### Accelerating molecular property calculations with nonorthonormal Krylov space methods

Here, we formulate Krylov space methods for large eigenvalue problems and linear equation systems that take advantage of decreasing residual norms to reduce the cost of matrix-vector multiplication. The residuals are used as subspace basis without prior orthonormalization, which leads to generalized eigenvalue problems or linear equation systems on the Krylov space. These nonorthonormal Krylov space (nKs) algorithms are favorable for large matrices with irregular sparsity patterns whose elements are computed on the fly, because fewer operations are necessary as the residual norm decreases as compared to the conventional method, while errors in the desired eigenpairs and solution vectors remain small. We consider real symmetric and symplectic eigenvalue problems as well as linear equation systems and Sylvester equations as they appear in configuration interaction and response theory. The nKs method can be implemented in existing electronic structure codes with minor modifications and yields speed-ups of 1.2-1.8 in typical time-dependent Hartree-Fock and density functional applications without accuracy loss. The algorithm can compute entire linear subspaces simultaneously which benefits electronic spectra and force constant calculations requiring many eigenpairs or solution vectors. The nKs approach is related to difference density methods in electronic ground state calculations, and particularly efficient for integral direct computationsmore »

- Publication Date:

- Grant/Contract Number:
- SC0008694

- Type:
- Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 144; Journal Issue: 17; Journal ID: ISSN 0021-9606

- Publisher:
- American Institute of Physics (AIP)

- Research Org:
- Univ. of California, Irvine, CA (United States)

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

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 74 ATOMIC AND MOLECULAR PHYSICS; eigenvalues; subspaces; linear equations; excitation energies; ground states

- OSTI Identifier:
- 1250890

- Alternate Identifier(s):
- OSTI ID: 1250585

```
Furche, Filipp, Krull, Brandon T., Nguyen, Brian D., and Kwon, Jake.
```*Accelerating molecular property calculations with nonorthonormal Krylov space methods*. United States: N. p.,
Web. doi:10.1063/1.4947245.

```
Furche, Filipp, Krull, Brandon T., Nguyen, Brian D., & Kwon, Jake.
```*Accelerating molecular property calculations with nonorthonormal Krylov space methods*. United States. doi:10.1063/1.4947245.

```
Furche, Filipp, Krull, Brandon T., Nguyen, Brian D., and Kwon, Jake. 2016.
"Accelerating molecular property calculations with nonorthonormal Krylov space methods". United States.
doi:10.1063/1.4947245. https://www.osti.gov/servlets/purl/1250890.
```

```
@article{osti_1250890,
```

title = {Accelerating molecular property calculations with nonorthonormal Krylov space methods},

author = {Furche, Filipp and Krull, Brandon T. and Nguyen, Brian D. and Kwon, Jake},

abstractNote = {Here, we formulate Krylov space methods for large eigenvalue problems and linear equation systems that take advantage of decreasing residual norms to reduce the cost of matrix-vector multiplication. The residuals are used as subspace basis without prior orthonormalization, which leads to generalized eigenvalue problems or linear equation systems on the Krylov space. These nonorthonormal Krylov space (nKs) algorithms are favorable for large matrices with irregular sparsity patterns whose elements are computed on the fly, because fewer operations are necessary as the residual norm decreases as compared to the conventional method, while errors in the desired eigenpairs and solution vectors remain small. We consider real symmetric and symplectic eigenvalue problems as well as linear equation systems and Sylvester equations as they appear in configuration interaction and response theory. The nKs method can be implemented in existing electronic structure codes with minor modifications and yields speed-ups of 1.2-1.8 in typical time-dependent Hartree-Fock and density functional applications without accuracy loss. The algorithm can compute entire linear subspaces simultaneously which benefits electronic spectra and force constant calculations requiring many eigenpairs or solution vectors. The nKs approach is related to difference density methods in electronic ground state calculations, and particularly efficient for integral direct computations of exchange-type contractions. By combination with resolution-of-the-identity methods for Coulomb contractions, three- to fivefold speed-ups of hybrid time-dependent density functional excited state and response calculations are achieved.},

doi = {10.1063/1.4947245},

journal = {Journal of Chemical Physics},

number = 17,

volume = 144,

place = {United States},

year = {2016},

month = {5}

}