Article pubs.acs.org/JPCA
Excess Electrons Bound to Molecular Systems with a Vanishing Dipole but Large Molecular Quadrupole Thomas Sommerfeld,* Katelyn M. Dreux,† and Robin Joshi Department of Chemistry and Physics, Southeastern Louisiana University, SLU 10878, Hammond, Louisiana 70402, United States S Supporting Information *
ABSTRACT: Electron attachment properties of covalent molecules and ion clusters with vanishing dipole moments but large quadrupoles are studied with coupled cluster ab initio methods. Selection of the molecules studied is driven by two goals, finding a paradigm quadrupole-bound anion and investigating whether there is a correlation between the magnitude of the molecular quadrupole and the vertical attachment energy. Out of all examined species, only the ion clusters and four of the covalent molecules are found to support bound anions. The shapes and spatial extents of the associated excess electron distributions are qualitatively and quantitatively characterized, respectively. Two of the four covalent systems are especially promising as paradigm systems because of advantageous trade-offs regarding the number of isomers and conformers as well as synthetic closeness to commercial sources. No correlation was found between the vertical attachment energy and molecular quadrupole in an analysis that included the newly identified bound anions, those molecules, which were found not to support bound anions, and succinonitrile, which had been studied before. Moreover, there is clearly no such thing as a “critical quadrupole moment”. There are, however, very strong electron correlation effects involved in the binding of the excess electrons, and similar to succinonitrile, for five out of six anions identified here, the molecular quadrupole of the neutral itself is too weak to bind an excess electron, and electron correlation in the form of dynamic polarization is required to do so.
1. INTRODUCTION Adding an excess electron to a stable closed-shell molecule can yield two classes of anions. On the one hand, the electron can be added to a compact valence orbital, that is, an orbital with a spatial extent similar to that of the molecular framework. The resulting valence anion can be stable, but it is often metastable by several eVs, at least at the geometry of the neutral, and typical lifetimes are often in the femtosecond range. On the other hand, the excess electron may be added to a diffuse Rydberg-like nonvalence orbital, provided that the molecule possesses one, where it is bound by comparatively weak longrange forces. Characteristic examples include dipole-bound anions such as CH3CN− or (H2O)−2 , quadrupole-bound anions such as the anti conformer of the succinonitrile anion, and correlation-bound anions such as (NaCl)−4 or xenon cluster anions.1−6 Nonvalence anions bridge the gap between stable valence states and the scattering continuum and can in this way aid in the process of capturing thermal electrons.7−9 They are a sensitive probe for electron correlation effects,5,10 and they have served as models for electrons solvated in or located on surfaces of molecular liquids, even though in these cases, the electron binding energies tend to increase into the eV range and then the states in question do acquire some valence character and their classification is less straightforward.2,11,12 © 2014 American Chemical Society
Considering nonvalence states of single molecules as a group, the main focus in the past has been on dipole-bound states,1,2,13 whereas for all other types, only a few examples have been investigated, and even fewer have been characterized well.1,9,14,15 The species studied here expand the group of nonvalence anions without a dipole, and the emphasis is on molecules that possess a large molecular quadrupole. For the time being, we will refer to these anions as “quadrupolebound”. Note, however, that there are two understandings of terms such as “dipole-bound”, “quadrupole-bound”, or “polarization-bound”, one stemming from the world of one-electron model Hamiltonians13,16 and one stemming from the ab initio world10,17 (see ref 5 for a detailed juxtaposition). Whereas for dipole-bound states the difference between the two meanings is more often than not insignificant, the opposite is true for quadrupole-bound states, and while the dipole moment is a good predictor for the existence of a bound anion,1 it will turn out that the molecular quadrupole moment is not (see section 5). Special Issue: Kenneth D. Jordan Festschrift Received: December 1, 2013 Revised: January 29, 2014 Published: February 12, 2014 7320
