Multi-configuration time-dependent Hartree

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.^[1]: 37

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.^[2]

MCTDH is suited for multi-dimensional problems, in particular for problems that are difficult or even impossible to solve in conventional ways.

${\displaystyle \Psi (q_{i},...,q_{f},t)=\sum _{j_{1}}^{n_{1}}...\sum _{j_{f}}^{n_{f}}A_{j_{1}...j_{f}}(t)\prod _{\kappa =1}^{f}\varphi _{j_{\kappa }}^{(\kappa )}(q_{\kappa },t)}$

Where the number of configurations is given by the product ${\displaystyle n_{1}...n_{f}}$. The single particle functions (SPFs), ${\displaystyle \varphi _{j_{\kappa }}^{(\kappa )}(q_{\kappa },t)}$, are expressed in a time-independent basis set:

${\displaystyle \varphi _{j_{\kappa }}^{(\kappa )}(q_{\kappa },t)=\sum _{i_{1}=1}^{N_{\kappa }}c_{i_{\kappa }}^{(\kappa ,j_{\kappa })}(t)\;\chi _{i_{\kappa }}^{(\kappa )}(q_{\kappa })}$

Where ${\displaystyle \chi _{i_{\kappa }}^{(\kappa )}(q_{\kappa })}$ is a primitive basis function, in general a Discrete Variable Representation (DVR) that is dependent on coordinate ${\displaystyle q_{\kappa }}$.^[1] If ${\displaystyle n_{1}...n_{f}=1}$, one returns to the Time Dependent Hartree (TDH) approach.^[3] In MCTDH, both the coefficients and the basis function are time-dependent and optimized using the variational principle.

${\displaystyle L=\langle \Psi |i{\frac {\partial }{\partial t}}-H|\Psi \rangle }$

Where:

${\displaystyle \delta \int _{t_{1}}^{t_{2}}L{\text{d}}t=0}$

Which is subject to the boundary conditions ${\displaystyle \delta L(t_{1})=\delta L(t_{2})=0}$. After integration, one obtains:

${\displaystyle {\text{Re}}\langle \delta \Psi |i{\frac {\partial }{\partial t}}-H|\Psi \rangle =0}$

${\displaystyle \delta ||i{\frac {\partial }{\partial t}}-H\Psi ||^{2}=0}$

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:

${\displaystyle {\begin{aligned}0&=\delta \langle i{\frac {\partial }{\partial t}}\Psi -H\Psi |i{\frac {\partial }{\partial t}}\Psi -H|\Psi \rangle \\&=\langle i\delta {\frac {\partial }{\partial t}}\Psi |i{\frac {\partial }{\partial t}}-H|\Psi \rangle +\langle (i{\frac {\partial }{\partial t}}-H)\Psi |i\delta {\frac {\partial }{\partial t}}\Psi \rangle \\&=-i\langle \delta \Psi |i{\frac {\partial }{\partial t}}-H|\Psi \rangle +i\langle (i{\frac {\partial }{\partial t}}-H)\Psi |\delta \Psi \rangle \\&=2{\text{Im}}\langle \delta \Psi |i{\frac {\partial }{\partial t}}-H|\Psi \rangle \end{aligned}}}$

If each variation of ${\displaystyle \delta \Psi ,i\delta \Psi }$ is an allowed variation, then both the Lagrangian and the McLanchlan Variational Principle turn into the Dirac-Frenkel Variational Principle:

${\displaystyle \langle \delta \Psi |i{\frac {\partial }{\partial t}}-H|\Psi \rangle =0}$

Which simplest and thus preferred method of deriving the equations of motion.^[1]

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 Manthe^[4] which was then generalized by Vendrell and Meyer.^[5]

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.^[6]

The generalized ML expansion of Meyer^[5] can be written as follows:

${\displaystyle {\begin{aligned}\varphi _{m}^{z-1,\kappa _{l-1}}(q_{\kappa _{l-1}}^{z-1})&=\sum _{j_{1}=1}^{n_{1}^{z}}...\sum _{j_{p^{z}}=1}^{n_{p^{z}}^{z}}A_{m;j_{1},...,j_{p^{z}}}^{z}\prod _{\kappa _{l}=1}^{p^{z}}\varphi _{j_{\kappa _{l}}}^{z,\kappa _{l}}(q_{\kappa _{l}}^{z})\\&=\sum _{J}A_{m;J}^{z}\cdot \Phi _{J}^{z}(q_{\kappa _{l-1}}^{z-1})\end{aligned}}}$

Where the coordinates are combined as

${\displaystyle q_{\kappa _{l-1}}^{z-1}=(q_{1}^{z},...,q_{p^{z}}^{z})}$

Where the equations of motion are now represented as follows:

${\displaystyle {\begin{aligned}i{\frac {\partial A_{J}^{1}}{\partial t}}&=\sum _{K}\langle \Phi _{J}^{1}|{\hat {H}}-\sum _{\kappa _{1}=1}^{p^{1}}{\hat {g}}^{1,\kappa _{1}}|\Phi _{K}^{1}\rangle A_{K}^{1}\\&=\sum _{K}\langle \Phi _{J}^{1}|{\hat {H}}|\Phi _{K}^{1}\rangle A_{K}^{1}-\sum _{\kappa _{1}=1}^{p^{1}}\sum _{i=1}^{n_{\kappa _{1}}^{1}}{\hat {g}}_{j_{\kappa _{1}}i}^{1,\kappa _{1}}A_{j_{1}...i...j_{p^{1}}}\end{aligned}}}$

The SPF EOMs are formally defined the same for all layers:

${\displaystyle i{\frac {\partial \varphi _{n}^{z,\kappa _{l}}}{\partial t}}=(1-P^{z,\kappa _{l}})\sum _{j,m=1}^{n_{\kappa _{l}}^{z}}(p^{z,\kappa _{l}})_{nj}^{-1}\cdot \langle {\hat {H}}\rangle _{jm}^{z,\kappa _{l}}\varphi _{m}^{z,\kappa _{l}}+\sum _{j=1}^{n_{\kappa _{l}}^{z}}{g_{jn}^{z,\kappa _{l}}\varphi _{j}^{z,\kappa _{l}}}}$

Where ${\displaystyle {\hat {g}}}$ is a Hermitian gauge operator defined as follows:

${\displaystyle \langle \varphi _{j}^{z,\kappa _{l}}|i{\frac {\partial }{\partial t}}\varphi _{k}^{z,\kappa _{l}}\rangle =\langle \varphi _{j}^{z,\kappa _{l}}|{\hat {g}}^{z,\kappa _{l}}|\varphi _{k}^{z,\kappa _{l}}\rangle =g_{jk}^{z,\kappa _{l}}}$

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.^[7]

The solvation of the hydronium ion is a topic of continued research. Researchers have been able to successfully use MCTDH to model the Zundel^[8] and Eigen^[9] ions in close agreement with experiment.

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.^[11] 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.^[12] These issues can also by circumvented by using the spin-boson or other similar bath models that do not pose the same assignment challenges.^[10]

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

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

Meyer, Hans-Dieter (2009). Multidimensional Quantum Dynamics: MCTDH Theory and Applications (PDF) (1 ed.). Hoboken: John Wiley & Sons, Incorporated. ISBN 978-3-527-32018-9.