# A reduced-scaling density matrix-based method for the computation of the vibrational Hessian matrix at the self-consistent field level

## Abstract

An analytical method to calculate the molecular vibrational Hessian matrix at the self-consistent field level is presented. By analysis of the multipole expansions of the relevant derivatives of Coulomb-type two-electron integral contractions, we show that the effect of the perturbation on the electronic structure due to the displacement of nuclei decays at least as r{sup −2} instead of r{sup −1}. The perturbation is asymptotically local, and the computation of the Hessian matrix can, in principle, be performed with O(N) complexity. Our implementation exhibits linear scaling in all time-determining steps, with some rapid but quadratic-complexity steps remaining. Sample calculations illustrate linear or near-linear scaling in the construction of the complete nuclear Hessian matrix for sparse systems. For more demanding systems, scaling is still considerably sub-quadratic to quadratic, depending on the density of the underlying electronic structure.

- Authors:

- Chair of Theoretical Chemistry, Department of Chemistry, University of Munich (LMU), Butenandtstr. 7, D-81377 München (Germany)

- Publication Date:

- OSTI Identifier:
- 22416199

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Chemical Physics; Journal Volume: 142; Journal Issue: 9; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CALCULATION METHODS; COULOMB FIELD; DENSITY; DENSITY MATRIX; DISTURBANCES; ELECTRONIC STRUCTURE; ELECTRONS; EXPANSION; IMPLEMENTATION; INTEGRALS; SELF-CONSISTENT FIELD

### Citation Formats

```
Kussmann, Jörg, Luenser, Arne, Beer, Matthias, and Ochsenfeld, Christian, E-mail: christian.ochsenfeld@uni-muenchen.de.
```*A reduced-scaling density matrix-based method for the computation of the vibrational Hessian matrix at the self-consistent field level*. United States: N. p., 2015.
Web. doi:10.1063/1.4908131.

```
Kussmann, Jörg, Luenser, Arne, Beer, Matthias, & Ochsenfeld, Christian, E-mail: christian.ochsenfeld@uni-muenchen.de.
```*A reduced-scaling density matrix-based method for the computation of the vibrational Hessian matrix at the self-consistent field level*. United States. doi:10.1063/1.4908131.

```
Kussmann, Jörg, Luenser, Arne, Beer, Matthias, and Ochsenfeld, Christian, E-mail: christian.ochsenfeld@uni-muenchen.de. Sat .
"A reduced-scaling density matrix-based method for the computation of the vibrational Hessian matrix at the self-consistent field level". United States.
doi:10.1063/1.4908131.
```

```
@article{osti_22416199,
```

title = {A reduced-scaling density matrix-based method for the computation of the vibrational Hessian matrix at the self-consistent field level},

author = {Kussmann, Jörg and Luenser, Arne and Beer, Matthias and Ochsenfeld, Christian, E-mail: christian.ochsenfeld@uni-muenchen.de},

abstractNote = {An analytical method to calculate the molecular vibrational Hessian matrix at the self-consistent field level is presented. By analysis of the multipole expansions of the relevant derivatives of Coulomb-type two-electron integral contractions, we show that the effect of the perturbation on the electronic structure due to the displacement of nuclei decays at least as r{sup −2} instead of r{sup −1}. The perturbation is asymptotically local, and the computation of the Hessian matrix can, in principle, be performed with O(N) complexity. Our implementation exhibits linear scaling in all time-determining steps, with some rapid but quadratic-complexity steps remaining. Sample calculations illustrate linear or near-linear scaling in the construction of the complete nuclear Hessian matrix for sparse systems. For more demanding systems, scaling is still considerably sub-quadratic to quadratic, depending on the density of the underlying electronic structure.},

doi = {10.1063/1.4908131},

journal = {Journal of Chemical Physics},

number = 9,

volume = 142,

place = {United States},

year = {Sat Mar 07 00:00:00 EST 2015},

month = {Sat Mar 07 00:00:00 EST 2015}

}