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center of mass or to the centers of positive charge.22 Basis set demands and system sizes for covalent systems and ion clusters differ, and therefore, somewhat different basis sets have been used. The set of covalent species includes fairly large molecules, and the VEA has been computed using the Aug-cc-pVDZ standard set augmented with an even-tempered 6s6p6d set (even scaling factor of 3.5, identical exponents between 0.1 and 1.55 × 10−5 for all angular momenta) centered at the molecule’s center of mass. Because we are searching for quadrupole-bound anions, adding more functions would be interesting; however, it leads to near-linear dependencies and numerical instabilities in the self-consistent field (SCF) and coupled cluster calculation for the larger molecules. For small molecules such as succinonitrile, adding more diffuse functions can be tested, and going from a 6s6p6d to a 8s8p8d set changes the VEA from 7.85 to 7.86 meV (for EA-EOM-CCSD results, see below). Note that this is only converged with respect to diffuse functions in angular momenta up to 2 and that it does not include convergence with respect to the valence set. Nevertheless, we expect this basis set to be sufficiently flexible to predict stability or nonstability within about 1 or 2 meV at threshold (cf. ref 22). The ion clusters studied contain alkali metal ions, and therefore, the core-polarized version of Dunning’s basis sets has been used. Moreover, the clusters are small, the interest is directed more to the threshold region, and there are atoms, O and Mg, in the symmetry center. Therefore, these atomic basis sets have been extended, and even-tempered basis functions with a scaling factor of √10 are used to do so such that the smallest exponent in all angular momenta is roughly 10−5, resulting in a 8s8p9d9f set for O and a 6s6p7d8f set for Mg. For the ion clusters, the VEA can be easily controlled (see below), and analysis of the natural orbitals shows that the most diffuse basis functions only mix significantly into the wave function once the VEA approaches the threshold region (1−2 meV). Last, the electronic structure method used to compute electron binding energies is the equation-of-motion coupled cluster with single and double excitations method for electron attachment energies (EA-EOM-CCSD).23 The EA-EOMCCSD approach is a so-called direct method, that is, it does not start from a SCF calculation for the anion, then addresses correlation effects in the anion, and then computes the difference between two total energies, but instead, it directly computes the electron attachment energies of the neutral molecule. The EA-EOM-CCSD offers three key advantages. First, a direct method is indispensable for searching for quadrupole ions, for systems such as the quadrupole-bound anion of succinonitrile,9 and all other quadrupole-bound anions that will be identified here are examples of correlation-bound anions,5 that is, anions where SCF treatments fail to predict a bound anion. In turn, indirect many-body methods started from SCF orbitals for an unbound anion will in all likelihood also describe or converge on an unbound state, in particular, methods that rely heavily on a starting point such as MP2 or CCSD. Second, regarding those nonvalence states for which such a comparison is possible, it has been shown that EA-EOMCCSD yields VEAs that are very close to results from coupled cluster with singles, doubles, and noniterative triples substitutions (e.g., refs 24 and 25). Third, EA-EOM-CCSD calculations yield natural orbitals, and again, for nonvalence states, it is typically possible to identify one natural orbital with an occupation number very close to one that describes the
Here, electron attachtment to a set of covalently bound molecular systems and to two ion clusters will be investigated. These systems have been selected, in the first place, in order to find a “paradigm” quadrupole anion, that is, a molecule that forms a quadrupole-bound anion but has no isomers or conformers that form dipole-bound or valence anions. Moreover, the farther the molecule is synthetically from a commercially available starting material, the slimmer the chances clearly are that experimental observation will even be attempted in the foreseeable future. In the second place, broadening the database of quadrupole-bound states should help to address some fundamental questions regarding electron binding to quadrupoles. It is well-known that in contrast to electron binding to monopoles and dipoles, no general statements about quadrupoles are possible and that all conclusions are tied to specific models.18 Still, the question whether there is a strong correlation between the molecular quadrupole and the vertical attachment energy (similar to what has been found for dipole moments1) and whether this trend holds only for related or even for diverse sets of molecules is a useful one. There is no lack of modeling activity in this direction;18 what is lacking are data for molecular systems. The paper is organized as follows. The next section describes the ab initio methods employed to find equilibrium structures and compute attachment energies and the distributions of excess electrons. In contrast to dipole moments, the molecular quadrupole is not quite as well-defined as there are several conventions about factors in different communities, molecular quadrupoles can depend on the origin, and there are different ways of defining its magnitude. These conventions and the one used here are described in section 3. Ab initio results are reported in section 4, and our conclusions are discussed in section 5.
