^{1,a)}, Manuel J. Louwerse

^{1,b)}, Evert Jan Baerends

^{1,c)}and Tomasz A. Wesolowski

^{2,d)}

### Abstract

We investigate the usefulness of a frozen-density embedding scheme within density-functional theory [J. Phys. Chem.97, 8050 (1993)] for the calculation of solvatochromic shifts. The frozen-density calculations, particularly of excitation energies have two clear advantages over the standard supermolecule calculations: (i) calculations for much larger systems are feasible, since the time-consuming time-dependent density functional theory (TDDFT) part is carried out in a limited molecular orbital space, while the effect of the surroundings is still included at a quantum mechanical level. This allows a large number of solvent molecules to be included and thus affords both specific and nonspecific solvent effects to be modeled. (ii) Only excitations of the system of interest, i.e., the selected embedded system, are calculated. This allows an easy analysis and interpretation of the results. In TDDFT calculations, it avoids unphysical results introduced by spurious mixings with the artificially too low charge-transfer excitations which are an artifact of the adiabatic local-density approximation or generalized gradient approximation exchange-correlation kernels currently used. The performance of the frozen-density embedding method is tested for the well-studied solvatochromic properties of the excitation of acetone. Further enhancement of the efficiency is studied by constructing approximate solvent densities, e.g., from a superposition of densities of individual solvent molecules. This is demonstrated for systems with up to 802 atoms. To obtain a realistic modeling of the absorption spectra of solvated molecules, including the effect of the solvent motions, we combine the embedding scheme with classical molecular dynamics (MD) and Car-Parrinello MD simulations to obtain snapshots of the solute and its solvent environment, for which then excitation energies are calculated. The frozen-density embedding yields estimated solvent shifts in the range of , in good agreement with experimental values of between 0.19 and .

J.N. gratefully acknowledges funding by a Forschungsstipendium of the Deutsche Forschungsgemeinschaft (DFG). M.J.L. acknowledges a grant from the Dutch National Research School Combination “Catalysis by Design” (NRSC-C). This work was supported by the Swiss National Science Foundation (SNSF).

I. INTRODUCTION

II. METHODOLOGY

III. CONSTRUCTION OF THE FROZEN ENVIRONMENT DENSITY

IV. CONVERGENCE WITH THE SIZE OF THE CLUSTER

A. Charge-transfer excitation problem

B. Efficiency of the KSCED excitation energy calculations

C. Convergence with the number of solvent molecules

V. (CP)MD SIMULATIONS

VI. CONCLUSION

### Key Topics

- Solvents
- 96.0
- Excitation energies
- 51.0
- Supramolecular assembly
- 34.0
- Molecular dynamics
- 11.0
- Excited states
- 9.0

## Figures

Example structure of acetone and its 88 nearest water molecules as obtained from a CPMD simulation, substructures of which have been used for a comparison of (partly) frozen-density embedding and supermolecule calculations.

Example structure of acetone and its 88 nearest water molecules as obtained from a CPMD simulation, substructures of which have been used for a comparison of (partly) frozen-density embedding and supermolecule calculations.

Excited states (SAOP/TZP/DZ) of acetone-water complexes with different numbers of water molecules. All water molecules have been explicitly taken into account in the TDDFT calculation (scheme A). The longer, dashed lines correspond to the and valence transitions.

Excited states (SAOP/TZP/DZ) of acetone-water complexes with different numbers of water molecules. All water molecules have been explicitly taken into account in the TDDFT calculation (scheme A). The longer, dashed lines correspond to the and valence transitions.

Excited states (SAOP/TZP/DZ) of acetone-water complexes with different numbers of water molecules. Two water molecules have been explicitly taken into account in the TDDFT calculation (except for the isolated molecule calculation), while all additional water molecules are included via the QM/QM embedding scheme (scheme B). The longer, dashed lines correspond to the and valence transitions.

Excited states (SAOP/TZP/DZ) of acetone-water complexes with different numbers of water molecules. Two water molecules have been explicitly taken into account in the TDDFT calculation (except for the isolated molecule calculation), while all additional water molecules are included via the QM/QM embedding scheme (scheme B). The longer, dashed lines correspond to the and valence transitions.

Excited states (SAOP/TZP/DZ) of acetone-water complexes with different numbers of water molecules. All water molecules are included via the QM/QM embedding scheme in the TDDFT calculation (scheme C). The longer, dashed lines correspond to the and valence transitions.

Excited states (SAOP/TZP/DZ) of acetone-water complexes with different numbers of water molecules. All water molecules are included via the QM/QM embedding scheme in the TDDFT calculation (scheme C). The longer, dashed lines correspond to the and valence transitions.

Excitation energies (SAOP/TZP, LDA/DZ for the frozen part; energies in units of eV) of the valence excitations of acetone in water using the QM/QM embedding scheme for a snapshot from a CPMD simulation, from which subsystems with different numbers of water molecules have been extracted. The frozen density in all cases was constructed as a sum of densities of water fragments. For clusters with more than 100 water molecules, rigid water molecules where assumed for all but the 20 water molecules nearest to the embedded acetone. For the largest clusters, the number also includes some acetone molecules from neighboring cells of the CPMD simulation.

Excitation energies (SAOP/TZP, LDA/DZ for the frozen part; energies in units of eV) of the valence excitations of acetone in water using the QM/QM embedding scheme for a snapshot from a CPMD simulation, from which subsystems with different numbers of water molecules have been extracted. The frozen density in all cases was constructed as a sum of densities of water fragments. For clusters with more than 100 water molecules, rigid water molecules where assumed for all but the 20 water molecules nearest to the embedded acetone. For the largest clusters, the number also includes some acetone molecules from neighboring cells of the CPMD simulation.

