SPFs contribute only slightly.

Large occupations of higher SPFs indicate that additional SPFs may be required for accurate propagation.

Computational Advantage¶

Suppose each node requires only a few SPFs.

Instead of storing one huge tensor,

\[ N^f \]

ML-MCTDH stores many small tensors,

\[ N_1N_2 + N_2N_3 + N_3N_4 +\cdots \]

whose total size grows much more slowly.

This reduction allows simulations involving

dozens of electronic states,
tens to hundreds of vibrational modes,
long femtosecond-to-picosecond dynamics,

that would be impossible using conventional MCTDH.

Connection to the Documentation¶

At this point we have constructed the complete ML-MCTDH wavefunction.

The next step is to determine how this wavefunction evolves with time.

This requires solving the Time-Dependent Schrödinger Equation using the Dirac–Frenkel Variational Principle, which derives the equations of motion for both the expansion coefficients and every SPF simultaneously.