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This chapter presents 20 problems covering the subject of bonding and reactivity of metal clusters, along with their corresponding solutions.

This chapter presents 23 problems covering the subject of atomic orbitals, along with the corresponding solutions.

This chapter presents 32 problems covering the subject of transition metal chemistry, along with their corresponding solutions.

This chapter presents 47 problems covering the subject of crystal structure, along with their corresponding solutions.

This chapter presents 22 problems covering the subject of vibrational spectroscopy, along with the corresponding solutions.

This chapter presents 52 problems covering the subject of molecular geometry and bonding, along with the corresponding solutions.

This chapter presents 41 problems covering the subject of crystal field theory, along with the corresponding solutions.

This chapter presents 19 problems covering the subject of bioinorganic chemistry, along with their corresponding solutions.

This chapter presents 23 problems covering the subject of molecular symmetry, along with the corresponding solutions.

The First Edition of this book, which appeared in 2013, serves as a problem text for Part I (Fundamentals of Chemical Bonding) and Part II (Symmetry in Chemistry) of the book Advanced Structural Inorganic Chemistry published by Oxford University Press in 2008. A Chinese edition was published by Peking University Press in August in the same year. Since then the authors have received much feedback from users and reviewers, which prompted them to prepare a Second Edition for students ranging from freshmen to senior undergraduates who aspire to attend graduate school after finishing their first degree in Chemistry. Four new chapters are added to this expanded Second Edition, which now contains over 400 problems and their solutions. The topics covered in 13 chapters follow the sequence: electronic states and configurations of atoms and molecules, introductory quantum chemistry, atomic orbitals, hybrid orbitals, molecular symmetry, molecular geometry and bonding, crystal field theory, molecular orbital theory, vibrational spectroscopy, crystal structure, transition metal chemistry, metal clusters: bonding and reactivity, and bioinorganic chemistry. The problems collected in this volume originate from examination papers and take-home assignments that have been part of the teaching program conducted by senior authors at The Chinese University of Hong Kong over nearly a half-century. Whenever appropriate, source references in the chemical literature are given for readers who wish to delve deeper into the subject. Eight Appendices and a Bibliography listing 157 reference books are provided to students and teachers who wish to look up comprehensive presentations of specific topics.

This chapter presents 33 problems covering the subject of introductory quantum chemistry, along with the corresponding solutions.

This chapter presents 29 problems covering the subject of electronic states and configurations of atoms and molecules, along with the corresponding solutions.

A calculation method for heats of formation (HOF, referred to as △Hf) based on the density functional theory (DFT) is presented in this work. Similar to Gaussian-3 theory, the atomic scheme is applied to calculate the heats of formation of the molecules. In this method, we have modified the formula for calculation of Gaussian-3 theory in several ways, including the correction for diffuse functions and the correction for higher polarization functions. These corrections are found to be significant. The average absolute deviation from experiment for the 164 calculated heats of formation is about 1.9 kcal·mol−1, while average absolute deviation from G3MP2 for the 149 (among the 164 molecules, 15 large-sized molecules can not be calculated at the G3MP2 level) calculated heats of formation is only about 1.9 kcal·mol−1. It indicates that the present method can be applied to predict the heats of formation of medium-sized and large-sized molecules, while the heats of formation of these molecules using Gaussian-3 theory are much difficult, even impossible, to calculate. That is, this method provides a choice in the calculation of △Hf for medium-sized and large-sized molecules.

The water-mediated neutral hydrolysis mechanism of carbonyl sulfide (OCS) has been re-examined using the hybrid supramolecule/continuum models with n = 2-8 explicit water cluster at the level of MP2(fc)(CPCM)/6-311++G(d,p)//MP2(fc)(CPCM)/6-31+G(d). Present calculations indicate that the potential energy surface in water solution is different from the one in the gas-phase, and only stepwise mechanism is observed in aqueous solution, i.e., monothiocarbonic acid (H2CO2S) is formed via monothiocarbonate (OCSOH(-), MTC) and its counterion, protonated water cluster, (H2O)nH3O(+). The predicted rate-determining step (RDS) barrier for the stepwise mechanism in water solution, about 90 kJ/mol, shows good agreement with the experimental values, 83.7-96.2 kJ/mol using six- or eight-water model including two cooperative water molecules. Moreover, two reaction pathways, the nucleophilic addition of water molecule across the C═O or the C═S bond of OCS are competitive.

