Pairing Theorem Research Articles

An extension of the pairing theorem

AbstractAn extended pairing scheme is presented which ensures the fulfillment of pairing conditions not only between the sets of occupied orbitals for spin α and for spin β, but also between their orthogonal complements, i.e., the sets of virtual orbitals for spin α and spin β, as well as between occupied orbitals for spin α and virtual orbitals for spin β and between virtual orbitals for spin α and occupied orbitals for spin β. It is shown that the extended pairing properties are suggested by some aspects of the construction of alternant molecular orbitals. The algorithm for singular value decomposition of rectangular matrices is proposed for use in practical implementations of the (extended) pairing scheme.

Some Features of Excited States Density Matrix Calculation and Their Pairing Relations in Conjugated Systems

Direct PPP-type calculations of self-consistent (SC) density matrices for excited states are described and the corresponding “thawn” molecular orbitals (MO) are discussed. Special atten­tion is addressed to particular solutions arising in conjugated systems of a certain symmetry, and to their chemical implications. The U(2) and U(3) algebras are applied, respectively, to the 4- electron and 6-electron cases; a natural separation of excited states in different cases follows. A simple approach to the convergence problem for excited states is given. The complementarity relations, an alternative formulation of the pairing theorem valid for heteromolecules and non-alternant systems, allow some fruitful experimental applications. Together with the extended pairing relations shown here, they may help to rationalize general trends.

Pairing theorem in the complex molecular orbital method

A significant feature of the closed-shell complex molecular orbital (CMO) wavefunction, recently discovered through numerical work, is proved formally as the pairing theorem. According to this theorem complex MOs in the CMO natural orbital (NO) representation may be brought into the pairing form in which each CMO has only two NO components. This result is essential for handling non-orthogonalities in the complex configuration mixing method.

ChemInform Abstract: TWO‐PHOTON SPECTROSCOPY OF DIPOLE‐FORBIDDEN TRANSITIONS. I. DIPOLE‐FORBIDDEN TRANSITIONS AND DOUBLE EXCITED CONFIGURATIONS IN THE CNDO‐CI METHODS

AbstractIn dieser 1. Mitt. wirdan Beispielen wie Butadien, Hexatrien, Benzol, Octatetraen, Naphthalin ,Anthracen und Stilben gezeigt, daß eine modifizierte CNDO/S‐Methode Anregungsenergien für Dipol‐erlaubte und ‐verbotene Übergänge mit gleicher Genauigkeit ergibt.

On the relationship between the gas phase, condensed phase and theoretical spectra of alternant hydrocarbon anions

Using electron affinities determined by electron transmission spectroscopy, electronic transition energies are derived for the anions of styrene and naphthalene. Because these anions are unstable against autoionization, transition energies in the gas phase have not been previously determined. These values are compared with those found by optical absorption on the stabilized anions in solution and hydrocarbon glasses as well as theoretical predictions and the results from cation spectra. The low lying transition energies are in excellent agreement in all methods. The higher transition energies derived from the gas phase measurements are in poorer agreement. The role of the anion lifetime in determining the degree of electronic reorganization in these short-lived states is discussed. Finally, the implications of our results for the pairing theorem in alternant hydrocarbons is also discussed.

MCD spectra of the triphenylene positive ion. A test of the pairing theorem for highly symmetrical hydrocarbons

The magnetic circular dichroism (MCD) spectra of the monovalent positive ion of triphenylene have been measured at different temperatures and compared with those of the corresponding negative ion of the same molecule. The results confirm the predicted pairing relations for the A and C parameters of MCD spectra of highly symmetrical hydrocarbons. The spectroscopic assignment of the bands, made with the aid of a π-electron calculation (PPP approximation), supports an earlier assignment for the triphenylene negative ion.

An improved Mclachlan type spin density calculation

A new method for the calculation of the spin density in π radicals is proposed within the McLachlan framework. Compared with previous methods, this new definition exhibits two main features: (i) for odd alternant neutral radicals, the computed spin density pattern is in better agreement with experiment (ii) for even alternant ionic radicals, the spin densities are identical in the cation and the anion, as required by the “ pairing theorem ”.

