Hund's rules

We review the present state of our undertanding of Hund's first and second rules, their domains of validity, and of generalizations in cases where the rules in their original form are invalid. These exceptions occur mainly in atomic configurations with more than one open shell withl ≥ 1, but also in cases with large orbital angular momentaL. We present a derivation of thealternating rule, which is, in some sense, a generalization of Hund's first rule, and present new rules, which generalize Hund's second rule. The importance of the concept ofunnatural parity states for an understanding of these rules is stressed. It is demonstrated that the lowest singlet-triplet average energy for the sameL corresponds to a pair of unnatural parity states (theunnatural parity rule). For sufficiently smalll ₁ andl ₂, the lowest average energy is that of the pair of states with the maximum possibleL (maximum-L rule), though exceptions are found already for moderately large values ofl ₁ andl ₂. Our analysis indicates that the validity of Hund's second rule is to some extent an accidental consequence of the minimisation of an elementary function ofl ₁,l ₂, andL — which does not depend monotonically onL — over states of unnatural parity, and that a more general and more fundamental rule should be formulated in terms of this function. We also discuss the generalisation of Hund's rules to molecules, as well as violations of them, with particular emphasis on the inversion of Hund's first rule by spin-polarization in molecules.     

Relation of the charge and spin correlation functions to the structures of π electron wave functions in alternant hydrocarbons

We analyze, in the Pariser–Parr–Pople (PPP ) model of alternant hydrocarbons, how the charge and spin correlation functions (CF 's) are related to the structure of a CI wave function and the MO 's of the systems. The analysis is based on the fact that an uncorrelated electron present in an orbital does not contribute to the linked dynamically correlated parts of the CF 's. By using the fact, simple rules are deduced predicting the covalent or ionic nature of the correlation structure in a low-lying state with two correlated electrons. The rules predict that the singlet minus and triplet plus states are covalent, while singlet plus and triplet minus states are ionic, where the plus and minus mean the alternancy symmetry. The rules also give a prediction for the unknown charge and spin correlation structures between different sites.     

Restricted and unrestricted Hartree–Fock approaches applied to spherical quantum dots in a magnetic field

The Roothaan and Pople–Nesbet approaches for real atoms are adapted to quantum dots in the presence of a magnetic field. Single‐particle Gaussian basis sets are constructed, for each dot radius, under the condition of maximum overlap with the exact functions. The chemical potential, charging energy, and total spin expected values are calculated, and we have verified the validity of the quantum dot energy shell structure as well as Hund's rule for electronic occupation at zero magnetic field. At finite field, we have observed the violation of Hund's rule and studied the influence of magnetic field on the closed and open energy shell configurations. We have also compared the present results with those obtained within the LS‐coupling scheme for low electronic occupation numbers. We focus only on ground‐state properties and consider quantum dots populated up to 40 electrons, constructed by GaAs or InSb nanocrystals. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Noncollinear magnetization density in VAu4

The spin and orbital magnetic moments of VAu₄ have been calculated using a first principles method that allows for noncollinear magnetic ordering. The large spin–orbit coupling of the Au atom is argued to induce large noncollinear components of the magnetization density as well as a parallel coupling between spin and orbital moments of the V atom, in contrast to expectations from Hund's third rule. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

August Wilhelm Hofmann—“Reigning Chemist-in-Chief”

One hundred and twenty-five years ago, on November 11th, 1867, the German Chemical Society of Berlin held its inaugural meeting. The main purpose of the Society was to unite pure and applied chemistry and to foster cooperation between academic research and the chemical industry. And, indeed, it soon became the major forum of German and even European chemistry. Its program clearly bears the hallmark of a single individual: August Wilhelm Hofmann, the Society's first president, who died 100 years ago. For his contemporaries, Hofmann represented a new type of chemistry professor. At no time since have professional chemists felt as abundantly endowed with potential for the future and with public esteem. Hofmann's portrait was monumental even then, and still today it would belong in any gallery devoted to our distinguished forebears. Anniversaries provide an opportunity to direct our attention toward the past—and thus to ourselves as well. We are, after all, heirs to that period from which the modern world derives its profile. Questions from our own time lead us to reacquaint ourselves with one of the founders of modern chemistry, but we may also benefit from a fresh look at an epoch which, beneath the surface of prosperity and progress, was as contradictory as our own, an epoch struggling to understand the role of science in the new industrial era.     

