Orbital correspondence analysis in maximum symmetry: Formulation and conceptual framework

A recently proposed method for the analysis of the course of chemical reactions, based on the maximal use of available symmetry, is formulated as a set of procedural rules. The application of these rules is illustrated with a simple prototype reaction: CH₂+C₂H₄ fcyclo-C₃H₆. They are then derived, using the formalism of time-dependent perturbation theory within the Born-Oppenheimer approximation, thus bringing out the method's underlying assumptions and its relation to the widely used Woodward-Hoffmann procedure.     

Symmetry-enthalpy correlations in diels-alder reactions

Woodward-Hoffmann (WH) rules provide strict symmetry selection rules: when they are obeyed, a reaction proceeds; when they are not obeyed, there is no reaction. However, the voluminous experimental literature provides ample evidence that strict compliance to symmetry requirements is not an obstacle for a concerted reaction to proceed, and therefore the idea has developed that it is enough to have a certain degree of the required symmetry to have reactivity. Here we provide quantitative evidence of that link, and show that as one deviates from the desired symmetry, the enthalpy of activation increases, that is, we show that concerted reactions slow down the further they are from the ideal symmetry. Specifically, we study the deviation from mirror symmetry (evaluated with the continuous symmetry measure (CSM)) of the [4+2] carbon skeleton of the transition state of a series of twelve Diels-Alder reactions in seven different solvents (and in the gas phase), in which the dienes are butadiene, cyclopentadiene, cyclohexadiene, and cycloheptadiene; the dienophiles are the 1-, 1,1-, and 1,1,2-cyanoethylene derivatives; the solvents were chosen to sample a range of dielectric constants from heptane to ethanol. These components provide twenty-four symmetry-enthalpy DFT-calculated correlation lines (out of which only one case is a relatively mild exception) that show the general trend of increase in enthalpy as symmetry decreases. The various combinations between the dienophiles, cyanoethylenes, and solvents provide all kinds of sources for symmetry deviations; it is therefore remarkable that although the enthalpy of activation is dictated by various parameters, symmetry emerges as a primary parameter. In our analysis we also bisected this overall picture into solvent effects and geometry variation effects to evaluate under which conditions the electronic effects are more dominant than symmetry effects.     

Woodward-Hoffmann rules in density functional theory: initial hardness response

The Woodward-Hoffmann rules for pericyclic reactions, a fundamental set of reactivity rules in organic chemistry, are formulated in the language of conceptual density functional theory (DFT). DFT provides an elegant framework to introduce chemical concepts and principles in a quantitative manner, partly because it is formulated without explicit reference to a wave function, on whose symmetry properties the Woodward-Hoffmann [J. Am. Chem. Soc. 87, 395 (1965)] rules are based. We have studied the initial chemical hardness response using a model reaction profile for two prototypical pericyclic reactions, the Diels-Alder cycloaddition of 1,3-butadiene to ethylene and the addition of ethylene to ethylene, both in the singlet ground state and in the first triplet excited state. For the reaction that is thermally allowed but photochemically forbidden, the initial hardness response is positive along the singlet reaction profile. (By contrast, for the triplet reaction profile, a negative hardness response is observed.) For the photochemically allowed, thermally forbidden reaction, the behavior of the chemical hardness along the initial stages of the singlet and triplet reaction profiles is reversed. This constitutes a first step in showing that chemical concepts from DFT can be invoked to explain results that would otherwise require invoking the phase of the wave function.     

Symmetry in the design of NMR multiple-pulse sequences

The symmetry principles of NMR pulse-sequence design are summarized. The discussion is guided by an analogy with tiling schemes in the decorative arts. The symmetry operations for NMR pulse sequences are discussed in terms of excitation field modifiers and temporal modifiers. The quantum operators which describe the effect of these modifiers on the excitation field spin Hamiltonian are provided. The symmetry transformations of spin propagators, and the different types of pulse-sequence elements are discussed. The common types of symmetry expansion are treated using the propagator transformations and the Euler angles for the excitation field propagators. The selection rules associated with symmetrical pulse sequences are discussed using average Hamiltonian theory.     

