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, in particular for problems that are difficult or even impossible to solve in conventional ways.
Methods
Basic algorithm
Wavefunction expansion
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 primitive basis function, in general a Discrete Variable Representation (DVR) that is dependent on coordinate q κ {\displaystyle q_{\kappa }} . If n 1 . . . n f = 1 {\displaystyle n_{1}...n_{f}=1} , 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
L = ⟨ Ψ | i ∂ ∂ t − H | Ψ ⟩ {\displaystyle L=\langle \Psi |i{\frac {\partial }{\partial t}}-H|\Psi \rangle }
Where:
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