Electronic structure of the ground state of the benzyl radical

Results from the -electron approximation are given for the total energy, spin-density distribution, and electron density, which have been derived by configuration interaction via self-consistent orbitals for closed and open shells and with allowance for all singly excited configurations and some doubly excited ones. Some singly excited configurations do not mix with the ground-state configuration in the first order of perturbation theory but make a contribution to the latter greater than do some of the configurations that do mix. Incorporation of all singly excited configurations results in a lower ground-state energy if orbitals for a closed shell are used instead of those for an open one. Calculations via closed-shell orbitals lead to an inhomogeneous electron-density distribution, which is gradually smoothed out as the set of configurations is expanded. The spin-density distribution is very much dependent on the number of configurations used and on the orbitals employed in them.     

Open-shell calculation on molecules with icosahedral symmetry

The ground and excited states of footballene C₆₀ and dodecahedral C₂₀ carbon species and their ions are calculated in the-electron approximation by means of Roothaan's open-shell method adjusted for systems of icosahedral symmetry. Ionization potentials and electron affinities are presented. The lowest singlet and triplet transitions of all possible symmetry types of the I_h-group are obtained for these molecules by a Cl method including all singly excited configurations.     

The configuration interaction method,and the triplet-singlet splitting in CH2

A configuration-interaction (CI) method in which the interaction matrix is never constructed has been investigated, following the original suggestion of Roos. Two methods have been used (1) for singlet states, which can be represented by a one determinant configuration of doubly occupied orbitals, CI with all singly and doubly excited configurations, (2) for states for which the restricted self-consistent field approximation is a single determinant, CI with all singly and doubly excited determinants. In case (2), the wavefunction may not be exactly an eigenfunction of S ². The methods were investigated using a double-zeta plus polarisation basis for CH₂. Both methods must give the same result for the lowest singlet ground state. Keeping the bond length fixed at 2.10 and 2.04 bohr respectively the bond angle for the singlet and triplet were found to be 100.8 ° and 132.0 °, with energies ?39.0312 a.u. and ?39.0563 a.u. respectively. These are the lowest variational energies obtained for these systems; the singlet-triplet splitting is thus predicted to be 15.4 kcal/mol.     

VESCF calculations of the spectra of some hydrocarbons containing triple bonds

The ultraviolet spectra of acetylene, diacetylene, triacetylene, allene, and their alkyl derivatives were calculated by a modified Pariser-Parr method using VESCF orbitals and including all singly and doubly excited states in the CI matrix, a method which has given good results in previous calculations on compounds containing only ethylenic carbon atoms. In the present work two new parameters were introduced, namely, the effective nuclear charge of an acetylenic carbon, and the integral (ie261-01). The calculated spectra agreed well with experiment for transitions observed above 190 m (< 6.53 eV). For transitions observed near or below 170 m (> 7.3 eV), the calculated energies were always too high. Agreement with experiment was improved by the inclusion of triply excited configurations, particularly in the case of acetylene.This research was supported by Grant GP 6763 from the National Science Foundation.     

Full configuration interaction for the benzyl radical

The electronic structure of the benzyl radical in its ground state has been computed using a model Hamiltonian due to Pariser–Parr with full configuration interaction as well as with different truncated configurational sets built on SCF open-shell orbitals. The correlation energy corresponding to this model was found to be equal to –0.929722 eV. With the singly excited configurations only 18% of this energy is taken into account. By extending the basis to include the doubly excited configurations one can account for 94% of the correlation energy. An analysis of the accuracy of the proton hyperfine splitting calculation caused by inaccurate computation of the wave function is given. If only singly and even doubly excited configurations are taken into account one cannot hope to obtain splittings with an accuracy of more than 0.5 g. Inclusion of triply excited configurations lowers this error by one order. In addition, the use of the simple McConnell relation may lead to an error in splitting calculations of no less than 1.5 g.     

A comparative calculation on excited state energies of some conjugated hydrocarbons

The energies of the single-configuration lowest π – π* singlet and triplet states of some conjugated hydrocarbons have been calculated by the MC-SCF method using the conjugate-gradient technique of minimisation. The results are compared with those calculated by other methods currently in use, like (a) single-configuration calculation with V_N?1 potential for virtual orbitals; (b) CI calculation involving singly excited states; and (c) TDHF method. It has been concluded that the results for the MC-SCF method are very good, considering that only a single open-shell configuration is involved.     

