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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 representing 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 generalization 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.
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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
for2nd‑row elements
IlyaV.Popov1,2· AndreiL.Tchougrée1,2,3
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 [46]. 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 ofPhysical Chemistry
andElectrochemistry ofRAS, Moscow, Russia
2 Independent University ofMoscow, Moscow, Russia
3 Chair ofSolid State andQuantum Chemistry, RWTH
- Aachen University, Aachen, Germany
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... At, quamquam orbitaliă Bungeniană breve monomialibus Slateri repraesentantur, nullum parametrum hōrum orbitalium -vel exponentes orbitales vel coëfficientes expansionum -significationem physicam habent. In dissertatiunculis nostris [6,7] formā orbitalium magis simplificatā, primōā V.Cl. V.A. Focke propositā [8], utebamur ad systemată orbitalium atomicōrum orthonormalium exstruendă solum unō parametrō per corticulam atomicam numeris quanticis n , demum exponenti orbitali ξ n , descriptă. ...
... esse, quod normalizatum sit, et ubi P n (x) sunt polynomiă in Adn. [6] descriptă. Pro n = + 1 solum unum membrum exsistat cujus coëfficiens numericus unitatem esse pōnȋmus: P +1 (x) ≡ 1; itaque functio R +1, (r) sola functio Slateri est. ...
... Ut suprā et in Adn. [6,7,12] ...
Article
Summarium Minimizatio angul¯orum Frobenian¯orum inter subspatiˇa functionaliˇa prognatˇa copiis differentibus functionum atomic¯arum est adhibita ad valores exponentium orbitalium � determinandos pro basibus minimalium atomic¯orum parametr¯orum (Moscovia-Aquisgrana-Lutetia Parisi¯orum – MAP) quae praebent optimam repraesentationem du¯abus copiis functionum atomic¯arum: alterae Bungenianae exsistenti ad elementa H–Xe, alterae Kogaensi porrectae ab H ad Lr (Z = 103). Valores exponentium ita inventi repraesentati ut functiones oneris nuclearis Z regulas lineares sequuntur in segminibus respectivis, praescriptiones regulis Slateri constitutas erg¯a exponentes Slaterianos simulantes. Exacte tamen regulas Slateri non sequuntur quia valores numeri quantici efficientis n� atque abstectionis incrementa � ab illis praescriptis differunt. Nihilominus ramos lineares dependeti ¯arum � ¯a Z juste structuram Tabulae Periodicae Element¯orum sequuntur et proprii sunt ad segmina respondentia p-, d- (transitiona) et f - (Lanthanoida ac Actinoida) elementis. Abstract The minimization of Frobenius angles between functional subspaces spanned by different sets of atomic functions is employed to determine the values of orbital exponents � characterizing minimum atomic parameters/Moscow-Aachen-Paris (MAP) basis