Two-photon spectroscopy of dipole-forbidden transitions

We have investigated the applicability of CNDO/S-type methods for the calculation of optical spectra of molecules with the special implication that the calculations should not only describe the intense, dipole-allowed transitions which dominate the one-photon absorption spectrum but also those transitions which are one-photon forbidden in first-order approximation. We show that such a method is well suited to predict dipole allowed and dipole forbidden transitions at a similar level of accuracy if double excited configurations are taken into account. In spite of the lack of perfect pairing in NDO methods there are still two types of states which exhibit a different sensitivity towards correlation effects. Therefore, the approximation by which we describe the R-dependence of the Coulomb repulsion gains much more importance than in cases where mainly dipole allowed transitions are of interest. These findings confirm results obtained earlier from theories for which the pairing theorem is valid. The calculated data show an excellent stability with respect to further increase of the number of configurations if at least about 200 energy selected configurations are taken into account.

A Characterization of Homology Manifolds

A homology «-manifold is a space which has the same local homology at each point as Euclidean n-space. The principal result of this paper is the characterization of triangulable homology manifolds by a global property: The rc-circuit X is a homology manifold if and only if the diagonal cycle A in X x X is Poincare dual to a cocycle with support A (Theorem 1). If X is a smooth manifold, this cocycle represents the Thorn class of the tangent bundle of X. The homological properties of Thorn classes have been studied by Milnor [13; §11] and Spanier [14; Chapter 6]. The proof is based on their techniques. A corollary of this proof is that an w-circuit X satisfies Poincare duality if and only if there is a class U dual to the diagonal which has a certain symmetry with respect to the canonical involution T on XxX; namely U-^V = U^T*V for all V (Proposition 1). Furthermore, for any ^-circuit X, the diagonal cycle is dual to some cocycle U, if coefficients are in a field (Proposition 2). Thus U\(XxX—A) is the obstruction to X being a homology manifold. Propositions 1 and 2 have been obtained independently by P. Holm [8]. The ideas of Lefschetz about intersection theory and the topology of algebraic varieties have been my constant guide (cf. [15]). Theorem 1 can be interpreted in terms of the intersection pairing (Theorem 3). This paper is a revised version of part of my doctoral thesis at Brandeis University [11], written under the supervision of Professor Jerome Levine. I have also been helped by the questions and suggestions of P. Lynch, D. Stone, A. Landman, and especially D. Sullivan. My viewpoint has recently been influenced by the work of I. Fary [3]. Homology will be singular homology throughout, with integer coefficients in §§1 and 2, and field coefficients in §§3 and 4. Sign conventions for products are those of [14].

Proof of the Hückel rule

The Hückel rule is proved for an arbitrary alternant system. Its validity is shown to be intimately related with the pairing theorem.

Magnetic Circular Dichroism of Cyclic π-Electron Systems

Abstract While the simple PPP model accounts for the absolute signs of low-energy π∗ transitions in cyclic π-electron chromophores (numerous new examples of application to heterocycles are given), its greatest contribution lies in the formulation of simple general rules. The previously known mirror-image pairing theorem for charged alternant hydrocarbons and the soft-hard rule for substituent and heteroatom effects on cyclic π-systems are now complemented by a ΔHOMO-ΔLUMO rule which permits prediction of the absolute sign of the lowest energy B term in cyclic π-systems which are formally related to an annulene containing 4N+2 electrons.

A topological Hückel model for organometallic complexes. Part II. Alternant bonding networks in η3- and η4-olefin complexes

Within the Huckel approximation, metal η3- and η4-olefin complexes are shown to have alternant bonding networks. Application of the pairing theorem to such systems indicates the electronic origins of the 18-electron rule and provides some insight into the chemical properties of these molecules. The topological implications of the bonding network in η-allyl complexes are also discussed.

Experimental study of the negative ion states of styrene. A test of the pairing theorem

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTExperimental study of the negative ion states of styrene. A test of the pairing theoremP. D. Burrow, J. A. Michejda, and K. D. JordanCite this: J. Am. Chem. Soc. 1976, 98, 20, 6392–6393Publication Date (Print):September 1, 1976Publication History Published online1 May 2002Published inissue 1 September 1976https://doi.org/10.1021/ja00436a059RIGHTS & PERMISSIONSArticle Views58Altmetric-Citations39LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (257 KB) Get e-Alerts Get e-Alerts

