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Autocorrelation Function

The propagated wavepacket contains all the information about the molecular dynamics. One of the most useful quantities that can be extracted from this wavepacket is the autocorrelation function, which measures how similar the wavefunction at a later time is to its initial state.

The autocorrelation function is the starting point for calculating absorption spectra, emission spectra, and other time-dependent spectroscopic observables. In MCTDH, it is generated automatically during wavepacket propagation and is written to a file named auto.

What is the Autocorrelation Function?

Suppose the initial wavepacket is

\[ \Psi(0). \]

As the propagation proceeds, the wavepacket evolves according to the time-dependent Schrödinger equation,

\[ \Psi(t) = e^{-i\hat Ht/\hbar} \Psi(0). \]

The autocorrelation function is defined as the overlap between the propagated wavepacket and the initial wavepacket,

\[ \boxed{ C(t) = \langle \Psi(0) | \Psi(t) \rangle. } \]

It is simply the inner product of the wavefunction at time \(t\) with the wavefunction at \(t=0\).

Physical Meaning

The value of \(C(t)\) describes how much of the original wavepacket remains after propagation.

If

\[ |C(t)|=1, \]

the propagated wavepacket is identical to the initial wavepacket.

If

\[ |C(t)|\approx0, \]

the wavepacket has evolved into a completely different region of configuration space.

Intermediate values indicate partial overlap due to vibrational motion, electronic population transfer, or quantum interference.

Why is the Autocorrelation Function Important?

Unlike the population analysis, which describes where the wavepacket is located, the autocorrelation function measures how the wavepacket evolves relative to its initial state.

It therefore contains information about

  • vibrational coherence,
  • recurrence of the wavepacket,
  • quantum interference,
  • electronic coherence,
  • spectroscopic transitions.

Most importantly,

the absorption spectrum is obtained directly from the Fourier transform of the autocorrelation function.

Generation During Propagation

When the wavepacket propagation starts,

the initial wavepacket is stored.

At every propagation step,

MCTDH computes

\[ C(t) = \langle \Psi(0) | \Psi(t) \rangle \]

and appends the result to the autocorrelation file.

Conceptually, the process is

Initial wavepacket Ψ(0)
          │
          ▼
Propagate to time t
          │
          ▼
Calculate overlap

<Ψ(0)|Ψ(t)>
          │
          ▼
Write value to auto
          │
          ▼
Repeat for every time step

Thus, one autocorrelation value is produced for every output time during the propagation.

The auto File

After a successful propagation, MCTDH generates a file named

auto

which contains the autocorrelation function as a function of time.

Although the exact formatting may differ slightly between MCTDH versions, the file generally contains

  • propagation time,
  • real part of the autocorrelation,
  • imaginary part of the autocorrelation,
  • and, in some versions, additional auxiliary quantities.

A typical structure is

Time(fs)      Re[C(t)]      Im[C(t)]
0.0000        1.000000      0.000000
0.5000        0.981245     -0.062114
1.0000        0.932571     -0.118503
...

The first point is always

\[ C(0)=1, \]

because the wavefunction is perfectly correlated with itself.

Interpretation

The magnitude

\[ |C(t)| \]

indicates how much of the initial wavepacket remains.

A rapidly decaying autocorrelation function usually indicates

  • rapid wavepacket spreading,
  • strong vibronic coupling,
  • fast nonadiabatic dynamics.

On the other hand,

periodic oscillations in \(C(t)\) indicate

  • coherent vibrational motion,
  • wavepacket recurrences,
  • periodic nuclear dynamics.

These oscillations later appear as peaks in the absorption spectrum.

Connection with the Output Files

During propagation,

the MCTDH output file reports quantities such as

Population

Natural weights

Expectation values

Norm

These quantities describe the instantaneous state of the wavepacket.

The auto file complements this information by recording how the entire wavepacket overlaps with its initial state throughout the propagation.

Thus,

  • the output file describes the dynamics at each time step,
  • the auto file stores the time-dependent quantum coherence needed for spectroscopy.

Relation to Absorption Spectra

The autocorrelation function is not usually analyzed directly.

Instead,

its Fourier transform is computed to obtain the absorption spectrum,

\[ \boxed{ I(E) \propto \int_0^\infty C(t) e^{\,iEt/\hbar} \,dt. } \]

This transformation converts the time-domain dynamics into the energy-domain spectrum.

In MCTDH, this is performed using the program

autospec86

which reads the auto file and generates the corresponding absorption spectrum.