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Absorption Spectrum

After propagating the wavepacket, MCTDH produces the autocorrelation function stored in the auto file. While this file contains the complete time-dependent information about the molecular dynamics, it is usually transformed into the frequency (or energy) domain to obtain the absorption spectrum.

The absorption spectrum describes how strongly a molecule absorbs light at different energies and is directly comparable with experimental UV-Visible spectra. MCTDH performs this transformation using the program autospec86, which Fourier transforms the autocorrelation function.

From the Autocorrelation Function to the Spectrum

The autocorrelation function,

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

contains the complete quantum evolution of the wavepacket.

To obtain the absorption spectrum, MCTDH computes its Fourier transform,

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

where

  • \(C(t)\) is the autocorrelation function,
  • \(E\) is the excitation energy,
  • \(I(E)\) is the absorption intensity.

This converts the wavepacket dynamics from the time domain into the energy domain.

Generating the Spectrum

The spectrum is generated using

autospec86 2.5 4.5 ev 33 1

For the present calculation, the program prints

The inputfile is :
/home/user/jitendra/dae/i-of-dynamics/wp3-pbf1-spf1/auto

The outputfile is :
/home/user/jitendra/dae/i-of-dynamics/wp3-pbf1-spf1/spectrum.pl

The program therefore

  1. reads the propagated autocorrelation function,
  2. performs the Fourier transform,
  3. applies exponential damping,
  4. writes the resulting spectrum to spectrum.pl.

Understanding the Command

The command

autospec86 2.5 4.5 ev 33 1

contains five user-defined parameters.

Argument Meaning
2.5 Minimum energy (eV)
4.5 Maximum energy (eV)
ev Energy unit
33 Damping time constant (\(\tau\)) in femtoseconds
1 Exponential damping function

Thus the spectrum will be calculated between

\[ 2.5 \le E \le 4.5 \text{ eV}. \]

Program Output

During execution, autospec86 prints information about the Fourier transform.

Emin, Emax :
2.500 4.500 ev

tau = 33.0 fs

No. of plot-points = 1000

No. of t-points = 600

t-final = 600.000 fs

These quantities describe how the spectrum is constructed.

Energy Window

Emin = 2.5 eV
Emax = 4.5 eV

Only transitions lying within this energy interval are included in the calculated spectrum.

Number of Time Points

No. of t-points = 600

This means the Fourier transform uses 600 values of the autocorrelation function.

Longer propagations generally produce higher spectral resolution because the Fourier transform contains more time information.

Number of Plot Points

No. of plot-points = 1000

The calculated spectrum is sampled at 1000 equally spaced energy values between 2.5 and 4.5 eV.

This determines the smoothness of the plotted spectrum.

Damping Constant

tau = 33.0 fs

Before performing the Fourier transform, the autocorrelation function is multiplied by

\[ e^{-t/\tau}. \]

This damping suppresses oscillations caused by the finite propagation length and produces smoother spectral lines.

A larger value of \(\tau\)

  • produces sharper peaks,
  • requires longer propagation times.

A smaller value

  • broadens the peaks,
  • reduces numerical noise.

Spectral Moments

The program also reports

Moments:

M0 = 1.4185E-02

M1 = 3.1743E+00

M2 = 1.0466E+01

width = 6.2463E-01

These quantities summarize the calculated spectrum.

Zeroth Moment (\(M_0\))

Represents the total integrated intensity of the spectrum,

\[ M_0=\int I(E)\,dE. \]

First Moment (\(M_1\))

Represents the average excitation energy,

\[ M_1 = \frac{\int EI(E)\,dE} {\int I(E)\,dE}. \]

Second Moment (\(M_2\))

Measures how widely the spectrum is distributed around its center.

Width

The reported width corresponds to the spectral broadening,

width = 0.6246 eV

A larger width generally indicates a broader distribution of vibronic transitions.

The Generated spectrum.pl File

After the Fourier transform, MCTDH creates

spectrum.pl

whose first few lines are

# Energy[ev]                  g0               g1               g2             g5 (g'1)

2.50000000E+00    1.07254956E+00    1.07265249E+00    1.07275469E+00    1.07294543E+00

2.50200000E+00    1.06060898E+00    1.06068017E+00    1.06075236E+00    1.06089647E+00

2.50400000E+00    1.04886099E+00    1.04890311E+00    1.04894725E+00    1.04904640E+00

Each row corresponds to one energy point in the calculated spectrum.

Interpreting the Columns

The columns are

Column Description
1 Excitation energy (eV)
2 Spectrum using damping function \(g_0\)
3 Spectrum using damping function \(g_1\)
4 Spectrum using damping function \(g_2\)
5 Spectrum using damping function \(g_5\) (first derivative form)

The first column is the energy axis, while the remaining columns contain spectral intensities calculated using different line-broadening functions available in autospec86.

For most applications, one of these columns (typically g1) is selected for plotting, depending on the desired broadening scheme.

Physical Interpretation

Each peak in the spectrum corresponds to a transition between vibronic states.

The

  • peak position gives the transition energy,
  • peak height reflects the transition intensity,
  • peak width is influenced by the damping parameter and propagation length.

Because the spectrum is obtained from the propagated wavepacket, it naturally includes

  • vibronic coupling,
  • quantum coherence,
  • nonadiabatic effects,
  • interference between different pathways,

all of which are difficult to capture using purely static electronic structure calculations.

The spectrum.pl file contains the complete absorption spectrum as numerical data.