The basis for the consistency between the woodward-hoffmann rule and fukui's frontier orbital theory

The Woodward-Hoffmann rule and Fukui's Frontier orbital theory are based on two rather different concepts, yet the two selection rules generally yield consistent conclusions concerning the feasibility of a concerted reaction. This paper suggests that their consistency can be understood in terms of the HOMO-HOMO interaction. First, one can show that the result of the HOMO-HOMO interaction reinforces that of the HOMO-LUMO interaction. Therefore it can be viewed that the HOMO-HOMO interaction is actually taken into account implicitly when the selection rule based solely on the HOMO-LUMO interaction is applied. Secondly, one can make an explicit interpretation of the Woodward-Hoffmann scheme in terms of the HOMO-HOMO interaction.     

Quantifying asymmetry in concerted reactions: solvents effect on a Diels-Alder cycloaddition

We propose the notion that if asymmetry characterizes a concerted reaction, a quantitative treatment in terms of continuous symmetry can bridge the gap between the Woodward-Hoffmann (WH) rules, originally formulated for symmetry-idealized unsubstituted reactants, and the fact that these rules hold for a much wider scope of reactions. Instead of focusing on symmetry conservation along the minimum energy path, we suggest that the distortion with respect to the original expected symmetry must attain a certain minimal value, not necessarily zero. To demonstrate this approach we studied the effect of solvents on the symmetry and reactivity of the classical [4 + 2] Diels-Alder cycloaddition of (E,E)-1,4-dimethoxy-1,3-butadiene with tetracyanoethylene, revealing the predictive value of this approach. Calculations of the enthalpy of activation and the charge separation at the transition state (TS) predict increased reactivity with the polarity of the solvent. The symmetry measure is in excellent correlation with the enthalpy of activation and the charge separation at the TS, indicating the higher reactivity of the more symmetric case, thus quantifying the main teaching of the WH rules. The advantages of using a global structural parameter that takes into account all geometrical parameters, i.e., the symmetry measure, over specific ones (e.g., asynchronicity) are discussed.     

Selection rules in alternant systems

The CI space X_n generated by n electrons moving over 2n spin orbitals is considered. It is shown that the transitions between different eigenstates ψ_i ? X_n of alternant and weakly alternant Hamiltonians are governed by some special selection rules. These selection rules are characteristic to alternant systems, and they do not apply to nonalternant systems. The set of all such selection rules can be easily derived from the splitting theorem. In particular, the selection rules associated with spin independent alternant systems are considered. As an example, the PPP Hamiltonian ?_p describing netural alternant hydrocarbons is treated. In the case of electron dipole transitions between eigenstates ψ_i ? X_n of the Hamiltonian ?_p, the selection rules obtained are in agreement with the selection rules derived previously by Pariser and McLachlan.     

Cyclohexadienyl — A homoaromatic radical?

A reinterpretation of the hyperfine coupling constants of the ESR spectrum of cyclohexadienyl radical is given in terms of a homoaromatic model in agreement with the Woodward-Hoffmann symmetry rules.This research was supported by Italian National Research Council, Chemistry Committee.     

Alternant systems and their properties

Neutral, ionic, and complete alternant systems are studied using the alternancy symmetry adapted (ASA ) approach. This approach is based on an explicit construction of ASA operators that are up to the sign invariant with respect to the particle-hole symmetry transformation. These operators serve as building blocks of alternant systems, and they determine their characteristic properties. All Hamiltonians describing neutral alternant systems are explicitly constructed. Up to some minor restrictions, all Hamiltonians describing ionic and complete alternant systems are also explicitly constructed. Inversely, given a Hamiltonian ? in a second quantization notation, one can easily check whether or not this Hamiltonian describes a neutral (ionic, complete) alternant system. All linear properties characteristic to neutral (ionic, complete) alternant systems are obtained. In particular, all one- and two-particle properties are derived in an explicit form. The properties obtained substantially generalize “classical” properties of alternant systems such as, in the case of neutral alternant systems, uniform charge density distribution, vanishing bond orders between atomic sites of the same parity, and alternancy selection rules for the electric dipole transitions.     

Selection rules and line-strength factors for multiphoton transitions in gas phase molecular spectroscopy

Multiphoton transitions are discussed in terms applicable to experimental spectroscopy. A simple tensor operator for the equations governing multiphoton transitions in molecules is obtained. Selection rules for these processes are derived by symmetry considerations and procedures for calculation of rotational line-strength factors are given. For the general transition, the rovibronic n-photon selection rules can be determined by the symmetry product of a n-photon operator with the rovibronic wavefunctions. The line-strength factors are found by considering the integral over rotations of the nth rank tensor operators. The line-strength evaluations do not depend on exact knowledge of the vibronic overlap, to within a constant factor.     

