Multi-configurational Time Dependant Hartree (MCTDH)

Nicholas J. Browning

Contents

  • Time Dependant Picture

  • Standard Method (SM)

  • Variational Principle (VP)

    •  SM Equations of Motion (EoM)

    • Pitfalls of SM

  • Multi-configurational TDH (MCTDH)

    • MCTDH EoM
    • MCTDH Memory Requirements
    • MCTDH Numerical Effort
    • MCTDH Limitations
  • MCTDH Mode Combination
  • MCTDH Example

Time Dependant Picture 

i\dot \Psi = H \Psi
iΨ˙=HΨi\dot \Psi = H \Psi

(TI) orthonormal basis functions

Goal: derive equations of motion for coefficients

the Standard Method

nuclear coordinates or DoF

TD expansion coefficients

number of basis functions for DoF k

number of DoF

{\left\langle{\delta\Psi}\right |}H - i\frac{\delta}{\delta t} {\left|{\Psi}\right\rangle} = 0
⟨δΨ∣H−iδδt∣Ψ⟩=0{\left\langle{\delta\Psi}\right |}H - i\frac{\delta}{\delta t} {\left|{\Psi}\right\rangle} = 0

Variational Principle

\Psi \in \{\delta\Psi\}
Ψ∈{δΨ}\Psi \in \{\delta\Psi\}
\delta_tH = 0
δtH=0\delta_tH = 0

Dirac-Frenkel Variational Principle

\langle \chi_{l_1} \ldots \chi_{l_f} | \sum\limits_{j_1 \ldots j_f} C_{j_1 \ldots j_f} H \chi_{j_1} \ldots \chi_{j_f} \rangle = i\langle \chi_{l_1} \ldots \chi_{l_f} | \sum\limits_{j_1 \ldots j_f} \dot C_{j_1 \ldots j_f}\chi_{j_1} \ldots \chi_{j_f} \rangle
⟨χl1…χlf∣∑j1…jfCj1…jfHχj1…χjf⟩=i⟨χl1…χlf∣∑j1…jfC˙j1…jfχj1…χjf⟩\langle \chi_{l_1} \ldots \chi_{l_f} | \sum\limits_{j_1 \ldots j_f} C_{j_1 \ldots j_f} H \chi_{j_1} \ldots \chi_{j_f} \rangle = i\langle \chi_{l_1} \ldots \chi_{l_f} | \sum\limits_{j_1 \ldots j_f} \dot C_{j_1 \ldots j_f}\chi_{j_1} \ldots \chi_{j_f} \rangle
\delta\Psi = \sum\limits_{l_1\ldots l_f}\frac{\delta\Psi}{\delta C_{l_1 \ldots l_f}}\delta C_{l_1 \ldots l_f}=\sum\limits_{l_1 \ldots l_f} \chi_{l_1}^{(1)}(q_1)\ldots \chi_{l_f}^{(f)}(q_f)\delta C_{l_1}\ldots l_f
δΨ=∑l1…lfδΨδCl1…lfδCl1…lf=∑l1…lfχl1(1)(q1)…χlf(f)(qf)δCl1…lf\delta\Psi = \sum\limits_{l_1\ldots l_f}\frac{\delta\Psi}{\delta C_{l_1 \ldots l_f}}\delta C_{l_1 \ldots l_f}=\sum\limits_{l_1 \ldots l_f} \chi_{l_1}^{(1)}(q_1)\ldots \chi_{l_f}^{(f)}(q_f)\delta C_{l_1}\ldots l_f
\dot\Psi=\sum\limits_{j_1 \ldots j_f} \dot C_{j_1 \ldots j_f}(t) \chi_{j_1}^{(1)}(q_1)\ldots \chi_{j_f}^{(f)}(q_f)
Ψ˙=∑j1…jfC˙j1…jf(t)χj1(1)(q1)…χjf(f)(qf)\dot\Psi=\sum\limits_{j_1 \ldots j_f} \dot C_{j_1 \ldots j_f}(t) \chi_{j_1}^{(1)}(q_1)\ldots \chi_{j_f}^{(f)}(q_f)
\sum\limits_{j_1 \ldots j_f} \langle \chi_{l_1} \ldots \chi_{l_f} | H \chi_{j_1} \ldots \chi_{j_f} \rangle C_{j_1 \ldots j_f} = i \dot C_{j_1 \ldots j_f}
∑j1…jf⟨χl1…χlf∣Hχj1…χjf⟩Cj1…jf=iC˙j1…jf \sum\limits_{j_1 \ldots j_f} \langle \chi_{l_1} \ldots \chi_{l_f} | H \chi_{j_1} \ldots \chi_{j_f} \rangle C_{j_1 \ldots j_f} = i \dot C_{j_1 \ldots j_f}
J=(j_1\ldots j_f)
J=(j1…jf)J=(j_1\ldots j_f)
\chi_J = \prod\limits_{\kappa=1}^f\chi_{j_\kappa}
χJ=∏κ=1fχjκ\chi_J = \prod\limits_{\kappa=1}^f\chi_{j_\kappa}
i \dot C_{L} = \sum\limits_{J} \langle \chi_{L} | H| \chi_{J} \rangle C_{J}
iC˙L=∑J⟨χL∣H∣χJ⟩CJi \dot C_{L} = \sum\limits_{J} \langle \chi_{L} | H| \chi_{J} \rangle C_{J}

