Density matrices and iterative natural modals in vibrational structure theory

Abstract

It is demonstrated how the second-quantization formulation of multi-mode dynamics leads to expressions for vibrational density matrices. The properties and different representations of these matrices are discussed. Diagonalizing the one-mode density matrices defines a set of natural modals for each vibrational mode. The theory and first implementation of the iterative natural modals (ItNaMo) method for correlated vibrational-structure models is presented. In the ItNaMo method, natural modals are used as basis functions for a subsequent correlated calculation. This optimization of the one-mode basis is repeated until the changes in the basis functions become sufficiently small. Ground-state and excited-state energy calculations are presented for water, formaldehyde, and ethylene. It is shown that using optimized coordinates for the water calculation makes the occupation numbers converge to zero much faster and thereby allows for large reductions in the required number of basis functions. For the higherorder wave-function models the ItNaMo method results in smaller energy errors compared to full vibrational configuration interaction (FVCI) and thereby facilitates a more accurate description of the wave function.

Systematic and variational truncation of the configuration space in the multiconfiguration time-dependent Hartree method: The MCTDH[n] hierarchy

“…If the propagation starts from a set of natural occupied modals [or perhaps iterative natural modals (ItNaMos) 43 ], we require,…”

Quasi-direct Quantum Molecular Dynamics: The Time-Dependent Adaptive Density-Guided Approach for Potential Energy Surface Construction

“…The one-mode density is a function of the coordinate of the mth mode (q m ) and of the time (t), and is obtained as an expectation value of the operator probing the presence of a certain mode at a point in space, δ qd m m . In first quantization, this operator is defined from the Dirac-delta function 43,44 = q q ( ) q m m m m m (5) and the one-mode density then becomes…”

Vibrational Coupled Cluster Computations in Polyspherical Coordinates with the Exact Analytical Kinetic Energy Operator

“…The vibrational coupled cluster (VCC) model ,,− accounts for dynamic correlation between modes, similar to the vibrational configuration interaction (VCI) model, ,− but with the possibility of faster convergence toward the full vibrational configuration interaction (FVCI) limit when utilizing truncated versions. The VCC wave function ansatz is a nonlinear exponential ansatz where the cluster operator T̂ controls excitations out of the reference state …”

“…These optimized coordinates are generated via vibrational self-consistent field (VSCF) 7,[19][20][21][22][23] calculations and have been demonstrated to facilitate faster convergence towards the full vibrational configuration interaction (FVCI) limit, in vibrational structure calculations. 18,[24][25][26][27][28][29] A different approach to reduce the delocalization of NCs is to employ a localization scheme, which relies on geometrical localization criteria, resulting in localized coordinates. [30][31][32][33][34][35][36][37][38][39][40] Localizing selected modes facilitates a faster convergence towards the FVCI limit for vibrational structure calculations and faster convergence in the n-mode expansion of the PES, as fewer high-order couplings are required.…”

“…The idea of variationally optimizing the NCs has been around for quite some years now. − The objective is to obtain a new set of coordinates that are linear combinations of the initial NCs but with a comparatively lesser degree of coupling between the modes. These optimized coordinates are generated via vibrational self-consistent field (VSCF) ,− calculations and have been demonstrated to facilitate faster convergence toward the full vibrational configuration interaction (FVCI) limit in vibrational structure calculations. ,− …”