Article pubs.acs.org/JPCA
Singly Occupied MOs in Mono- and Diradical Conjugated Hydrocarbons: Comparison between Variational Single-Reference, π‑Fully Correlated and Hückel Descriptions Nicolas Suaud, Renaud Ruamps, Jean-Paul Malrieu, and Nathalie Guihéry* Laboratoire de Chimie et Physique Quantiques, UMR 5626 CNRS, IRSAMC, Université de Toulouse III Paul Sabatier, 118 Rte de Narbonne, 31062 Toulouse Cedex, France ABSTRACT: This work compares three descriptions of the unpaired electrons distribution in conjugated monoradical and diradical hydrocarbons involving one or two methylene groups attached to an aromatic skeleton. The first one is the simple Hückel topological Hamiltonian, the singly occupied molecular orbitals (SOMO) of which may be analytically obtained. The second one is the restricted open-shell self-consistent field (ROHF-SCF) method. The soobtained distribution of the unpaired electrons on the skeleton appears deeply different from that predicted by the Hückel Hamiltonian, being more strongly localized on the external methylene groups. More elaborate methods treat all π electrons in the π valence molecular orbitals (MOs) through a full valence π complete active space self-consistent field (CASSCF) treatment. The distributions of the unpaired electrons (given by the natural MOs of occupation number close to 1) are surprisingly similar to those predicted by the Hückel model. The spin density distributions, including spin polarization effects, can be improved by further configuration interactions involving one hole−one particle excitations and compared with the experimental hyperfine coupling constant ratios. This comparison confirms the lack of delocalization of the magnetic orbitals defined from the self-consistent single-reference treatment. We show that, provided correct SOMO are used, a single excitation CI performed on top of a single reference gives accurate spin densities. Finally, a rationalization of the role of the dynamic correlation in correcting the excessive localization of the unpaired electron(s) at the ROHF level on the exocyclic methylene group(s) is given, attributing it to the dynamic charge polarization of the charge transfer configurations between methylene and the aromatic frame.
I. INTRODUCTION
of these simple pictures, compared to more sophisticated descriptions, may be found in reviews by Borden et al.5,6 Among the successes of the Hückel model one must quote the prediction7 of the spin-density distribution in radical conjugated hydrocarbons. The contact terms on the in-plane protons responsible for the interaction between the electron spin and the nuclear spins are observable by electron spin resonance (ESR) spectroscopy.8 These contact terms are due to the spin-polarization7 of the σ electrons in the CH bonds and they depend on the spin density in the π atomic orbital (AO) of the carbon to which the hydrogen atom is attached. These radical systems possess an unpaired number of electrons (2n + 1), and the (n + 1)th MO is considered as being singly occupied and is labeled SOMO (singly occupied MO). This spin density was approximated by the square of the coefficient of this SOMO, and the proportionality of the observed contact terms with these calculated quantities had led some
Conjugated hydrocarbons were the first and privileged training ground of quantum chemistry until the mid sixties. The delocalized character of these systems led to some properties that could not be understood from chemical intuition, for instance, the spacing between the ionization potentials, some spectroscopic features, and some specific reactivity preferences.1 The quantum origin of these properties was not accessible to the dot-and-arrow models with which the chemists used to interpret the properties of the molecules they handled. The rather primitive monoelectronic model Hamiltonians provided a rich harvest of results and rationalizations. The simplest model is the topological Hückel Hamiltonian, which only deals with the π system and only considers equal hopping integrals between adjacent carbons. The considerable amount of results obtained from this simple model has been, for instance, nicely presented in Salem’s book.1 One should also mention that beyond this model, the electron−electron correlation problem was already considered through the introduction of the Pariser−Parr−Pople (PPP) Hamiltonian,2,3 a sophisticated version of the Hubbard Hamiltonian,4 proposed in solid state physics a few years later, which remains a milestone and a reference model in this domain. The relevance © XXXX American Chemical Society
Special Issue: Energetics and Dynamics of Molecules, Solids, and Surfaces - QUITEL 2012 Received: December 10, 2013 Revised: February 27, 2014
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rational way to generate ferromagnetic lattices12,13 or bistable architectures. The possibility of starting from graphene patches or carved graphenes attracts special interest,21−25 especially in the context of new materials for spintronics.18 For such large architectures, tight-binding methods constitute natural computational approaches and their relevance is of course crucial.26,27
