P.‐O. Löwdin and the International Journal of Quantum Chemistry: A kaleidoscopic agenda for quantum chemistry

This is a article about P.‐O. Löwdin's life, his work in shaping quantum chemistry into a mature discipline at the intersection of mathematics, physics, chemistry, and biology, and his founding of the International Journal of Quantum Chemistry in 1967. Unavoidably, it is, also, a article reflecting our views about the history of quantum chemistry. We attempt to convey the complexities in the becoming of a subdiscipline, like quantum chemistry, where a variety of factors will have to be taken into consideration for a comprehensive understanding of its historical developments: the relations of chemists to the Heisenberg‐Schrödinger formulation of quantum mechanics after 1926, the institutional dynamics centered around the establishment of new courses and chairs, the research agendas and the vying for dominance within the community of quantum chemists, the methodological, and philosophical issues that have never left the quantum chemists indifferent, and, of course, the dramatic role of the computer in transforming the culture for actually practicing quantum chemistry. Furthermore, attracted by American history, culture, and ways of life, Löwdin suggested in the late 1970s that the post‐WWII character of quantum chemistry was dependent on its ability to hub a “scientific melting pot,” much like the United States of America which he viewed as a fusion of people from diverse provenances and cultures. In this article, we attempt to investigate another metaphor, that of the “kaleidoscope.” Löwdin believed that quantum chemistry's strength arose from its ability to nurture a multiplicity of heterogeneous cultural elements/subcultures and practices, interacting with each other, exchanging perspectives and modes of action, which circulated in an increasingly extended network of actors and institutional frameworks. © 2013 Wiley Periodicals, Inc.     

Pipek–Mezey localization of occupied and virtual orbitals

Recent advances in orbital localization algorithms are used to minimize the Pipek–Mezey localization function for both occupied and virtual Hartree–Fock orbitals. Virtual Pipek–Mezey orbitals for large molecular systems have previously not been considered in the literature. For this work, the Pipek–Mezey (PM) localization function is implemented for both the Mulliken and a Löwdin population analysis. The results show that the standard PM localization function (using either Mulliken or Löwdin population analyses) may yield local occupied orbitals, although for some systems the occupied orbitals are only semilocal as compared to state‐of‐the‐art localized occupied orbitals. For the virtual orbitals, a Löwdin population analysis shows improvement in locality compared to a Mulliken population analysis, but for both Mulliken and Löwdin population analyses, the virtual orbitals are seen to be considerably less local compared to state‐of‐the‐art localized orbitals. © 2013 Wiley Periodicals, Inc.     

Application of Löwdin's canonical orthogonalization method to the Slater‐type orbital configuration‐interaction basis set

We apply Löwdin's canonical orthogonalization method to investigate the linearly dependent problem arising from the variational calculation of atomic systems using Slater‐type orbital configuration‐interaction (STO‐CI) basis functions. With a specific arithmetic precision used in numerical computations, the nonorthogonal STO‐CI basis is easily linearly dependent when the number of basis functions is sufficiently large. We show that Löwdin's canonical orthogonalization method can successfully overcome such problem and simultaneously reduce the dimension of basis set. This is illustrated first through an S‐wave model He atom, and then the real two‐electron atoms in both the ground and excited states. In all of these calculations, the variational bound state energies of the two‐electron systems are obtained in reasonably high accuracy using over‐redundant STO‐CI bases, however, without using extended high‐precision technique. © 2015 Wiley Periodicals, Inc.     