2. COMPUTATIONAL METHODS Computing vertical electron affinities associated with nonvalence states is a nontrivial task and requires nonstandard basis sets and electronic structure methods. In the following, first, the standard computational methods used to find the minimal energy structures of the neutral molecules are described. Second, the basis sets used for the anions and tests regarding basis set saturation are reported. Last, the electronic structure methods needed in this context are discussed. Four different valence basis sets were used: Dunning’s augmented correlation-consistent double-ζ set (Aug-cc-pVDZ), the related core-polarized augmented correlation-consistent double-ζ and triple-ζ sets (Aug-cc-pCVDZ and Aug-ccpCVTZ),19 and Ahlrich’s redefined triple-ζ set (Def2TZVP).20 All neutral molecules were initially optimized with the Tao−Perdew−Staroverov−Scuseria density functional21 and the Def2-TZVP basis set, and all structures were confirmed to be minima by computing second derivatives of their energy. Then, their geometry was reoptimized using second-order perturbation theory (MP2), again with the Def2-TZVP basis set, and the MP2 geometries were subsequently used for computing properties including the vertical electron affinity (VEA). For computing VEAs corresponding to valence anions, basis sets need to be augmented with diffuse functions, and standard sets exist to do so. If nonvalence states are considered, additional functions with even smaller exponents are needed; however, depending on the state considered, these functions are not needed on every atom but can instead be added to the 7321
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quadrupoles. Here, the so-called Frobenius norm is used to define the quadrupole magnitude
excess electron similar to a Dyson orbital from Green’s functions calculations.5,24−27 Version 2.9 of the Orca package was used for density functional calculations and MP2 optimizations, and version 1 of the CFOUR package was used for EA-CCSD-EOM calculations and all property calculations.28,29
Q≡
1 2
∫ ρ( r ⃗)(3xixj − r 2δij) d3x
(2)
where θ̲ is the quadrupole tensor of the molecule, and taking its transpose is actually not needed owing to the symmetry of the tensor. The advantages of this definition is that the Frobenius norm is related to expressions in electromagnetism such as the power emitted by a radiating quadrupole, and that in contrast to induced norms, which use the maximum eigenvalue of θ̲Tθ̲, Q averages over the whole tensor and in this way represents the overall polarity of the molecule in a more balanced way. A disadvantage is that the quadrupole magnitude of the two textbook examples, four-charges-in-a-square and three-chargesin-a-line, are larger than their quadrupole moments, Q̃ , by factors of √2 and √1.5. However, this can hardly be called a disadvantage because Q̃ is analogous to a repeated component of a vector, and the magnitude of that vector should naturally be larger than its component. Be that as it may, apart from pathological cases, norm definitions should give results that correlate fairly well with each other, and for the set of molecules studied here, the correlation between the Frobenius norm and an induced norm was tested and was found to be indeed good.
3. QUADRUPOLE MOMENT AND QUADRUPOLE MAGNITUDE There is some confusion in the literature related to conventions for reporting quadrupole moments (cf. ref 18), and the first point of this section is to clearly lay out the conventions used here. Moreover, the quadrupole moment of a molecule is a tensor, which can be diagonalized by proper choice of coordinate system, but if the goal is to characterize the size of a molecular quadrupole by a single number, then a norm or magnitude is needed. For tensors, several norms can be used, and again, no common convention exists for a quadrupole magnitude, and the convention used here needs to be specified. First, in different physics and chemistry communities, there are different conventions of splitting the factor of 1/6, which emerges in the perturbation expansion of the field of a charge distribution, into factors belonging to the quadrupole moment and factors belonging to the expansion. Even focusing on just one area such as quantum chemistry, different conventions are used, and thus, a reported value can be easily misread for, say, 2/3 or 3/2 of its value, unless the convention is explicitly stated (cf. ref 18). Here, we use Stone’s “molecular physics” definition θij =
tr( θ̲ T θ̲ )
4. AB INITIO RESULTS In this section, the possibility of electron attachment to a series of closed-shell molecules with a large quadrupole moment but vanishing dipole is studied computationally. First, the molecules selected for study will be discussed at some length, then, for those systems supporting bound anions, the distribution of the excess electrons will be analyzed, and last, the correlation of the VEA and the quadrupole magnitude will be investigated. The selection of systems studied was mainly motivated by two questions. First, it would be advantageous to have a paradigm quadrupole-bound anion,4 that is, a molecule that supports only a quadrupole-bound state, while none of its conformers form any valence or dipole-bound anions whatsoever. More specifically, this translates into a molecule with a saturated, rigid framework to which two or more electronwithdrawing or -donating groups can be attached. Rigidity is required to avoid complications with isomers and conformers, and saturation is required to avoid bound valence states. In addition, the synthetic route to the molecule should not be too long to make it accessible not only to theoreticians, and it is already clear that some trade-offs between these requirements will be necessary. The second reason for selecting certain systems involves a number of questions connected with the correlation between the quadrupole magnitude and the VEA. To this end, molecules with different quadrupole magnitudes are needed, and while for the ion clusters the quadrupole moment can be changed systematically by changing the bond lengths, for the molecular species, the number of substituents and the substituents themselves are varied. One molecular framework offering a good compromise between commercial availability and the existence of a small number of isomers and conformers are 1,4-substituted cyclohexanes, and a series of di- and tetra-substituted cyclohexanes have been investigated. While for the tetrasubstituted species only a single isomer exists, the disubstituted species have a cis isomer and a trans isomer, and the trans isomer shows two chair conformers (Figure 1), referred to as trans(ee) and trans(aa) depending on whether the two