SM Equations of Motion

Pitfalls of the Standard Method

Need at least 10 basis functions

per degree of freedom

10^f
10f10^f

coupled equations to be solved

f=3N-6 =12=10^{12}
f=3N−6=12=1012f=3N-6 =12=10^{12}

E.g. 6 atom molecule

J=(j_1\ldots j_f)
J=(j1…jf)J=(j_1\ldots j_f)
\chi_J = \prod\limits_{\kappa=1}^f\chi_{j_\kappa}
χJ=∏κ=1fχjκ\chi_J = \prod\limits_{\kappa=1}^f\chi_{j_\kappa}
i \dot C_{L} = \sum\limits_{J} \langle \chi_{L} | H |\chi_{J} \rangle C_{J}
iC˙L=∑J⟨χL∣H∣χJ⟩CJi \dot C_{L} = \sum\limits_{J} \langle \chi_{L} | H |\chi_{J} \rangle C_{J}

Time Dependant Hartree (TDH)

constraints:

variation in

variation in

TDH EoM

choice of gk is arbitrary - simply shifts phase factors between a and \psi_k

TDH Pitfalls

20^{12}
201220^{12}
12 \times 20
12×2012 \times 20

TDH reduces an f-dimensional PDE to a set of f one-dimensional PDEs

SM

TDH

visualisation of v - <v>

MCTDH

\Psi (q_1, \ldots, q_f, t) = \sum\limits_{j_1}^{n_1}\ldots\sum\limits_{j_f}^{n_f}A_{j_1 \ldots j_f}(t) \prod\limits_{k=1}^{f}\psi_{j_k}^{(k)}(q_k, t)
Ψ(q1,…,qf,t)=∑j1n1…∑jfnfAj1…jf(t)∏k=1fψjk(k)(qk,t)\Psi (q_1, \ldots, q_f, t) = \sum\limits_{j_1}^{n_1}\ldots\sum\limits_{j_f}^{n_f}A_{j_1 \ldots j_f}(t) \prod\limits_{k=1}^{f}\psi_{j_k}^{(k)}(q_k, t)
=\sum\limits_JA_J\Phi_J
=∑JAJΦJ=\sum\limits_JA_J\Phi_J
\psi_{j_k}^{(k)}(q_k, t) = \sum\limits_{i_k=1}^{N_k}c_{i_kj_k}^{(k)}(t)\chi_{i_k}^{(k)}(q_k)
ψjk(k)(qk,t)=∑ik=1Nkcikjk(k)(t)χik(k)(qk)\psi_{j_k}^{(k)}(q_k, t) = \sum\limits_{i_k=1}^{N_k}c_{i_kj_k}^{(k)}(t)\chi_{i_k}^{(k)}(q_k)
J=(j_1\ldots j_f)
J=(j1…jf)J=(j_1\ldots j_f)

