## This content will become publicly available on April 18, 2020

## On the numerical solution of the exact factorization equations

## Abstract

The exact factorization (EF) approach to coupled electron-ion dynamics recasts the time-dependent molecular Schrödinger equation as two coupled equations, one for the nuclear wavefunction and one for the conditional electronic wavefunction. The potentials appearing in these equations have provided insight into non-adiabatic processes, and new practical non-adiabatic dynamics methods have been formulated starting from these equations. We provide a first demonstration of a self-consistent solution of the exact equations, with a preliminary analysis of their stability and convergence properties. The equations have an unprecedented mathematical form, involving a Hamiltonian outside the class of Hermitian Hamiltonians usually encountered in time-propagation, and so the usual numerical methods for time-dependent Schrödinger fail when applied in a straightforward way to the EF equations. We find an approach that enables stable propagation long enough to witness non-adiabatic behavior in a model system before non-trivial instabilities take over. Implications for the development and analysis of EF-based methods are discussed.

- Authors:

- Hunter College, New York, NY (United States). Dept. of Physics and Astronomy
- Hunter College, New York, NY (United States). Dept. of Physics and Astronomy; City Univ. of New York, NY (United States). Graduate Center. Physics Program. Chemistry Program

- Publication Date:

- Research Org.:
- Hunter College, New York, NY (United States)

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

- OSTI Identifier:
- 1507622

- Alternate Identifier(s):
- OSTI ID: 1507620

- Grant/Contract Number:
- SC0015344

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 150; Journal Issue: 15; Journal ID: ISSN 0021-9606

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; quantum chemical dynamics; time-dependent Schrodinger equation; non-adiabatic molecular dynamics; potential energy surfaces

### Citation Formats

```
Gossel, Graeme H., Lacombe, Lionel, and Maitra, Neepa T.. On the numerical solution of the exact factorization equations. United States: N. p., 2019.
Web. doi:10.1063/1.5090802.
```

```
Gossel, Graeme H., Lacombe, Lionel, & Maitra, Neepa T.. On the numerical solution of the exact factorization equations. United States. doi:10.1063/1.5090802.
```

```
Gossel, Graeme H., Lacombe, Lionel, and Maitra, Neepa T.. Thu .
"On the numerical solution of the exact factorization equations". United States. doi:10.1063/1.5090802.
```

```
@article{osti_1507622,
```

title = {On the numerical solution of the exact factorization equations},

author = {Gossel, Graeme H. and Lacombe, Lionel and Maitra, Neepa T.},

abstractNote = {The exact factorization (EF) approach to coupled electron-ion dynamics recasts the time-dependent molecular Schrödinger equation as two coupled equations, one for the nuclear wavefunction and one for the conditional electronic wavefunction. The potentials appearing in these equations have provided insight into non-adiabatic processes, and new practical non-adiabatic dynamics methods have been formulated starting from these equations. We provide a first demonstration of a self-consistent solution of the exact equations, with a preliminary analysis of their stability and convergence properties. The equations have an unprecedented mathematical form, involving a Hamiltonian outside the class of Hermitian Hamiltonians usually encountered in time-propagation, and so the usual numerical methods for time-dependent Schrödinger fail when applied in a straightforward way to the EF equations. We find an approach that enables stable propagation long enough to witness non-adiabatic behavior in a model system before non-trivial instabilities take over. Implications for the development and analysis of EF-based methods are discussed.},

doi = {10.1063/1.5090802},

journal = {Journal of Chemical Physics},

number = 15,

volume = 150,

place = {United States},

year = {2019},

month = {4}

}