Aufbau Research Articles

A Periodic Table for Benzenoid Hydrocarbon Isomer Classes and Beyond

AbstractFor Abstract see ChemInform Abstract in Full Text.

Accurate energy eigenvalues and eigenfunctions for the two‐dimensional confined hydrogen atom

AbstractA study of the two‐dimensional hydrogen atom confined within a circle of impenetrable walls is presented. The potential inside the box is Coulomb type, whereas outside it is infinite. The energy eigenvalues and some radial wave function properties are computed with high accuracy for different box sizes. We derive the polarizability in the Kirkwood approximation, calculate the Fermi contact term as a function of the confinement radius, and investigate the filling order of the one‐electron states. When the electronic configuration of many electrons is constructed by means of the Aufbau principle, the model predicts the inversion 2s–3d levels in the N atom. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

The spin-unrestricted molecular Kohn–Sham solution and the analogue of Koopmans’s theorem for open-shell molecules

Spin-unrestricted Kohn-Sham (KS) solutions are constructed from accurate ab initio spin densities for the prototype doublet molecules NO(2), ClO(2), and NF(2) with the iterative local updating procedure of van Leeuwen and Baerends (LB). A qualitative justification of the LB procedure is given with a "strong" form of the Hohenberg-Kohn theorem. The calculated energies epsilon(isigma) of the occupied KS spin orbitals provide numerical support to the analogue of Koopmans' theorem in spin-density functional theory. In particular, the energies -epsilon(ibeta) of the minor spin (beta) valence orbitals of the considered doublet molecules correspond fairly well to the experimental vertical ionization potentials (VIPs) I(i) (1) to the triplet cationic states. The energy -epsilon(Halpha) of the highest occupied (spin-unpaired) alpha orbital is equal to the first VIP I(H) (0) to the singlet cationic state. In turn, the energies -epsilon(ialpha) of the major spin (alpha) valence orbitals of the closed subshells correspond to a fifty-fifty average of the experimental VIPs I(i) (1) and I(i) (0) to the triplet and singlet states. For the Li atom we find that the exact spin densities are represented by a spin-polarized Kohn-Sham system which is not in its ground state, i.e., the orbital energy of the lowest unoccupied beta spin orbital is lower than that of the highest occupied alpha spin orbital ("a hole below the Fermi level"). The addition of a magnetic field in the -z direction will shift the beta levels up so as to restore the Aufbau principle. This is an example of the nonuniqueness of the mapping of the spin density on the KS spin-dependent potentials discussed recently in the literature. The KS potentials may no longer go to zero at infinity, and it is in general the differences nu(ssigma)( infinity )-epsilon(isigma) that can be interpreted as (averages of) ionization energies. In total, the present results suggest the spin-unrestricted KS theory as a natural one-electron independent-particle model for interpretation and assignment of the experimental photoelectron spectra of open-shell molecules.

Cover Picture: 5‐Dehydro‐1,3‐quinodimethane: A Hydrocarbon with an Open‐Shell Doublet Ground State (Angew. Chem. Int. Ed. 6/2004)

Contrary to the accepted guidelines for the occupancy of orbitals (such as Hund's rule or the Aufbau principle), three unpaired electrons are coupled antiferromagnetically in the 5‐dehydro‐m‐xylylene triradical. This first example of an organic molecule with an “open‐shell” doublet ground state is described by P. G. Wenthold, A. I. Krylov, and co‐workers on page 742 ff. The cover pictures shows the singly occupied orbitals in the triradical and highlights the correlation between theoretical and experimental results.

Just how ab initio is ab initio quantum chemistry?

Quantum Mechanics has been the most spectacularly successful theory in the history of science. As is often mentioned the accuracy to which the gyromagnetic ratio of the electron can be calculated is a staggering nine decimal places. Quantum Mechanics has revolutionized the study of radiation and matter since its inception just overonehundredyears ago. Theimpactof thetheoryhas been felt in such fields as solid state physics, biochemistry, astrophysics, materials science and electronic engineering, not to mention chemistry, the subject of this conference. Quantum Mechanics offers the most comprehensive and most successful explanation of many chemical phenomena such as the nature of valency and bonding as well as chemical reactivity. It has also provided a fundamental explanation of the periodic system of the elements that summarizes a vast amount of empirical chemical knowledge. Quantum Mechanics has become increasingly important in the education of chemistry students. The general principles provided by the theory mean that students can now spend less time memorizing chemical facts and more time in actually thinking about chemistry. I hope that with these opening words I have succeeded in convincingtheaudiencethat Ido notcomebefore youto denythepower and influence of Quantum Mechanics in the field of chemistry. � A previous version of this article appeared as ‘Lowdin’s Remarks on the Aufbau Principle and a Philosopher’s View of Ab Initio Quantum Chemistry’ in E.J. Brandas, E.S. Kryachko (Eds.) Fundamental World of Quantum Chemistry, Vol. II, 675‐694, Kluwer, Dordrecht, 2003.