Quantum chemistry and Dick Stufkens photochemistry

A survey of the quantum chemical results obtained for several MLCT complexes studied in Amsterdam is presented in order to illustrate the progress made in this field from the beginning of the 1980s when excited states properties were analyzed in terms of bonding and antibonding molecular orbitals (MO). Nowadays the photoactive states and the main features of the absorption spectra can be determined without any ambiguity. The time scales of elementary processes such as direct ultra-fast dissociations or intersystem crossing processes, are also readily available.     

What Will Chemistry Do in the Next Twenty Years?

The path of chemistry in the future will be determined both by its participation in solving large-scale societal problems and by its generation of new ideas through basic research. This article sketches four of the areas of societal “pull” in which chemistry will play a role in solving applied problems—national security, health care, the environment, and energy—and four areas in which basic research will be especially fruitful—materials chemistry, biological chemistry, computational chemistry, and chemistry exploring the limits of size and speed in chemical phenomena.     

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.     

The structural theory and physical organic chemistry

Woolley's revolutionary proposal that quantum mechanics does not sanction the concept of “molecular structure”—which is but only a “metaphor”—has fundamental implications for physical organic chemistry. On the one hand, the Uncertainty Principle limits the precision with which transition state structures may be defined; on the other, extension of the structure concept to the transition state may be unviable. Attempts to define transition states have indeed caused controversy. Consequences for molecular recognition, and a mechanistic classification, are also discussed.     

Symmetry-Induced Singlet-Triplet Inversions in Non-Alternant Hydrocarbons**

Molecules with inversion of the singlet and triplet excited-state energies are highly promising for the development of organic light-emitting diodes (OLEDs). To date, azaphenalenes are the only class of molecules where these inversions have been identified. Here, we screen a curated database of organic crystal structures to identify existing compounds for violations of Hund's rule in the lowest excited states. We identify two further classes with this behavior. The first, a class of zwitterions, has limited relevance to molecular emitters as the singlet-triplet inversions occur in the third excited singlet state. The second class consists of two D_2h-symmetry non-alternant hydrocarbons, a fused azulene dimer and a bicalicene, whose lowest excited singlet states violate Hund's rule. Due to the connectivity of the polycyclic structure, they achieve this symmetry through aromatic stabilization. These hydrocarbons show promise as the next generation of building blocks for OLED emitters.     

Photoionization of oriented molecules

A general expression for the angular distribution of photoelectrons with defined spin polarization ejected from oriented molecules is derived in the electric-dipole approximation in the limit of a weak radiation field. An analysis of its geometrical part permits to draw definite conclusions without calculating matrix elements. For linear molecules it is shown that in Hund's cases (a) and (b) photoelectrons may be polarized only parallel to the molecular axis, while in Hund's case (c) they may be polarized in any direction. Appearance of a circular dichroism for nonchiral oriented molecules of relatively low symmetry is predicted. Dependence of a circular dichroism in the angular distribution of photoelectrons on the symmetry of molecules is demonstrated. The results may serve as the framework for studying molecules oriented on surfaces, in liquid crystals, or by molecular beam techniques.     

Intuitive local picture for spin polarization of chemical bonds

The spin polarization of chemical bonds near radical centers is investigated by an analytical spin unrestricted Hartree–Fock model with numerical examples. The centroid analysis of localized molecular orbitals is also introduced to obtain an intuitive local picture for the spin polarization. The alternation of spin alignments in molecules are discussed with orbital symmetries and Hund's rule through chemical bonds. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Spin quenching of transition metals deposited on MgO insulator and CdO semiconductor density functional calculations

We have analyzed spin quenching of first row transition metals deposited on (001) defect‐free and defect‐containing surfaces of MgO insulator and CdO semiconductor by means of density functional calculations and embedded cluster model. Clusters of moderate sizes were embedded in the simulated Coulomb fields that closely approximate the Madelung fields of the host surfaces, and relaxation of ions that surround the defect sites was taken into account. Spin states of metals deposited on the defect free surfaces were maintained as in the isolated metals except for Ti, V, and Co on MgO, and Ti, V, and Cr on CdO. On the defect containing surfaces, spin states were maintained too except for Fe on MgO, and V and Cr on CdO. The metal‐support interactions stabilize the low spin state of the adsorbed metal with respect to the isolated metal, but the effect was not in general enough to quench the spin. Spin polarization effects tend to preserve the spin states of the adsorbed metals relative to those of the isolated metals. Although charge transfer took place from the adsorbed metal to the insulator surface, it took place the other way round from the semiconductor surface to the adsorbed metal. The encountered variations in magnetic properties were attributed to the smaller band gap of the semiconductor, and the behavior of a single metal atom adsorbed on a particular surface was a result of a competition between Hund's rule for the adsorbed metal and the formation of a chemical bond at the interface. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Fundamental theories and their empirical patches

Many theories require empirical patches or ad hoc assumptions to work properly in application to chemistry. Some examples include the Bohr quantum theory of atomic spectra, the Pauli exclusion principle, the Marcus theory of the rate-equilibrium correlation, Kekule’s hypothesis of bond oscillation in benzene, and the quantum calculation of reaction pathways. Often the proposed refinements do not grow out of the original theory but are devised and added ad hoc. This brings into question the goal of constructing theories derived from first principles and the concept of ranking the merit of theories according to their freedom from empirical contamination.     

