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.
MCTDH can predict the motion of the nuclei of a molecular system evolving on one or several coupled electronic potential energy surfaces. It is an approximate method whose numerical efficiency decreases with growing accuracy.
MCTDH is suited for multi-dimensional problems,
Methods Basic algorithm Wavefunction expansion
Where the number of configurations is given by the product . The single particle functions (SPFs), , are expressed in a time-independent basis set:
Where is a primitive basis function, in general a Discrete Variable Representation (DVR) that is dependent on coordinate . If , 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
Where:
Which is subject to the boundary conditions . After integration, one obtains:
McLachlan Variational Principle
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:
Dirac-Frenkel Variational Principle
If each variation of is an allowed variation, then both the Lagrangian and the McLanchlan Variational Principle turn into the Dirac-Frenkel Variational Principle:
Which simplest and thus preferred method of deriving the equations of motion.
Multi-layer extension Motivation The original ansatz of MCTDH generates a single layer tensor tree; however, there is a limit to the size and complexity this single layer can handle. This prompted the development of a multilayer (ML)-MCTDH ansatz by Wang and Thoss, which was generalized by Manthe implemented in the Heidelberg package by Vendrell and Meyer.
Tensor Tree Formalism Multiple layers are generated through the creation of a tensor tree of nodes linking the modes (DOFs). Solving the tree layout is an NP-hard problem, but strides have been taken to automate this process through mode correlations by Mendive-Tapia.
Wave function expansion
The generalized ML expansion of Meyer can be written as follows:
Where the coordinates are combined as
Equations of motion
Where the equations of motion are now represented as follows:
The SPF EOMs are formally defined the same for all layers:
Where is a Hermitian gauge operator defined as follows:
Examples of uses in literature
NOCl
The first verification of the MCTDH method was with the NOCl molecule. Its size and asymmetry makes it a perfect test bed for MCTDH: it is small and simple enough for its numerics to be manually verified, yet complicated enough for it to already squeeze advantages against conventional product-basis methods.
Water clusters
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
+ 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.
Software packages implementing the MCTDH method
Package Name Group University Link Heidelberg MCTDH TC Group Heidelberg University Linkhttp://mctdh.uni-hd.de/ QUANTICS Worth UCL Linkhttps://www2.chem.ucl.ac.uk/quantics/doc/index.html MCTDH-X N/A ETH Zurich Linkhttp://ultracold.org/menu/
Example Usage of the Heidelberg Package for NOCl Input and Operator File nocl0.inp nocl0.opRUN-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 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
Output absorption spectrum
References
Further reading