MCD spectra of diphenylmethyl cation and anion. A test of the pairing theorem

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTMCD spectra of diphenylmethyl cation and anion. A test of the pairing theoremKarl L. Tseng and Josef MichlCite this: J. Am. Chem. Soc. 1976, 98, 20, 6138–6141Publication Date (Print):September 1, 1976Publication History Published online1 May 2002Published inissue 1 September 1976https://doi.org/10.1021/ja00436a012RIGHTS & PERMISSIONSArticle Views48Altmetric-Citations10LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (495 KB) Get e-Alerts Get e-Alerts

Graph theory of free radicals. Validation of a recent assertion and its relation to the pairing theorem

The criteria for the occurrence of a symmetric eigenvector in odd graphs are deduced; these confirm the validity of a recent assertion concerning the cause of such an eigenvector. The relationship to the Coulson–Rushbrooke Pairing Theorem is noted.

Untersuchung einer Resonanz des karzinogenen Benzpyren. Studies on a Resonance of Carcinogenic Benzpyrene

The alpha- and p-states of 3.4-benzpyrene are calculated by use of common parameters and of a parameter set which predicts the quasi-degeneracy of these states. The quasi-degeneracy will be represented by the breadage of pairing theorem and is to be described as a correlation effect, caused by an asymmetric potential. An external field represents an external perturbation, introduced by interaction with the bioreceptor. The calculation shows a sensitive dependence of probability amplitudes for different configurations which participate in the quasi-degenerated resonance states. Thereby the dipole moment may quickly be changed so that excitons are produced. The resonance phenomenon may lead to the carcinogenic effect.

Erläuterungen zum Resonanzmodell der Karzinogenese * / Some Remarks on a Resonance Hypothesis of Carcinogenesis

Abstract Resonance Hypothesis, Carcinogenesis, Polycyclic Hydrocarbons Three criteria for carcinogenic polycyclic hydrocarbons, which are to be satisfied, are discussed in regard to a resonance-hypothesis of carcinogenesis. According to this hypo­ thesis, the carcinogenic effect is caused by a Fermi-resonance between α-states and p-states which possibly leads to the breakage of pairing theorem by an asymmetric potential, in certain cases caused by vibronic coupling.

The use of the CNDO method in spectroscopy: Naphthalene and the breakdown of the pairing theorems

It is shown that CNDO S calculations provide 6 significant figures of accuracy in the energies of configurations, and 4 significant figures after configuration interaction. The pairing theorem is shown not to hold in the CNDO S method since all F μν are calculated and since the F μμ are not constant. It is further shown that the degeneracies required by the pairing theorem are not even approximated in naphthalene. Finally, using absorption and emission spectroscopic measurements, including triplet-triplet and excited singlet-singlet absorptions, the spectroscopic states of naphthalene are discussed in detail.

Unrestricted Hartree‐Fock calculations of spin densities with orthogonalized atomic orbitals: Proton and 13C hyperfine splittings in some pi hydrocarbon radicals

AbstractThe spin density distribution in a few hydrocarbon radicals has been calculated using orthogonalized atomic orbitals in the Unrestricted Hartree‐Fock formalism of Amos and Snyder and including certain more important two‐electron hybrid and exchange integrals and all the core‐resonance integrals. Our calculated spin densities for the cation and anion radicals of alternant hydrocarbons, which are now different due to the breakdown of the pairing theorem, are, in general, of the right relative order so that even the simple McConnell type of relation can account partly for the observed differences in the proton splittings between cations and anions. The proton splittings for position 2 of naphthalene and anthracene radical ions are correctly predicted, thus clearing up the well‐known cation‐anion anomaly for this position. Comparative calculations have been made to show that the spin density results are worsened with the neglect of the integrals of the type mentioned before. An empirical analysis correlating the observed 13C splittings and the spin density results over a non‐orthogonal basis set shows that the available 13C splittings in alternant hydrocarbon radical ions can be explained with a set of sigma‐pi parameters which are consistent with the theory. It is shown that even though the spin densities in cations and anions may be different, these can lead to similar 13C splittings.

The calculation of π-electron spin densities in even-membered radical ions

The even alternant hydrocarbon pairing theorem is used to justify a new recommended choice for electron placement in radical cations when using the McLachlan α-correction method. McLachlan-type calculations and those based on annihilation of the major contaminating higher-spin multiplet from the unrestricted Hartree-Fock wave function are presented for a number of new conjugated radical ions to justify the claim that the application of the former method to non-annular or heteroatomic systems is not satisfactory.