(1)
where θij is an element of the Cartesian quadrupole tensor, θ̲,30 and ρ(r)⃗ is the charge distribution with r ⃗ = (x1,x2,x3). Moreover, provided that a charge distribution has a nonvanishing dipole moment, the quadrupole tensor including its magnitude is origin-dependent. In this context, this dependence is largely irrelevant, but for comparison, VEAs and molecular quadrupoles of conformers with dipoles have been computed, and for these conformers the origin-dependence matters. In quantum chemistry, two conventions are commonly used, the center of mass and center of charge origin; the quadrupole moments reported here are with respect to the center of mass origin. The second issue is the matter of ranking a set of molecules by their quadrupole moment, which is not simply a number but a symmetric, traceless tensor. In highly symmetric cases, for example, a linear symmetric arrangement of three point charges, (−)(2+)(−), the three-charges-on-a-line case, or a square of alternating positive and negative charges (fourcharges-in-a-square), the quadrupole tensor can indeed be expressed in terms of a single number, Q̃ , which is often referred to as the quadrupole moment. However, for a molecule such as the anti conformer of succinonitrile (C2h symmetry), in the coordinate system aligned with the symmetry axes, θ̲ contains three different values on its diagonal, which of course sum up to zero, and there are two identical nonvanishing offdiagonal elements. Even if θ̲ is diagonalized, at least two numbers are needed to characterize it, and thus, some type of norm is needed to establish a magnitude for the quadrupole tensor. In comparison with vectors, there exist many more matrix and tensor norms, and to the best of our knowledge, no common convention has been established for molecular 7322
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Figure 1. Cis−trans isomers of 1,4-substituted cyclohexane. The two structures on the left show two conformers of the trans isomer; the structure on the right shows shows one conformer of the cis isomer. In Table 2, the trans conformers in the middle and on the left are referred to as trans(ee) and trans(aa) to indicate the position of the two X groups, which are both equatorial or axial, respectively.
Figure 3. One of the C3 symmetrical minimal energy structures of 1,4diazabicyclo[2.2.2]octane di-N-oxide (left) and the associated D3h symmetrical transition state (right) connecting the C3 structures. The energy difference between the two structures shown is predicted to be 1.2 kJ/mol, and the dipole moment induced by the distortion to C3 is predicted to be negligible (Table 2).
substituents occupy equatorial or axial positions, respectively. 1,4-Dimethylpiperazine di-N-oxide is another compound that follows the same building principle and possesses the same cis/ trans isomers and conformer pairs (Figure 2). A different construction principle employs more rigid, polycyclic frameworks, which avoids the isomer−conformer issues at the price of more involved synthetic routes typically associated with saturated polycyclic molecules having functional groups in specific positions. Here, one member of this class is studied, 1,4-diazabicyclo[2.2.2]octane di-N-oxide (BNO) (Figure 3). The advantage of this molecule is that the tertiary amine, 1,4diazabicyclo[2.2.2]octane, a direct precursor of BNO, is commercially available. Moreover, three additional covalent molecules were included for comparison. Succinonitrile was reinvestigated with the same methods and basis sets for the sake of a consistent data set; the formamide dimer and the Nmethyl formamide dimer have been included because they have been identified as small systems with a large quadrupole and have been put forward as possible candidates for quadrupolebound states based on results from a model Hamiltonian.14,31 Last, two ion clusters have been included in the set. Ion clusters are perhaps the most straightforward way to build molecular systems without dipole moments but large quadrupole magnitudes. In fact, Be2O−2 , the earliest anion, which has been referred to as quadrupole-bound, is clearly derived from the textbook four-point-charges-in-a-square example, and other analogues following the same building principle have been investigated in this and other contexts.2,5,32,33 It seems surprising, therefore, that quadrupole-bound anions of ion clusters based on the second textbook example, three-pointcharges-on-a-line, have not been investigated so far. Here, the VEA of the clusters NaONa modeling a (+)(2−)(+) and FMgF modeling a (−)(2+)(−) point charge arrangement will be studied, where specifically NaONa and FMgF have been chosen because they represent isoelectronic species. It should be mentioned that the FMgF− anion has been investigated before34 and found to show a bent structure, which makes
FMgF a priori unsuitable as a candidate for a paradigm quadrupole system, but it is of course nevertheless useful regarding the quadrupole magnitude VEA correlation question. Moreover, the quadrupole magnitudes of FMgF and NaONa can be easily modified by changing the respective bond lengths, making these two ion clusters ideal model system (see below). Before turning to the VEA of the covalent molecules and ion clusters, let us briefly consider two issues regarding the geometrical structure of the neutral systems, the relative energies for molecules that exist as different isomers and conformers and the minimum-energy structure of BNO. The first issue regards 1,4 doubly substituted six-membered ring systems (cf. Figures 1 and 2), and the associated relative energies of the different isomers and conformers are listed in Table 1. While the bulky substituents (CH3, CF3) follow the textbook rule in that they prefer the equatorial position, the opposite is true for F, Cl, and CN, even though for the latter, the energy differences are much smaller (