Selected configuration interaction with truncation energy error and application to the Ne atom

Selected configuration interaction (SCI) for atomic and molecular electronic structure calculations is reformulated in a general framework encompassing all CI methods. The linked cluster expansion is used as an intermediate device to approximate CI coefficients B(K) of disconnected configurations (those that can be expressed as products of combinations of singly and doubly excited ones) in terms of CI coefficients of lower-excited configurations where each K is a linear combination of configuration-state-functions (CSFs) over all degenerate elements of K. Disconnected configurations up to sextuply excited ones are selected by Brown's energy formula, Delta E(K) = (E-H(KK))B(K)2/(1-B(K)2), with B(K) determined from coefficients of singly and doubly excited configurations. The truncation energy error from disconnected configurations, Delta E(dis), is approximated by the sum of Delta E(K)s of all discarded Ks. The remaining (connected) configurations are selected by thresholds based on natural orbital concepts. Given a model CI space M, a usual upper bound E(S) is computed by CI in a selected space S, and E(M) = E(S) + Delta E(dis) + delta E, where delta E is a residual error which can be calculated by well-defined sensitivity analyses. An SCI calculation on Ne ground state featuring 1077 orbitals is presented. Convergence to within near spectroscopic accuracy (0.5 cm(-1)) is achieved in a model space M of 1.4 x 10(9) CSFs (1.1 x 10(12) determinants) containing up to quadruply excited CSFs. Accurate energy contributions of quintuples and sextuples in a model space of 6.5 x 10(12) CSFs are obtained. The impact of SCI on various orbital methods is discussed. Since Delta E(dis) can readily be calculated for very large basis sets without the need of a CI calculation, it can be used to estimate the orbital basis incompleteness error. A method for precise and efficient evaluation of E(S) is taken up in a companion paper.     

Effect of configurational composition of the wave function on calculated dipole polarizability of molecules

On the basis of the -electron approximation we have investigated the dependence of the calculated dipole polarizability of molecules on the configurational composition of the variational wave function. The values which were closest to the results of complex configurational interaction were obtained for sets including the single-determinant Hartree-Fock function and all the bonding doubly excited configurations constructed from the localized MO, and also from the complete set of singly and doubly excited configurations.Translated from Teoreticheskaya i Eksperimental'naya Khimiya, Vol. 26, No. 1, pp. 70–73, January–February, 1990.     

Role of the low-lying two-electron excited Ag states of C60 in contributing to its hyperpolarizability γ

The role of the low-lying two-electron excited ¹A_g states of C₆₀ in contributing to its hyperpolarizability γ is investigated by singly and doubly excited CI calculations in the semi-empirical CNDO/S approximation. It is found that inclusion of the doubly excited configurations is essential in evaluating γ. The calculated values of γ_zzzz(−3ω; ω, ω, ω) at wavelengths of ∞, 1.9, 1.83 μm sufficiently far from the first resonance are in reasonable agreement with experiment.     

Study of the electronic structure of radicals by the CI method

Results are presented which were obtained in the -electronic approximation for the ground state properties of the benzyl radical (total energy, distribution of the electronic and spin density) by the CI method on closed and open shell SCF orbitals taking into account all singly and some doubly excited configurations. Some of the singly excited configurations which in the first order perturbation theory do not interact with the ground state configuration contribute to the ground state more than some of those which interact with the latter. Certain doubly excited configurations contribute more than some singly ones. The consideration of all singly excited configurations leads to a lower ground state energy if the closed shell orbitals are used compared with the open shell ones. The former bring about the non-uniform electron density distribution becoming more smooth as the set of the basis configurations becomes larger. The spin density distribution strongly depends on the number of the configurations as well as on the orbitals used for their construction. The computation of the benzyl radical done by Hinchliffe by the CI method on open shell orbitals is believed to be wrong.

