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Generated Operator File

After all required information has been collected, operator-make.sh automatically generates an MCTDH operator file (*.op). This file contains the complete vibronic Hamiltonian in the format required by the MCTDH package and serves as the primary input for both wavepacket propagation and Hamiltonian diagonalization.

Download Operator File

The operator file consists of several sections, each defining a different component of the Hamiltonian.

Overall Structure

A typical operator file generated by the script has the following structure.

OP_DEFINE-SECTION
    Title
END-OP_DEFINE-SECTION

PARAMETER-SECTION
    Vibrational frequencies
    Electronic energies
    Interstate couplings
    Linear coupling constants
    Quadratic coupling constants
END-PARAMETER-SECTION

HAMILTONIAN-SECTION
    Kinetic energy
    Harmonic potentials
    Electronic energies
    Linear vibronic couplings
    Quadratic vibronic couplings
END-HAMILTONIAN-SECTION

hamiltonian-section_h1
hamiltonian-section_h2
...
hamiltonian-section_hN

end-operator

Each of these sections is discussed below.

OP_DEFINE-SECTION

The operator file begins with the OP_DEFINE-SECTION.

OP_DEFINE-SECTION
title
...
end-title
end-op_define-section

This section contains descriptive information about the calculation, including the title specified by the user during execution of the Bash script.

Although it does not influence the Hamiltonian itself, it provides useful metadata for identifying the operator file.

PARAMETER-SECTION

The Parameter Section contains every numerical quantity required to define the Hamiltonian.

These include

  • vibrational frequencies,
  • electronic state energies,
  • interstate electronic couplings,
  • linear vibronic coupling constants,
  • quadratic vibronic coupling constants.

All subsequent Hamiltonian definitions reference these parameters by name.

Vibrational Frequencies

The first parameters define the harmonic frequencies of the selected vibrational modes.

w1 = ...
w2 = ...
w3 = ...

Each frequency corresponds to one active vibrational coordinate selected during the mode-selection step.

The frequencies are converted from cm⁻¹ to electron volts (eV) before being written to the operator file.

Electronic Energies

The diagonal electronic energies are written as

e11
e22
e33
...

The energy of the lowest electronic state is set to zero, while the remaining state energies are reported relative to this reference.

This convention simplifies the Hamiltonian without affecting the resulting dynamics.

Interstate Couplings

Electronic couplings between different diabatic states are represented by

e12
e13
e23
...

These quantities describe the interaction between electronic states in the absence of nuclear motion and are extracted directly from the reference diabatic GAMESS calculation.

Linear Vibronic Couplings

The first-order vibronic coupling constants are written using parameters of the form

h11_k1
h12_k1
h22_k1

where

  • the first two indices denote the interacting electronic states,
  • k indicates a linear coupling,
  • the final number identifies the vibrational mode.

These parameters describe the linear dependence of the Hamiltonian on the normal coordinate.

Quadratic Vibronic Couplings

Second-order vibronic coupling constants are represented as

h11_g1
h12_g1
h22_g1

Here,

  • g denotes a quadratic coupling term,
  • the remaining indices follow the same convention used for the linear couplings.

These parameters account for quadratic variations of the electronic Hamiltonian with respect to the vibrational coordinates.

HAMILTONIAN-SECTION

The Hamiltonian section combines all previously defined parameters to construct the complete vibronic Hamiltonian used by MCTDH.

Its components include

  • vibrational kinetic energy,
  • harmonic potential energy,
  • electronic state energies,
  • linear vibronic coupling,
  • quadratic vibronic coupling.

Vibrational Hamiltonian

Each vibrational mode contributes a harmonic oscillator Hamiltonian consisting of

w_i KE

representing the kinetic energy operator, and

0.5*w_i q^2

representing the harmonic potential energy.

Together these define the uncoupled vibrational motion of each normal mode.

Electronic Hamiltonian

The electronic contribution consists of the diabatic state energies

e11
e22
...

together with the interstate coupling terms

e12
e13
...

These terms define the purely electronic part of the Hamiltonian.

Linear Coupling Terms

The interaction between electronic and vibrational motion is introduced through the linear coupling operators

hij_kv | q

Each term couples one vibrational coordinate to a pair of electronic states.

These terms are responsible for the first-order vibronic interactions that drive nonadiabatic dynamics.

Quadratic Coupling Terms

Second-order vibronic interactions are represented by

0.5*hij_gv | q²

These terms describe quadratic corrections to the electronic Hamiltonian and improve the accuracy of the vibronic model, particularly for strongly distorted potential energy surfaces.

One-Dimensional Hamiltonians

After the full Hamiltonian has been defined, the script automatically generates separate Hamiltonian sections for every selected vibrational mode.

For example,

hamiltonian-section_h1
hamiltonian-section_h2
...

Each section contains

  • kinetic energy,
  • harmonic potential,
  • linear coupling,
  • quadratic coupling,

for a single vibrational coordinate.

These one-dimensional Hamiltonians are primarily used during the diagonalization stage to obtain the single-particle functions (SPFs) employed in MCTDH calculations.

End of the Operator File

The operator file concludes with

end-operator

which signals the end of the Hamiltonian definition and completes the operator file.

Summary

The generated operator file provides a complete mathematical description of the vibronic Hamiltonian required by MCTDH. It combines the harmonic vibrational modes, diabatic electronic energies, interstate couplings, and both linear and quadratic vibronic coupling terms into a single file that can be used directly for wavepacket propagation or Hamiltonian diagonalization.

The next section demonstrates how this operator file is used together with the wavefunction and input files to perform an MCTDH calculation.