Skip to content

Dirac–Frenkel Variational Principle

The previous sections introduced the vibronic Hamiltonian, the wavefunction expansion, the Single Particle Functions (SPFs), and the Multi-Layer MCTDH (ML-MCTDH) representation. The remaining question is:

How are all of these quantities propagated in time?

MCTDH answers this using the Dirac–Frenkel Time-Dependent Variational Principle (TDVP).

Instead of solving the full time-dependent Schrödinger equation directly—which is impossible for systems with dozens of coupled vibrational modes—MCTDH searches for the best possible wavefunction within the chosen ML-MCTDH ansatz at every instant of time.

The Dirac–Frenkel principle provides the equations of motion for:

  • the configuration coefficients,
  • every Single Particle Function (SPF),
  • and, in ML-MCTDH, every layer of the tree.

Consequently, both the coefficients and the basis functions evolve together during propagation, making MCTDH far more efficient than propagation in a fixed basis.

The Time-Dependent Schrödinger Equation

The exact quantum dynamics is governed by

\[ i\hbar\frac{\partial}{\partial t}\Psi(t) = \hat H\Psi(t). \]

If the wavefunction could be represented exactly, solving this equation would produce the exact molecular dynamics.

However, for a system containing tens of vibrational modes,

\[ \Psi(Q_1,Q_2,\ldots,Q_f,t) \]

contains an astronomical number of coefficients, making direct propagation computationally impossible.

The Variational Idea

Instead of solving the exact equation,

MCTDH assumes that the wavefunction always remains inside the ML-MCTDH manifold,

\[ \Psi(t)=\Psi(\mathbf{A}(t),\varphi(t)), \]

where

  • \(\mathbf A(t)\) are the configuration coefficients,
  • \(\varphi(t)\) denotes all SPFs in every layer.

The goal is therefore

Find the evolution of these parameters that makes the approximate wavefunction satisfy the Schrödinger equation as closely as possible.

Dirac–Frenkel Condition

The Dirac–Frenkel variational principle states that the residual error

\[ i\hbar\dot\Psi-\hat H\Psi \]

must be orthogonal to every allowed variation of the wavefunction,

\[ \boxed{ \left\langle \delta\Psi \left| i\hbar\frac{\partial}{\partial t} - \hat H \right| \Psi \right\rangle = 0. } \]

This single equation is the mathematical foundation of MCTDH.

Rather than solving the Schrödinger equation exactly,

MCTDH solves the best possible approximation inside the chosen wavefunction manifold.

Physical Interpretation

Imagine the exact quantum wavefunction moving through an infinitely large Hilbert space.

The ML-MCTDH wavefunction occupies only a tiny curved surface inside that enormous space.

The Dirac–Frenkel principle ensures that

  • the approximate wavefunction always remains on that surface,
  • while following the exact trajectory as closely as possible.

This minimizes the instantaneous propagation error.

What Does It Determine?

Applying the variational principle automatically generates

  1. equations of motion for the expansion coefficients,
  2. equations of motion for every SPF,
  3. equations for every node of the multilayer tree.

Thus,

  • coefficients change with time,
  • basis functions also change with time,
  • the entire basis adapts continuously to the evolving wavepacket.

This adaptive basis is the main reason MCTDH can describe quantum dynamics with relatively few SPFs.

Relation to the MCTDH Output

During propagation,

Time = 0.50 fs

Natural weights

population

Mode expectation values

all reported quantities are obtained from the wavefunction generated by solving the Dirac–Frenkel equations of motion.

The adaptive evolution of SPFs allows the program to maintain high accuracy while keeping the number of configurations manageable.

Connection to the Next Section

The Dirac–Frenkel principle establishes how the parameters should evolve, but it does not explicitly show what equations are solved.

In the next section, we derive the Equations of Motion (EOMs) for

  • the configuration coefficients,
  • the Single Particle Functions (SPFs),
  • and the ML-MCTDH tree,

which are integrated numerically during every propagation step.