theoreticians to think that molecular orbitals could be “experimentally seen”.9 The extension to diradicals, where two electrons occupy two SOMOs, was straightforward. Despite the simplicity of the Hückel model and in particular the lack of negative spin densities (for a positive Ms total spin component) in the π systems, the results were in reasonable agreement with those of the more sophisticated two-electron PPP Hamiltonian. With the advances of ab initio quantum chemistry, these elementary pictures were partially forgotten. The present work will first recall that the spatial distribution of the unpaired electron given by the topological Hückel Hamiltonian can be determined analytically. This study focuses on a series of molecules in which one or two CH2 methylene groups are attached to benzene or naphthalene skeletons. Because these mono- and diradicals are supposed to bear one or two unpaired electrons, the natural way to study them using ab initio wave function theory (WFT) methods consists of starting from a restricted open-shell Hartree−Fock (ROHF) having one unpaired electron for monoradicals and two unpaired electrons coupled in a triplet state for diradicals. As already reported by Borden and Davidson,10 the resulting spin distributions happen to be deeply different from those predicted by the Hückel model, being much more concentrated on the extra cyclic methylene groups. As shown by the same authors, one may get an improved description of the π electron system by performing a full CASSCF treatment of all π electrons in optimized π valence MOs.11 This description accounts for the spin polarization of the π electrons and therefore introduces negative spin densities. However, the natural molecular orbitals (NOs), which have an occupation number equal or close to 1 exhibit spin densities that happen to be surprisingly similar to those predicted by the topological Hückel Hamiltonian. The present paper will first illustrate the unexpected quality of the Hückel picture. Then it will exhibit an apparent contradiction between the facts that the full π CASSCF calculation definitely confirms that only one or two MOs may be considered as singly occupied, whereas the corresponding ROHF treatment provides a completely biased description of the π system. As a confirmation of this paradox, we shall demonstrate that starting from the single determinant built from the NOs of the full π CASSCF calculation, and introducing the excitations from doubly occupied to virtual MOs (single reference + single excitations, SR+S) to account for the spin polarization phenomenon, give the same spin density distribution as a CAS(Full valence π)+S calculation. These spin distributions are in good agreement with the experimental hyperfine coupling constants and at variance with the ROHF+S ones. An analytic derivation proposes a rationalization of the defects of the ROHF or minimal CASSCF calculations as missing the dynamic charge polarization effects brought by the one hole (1h)−one particle (1p) excitations on top of the charge transfer configurations between methylene and the aromatic frame. Although opened a long time ago, this domain is no longer an academic one. Actually, the interest for magnetic properties of conjugated hydrocarbons, and in particular fused polybenzenic architectures, is rapidly increasing. Until recently, the attempts to conceive and synthesize molecular or periodic materials presenting challenging magnetic properties, such as ferromagnetism, antiferromagnetism, spin crossover, and high magnetic anisotropy, essentially involved transition metal ions or lanthanides as spin carriers and belonged to coordination chemistry. It is nowadays tempting to transfer these concepts to organic compounds.12−20 Conjugated radical or polyradical conjugated hydrocarbons may play the same role as magnetic ions12 and may be assembled in a
II. ANALYTICAL DERIVATION OF THE SOMO COEFFICIENTS OF THE TOPOLOGICAL HÜ CKEL HAMILTONIAN The here-after considered conjugated hydrocarbons are “alternant” (solid state physicists say “bipartite”). They do not involve odd-membered rings, and consequently the carbon atoms can be separated into two categories (starred or nonstarred) in such a way that each starred atom is chemically connected to nonstarred atoms and reciprocally. An illustration of starred and nonstarred atoms is shown for the here-studied compounds in Figure 1.
Figure 1. Representation of the methylenebenzene (compound 1), two methylenenaphthalene isomers (compounds 2 and 3), metaxylylene (compound 4), and two dimethylenenaphthalene isomers (compounds 5 and 6). The starred atoms (see text) are indicated as well as the no-starred atoms for compounds 1, 2, and 4.
The expression of the Hückel Hamiltonian28−30 is Ĥ =
∑ tpq(ap+aq + aq+ap) p,q
where the sum runs over all couples of chemically bonded atoms. During the 1940s the “mirror theorem” has been proposed by Longuett−Higgins31 and Hall.32 For systems in which the number (na) of starred atoms is larger than the number (nb) of nonstarred ones, the mirror theorem says that the Hückel Hamiltonian presents na − nb eigenvalues of energy zero. In other words, the SOMO satisfies the eigenequation
Ĥ ϕa = 0ϕa = 0 The theorem also says that the coefficients of the SOMO on the minor (nonstarred) sites are zero. Contrarily to what is sometimes said,33 this property is also satisfied when the hopping integrals tpq are different. If they are equal, it is easier to establish the corresponding amplitudes of the nonzero coefficients of the SOMO of monoradicals. They are given by projecting the eigenequation on the q AOs of the nonstarred atoms, according to ⟨q|Ĥ |ϕa⟩ =
∑ tqp *cp * = 0 p*