MCTDH

\Psi_l^{(k)} = \langle\psi_l^{(k)}|\Psi\rangle = \sum\limits_{J^k}A_{J_l^k}\prod\limits_{v\ne k}\psi_{j_v}^{(v)}
Ψl(k)=⟨ψl(k)∣Ψ⟩=∑JkAJlk∏v≠kψjv(v)\Psi_l^{(k)} = \langle\psi_l^{(k)}|\Psi\rangle = \sum\limits_{J^k}A_{J_l^k}\prod\limits_{v\ne k}\psi_{j_v}^{(v)}
\Psi (q_1, \ldots, q_f, t) = \sum\limits_JA_J\Phi_J =\sum\limits_{j=1}^{n_k}\psi_j^{(k)}\Psi_j^{(k)}
Ψ(q1,…,qf,t)=∑JAJΦJ=∑j=1nkψj(k)Ψj(k)\Psi (q_1, \ldots, q_f, t) = \sum\limits_JA_J\Phi_J =\sum\limits_{j=1}^{n_k}\psi_j^{(k)}\Psi_j^{(k)}
\langle H \rangle_{jl}^{(k)} = \langle \Psi_j^{(k)}| H| \Psi_l^{(k)} \rangle
⟨H⟩jl(k)=⟨Ψj(k)∣H∣Ψl(k)⟩\langle H \rangle_{jl}^{(k)} = \langle \Psi_j^{(k)}| H| \Psi_l^{(k)} \rangle
\rho_{jl}^{(k)} = \langle\Psi_j^{(k)}|\Psi_l^{(k)}\rangle = \sum\limits_{J^k}A_{J_j^k}^*A_{J_l^k}
ρjl(k)=⟨Ψj(k)∣Ψl(k)⟩=∑JkAJjk∗AJlk\rho_{jl}^{(k)} = \langle\Psi_j^{(k)}|\Psi_l^{(k)}\rangle = \sum\limits_{J^k}A_{J_j^k}^*A_{J_l^k}
P^{(k)} = \sum\limits_{j=1}^{n_k}|\psi_j^{(k)}\rangle\langle\psi_j^{(k)}|
P(k)=∑j=1nk∣ψj(k)⟩⟨ψj(k)∣P^{(k)} = \sum\limits_{j=1}^{n_k}|\psi_j^{(k)}\rangle\langle\psi_j^{(k)}|

DFVP

i\dot A_J = \sum\limits_L\langle\Phi_J|H|\Phi_L\rangle A_L- i\sum\limits_{k=1}^f\sum\limits_{l=1}^{n_k}g_{j_kl}^{(k)}A_{J^k_l}
iA˙J=∑L⟨ΦJ∣H∣ΦL⟩AL−i∑k=1f∑l=1nkgjkl(k)AJlki\dot A_J = \sum\limits_L\langle\Phi_J|H|\Phi_L\rangle A_L- i\sum\limits_{k=1}^f\sum\limits_{l=1}^{n_k}g_{j_kl}^{(k)}A_{J^k_l}
i \dot {\mathbf{\psi}}^{(k)}= (\hat g^{(k)}\mathbf{1})\mathbf{\psi}^{(k)}+ (1 -P^{(k)})\{ (\mathbf{\rho}^{(k)})^{-1}\langle\mathbf H \rangle^{(k)} - \hat g ^{(k)} \mathbf{1}\}\mathbf{\psi}^{(k)}
iψ˙(k)=(g^(k)1)ψ(k)+(1−P(k)){(ρ(k))−1⟨H⟩(k)−g^(k)1}ψ(k)i \dot {\mathbf{\psi}}^{(k)}= (\hat g^{(k)}\mathbf{1})\mathbf{\psi}^{(k)}+ (1 -P^{(k)})\{ (\mathbf{\rho}^{(k)})^{-1}\langle\mathbf H \rangle^{(k)} - \hat g ^{(k)} \mathbf{1}\}\mathbf{\psi}^{(k)}
i\dot A_J = \sum\limits_L\langle\Phi_J|H|\Phi_L\rangle A_L
iA˙J=∑L⟨ΦJ∣H∣ΦL⟩ALi\dot A_J = \sum\limits_L\langle\Phi_J|H|\Phi_L\rangle A_L
i \dot {\mathbf{\psi}}^{(k)}=(1 -P^{(k)})\{h^{(k)} \mathbf{1} + (\mathbf{\rho}^{(k)})^{-1}\langle\mathbf H_R \rangle^{(k)} \}\mathbf{\psi}^{(k)}
iψ˙(k)=(1−P(k)){h(k)1+(ρ(k))−1⟨HR⟩(k)}ψ(k)i \dot {\mathbf{\psi}}^{(k)}=(1 -P^{(k)})\{h^{(k)} \mathbf{1} + (\mathbf{\rho}^{(k)})^{-1}\langle\mathbf H_R \rangle^{(k)} \}\mathbf{\psi}^{(k)}
H=\sum\limits_kh^{(k)} + H_R
H=∑kh(k)+HRH=\sum\limits_kh^{(k)} + H_R
\hat g^{(k)} \equiv 0
g^(k)≡0\hat g^{(k)} \equiv 0
\mathbf{\psi}^{(k)}= (\psi_1^{(k)}, \ldots, \psi_{n_k}^{(k)})^T
ψ(k)=(ψ1(k),…,ψnk(k))T\mathbf{\psi}^{(k)}= (\psi_1^{(k)}, \ldots, \psi_{n_k}^{(k)})^T