Radial distribution function and integral from a MD simulation (GAFF/TIP3P) in comparison to a CPMD simulation of the same system. Top, O(carbonyl)–H(water) distribution function; bottom, O(carbonyl)–O(water) distribution function.

Radial distribution function and integral from a MD simulation (GAFF/TIP3P) in comparison to a CPMD simulation of the same system. Top, O(carbonyl)–H(water) distribution function; bottom, O(carbonyl)–O(water) distribution function.

Simulated (SAOP/TZP/DZ) absorption spectrum of acetone in water. Excitation energies have been calculated for snapshots from a CPMD simulation for acetone vapor (dashed line) or acetone in water (solid line), respectively; all water molecules within a radius of from the acetone molecule have been considered in the latter calculations, and have been treated in a frozen-density fashion. In total, 300 (vapor) or 220 (solution) configurations have been sampled. A Gaussian smearing of has been applied to the peaks in the spectrum. Additionally, the curves of a Gaussian fit to the simulated absorption bands are shown for the gas-phase (dotted line) or solvated (dashed-dotted line) molecule, respectively.

Simulated (SAOP/TZP/DZ) absorption spectrum of acetone in water. Excitation energies have been calculated for snapshots from a CPMD simulation for acetone vapor (dashed line) or acetone in water (solid line), respectively; all water molecules within a radius of from the acetone molecule have been considered in the latter calculations, and have been treated in a frozen-density fashion. In total, 300 (vapor) or 220 (solution) configurations have been sampled. A Gaussian smearing of has been applied to the peaks in the spectrum. Additionally, the curves of a Gaussian fit to the simulated absorption bands are shown for the gas-phase (dotted line) or solvated (dashed-dotted line) molecule, respectively.

## Tables

Excitation energies (SAOP/TZP; in units of eV) of the and valence excitations of acetone in water using the QM/QM embedding scheme for clusters with two water molecules (optimized structure) or 52 water molecules (structure from arbitrary snapshot of a MD simulation). (N.B. Due to the different structures, the 2 and 52 results are not comparable.) For the preparation of the frozen densities, either SAOP/TZP and LDA/TZP (two water molecules) or SAOP/DZ and LDA/DZ (52 water molecules) were applied in combination with different SCF convergence parameters (see text for explanation). Additionally, approximate densities from superpositions of molecular fragments (“mol. frags.”), either taking the sum of the fragment densities (“sumf.”) or the density obtained after one diagonalization of the Fock matrix based on these fragment densities (“diag.”) are employed.

Excitation energies (SAOP/TZP; in units of eV) of the and valence excitations of acetone in water using the QM/QM embedding scheme for clusters with two water molecules (optimized structure) or 52 water molecules (structure from arbitrary snapshot of a MD simulation). (N.B. Due to the different structures, the 2 and 52 results are not comparable.) For the preparation of the frozen densities, either SAOP/TZP and LDA/TZP (two water molecules) or SAOP/DZ and LDA/DZ (52 water molecules) were applied in combination with different SCF convergence parameters (see text for explanation). Additionally, approximate densities from superpositions of molecular fragments (“mol. frags.”), either taking the sum of the fragment densities (“sumf.”) or the density obtained after one diagonalization of the Fock matrix based on these fragment densities (“diag.”) are employed.

Excitation energies (SAOP/TZP/DZ, LDA/DZ for the frozen part; energies in units of eV) of the and valence excitations of acetone in water using the QM/QM embedding scheme for a snapshot from a CPMD simulation, from which subsystems with different numbers of water molecules have been extracted. Scheme A, supermolecule calculations; scheme B, the two nearest water molecules are included in the embedded system, the other water molecules are frozen; scheme C, all water molecules are frozen. We also show the values for the optimized isolated structure for comparison. For the largest cluster with , the frozen density was constructed as a superposition of molecular densities, see text. For the cluster with , both methods to construct the density are compared.

Excitation energies (SAOP/TZP/DZ, LDA/DZ for the frozen part; energies in units of eV) of the and valence excitations of acetone in water using the QM/QM embedding scheme for a snapshot from a CPMD simulation, from which subsystems with different numbers of water molecules have been extracted. Scheme A, supermolecule calculations; scheme B, the two nearest water molecules are included in the embedded system, the other water molecules are frozen; scheme C, all water molecules are frozen. We also show the values for the optimized isolated structure for comparison. For the largest cluster with , the frozen density was constructed as a superposition of molecular densities, see text. For the cluster with , both methods to construct the density are compared.

Shifts in the excitation energies compared to the optimized structure of isolated acetone, calculated from the excitation energies obtained with schemes A, B, and C shown in Table II.

Shifts in the excitation energies compared to the optimized structure of isolated acetone, calculated from the excitation energies obtained with schemes A, B, and C shown in Table II.

Estimations of the solvent shift of acetone in water (compared to the vapor spectrum) using different ways to estimate the peak maxima. All energies given in units of eV. Also given is the number of configurations used to extract the values. The labels have the following meanings: mte, mean transition energy; avg, average transition energy; max, peak maximum in simulated spectrum; max, fit, peak maximum in Gaussian fit to simulated spectrum (recommended).

Estimations of the solvent shift of acetone in water (compared to the vapor spectrum) using different ways to estimate the peak maxima. All energies given in units of eV. Also given is the number of configurations used to extract the values. The labels have the following meanings: mte, mean transition energy; avg, average transition energy; max, peak maximum in simulated spectrum; max, fit, peak maximum in Gaussian fit to simulated spectrum (recommended).

Article metrics loading...

Full text loading...

Commenting has been disabled for this content