sets providing the best representation of two Hartree-Fock based atomic basis sets: that of Bunge et al. available for elements H–Xe and that of Koga and Thakkar spanning H to Lr (Z = 103). So-extracted values of exponents follow piecewise linear laws as functions of the nuclear charge Z resembling the prescriptions set by Slater’s rules for the orbital exponents. In details, however, the rules proposed by Slater are not precisely followed, neither for effective principal quantum numbers n� nor screening increments �. Nevertheless, the linear pieces of the � vs Z follow the structure of the Periodic Table being specific for the segments corresponding to p-, d- (transition) and f - (Lanthanides and Actinides) elements, respectively. Ðåçþìå Ñ ïîìîùüþ ìèíèìèçàöèè ôðîáåíèóñîâñêèõ óãëîâ ìåæäó ôóíêöèîíàëüíûìè ïîä- ïðîñòðàíñòâàìè, ðàñòÿíóòûìè ðàçëè÷íûìè íàáîðàìè àòîìíûõ ôóíêöèé, ïîëó÷åíû çíà÷åíèÿ îðáèòàëüíûõ ýêñïîíåíò �, õàðàêòåðíûõ äëÿ ôóíêöèé òèïà ÌÀÏ (ìèíèìàëüíî àòîìíî ïàðàìå- òðèçîâàííûõ / ìîñêîâñêî-àõåíñêî-ïàðèæñêèõ), äàþùèå íàèëó÷øåå ïðèáëèæåíèå ïîñëåäíèõ ê äâóì íàáîðàì àòîìíûõ ôóíêöèé: Áóíãå, èçâåñòíûõ äëÿ ýëåìåíòîâ H–Xe, è Êîãà, ïîêðûâàþùèõ èíòåðâàë ýëåìåíòîâ îò H äî Lr (Z = 103). Ïîëó÷åííûå òàêèì îáðàçîì çíà÷åíèÿ ýêñïîíåíò, êàê ôóíêöèè Z, ïîä÷èíÿþòñÿ êóñî÷íî-ëèíåéíûì çàêîíàì, íàïîìèíàþùèì ïðåäïèñàíûå ïðàâèëàìè Ñëýòåðà äëÿ åãî îðáèòàëüíûõ ýêñïîíåíò.  äåòàëÿõ, îäíàêî, ïðàâèëà Ñëýòåðà äëÿ ÌÀÏ-ýêñïîíåíò íå âûïîëíÿþòñÿ, òàê êàê çíà÷åíèÿ ýôôåêòèâíîãî ãëàâíîãî êâàíòîâîãî ÷èñëà n� òàê è èíêðåìåíòû ýêðàíèðîâàíèÿ � îòëè÷àþòñÿ îò ïðåäëîæåííûõ Ñëýòåðîì çíà÷åíèé. Òåì íå ìåíåå, îòðåçêè ëèíåéíûõ çàâèñèìîñòåé � îò Z õîðîøî ñîãëàñóþòñÿ ñî ñòðóêòóðîé Ïåðèîäè÷åñêîé Ñèñòåìû Ýëåìåíòîâ è ñïåöèôè÷íû äëÿ îòðåçêîâ çíà÷åíèé Z, îòâå÷àþùèõ, ñîîòâåòñòâåííî, p-, d- (ïåðåõîäíûå) è f - (ëàíòàíîèäû è àêòèíîèäû) ýëåìåíòîâ. �Ýòà ñòàòüÿ ïóáëèêóåòñÿ íà ëàòûíè â îçíàìåíîâàíèå ñëóæáû ïðîôåññîðà Îéãåíà Øâàðöà â êà÷åñòâå ðåäàêòîðà ñòàòåé, ïóá- ëèêîâàâøèõñÿ íà ýòîì ÿçûêå â Theoretica Chemica Acta â 60-ûå ãîäû ïðîøëîãî âåêà. yElectronic address: tchougreeff@phyche.ac.ru I. INTRODUCTIO
... At, quamquam orbitaliă Bungeniană breve monomialibus Slateri repraesentantur, nullum parametrum hōrum orbitalium -vel exponentes orbitales vel coëfficientes expansionum -significationem physicam habent. In dissertatiunculis nostris [6,7] formā orbitalium magis simplificatā, primōā V.Cl. V.A. Focke propositā [8], utebamur ad systemată orbitalium atomicōrum orthonormalium exstruendă solum unō parametrō per corticulam atomicam numeris quanticis n , demum exponenti orbitali ξ n , descriptă. ...
... esse, quod normalizatum sit, et ubi P n (x) sunt polynomiă in Adn. [6] descriptă. Pro n = + 1 solum unum membrum exsistat cujus coëfficiens numericus unitatem esse pōnȋmus: P +1 (x) ≡ 1; itaque functio R +1, (r) sola functio Slateri est. ...
... Ut suprā et in Adn. [6,7,12] ...
