^{1,2}, Robert Balawender

^{2}and Janusz Zachara

^{1,a)}

### Abstract

We present an analytical approach to treat higher order derivatives of Hartree-Fock (HF) and Kohn-Sham (KS) density functional theory energy in the Born-Oppenheimer approximation with respect to the nuclear charge distribution (so-called *alchemical derivatives*). Modified coupled perturbed self-consistent field theory is used to calculate molecular systems response to the applied perturbation. Working equations for the second and the third derivatives of HF/KS energy are derived. Similarly, analytical forms of the first and second derivatives of orbital energies are reported. The second derivative of Kohn-Sham energy and up to the third derivative of Hartree-Fock energy with respect to the nuclear charge distribution were calculated. Some issues of practical calculations, in particular the dependence of the basis set and Becke weighting functions on the perturbation, are considered. For selected series of isoelectronic molecules values of available alchemical derivatives were computed and Taylor series expansion was used to predict energies of the “surrounding” molecules. Predicted values of energies are in unexpectedly good agreement with the ones computed using HF/KS methods. Presented method allows one to predict orbital energies with the error less than 1% or even smaller for valence orbitals.

This work was partially supported by the Ministry of Science and Higher Education of Poland through Grant No. N204275939 and by the Warsaw University of Technology. The computations were carried out at the Computer Center of the Faculty of Chemistry, University of Warsaw. Valuable suggestions on the manuscript and fruitful discussions with P. A. Guńka are gratefully acknowledged.

I. INTRODUCTION

II. THEORY

A. Notation

B. Alchemical derivatives

C. Derivatives of Kohn-Sham DFT energy

D. Application of coupled perturbed Kohn-Sham theory

E. The derivatives of selected properties

III. WEIGHTS’ DERIVATIVES

IV. NUMERICAL EXAMPLES

A. Computational details

B. Results and discussion

V. CONCLUSIONS

### Key Topics

- Density functional theory
- 16.0
- General molecular properties
- 7.0
- Matrix equations
- 7.0
- Perturbation methods
- 6.0
- Basis sets
- 5.0

## Tables

Values of Bragg-Slater radii and its derivatives for hydrogen and 2nd and 3rd rows of periodic table.

Values of Bragg-Slater radii and its derivatives for hydrogen and 2nd and 3rd rows of periodic table.

Alchemical derivatives computed for Li_{3}N molecule at PBE/aug-cc-pVDZ level of theory, using numerical atomic grids of different quality. Geometry of Li_{3}N was optimized using the biggest grid. Results computed *with* the inclusion of weighting functions' derivatives are compared to those *without* them. Values in boldface type come from very dense grid and are considered exact.

Alchemical derivatives computed for Li_{3}N molecule at PBE/aug-cc-pVDZ level of theory, using numerical atomic grids of different quality. Geometry of Li_{3}N was optimized using the biggest grid. Results computed *with* the inclusion of weighting functions' derivatives are compared to those *without* them. Values in boldface type come from very dense grid and are considered exact.

PBE0 ground state energies (*E* _{0}) together with the values of the first and the second derivatives with respect to charge of the central atom. Predicted values of energies of molecules with the charge of central nuclei changed by ±1. All values are given in the atomic units.

PBE0 ground state energies (*E* _{0}) together with the values of the first and the second derivatives with respect to charge of the central atom. Predicted values of energies of molecules with the charge of central nuclei changed by ±1. All values are given in the atomic units.

Hartree-Fock ground state energies (*E* _{0}) together with the values of the first, the second, and the third derivatives with respect to charge of the central atom. Predicted values of energies of molecules with the charge of central nuclei changed by ±1. All values are given in the atomic units.

Hartree-Fock ground state energies (*E* _{0}) together with the values of the first, the second, and the third derivatives with respect to charge of the central atom. Predicted values of energies of molecules with the charge of central nuclei changed by ±1. All values are given in the atomic units.

Orbital energies of CH_{4} molecule along with their derivatives (*T* _{d} symmetry point group). Predicted values of orbital energies of NH_{4} ^{+} cations are compared to ones computed with DFT using the geometry fixed to CH_{4}. Only symmetry unique orbitals are included. Results from PBE0/aug-cc-pVTZ level of theory. All values are given in atomic units, unless stated otherwise.

Orbital energies of CH_{4} molecule along with their derivatives (*T* _{d} symmetry point group). Predicted values of orbital energies of NH_{4} ^{+} cations are compared to ones computed with DFT using the geometry fixed to CH_{4}. Only symmetry unique orbitals are included. Results from PBE0/aug-cc-pVTZ level of theory. All values are given in atomic units, unless stated otherwise.

Orbital energies of CO_{2} molecule along with their derivatives (*D* _{∞h } symmetry point group). Predicted values of orbital energies of NO_{2} ^{+} cations are compared to ones computed with DFT using the geometry fixed to CO_{2}. Only symmetry unique orbitals are included. Results from PBE0/aug-cc-pVTZ level of theory. All values are given in atomic units, unless stated otherwise.

Orbital energies of CO_{2} molecule along with their derivatives (*D* _{∞h } symmetry point group). Predicted values of orbital energies of NO_{2} ^{+} cations are compared to ones computed with DFT using the geometry fixed to CO_{2}. Only symmetry unique orbitals are included. Results from PBE0/aug-cc-pVTZ level of theory. All values are given in atomic units, unless stated otherwise.

Orbital energies of CN^{−} anions along with their derivatives (*C* _{∞v } symmetry point group). Predicted values of orbital energies of CO molecules are compared to ones computed with DFT using the geometry fixed to CN^{−}. Only symmetry unique orbitals were included. Results from PBE0/aug-cc-pVTZ level of theory. All values are given in atomic units, unless stated otherwise.

Orbital energies of CN^{−} anions along with their derivatives (*C* _{∞v } symmetry point group). Predicted values of orbital energies of CO molecules are compared to ones computed with DFT using the geometry fixed to CN^{−}. Only symmetry unique orbitals were included. Results from PBE0/aug-cc-pVTZ level of theory. All values are given in atomic units, unless stated otherwise.

Orbital energies of cyclopentadienyl anions (C_{5}H_{5} ^{−}≡Cp^{−}) along with their derivatives (*D* _{5h } symmetry point group). Predicted values of orbital energies of pirole molecule (C_{4}H_{4}NH, *C* _{2v } symmetry point group) are compared to ones computed with DFT using the geometry fixed to Cp^{−}. Only those orbitals for which orbital energies splitting occurs were included. Results from PBE0/aug-cc-pVTZ level of theory. All values are given in atomic units, unless stated otherwise.

Orbital energies of cyclopentadienyl anions (C_{5}H_{5} ^{−}≡Cp^{−}) along with their derivatives (*D* _{5h } symmetry point group). Predicted values of orbital energies of pirole molecule (C_{4}H_{4}NH, *C* _{2v } symmetry point group) are compared to ones computed with DFT using the geometry fixed to Cp^{−}. Only those orbitals for which orbital energies splitting occurs were included. Results from PBE0/aug-cc-pVTZ level of theory. All values are given in atomic units, unless stated otherwise.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content