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Dunning basis sets37 were used: C, 9s4p1d contracted in 3s2p1d; H, 4s1p contracted 2s1p. In UDFT calculations the B3LYP functional is used. In these calculations the difference to the expected values of ⟨S2⟩ never exceeds 0.04 for monoradicals and 0.12 for diradicals. Some BLYP calculations have also been performed. All the geometries were optimized in an unrestricted formalism, using the B3LYP functional for the Msmax solution. First, the orbitals were optimized at the ROHF level for the doublet ground state for monoradicals and triplet ground state for diradicals. Second, the orbitals of the ground state were optimized using a CAS(Full valence π)SCF calculation. The Hückel orbitals were analytically determined. Although the spin population of the SOMOs is the most important contribution to the spin density, the spin polarization of both σ and other π orbitals contributes quantitatively to the whole spin density. To introduce the spin polarization, singleexcitations should be accounted for, and this can be achieved using the restricted active space (RAS) program of the Molcas package. Notice that these excitations act on all the determinants of the active space, including those that, for symmetry reasons, have a zero coefficient in the CAS function. Unfortunately, the total number of orbitals that can be introduced in the RAS is limited to 50. To compare the spin densities obtained from the ROHF orbitals and those obtained from the CAS(Full valence π)SCF orbitals, we have technically proceeded as follows (Figure 3 for the notation of the π MO subspace): (i) Using the ROHF orbitals, we have performed a RASCI calculation involving all single excitations of an electron from any of the singly or doubly occupied π MOs (spaces 1 and 2) to either the SOMOs or any empty π orbitals (spaces 2, 3, and 4) (according to the Molcas conventions, all doubly occupied π orbitals belong to RAS1, the SOMOs belong to RAS2, and the empty π* orbitals belong to RAS3). This calculation is noted SR+S(Full π) in Table 2. (ii) The same calculation has been performed using the CAS(Full valence π)SCF NOs: it only differs from (i) by the set of MOs that is used. This SR+S calculation is noted SR+S(Full π) with CAS(Full valence π)SCF NOs. (iii) Using the CAS(Full valence π)SCF orbitals and wave function as reference, we have performed a RAS calculation. All the valence π MO belong to the CAS (spaces 1, 2, and 3) and the single excitation toward the nonvalence π* MO (space 4) are considered (all the valence π orbitals belong to the RAS2 subspace whereas all empty π orbitals belong the RAS3 subspace). This calculation is a multireference+Single excitations (MR+S) and is noted CAS(Full valence π)+S(Full π) with CAS(Full valence π)SCF NOs. Because such calculations may be computationally expensive, we have focused our study on the two smallest systems, compounds 1 and 4, for which (i), (ii), and (iii) results are reported in Table 2. The results of unrestricted single determinant calculations have also been reported. The spin densities in UDFT calculations are known to be sensitive to the amount of Fock exchange in the hybrid functional. We have therefore performed UBLYP, UB3LYP, and UHF calculations. As already noticed in the literature, the UHF results are absolutely irrelevant, the spin density being exceedingly delocalized, with
p and q being located on neighbor atoms. If all hopping integrals are equal, the equation
∑ cp * = 0 p*
is satisfied for all nonstarred sites q (p* and q being first neighbors). In practice, one may first give a coefficient 1 to the AO of a starred atom that has only two second neighbor sites. The coefficient of the SOMO on the AO of these atoms will be equal to −1. Progressing along the molecular graph, the satisfaction of the above equation leads to integer coefficients either positive or negative on the starred atoms. The norm of this vector is easily calculated and so is the spin density on the starred atoms, which is obtained by dividing the square of the coefficient by the norm. This means that the spin densities are necessarily rational numbers, a feature which, to our knowledge, had not been noticed. Figure 2 presents the coefficients of the
Figure 2. Hückel coefficients of the SOMOs for compound 1 and 2 (top) and 4 (bottom). The norm (N) and spin densities (SP) on the external atoms are N = 7 = (2)2 + (−1)2 + (−1)2 + 12, SP = 4/7 = 22/ N for 1, N = 20, SP = 9/20 for 2, and N = 3 (left MOs) and N = 4 (right MOs) whereas the total spin density is 7/12 for 4.
SOMO of the hereafter studied monoradicals and the corresponding spin densities calculated on the external atom. The same reasoning could be applied to diradicals and would provide the spin density contribution of the two SOMOs. Diradicals involve two SOMOs, which are degenerate and not uniquely defined. When the geometry exhibits a symmetry element for which the two SOMOs belong to a different irreducible representation, as is the case for compounds 4 and 5, one may determine the coefficients of the symmetry-adapted SOMOs by fixing in-phase and out-of-phase equal coefficients on the external carbons. In such cases, Hückel SOMOs may be compared one by one to the symmetry-adapted SOMOs obtained using the exact electronic Hamiltonian. In the absence of symmetry, one may find one SOMO identical to that of a parent monoradical and define the second one by orthogonality to the first one. The overall spin densities on the external methylene groups are 7/12 in compound 4, and 17/32 in compound 5. For compound 6 the spin densities are respectively 5/9 and 17/36 on carbons numbered 1 and 12, respectively.
III. COMPUTATIONAL INFORMATION Wave function theory and unrestricted density functional theory (UDFT) calculations have been performed using the Molcas7.2 package34−36 version 7.2. The following cc-pVDZ C
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coefficients on nonstarred (minor) atoms. Nevertheless, the coefficients on these sites are very small (
