Basic algorithm Wavefunction expansion \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^{(\kappa)}_{j_{\kappa}}(q_{\kappa},t) Where the number of configurations is given by the product n_1 ... n_f. The single particle functions (SPFs), \varphi^{(\kappa)}_{j_{\kappa}}(q_{\kappa},t), are expressed in a time-independent basis set: \varphi^{(\kappa)}_{j_{\kappa}}(q_{\kappa},t) = \sum_{i_1 = 1}^{N_\kappa}c_{i_\kappa}^{(\kappa, j_\kappa)}(t) \; \chi_{i_\kappa}^{(\kappa)}(q_\kappa) Where \chi_{i_\kappa}^{(\kappa)}(q_\kappa) is a primitive
basis function, in general a
Discrete Variable Representation (DVR) that is dependent on coordinate q_\kappa. If 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 = \langle\Psi | i \frac{\partial}{\partial t} - H | \Psi \rangle Where: \delta \int_{t_1}^{t_2} L \text{d}t = 0 Which is subject to the boundary conditions \delta L (t_1) = \delta L (t_2) = 0 . After integration, one obtains: \text{Re} \langle \delta \Psi | i \frac{\partial}{\partial t} - H | \Psi \rangle = 0
McLachlan Variational Principle \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: \begin{align} 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{align}
Dirac-Frenkel Variational Principle If each variation of \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: \langle \delta \Psi | i \frac{\partial}{\partial t} - H | \Psi \rangle = 0 Which simplest and thus preferred method of deriving the equations of motion. which was then generalized by Vendrell and Meyer.
Wave function expansion The generalized ML expansion of Meyer can be written as follows: \begin{align} \varphi^{z-1,\kappa_{l-1}}_{m}(q^{z-1}_{\kappa_{l-1}}) &= \sum^{n_1^z}_{j_1 = 1} ... \sum^{n_{p^z}^z}_{j_{p^z} = 1} A^z_{m;j_1, ..., j_{p^z}} \prod^{p^z}_{\kappa_l = 1}\varphi^{z,\kappa_l}_{j_{\kappa_l}} (q^z_{\kappa_l}) \\ &= \sum_{J}A^z_{m;J}\cdot \Phi^z_J (q^{z-1}_{\kappa_{l-1}}) \end{align} Where the coordinates are combined as q^{z-1}_{\kappa_{l-1}}= (q^z_1, ..., q^z_{p^z})
Equations of motion Where the equations of motion are now represented as follows: \begin{align} i\frac{\partial A^1_J}{\partial t} &= \sum_K \langle \Phi^1_J | \hat{H} - \sum^{p^1}_{\kappa_1 = 1} \hat{g}^{1,\kappa_1}| \Phi^1_K \rangle A^1_K\\ &= \sum_K \langle \Phi^1_J | \hat{H} | \Phi^1_K \rangle A^1_K - \sum^{p^1}_{\kappa_1 = 1}\sum_{i=1}^{n^1_{\kappa_1}}\hat{g}^{1,\kappa_1}_{j_{\kappa_1}i}A_{j_1 ... i ... j_{p^1}} \end{align} The SPF EOMs are formally defined the same for all layers: i\frac{\partial \varphi^{z,\kappa_l}_n}{\partial t} = (1 - P^{z,\kappa_l})\sum^{n^z_{\kappa_l}}_{j,m = 1} (p^{z,\kappa_l})^{-1}_{nj}\cdot \langle\hat{H}\rangle^{z, \kappa_l}_{jm}\varphi^{z, \kappa_l}_m + \sum^{n^z_{\kappa_l}}_{j=1}{g^{z, \kappa_l}_{jn} \varphi^{z,\kappa_l}_j} Where \hat{g} is a
Hermitian gauge operator defined as follows: \langle \varphi^{z,\kappa_l}_{j}| i\frac{\partial}{\partial t}\varphi^{z, \kappa_l}_k \rangle = \langle \varphi^{z,\kappa_l}_{j} | \hat{g}^{z, \kappa_l} | \varphi^{z,\kappa_l}_{k} \rangle = g^{z, \kappa_l}_{jk} ==Examples of uses in literature==