In order to explore the existence of α-effect in gas-phase SN2@N reactions, and to compare its similarity and difference with its counterpart in SN2@C reactions, we have carried out a theoretical study on the reactivity of six α-oxy-Nus (FO−, ClO−, BrO−, HOO−, HSO−, H2NO−) in the SN2 reactions toward NR2Cl (R = H, Me) and RCl (R = Me, i-Pr) using the G2(+)M theory. An enhanced reactivity induced by the α-atom is found in all examined systems. The magnitude of the α-effect in the reactions of NR2Cl (R = H, Me) is generally smaller than that in the corresponding SN2 reaction, but their variation trend with the identity of α-atom is very similar. The origin of the α-effect of the SN2@N reactions is discussed in terms of activation strain analysis and thermodynamic analysis, indicating that the α-effect in the SN2@N reactions largely arises from transition state stabilization, and the “hyper-reactivity” of these α-Nus is also accompanied by an enhanced thermodynamic stability of products from the n(N) σ*(O−Y) negative hyperconjugation. Meanwhile, it is found that the reactivity of oxy-Nus in the SN2 reactions toward NMe2Cl is lower than toward i-PrCl, which is different from previous experiments, that is, the SN2 reactions of NH2Cl is more facile than MeCl. © 2013 Wiley Periodicals, Inc.

A simple method is presented which allows us to estimate the binding energies and bonding capabilities of exotic species formed by electrons, positrons, and protons, namely, the positronium atom (Ps) and the dipositronium (Ps2) and positronium hydride (PsH) molecules. These exotic species are in fact the simplest clusters formed by the aforementioned fundamental particles and antiparticles. The simple Bohr atom model for hydrogen is extended to calculate the binding energies and radii of atomic orbitals of the Ps atom, as well as its atomic and covalent sizes and electronegativity. Based on these properties, the bond lengths and bond energies, including the covalent and ionic contributions, of the Ps2 and the PsH molecules can be estimated from the corresponding (known) values of the dihydrogen molecule (H2) using a Huckel-like molecular orbital calculation.

The reactivity order of 12 anions toward ethyl chloride has been investigated by using the G2(+) method, and the competitive E2 and SN2 reactions are discussed and compared. The reactions studied are X(-) + CH3CH2Cl → HX + CH2═CH2 + Cl(-) and X(-) + CH3CH2Cl → CH3CH2X + Cl(-), with X = F, Cl, Br, HO, HS, HSe, NH2 PH2, AsH2, CH3, SiH3, and GeH3. Our results indicate that there is no general and straightforward relationship between the overall barriers and the proton affinity (PA) of X(-); instead, discernible linear correlations only exist for the X's within the same group of the periodic table. Similar correlations are also found with the electronegativity of central atoms in X, deformation energy of the E2 transition state (TS), and the overall enthalpy of reaction. It is revealed that the electronegativity will significantly affect the barrier height, and a more electronegative X will stabilize the E2 and SN2 transition states. Multiple linear regression analysis shows that there is a reasonable linear correlation between E2 (or SN2) overall barriers and the linear combination of PA of X(-) and electronegativity of the central atom.

Doping effects on the structural and electronic properties of ladderanes and ladder polysilanes have been studied using density functional theory. Two types of doping: substitution with isoelectronic atoms or heteroatoms (or radicals), have been used to design low band gap ladderanes. It is found that the B-doped [n]-ladderanes and 1,2 P-doped [n]-silaladderanes exhibit a very noticeable bent conformation, whereas the 1, 2 and 1,3 N-doped ladderanes, P-doped ladderanes, and silaladderanes keep the relatively straight ladder shapes. The isoelectronic atom doping reduces the HOMO―LUMO (H-L) gaps of [n]-ladderanes but increases those of [n]-silaladderanes with n > 5. The present results show that isoelectronic atom doping is not an effective way to decrease the H-L gaps of ladderanes and silaladderanes. Heteroatom doping has a more pronounced effect than the isoelectronic atom doping. The HOMOs of heteroatom-doped ladderanes and silaladderanes are destabilized and LUMOs are stabilized, leading to significant reduction of H-L gaps. Most of the B-, N-, and P-doped [n]-silaladderanes we designed have H-L gaps below 1.5 eV. Therefore, it is expected that these silaladderanes are promising candidates of conductive or semiconductive materials. The heteroatom doping is a viable approach to reduce H-L gaps for the silaladderanes. In addition, it is found that nine different density functionals, including B3LYP, SVWN LDA, four pure GGAs, and three hybrid GGAs, as well as the time-dependent B3LYP method, all lead to the same predictions on the H-L gaps of ladderanes, silaladderanes, as well as their doped derivatives.

