Molecular Hamiltonian¶
The Time-Dependent Schrödinger Equation introduced in the previous chapter,
describes how the molecular wavefunction evolves with time. The central quantity in this equation is the molecular Hamiltonian \(\hat H\), which contains all of the physical information governing the dynamics.
In MCTDH, the Hamiltonian determines how the wavepacket moves across one or more coupled potential energy surfaces. Every quantity computed during propagation—including electronic populations, vibrational expectation values, autocorrelation functions, and absorption spectra—is determined by this operator.
The Total Molecular Hamiltonian¶
For a molecule consisting of nuclei and electrons, the complete Hamiltonian can be written as
where
- \(\hat T_\mathrm{nuc}\) is the nuclear kinetic energy,
- \(\hat T_\mathrm{elec}\) is the electronic kinetic energy,
- \(\hat V_{ee}\) is the electron–electron repulsion,
- \(\hat V_{nn}\) is the nucleus–nucleus repulsion,
- \(\hat V_{en}\) is the electron–nucleus attraction.
This Hamiltonian describes the complete quantum mechanical motion of both electrons and nuclei.
Applying the Born–Oppenheimer Approximation¶
Directly solving the full molecular Hamiltonian is computationally impossible for realistic molecular systems.
Instead, MCTDH adopts the Born–Oppenheimer approximation, which separates the much faster electronic motion from the slower nuclear motion.
The electronic Schrödinger equation is solved first using quantum chemistry programs such as GAMESS or Gaussian, yielding
- electronic energies,
- equilibrium geometries,
- normal-mode frequencies,
- electronic couplings,
- vibronic coupling constants.
These quantities become the input for constructing the MCTDH Hamiltonian.
After this separation, only the nuclear degrees of freedom are propagated explicitly.
Nuclear Hamiltonian¶
Following the Born–Oppenheimer approximation, the Hamiltonian used for nuclear dynamics becomes
where
- \(\hat T_\mathrm{nuc}\) is the nuclear kinetic-energy operator,
- \(\hat V(\mathbf Q)\) is the potential-energy operator,
- \(\mathbf Q\) denotes the set of normal-mode coordinates.
This is the Hamiltonian that MCTDH propagates.
Nuclear Kinetic Energy¶
Expressed in mass-weighted normal coordinates, the kinetic-energy operator is
where
- \(f\) is the number of vibrational modes,
- \(Q_i\) is the \(i^\text{th}\) normal coordinate.
This operator determines how the wavepacket spreads and moves through coordinate space during propagation.
Potential Energy Operator¶
The second part of the Hamiltonian is the potential-energy operator,
which specifies the energy of the molecule at every point in nuclear coordinate space.
For a single electronic state,
while for multiple coupled electronic states,
The diagonal elements represent individual potential-energy surfaces, whereas the off-diagonal elements describe interstate couplings that allow population transfer between electronic states.
Relation to the Operator File¶
The operator file generated in the previous tutorial is a numerical representation of this Hamiltonian.
It contains
- kinetic-energy operators,
- harmonic vibrational potentials,
- electronic state energies,
- interstate coupling terms,
- and higher-order vibronic interactions.
During propagation, MCTDH repeatedly applies this Hamiltonian to the evolving wavepacket to solve the Time-Dependent Schrödinger Equation.
From Molecular Hamiltonian to Vibronic Hamiltonian¶
Although the molecular Hamiltonian provides the general framework for quantum dynamics, practical MCTDH calculations require a Hamiltonian expressed in terms of normal-mode coordinates and electronic states.
This leads to the vibronic Hamiltonian, where the potential-energy surfaces are expanded around a reference geometry and expressed using linear and quadratic vibronic coupling terms. The resulting Hamiltonian is precisely the form written into the MCTDH operator file.
The next chapter derives this vibronic Hamiltonian and explains how electronic energies, vibrational frequencies, and coupling constants obtained from quantum chemistry calculations are transformed into the operator used during MCTDH propagation.