Surprising Electronic Structure of the BeH– Dimer: a Full-Configuration

Article pubs.acs.org/JPCA

Surprising Electronic Structure of the BeH− Dimer: a FullConfiguration-Interaction Study Marco Verdicchio,†,‡ Gian Luigi Bendazzoli,§ Stefano Evangelisti,† and Thierry Leininger*,† †

Laboratoire de Chimie et Physique Quantiques - IRSAMC, Université de Toulouse et CNRS, 118, Route de Narbonne, F-31062 Toulouse Cedex, France ‡ Dipartimento di Chimica, Università di Perugia, 8, Via Elce di Sotto, 06123 Perugia, Italy § Dipartimento di Chimica Industriale “Toso Montanari”, P.A.M. Università di Bologna, Viale del Risorgimento 4, 40136 Bologna, Italy S Supporting Information *

ABSTRACT: The electronic structure of the beryllium hydride anion, BeH−, was investigated at valence fullconfiguration-interaction (FCI) level, using large cc-pV6Z basis sets. It appears that there is a deep change of the wave function nature as a function of the internuclear distance: the ion structure goes from a weakly bonded Be···H− complex, at long distance, to a rather strongly bonded system (more than 2 eV) at short distance, having a (:Be−H)− Lewis structure. In this case, it is the beryllium atom that formally bears the negative charge, a surprising result in view of the fact that it is the hydrogen atom that has a larger electronegativity. Even more surprisingly, at very short distances the average position of the total electronic charge is close to the beryllium atom but on the opposite side with respect to the hydrogen position.

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INTRODUCTION Beryllium is a four-electron atom having a 1s2 2s2 closed-shell electronic structure, a fact that explains why this species is not particularly reactive. This is related to a “negative” electron affinity (EA) (isolated Be− anions do not exist as stable species), and a large ionization potential (IP), larger than for both lithium and boron. However, it is well-known that the Be atom is able to form two sp hybrid orbitals, which are placed in a linear arrangement at the opposite sides with respect to the Be atom. A system showing this feature is the BeH2 molecule, often quoted in textbooks as a prototype for the sp molecularorbital hybridization. This possibility is due to the quasidegenerate nature of the 2s and 2p atomic orbitals in the isolated Be atom.1 Indeed, the first excited triplet state (3P) is only 2.725 192 eV above the ground state, whereas the corresponding singlet (1P) lies 5.277 430 eV above the fundamental.2 Experimentally, despite its popularity in textbooks, BeH2 is a relatively poorly known system. In fact, the synthesis and accurate measurement of the isolate-molecule spectroscopic constants have been the object of a recent study.3 At equilibrium, the Be−H bond length in this species is found to be 1.3324 Å, whereas from the computed energetics of the reaction Be(g) + H2(g) → BeH2(g) one can deduce a bond energy of 2.55 eV. Both values are very close to those of the BeH free radical, whose bond length is 1.3417 Å. The bond in beryllium dimer, whose existence as a relatively stable molecule has been a surprise among the chemistry © 2012 American Chemical Society

community, is also a result of sp hybridization. Moreover, in some recent theoretical works,4−8 we have shown that, because of sp hybridization, beryllium atoms are able to form linear chains, which are local minima on the potential energy surface (PES) of the system. Each atom in the inner part of the chain uses its two valence electrons to participate to two sp bonds. Each one of the two terminal Be atoms, on the other hand, has a partly occupied sp hybrid orbital (edge orbital), hosting only one electron. The four-valence-electron Be2 species can then be seen as the smallest of these structures. Another interesting case is the anionic species BeH−, having also four valence electrons. Because the beryllium atom is unable to accept an extra electron (accordingly to the previously reported negative EA), whereas the H− ion is a well-known stable species (EA = 0.75419 eV),9 the BeH− anion will dissociate as Be + H−. This means that, for large values of the interatomic distance R, we are in the presence of the interaction between two closed-shell systems, one of which bears a negative charge. The large-R situation is therefore very much similar to the structure of the HeH− anion, which is a weakly bonded system with an interatomic potential going to the asymptote as R−4 (R being the internuclear distance).10 At short distance, on the other hand, the Be atom is expected to form the two previously mentioned sp hybrid. One of them Received: October 16, 2012 Revised: December 1, 2012 Published: December 3, 2012 192