This article is divided into two parts. In the first part, the atomic unit (au) system is introduced and the scales of time, space (length), and speed, as well as those of mass and energy, in the atomic world are discussed. In the second part, the utility of atomic units in quantum mechanical and spectroscopic calculations is illustrated with examples. It is shown that the system of atomic units can greatly facilitate calculations and unit conversions. In addition, a number of topics that can be fruitfully treated with the au system, including photoelectric effect, hydrogen atom and different Rydberg constants, positronium atom, atomic mass unit, nuclear binding energy, nuclear reaction energetics, annihilation and mass-energy interconversions, femtosecond and attosecond spectroscopies, quantum-size effects, excitons in semiconductors, and so forth are presented and discussed.

Computational investigations on the gas-phase homodimerization and the keto-to-enol tautomerism of monochalcogenocarboxylic acids, CH3C(=O)XH (X = S, Se, Te) , were performed using ab initio molecular orbital methods (HF and MP2) with the 6-311+G(d, p) and 6-311+G(2df, 2df) basis sets. Calculated results indicate that the dimerization enthalpy values, ΔH, for the enol-dimers, [CH3C(=X)OH]2(X = S, Se, Te) , are notably higher than those for the corresponding keto-dimers, [CH3C(=O)XH]2(X = S, Se, Te) , while the ΔH values decrease as the electronegativity of chalcogen atom is lowered, for both the keto- and enol-dimers. It is found that the homodimerization of monochalcogenocarboxylic acids is thermodynamically unfavorable because the releasing dimerization heat cannot overcome the loss of entropy. This is contrasted with the case in the carboxylic acid, where the much higher dimerization enthalpy is responsible for the favorable dimerization in the gas phase. Our results also suggest that the tautomeric reactions of the monochalcogenocarboxylic acids in the gas phase may proceed by an eight-membered ring TS with intermolecular double proton transfer and lower tautomeric barrier, instead of by direct intramolecular proton transfer in monomer with a highly strained four-membered TS. The geometrical and energetic characteristics of the homodimers and tautomeric TSs are also further elucidated by NBO analysis.

The enhanced reactivity exhibited by six pseudo-α-bases, RC≡CZ− (R = H and Cl; Z = O, S, and Se) in gas-phase E2 reactions with ethyl chloride was examined at the G2(+) level. It is found that anomalous reactivity is observed despite the fact that these chalcogen bases do not possess adjacent lone-pair electrons. The influence of the halide leaving groups on the α-effect and the origin of the α-effect in the E2 reactions of ethyl halides are investigated and discussed.

The results of a theoretical study of the reaction mechanism for the neutral hydration of ketene, H2C=C=O + (n + 1) H2O → CH3COOH + nH2O (n = 0–4), in solution are presented. All structures were optimized and characterized at the MP2(fc)/6-31 + G* level of theory, and then re-optimized by MP2(fc)/6-311 ++G**, and the effect of the bulk solvent is taken into account according to the conductor-like polarized continuum model (CPCM) using the gas MP2(fc)/6-311 ++G** geometries. Energies were refined for five-water hydration at higher level of theory, QCISD(T)(fc)/6-311 ++G**//MP2(fc)/6-311 ++G**. In the combined supermolecular/continuum model, one water molecule directly attacks the central C-atom, and the other four explicit water molecules are divided into two groups, one acting as catalyst(s) by participating in the proton transfer to reduce the tension of proton transfer ring, and the other being placed near the non-reactive oxygen or carbon atom in order to catalyze the hydration by engaging in hydrogen-bonding to the substrate (the so-called cooperative effect). Between the two possible nucleophilic addition reactions of water molecule, across the C=O bond or the C=C bond, the former one is preferred. Our calculations suggest that the favorable hydrolysis mechanism of ketene involves a sort of eight-membered ring transition structure formed by a three-water proton transfer loop, and a cooperative dimeric water near the non-reactive carbon-atom. The best-estimated in the present paper for the rate-determining barrier in solution, $$ \Updelta G_{\text{sol}}^{ \ne } $$ (298 K), is about 58 kJ/mol, reasonably close to the available experimental result.