Nonreactivity of rydberg states woodward-hoffmann forbidden [2s+2s] photochemical cycloadditions

A theoretical analysis is presented for the [2_s + 2_s] cyclodimerisation of molecules in which the lowest unoccupied MO is of Rydberg nature. The concerted mechanism is forbidden by the Woodward-Hoffmann rules for the thermal and the photochemical path. However, the frontier MO approach predicts an enhancement of these reactions. Experiments related to this analysis are discussed.     

Symmetry‐Enthalpy Correlations in Diels–Alder Reactions

Woodward–Hoffmann (WH) rules provide strict symmetry selection rules: when they are obeyed, a reaction proceeds; when they are not obeyed, there is no reaction. However, the voluminous experimental literature provides ample evidence that strict compliance to symmetry requirements is not an obstacle for a concerted reaction to proceed, and therefore the idea has developed that it is enough to have a certain degree of the required symmetry to have reactivity. Here we provide quantitative evidence of that link, and show that as one deviates from the desired symmetry, the enthalpy of activation increases, that is, we show that concerted reactions slow down the further they are from the ideal symmetry. Specifically, we study the deviation from mirror symmetry (evaluated with the continuous symmetry measure (CSM)) of the [4+2] carbon skeleton of the transition state of a series of twelve Diels–Alder reactions in seven different solvents (and in the gas phase), in which the dienes are butadiene, cyclopentadiene, cyclohexadiene, and cycloheptadiene; the dienophiles are the 1‐, 1,1‐, and 1,1,2‐cyanoethylene derivatives; the solvents were chosen to sample a range of dielectric constants from heptane to ethanol. These components provide twenty‐four symmetry–enthalpy DFT‐calculated correlation lines (out of which only one case is a relatively mild exception) that show the general trend of increase in enthalpy as symmetry decreases. The various combinations between the dienophiles, cyanoethylenes, and solvents provide all kinds of sources for symmetry deviations; it is therefore remarkable that although the enthalpy of activation is dictated by various parameters, symmetry emerges as a primary parameter. In our analysis we also bisected this overall picture into solvent effects and geometry variation effects to evaluate under which conditions the electronic effects are more dominant than symmetry effects.     

Collinear collision of an atom with a homonuclear diatomic molecule

The problem of collinear scattering of an atom from a homonuclear diatomic molecule is formulated in terms of a first-order nonlinear matrix differential equation for the variable coefficient of reflection. For a homonuclear molecule when the target Hamiltonian is invariant under the parity transformation, only transitions between even states or odd states are possible. This selection rule reduces the number of open or closed channels that contribute to the reflection and transmission coefficients. But for numerical calculation, under the conditions of the problem, one can approximate the target Hamiltonian by the Hamiltonian of a displaced harmonic oscillator. In this approximation, the reflectional symmetry of the Hamiltonian is not preserved and transitions between any two levels of the target are possible. To simplify the problem further, the interaction between the projectile and the target is assumed to be a sum of two Gaussian terms. For this combination of the potentials the many-channel interaction can be expressed analytically. By fitting the Lennard–Jones potential with a sum of two Gaussian potentials and solving the matrix differential equation, transition probabilities are obtained for the He? H₂ collision. The numerical results are compared with the results found by Secrest and Johnson, and by Clark and Dickinson.     

The Perturbation Treatment of Chemical Reactivity

The perturbation method of treating chemical reactions is discussed in a simple manner, by invoking the general hypothesis that the initial perturbation determines the course of a reaction. This approach is used to develop a general theory, as an alternative to the transition state theory. Such apparently diverse concepts as the symmetry rules for cyclic processes, and hard and soft acids and bases can be derived by this treatment, which is applied to radical reactions and aliphatic and aromatic substitution.     

Comparison between the symmetry-correlation examination and the perturbation method

A comparison between the construction of symmetry-correlation diagrams and the perturbation method for studying chemical reactions is carried out. The perturbation method consists of decomposing the system Hamiltonian H into a sum, H = H₀ + H′. Various symmetry correlation schemes appearing in the literature may be explained by the nonuniqueness of the decomposition scheme. All symmetry selection rules may be viewed as the varieties. By examining the symmetry-correlation diagrams, processes under investigation may be called “forbidden” or “allowed,” depending on the topological feature. Of particular importance is the topology associated with the “avoided crossing.” By making the comparison, we can establish the correspondence of the two methods and conclude that the perturbation order furnishes the origin of the “forbiddenness” of a process.     

