Equations of Motion (EOM)¶

In the previous section, we introduced the Dirac–Frenkel variational principle, which states that the ML-MCTDH wavefunction should evolve in such a way that it remains the best possible approximation to the exact solution of the time-dependent Schrödinger equation.

Applying this variational principle to the ML-MCTDH wavefunction produces a set of coupled Equations of Motion (EOMs). These equations determine how

the expansion coefficients evolve,
the Single Particle Functions (SPFs) evolve,
and, in ML-MCTDH, how every layer of the tree evolves with time.

Unlike conventional quantum dynamics, where only the expansion coefficients change, MCTDH allows both the coefficients and the basis itself to evolve simultaneously.

The ML-MCTDH Wavefunction¶

The wavefunction is written as

\[ \Psi(t) = \sum_J A_J(t) \, \prod_{\kappa=1}^{f} \varphi_{j_\kappa}^{(\kappa)}(Q_\kappa,t). \]

where

\(A_J(t)\) are the configuration coefficients,
\(\varphi_j^{(\kappa)}\) are the time-dependent SPFs,
\(Q_\kappa\) are the nuclear coordinates.

Since both quantities depend on time,

the total derivative of the wavefunction becomes

\[ \frac{\partial\Psi}{\partial t} = \sum_J \dot A_J \Phi_J + \sum_{\kappa} A_J \frac{\partial\Phi_J}{\partial t}, \]

where

\[ \Phi_J= \prod_\kappa \varphi_{j_\kappa}^{(\kappa)}. \]

Thus the wavefunction changes because

the coefficients evolve,
and the basis functions themselves also evolve.

1. Equation of Motion for the Coefficients¶

Projecting the Dirac–Frenkel equation onto the configuration basis gives

\[ \boxed{ i\hbar \dot{\mathbf A} = \mathbf H \mathbf A } \]

where

\(\mathbf A\) is the vector of configuration coefficients,
\(\mathbf H\) is the Hamiltonian matrix constructed in the current SPF basis.

This equation resembles the ordinary time-dependent Schrödinger equation,

except that the Hamiltonian matrix changes continuously because the SPFs are themselves changing.

Interpretation¶

At every propagation step,

the Hamiltonian matrix is evaluated in the current adaptive basis,

and the coefficients are updated accordingly.

This determines

electronic populations,
vibrational amplitudes,
coherence between states.

The populations printed in the MCTDH output

population :
0.16042
0.00321
0.82544
0.01093

are obtained directly from these evolving coefficients.

2. Equation of Motion for the SPFs¶

The basis functions also satisfy their own time-dependent equation,

\[ \boxed{ i\hbar |\dot\varphi_j\rangle = (1-P) \rho^{-1} \langle H\rangle |\varphi_j\rangle } \]

where

\(P\) is the projector onto the current SPF space,
\(\rho\) is the density matrix,
\(\langle H\rangle\) is the mean-field Hamiltonian.

This equation allows the basis to adapt continuously to the changing wavepacket.

Why the Projector Appears¶

Without the projector,

the SPF evolution would simply rotate the basis functions without improving the wavefunction.

The projector

\[ (1-P) \]

forces the evolution only in directions that expand the variational space, preventing redundant motion.

Mean-Field Hamiltonian¶

The operator

\[ \langle H\rangle \]

is called the mean-field Hamiltonian.

Rather than using the full multidimensional Hamiltonian,

each SPF feels an effective Hamiltonian obtained after averaging over all remaining modes.

Consequently,

each mode evolves while accounting for its coupling to every other mode.

3. Coupled Evolution¶

The coefficient equation requires the current SPFs,

while the SPF equation requires the current coefficients.

Therefore,

the two equations are solved simultaneously.

The workflow is

Current wavefunction
        │
        ▼
Construct Hamiltonian
        │
        ▼
Update coefficients
        │
        ▼
Construct density matrices
        │
        ▼
Update SPFs
        │
        ▼
Generate new wavefunction
        │
        ▼
Proceed to next time step

This self-consistent procedure is repeated throughout the propagation.

Extension to ML-MCTDH¶

In standard MCTDH,

only one set of SPFs is propagated.

In ML-MCTDH,

every node of the multilayer tree has

its own coefficients,
its own density matrix,
its own mean-field Hamiltonian,
and its own equations of motion.

Therefore,

the Dirac–Frenkel principle is applied recursively throughout the tree.

The complete propagation consists of solving the equations of motion for every layer simultaneously.

Relation to the Output File¶

The quantities printed during propagation

Natural weights

Mode expectation values

Population

Norm

Energy

are all computed from the wavefunction obtained by integrating these coupled equations of motion.

For example,

Norm verifies that the numerical integration preserves normalization.
Population comes from the evolving coefficients.
Expectation values are calculated from the propagated SPFs.
Natural weights are obtained from density matrices constructed during the SPF evolution.

Thus, every line of the MCTDH output is ultimately a consequence of solving these equations of motion.

Connection to the Next Section¶

The equations of motion determine how the wavefunction evolves, but they do not specify how these differential equations are integrated numerically.

In the next section, we discuss the Numerical Propagation Algorithms, including the variable-step integrators and predictor–corrector techniques used by MCTDH to advance the wavefunction from one time step to the next.