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combines with the 1s hydrogen orbital to form a σ Be−H orbital. The two electrons located in this bonding orbital come (formally) one from Be and the other one from H. The other sp hybrid orbital, the one located at the opposite side with respect to the H atom, can host the second Be valence electron and the anionic negative charge (again, we stress the fact that this partition of the electrons is purely formal). If this scenario is correct, we can predict that, for the distances that give rise to the formation of sp hybrids, the Lewis structure of the anionic dimer will be (:Be−H)−, in which the negative charge in excess is formally located on the beryllium atom. On the other hand, for large values of the interatomic distance R, we will have a Be···H− weakly bonded system, with the negative charge located on the hydrogen atom. Notice that the short-distance result is rather surprising, in view of the large difference between the electronegativity of the two atoms (2.20 for hydrogen and only 1.57 for beryllium in the Pauling scale11). To confirm the previously presented qualitative picture, we decided to perform a high-quality full-configuration-interaction investigation on this system. We used cc-pV6Z basis sets, which contains up to h function for hydrogen and i functions for beryllium (see Table 1 for details). The center of charge of the

The FCI calculations presented in this work have been performed by keeping the 1s orbital on the beryllium atom doubly occupied and frozen at the SCF level (fc-FCI), with the 4 valence electrons correlated in the FCI space without taking into account core−core and core−valence contributions. In fact, the contribution of core-related effects to the bond structure is expected to be very small. Relativistic effects and spin−orbit coupling have also been neglected, a fact justified because of the weakness of such effects for light atoms like beryllium and hydrogen. The size of the FCI space in terms of symmetry adapted determinants, as well as the total number of orbitals for each basis set, are reported in Table 1. Finally, to evaluate the charge distribution of the system, an analysis of the electric dipole moment, as well as the contribution of each natural orbital, have been calculated at different internuclear distances. The PES of the system was sampled at different values of the interatomic distance (R) up to the total dissociation, for the D, G, and F basis-sets. In particular from R = 1.0−2.0 bohr with a step size of 0.2, from 2.0 to 5.0 in steps of 0.1, from 5.0 to 6.0 in steps of 0.2, from 6.0 to 10.0 in steps of 0.5, from 10 to 15 in steps of 1, and finally from 20 to 100 with steps of 20 bohr. For the two largest basis sets, H and I, only five points around the minimum, the R = 100 asymptotic point and a few points in the region where the dipole momentum changes sign, have been calculated. The calculated potential energy curves, in the region close to the minima, have been finally interpolated for all the employed basis set to obtain the equilibrium geometries and dissociation energies using a cubic function. Finally, to study the effect of the inclusion of augmented basis functions and to have a better insight into our FCI results, we performed additional MRCI calculations with the aug-cc-PVQZ22−24

Table 1. Total Number of Basis Functions and FCI Space Size for the Five Employed Basis Sets contractions basis set

Be

H

total no. of functions

no. of FCI determinants

D F G H I

7s6p5d 7s6p5d4f 7s6p5d4f3g 7s6p5d4f3g2h 7s6p5d4f3g2h1i

6s5p 6s5p4d 6s5p4d3f 6s5p4d3f2g 6s5p4d3f2g1h

71 119 167 207 231

1 521 853 12 086 733 47 208 513 111 918 625 173 923 093

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RESULTS AND DISCUSSION At the beginning of this section the analysis of the HF results with the above-mentioned basis set is presented. In the second part we have reported the dissociation energies (De) and the obtained equilibrium distances (Req) as well as the analysis of center of charge of the system and its decomposition as contributions of the most relevant natural orbitals obtained with the fc-FCI calculations. In the ending subsection the MRCI analysis is presented. Hartree−Fock Description. We performed closed-shell Hartree−Fock calculations with the different truncations of the basis set. It appears that the well in the PES is already present at this level of description of the system, showing that the bond is not an effect of electron correlation. This is confirmed by the depth of the well, which is of the order of 2 eV. This relatively large value should be compared with the much shallower minimum of beryllium dimer.25 In Table 2, the equilibrium distance and well depth obtained in the case of the different truncations are reported and the curves obtained with these results are depicted in Figure 1. As can be noticed, the truncation level of the basis set does not affect in a significant way the calculated equilibrium distances and dissociation energies which remain practically unchanged with the growing of the number of basis functions (except if only s orbitals are included in the basis, in which case there is a very shallow minimum at R = 3.2 au following a potential energy barrier of the same magnitude, see Supporting Information). This is confirmed also by the plotted potential energy curves which overlap almost perfectly for all the range of examined internuclear distances.

system was computed, to have an indication about the average position of the electrons. We also computed the fullconfiguration-interaction (FCI) natural orbitals (NO), their occupation numbers, and the center of charge associated to each single orbital. In this way, we were able to decompose the total center of charge as a sum of contributions coming from each NO. The occupations of the NO’s give a good indication of the character (single reference versus multireference) of the wave function. In this case, as discussed later, it is possible to see that at long and short distances the wave function is dominated by a single Slater determinant, whereas there is an intermediate region (at about 6 bohr) showing a more pronounced multireference nature of the system.

