^{1,a)}, Andreas Krapp

^{1}, Maria Francesca Iozzi

^{1}, Vebjørn Bakken

^{1}, Trygve Helgaker

^{1}, Filip Pawłowski

^{2,b)}and Pawel Sałek

^{3,c)}

### Abstract

An efficient, linear-scaling implementation of Kohn–Sham density-functional theory for the calculation of molecular forces for systems containing hundreds of atoms is presented. The density-fitted Coulomb force contribution is calculated in linear time by combining atomic integral screening with the continuous fast multipole method. For higher efficiency and greater simplicity, the near-field Coulomb force contribution is calculated by expanding the solid-harmonic Gaussian basis functions in Hermite rather than Cartesian Gaussians. The efficiency and linear complexity of the molecular-force evaluation is demonstrated by sample calculations and applied to the geometry optimization of a few selected large systems.

We would like to acknowledge the financial support from the Norwegian Research Council through a Strategic University Program in Quantum Chemistry (Grant No. 154011/420), through the Nanomat “Molecular Modeling in Nanotechnology” program (Grant No. 158538/431), and through the CoE Centre for Theoretical and Computational Chemistry (Grant No. 179568/V30). We would like to further acknowledge the NOTUR computing facilities which have been used to conduct the calculations presented in this paper. F.P. would like to acknowledge the financial support from the Foundation for Polish Science (FNP) via Homing program (Grant No. HOM/2008/10B) within EEA Financial Mechanism, which allowed for short visits to Oslo.

I. INTRODUCTION

II. THEORY AND IMPLEMENTATION

A. The density-fitted Coulomb contributions to the energy and gradient

B. The NF Coulomb contribution to the molecular gradient

C. The FF Coulomb contribution to the molecular gradient

D. One-electron contributions to the molecular gradient

E. The exchange-correlation contribution to the molecular gradient

F. Geometry optimization

III. RESULTS AND DISCUSSION

A. Linear polyene chains

B. Sample molecular gradient evaluations

C. Geometry optimizations

IV. CONCLUSIONS

### Key Topics

- Integration
- 9.0
- Density functional theory
- 5.0
- Linear equations
- 5.0
- Kinematics
- 3.0
- Matrix equations
- 3.0

## Figures

(a) The titin-I27 domain highlighting the disulfide bridging bond and a schematic representation of the stretching of the polyprotein strain by the aid of the atomic force microscopy; (b) the titin- model, designed to model the redox-active site in the titin-I27 domain, created from one snapshot (Ref. 58) of titin-I27 during a molecular-dynamics simulation of the force induced unfolding.

(a) The titin-I27 domain highlighting the disulfide bridging bond and a schematic representation of the stretching of the polyprotein strain by the aid of the atomic force microscopy; (b) the titin- model, designed to model the redox-active site in the titin-I27 domain, created from one snapshot (Ref. 58) of titin-I27 during a molecular-dynamics simulation of the force induced unfolding.

Timings at the BP86/6- level of theory for a single construction of the XC and the density-fitted-Coulomb contribution to the Kohn–Sham matrix as a function of the number of carbon atoms for linear polyene chains . The Coulomb timings are separated into timings for the near-field contribution (NF-J), the far-field contributions (FF-J), and the linear equation solver (Linsol). The auxiliary basis set in Ref. 59 was used as the density-fitting basis.

Timings at the BP86/6- level of theory for a single construction of the XC and the density-fitted-Coulomb contribution to the Kohn–Sham matrix as a function of the number of carbon atoms for linear polyene chains . The Coulomb timings are separated into timings for the near-field contribution (NF-J), the far-field contributions (FF-J), and the linear equation solver (Linsol). The auxiliary basis set in Ref. 59 was used as the density-fitting basis.

Timings at the BP86/6- level of theory for the calculation of the molecular gradient as a function of the number of carbon atoms for linear polyene chains . Timings are for the non-Coulomb one-electron contribution (1el), the XC contribution, and the (one- and two-electron) near-field (NF-J) and far-field (FF-J) Coulomb contributions to the gradient. The auxiliary basis set in Ref. 59 was used as the density-fitting basis.

Timings at the BP86/6- level of theory for the calculation of the molecular gradient as a function of the number of carbon atoms for linear polyene chains . Timings are for the non-Coulomb one-electron contribution (1el), the XC contribution, and the (one- and two-electron) near-field (NF-J) and far-field (FF-J) Coulomb contributions to the gradient. The auxiliary basis set in Ref. 59 was used as the density-fitting basis.

## Tables

Timings in seconds for a single construction of the Kohn–Sham matrix and for the calculation of the molecular gradient for a range of molecules, ordered by increasing number of atoms (Atoms), at the BP86 level of theory. The timings are split into contributions from the XC and the density-fitted NF and FF Coulomb contributions. For the force evaluation, we also give the one-electron kinetic-energy and reorthonormalization contribution (1el). Also listed are the number of primitive (Prim) and contracted (Cont) basis functions. The auxiliary basis set in Ref. 59 was used in all cases, except otherwise stated.

Timings in seconds for a single construction of the Kohn–Sham matrix and for the calculation of the molecular gradient for a range of molecules, ordered by increasing number of atoms (Atoms), at the BP86 level of theory. The timings are split into contributions from the XC and the density-fitted NF and FF Coulomb contributions. For the force evaluation, we also give the one-electron kinetic-energy and reorthonormalization contribution (1el). Also listed are the number of primitive (Prim) and contracted (Cont) basis functions. The auxiliary basis set in Ref. 59 was used in all cases, except otherwise stated.

Average timings in seconds for the electronic energy, the molecular gradient, and the geometry step for the geometry optimizations of the taxol, valinomycin, and titin-. All calculations were carried out at the BP86/6-31G level of theory with the auxiliary basis set in Ref. 59. Also reported are the average number of SCF iterations in each energy optimization, with the root-mean-square SCF gradient norm converged to , and the number of geometry steps needed to converge the geometry, with threshold (see text).

Average timings in seconds for the electronic energy, the molecular gradient, and the geometry step for the geometry optimizations of the taxol, valinomycin, and titin-. All calculations were carried out at the BP86/6-31G level of theory with the auxiliary basis set in Ref. 59. Also reported are the average number of SCF iterations in each energy optimization, with the root-mean-square SCF gradient norm converged to , and the number of geometry steps needed to converge the geometry, with threshold (see text).

Article metrics loading...

Full text loading...

Commenting has been disabled for this content