A Periodic Table for Benzenoid Hydrocarbon Isomer Classes and Beyond

The Periodic Table of Atomic Elements is a two-dimensional array (spreadsheet) generated according to an aufbau principle. As will be illustrated by a Formula Periodic Table for Benzenoid Hydrocarbons, this construction turns out to be more general than heretofore realized. These ‘super spreadsheets’ have a unique structure of their own, which contains chemical information in compact form. The concept of a periodic table set is like the concept of a group in mathematics. Once their respective criteria are fulfilled, one can anticipate their characteristic set properties. As such, a periodic table set is a useful mnemonic device for organizing, studying, and understanding property trends in large classes of related molecules. The Periodic Table for Benzenoid Hydrocarbons has organized this class of hydrocarbons into a systematic framework and has led to the identification of a new topological paradigm.

Ab Initio Investigation of Structures and Stability of SinCm Clusters

Various structural possibilities for Si n C and Si n C 2 (n = 1-7) neutral and anionic isomers were investigated using the second-order Miller-Plesset (MP2) approximation at the 6-311+G(d) level. The calculations predicted the existence of a number of previously unknown isomers (i.e., Si 5 C 2 and Si 6 C 2 ). We found that the basis set [6-31G(d)] with the MP2 approximation was too small to explain the photoelectron spectra of Si n C m clusters reasonably. Our present results agree satisfactorily with the photoelectron spectroscopy. The strong C-C bond is no longer the dominant factor in the building-up principle of mixed Si 6 C 2 neutral and anionic clusters. The calculated adiabatic electron affinities in their ground states showed that Si n C m (n + m = 3, 6) clusters are more stable than any others in Si n C and Si n C 2 (n = 1-7) species, being consistent with the observed TOF signal intensities. Their stability tends to decrease with the increase in the size of these clusters.

Computational Study of the Geometry and Properties of the Metcars Ti8C12 and Mo8C12

We report the results of extensive ab initio HF and post-HF (as well as DFT) studies of the “magic number” metallocarbohedrene (“metcar”) clusters Ti8C12 and Mo8C12 in various electronic states of Td symmetry and the Jahn−Teller-distorted D2d, C3v, and C1 symmetries. An essential feature of the present work is that it is a systematic study employing a hierarchy of theoretical methods to explore the effect of refining the treatment of electron correlation in determining the geometry and electronic ground state of these species. For Ti8C12, we show using relatively high-level theories such as MP2, MP4, and QCISD that the Aufbau principle for the occupation of the molecular orbitals is obeyed, resulting in a Jahn−Teller distortion of the proposed Td symmetry. These higher-level calculations identify a D2d structure close to Td symmetry for the electronic ground state and allow some of its chemical properties to be explored with confidence using a lower level of theory. The reactivity of Ti8C12 toward H2O, CO, and Cl is also investigated. It is found that Ti8C12 can act as a Lewis acid to accept lone pairs of electrons from H2O (Lewis base) and that it can also be oxidized by Cl atoms through electron donation from C2 units in Ti8C12 to the Cl mediated by a Ti dz2 orbital. Thus, a relationship among structure, electronic properties, and reactivity is established. For Mo8C12, we find that the Td structure is not subject to a Jahn−Teller effect, and it is a true minimum at the HF level; B3LYP DFT calculations prefer a lower-symmetry (near-D2) structure. The results of ab initio and DFT methods are compared.

The use of Kitaigorodskii's Aufbau principle in the solid-state study of crystalline borane compounds. A preliminary account

Very many molecular structures of polyhedral borane compounds have been established by solid-state crystallographic analysis, but their intermolecular interactions, and the factors that dictate their supramolecular arrangements in the crystal lattice, are essentially unexamined. Kitaigorodskii's Aufbau principle (KAP) forms the basis of a technique for such an examination and simplifies the visualisation of molecular packing by breaking it down into a series of symmetry-generated steps, of which each permits even the weakest intermolecular interactions to be readily identified. KAP is based on close-packing principles, and constructs the crystal of a molecular compound in a three-step process: first, the molecular units pack to form one-dimensional chain substructures, these then pack together to yield two-dimensional sheet substructures; these, in turn, stack on top of one another to yield the observed three-dimensional structure. The method is exemplified by consideration of the molecular arrangements in the ‘wrap-around’ encapsulation of 4,4′-bipyridyl with [6,9-(4,4′-bipyridyl) 2- arachno-B 10H 12] in their 1:1 co-crystal, the complex three-dimensional dihydrogen-bonding network in the crystal structure of H 3BNH 3, solvent channels in the structure of the co-crystal of n-hexane with [C 5H 5NMe][1-(SMe)-10-(SMe 2)- closo-B 10H 8], and systematic variations within the crystal structures of [6,9-(NC 5H 4-4-R) 2- arachno-B 10H 12], where R is Me, Et and n-Pr. It is concluded in this preliminary account that KAP provides a simple yet rigorous tool for studying a whole range of both gross and subtle molecular packing effects in the solid-state.

