MCTDH Wavefunction Expansion¶

The Time-Dependent Schrödinger Equation describes how the molecular wavefunction evolves with time,

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

The Hamiltonian was developed in the previous sections. The remaining challenge is to represent the multidimensional wavefunction efficiently.

For a molecule with many vibrational modes, a direct numerical representation of the wavefunction becomes prohibitively expensive because the number of basis functions grows exponentially with the number of coordinates. This phenomenon is known as the curse of dimensionality.

The Multi-Configuration Time-Dependent Hartree (MCTDH) method overcomes this problem by expanding the wavefunction in a basis that evolves together with the dynamics.

The Curse of Dimensionality¶

Suppose each vibrational mode is represented by \(N\) primitive basis functions.

For a molecule containing \(f\) vibrational modes, the total number of direct-product basis functions is

\[ N_{\mathrm{total}} = N^f. \]

For example,

Modes	Primitive functions per mode	Total basis functions
10	20	\(20^{10}=1.02\times10^{13}\)
20	20	\(20^{20}=1.05\times10^{26}\)
32	20	\(20^{32}\approx4.3\times10^{41}\)

Clearly, storing or propagating such a wavefunction is impossible.

Direct Product Representation¶

A conventional quantum dynamics calculation expands the wavefunction as

\[ \Psi(\mathbf Q,t) = \sum_{j_1} \sum_{j_2} \cdots \sum_{j_f} C_{j_1j_2\cdots j_f}(t) \, \chi_{j_1}(Q_1) \chi_{j_2}(Q_2) \cdots \chi_{j_f}(Q_f), \]

where

\(C_{j_1j_2\cdots j_f}(t)\) are expansion coefficients,
\(\chi_j(Q)\) are fixed primitive basis functions.

The primitive basis never changes during propagation.

As the wavepacket moves across the potential energy surface, increasingly many basis functions are required to describe its evolution accurately.

The MCTDH Idea¶

Instead of using fixed basis functions, MCTDH allows the basis itself to evolve with time.

The wavefunction is written as

\[ \boxed{ \Psi(\mathbf Q,t) = \sum_{j_1} \sum_{j_2} \cdots \sum_{j_f} A_{j_1j_2\cdots j_f}(t) \prod_{\kappa=1}^{f} \varphi^{(\kappa)}_{j_\kappa}(Q_\kappa,t) } \]

where

\(A_{j_1j_2\cdots j_f}(t)\) are the configuration coefficients,
\(\varphi_j^{(\kappa)}(Q_\kappa,t)\) are the Single Particle Functions (SPFs),
both quantities depend on time.

This is the defining equation of the MCTDH method.

Configuration Coefficients¶

The coefficients

\[ A_{j_1j_2\cdots j_f}(t) \]

describe the contribution of each multidimensional configuration to the total wavefunction.

Unlike conventional approaches, these coefficients evolve simultaneously with the basis functions.

During propagation,

electronic populations change,
vibrational amplitudes evolve,
quantum interference develops,

and the coefficients continuously adapt to describe the changing wavepacket.

Single Particle Functions¶

Each coordinate is described by a small number of adaptive basis functions,

\[ \varphi_j^{(\kappa)}(Q_\kappa,t). \]

The superscript \(\kappa\) identifies the vibrational mode, while the index \(j\) labels the different SPFs associated with that coordinate.

Unlike primitive harmonic oscillator functions, SPFs change continuously throughout the propagation.

They automatically follow the motion of the nuclear wavepacket.

Product Basis¶

The multidimensional basis functions are products of SPFs,

\[ \Phi_J(\mathbf Q,t) = \prod_{\kappa=1}^{f} \varphi^{(\kappa)}_{j_\kappa}(Q_\kappa,t). \]

Instead of storing every possible primitive basis function, MCTDH constructs only those configurations needed to describe the evolving dynamics.

This dramatically reduces the computational cost.

Vibronic Wavefunction¶

For vibronic dynamics, the nuclear wavepacket exists on several electronic states simultaneously.

The total wavefunction becomes

\[ \boxed{ \Psi(\mathbf Q,t) = \sum_{\alpha} \chi_\alpha(\mathbf Q,t) |\alpha\rangle } \]

where

\(|\alpha\rangle\) represents a diabatic electronic state,
\(\chi_\alpha(\mathbf Q,t)\) is the nuclear wavepacket associated with that state.

Initially, only one electronic state may be populated.

During propagation, the off-diagonal vibronic couplings introduced in the Hamiltonian transfer population between these nuclear wavepackets.

Connection with the Propagation Input¶

The mathematical wavefunction is defined in the propagation input through the basis sections.

PBASIS-SECTION

SPF-BASIS-SECTION

INIT_WF-SECTION

These sections specify

the primitive basis functions,
the number of SPFs,
the initial wavepacket,
the occupied electronic state.

Thus, the propagation input file provides the numerical realization of the MCTDH wavefunction expansion.

Physical Interpretation¶

The wavefunction can be viewed as a multidimensional quantum cloud moving on coupled potential energy surfaces.

As time progresses,

the wavepacket spreads,
oscillates along vibrational coordinates,
transfers between electronic states,
changes its shape continuously.

Rather than forcing this evolving wavepacket into a rigid basis, MCTDH allows the basis itself to move together with the dynamics.

This adaptive representation is the key reason why MCTDH can efficiently simulate systems containing dozens of vibrational degrees of freedom.

Key Points¶

Conventional methods use a fixed basis throughout the propagation.
MCTDH employs a time-dependent basis.
Both the expansion coefficients and basis functions evolve with time.
The adaptive basis drastically reduces the number of configurations required.
This wavefunction representation forms the foundation of all subsequent MCTDH equations.

Next Step¶

The adaptive basis functions introduced here are known as Single Particle Functions (SPFs). They are the central building blocks of the MCTDH method and are responsible for its remarkable computational efficiency.

The next section explains how SPFs are constructed and why they provide such a compact representation of multidimensional wavepackets.