^{1}and Johannes Neugebauer

^{1,a)}

### Abstract

Standard implementations of time-dependent density-functional theory (TDDFT) for the calculation of excitation energies give access to a number of the lowest-lying electronic excitations of a molecule under study. For extended systems, this can become cumbersome if a particular excited state is sought-after because many electronic transitions may be present. This often means that even for systems of moderate size, a multitude of excited states needs to be calculated to cover a certain energy range. Here, we present an algorithm for the selective determination of predefined excited electronic states in an extended system. A guess transition density in terms of orbital transitions has to be provided for the excitation that shall be optimized. The approach employs root-homing techniques together with iterative subspace diagonalization methods to optimize the electronic transition. We illustrate the advantages of this method for solvated molecules, core-excitations of metal complexes, and adsorbates at cluster surfaces. In particular, we study the local excitation of a pyridine molecule adsorbed at a silver cluster. It is shown that the method works very efficiently even for high-lying excited states. We demonstrate that the assumption of a single, well-defined local excitation is, in general, not justified for extended systems, which can lead to root-switching during optimization. In those cases, the method can give important information about the spectral distribution of the orbital transition employed as a guess.

This work was supported by a TOP grant of the Netherlands Organization for Scientific Research (NWO) and a computer time grant from the Netherlands National Computing Facilities Foundation (NCF).

I. INTRODUCTION

II. THEORY

III. COMPUTATIONAL DETAILS

IV. TEST CASE:

V. CORE ELECTRON EXCITATIONS IN

VI. ADSORBATE EXCITATIONS: CLUSTER

VII. LOCAL EXCITATIONS IN SOLVATED SYSTEMS

VIII. CONCLUSIONS

### Key Topics

- Eigenvalues
- 69.0
- Excited states
- 39.0
- Excitation energies
- 35.0
- Adsorbates
- 9.0
- Silver
- 8.0

## Figures

Convergence of the excitation energy (BP86/TZP) for excitation no. 95 (dominant orbital transition ) with the selective Davidson procedure and a conventional Davidson method for the lowest 500 roots of the complex shown in the inset (optimized with BP86/TZP).

Convergence of the excitation energy (BP86/TZP) for excitation no. 95 (dominant orbital transition ) with the selective Davidson procedure and a conventional Davidson method for the lowest 500 roots of the complex shown in the inset (optimized with BP86/TZP).

Convergence of the excitation energy (BP86/TZP) for excitation nos. 93–96 (numbers refer to the reference calculation) with the selective Davidson procedure of the system shown in Fig. 1. The left plot shows the excitation energy convergence for calculations of individual excitations, while the right plot shows result for the simultaneous optimization of the set of four excitations.

Convergence of the excitation energy (BP86/TZP) for excitation nos. 93–96 (numbers refer to the reference calculation) with the selective Davidson procedure of the system shown in Fig. 1. The left plot shows the excitation energy convergence for calculations of individual excitations, while the right plot shows result for the simultaneous optimization of the set of four excitations.

Error in the selective Davidson optimization of excited-state no. 95 for the system. Shown are the error in the energy (top; energy difference in two subsequent cycles), the error in the eigenvector (middle; measured by the last expansion coefficient ), and the absolute value of the largest element of the residual vector (bottom). Two different fit basis sets have been used; for comparison, we also show results with a fit-free reference calculation in which a precalculated matrix is employed in the matrix-vector products (see main text for details). Note that the scale for the errors is logarithmic.

Error in the selective Davidson optimization of excited-state no. 95 for the system. Shown are the error in the energy (top; energy difference in two subsequent cycles), the error in the eigenvector (middle; measured by the last expansion coefficient ), and the absolute value of the largest element of the residual vector (bottom). Two different fit basis sets have been used; for comparison, we also show results with a fit-free reference calculation in which a precalculated matrix is employed in the matrix-vector products (see main text for details). Note that the scale for the errors is logarithmic.

Left: structure of employed in this work (based on the experimental parameters given in Ref. 23); right: isosurface plots of two representative orbitals from the -manifold of Ti and Cl in (isosurface value: ±0.003).

