On the use of reduced-density matrices for the semi-automatic assignment of vibrational states

ABSTRACT

The use of one- and two-mode reduced-density matrices (RDM), ${\mathrm{\Gamma }}_1(q_i{}^{\prime },q_i)$ and ${\mathrm{\Gamma }}_2(q_i{}^{\prime },q_j{}^{\prime },q_i,q_j)$, respectively, and in particular the use of their diagonal elements, ${\mathrm{\Gamma }}_1(q_i,q_i)$ and ${\mathrm{\Gamma }}_2(q_i,q_j,q_i,q_j)$, is suggested for the assignment of normal-mode-like quantum numbers to variationally computed vibrational wave functions of semirigid molecules when the computation is based on a nuclear-motion Hamiltonian expressed in curvilinear internal coordinates $q_i$. The use of RDMs for the semi-automatic assignment of vibrational states is tested on the H${}_2{}^{16}$O molecule, whereby about the first 250 states, in the energy range of 0–25,000 cm⁻¹, are assigned. The proposed semi-automatic assignment procedure takes advantage of the fact that (a) for semirigid molecules it is often possible to define internal coordinates which mimick normal coordinates defined by the harmonic counterparts of the anharmonic vibrations, (b) overlaps between already assigned and yet unassigned RDMs provide outstanding and often unambiguous suggestions for the quantum numbers, and (c) an energy-decomposition scheme helps to decide among possible assignment possibilities suggested by the computed density overlaps.

GRAPHICAL ABSTRACT

Keywords:

• Wave-function analysis
• one- and two-mode reduced-density matrices
• vibrational assignment
• quantum numbers

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors gratefully acknowledge the financial support they received from NKFIH [grant number K119658].