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Zusammenfassung Rechnungen im Rahmen der -Elektronen-NÄherung für den Grundzustand des Benzyl-Radikals (Gesamtenergie, Verteilung von Elektronen- und Spin-Dichte) mittels des SCF-CI-Verfahrens für geschlossene und offene Schalen unter Einschlu\ aller einfach und einiger zweifach angeregten Konfigurationen werden mitgeteilt. Dabei zeigt sich, da\ einfach angeregte Konfigurationen, die in der Störungsrechnung erster Ordnung nicht mit der Grundfiguration kombinieren, zum Grundzustand mehr als solche, die kombinieren, beitragen.Zusammenfassung Die Berücksichtigung aller einfach angeregten Konfigurationen führt zu einer niedrigeren Grundzustandsenergie, wenn die Einteilchenfunktionen, die sich für geschlossene Schalen ergeben, an Stelle derer für offene Schalen verwendet werden. Erstere führen zu ungleichförmiger Verteilung der Elektronendichte, die um so mehr geglÄttet ist, je grö\er die Anzahl der Konfigurationen wird. Die Verteilung der Spin-Dichte hÄngt stark von der Anzahl der Konfigurationen wie auch von den benutzten Orbitalen ab. Die Berechnung des Benzyl-Radikals von Hinchliffe mittels des CI-Verfahrens für offene Schalen wird als unrichtig angesehen.

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Résumé Propriétés de l'état fondamental du radical benzyle (énergie totale, distribution des densités électroniques et de spin) dans l'approximation par la méthode d'I.C. sur les orbitales SCF des couches complètes et incomplètes avec introduction de toutes les configurations monoexcitées et de certaines configurations diexcitées. Certaines configurations monoexcitées qui n'intéragissent pas avec la configuration fondamentale au premier ordre de la théorie des perturbations apportent à l'état fondamental une contribution plus élevée que certains de ceux qui interagissent à ce stade là. Certaines configurations diexcitées ont une contribution plus forte que certaines monoexcitées. Si l'on tient compte de toutes les configurations monoexcitées, on obtient une énergie plus basse en utilisant les orbitales des couches complètes et non celles des couches ouvertes. On obtient ainsi une distribution électronique dont la non uniformité va en s'affaiblissant lorsque la base de configurations s'élargit. La distribution de densité de spin dépend fortement du nombre de configurations employé ainsi que des orbitales utilisées. Le calcul d'I.C. d'Hinchliffe sur le radical benzyle à l'aide des orbitales des couches ouvertes semble faux.

     

Spectrum and polarizability of polyphenyls in the Pariser-Parr-Pople approximation

The Pariser-Parr-Pople approximation is used to calculate the electronic structure, electronic absorption spectrum, and polarizability for biphenyl, terphenyl, and quaterphenyl. The self-consistent MO are taken as basis functions for the multiconfiguration approximation. The minimum number of singly excited configurations is included. Bond orders, bond lengths, transition energies, oscillator strengths, and wave functions of the excited states are given in that approximation. The results are compared with published data and, where possible, with experiment. Calculation of the -electron polarizability with allowance for configuration interaction is discussed. The agreement with experiment is satisfactory for all the characteristics calculated.     

Nonstationary perturbation theory for open-shell molecules

Nonstationary perturbation theory equations have been obtained for open-shell molecules. The equations were formulated in terms of a density matrix in the MO-LCAO method. The first variant is coupled perturbation theory in the framework of the restricted Hartree-Fock method for open shells, and the second variant is variational perturbation theory for ground and excited electronic states of molecules, in which the perturbed wave function of the system is constructed in the form of a superposition of the ground and singly excited configurations composed of the Hartree-Fock orbitals of the open shell. A calculation of the Cauchy moments of the dynamic dipole polarizability of several molecules of conjugated open-shell hydrocarbons, viz., doublet states of odd alternant hydrocarbons, as well as triplet excited states and doublet states of radical ions of even alternant hydrocarbons, has been carried out in the framework of both methods.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 21, No. 1, pp. 18–27, January–February, 1985.     

