^{1,a)}, Oleg Gritsenko

^{1,b)}and Evert Jan Baerends

^{1,c)}

### Abstract

The failure of the time-dependent density-functional theory to describe long-range charge-transfer (CT) excitations correctly is a serious problem for calculations of electronic transitions in large systems, especially if they are composed of several weakly interacting units. The problem is particularly severe for molecules in solution, either modeled by periodic boundary calculations with large box sizes or by cluster calculations employing extended solvent shells. In the present study we describe the implementation and assessment of a simple physically motivated correction to the exchange-correlation kernel suggested in a previous study [O. Gritsenko and E. J. Baerends J. Chem. Phys.121, 655 (2004)]. It introduces the required divergence in the kernel when the transition density goes to zero due to a large spatial distance between the “electron” (in the virtual orbital) and the “hole” (in the occupied orbital). A major benefit arises for solvated molecules, for which many CT excitations occur from solvent to solute or vice versa. In these cases, the correction of the exchange-correlation kernel can be used to automatically “clean up” the spectrum and significantly reduce the computational effort to determine low-lying transitions of the solute. This correction uses a phenomenological parameter, which is needed to identify a CT excitation in terms of the orbital density overlap of the occupied and virtual orbitals involved. Another quantity needed in this approach is the magnitude of the correction in the asymptotic limit. Although this can, in principle, be calculated rigorously for a given CT transition, we assess a simple approximation to it that can automatically be applied to a number of low-energy CT excitations without additional computational effort. We show that the method is robust and correctly shifts long-range CT excitations, while other excitations remain unaffected. We discuss problems arising from a strong delocalization of orbitals, which leads to a breakdown of the correction criterion.

One of the authors (J.N.) acknowledges funding by a Forschungsstipendium of the Deutsche Forschungsgemeinschaft (DFG).

I. INTRODUCTION

II. METHODOLOGY

III. A SIMPLE TEST SYSTEM: He⋯Be

IV. ETHYLENE-TETRAFLUOROETHYLENE

V. SOLVATED ACETONE

VI. CONCLUSIONS

### Key Topics

- Excitation energies
- 68.0
- Density functional theory
- 10.0
- Excited states
- 9.0
- Ground states
- 5.0
- Basis sets
- 4.0

## Figures

Schematic representation of the (uncorrected) coupling matrix for a system consisting of two fragments with a large separation. Left: full coupling matrix in the basis of all occupied (o1/o2)-virtual (v1/v2) orbital pairs for fragments 1 and 2. Right: coupling matrix after the removal of the orbital pairs corresponding to CT excitations. The white areas correspond to matrix elements that will be (close to) zero due to a zero differential overlap.

Schematic representation of the (uncorrected) coupling matrix for a system consisting of two fragments with a large separation. Left: full coupling matrix in the basis of all occupied (o1/o2)-virtual (v1/v2) orbital pairs for fragments 1 and 2. Right: coupling matrix after the removal of the orbital pairs corresponding to CT excitations. The white areas correspond to matrix elements that will be (close to) zero due to a zero differential overlap.

Excitation energies for the system He⋯Be as a function of the internuclear distance from SAOP/TZ2P calculations [fcorr: corrected according to Eq. (4)]. CISD data from Ref. 9 are given for comparison.

Excitation energies for the system He⋯Be as a function of the internuclear distance from SAOP/TZ2P calculations [fcorr: corrected according to Eq. (4)]. CISD data from Ref. 9 are given for comparison.

Isosurface plots of orbitals around the HOMO-LUMO gap involved in some of the low-lying CT excitations of the ethylene-tetrafluoroethylene complex (ascending orbital energies from left to right; distance: ).

Isosurface plots of orbitals around the HOMO-LUMO gap involved in some of the low-lying CT excitations of the ethylene-tetrafluoroethylene complex (ascending orbital energies from left to right; distance: ).

Adiabatic excited-state potential energy curves (solid lines) for irrep of the ethylene-tetrafluoroethylene complex (SAOP/TZP; zero point: ground-state energy at ). Top: no kernel correction; bottom: kernel correction applied. Labels correspond to the character of the excitation at a distance of ; the character of the excitations may change due to avoided crossings. In the lower diagram, also a pure -like curve for the state (dotted line; shifted by for clarity of presentation) as well as “intuitive” diabatic states are shown. The latter curves connect data points of states with similar characters (dashed lines; shifted by for clarity of presentation).

Adiabatic excited-state potential energy curves (solid lines) for irrep of the ethylene-tetrafluoroethylene complex (SAOP/TZP; zero point: ground-state energy at ). Top: no kernel correction; bottom: kernel correction applied. Labels correspond to the character of the excitation at a distance of ; the character of the excitations may change due to avoided crossings. In the lower diagram, also a pure -like curve for the state (dotted line; shifted by for clarity of presentation) as well as “intuitive” diabatic states are shown. The latter curves connect data points of states with similar characters (dashed lines; shifted by for clarity of presentation).

