Excitation energies of dissociating H2: A problematic case for the adiabatic approximation of time-dependent density functional theory

Abstract

Time-dependent density functional theory (TDDFT) is applied for calculation of the excitation energies of the dissociating H2 molecule. The standard TDDFT method of adiabatic local density approximation (ALDA) totally fails to reproduce the potential curve for the lowest excited singlet Σu+1 state of H2. Analysis of the eigenvalue problem for the excitation energies as well as direct derivation of the exchange-correlation (xc) kernel fxc(r,r′,ω) shows that ALDA fails due to breakdown of its simple spatially local approximation for the kernel. The analysis indicates a complex structure of the function fxc(r,r′,ω), which is revealed in a different behavior of the various matrix elements K1c,1cxc (between the highest occupied Kohn–Sham molecular orbital ψ1 and virtual MOs ψc) as a function of the bond distance R(H–H). The effect of nonlocality of fxc(r,r′) is modeled by using different expressions for the corresponding matrix elements of different orbitals. Asymptotically corrected ALDA (ALDA-AC) expressions for the matrix elements K12,12xc(στ) are proposed, while for other matrix elements the standard ALDA expressions are retained. This approach provides substantial improvement over the standard ALDA. In particular, the ALDA-AC curve for the lowest singlet excitation qualitatively reproduces the shape of the exact curve. It displays a minimum and approaches a relatively large positive energy at large R(H–H). ALDA-AC also produces a substantial improvement for the calculated lowest triplet excitation, which is known to suffer from the triplet instability problem of the restricted KS ground state. Failure of the ALDA for the excitation energies is related to the failure of the local density as well as generalized gradient approximations to reproduce correctly the polarizability of dissociating H2. The expression for the response function χ is derived to show the origin of the field-counteracting term in the xc potential, which is lacking in the local density and generalized gradient approximations and which is required to obtain a correct polarizability.