*in situ*using a psinc basis set

^{1}, Nicholas D. M. Hine

^{2}and Chris-Kriton Skylaris

^{1,a)}

### Abstract

*In situ*optimization of a set of localized orbitals with respect to a systematically improvable basis set independent of the position of the atoms, such as psinc functions, would theoretically eliminate the correction due to Pulay forces from the total ionic forces. We demonstrate that for strict localization constraints, especially with small localization regions, there can be non-negligible Pulay forces that must be calculated as a correction to the Hellmann-Feynman forces in the ground state. Geometry optimization calculations, which rely heavily upon accurate evaluation of the total ionic forces, show much better convergence when Pulay forces are included. The more conventional case, where the local orbitals remain fixed to pseudo-atomic orbital multiple-ζ basis sets, also benefits from this implementation. We have validated the method on several test cases, including a DNA fragment with 1045 atoms.

A.R.S. acknowledges the support of the Engineering and Physical Sciences Research Consul (U.K.) (EPSRC (GB)) (Grant No. EP/F038038/1) for a high end computing studentship through the UK Car-Parrinello consortium. N.D.M.H. acknowledges the support EPSRC (Grant No. EP/G05567X/1) for postdoctoral funding through the HPC software development call 2008/2009. C.-K.S. acknowledges support from the Royal Society in the form of a University Research Fellowship. The authors are grateful for the computing resources provided by Southampton University's iSolutions unit (Iridis3 supercomputer) which have enabled all the calculations presented here.

I. INTRODUCTION

II. ENERGY MINIMIZATION

III. IONIC FORCES

IV. RESULTS

A. Convergence of the forces

B. Geometry optimization using NGWFs

C. Geometry optimization using PAOs

D. Geometry optimization of large systems

V. CONCLUSIONS

### Key Topics

- Basis sets
- 13.0
- Ground states
- 8.0
- Hydrogen bonding
- 8.0
- Density functional theory
- 6.0
- DNA
- 6.0

## Figures

Convergence of the forces with respect to the kinetic energy cutoff and the NGWF radius. Left: force acting on an oxygen atom in CO_{2} along the covalent bond. Right: force acting on a hydrogen atom in H_{2}O dimer along the hydrogen bond. The first row corresponds to the Hellmann-Feynman force, the second to the Pulay force, and the third are the Pulay-corrected (PC) total forces (Hellmann-Feynman forces plus Pulay). *R* refers to the NGWF radii.

Convergence of the forces with respect to the kinetic energy cutoff and the NGWF radius. Left: force acting on an oxygen atom in CO_{2} along the covalent bond. Right: force acting on a hydrogen atom in H_{2}O dimer along the hydrogen bond. The first row corresponds to the Hellmann-Feynman force, the second to the Pulay force, and the third are the Pulay-corrected (PC) total forces (Hellmann-Feynman forces plus Pulay). *R* refers to the NGWF radii.

Egg-box effect on the energy (top) and force along the covalent bond (bottom) of a CO_{2} molecule with SZ (red circles), DZP (blue squares), and TZDP (green diamonds) PAO basis sets and NGWFs (black triangles).

Egg-box effect on the energy (top) and force along the covalent bond (bottom) of a CO_{2} molecule with SZ (red circles), DZP (blue squares), and TZDP (green diamonds) PAO basis sets and NGWFs (black triangles).

Convergence of the maximum absolute value of the force during geometry optimization of the adenine-thymine DNA base pair with NGWF radii of 3.70 Å (top) and 7.94 Å (bottom). The convergence threshold was 0.015 eV/Å.

Convergence of the maximum absolute value of the force during geometry optimization of the adenine-thymine DNA base pair with NGWF radii of 3.70 Å (top) and 7.94 Å (bottom). The convergence threshold was 0.015 eV/Å.

Convergence of the binding energy in the adenine-thymine complex with increasing NGWF radii. The calculations that use structures obtained with Pulay-corrected forces show systematic convergence (blue squares) with increasing NGWF radius. In contrast, the calculations with structures obtained with uncorrected Hellmann-Feynman forces (red circles) lack of convergence and show large variations of the binding energy. The calculations represented with black crosses show convergence of the binding energy with increasing NGWF radius. They are based on fixed molecular geometries obtained with Pulay-corrected forces and NGWF radius of 3.70 and 7.94 Å.

Convergence of the binding energy in the adenine-thymine complex with increasing NGWF radii. The calculations that use structures obtained with Pulay-corrected forces show systematic convergence (blue squares) with increasing NGWF radius. In contrast, the calculations with structures obtained with uncorrected Hellmann-Feynman forces (red circles) lack of convergence and show large variations of the binding energy. The calculations represented with black crosses show convergence of the binding energy with increasing NGWF radius. They are based on fixed molecular geometries obtained with Pulay-corrected forces and NGWF radius of 3.70 and 7.94 Å.

Convergence of the maximum force during the BFGS geometry optimization of the tennis-ball dimer using Hellmann-Feynman forces (top left molecule) and Pulay-corrected forces (top right).

Convergence of the maximum force during the BFGS geometry optimization of the tennis-ball dimer using Hellmann-Feynman forces (top left molecule) and Pulay-corrected forces (top right).

Convergence of the maximum value of the force during the geometry optimization of the DNA molecule using NGWFs and DZP and TZDP PAO basis sets.

Convergence of the maximum value of the force during the geometry optimization of the DNA molecule using NGWFs and DZP and TZDP PAO basis sets.

## Tables

Geometry optimization of adenine-thymine using ONETEP for different localization radii *R* _{α} with HF forces and PC forces. Results show the hydrogen bond lengths, the maximum absolute value of the force, |*F*|_{max}, and the number of BFGS steps required.

Geometry optimization of adenine-thymine using ONETEP for different localization radii *R* _{α} with HF forces and PC forces. Results show the hydrogen bond lengths, the maximum absolute value of the force, |*F*|_{max}, and the number of BFGS steps required.

Length of the two hydrogen bonds in the adenine-thymine complex obtained with different PAO multiple-ζ basis sets of different localization radius, *R*, given in Å. The total forces converged to 0.015 Å/eV.

Length of the two hydrogen bonds in the adenine-thymine complex obtained with different PAO multiple-ζ basis sets of different localization radius, *R*, given in Å. The total forces converged to 0.015 Å/eV.

Structural parameters of the optimized 1045 atom DNA fragment as optimized by ONETEP. N⋯H(4), N⋯H(8), and N⋯H(12) correspond to the hydrogen bond of the fourth, eighth, and twelfth pairs, respectively, that involve a nitrogen atom.

Structural parameters of the optimized 1045 atom DNA fragment as optimized by ONETEP. N⋯H(4), N⋯H(8), and N⋯H(12) correspond to the hydrogen bond of the fourth, eighth, and twelfth pairs, respectively, that involve a nitrogen atom.

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