An analysis of the zero differential overlap approximation. Towards an improved semiempirical MO method beyond it

Summary Some systematic errors of the zero differential overlap (ZDO) approximation in semiempirical molecular orbital (MO) methods are discussed. In electron methods, a power series expansion of the inverse square rootS ^–1/2 of the overlap matrix and application of the Mulliken approximation to the two-electron integrals show that the ZDO Hamiltonian coincides with the Hamiltonian obtained by explicit performance of the Löwdin transformation up to first-order terms of diatomic overlap densities. Higher than first-order terms lead to a systematic up-shift of the canonical MO energies. Although a power series expansion ofS ^–1/2 is no longer possible in all-valence-electron methods, the MO levels resulting from the ZDO approximation are also systematically placed at too low energies, especially the low-lying occupied and the virtual MOs. A method based on explicit performance of the Löwdin transformation and retaining the simplicity of the ZDO approach for the calculation of Fock matrix elements is developed. The parameters of this method are obtained by very simple manipulations of the original ZDO parameters. Numerical calculations show that a considerable improvement of the MO energy spectrum in the inner valence region can be obtained in this way     

Charge-conserving electron density averaging for a set of nuclear configurations

The Löwdin-Inverse Löwdin Transformation (LILT) technique combined with the Diophantine Density Matrix Purification (DDMP) method is suggested for a charge-preserving density averaging approach for a range of nuclear configurations.     

The connected‐moments polynomial approach for Hamiltonian eigenvalues calculation and its application to the one‐particle systems

The new connected‐moments polynomial approach (CMP) is developed for evaluation of Hamiltonian eigenvalues. It is based on properties of specially designed polynomial and does not use any basis set and variational procedure. Like all the methods based on hamiltonain moments knowledge, the CMP is conceptually simple but is less tedious and is usually convergent even for very “crude” trial functions. This method is applicable not only to the ground state energy calculation but also to the excited states. The formalism is presented in two modifications: non‐local (integral) and local (integral‐free) ones. An accuracy of both versions is illustrated by numerical examples of Hamiltonian eigenvalues calculations for harmonic and anharmonic oscillators. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Simplified ab-initio calculations for molecular systems

A method is described for reducing a large part of the arithmetic of exact ab-initio SCF molecularorbital calculations based on Slater-type-orbitals without noticeable loss of numerical accuracy. The procedure involves the transformation to Löwdin orthogonalized orbitals and then invoking the NDDO approximation. The three- and four-centre two-electron integrals required are estimated by a truncated Ruedenberg expansion. All one-electron integrals are evaluated exactly. No empirical parameters are employed. Numerical tests on CO, OF₂, O₃ and ONF show that the NDDO approximation is very accurate for Löwdin functions and that the Ruedenberg expansion is arithmetically satisfactory for the SCF MO calculations.     

Classical chemical concepts from ab initio SCF calculations

Alternative definitions of bond order, valency, gross orbital populations and total atomic charge for SCF wavefunctions are compared. It is found that there are sound theoretical and numerical reasons for preferring definitions based on the Löwdin density matrix.     

Calculation of two-center overlap integral in molecular coordinate system over Slater type orbital using Löwdin α-radial and Guseinov rotation–angular functions

By use of Löwdin and Guseinov relations for the radial and angular part of two-center overlap integrals, respectively, the computer calculations of overlap integrals over Slater type orbitals (STOs) in molecular coordinate system are performed. The results of calculations are valid for arbitrary principal quantum numbers, screening constants and location of STOs. Excellent agreement with benchmark results and stability of the technique are demonstrated.     

Relation between the Slater–Kohn–Sham orbitals generated by the one-body potential V(r) of DFT and the Löwdin natural orbitals for inhomogeneous electron liquids

The Löwdin natural orbitals (NO) are defined as those orbitals which bring the first-order density matrix of a correlated electron assembly into diagonal form. Another one-particle density matrix with the same diagonal elements is the single-particle idempotent Dirac density matrix generated by the one-body potential of density functional theory. Here, we compare the off diagonal form of γ expanded in terms of the Slater–Kohn–Sham (SKS) orbitals generated by V(r) with the NO expansion of Löwdin for general inhomogeneous electron liquids. In particular, the equation of motion of the correlated γ is corrected from that of γ _s , both now containing the one-body potential V(r). To illustrate the theory presented here, we first construct an approximate, albeit accurate, correlated 1DM for the ground state of the He atom and display connections between the resulting NOs and the SKS orbitals. The second example we discuss, but now quite briefly, is that of the inhomogeneous electron liquid in crystalline Si, where the NO expansion is available from the literature.     

