# LIBGRPP: A Library for the Evaluation of Molecular Integrals of the Generalized Relativistic Pseudopotential Operator over Gaussian Functions

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Generalized Relativistic Pseudopotentials

- An approximate nature of the many-electron Hamiltonian used to evaluate atomic spinors, which in turn define the potentials ${U}_{nlj}\left(r\right)$. The construction of modern GRPPs is based on atomic four-component all-electron calculations with the Dirac–Coulomb–Breit Hamiltonian, employing Fermi nuclear charge distribution, and accounting for the quantum electrodynamic correction [28] by means of the Lamb shift model potential [18,64];
- The neglect of correlations between excluded and explicitly treated electrons and inner core polarization and smoothing of pseudo wavefunctions in the inner core area. The corresponding errors naturally decrease while reducing the number of excluded electronic shells (so-called tiny-core and empty-core versions of GRPPs [28,58]);
- A roughly approximate mean-field-like simulation of Breit interactions between the explicitly treated electrons by the corresponding contributions to one-electron GRPPs. In principle, this factor can limit the feasibility of core size reduction for heavy atoms.

#### 2.2. Scalar-Relativistic Part: Integrals over the Local Potential

#### 2.3. Scalar-Relativistic Part: Integrals with Angular Projectors

#### 2.4. Integrals over the Effective Spin-Orbit Interaction Operator

#### 2.5. Integrals over Non-Local Terms of GRPP

## 3. The LIBGRPP Library

`libgrpp_shell_t`type (see Figure 5a). Each shell is attached to some point in space, normally coinciding with the atom to which this batch of basis functions belongs. Quite a similar data structure

`libgrpp_potential_t`is provided to represent components of a pseudopotential (see Figure 5b). LIBGRPP also contains “constructor” and “destructor” routines to simplify, respectively, construction and deallocation of objects of these two basic data types. All data structures and subroutines of LIBGRPP start with the

`libgrpp_`prefix. After the objects representing atom-centered shells of basis functions and a pseudopotential operator have been created, integrals for a given shell pair are to be calculated. For this purpose, special subroutines representing different terms in Equation (6) are provided. The resulting integrals between Cartesian components are packed into a one-dimensional array, which is assumed to be pre-allocated (see Figure 6).

## 4. Pilot Applications

#### 4.1. Electronic States of the ThO Molecule

#### 4.2. Electronic States of the UO${}_{2}$ Molecule

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FS-RCCSD | Fock space relativistic coupled cluster method with single and double excitations |

GRPP | Generalized relativistic pseudopotential |

IH | Intermediate Hamiltonian |

QED | Quantum electrodynamics |

SO | Spin-orbit |

v-RPP | valence (semilocal) part of GRPP |

## Appendix A. Analytic Gradients of GRPP Integrals

## Appendix B. Obara-Saika Recurrence Relations for the Local Part of the GRPP Operator

## Appendix C. Analytic Evaluation of One-Center RPP Integrals

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**Figure 1.**Coordinate system used to evaluate pseudopotential integrals. Gaussian basis functions ${\varphi}_{A}$ and ${\varphi}_{B}$ are re-expanded at the point $\mathit{C}$, where the center of a pseudopotential is located.

**Figure 2.**Plots of modified spherical Bessel function (

**a**) and its exponentially scaled counterpart (

**b**). ${I}_{n+1/2}\left(x\right)$ stands for a modified Bessel function of the first kind.

**Figure 3.**Flowchart of the algorithm used for integration of the semilocal scalar part (type 2 integrals).

**Figure 5.**(

**a**) Data structure representing a shell of contracted Gaussian basis functions. The

`cart_list`field contains a pointer to an array in which all possible Cartesian combinations with the given angular momentum L are stored. (

**b**) Data structure representing the component ${U}_{nlj}\left(r\right)$ of the GRPP operator. The field

`J`is not used for local and semilocal terms of GRPP;

`L`is not used for local terms.

**Figure 6.**(

**a**) Declaration of the LIBGRPP subroutine designed to evaluate type 1 integrals (over the local part of RPP). Other LIBGRPP subroutines have essentially the same interface. Matrix elements between primitive Gaussians with different Cartesian parts are packed into a one-dimensional array

`matrix`of type

`double`(linear indices of each matrix element are given inside the cells). (

**b**) The array of calculated RPP matrix elements exemplified for the case of the d-f shell pair.

**Figure 7.**

**Top**: Deviations of IH-FS-RCC vertical excitation energies (${T}_{v}$) in ThO computed within the GRPP/Gaunt model and its semilocal “valence’’ component (v-RPP) from their counterparts obtained with all-electron Dirac–Coulomb–Gaunt Hamiltonian, ${T}_{v}\left(\mathrm{AE}\phantom{\rule{0.277778em}{0ex}}\mathrm{DCG}\right)$.

**Bottom**: contributions of Gaunt interactions ($\Delta $(Gaunt)) and retardation plus QED effects ($\Delta $(R+QED)) to ${T}_{v}$.

**Figure 8.**

**Top**: Deviations of IH-FS-RCC vertical excitation energies (${T}_{v}$) in UO${}_{2}$ computed within the GRPP/Gaunt model and its semilocal “valence’’ component (v-RPP) from their counterparts obtained with all-electron Dirac–Coulomb–Gaunt Hamiltonian, ${T}_{v}\left(\mathrm{AE}\phantom{\rule{0.277778em}{0ex}}\mathrm{DCG}\right)$.

**Bottom**: contributions of Gaunt interactions ($\Delta $(Gaunt)) and retardation plus QED effects ($\Delta $(R+QED)) to ${T}_{v}$.

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## Share and Cite

**MDPI and ACS Style**

Oleynichenko, A.V.; Zaitsevskii, A.; Mosyagin, N.S.; Petrov, A.N.; Eliav, E.; Titov, A.V. LIBGRPP: A Library for the Evaluation of Molecular Integrals of the Generalized Relativistic Pseudopotential Operator over Gaussian Functions. *Symmetry* **2023**, *15*, 197.
https://doi.org/10.3390/sym15010197

**AMA Style**

Oleynichenko AV, Zaitsevskii A, Mosyagin NS, Petrov AN, Eliav E, Titov AV. LIBGRPP: A Library for the Evaluation of Molecular Integrals of the Generalized Relativistic Pseudopotential Operator over Gaussian Functions. *Symmetry*. 2023; 15(1):197.
https://doi.org/10.3390/sym15010197

**Chicago/Turabian Style**

Oleynichenko, Alexander V., Andréi Zaitsevskii, Nikolai S. Mosyagin, Alexander N. Petrov, Ephraim Eliav, and Anatoly V. Titov. 2023. "LIBGRPP: A Library for the Evaluation of Molecular Integrals of the Generalized Relativistic Pseudopotential Operator over Gaussian Functions" *Symmetry* 15, no. 1: 197.
https://doi.org/10.3390/sym15010197