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Single Particle Functions (SPFs)

The defining feature of the Multi-Configuration Time-Dependent Hartree (MCTDH) method is that the basis functions themselves evolve during the propagation. These adaptive basis functions are known as Single Particle Functions (SPFs).

Instead of representing each vibrational coordinate using hundreds of fixed primitive basis functions, MCTDH represents each coordinate by a much smaller set of optimized SPFs. As the nuclear wavepacket evolves, these SPFs evolve with it, allowing an accurate description of the dynamics using a compact basis.

From Primitive Basis to SPFs

Each vibrational coordinate is first represented by a primitive basis set,

\[ \{\chi_1(Q),\chi_2(Q),\ldots,\chi_{N}(Q)\}, \]

which usually consists of

  • Harmonic Oscillator DVR,
  • Hermite DVR,
  • Sine DVR,
  • FFT basis,
  • or other primitive basis functions.

These primitive basis functions remain fixed throughout the calculation.

An SPF is constructed as a time-dependent linear combination of these primitive basis functions,

\[ \boxed{ \varphi_j^{(\kappa)}(Q_\kappa,t) = \sum_{l=1}^{N_\kappa} c_{jl}^{(\kappa)}(t) \chi_l(Q_\kappa) } \]

where

  • \(N_\kappa\) is the number of primitive basis functions,
  • \(c_{jl}^{(\kappa)}(t)\) are time-dependent expansion coefficients,
  • \(\chi_l(Q_\kappa)\) are fixed primitive basis functions.

Thus, the primitive basis never changes, but the SPFs continuously adapt by changing the coefficients \(c_{jl}(t)\).

Role of SPFs

Each SPF represents one optimized one-dimensional wavefunction for a particular vibrational coordinate.

Rather than storing hundreds of primitive basis functions,

\[ \chi_1,\chi_2,\ldots,\chi_{100}, \]

MCTDH may require only

\[ \varphi_1,\varphi_2,\varphi_3,\varphi_4. \]

These few adaptive functions are sufficient because they evolve to follow the actual nuclear dynamics.

MCTDH Wavefunction Using SPFs

The complete wavefunction is written as

\[ \Psi(Q,t) = \sum_J A_J(t) \Phi_J(Q,t), \]

where each product basis function is

\[ \boxed{ \Phi_J(Q,t) = \prod_{\kappa=1}^{f} \varphi_{j_\kappa}^{(\kappa)}(Q_\kappa,t) } \]

Every multidimensional basis function is therefore built entirely from SPFs.

Time Evolution of SPFs

Unlike conventional basis functions,

\[ \chi(Q), \]

which remain unchanged,

the SPFs satisfy their own equations of motion,

\[ \frac{\partial}{\partial t} \varphi_j^{(\kappa)} \neq 0. \]

As the wavepacket propagates,

  • the SPFs change shape,
  • shift their centers,
  • redistribute their amplitudes,
  • adapt to the evolving nuclear motion.

Consequently, the basis continuously follows the wavepacket.

Why Are SPFs Efficient?

Suppose a primitive basis contains

  • 60 harmonic oscillator functions.

A conventional propagation must retain all 60 basis functions.

However, if the wavepacket occupies only a small region of coordinate space, perhaps only

  • 6–10 SPFs

are needed to describe it accurately.

This reduces the computational cost dramatically.

For example,

Primitive basis Number of SPFs
64 DVR functions 8 SPFs
80 DVR functions 10 SPFs
128 DVR functions 12 SPFs

Although the primitive basis remains large enough to span the coordinate space, only a small adaptive subspace is propagated.

Connection with the Propagation Input

The number of SPFs is specified in the propagation input file.

For example,

SPF-BASIS-SECTION

v01      = 6
v02      = 6
v03      = 8
...
el        = 4

Each value specifies how many SPFs are assigned to a particular degree of freedom.

Increasing the number of SPFs generally

  • improves the accuracy,
  • increases the computational cost,
  • requires more memory.

Choosing an appropriate number of SPFs is therefore one of the most important aspects of setting up an MCTDH calculation.

Natural Weights and SPF Quality

One advantage of MCTDH is that it provides a direct measure of how well the chosen SPFs represent the wavefunction.

During propagation, the output file prints the Natural Weights,

Natural weights *1000

node: 1

m1:
988.137
5.625
4.684
1.553

These numbers indicate how important each SPF is.

If only the first few natural weights are significant while the remaining values are extremely small,

the chosen SPF basis is sufficient.

However, if many SPFs have large natural weights,

additional SPFs should be included to improve the accuracy.

Thus, the Natural Weights printed in the output file serve as an automatic convergence indicator for the SPF basis.

Physical Interpretation

The SPFs may be viewed as adaptive vibrational orbitals.

Just as molecular orbitals in Hartree–Fock adapt to describe the electronic wavefunction,

the SPFs adapt to describe the evolving nuclear wavepacket.

Instead of forcing the wavepacket into a rigid basis,

the basis itself follows the quantum dynamics.

This adaptive representation is the primary reason why MCTDH can efficiently simulate systems containing dozens of coupled vibrational modes.

Key Points

  • SPFs are time-dependent basis functions.
  • Each SPF is expanded in a fixed primitive basis.
  • Only the expansion coefficients evolve with time.
  • SPFs greatly reduce the number of basis functions required.
  • Their quality is monitored using the Natural Weights printed in the output file.
  • The number of SPFs is controlled through the SPF-BASIS-SECTION of the propagation input.

Next Step

Although SPFs significantly reduce the computational cost, very large molecular systems may still require thousands of configurations.

The Multi-Layer MCTDH (ML-MCTDH) method overcomes this limitation by organizing the SPFs into a hierarchical tree structure, enabling efficient simulations of systems containing hundreds of coupled degrees of freedom.