Multi-configuration time-dependent Hartree: Difference between revisions

[[File:Absorption spectrum NOCl.png|thumb|center|upright=2.0|The absorption spectrum for the NOCl molecule on excitation to the S1 state]]

[[File:Absorption spectrum NOCl.png|thumb|center|upright=2.0|The absorption spectrum for the NOCl molecule on excitation to the S1 state]]

==Further Reading==

* {{cite journal | last1=Meyer | first1=H.-D. | last2=Manthe | first2=U. | last3=Cederbaum | first3=L.S. | title=The multi-configurational time-dependent Hartree approach | journal=Chemical Physics Letters | publisher=Elsevier BV | volume=165 | issue=1 | year=1990 | issn=0009-2614 | doi=10.1016/0009-2614(90)87014-i | pages=73–78| bibcode=1990CPL…165…73M }}

* {{cite journal | last1=Manthe | first1=U. | last2=Meyer | first2=H.‐D. | last3=Cederbaum | first3=L. S. | title=Wave‐packet dynamics within the multiconfiguration Hartree framework: General aspects and application to NOCl | journal=The Journal of Chemical Physics | publisher=AIP Publishing | volume=97 | issue=5 | year=1992 | issn=0021-9606 | doi=10.1063/1.463007 | pages=3199–3213| bibcode=1992JChPh..97.3199M }}

*{{cite journal | last1=Beck | first1=M. H. |last2=Jäckle|first2=A.|last3=Worth|first3=G. A.|last4=Meyer|first4=H.-D.| title=The multiconfiguration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets | journal=Physics Reports | publisher=Elsevier BV | volume=324 | issue=1 | year=2000 | issn=0370-1573 | doi=10.1016/s0370-1573(99)00047-2 | pages=1–105| bibcode=2000PhR…324….1B }}

==External links==

*[http://mctdh.uni-hd.de The Heidelberg MCTDH Homepage]

[[Category:Quantum chemistry]]

[[Category:Quantum chemistry]]

[[Category:Scattering]]

[[Category:Scattering]]

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==References==

Multi-configuration time-dependent Hartree (MCTDH) is a general algorithm to solve the time-dependent Schrödinger equation for multidimensional dynamical systems consisting of distinguishable particles.

MCTDH can thus determine the quantal 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 ^[1].

MCTDH is suited for multi-dimensional problems, in particular for problems that are difficult or even impossible to solve in conventional ways ^{[citation needed]}.

Ψ ( q i , . . . , q f , t ) = ∑ j 1 n 1 . . . ∑ j f n f A j 1 . . . j f ( t ) ∏ κ = 1 f φ j κ ( κ ) ( q κ , t ) {\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 n 1 . . . n f {\displaystyle n_{1}…n_{f}} . The single particle functions (SPFs), φ j κ ( κ ) ( q κ , t ) {\displaystyle \varphi _{j_{\kappa }}^{(\kappa )}(q_{\kappa },t)} , are expressed in a time-independent basis set:

φ j κ ( κ ) ( q κ , t ) = ∑ i 1 = 1 N κ c i κ ( κ , j κ ) ( t ) χ i κ ( κ ) ( q κ ) {\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 χ i κ ( κ ) ( q κ ) {\displaystyle \chi _{i_{\kappa }}^{(\kappa )}(q_{\kappa })} is a primative basis function, in general a Discrete Variable
Representation (DVR) that is dependent on coordinate q κ {\displaystyle q_{\kappa }} ^[2]. If n 1 . . . n f = 1 {\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.

L = ⟨ Ψ | i ∂ ∂ t − H | Ψ ⟩ {\displaystyle L=\langle \Psi |i{\frac {\partial }{\partial t}}-H|\Psi \rangle }

Where:

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

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

Re ⟨ δ Ψ | i ∂ ∂ t − H | Ψ ⟩ = 0 {\displaystyle {\text{Re}}\langle \delta \Psi |i{\frac {\partial }{\partial t}}-H|\Psi \rangle =0}

δ | | i ∂ ∂ t − H Ψ | | 2 = 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:

0 = δ ⟨ i ∂ ∂ t Ψ − H Ψ | i ∂ ∂ t Ψ − H | Ψ ⟩ = ⟨ i δ ∂ ∂ t Ψ | i ∂ ∂ t − H | Ψ ⟩ + ⟨ ( i ∂ ∂ t − H ) Ψ | i δ ∂ ∂ t Ψ ⟩ = − i ⟨ δ Ψ | i ∂ ∂ t − H | Ψ ⟩ + i ⟨ ( i ∂ ∂ t − H ) Ψ | δ Ψ ⟩ = 2 Im ⟨ δ Ψ | i ∂ ∂ t − H | Ψ ⟩ {\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 δ Ψ , i δ Ψ {\displaystyle \delta \Psi ,i\delta \Psi } is an allowed variation, then both the the Lagrangian and the McLanchlan Variational Principal turn into the Dirac-Frenkel Variational Principle:

⟨ δ Ψ | i ∂ ∂ t − H | Ψ ⟩ = 0 {\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^[2].

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:

φ m z − 1 , κ l − 1 ( q κ l − 1 z − 1 ) = ∑ j 1 = 1 n 1 z . . . ∑ j p z = 1 n p z z A m ; j 1 , . . . , j p z z ∏ κ l = 1 p z φ j κ l z , κ l ( q κ l z ) = ∑ J A m ; J z ⋅ Φ J z ( q κ l − 1 z − 1 ) {\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

q κ l − 1 z − 1 = ( q 1 z , . . . , q p z z ) {\displaystyle q_{\kappa _{l-1}}^{z-1}=(q_{1}^{z},…,q_{p^{z}}^{z})}

Where the equations of motion are now represented as follows:

i ∂ A J 1 ∂ t = ∑ K ⟨ Φ J 1 | H ^ − ∑ κ 1 = 1 p 1 g ^ 1 , κ 1 | Φ K 1 ⟩ A K 1 = ∑ K ⟨ Φ J 1 | H ^ | Φ K 1 ⟩ A K 1 − ∑ κ 1 = 1 p 1 ∑ i = 1 n κ 1 1 g ^ j κ 1 i 1 , κ 1 A j 1 . . . i . . . j p 1 {\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:

i ∂ φ n z , κ l ∂ t = ( 1 − P z , κ l ) ∑ j , m = 1 n κ l z ( p z , κ l ) n j − 1 ⋅ ⟨ H ^ ⟩ j m z , κ l φ m z , κ l + ∑ j = 1 n κ l z g j n z , κ l φ j z , κ l {\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 g ^ {\displaystyle {\hat {g}}} is a Hermitian gauge operator defined as follows:

⟨ φ j z , κ l | i ∂ ∂ t φ k z , κ l ⟩ = ⟨ φ j z , κ l | g ^ z , κ l | φ k z , κ l ⟩ = g j k z , κ l {\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. It’s 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