Hamiltonian equations numerical convergence

In closing this section we emphasize that the MCTDH equations of motion conserve the norm of the wave function and, for time-independent Hamiltonians, the mean energy. This follows directly from the variational principle. Moreover, the MCTDH wave function converges toward the numerically exact wave function with increasing numbers of single-particle functions. 

As implied by Equation (4.11) and Equation (4.12), the computational cost of LR-TDDFT grows rapidly with increasing system size, because the size of the response matrices is proportional to the product of the numbers of occupied and unoccupied molecular orbitals, that is, (Voce x Aunocc- Moreover, the diffuse unoccupied orbitals usually require very large basis sets to satisfactorily converge the results. To circumvent these numerical challenges, a RT-TDDFT method was derived to track the response of an optical system upon external perturbation. In RT-TDDFT, an additional term, E r) x ]i r), is added to the system s quantum Hamiltonian, Hq, to reflect Kght-matter interactions ... 

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