# Development of efficient time-evolution method based on three-term recurrence relation

## Abstract

The advantage of the real-time (RT) propagation method is a direct solution of the time-dependent Schrödinger equation which describes frequency properties as well as all dynamics of a molecular system composed of electrons and nuclei in quantum physics and chemistry. Its applications have been limited by computational feasibility, as the evaluation of the time-evolution operator is computationally demanding. In this article, a new efficient time-evolution method based on the three-term recurrence relation (3TRR) was proposed to reduce the time-consuming numerical procedure. The basic formula of this approach was derived by introducing a transformation of the operator using the arcsine function. Since this operator transformation causes transformation of time, we derived the relation between original and transformed time. The formula was adapted to assess the performance of the RT time-dependent Hartree-Fock (RT-TDHF) method and the time-dependent density functional theory. Compared to the commonly used fourth-order Runge-Kutta method, our new approach decreased computational time of the RT-TDHF calculation by about factor of four, showing the 3TRR formula to be an efficient time-evolution method for reducing computational cost.

- Authors:

- Department of Materials and Life Science, Faculty of Science and Technology, Sophia University, Tokyo 102-8554 (Japan)

- Publication Date:

- OSTI Identifier:
- 22415858

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 142; Journal Issue: 20; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9606

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; CHEMISTRY; COMPARATIVE EVALUATIONS; DENSITY FUNCTIONAL METHOD; ELECTRONS; HARTREE-FOCK METHOD; PERFORMANCE; QUANTUM MECHANICS; RUNGE-KUTTA METHOD; SCHROEDINGER EQUATION; TIME DEPENDENCE

### Citation Formats

```
Akama, Tomoko, E-mail: a.tomo---s-b-l-r@suou.waseda.jp, Kobayashi, Osamu, and Nanbu, Shinkoh, E-mail: shinkoh.nanbu@sophia.ac.jp.
```*Development of efficient time-evolution method based on three-term recurrence relation*. United States: N. p., 2015.
Web. doi:10.1063/1.4921465.

```
Akama, Tomoko, E-mail: a.tomo---s-b-l-r@suou.waseda.jp, Kobayashi, Osamu, & Nanbu, Shinkoh, E-mail: shinkoh.nanbu@sophia.ac.jp.
```*Development of efficient time-evolution method based on three-term recurrence relation*. United States. doi:10.1063/1.4921465.

```
Akama, Tomoko, E-mail: a.tomo---s-b-l-r@suou.waseda.jp, Kobayashi, Osamu, and Nanbu, Shinkoh, E-mail: shinkoh.nanbu@sophia.ac.jp. Thu .
"Development of efficient time-evolution method based on three-term recurrence relation". United States. doi:10.1063/1.4921465.
```

```
@article{osti_22415858,
```

title = {Development of efficient time-evolution method based on three-term recurrence relation},

author = {Akama, Tomoko, E-mail: a.tomo---s-b-l-r@suou.waseda.jp and Kobayashi, Osamu and Nanbu, Shinkoh, E-mail: shinkoh.nanbu@sophia.ac.jp},

abstractNote = {The advantage of the real-time (RT) propagation method is a direct solution of the time-dependent Schrödinger equation which describes frequency properties as well as all dynamics of a molecular system composed of electrons and nuclei in quantum physics and chemistry. Its applications have been limited by computational feasibility, as the evaluation of the time-evolution operator is computationally demanding. In this article, a new efficient time-evolution method based on the three-term recurrence relation (3TRR) was proposed to reduce the time-consuming numerical procedure. The basic formula of this approach was derived by introducing a transformation of the operator using the arcsine function. Since this operator transformation causes transformation of time, we derived the relation between original and transformed time. The formula was adapted to assess the performance of the RT time-dependent Hartree-Fock (RT-TDHF) method and the time-dependent density functional theory. Compared to the commonly used fourth-order Runge-Kutta method, our new approach decreased computational time of the RT-TDHF calculation by about factor of four, showing the 3TRR formula to be an efficient time-evolution method for reducing computational cost.},

doi = {10.1063/1.4921465},

journal = {Journal of Chemical Physics},

issn = {0021-9606},

number = 20,

volume = 142,

place = {United States},

year = {2015},

month = {5}

}