Left: structure of employed in this work (based on the experimental parameters given in Ref. 23); right: isosurface plots of two representative orbitals from the -manifold of Ti and Cl in (isosurface value: ±0.003).

Structures of the cluster (taken from Ref. 36) and of isolated pyridine (optimized with BP86/TZP).

Structures of the cluster (taken from Ref. 36) and of isolated pyridine (optimized with BP86/TZP).

Convergence of excitation energies (BP86/TZP; the to core orbitals were kept frozen for Ag atoms) of the cluster shown in Fig. 5 with the selective Davidson procedure. The left plot indicates the convergence of the transition dominated by the orbital transition. The right plot shows the convergence for the simultaneous optimization of two excited states, in which the transition was included as a guess in addition.

Convergence of excitation energies (BP86/TZP; the to core orbitals were kept frozen for Ag atoms) of the cluster shown in Fig. 5 with the selective Davidson procedure. The left plot indicates the convergence of the transition dominated by the orbital transition. The right plot shows the convergence for the simultaneous optimization of two excited states, in which the transition was included as a guess in addition.

Isosurface plots of the and orbitals of isolated pyridine and pyridine adsorbed on the cluster (isosurface value: ±0.03).

Isosurface plots of the and orbitals of isolated pyridine and pyridine adsorbed on the cluster (isosurface value: ±0.03).

Top: convergence of the excitation energy (BP86/TZP; the to core orbitals were kept frozen for Ag atoms) of pyridine adsorbed on the cluster with the selective Davidson method. The norm of the residual was converged to less than , and the energy to less than . The two plateau energies shown in Fig. 9 are marked with labels A and B. Bottom: convergence of the spectral distribution of the contribution weight of the same orbital transition for several iteration steps (broadened with a half-width of 0.3 eV).

Top: convergence of the excitation energy (BP86/TZP; the to core orbitals were kept frozen for Ag atoms) of pyridine adsorbed on the cluster with the selective Davidson method. The norm of the residual was converged to less than , and the energy to less than . The two plateau energies shown in Fig. 9 are marked with labels A and B. Bottom: convergence of the spectral distribution of the contribution weight of the same orbital transition for several iteration steps (broadened with a half-width of 0.3 eV).

Dependence of the contribution weight of the orbital transition to excitations as a function of the excitation energy obtained in a reference calculation on 3500 excited states. The contributions are represented as Gaussian functions with two different half-widths (hw). Also indicated are the energies of two of the plateaus in the selective Davidson calculation (A and B), see Fig. 8 (upper plot).

Dependence of the contribution weight of the orbital transition to excitations as a function of the excitation energy obtained in a reference calculation on 3500 excited states. The contributions are represented as Gaussian functions with two different half-widths (hw). Also indicated are the energies of two of the plateaus in the selective Davidson calculation (A and B), see Fig. 8 (upper plot).

Structure of acetone surrounded with 32 water molecules optimized with BP86/TZP, where core orbitals on oxygen and carbon atoms were kept frozen.

Structure of acetone surrounded with 32 water molecules optimized with BP86/TZP, where core orbitals on oxygen and carbon atoms were kept frozen.

Isosurface plots of orbitals of the complex (isosurface value: ±0.03). The orbitals were labeled according the symmetry of similar orbitals of isolated acetone (shown in the insets).

Isosurface plots of orbitals of the complex (isosurface value: ±0.03). The orbitals were labeled according the symmetry of similar orbitals of isolated acetone (shown in the insets).

Residual norms for the excitations of the complex shown in Table V during the iterative optimization. Note that the scale for the residual norm is logarithmic.

Residual norms for the excitations of the complex shown in Table V during the iterative optimization. Note that the scale for the residual norm is logarithmic.

## Tables

Excitation energies (BP86/TZP) of the complex in the energy region from 11.50 to 11.60 eV. : reference energies, obtained with a standard block Davidson method (500 states optimized); : excitation energies taken from the selective Davidson calculation on the set of four states; : excitation energies calculated individually with the selective Davidson algorithm. The orbital transitions chosen as a guess are listed as well.

