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Understanding the Propagation Output

Once the propagation starts, MCTDH periodically writes the status of the simulation to the output file. The interval between two consecutive output blocks is controlled by the tout keyword specified in the propagation input file.

Each time block contains

  • propagation time,
  • CPU time,
  • wavefunction norm,
  • total energy,
  • natural weights,
  • vibrational expectation values,
  • electronic populations.

Since every time step has the same structure, understanding one output block is sufficient to interpret the complete propagation.

Throughout this chapter, we use the output corresponding to 0.50 fs as an example.

Overall Structure

Each propagation block consists of the following sections.

  1. Time and Energy Information
  2. Natural Weights
  3. Mode Expectation Values
  4. Electronic Populations

Time and Energy Information

The first lines summarize the numerical status of the propagation.

Snippet 1 — Time and Energy

Time  =       0.50 fs,       CPU =      243.25 s,    Norm    = 0.99999979

E-tot =   5.667911 eV,
E-corr =   5.667911 eV,
Delta-E =     0.0000 meV

Time

Time = 0.50 fs

This indicates that the wavepacket has propagated for 0.50 femtoseconds from the initial state.

Each subsequent block corresponds to the next output interval specified by

tout = 0.5

in the input file.

CPU Time

CPU = 243.25 s

This is the cumulative processor time required to reach the current propagation step.

The CPU time depends on

  • number of electronic states,
  • number of vibrational modes,
  • multilayer tree,
  • number of SPFs,
  • integration tolerances.

Wavefunction Norm

Norm = 0.99999979

The propagated wavefunction should satisfy

\[ \langle\Psi|\Psi\rangle = 1. \]

A value very close to unity indicates that the numerical propagation remains stable.

For a well-converged calculation,

Norm ≈ 1

throughout the entire simulation.

Total Energy

E-tot = 5.667911 eV

This is the expectation value

\[ \langle\Psi(t)|\hat H|\Psi(t)\rangle. \]

For a time-independent Hamiltonian, the total energy should remain nearly constant during propagation.

Energy Drift

Delta-E = 0.0000 meV

The energy drift measures the deviation of the total energy from its initial value.

For this calculation,

Delta-E = 0.0000 meV

showing excellent energy conservation throughout the propagation.

Natural Weights

The next section prints the natural weights of every node in the multilayer wavefunction.

Snippet 2 — Root Node

node:   1    layer:  0

m1:
988.137
5.625
4.684
1.553

m2:
988.137
5.625
4.684
1.553

Natural weights are the eigenvalues of the reduced density matrix for each node.

They indicate how efficiently the wavefunction is represented by the chosen Single Particle Functions (SPFs).

Because the values are printed multiplied by 1000, the first weight corresponds to

\[ 988.137/1000 = 0.988137. \]

Thus,

Weight Contribution
988.137 98.81 %
5.625 0.56 %
4.684 0.47 %
1.553 0.16 %

The first SPF therefore contains almost the entire wavefunction.

Higher Layer Nodes

Each branch of the multilayer tree is printed separately.

Snippet 3 — Layer 1

node: 2

layer: 1

m1:
994.516
2.911
1.627
0.936
...

Lower natural weights indicate that higher SPFs contribute only small corrections.

If the last few natural weights become large, additional SPFs may be required for convergence.

Monitoring these values throughout the propagation is one of the standard convergence tests in ML-MCTDH calculations.

Mode Expectation Values

The next section reports the expectation values for every active vibrational mode.

Snippet 4 — Mode v32

v32 :

<q>=   0.0095

<dq>=  0.6921

<n>=   0.0107

<dn>=  0.1143

Each quantity has a direct physical interpretation.

Quantity Meaning
<q> Average displacement along the normal coordinate
<dq> Width (standard deviation) of the wavepacket
<n> Average vibrational occupation number
<dn> Fluctuation in vibrational occupation

Coordinate Expectation Value

<q> = 0.0095

represents

\[ \langle Q\rangle. \]

A positive value indicates that the wavepacket has moved slightly along the positive direction of the normal coordinate.

Wavepacket Width

<dq> = 0.6921

corresponds to

\[ \sqrt{\langle Q^2\rangle-\langle Q\rangle^2}. \]

This quantity measures the spatial spread of the nuclear wavepacket.

Vibrational Excitation

<n> = 0.0107

is the expectation value

\[ \langle a^\dagger a\rangle, \]

which gives the average vibrational excitation of the mode.

Occupation Fluctuation

<dn> = 0.1143

measures the fluctuation in vibrational occupation,

\[ \sqrt{\langle n^2\rangle-\langle n\rangle^2}. \]

Electronic Populations

The final line of each output block gives the electronic populations.

Snippet 5 — Electronic Populations

population :

0.16042
0.00321
0.82544
0.01093

These numbers correspond to the probability of finding the wavepacket on each diabatic electronic state.

Electronic State Population
State 1 0.16042
State 2 0.00321
State 3 0.82544
State 4 0.01093

The populations always satisfy

\[ \sum_i P_i = 1, \]

apart from small numerical errors.

Comparing these values at successive time steps reveals the transfer of population between electronic states caused by vibronic coupling.

Successful Completion

After the final propagation step, MCTDH prints a summary indicating that the calculation completed successfully.

Snippet 6 — End of Propagation

Propagation was successful.

total time/wall time   :          7.53

Total time [h:m:s] :    8 : 23 : 50.33

Wall  time [h:m:s] :    1 :  6 : 55.78

Interpretation

The message

Propagation was successful.

confirms that the propagation reached the requested final time without numerical failure.

The timing information reports

  • Total CPU time — the cumulative processor time used by all computing cores.
  • Wall time — the actual elapsed time experienced by the user.
  • CPU/Wall ratio — an indication of the parallel efficiency of the calculation.

Finally, MCTDH prints the host name, working directory, and calculation title, providing a complete record of where the propagation was executed.

Summary

The propagation output provides a complete snapshot of the quantum dynamics at each output time. The Time and Energy section monitors numerical stability, the Natural Weights assess the convergence of the multilayer wavefunction, the Mode Expectation Values describe the evolution of the nuclear wavepacket, and the Electronic Populations track the redistribution of probability among the electronic states. Together, these quantities allow both the numerical quality and the physical behavior of the MCTDH propagation to be evaluated throughout the simulation.