Among various nanomaterials, one-dimensional silicon-based nanomaterials occupy a very important place because of their unique mechanical, thermal, optical, electrical, and optoelectronic properties, and potential technological applications. Herein, we review the recent theoretical calculations of the structures and properties of silicon-based nanowires and nanotubes, with special emphasis on the particularities and peculiarities of silicon and silicon-containing nanowires and nanotubes. These calculations allow not only rationalization of the observed experimental results, but also predictions of the yet-unknown nanostructures and their unusual properties.

Due to the poor water solubility of andrographolide (andro), an inclusion technique has been developed to modify its physical and chemical properties so as to improve its bioavailability. In contrast with the immense experimental studies on the inclusion complexes of andro:cyclodextrin, no computational study has so far been carried out on this system. In this work, preliminary docking experiments with AutoDock were performed. Density Functional Theory (DFT) and Austin Model 1 (AM1) calculations upon the docking instances were applied to investigate the two possible modes of molecular inclusions between andro and x-cyclodextrin (xCD, where x is alpha, beta or gamma). Atoms-in-Molecules (AIM) analysis based on the B3LYP/cc-pVDZ wavefunction was applied to verify the existence of the intermolecular hydrogen bonds. It was found that the most stable complex among the six possible inclusion complexes was the one formed between andro and betaCD with andro's decalin ring moiety wrapped by CD at a ratio of 1:1. The hydrogen bonds between andro and CD were responsible for the stability of the inclusion complexes. The calculated data were found to be consistent with the experimental results. Thus, the results of this study can aid new drug design processes.

A novel species, carbon-centered tri- s -tetrazine (C 4 N 9 H 3 ), and its 10 derivatives (C 4 N 9 R 3 , where R = OH, F, CN, N 3 , NH 2 , NO 2 , N NH, N 2 H 3 , C CH, and CH CH 2 ) have been studied computationally. Density functional theory (DFT) has been used to study the geometries, electronic structure, harmonic vibrational frequencies, ionization energies of the 11 compounds at the restricted (for neutrals) and the unrestricted (for cations) B3LYP/cc-pVDZ level of theory. Atoms in molecule (AIM) and natural bond orbital (NBO) analyses have been used to obtain the bonding properties. Valence bond (VB) theory is applied to explain the unusual pyramidal structure around the carbon-center and electron arrangements of orbitals. We found: (1) All the species possess novel bonding features and geometrical structures. The atoms on the periphery of each species are sp 2 hybridized. Each of these atoms offers an orbital to form an extensive conjugation system 12 π 15 (a π system consisting of 12 centers and 15 electrons). The central carbon atom C13 is sp 3 hybridized, which makes the non-planar molecule shape like a straw–hat. Atom C13 also participates in the conjugated π system with its sp 3 hybridized orbital, thus forming an extensive 13 π 16 conjugate π system covering the whole C 4 N 9 framework. (2) The change of charge on C13 is the largest among all the atoms when the species is ionized and the atomic charges are redistributed. In other words, C13 is the attack center for electrophilic agents. Thus, the species is carbanion-like. (3) All the species have low ionization energies (IEs). The electron ionized mainly comes from C13. They may have wide applications in organic chemistry, in organometallic chemistry and in alkyl lithium chemistry once they are synthesized.

In this work, the geometries, harmonic vibrational frequencies, and high-energy density material (HEDM) properties of a novel species and its six derivatives with the general formula C14N12-R6 (R = H, OH, F, CN, N3, NH2, and NO2) have been investigated at the restricted and unrestricted B3LYP/cc-pVDZ levels of theory. Natural bond orbital (NBO), natural orbital (NO), and atoms in molecules (AIM) analyses are applied to examine their electronic topologies. It is found that for the four species of R = H, CN, N3, and NO2, (1) there exist high LUMO occupation numbers, (2) there is considerable spin density congregated on the two central carbon atoms, (3) there exists through space interaction (or intramolecular interaction, which is one of the stabilizing factors of a diradicaloid) between the two central carbon atoms, (4) the distance (about 3 A) between the two central carbon atoms (as the apexes of two trigonal pyramids with their bases facing each other) is suitable and favorable for diradical formation. All the results support that these four species are diradicals or diradicaloids. Furthermore, the appreciable singlet-triplet energy gaps indicate that these four diradicals tend to have a singlet ground state. There is a moderate HOMO-LUMO gap (on the order of 1.5 to 2.1 eV) for these four species. These four singlet diradicals may be novel organic semiconductor materials or nonlinear optical materials. On the other hand, the remaining three species, with R = OH, F, and NH2, are not diradicaloids.