Time-Dependent Schrödinger Equation¶
Every MCTDH calculation begins with the Time-Dependent Schrödinger Equation (TDSE). Unlike electronic structure methods such as Hartree–Fock, CASSCF, or XMCQDPT, which solve for stationary electronic states, MCTDH follows the real-time evolution of a quantum wavepacket.
Instead of asking
"What is the energy of this state?"
MCTDH asks
"How does the quantum state evolve with time?"
The answer is provided by the Time-Dependent Schrödinger Equation.
The Fundamental Equation¶
The evolution of the molecular wavefunction is governed by
where
- \(i=\sqrt{-1}\) is the imaginary unit,
- \(\hbar\) is the reduced Planck constant,
- \(t\) is time,
- \(\mathbf Q=(Q_1,Q_2,\ldots,Q_f)\) represents all nuclear coordinates,
- \(\hat H\) is the molecular Hamiltonian,
- \(\Psi(\mathbf Q,t)\) is the time-dependent molecular wavefunction.
Unlike static quantum chemistry methods, the wavefunction is explicitly a function of both nuclear coordinates and time.
Physical Meaning¶
At time
the wavepacket is prepared using the initial conditions specified in the propagation input file.
For example,
- excitation onto an excited electronic state,
- a displaced vibrational ground state,
- or a superposition of several electronic states.
MCTDH then computes how this wavepacket evolves under the action of the Hamiltonian.
Time Evolution¶
The Hamiltonian acts continuously on the wavefunction,
changing both its amplitude and phase.
As propagation proceeds,
- probability flows between electronic states,
- vibrational energy redistributes among normal modes,
- coherent oscillations appear,
- quantum interference develops.
This continuous evolution produces the changing state populations and vibrational expectation values observed in the propagation output.
Why Time Dependence is Necessary¶
Electronic structure programs usually solve
which provides stationary eigenstates and their corresponding energies.
This equation cannot describe
- ultrafast dynamics,
- wavepacket motion,
- electronic population transfer,
- decoherence,
- or photoinduced reactions.
The TDSE is required whenever the molecular system changes with time.
Connection to the Operator File¶
The Hamiltonian appearing in the TDSE,
is exactly the Hamiltonian constructed during the Operator File Generation step.
The operator file contains
- electronic energies,
- vibrational kinetic energy,
- harmonic potentials,
- linear vibronic couplings,
- quadratic vibronic couplings,
- interstate coupling operators.
During propagation, MCTDH repeatedly applies this Hamiltonian to the evolving wavefunction.
Connection to the Propagation Input¶
The propagation input file defines
- the initial wavefunction,
- the simulation time,
- the numerical integrator,
- the ML-tree,
- the SPF expansion,
- the primitive basis.
Together with the Hamiltonian from the operator file, these quantities completely define the TDSE that MCTDH solves.
Connection to the Output¶
Every quantity printed during propagation originates from the evolving solution of the TDSE.
For example,
-
Electronic populations are obtained by projecting the wavefunction onto each electronic state.
-
Expectation values are calculated from the propagated wavefunction.
-
Natural weights measure how efficiently the SPFs represent the evolving state.
-
Autocorrelation functions compare the propagated wavepacket with its initial state.
-
Absorption spectra are obtained by Fourier transforming the autocorrelation function.
Thus, every file generated during propagation is ultimately derived from the numerical solution of the Time-Dependent Schrödinger Equation.
What Comes Next?¶
The TDSE defines the evolution of the wavefunction, but it does not specify the form of the Hamiltonian.
The next chapter constructs the Molecular Hamiltonian, showing how kinetic energy, potential energy, and electronic couplings combine to produce the operator used throughout an MCTDH calculation.