On the “Wolfsberg–Helmholz Conjecture” of “Extended‐Hückel Theory”

After having reviewed some pioneer integral approximations closely related to Rüdenberg's expansions of one‐ and two‐electron orbital products, we apply the previously described “Implicit Multi‐Center Integration” techniques on Roothaan's “restricted” Fock‐matrix components over standard atomic orbital bases. The resulting compact forms are very similar to the well‐known “Wolfsberg–Helmholz Conjecture” of “Extended‐Hückel Theory,” which relates the various off‐diagonal matrix elements of “restricted” Fock‐type to their corresponding diagonal counterparts. In this way, a “nonempirical Extended‐Hückel Theory” can be created. © 2012 Wiley Periodicals, Inc.     

A Constrained Disorder Refinement: “Reinvestigation of “Single‐Crystal X‐ray Structure of 1,3‐Dimethylcyclobutadiene” by M. Shatruk and I. V. Alabugin”

Shatruk and Alabugin propose an alternative structural model for the observed electron density that we have attributed to the photochemical formation of 1,3‐dimethylcyclobutadiene in a protective solid crystalline matrix. The main criticism from Shatruk and Alabugin concerns the modeling of the disorder in the calixarene cavity and in particular the neglect of a residual electron density close to the O1 atom. We published (Chem. Eur. J. 2011 , 17, 10021) our opinion concerning this “ignored peak” in the Supporting Information of the paper. The current response to the Correspondence demonstrates that Shatruk and Alabugin have over‐modeled our data by assigning a small electron density peak, which is hardly more than the density corresponding to a hydrogen atom, to an under‐occupied oxygen site, using inappropriate refinement contraints.     

Inducing Axial Chirality in a “Geländer” Oligomer by Length Mismatch of the Oligomer Strands

Helical molecules are not only esthetically appealing due to their structural beauty, they also display unique physical properties as a result of their chirality. We describe herein a new approach to “Geländer” oligomers by interlinking two oligomer strands of different length. To compensate for the dimensional mismatch, the longer oligo(benzyl ether) oligomer wraps around the oligophenyl backbone. The new “Geländer” oligomer 1 was assembled in a sequence of functional‐group transformations and cross‐coupling steps followed by final cyclizations based on nucleophilic substitution reactions, and was fully characterized, including X‐ray diffraction analysis. The isolation of pure enantiomers enabled the racemization process to be studied by circular dichroism spectroscopy.     

“On‐Droplet” Chemistry: The Cycloaddition of Diethyl Azodicarboxylate and Quadricyclane

Sharpless and co‐workers previously studied the [2σ+2σ+2π] cycloaddition of diethyl azodicarboxylate (DEAD) and quadricyclane and reported that the addition of water to the neat reagents caused an acceleration in the reaction rate, giving birth to what has been called “on‐water” chemistry. We have examined the same reaction in aqueous microdroplets (ca. 5 μm diameter) and find that the cycloaddition reaction is accelerated even further (by a factor of 10²) compared to that of the “on‐water” reaction reported previously. The trends of acceleration in solvents other than water demonstrated by Sharpless and colleagues were replicated in the corresponding microdroplet experiments. We also find that DEAD reacts with itself to form a variety of hydrazine carboxylates and intercept intermediates of this reaction in microdroplets to validate a mechanism proposed herein. We suggest that “on‐droplet” chemistry, similar to “on‐water” chemistry, may be a general process of synthetic interest.     