Spin—orbit and vibronic perturbations on the chiroptical spectra of dissymmetric pseudo-tetragonal metal complexes

The influence of spin—orbit and vibronic interactions upon the chiroptical properties of nearly degenerate dd transitions in metal complexes of pseudo-tetragonal symmetry is investigated. A model system is considered in which three nearly degenerate dd excited states are coupled via both spinorbit and vibronic interactions. Vibronic interactions among the three nearly degenerate dd electronic states are assumed to arise from a pseudo-Jahn—Teller (PJT) mechanism involving three different vibrational modes (each nontotally symmetric in the point group of the undistorted model system).A vibronic hamiltonian is constructed (for the excited states of the model system) which includes linear coupling terms in each of the three PJT-active vibrational modes as well as a linear coupling term in one totally symmetric mode of the system and a spin—orbit interaction term. Wavefunctions and eigenvalues for the spin—orbit/vibronic perturbed excited states. of the model system are obtained by diagonalizing this hamiltonian in a basis constructed of uncoupled vibrational and electronic (orbital and spin) wavefunctions.Rotatory strengths associated with transitions to vibronic levels of the perturbed system are calculated and “rotatory strength spectra” are computed assuming gaussian shaped vibronic spectral components. Calculations are carried out for a number of vibronic and spin—orbit coupling parameters and for various splitting energies between the interacting electronic states. The calculated results suggest that chiroptical spectra associated with transitions to a set of nearly degenerate dd excited states of a chiral transition metal complex cannot be interpreted directly without some consideration of the effects introduced by spin—orbit and vibronic perturbations. These perturbations can lead to substantial alterations in the sign patterns and intensity distributions of rotatory strength among vibronic levels derived from the interacting electronic states and it is generally not valid to assign specific features in the observed circular dichroism spectra to transitions between states with well-defined electronic (orbital and spin) identities.Our theoretical model is conservative with respect to the total (or net) rotatory strength associated with transitions to levels derived from the three interacting electronic states; the vibronic and spin—orbit coupling operators are operative only within this set of states. That is, the total (or net) rotatory strength associated with these transitions remains invariant to the vibronic and spin—orbit coupling parameters of the model.     

计算化学在分子轨道教学中的应用

分子轨道理论是理解分子电子结构与微观性质的重要理论之一,也是本科生与研究生结构化学教学中的重点与难点。学生对原子轨道组合形成分子轨道、分子轨道能级交叉混合等知识的理解缺乏形象直观、定量的认识。本文通过基于量子化学或密度泛函理论的Gaussian 03计算软件,计算、绘制并分析了F_2、O_2、N_2、HF、CO等的分子轨道能级图,将学生较难理解的内容定量、直观地呈现出来,形象地解释了分子轨道成键原则与电子填充原则等分子轨道理论中的重难点,加深了学生对分子轨道理论的理解,特别是sp轨道混杂导致的σ_(2p_z)与π_(2p)轨道能级交叉这一难点,激发了学生学习的主动性和积极性,提高了教学质量。在此基础上,利用分子轨道理论分析了CO_2的电子结构,使学生学会应用分子轨道理论解决实际问题,巩固了相关课堂理论知识。     

Correlation effects in the low–lying excited states of the PPP models of alternant hydrocarbons. I. Qualitative rules for the effect of limited configuration interaction

Simple rules for an estimate of the correlation effects in the low-lying states of alternant hydrocarbons, as described by the Pariser–Parr–Pople Hamiltonian, are formulated. These rules are based on the alternancy and spin symmetry classification of states in both strongly and weakly correlated limits and on the valence bond characteristics of those states in the fully correlated limit. It is shown that the largest effect of the electron correlation will be found for the singlet “minus” states (using Pariser's classification of the alternancy symmetry species), a smaller effect for the triplet “plus” states, and a much smaller effect for the remaining states. These rules are exemplified by limited CI calculations including all monoexcited and all mono- and bi-excited configurations, respectively, for a number of π-electronic systems. In view of these rules the success of the PPP model in the monoexcited CI approximation may be understood.