On the required shape corrections to the local density and generalized gradient approximations to the Kohn–Sham potentials for molecular response calculations of (hyper)polarizabilities and excitation energies

Abstract

It is well known that shape corrections have to be applied to the local-density ͑LDA͒ and generalized gradient ͑GGA͒ approximations to the Kohn-Sham exchange-correlation potential in order to obtain reliable response properties in time dependent density functional theory calculations. Here we demonstrate that it is an oversimplified view that these shape corrections concern primarily the asymptotic part of the potential, and that they affect only Rydberg type transitions. The performance is assessed of two shape-corrected Kohn-Sham potentials, the gradient-regulated asymptotic connection procedure applied to the Becke-Perdew potential ͑BP-GRAC͒ and the statistical averaging of ͑model͒ orbital potentials ͑SAOP͒, versus LDA and GGA potentials, in molecular response calculations of the static average polarizability ␣, the Cauchy coefficient S Ϫ4 , and the static average hyperpolarizability ␤. The nature of the distortions of the LDA/GGA potentials is highlighted and it is shown that they introduce many spurious excited states at too low energy which may mix with valence excited states, resulting in wrong excited state compositions. They also lead to wrong oscillator strengths and thus to a wrong spectral structure of properties like the polarizability. LDA, Becke-Lee-Yang-Parr ͑BLYP͒, and Becke-Perdew ͑BP͒ characteristically underestimate contributions to ␣ and S Ϫ4 from bound Rydberg-type states and overestimate those from the continuum. Cancellation of the errors in these contributions occasionally produces fortuitously good results. The distortions of the LDA, BLYP, and BP spectra are related to the deficiencies of the LDA/GGA potentials in both the bulk and outer molecular regions. In contrast, both SAOP and BP-GRAC potentials produce high quality polarizabilities for 21 molecules and also reliable Cauchy moments and hyperpolarizabilities for the selected molecules. The analysis for the N 2 molecule shows, that both SAOP and BP-GRAC yield reliable energies i and oscillator strengths f i of individual excitations, so that they reproduce well the spectral structure of ␣ and S Ϫ4 .

Vibronic coupling and double excitations in linear response time-dependent density functional calculations: Dipole-allowed states of N2

“…The experimental determination of the latter value seems to be quite difficult in view of the complicated structure of the N 2 absorption spectrum in this energy range and the low oscillator strength of this transition, making a deconvolution into contributions of at least three different electronic states necessary. We also quote in the table the alternative assignment of the orbital transitions to the experimentally observed states, which as noted by Grün-ing et al 13 and mentioned in the Introduction, would considerably improve the agreement of the calculations with the experimental vertical excitation energies from the more recent Ref. 1.…”

“…Details of the basis can be found in Ref. 13. For ground-state structure optimization, frequency analysis, and reference energy calculations, we employed the Becke-Perdew-Wang exchange-correlation functional, dubbed BPW91; 29,30 test calculations showed that other generalized-gradient approximations to the exact exchangecorrelation functional yield very similar results.…”

“…As pointed out in Ref. 13 this would mean that SAOP and other asymptotically correct potentials like BP-GRAC and HCTH͑AC͒ as well as many ab initio methods all overestimate the vertical excitation energy for this transition by 0.3-0.5 eV if the experimental assignment is correct.…”

“…In particular the results from calculations using the statistical averaging of ͑model͒ orbital potentials [15][16][17] ͑SAOP͒ in Ref. 13 raised the question whether a different assignment should be made, the calculated vertical excitations being in very good agreement ͑errors р0.16 eV for all three investigated 1 ⌸ u states and all but one of the three 1 ⌺ u ϩ states in comparison to the experimental values from Ref. 1͒ when it is assumed that the 2 u →1 g valence excitation corresponds, at R eq , to the highest of the three states of 1 ⌸ u symmetry, o 1 ⌸ u .…”

“…Besides these attempts to model the fine structure of the absorption spectrum, several first-principles calculations of vertical excitation energies for N 2 have been performed, including polarization-propagator and equation-of-motion techniques, 8 multireference coupled-cluster ͑MR-CC͒, 9 equation-of-motion coupled-cluster ͑EOM-CC͒, 10 and density functional theory ͑DFT͒ calculations employing asymptotically corrected potentials, [11][12][13] as well as calculations ushave been obtained from the set of recommended parameters given in Ref. 7 is shown in Fig.…”