Multilayer Formulation of the Multi-Configuration Time-Dependent Hartree Theory

Haobin WangDepartment of Chemistry and

BiochemistryNew Mexico State UniversityLas Cruces, New Mexico, USA

Collaborator: Michael ThossSupport: NSF, NERSC

• From convention wave packet propagation to MCTDH: a variational perspective

• The multilayer formulation of MCTDH (ML-MCTDH)

• Scaling of the ML-MCTDH theory

• Generalization to treat identical particles: ML-MCTDH with Second Quantization (ML-MCTDH-SQ)

Outline

Conventional Wave Packet Propagation

• Dirac-Frenkel variational principle

• Conventional Full CI Expansion (orthonormal basis)

• Equations of Motion

• Capability: <10 degrees of freedom (<~n10 configurations)

Multi-Configuration Time-Dependent Hartree

• Multi-configuration expansion of the wave function

• Variations

• Both expansion coefficients and configurations are time-dependent

Meyer, Manthe, Cederbaum, Chem. Phys. Lett. 165 (1990) 73

MCTDH Equations of Motion (Meyer, Manthe, Cederbaum)

• Reduced density matrices and mean-field operators

The “single hole” function

Manthe, Meyer, Cederbaum, J.Chem.Phys. 97, 3199 (1992).

Meyer, Manthe, Cederbaum, Chem. Phys. Lett. 165, 73 (1990)

Variational Grouping of the Subspaces

• Single particle functions (SPFs): full CI expansion within the subspace (and adiabatic basis contraction)

• Only a few SPFs are selected among the full CI subspace, and then build the approximation for the whole space

• Thus, the philosophy is different!

• The “complete active space” strategy: first defines the whole space, then selects a subset

The MCTDH Theory

• Capability of the MCTDH theory: ~10×10 = 100 degrees of freedom

Worth, Meyer, Cederbaum, J. Chem. Phys. 105, 4412 (1996)Worth, Meyer, Cederbaum, J. Chem. Phys. 109, 3518 (1998)Raab, Worth, Meyer, Cederbaum, J. Chem. Phys. 110, 936 (1999)Mahapatra, Worth, Meyer, Cederbaum. Koppel, J. Phys.Chem. A 105, 5567 (2001)H. Koppel, Doscher, Baldea, Meyer, Szalay, J. Chem. Phys. 117, 2657 (2002)Nest, Meyer, J. Chem. Phys. 117, 10499 (2002)Huarte-Larranaga. Manthe, J. Chem. Phys. 113, 5115 (2000)Huarte-Larranaga, U. Manthe, J. Chem. Phys. 117, 4653 (2002)McCurdy, Isaacs, Meyer, Rescigno, Phys.Rev. A 67, 042708 (2003) Gatti, Meyer, Chem.Phys. 304, 3 (2004)Wu, Werner, Manthe, Science 306, 2227 (2004)

Kühn, Chem.Phys.Lett. 402, 48-53 (2005) Markmann, Worth, Mahapatra, Meyer, H. Köppel, Cederbaum. J.Chem.Phys. 123, 204310,

(2005) Viel, Eisfeld, Neumann, Domcke, Manthe, J.Chem.Phys.,124, 214306, (2006) Vendrell, Gatti, Meyer, Angewandte Chemie 46, 6918 (2007)

••••••

Multilayer Formulation of the MCTDH Theory

• Another multi-configuration expansion of the SP functions

• More complex way of expressing the wave function

Wang, Thoss, J. Chem. Phys. 119 (2003) 1289

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ML-MCTDH Equations of Motion

Wang, Thoss, J. Chem. Phys. 119 (2003) 1289

Exploring Dynamical Simplicity Using ML-MCTDH

• Capability of the two-layer ML-MCTDH: ~10×10×10 = 1000 degrees of freedom

• Capability of the three-layer ML-MCTDH: ~10×10×10×10 = 10000 degrees of freedom

Conventional

MCTDH

ML-MCTDH

The Scaling of the ML-MCTDH Theory

• f: the number of degrees of freedom • L: the number of layers• N: the number of (contracted) basis functions• n: the number of single-particle functions

• The Spin-Boson Model

The Scaling of the ML-MCTDH Theory

electronic

nuclear

coupling

• Hamiltonian

• Bath spectral density

Model Scaling of the ML-MCTDH Theory

Model Scaling of the ML-MCTDH Theory

Model Scaling of the ML-MCTDH Theory

Simulating Time Correlation Functions

• Examples

• Imaginary Time Propagation and Monte Carlo Sampling

Simulating Electric Current

V

M. Galperin, M.A. Ratner, A. Nitzan, J. Phys. Condens. Matter, 19, 103201 (2007)

F

F

F

F

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Vibrationally inelastic electron transport

Modeling:

•Tight-binding approximation, Wannier states of each lead transformed to Bloch states

•additional (or missing) electron in the bridge state results in a change of the potential energy surface

Calculation of the current:

Δq M-

M

Čižek, Thoss, Domcke, Phys. Rev. B 70 (2004) 125406

The MCTDHF Approach?

Fermi-Dirac Statistics: Anti-symmetric wave function

J. Caillat, J. Zanghellini, M. Kitzler, O. Koch, W. Kreuzer, and A. Scrinzi, Phys. Rev. A 71, 012712 (2004)

M. Nest, T. Klammroth, and P. Saalfrank, JCP 122, 124102 (2005)

T. Kato and H. Kono, CPL 392, 533 (2004)

What Strategy?

Active Space

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But how to put identical particles into different groups (and try todistinguish them)?

The Concept of Second Quantization

Fock Space

Each determinant is represented by an occupation-number vector

which can be represented by actions of creation operators

Creation

Empty orbital exposed to creation operator

Annihilation

0

Empty orbital exposed to annihilation operator

Annihilation

Filled orbital exposed to annihilation operator

Creation

0

Filled orbital exposed to creation operator

The ML-MCTDH-SQ Theory

Fock sub-space within one “single particle” for several states/electronsThe multi-configuration combination of the Fock sub-space to form the whole Fock space

The multilayer formulation

The ML-MCTDH-SQ Theory

Change the identical particle system to “distinguishable particles”Each “particle” defines a Fock subspace with all possible occupations

Second Quantization vs. Slater Determinant: two formal ways of enforcing permutation/exchange symmetry

•Slater Determinant: wave function approach, valid for any form of Hamiltonian operators

•Second Quantization: operator approach, superior for special form of Hamiltonian

The occupation for each “particle”/subspace is not conserved. However, the total occupation within the whole Fock space is of course conserved.

The formulation for Bosons is simpler than Fermions

Simulating Current Without Nuclear Motion

Without Nuclear Motion

Absorbing Boundary Condition

Effect of Nuclear Motion

Summary of the ML-MCTDH Theory

• Powerful tool to propagate wave packet in “complex” systems

• Can reveal various dynamical information– population dynamics and rate constant– wave packet motions – time-resolved nonlinear spectroscopy

• Has been generalized to handle indistinguishable particles

• Limitation: can only be implemented for certain class of models– Potentials: two-body, three-body, etc. (but cf. the CDVR)– Product form of the Hamiltonian

• Difficulties:– Implementation: somewhat challenging– Long time dynamics: “chaos”