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Full-text available
Summarium Minimizatio angulōrum Frobenianōrum inter subspatiǎ functionaliǎ prognatǎ copiis differentibus functionum atomicārum est adhibita ad valores exponentium orbitalium ξ determinandos pro ba-sibus minimalium atomicōrum parametrōrum (M oscovia-Aquisgrana-Lutetia Parisiōrum-MAP) quae praebent optimam repraesentationem duābus copiis functionum atomicārum: alterae Bunge-nianae exsistenti ad elementa H-Xe, alterae Kogaensi porrectae ab H ad Lr (Z = 103). Valores exponentium ita inventi repraesentati ut functiones oneris nuclearis Z regulas lineares sequuntur in segminibus respectivis, praescriptiones regulis Slateri constitutas ergā exponentes Slaterianos si-mulantes. Exacte tamen regulas Slateri non sequuntur quia valores numeri quantici efficientis n * atque abstectionis incrementa σ ab illis praescriptis differunt. Nihilominus ramos lineares depen-detiārum ξā Z juste structuram Tabulae Periodicae Elementōrum sequuntur et proprii sunt ad segmină respondentiă p-, d-(transitionă) et f-(Lanthanoidă ac Actinoidă) elementis. Abstract The minimization of Frobenius angles between functional subspaces spanned by different sets of atomic functions is employed to determine the values of orbital exponents ξ characterizing minimum atomic parameters/Moscow-Aachen-Paris (MAP) basis sets providing the best representation of two Hartree-Fock based atomic basis sets: that of Bunge et al. available for elements H-Xe and that of Koga and Thakkar spanning H to Lr (Z = 103). So-extracted values of exponents follow piecewise linear laws as functions of the nuclear charge Z resembling the prescriptions set by Slater's rules for the orbital exponents. In details, however, the rules proposed by Slater are not precisely followed, neither for effective principal quantum numbers n * nor screening increments σ. Nevertheless, the linear pieces of the ξ vs Z follow the structure of the Periodic Table being specific for the segments corresponding to p-, d-(transition) and f-(Lanthanides and Actinides) elements, respectively. Резюме С помощью минимизации фробениусовских углов между функциональными под-пространствами, растянутыми различными наборами атомных функций, получены значения орбитальных экспонент ξ, характерных для функций типа МАП (минимально атомно параме-тризованных / московско-ахенско-парижских), дающие наилучшее приближение последних к двум наборам атомных функций: Бунге, известных для элементов H-Xe, и Кога, покрывающих интервал элементов от H до Lr (Z = 103). Полученные таким образом значения экспонент, как функции Z, подчиняются кусочно-линейным законам, напоминающим предписаные правилами Слэтера для его орбитальных экспонент. В деталях, однако, правила Слэтера для МАП-экспонент не выполняются, так как значения эффективного главного квантового числа n * так и инкременты экранирования σ отличаются от предложенных Слэтером значений. Тем не менее, отрезки линейных зависимостей ξ от Z хорошо согласуются со структурой Периодической Системы Элементов и специфичны для отрезков значений Z, отвечающих, соответственно, p-, d-(переходные) и f-(лантаноиды и актиноиды) элементов. * Эта статья публикуется на латыни в ознаменование службы профессора Ойгена Шварца в качестве редактора статей, пуб-ликовавшихся на этом языке в Theoretica Chemica Acta в 60-ые годы прошлого века. † Electronic address: tchougreeff@phyche.ac.ru
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1. Introduction 2. The quantum mechanical method 3. Angular momentum 4. The theory of radiation 5. One-electron spectra 6. The central-field approximation 7. The Russell-Saunders case: energy levels 8. The Russell-Saunders case: eigenfunctions 9. The Russell-Saunders case: line strengths 10. Coupling 11. Intermediate coupling 12. Transformations in the theory of complex spectra 13. Configurations containing almost closed shells. X-rays 14. Central fields 15. Configuration interaction 16. The Zeeman effect 17. The Stark effect 18. The nucleus in atomic spectra Appendix. Universal constants and natural atomic units.
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