Premium
Uncommonly accurate energies for the general quartic oscillator
Author(s) -
Okun Pavel,
Burke Kieron
Publication year - 2021
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.26554
Subject(s) - quartic function , eigenvalues and eigenvectors , virial theorem , mathematics , basis (linear algebra) , benchmark (surveying) , energy (signal processing) , virial coefficient , asymptotic expansion , physics , mathematical physics , mathematical analysis , quantum mechanics , pure mathematics , geometry , geodesy , galaxy , geography
Recent advances in the asymptotic analysis of energy levels of potentials produce relative errors in eigenvalue sums of order 10 −34 , but few non‐trivial potentials have been solved numerically to such accuracy. We solve the general quartic potential (arbitrary linear combination of x 2 and x 4 ) beyond this level of accuracy using a basis of several hundred oscillator states. We list the lowest 20 eigenvalues for 9 such potentials. We confirm the known asymptotic expansion for the levels of the pure quartic oscillator, and extract the next two terms in the asymptotic expansion. We give analytic formulas for expansion in up to three even basis states. We confirm the virial theorem for the various energy components to similar accuracy. The sextic oscillator levels are also given. These benchmark results should be useful for extreme tests of approximations in several areas of chemical physics and beyond.