Violations of the noncrossing rule due to special symmetry and alternance

This paper reports a study on which behavior of the Hamiltonian gives rise to violation of the noncrossing rule. In principle, the noncrossing rule may be violated when a special symmetry other than spatial and spin symmetries is present or there exists the so-called alternance, which corresponds to a Hamiltonian in a real vector space anticommuting with a Hermitian operator. In the HMO models for pericyclic reactions, violations due to special symmetry or alternance have been found. The [m,n] supra–antara cycloadditions have no symmetry in the traditional sense, but have special symmetry leading to the existence of crossings in the correlation diagram. Alternance results in one crossing in the middle of the correlation diagram of a forbidden pericyclic reaction with intermediate states in the form of even alternant hydrocarbon. For the reactions with intermediate states in the form of odd alternant hydrocarbon such as [2,4]-cycloaddition of an allyl cation or an allyl anion to butadiene, there should be no crossing in the correlation diagrams, and both the supra–supra and the supra–antara processes are predicted to be allowed. Such a prediction is beyond the Woodward–Hoffmann rule.     

Semiempirical MO calculations on symmetry governed reactions

Two symmetry governed reactions, the electrocyclic transformation of planar cyclopropyl cation to allyl cation and the dimerization of ethylene to cyclobutane, are examined using a modified INDO method. Results for the cyclopropyl-allyl cation reaction agree well with previously publishedab initio results, and are much improved over previously published CNDO results. The symmetry-allowed disrotatory path is predicted to be significantly favored over the forbidden conrotatory transition. For the ethylene-cyclobutane system two surprising results are predicted within the constraints imposed upon the reaction path: first, that the entire reaction should occur within a small range in the separation of the two ethylene molecules as they approach one another, and second, that the symmetry-forbidden [2 _s +2 _s addition should be slightly favored over the symmetry-allowed [2 _s +2 _a addition. Since the Woodward-Hoffmann rules deal exclusively with changes in electronic energy, it is suggested that they should be applied with some caution to reactions in which changes in nuclear repulsion are quite large during the reaction process.     

Electrocyclic Ring Opening Reactions of Ethylene Oxides

Substituents capable of stabilizing negative charge endow ethylene oxides with the ability to undergo electrocyclic ring opening at the CC bond; heating or irradiation generates small equilibrium concentrations of carbonyl ylides which can engage in 1,3-dipolar cycloadditions. Apart from the normal conrotatory mode of ring opening, predicted by the Woodward-Hoffmann rules, the disrotatory process, forbidden by orbital symmetry, also appears to occur in the case of α-cyano-cis-stilbene oxide. Kinetic studies on α-cyano-trans -and -cis-stilbene oxide permit construction of the energy profile for electrocyclic ring opening to give the stereoisomeric carbonyl ylides and for their recyclization and rotation.     

螯变反应中的轨道对称守恒原理及奇偶选择定则

本文在唐敖庆等人提出的化学反应局部对称性新概念的基础上, 按Woodward-Hoffmann对螯变反应的约定, 利用HMO法对含奇数或偶数个碳原子的反应体系推导出各类螯变反应的MO特征根方程。进而做出了能量相关曲线和电子总能量曲线, 证明了Woodward-Hoffmann的选择定则完全适用于含奇数或偶数个碳的螫变反应体系, 并得到了中性自由基体系的基态反应都是对称性禁阻的结论。     

Mechanistic pathways for ketene dimerization

A semiempirical MO method was used to calculate potential energy surfaces for ketene dimerization. One supra-supra and two supra-antara pathways were investigated. Six parameters were used to characterize the relative motion of individual groups. Two levels of approximation for the motion were considered. First, synchronous reaction pathways were followed, and a set of single-dimensional potential curves generated. Then, deviations from these synchronous reaction pathways were considered for each parameter and five two-dimensional potential surfaces for each pathway generated. Agreement with the available experimental data is satisfactory. However, we conclude that contrary to the Woodward-Hoffmann rules [1] the supra-supra pathway is allowed and the supra-antara pathways essentially forbidden. In the former case the local symmetry of the orbitals involved in the ring formation is not conserved during the reaction, in the latter case the nuclear repulsion is dominating.