Multi-configuration time-dependent Hartree (MCTDH) is an approach to quantum molecular dynamics, an algorithm to solve the time-dependent Schrödinger equation for multidimensional dynamical systems consisting of distinguishable particles. The nuclei of molecules is one example of such particles and their vibrational motion is a form of time-dependence. The method uses an overall wavefunction composed of products of single-particle wavefunctions as first proposed by Douglas Hartree in 1927. The "multiconfiguration" part of the method refers to combining multiple such products. If <math>n_1 ... n_f = 1</math>, one returns to the Time Dependent Hartree (TDH) approach. In MCTDH, both the coefficients and the basis function are time-dependent and optimized using the variational principle.
Equations of motion
Lagrangian Variational Principle
<math>L = \langle\Psi | i \frac{\partial}{\partial t} - H | \Psi \rangle</math>
Where:
<math>\delta \int_{t_1}^{t_2} L \text{d}t = 0</math>
Which is subject to the boundary conditions <math>\delta L (t_1) = \delta L (t_2) = 0 </math>. After integration, one obtains:
<math>\text{Re} \langle \delta \Psi | i \frac{\partial}{\partial t} - H | \Psi \rangle = 0</math>
McLachlan Variational Principle
<math>\delta || i \frac{\partial}{\partial t} - H \Psi ||^2 = 0</math>
Where only the time derivative is to be varied. We can rewrite this norm squared term as a scalar product, and vary the bra and ket side of the product:
<math>
\begin{align}
0 &= \delta\langle i \frac{\partial}{\partial t} \Psi - H \Psi |i \frac{\partial}{\partial t}\Psi - H | \Psi \rangle \\
&= \langle i \delta \frac{\partial}{\partial t} \Psi | i \frac{\partial}{\partial t} - H | \Psi \rangle + \langle (i\frac{\partial}{\partial t} - H)\Psi | i\delta \frac{\partial}{\partial t}\Psi \rangle\\
&= -i \langle \delta \Psi | i \frac{\partial}{\partial t} - H | \Psi \rangle + i\langle (i\frac{\partial}{\partial t} - H)\Psi | \delta \Psi \rangle \\
&= 2 \text{Im} \langle \delta \Psi | i \frac{\partial}{\partial t} - H | \Psi \rangle
\end{align}
</math>
Dirac-Frenkel Variational Principle
If each variation of <math>\delta \Psi, i\delta\Psi</math> is an allowed variation, then both the Lagrangian and the McLanchlan Variational Principle turn into the Dirac-Frenkel Variational Principle:
<math>\langle \delta \Psi | i \frac{\partial}{\partial t} - H | \Psi \rangle = 0</math>
Which simplest and thus preferred method of deriving the equations of motion., which was generalized by Manthe implemented in the Heidelberg package by Vendrell and Meyer.
thumb|center|Example MCTDH tree with l representing layers and q1-6 being the modes.
Wave function expansion
The generalized ML expansion of Meyer
Water clusters
thumb
The solvation of the hydronium ion is a topic of continued research. Researchers have been able to successfully use MCTDH to model the Zundel and Eigen ions in close agreement with experiment.
Limitations
{| class="wikitable"
|+ Approximate Degree of Freedom Allowance for Each Computational Method
|-
! Method !! Degrees of Freedom Possible
|-
| Conventional Methods (e.g. TDH) || 6
|-
| MCTDH || 12
|-
| ML-MCTDH || 24+
|-
|ML-MCTDH with the Spin-Boson Model || 1000+
|}
For a typical input in ML-MCTDH to be run, a node tree, potential energy surface, and equations of motion must be generated by the user. These prerequisites—along with total compute time—soft-cap the size of systems able to be studied with ML-MCTDH; however, advances in neural networks have been shown to address the difficulty of the generation of potential energy surfaces. These issues can also by circumvented by using the spin-boson or other similar bath models that do not pose the same assignment challenges.
|-
| QUANTICS || Worth || UCL || Link
|-
| MCTDH-X || N/A || ETH Zurich || Link
|}
Example Usage of the Heidelberg Package for NOCl
Input and Operator File
{| class="wikitable"
|-
! nocl0.inp !! nocl0.op
|-
|<pre>
RUN-SECTION
relaxation
tfinal= 50.0
tout= 10.0
name = nocl0
overwrite
output psi=double timing
end-run-section
OPERATOR-SECTION
opname = nocl0
end-operator-section
SBASIS-SECTION
rd = 5
rv = 5
theta = 5
end-sbasis-section
pbasis-section
- Label DVR N Parameter
rd sin 36 3.800 5.600
rv HO 24 2.136 0.272,ev 13615.5
theta Leg 60 0 0
end-pbasis-section
INTEGRATOR-SECTION
CMF/var = 0.50 , 1.0d-5
BS/spf = 10 , 1.0d-7
SIL/A = 12 , 1.0d-7
end-integrator-section
INIT_WF-SECTION
build
rd gauss 4.315 0.0 0.0794
rv HO 2.151 0.0 0.218,eV 13615.5
theta gauss 2.22 0.0 0.0745
end-build
end-init_wf-section
ALLOC-SECTION
maxkoe=160
maxhtm=220
maxhop=220
maxsub=60
maxLMR=1
maxdef=85
maxedim=1
maxfac=25
maxmuld=1
maxnhtmshift=1
end-alloc-section
end-input</pre>
|| <pre>
OP_DEFINE-SECTION
title
NOCl S0 surface
end-title
end-op_define-section
PARAMETER-SECTION
mass_rd = 16.1538, AMU
mass_rv = 7.4667, AMU
end-parameter-section
HAMILTONIAN-SECTION
---------------------------------------------------------
modes | rd | rv | theta
---------------------------------------------------------
0.5/mass_rd | q^-2 | 1 | j^2
0.5/mass_rv | 1 | q^-2 | j^2
1.0 | KE | 1 | 1
1.0 | 1 | KE | 1
1.0 |1&2&3 V
---------------------------------------------------------
end-hamiltonian-section
LABELS-SECTION
V = srffile {nocl0um, default}
end-labels-section
end-operator
</pre>
|}
Output absorption spectrum
thumb|center|upright=2.0|The absorption spectrum for the NOCl molecule on excitation to the S1 state