Azulene: polarized absorption in stretched polymers and magnetic circular dichroism

Linear dichroic and magnetic circular dichroic spectroscopy is used to characterize the excited states of azulene. Comparison with crystal spectra underlines the necessity of using two-parameter equations for evaluations of stretched sheet spectra despite some previous claims to the contrary. The MCD spectrum reveals strong vibronic interactions in the second transition.The results are interpreted on the basis of Pariser—Parr—Pople (PPP) type calculations including some with extensive configuration interaction (CI). Calculated signs of the first three B terms spectrum agree with experimental signs irrespective of the details of the PPP model, and even the numerical values are in an order-of-magnitude agreement. Only the calculation with extensive CI gives agreement for signs of higher B terms. It suggest the possible presence of two excited states, but the present experimental evidence for these is insufficient. One of these new states would be best described as a predominantly doubly excited state.The origin of the B terms is traced to contributions due to magnetic mixing of individual zero-order excited states and the results permit determination of the absolute sense of magnetic dipole transition moments between excited states with respect to the sense of the vector product of the electronic dipole transition moments from the ground state to these excited states. A comparison is performed with B terms calculated using finite perturbation technique and gauge-invariant orbitals.     

Excited state geometries within time-dependent and restricted open-shell density functional theories

Singlet excited state geometries of a set of medium sized molecules with different characteristic lowest excitations are studied. Geometry optimizations of excited states are performed with two closely related restricted open-shell Kohn–Sham methods and within linear response to time-dependent density functional theory. The results are compared to wave-function based methods. Excitation energies (vertical and adiabatic) calculated from the open-shell methods show systematic errors depending on the type of excitation. However, for all states accessible by the restricted methods a good agreement for the geometries with time-dependent density functional theory and wave-function based methods is found. An analysis of the energy with respect to the mixing angle for the singly occupied orbitals reveals that some states (mostly [n→π^*]) are stable when symmetry constraints are relaxed and others (mostly [π→π^*]) are instable. This has major implications on the applicability of the restricted open-shell methods in molecular dynamics simulations.     

Ab initio SCF-CI studies of the intermolecular interaction between two hydrogen molecules near the Van der Waals minimum

SCF-CI calculations have been used to study the intermolecular energy between two hydrogen molecules in four different geometrical configurations. The CI matrix was diagonalized using perturbation techniques. The importance of the perturbation expansion order upon the intermolecular energy could therefore be studied. The wave function includes all singly and doubly excited configurations. The natural orbitals have been determined and their relative importance on the intermolecular energy is considered.     

Free-electron model for electronic spectra of benzene

Using a three dimensional free-electron model for the π-electron system the energies of the various singly and doubly excited configurations of benzene have been calculated. It has been found that the energies of the various singly excited configurations are in good agreement with the values obtained by Craig; whereas the energies of the doubly excited configurations are higher than those obtained by Craig. Therefore the configurational mixing for the doubly excited configurations is not necessary in this kind of model.     

The multiconfiguration SCF method and large-scale configuration interaction in calculations for the A1 and B1 states of the water molecule

The multiconfiguration Hartree-Fock method (MC SCF) is examined for a superposition of a reference determinant ¦> containing all singly and doubly excited configurations. The gradients of the energy functional have been calculated for the singlet states. Pulay's method has been used in identifying the minimum. Calculations on the ground term A₁ and excited term B₁ for H₂O in various bases indicate that the MC SCF procedure of the type considered is the most important when the contribution from ¦> is predominant. A realization is given for the superoperator configuration-interaction algorithm, which enables one to use an ES-1035 computer to calculate the ground state of the water-molecule dimer with 38,700 configurations.Read at the Seventh All-Union Seminar on Nonempirical Computational Quantum Chemistry (Chernogolovka, 1985).Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 23, No. 1, pp. 1–10, January–February, 1987.We are indebted to A. V. Nemukhin for discussion and advice.     

Conjugated radicals

The naphthalene radical anion was calculated by the open-shell SCF method due to Roothaan combined with a configuration interaction treatment. The effect of the doubly excited configurations on the calculated transition energies was found to be unimportant in view of the interpretation of its electronic spectrum.Part V: Collect. Czechoslov. Chem. Commun. 35, 892 (1970).     

Effective potential energy barriers and X-ray transitions of sulphate ion

X-ray transition energies and intensities of sulphate ion are computed from limited CI wavefunctions including singly excited configurations. The results are interpreted in terms of one-electron promotions by using the improved virtual orbitals introduced first by Hunt and Goddard III [11]. The effective potential for virtual orbitals shows a barrier at large distance from sulphur nucleus.