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Theoretical Chemistry Accounts (2019) 138:9

https://doi.org/10.1007/s00214-018-2386-x

REGULAR ARTICLE

Atomic orbitals revisited: generalized hydrogen‑like basis sets

for2nd‑row elements

IlyaV.Popov1,2· AndreiL.Tchougrée1,2,3

Received: 16 June 2018 / Accepted: 15 November 2018 / Published online: 4 December 2018

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract

In the present work, we revisit the problem of atomic orbitals from the positions mostly dictated by semiempirical approaches

in quantum chemistry. To construct basis set, having proper nodal structure and simple functional form of orbitals and repre-

senting atomic properties with reasonable accuracy, authors propose an Ansatz based on gradual improvement of hydrogen

atomic orbitals. According to it, several basis sets with diﬀerent numbers of variable parameters are considered and forms

of orbitals are obtained for the 2nd-row elements either by minimization of their ground state energy (direct problem) or by

extracting from atomic spectra (inverse problem). It is shown that so-derived three- and four-parametric basis sets provide

accurate description of atomic properties, being, however, substantially provident for computational requirements and, what

is more important, simple to handle in analytic models of quantum chemistry. Since the discussed Ansatz allows a generaliza-

tion for heavier atoms, our results may be considered not only as a solution for light elements, but also as a proof of concept

with possible further extension to a wider range of elements.

Keywords Atomic orbitals· Atoms· Analytic models· Semiempirical methods

1 Introduction

Quantum chemical description of molecules involves one-

electron states, expanded against ﬁnite sets of basis func-

tions. Quality and eﬃciency of electronic structure calcula-

tions and f eatures of their computational implementation

substantially rely on the properties of the underlying basis

set. This fact is emphasized in almost each handbook on

quantum chemistry (see, e.g., [1]) and reﬂected in the huge

number of basis sets available in the literature for description

of various objects and properties [2, 3]. Two major classes

of basis functions (coming from two main types of objects in

theoretical chemistry) are local (atomic) basis functions and

plane waves. Although using the latter is extremely eﬃcient

numerically, the incurred loss of local chemical information

generated demand for a posteriori analysis tools projecting

the results obtained in the plane wave basis onto local basis

sets as successfully implemented, e.g., in the LOBSTER

package [4–6]. In this work, we focus only on local atomic

orbitals, and thus our further discussion will be restricted

to them.

Atomic functions appear in either numerical (tabular) or

analytic form [1]. Numerical atomic orbitals (for example,

Ref.[7]) come from accurate ab initio calculations on many-

electron atoms, but their actual application is restricted to

very simple and highly symmetric (usually linear) mol-

ecules. By contrast, analytic atomic orbitals (AOs) are the

main tool of quantum chemistry. Analytic AOs are in their

turn linear combinations of primitives, the latter being either

Slater-type (STO) [8] or Gaussian-type functions [9]. The

numbers of primitives and variable parameters are deter-

mined by two target characteristics of a basis set: ﬂexibil-

ity (growing with number of primitives and parameters)

and eﬃciency in computational and analytic applications

Published as part of the special collection of articles In Memoriam

of János Ángyán.

Electronic supplementary material The online version of this

article (https ://doi.org/10.1007/s0021 4-018-2386-x) contains

supplementary material, which is available to authorized users.

* Andrei L. Tchougréeﬀ

andrei.tchougreeﬀ@ac.rwth-aachen.de

1 A.N. Frumkin Institute ofPhysical Chemistry

andElectrochemistry ofRAS, Moscow, Russia

2 Independent University ofMoscow, Moscow, Russia

3 Chair ofSolid State andQuantum Chemistry, RWTH

- Aachen University, Aachen, Germany

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