# Multiresolution quantum chemistry in multiwavelet bases: excited states from time-dependent Hartree–Fock and density functional theory via linear response

## Abstract

Using the fully numerical method for time-dependent Hartree–Fock and density functional theory (TD-HF/DFT) with the Tamm–Dancoff (TD) approximation we use a multiresolution analysis (MRA) approach to present our findings. From a reformulation with effective use of the density matrix operator, we obtain a general form of the HF/DFT linear response equation in the first quantization formalism. It can be readily rewritten as an integral equation with the bound-state Helmholtz (BSH) kernel for the Green's function. The MRA implementation of the resultant equation permits excited state calculations without virtual orbitals. Moreover, the integral equation is efficiently and adaptively solved using a numerical multiresolution solver with multiwavelet bases. Our implementation of the TD-HF/DFT methods is applied for calculating the excitation energies of H _{2}, Be, N _{2}, H _{2}O, and C _{2}H _{4} molecules. The numerical errors of the calculated excitation energies converge in proportion to the residuals of the equation in the molecular orbitals and response functions. The energies of the excited states at a variety of length scales ranging from short-range valence excitations to long-range Rydberg-type ones are consistently accurate. It is shown that the multiresolution calculations yield the correct exponential asymptotic tails for the response functions, whereas those computedmore »

- Authors:

- Inst. for Molecular Science, Aichi (Japan)
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- Univ. of Colorado, Boulder, CO (United States)
- Stony Brook Univ., NY (United States); Brookhaven National Lab. (BNL), Upton, NY (United States)

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1265805

- Grant/Contract Number:
- AC05-00OR22725; MDA972-00-1-0016; ACI-0082982; DMS-0219326; AC03-76SF0098

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physical Chemistry Chemical Physics. PCCP (Print)

- Additional Journal Information:
- Journal Name: Physical Chemistry Chemical Physics. PCCP (Print); Journal Volume: 17; Journal Issue: 47; Journal ID: ISSN 1463-9076

- Publisher:
- Royal Society of Chemistry

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY

### Citation Formats

```
Yanai, Takeshi, Fann, George I., Beylkin, Gregory, and Harrison, Robert J.
```*Multiresolution quantum chemistry in multiwavelet bases: excited states from time-dependent Hartree–Fock and density functional theory via linear response*. United States: N. p., 2015.
Web. doi:10.1039/C4CP05821F.

```
Yanai, Takeshi, Fann, George I., Beylkin, Gregory, & Harrison, Robert J.
```*Multiresolution quantum chemistry in multiwavelet bases: excited states from time-dependent Hartree–Fock and density functional theory via linear response*. United States. doi:10.1039/C4CP05821F.

```
Yanai, Takeshi, Fann, George I., Beylkin, Gregory, and Harrison, Robert J. Wed .
"Multiresolution quantum chemistry in multiwavelet bases: excited states from time-dependent Hartree–Fock and density functional theory via linear response". United States.
doi:10.1039/C4CP05821F. https://www.osti.gov/servlets/purl/1265805.
```

```
@article{osti_1265805,
```

title = {Multiresolution quantum chemistry in multiwavelet bases: excited states from time-dependent Hartree–Fock and density functional theory via linear response},

author = {Yanai, Takeshi and Fann, George I. and Beylkin, Gregory and Harrison, Robert J.},

abstractNote = {Using the fully numerical method for time-dependent Hartree–Fock and density functional theory (TD-HF/DFT) with the Tamm–Dancoff (TD) approximation we use a multiresolution analysis (MRA) approach to present our findings. From a reformulation with effective use of the density matrix operator, we obtain a general form of the HF/DFT linear response equation in the first quantization formalism. It can be readily rewritten as an integral equation with the bound-state Helmholtz (BSH) kernel for the Green's function. The MRA implementation of the resultant equation permits excited state calculations without virtual orbitals. Moreover, the integral equation is efficiently and adaptively solved using a numerical multiresolution solver with multiwavelet bases. Our implementation of the TD-HF/DFT methods is applied for calculating the excitation energies of H2, Be, N2, H2O, and C2H4 molecules. The numerical errors of the calculated excitation energies converge in proportion to the residuals of the equation in the molecular orbitals and response functions. The energies of the excited states at a variety of length scales ranging from short-range valence excitations to long-range Rydberg-type ones are consistently accurate. It is shown that the multiresolution calculations yield the correct exponential asymptotic tails for the response functions, whereas those computed with Gaussian basis functions are too diffuse or decay too rapidly. Finally, we introduce a simple asymptotic correction to the local spin-density approximation (LSDA) so that in the TDDFT calculations, the excited states are correctly bound.},

doi = {10.1039/C4CP05821F},

journal = {Physical Chemistry Chemical Physics. PCCP (Print)},

number = 47,

volume = 17,

place = {United States},

year = {Wed Feb 25 00:00:00 EST 2015},

month = {Wed Feb 25 00:00:00 EST 2015}

}

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