 MCTDH - Memory

f Standard Method MCTDH
3 1.54MB 190KB 343 672 32768
4 48MB 620KB 2401 896 1.05E+06
6 48GB 22MB 117E+03 1344 1.07E+09
9 1.54PB 7.2GB 40E+06 2016 3.5E+13
\text{mem}_{\text{St.Mthd}} = 3 \times N^f \times \text{complex16 bytes}
memSt.Mthd=3×Nf×complex16 bytes\text{mem}_{\text{St.Mthd}} = 3 \times N^f \times \text{complex16 bytes}
\text{mem}_{\text{MCTDH}}=12\times( n^f + fnN ) \times \text{complex16 bytes}
memMCTDH=12×(nf+fnN)×complex16 bytes\text{mem}_{\text{MCTDH}}=12\times( n^f + fnN ) \times \text{complex16 bytes}

n=7, N=32

gain_{mem} = \frac {1}{4}(\frac{N}{n})^f
gainmem=14(Nn)fgain_{mem} = \frac {1}{4}(\frac{N}{n})^f
n^f
nfn^f
fnN
fnNfnN
N^f
NfN^f

 MCTDH - CPU EFFORT

\text{effort}_\text{St.Method} \approx c_0.f.N^{f+1}
effortSt.Method≈c0.f.Nf+1\text{effort}_\text{St.Method} \approx c_0.f.N^{f+1}
gain_{CPU} = \frac{c_0}{c_2}\frac{1}{sf}(\frac{N}{n})^{f+1}
gainCPU=c0c21sf(Nn)f+1gain_{CPU} = \frac{c_0}{c_2}\frac{1}{sf}(\frac{N}{n})^{f+1}
\text{effort}_\text{MCTDH} \approx c_1.s.f.n.N^2 + c_2.s.f^2.n^{f+1}
effortMCTDH≈c1.s.f.n.N2+c2.s.f2.nf+1\text{effort}_\text{MCTDH} \approx c_1.s.f.n.N^2 + c_2.s.f^2.n^{f+1}

Limiting factor for both memory and CPU is A-vector length 

 MCTDH - Limitations

n^f
nfn^f
\text{mem}_{\text{MCTDH}}=12\times( n^f + fnN ) \times \text{complex16 bytes}
memMCTDH=12×(nf+fnN)×complex16 bytes\text{mem}_{\text{MCTDH}}=12\times( n^f + fnN ) \times \text{complex16 bytes}
\text{effort}_\text{MCTDH} \approx c_1.s.f.n.N^2 + c_2.s.f^2.n^{f+1}
effortMCTDH≈c1.s.f.n.N2+c2.s.f2.nf+1\text{effort}_\text{MCTDH} \approx c_1.s.f.n.N^2 + c_2.s.f^2.n^{f+1}