Excitation energies (BP86/TZP) of the complex in the energy region from 11.50 to 11.60 eV. : reference energies, obtained with a standard block Davidson method (500 states optimized); : excitation energies taken from the selective Davidson calculation on the set of four states; : excitation energies calculated individually with the selective Davidson algorithm. The orbital transitions chosen as a guess are listed as well.

Excitation energies (in units of eV; LB94/QZ3P for Cl and QZ3P-3DIF for Ti) of ; : reference energies obtained with a standard block Davidson method; : excitation energies taken from the selective Davidson calculation on the set of states; : excitation energies from the work of Stener *et al.* (Ref. 23); : experimental energies (Ref. 23). Also shown are the energies of the first Davidson iteration and the corresponding error norms together with final error norms . The orbital transitions (“orb. trans.”) chosen as a guess are listed as well.

Excitation energies (in units of eV; LB94/QZ3P for Cl and QZ3P-3DIF for Ti) of ; : reference energies obtained with a standard block Davidson method; : excitation energies taken from the selective Davidson calculation on the set of states; : excitation energies from the work of Stener *et al.* (Ref. 23); : experimental energies (Ref. 23). Also shown are the energies of the first Davidson iteration and the corresponding error norms together with final error norms . The orbital transitions (“orb. trans.”) chosen as a guess are listed as well.

Excitation energies (BP86/TZP; the to core orbitals were kept frozen for Ag atoms) of the cluster. : reference energies obtained with a standard block Davidson method (300 states optimized); : excitation energies taken from the selective Davidson calculation. : contribution weight of the guess transition from the selective Davidson calculation; : contribution weight from the reference calculation. Also shown are the norms of the residual vectors (; in a.u.) at the end of the calculation. The orbital transitions chosen as a guess are listed in column “orbital transition.”

Excitation energies (BP86/TZP; the to core orbitals were kept frozen for Ag atoms) of the cluster. : reference energies obtained with a standard block Davidson method (300 states optimized); : excitation energies taken from the selective Davidson calculation. : contribution weight of the guess transition from the selective Davidson calculation; : contribution weight from the reference calculation. Also shown are the norms of the residual vectors (; in a.u.) at the end of the calculation. The orbital transitions chosen as a guess are listed in column “orbital transition.”

Excitation energy (BP86/TZP; the to core orbitals were kept frozen for Ag atoms) of excitation no. 21 of the cluster obtained from standard block Davidson calculations with different update schemes and different numbers of excited states. Also shown are the contribution weights for the and orbital transitions to these excitations.

Excitation energy (BP86/TZP; the to core orbitals were kept frozen for Ag atoms) of excitation no. 21 of the cluster obtained from standard block Davidson calculations with different update schemes and different numbers of excited states. Also shown are the contribution weights for the and orbital transitions to these excitations.

Excitation energies (BP86/TZP; the core orbitals on oxygen and carbon atoms were kept frozen) of the system for excitations with large contributions of the orbital transitions listed below. : reference energies; energies in the first cycle of the selective Davidson calculation; : final excitation energies taken from the selective Davidson calculations (all energies are given in units of eV); “orbital transition” refers to the transition used as a guess in the selective Davidson procedure, and labels in parentheses refer to the symmetry labels of the related orbitals of isolated acetone in symmetry; : contribution weight of the orbital transition obtained from the selective Davidson calculation; : contribution weight of the orbital transition obtained from the reference calculation. Also shown are the norms of the residual vectors in the first (; all residual norms are in a.u.) and final iteration .

Excitation energies (BP86/TZP; the core orbitals on oxygen and carbon atoms were kept frozen) of the system for excitations with large contributions of the orbital transitions listed below. : reference energies; energies in the first cycle of the selective Davidson calculation; : final excitation energies taken from the selective Davidson calculations (all energies are given in units of eV); “orbital transition” refers to the transition used as a guess in the selective Davidson procedure, and labels in parentheses refer to the symmetry labels of the related orbitals of isolated acetone in symmetry; : contribution weight of the orbital transition obtained from the selective Davidson calculation; : contribution weight of the orbital transition obtained from the reference calculation. Also shown are the norms of the residual vectors in the first (; all residual norms are in a.u.) and final iteration .

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