ArticlePublisher preview available
To read the full-text of this research, you can request a copy directly from the authors.

Abstract and Figures

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.
This content is subject to copyright. Terms and conditions apply.
Vol.:(0123456789)
1 3
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] ...
Preprint
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
... A method of comparing the formal rules of ascribing exponents' with the pragmatic basis sets has been proposed in Ref [12]. Namely, following an early suggestion dating back to V.A. Fock [13], we proposed a minimal atomic parameter (MAP) form [14] of the radial function: R n (r) ∝ (2ξ n r) P n (2ξ n r) exp (−ξ n r) , which permits to construct sets of AOs which like the STOs are characterized by the unique subshell specific parameter ξ n -orbital exponent, but through its construction assure the orthogonality of the radial parts for the different values of n at the same value of thanks to the polynomial multipliers P n . ...
Preprint
Full-text available
The piecewise linear dependence of orbital exponents ξ characterizing either nonorthogonal Slater or orthogonal minimum atomic parameters/Moscow-Aachen-Paris (MAP) radial parts of atomic orbitals is theoretically derived from a plausible model of electronic subshell energy and compared with their values derived from a pragmatic Koga basis set covering the elements from H to Lr (Z = 103). So derived values of exponents as well follow piecewise linear laws as functions of the nuclear charge Z. The linear branches of the ξ vs Z fairly follow the structure of the Periodic Table being specific for the segments of the Z values corresponding to p-, d-and f-elements, respectively. In details, however, the parameters of the theoretical linear dependencies of orbital exponents ξ on Z and those derived from the pragmatic basis set (referred to as experimental) differ from each other which will be addressed elswhere.
Article
Full-text available
The piecewise linear dependence of orbital exponents ξ\xi characterizing either nonorthogonal Slater or orthogonal minimum atomic parameters/Moscow–Aachen–Paris (MAP) radial parts of atomic orbitals is theoretically derived from a plausible model of electronic subshell energy and compared with their values derived from a pragmatic Koga basis set covering the elements from H to Lr ( Z=103Z = 103 ). So derived values of exponents as well follow piecewise linear laws as functions of the nuclear charge ZZ . The linear branches of the ξ\xi vs. ZZ fairly follow the structure of the Periodic Table being specific for the segments of the ZZ values corresponding to p -, d - and f -elements, respectively. In details, however, the parameters of the theoretical linear dependencies of orbital exponents ξ\xi on ZZ and those derived from the pragmatic basis set (referred to as experimental) differ from each other which will be addressed elsewhere.
Article
Full-text available
Exponential type orbital with hyperbolic cosine basis functions, proposed recently for Hartree–Fock–Roothaan calculations of neutral atoms, are studied in detail for the calculations of isoelectronic series of atoms from Be to Ne. Calculations are performed for the neutral and the first 20 cationic members of the isoelectronic series of each atom in its ground state. Three of the most popular exponential type orbitals (Slater type functions, B functions and ψ(α) functions with α = 2) are combined with modified hyperbolic cosine function cosh(βr + γ) to improve the basis function quality within the minimal basis sets framework. Performances of the basis functions are compared with each other by using the same number of variational parameters in them. The obtained results are also compared with numerical Hartree–Fock and extended Slater type basis set results. The presented accuracy of the minimal basis descriptions of atomic systems supports the usage of these unconventional basis functions in electronic structure and property calculations.
Article
Full-text available
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 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
Article
We present a software package GoGreenGo—an overlay aimed to model local perturbations of periodic systems due to either chemisorption or point defects. The electronic structure of an ideal crystal is obtained by worldwide‐distributed standard quantum physics/chemistry codes, and then processed by various tools performing projection to atomic orbital basis sets. Starting from this, the perturbation is addressed by GoGreenGo with use of the Green's functions formalism, which allows evaluating its effect on the electronic structure, density matrix, and energy of the system. In the present contribution, the main accent is made on processes of chemical nature, such as chemisorption or doping. We address a general theory and its computational implementation supported by a series of test calculations of the electronic structure perturbations for benchmark model solids: simple, face‐centered, and body‐centered cubium systems. In addition, more realistic problems of local perturbations in graphene lattice, such as lattice substitution, vacancy, and “on‐top” chemisorption, are considered. Point defects in crystals form a wide class of processes being of great importance in solid‐state chemistry. Only by considering surface chemistry one can propose a numerous examples ‐ from formation of isolated surface defects to single particle chemisorption and elementary reactions on catalysts' surfaces. Theoretical investigation of these processes, aiming to understand their mechanisms from the electronic structure perspective, presents one of many important branches of solid‐state chemistry deserving close attention. In this work we present a new software package GoGreenGo specifically designed to perform computationally effective quantum chemical calculations of local processes in solids and to provide results in “chemical” terms.
Article
Basis sets featuring single‐exponent radial functions for each of the nℓ subshells and orthogonality of the radial parts for different values of n within the same ℓ have been generated for elements 1–54 of the periodic table, by minimizing the total energy for different spectroscopic states. The derived basis sets can be fairly dubbed as MAP (minimal atomic parameter/Moscow–Aachen–Paris) basis sets. We show that fundamental properties (total energy, radial expectation values, node positions, etc.) of the generated MAP orbital sets are astonishingly close to those obtained with much larger basis sets known in the literature, without numerical inconsistencies. The obtained exponents follow simple relations with respect to the nuclear charge Z.