MCTDH - Mode Combination

Q_k\equiv(q_{k,1},q_{k,2 }, \ldots, q_{k, d})
Qk≡(qk,1,qk,2,…,qk,d)Q_k\equiv(q_{k,1},q_{k,2 }, \ldots, q_{k, d})
\psi_j^{(k)}(Q_k, t)=\psi_j^{(k)}(q_{k,1},q_{k,2 }, \ldots, q_{k, d})
ψj(k)(Qk,t)=ψj(k)(qk,1,qk,2,…,qk,d)\psi_j^{(k)}(Q_k, t)=\psi_j^{(k)}(q_{k,1},q_{k,2 }, \ldots, q_{k, d})
\Psi(Q_1, \ldots, Q_p, t) = \sum\limits_{j_1}^{n_1}\ldots\sum\limits_{j_p}^{n_p}A_{j_1 \ldots j_p}(t) \prod\limits_{k=1}^{p}\psi_{j_k}^{(k)}(Q_k, t)
Ψ(Q1,…,Qp,t)=∑j1n1…∑jpnpAj1…jp(t)∏k=1pψjk(k)(Qk,t)\Psi(Q_1, \ldots, Q_p, t) = \sum\limits_{j_1}^{n_1}\ldots\sum\limits_{j_p}^{n_p}A_{j_1 \ldots j_p}(t) \prod\limits_{k=1}^{p}\psi_{j_k}^{(k)}(Q_k, t)
\equiv\Psi (q_1, \ldots, q_f, t)
≡Ψ(q1,…,qf,t)\equiv\Psi (q_1, \ldots, q_f, t)
\psi_j^{(k)} (Q_k, t)=\sum\limits_{i_1\ldots i_d}C_{i_1\ldots i_d}^{(k,j)} \chi^{(k,1)}(q_{k, 1})\ldots\chi^{(k,d)}(q_{k, d})
ψj(k)(Qk,t)=∑i1…idCi1…id(k,j)χ(k,1)(qk,1)…χ(k,d)(qk,d)\psi_j^{(k)} (Q_k, t)=\sum\limits_{i_1\ldots i_d}C_{i_1\ldots i_d}^{(k,j)} \chi^{(k,1)}(q_{k, 1})\ldots\chi^{(k,d)}(q_{k, d})

 MCTDH-MC Pictoral Representation

a) Standard method

b) MCTDH

c) MCTDH with Mode Combination

d) ML-MCTDH (3 layer)

e) ML-MCTDH (4 layer)

 MCTDH Mode Combination - Memory

f St. Method MCTDH 2-mode MCTDH 3-mode MCTDH
2 48KB 282KB - -
4 48MB 620KB 6MB -
6 48GB 22MB 10MB 290MB
10 48PB 51GB 160MB 310MB
15 - - 210GB 1.90GB
 18 - - ​7.38TB ​29.3GB
\text{mem}_{\text{MCTDH-MC}} = 12\times( \tilde {n}^{p} + p\tilde{n}N^d ) \times \text{complex16 bytes}
memMCTDH-MC=12×(n~p+pn~Nd)×complex16 bytes\text{mem}_{\text{MCTDH-MC}} = 12\times( \tilde {n}^{p} + p\tilde{n}N^d ) \times \text{complex16 bytes}
\text{mem}_{\text{St.Mthd}} = 3 \times N^f \times \text{complex16 bytes}
memSt.Mthd=3×Nf×complex16 bytes\text{mem}_{\text{St.Mthd}} = 3 \times N^f \times \text{complex16 bytes}
\text{mem}_{\text{MCTDH}}=12\times( n^f + fnN ) \times \text{complex16 bytes}
memMCTDH=12×(nf+fnN)×complex16 bytes\text{mem}_{\text{MCTDH}}=12\times( n^f + fnN ) \times \text{complex16 bytes}
\tilde {n}_2 = 15
n~2=15\tilde {n}_2 = 15
\tilde {n}_3= 23
n~3=23\tilde {n}_3= 23
N=32
N=32N=32
n = 7
n=7n = 7

Example: Butatriene PhotoIonisation

C. Cattarius, G. A. Worth, H. D. Meyer, L. S. Cederbaum, J. Chem. Phys, 115 (2001), 2088-2100.

18f, 2 electronic states

Thanks.

MCTDH

By Nick Browning

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MCTDH

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