Configuration irregularities: deviations from the Madelung rule and inversion of orbital energy levels

(a) The electron configurations of many atoms that do not obey the Madelung rule of orbital occupancy can be explained by minimum-energy configurations obtained from the Dirac Hyper Hartree–Fock equations. The resulting non-integral occupation of ns and ( n−1)d orbitals (the usual interpretation for energy band calculations in solids) explains eight of the ten `anomalous' configurations. (b) All four d- and f-block atoms whose singly charged cations result from the loss of an ( n−1)d electron rather than an ns electron are shown to have ( n−1)d rather than ns as their highest occupied energy levels.

A full-potential linear-muffin-tin-orbitalmolecular-dynamics study of B7, B10 and B13clusters

The structures of B7, B10 and B13 boron clusters arestudied using the full-potential linear-muffin-tin-orbitalmolecular-dynamics method. Seven stable structures for B7 andfifteen for B10 have been obtained. C2h-B10 isthe most stable among the 15 structures, but C2v-B10is not stable. For B13, three degenerate ground-statestructures have been found. The potential surface near C2v-B7 (ground state) and D6h-B7 is very flat. As afundamental unit in constructing bigger clusters, C2v-B7 will change its form easily. The most stable structuresfor B7, B10 and B13 clusters are two-dimensional(quasi-) planar clusters, rather than the three-dimensional ones.General speaking, these clusters obey the `Aufbau principle'.

Aufbau principle of complex open-framework structures of metal phosphates with different dimensionalities.

Open-framework metal phosphates occur as one-dimensional (1D) chains or ladders, two-dimensional (2D) layers, and complex three-dimensional (3D) structures. Zero-dimensional monomers have also been isolated recently. These materials are traditionally prepared by hydrothermal means, in the presence of organic amines, but the reactions of amine phosphates with metal ions provide a facile route for the synthesis, and also throw some light on the mode of formation of these fascinating architectures. Careful studies of the transformations of monophasic zinc phosphates of well-characterized structures show that the 1D structures transform to 2D and 3D structures, while the 2D structures transform to 3D structures. The zero-dimensional monomers transform to 1D, 2D, and 3D structures. There is reason to believe that the 0D monomers, comprising four-membered rings, are the most basic structural units of the open-framework phosphates and that after an optimal precursor state, such as the ladder structure, is formed, further building may occur spontaneously. Evidence for the occurrence of self-assembly in the formation of complex structures is provided by the presence of the structural features of the one-dimensional starting material in the final products. These observations constitute the beginning of our understanding of the building-up principle of such complex structures.

Multiplicity, instability, and SCF convergence problems in Hartree-Fock solutions

We present a study of the instability and convergence of Hartree–Fock (HF) ab initio solutions for the diatomic systems H2, LiH, CH, C2, and N2. In our study, we consider real molecular orbitals (MOs) and analyze the classes of single-determinant functions associated to Hartree–Fock–Roothaan (HFR) and Hartree–Fock–Pople–Nesbet (HFPN) equations. To determine the multiple HF solutions, we used either an SCF iterative procedure with aufbau and non-aufbau ordering rules or the algebraic method (AM). Stability conditions were determined using TICS and ASDW stability matrices, derived from the maximum and minimum method of functions (MMF). We examined the relationship between pure SCF convergence criterion with the aufbau ordering rule, and the classification of the HF solution as an extremum point in its respective class of functions. Our results show that (i) in a pure converged SCF calculation, with the aufbau ordering rule, the solutions are not necessarily classified as a minimum of the HF functional with respect to the TICS or ASDW classes of solutions, and (ii) for all studied systems, we obtained local minimum points associated only with the aufbau rule and the solutions of lower energies. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 600–610, 2000

Density functional theory with fractionally occupied frontier orbitals and the instabilities of the Kohn–Sham solutions for defining diradical transition states: Ring-opening reactions