Adiabatic excited-state potential energy curves for irrep of the ethylene-tetrafluoroethylene complex (SAOP/TZP; zero point: ground-state energy at ). Top: no kernel correction; bottom: kernel correction applied. Labels correspond to the characters of the excitation at a distance of ; the character of the excitations may change due to avoided crossings.

Adiabatic excited-state potential energy curves for irrep of the ethylene-tetrafluoroethylene complex (SAOP/TZP; zero point: ground-state energy at ). Top: no kernel correction; bottom: kernel correction applied. Labels correspond to the characters of the excitation at a distance of ; the character of the excitations may change due to avoided crossings.

Adiabatic excited-state potential energy curves (solid lines) for irrep of the ethylene-tetrafluoroethylene complex (CC2/TZVP; zero point: ground-state energy at ). We also show a pure -like curve for the CT-state (dotted line; shifted by for clarity of presentation) as well as the “intuitive” diabatic potential energy curve for the lowest CT-like transition (dashed lines; shifted by for clarity of presentation). For short distances, the character of this excitation spreads over the three lowest excitations in this irrep (indicated by additional dashed lines).

Adiabatic excited-state potential energy curves (solid lines) for irrep of the ethylene-tetrafluoroethylene complex (CC2/TZVP; zero point: ground-state energy at ). We also show a pure -like curve for the CT-state (dotted line; shifted by for clarity of presentation) as well as the “intuitive” diabatic potential energy curve for the lowest CT-like transition (dashed lines; shifted by for clarity of presentation). For short distances, the character of this excitation spreads over the three lowest excitations in this irrep (indicated by additional dashed lines).

Isosurface plots of the orbitals of the ethylene-tetrafluoroethylene complex showing a pronounced mixing for a distance of .

Isosurface plots of the orbitals of the ethylene-tetrafluoroethylene complex showing a pronounced mixing for a distance of .

Excitation energies obtained for different numbers of optimized states in irrep of the ethylene-tetrafluoroethylene complex. Left: default guess (orbital energy differences) used to construct guesses for the lowest excitations; right: corrected guess [Eq. (16)] applied.

Excitation energies obtained for different numbers of optimized states in irrep of the ethylene-tetrafluoroethylene complex. Left: default guess (orbital energy differences) used to construct guesses for the lowest excitations; right: corrected guess [Eq. (16)] applied.

Structure of the acetone-water cluster and isosurface plot of one of the orbitals with a partial lone pair character .

Structure of the acetone-water cluster and isosurface plot of one of the orbitals with a partial lone pair character .

Spectra (SAOP/TZP/DZ) of the acetone∙20 cluster shown in Fig. 9 from a conventional TDDFT calculation (“no correction”) as well as from two calculations using the asymptotic correction to the coupling matrix with different values of the switching parameter . The spectra are modeled by applying a Gaussian broadening of (dotted lines) and (solid lines). For the spectra with a half width of , also the positions of the maxima are indicated.

Spectra (SAOP/TZP/DZ) of the acetone∙20 cluster shown in Fig. 9 from a conventional TDDFT calculation (“no correction”) as well as from two calculations using the asymptotic correction to the coupling matrix with different values of the switching parameter . The spectra are modeled by applying a Gaussian broadening of (dotted lines) and (solid lines). For the spectra with a half width of , also the positions of the maxima are indicated.

## Tables

Number of matrix-vector products needed to converge roots (irrep ) in the TDDFT calculation. A: default zero-order guess used to construct lowest-energy eigenvectors and preconditioner; B: guess vectors based on corrected guess energies [Eq. (16)]; and C: guess vectors and preconditioner based on Eq. (16). Note that scheme A converges to eigenvalues different from those obtained in schemes B and C for small (see Fig. 8).

Number of matrix-vector products needed to converge roots (irrep ) in the TDDFT calculation. A: default zero-order guess used to construct lowest-energy eigenvectors and preconditioner; B: guess vectors based on corrected guess energies [Eq. (16)]; and C: guess vectors and preconditioner based on Eq. (16). Note that scheme A converges to eigenvalues different from those obtained in schemes B and C for small (see Fig. 8).

Excitation energies (SAOP/TZP/DZ; in units of eV) of the lowest transitions of the acetone-water cluster shown in Fig. 9 from a conventional TDDFT calculation (“conv.”) and calculations with the asymptotic correction. In the latter case, we either used the default switching parameter or a larger value of . Also given are the oscillator strengths (in a.u.) from the conventional calculation and the dominant orbital contributions; the orbitals are characterized in Table III.

Excitation energies (SAOP/TZP/DZ; in units of eV) of the lowest transitions of the acetone-water cluster shown in Fig. 9 from a conventional TDDFT calculation (“conv.”) and calculations with the asymptotic correction. In the latter case, we either used the default switching parameter or a larger value of . Also given are the oscillator strengths (in a.u.) from the conventional calculation and the dominant orbital contributions; the orbitals are characterized in Table III.

Characterization of the orbitals (SAOP/TZP/DZ) of the acetone-water cluster shown in Fig. 9.

Characterization of the orbitals (SAOP/TZP/DZ) of the acetone-water cluster shown in Fig. 9.

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