Article
The minimum atomic parameters/Moscow–Aachen–Paris (MAP) basis sets—reintroduced in the previous paper—are analyzed with respect to spatial features as orbital shape, possible fits to alternative orbital sets (numerical or quasi‐numerical orbitals, nodeless Slater orbitals), respect of Kato's condition and radial distribution of energy components. For comparing orbital spaces the Frobenius angle between the orbital subspaces they span is introduced as numerical tool. It is shown that the electronic density of the MAP states is depleted around the nucleus with respect to the other orbital sets. Despite this, the similarity between the respective subspaces in all cases (except a unique case of the Pd atom) as measured by the cosine of the Frobenius angle amounts above 0.96 for all atoms. Deviations from the perfect value of Kato's condition amounts systematically to 0.3 and 0.5 for all elements considered. Integrating one‐electron energy contributions from r = ∞ to a finite radius, MAP and Bunge orbitals show about the same values, but for the inner region governed by the polynomial oscillations.
Conference Paper
Full-text available
The deductive molecular mechanics (DMM) directly expresses the energy of a system of atoms in terms of the chemical bonds built upon hybrid atomic orbitals. Here we apply it to write down the total energy of the most symmetric crystalline water polymorph – ice X and to study its stability with respect to transition to the paraelectric ice VII with partial occupancy of hydrogen position and/or to the antiferroelectric ice VIII and the transition between the two latter phases. On this route we could establish several statements which can be qualified as theorems of DMM about ices. Further application of the proposed approach can eventually lead to the analytic description of the entire phase diagram of crystalline water (ices).
Article
Full-text available
The computer program LOBSTER (Local Orbital Basis Suite Towards Electronic-Structure Reconstruction) enables chemical-bonding analysis based on periodic plane-wave (PAW) density-functional theory (DFT) output and is applicable to a wide range of first-principles simulations in solid-state and materials chemistry. LOBSTER incorporates analytic projection routines described previously in this very journal [J. Comput. Chem. 2013, 34, 2557] and offers improved functionality. It calculates, among others, atom-projected densities of states (pDOS), projected crystal orbital Hamilton population (pCOHP) curves, and the recently introduced bond-weighted distribution function (BWDF). The software is offered free-of-charge for non-commercial research. © 2016 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.
Article
Full-text available
The performance of basis sets made of numerical atomic orbitals is explored in density-functional calculations of solids and molecules. With the aim of optimizing basis quality while maintaining strict localization of the orbitals, as needed for linear-scaling calculations, several schemes have been tried. The best performance is obtained for the basis sets generated according to a new scheme presented here, a flexibilization of previous proposals. Strict localization is maintained while ensuring the continuity of the basis-function derivative at the cutoff radius. The basis sets are tested versus converged plane-wave calculations on a significant variety of systems, including covalent, ionic, and metallic. Satisfactory convergence is obtained for reasonably small basis sizes, with a clear improvement over previous schemes. The transferability of the obtained basis sets is tested in several cases and it is found to be satisfactory as well.
Article
The story of C_2 continues to fascinate chemists spinning around a possibility of quadruple bonds for p-elements and discussing a wealth of options for the nature of an unconventional fourth bond. This led to lively discussions about the ways of counting or measuring bonds, and the interplay between the bond strength (energy), its length, and rigidity/stiffness (elasticity). Even old concerns about the possibility of theorems in chemistry and thus of the place of chemistry among ‘true’ sciences had been revived. We show that under some mild conditions certain exact statements (lemmas and theorems) about the C_2 molecule can be relatively easily proven which, if not resolves the controversy entirely, at least provides some theoretical reference points to the discussion. Some more general consequences from this experience are discussed as well.
Article
Electronic structure methods for molecular systems rely heavily on using basis sets composed of Gaussian functions for representing the molecular orbitals. A number of hierarchical basis sets have been proposed over the last two decades, and they have enabled systematic approaches to assessing and controlling the errors due to incomplete basis sets. We outline some of the principles for constructing basis sets, and compare the compositions of eight families of basis sets that are available in several different qualities and for a reasonable number of elements in the periodic table. © 2012 John Wiley & Sons, Ltd. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods
Article
Quantum-chemical computations of solids benefit enormously from numerically efficient plane-wave (PW) basis sets, and together with the projector augmented-wave (PAW) method, the latter have risen to one of the predominant standards in computational solid-state sciences. Despite their advantages, plane waves lack local information, which makes the interpretation of local densities-of-states (DOS) difficult and precludes the direct use of atom-resolved chemical bonding indicators such as the crystal orbital overlap population (COOP) and the crystal orbital Hamilton population (COHP) techniques. Recently, a number of methods have been proposed to overcome this fundamental issue, built around the concept of basis-set projection onto a local auxiliary basis. In this work, we propose a novel computational technique toward this goal by transferring the PW/PAW wavefunctions to a properly chosen local basis using analytically derived expressions. In particular, we describe a general approach to project both PW and PAW eigenstates onto given custom orbitals, which we then exemplify at the hand of contracted multiple-ζ Slater-type orbitals. The validity of the method presented here is illustrated by applications to chemical textbook examples-diamond, gallium arsenide, the transition-metal titanium-as well as nanoscale allotropes of carbon: a nanotube and the C60 fullerene. Remarkably, the analytical approach not only recovers the total and projected electronic DOS with a high degree of confidence, but it also yields a realistic chemical-bonding picture in the framework of the projected COHP method. Copyright © 2013 Wiley Periodicals, Inc.
Book
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.
Article
In analogy with the method of Zener for the atoms from Li to F, simple rules are set up giving approximate analytic atomic wave functions for all the atoms, in any stage of ionization. These are applied to x-ray levels, sizes of atoms and ions, diamagnetic susceptibility, etc. In connection with ferromagnetism it is shown that if this really depends on the existence of incomplete shells within the atoms, rather far apart in the crystal, then the metals most likely to show it would be Fe, Co, Ni, and alloys of Mn and Cu (Heusler alloys).