Density functional theory with fractionally occupied frontier orbitals was combined with an analysis of the instability of the Kohn–Sham solutions and applied to the study of the ring-opening reactions of 1,2-dioxetene, 1,2-dithiete, 1,2-diselenete, and 1,2-ditellurete. These molecules possess transition structures and reactive intermediates for the ring-opening reactions which exhibit strong nondynamical electron correlation. All restricted density functional theory (DFT) solutions for the closed shell transition state structures for all these species are unstable. The solutions with the hybrid Hartree–Fock DFT functionals, B3LYP and B3PW91, are triplet unstable, while for the pure DFT functional BLYP the instability is due to a violation of the Aufbau principle. The same types of instabilities were found for the 1,2-diselenete and 1,2-ditellurete intermediates. Lower energy stable solutions for the diradical transition structures were found with unrestricted DFT methods allowing fractionally occupied orbitals. This DFT approach indicates a decrease in the active orbital space from four fractionally occupied natural orbitals in earlier multireference predictions to two fractionally occupied Kohn–Sham orbitals.

Identification of atomic-like electronic states in indium arsenide nanocrystal quantum dots

Semiconductor quantum dots, due to their small size, mark the transition between molecular and solid-state regimes, and are often described as ‘artificial atoms’ (13). This analogy originates from the early work on quantum confinement effects in semiconductor nanocrystals, where the electronic wavefunctions are predicted4 to exhibit atomic-like symmetries, for example ‘s ’ and ‘p ’. Spectroscopic studies of quantum dots have demonstrated discrete energy level structures and narrow transition linewidths5,6,7,8,9, but the symmetry of the discrete states could be inferred only indirectly. Here we use cryogenic scanning tunnelling spectroscopy to identify directly atomic-like electronic states with s and p character in a series of indium arsenide nanocrystals. These states are manifest in tunnelling current–voltage measurements as two- and six-fold single-electron-charging multiplets respectively, and they follow an atom-like Aufbau principle of sequential energy level occupation10.

Radioactive Decay and the Structure of e (∞) Quantum Spacetime

Some theoretical arguments and some experimental evidence are given for the hypothesis that the form of the spectra of the Beta radioactive decay, as well as the micro background radiation and related phenomena, can be traced back to the geometrical–topological structure of spacetime itself. It is argued that this structure obeys the same Aufbau principle upon which Plancks black body radiation law is based. The main conclusion here is that the universe is four-dimensional but every point in this space, however small, is again also four-dimensional and that this accounts for the form of classical and quantum behaviour observed.

Analysis of π‐electronic structures of small alternant hydrocarbons to infinitely large polymeric strips: The aufbau principle and end‐group effects

Paired series of strongly subspectral molecular graphs which possess a preponderance of common eigenvalues can be constructed by successive attachment of specific aufbau units (repeated subgraphs) to certain seed graphs. Recursion relations for the characteristic polynomials of such series are identical and differ only in the input of initial characteristic polynomials. A new operator method for the derivation of recursion relations for singly connected polymeric series is presented. Strongly subspectral series devolve to the same infinite-limit member and density of states, and two types of end-group effects are demonstrated. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 721–733, 1999

A Critique of Atkins' Periodic Kindom and Some Writings on Electronic Structure

This article consists of a critique of the writings of Peter Atkins. The topics discussed include the quantum mechanical explanation of the periodic system, the aufbau principle and the order of occupation of orbitals by electrons. It is also argued that Atkins fails to appreciate the philosophical significance of the more general version of the Pauli Exclusion Principle and that this omission has ramifications in the popular presentation of chemistry as well as chemical education and philosophy of chemistry in general.

Analysis of ?-electronic structures of small alternant hydrocarbons to infinitely large polymeric strips: The aufbau principle and end-group effects

Paired series of strongly subspectral molecular graphs which possess a preponderance of common eigenvalues can be constructed by successive attachment of specific aufbau units (repeated subgraphs) to certain seed graphs. Recursion relations for the characteristic polynomials of such series are identical and differ only in the input of initial characteristic polynomials. A new operator method for the derivation of recursion relations for singly connected polymeric series is presented. Strongly subspectral series devolve to the same infinite-limit member and density of states, and two types of end-group effects are demonstrated. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 721–733, 1999

Directed toward the Development of a Unified Structure Theory of Polycyclic Conjugated Hydrocarbons: The Aufbau Principle in Structure/Similarity Studies

This work introduces two new concepts applicable to structure/similarity studies of conjugated hydrocarbons: (1) The first concept deals with orthogonal strongly subspectral series of molecular graphs and their properties. (2) In the second concept, if a relationship exists among set of strongly subspectral molecular graphs, then an equivalent relationship exists among the complementary molecular graphs of that set. Two sets of triplet series having strongly subspectral molecular graphs are presented. Collections of subspectral molecular graphs and their eigenvalues are tabulated for the first time.