“Synthetic Metals”: A Novel Role for Organic Polymers (Nobel Lecture)

Since the initial discovery in 1977, that polyacetylene (CH)_x, now commonly known as the prototype conducting polymer, could be p‐ or n‐doped either chemically or electrochemically to the metallic state, the development of the field of conducting polymers has continued to accelerate at an unexpectedly rapid rate and a variety of other conducting polymers and their derivatives have been discovered. Other types of doping are also possible, such as “photo‐doping” and “charge‐injection doping” in which no counter dopant ion is involved. One exciting challenge is the development of low‐cost disposable plastic/paper electronic devices. Conventional inorganic conductors, such as metals, and semiconductors, such as silicon, commonly require multiple etching and lithographic steps in fabricating them for use in electronic devices. The number of processing and etching steps involved limits the minimum price. On the other hand, conducting polymers combine many advantages of plastics, for example, flexibility and processing from solution, with the additional advantage of conductivity in the metallic or semiconducting regimes; however, the lack of simple methods to obtain inexpensive conductive polymer shapes/patterns limit many applications. Herein is described a novel, simple, and cheap method to prepare patterns of conducting polymers by a process which we term, “Line Patterning”.     

Derivation of the Clebsch–Gordan coefficients by means of projection operators

The Clebsch–Gordan coefficients for the coupling of two angular momenta are derived by using the projection operator technique, developed by Löwdin. The derivation is done in two steps; first for the so-called principal case, i.e. k = m, then for the general case with an arbitrary m. Two different derivations are given for the principal case, a direct one and one based on a recursion procedure. The general case is obtained from the principal case with a step-down operator.     

The Model of “Minimal Oscillator” Derived from the Kinetic Mechanism of the H2O2–NaSCN–NaOH–CuSO4 Dynamical System

Dissipative chemical reactions, which involve oscillatory variations of the concentrations of the intermediates in time, are usually characterized with complicated kinetic mechanisms. However, the essential source of the oscillations can often be reduced to only a few reaction steps providing the alternative domination of the positive and negative feedback loops. In an extreme case such a reduction leads to the so–called “minimal oscillator,” the concept used in the past for the well‐known Belousov‐Zhabotinsky (BZ) reaction. In the present work, we construct such a minimal system for the (discovered by M. Orbán) H₂O₂–NaSCN–NaOH–CuSO₄ homogeneous oscillator, in which instabilities originate from kinetic mechanism substantially different from that proposed for the BZ system. The methodology involves intuitive analysis of the reaction mechanism, supported by numerical calculations and spectrophotometric measurements. We show how the actual, only three‐variable model evolves from our previously elaborated: nine‐ and five–variable mechanisms and prove that its further reduction to two–variable one is not possible. Thus the present work is a final step in our searches for the “minimal Orbán oscillator”.     

Charge distribution in Mn(salen) complexes

The charge and spin distribution in manganese‐salen complexes were analyzed using different basis sets and density functionals. Five population analysis methods [Mulliken, Löwdin, Natural population analysis (NPA), atoms in molecules (AIM), and CHelpG] were used to characterize the charge distribution. Results show that NPA and AIM were the only methods capable of giving charges with the correct sign for all cases under study. According to the analysis of the natural charge and spin distributions, the salen ligand shows a complex behavior, counteracting the effect of the chloro and oxo ligands on the metal center. Furthermore, the presence of a chloride counter ion increases the oxo‐radical character of Oxo‐Mn(salen) complexes, which may play an important role in the rationalization of the catalytic properties of Mn(salen) complexes. © 2014 Wiley Periodicals, Inc.     

Designing Homogeneous Bromine Redox Catalysis for Selective Aliphatic C−H Bond Functionalization

The potential of homogeneous oxidation catalysis employing bromine has remained largely unexplored. We herein show that the combination of a tetraalkylammonium bromide and meta‐chloroperbenzoic acid offers a unique catalyst system for the convenient and selective oxidation of saturated C(sp³)−H bonds upon photochemical initiation with day light. This approach enables remote, intramolecular, position‐selective C−H amination as demonstrated for 20 different examples. For the first time, an N‐halogenated intermediate was isolated as the active catalyst state in a catalytic Hofmann–Löffler reaction. In addition, an expeditious one‐pot synthesis of N‐sulfonyl oxaziridines from N‐sulfonamides was developed and exemplified for 15 transformations. These pioneering examples provide a change in paradigm for molecular catalysis with bromine.