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COMPUTATIONAL DETAILS The electronic energies for the BeH− dimer have been evaluated at the full configuration interaction level of theory using the NEPTUNUS code12 developed by our group (details of the method and the algorithm could be found elsewhere in the literature13−16). The Hartree−Fock preliminary steps for the evaluation of the one- and two-electron integrals have been carried out with the DALTON quantum chemistry package,17 whereas the following four-index transformation has been performed with the Ferrara transformation code.18 The interface between the two codes as well as with the FCI code has been realized using the Q5Cost data format and libraries.19−21 193

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so the physical meaning of the excited RHF orbitals is not particularly relevant. fc-FCI Results. Figure 3 shows the PES of the ground X Σ+g electronic state of the BeH− dimer for the five adopted basis

Table 2. HF Interpolated Dissociation Energies and Equilibrium Internuclear Distances for the Different Basis Sets basis set

Req (bohr)

De (eV)

d f g h i

2.6861 2.6848 2.6844 2.6846 2.6847

2.0545 2.0584 2.0603 2.0612 2.0614

Figure 3. Potential energy curves of BeH− with five different basis set of increasing dimension (see text for more details). In the inset a zoom of the minima region has been reported (the lines corresponds to the fitted cubic functions).

sets (only five points around the minimum have been computed for the H and I sets). The equilibrium distance is close to 2.7 bohr, with a dissociation energy of about 2.1 eV (see Table 3 for accurate

Figure 1. HF potential energy curve for the cc-PV6Z basis set and its truncations (see text for more details).

We also report, in Figure 2, the two valence occupied-orbital energies at RHF level, obtained with the I and an s-only basis

Table 3. FCI Interpolated Dissociation Energies and Equilibrium Internuclear Distances for the Different Basis Sets basis set

Req (bohr)

De (eV)

D F G H I ∞

2.7034 2.7028 2.7020 2.7022 2.7021

2.0534 2.1012 2.1115 2.1154 2.1164 2.1183

values). These values indicate that at short distance we have the formation of a fairly stable true chemical bond. The size of the basis sets is large enough to give almost converged values for the spectroscopic constants (disregarding core effects). This is confirmed, for instance, by the results of our recent study of beryllium dimer.26 However, to estimate the contribution of higher angular momenta function in the basis set, we have extrapolated the dissociation energies to the infinite basis-set limit with the interpolation expression:

Figure 2. Orbital energies of the σ1 (line) and σ2 (dots) orbitals for the I (filled) and s-only basis sets (empty).

set as a function of the internuclear distance. It appears that, in the region of R close to 5.0 bohr, there is an avoided crossing between these two valence orbitals in the case of the I basis set. It can be seen that the s−p mixing stabilizes the highest occupied orbital; when only s functions are used, the orbital energy rapidly decreases to reach the asymptotic ε1s(H−) = −0.0459 au value in the avoided crossing region. There is no evidence, on the other hand, of avoided crossings between occupied and virtual orbitals in the considered R region. Furthermore, as will be discussed in the last subsection, all the excited states of this system have an auto ionizing nature,

De(l) = D∞ + Al −β

(1)

where De(l) represents the dissociation energy calculated with the basis-set truncated to the lth angular momentum, A and β are suitable constants, and D∞ is the extrapolated dissociation energy for the infinite basis set. This equation is a generalization of an extrapolation expression proposed in refs 27 and 28 to extrapolate CI energies as a function of the maximum basis-set angular momentum. The interpolating function has been reported in Figure 4, showing an extremely accurate reproduction of the fitted values. 194

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α = −2k and using the same basis set, is 37.932 (au). The agreement between the two values is remarkable. For large values of the interatomic distance, therefore, the BeH− anion can be described as two interacting closed-shells, with a negative charge concentrated around the hydrogen atom. However, when the two atoms approach each other, the negative charge moves gradually from H to Be, despite the difference in electronegativity between the two atoms, which should favor a negative hydrogen. To follow this movement of the charge, we defined the charge shift (CS), η, as the ratio between the electronic center of charge, calculated at the FCI level, and the interatomic distance R. In all the calculations performed in this work, the Be atom is kept fixed at the origin of the coordinate system, while the H moves along the z axis. For this reason, at large value of R the contribution of the electrons on beryllium is practically zero, while the two electrons on the hydrogen give the largest contribution to the total electronic dipole moment. For our system, therefore, a value η = 2 is obtained when two negative charges are located on the H nucleus; a value of η = 1 is obtained when the negative charge in excess is located at the middle of the distance between the two nuclei (the other electrons being on the nuclei); η = 0 is obtained if all the electrons are located symmetrically around the Be atom; finally, a negative η means that there is an excess of negative charge on the opposite side with respect to he hydrogen position. In Figure 6 the η and the system energy (with respect to dissociation) as a function of the interatomic distance R for the

Figure 4. Extrapolated dissociation energy using eq 1 for the BeH− dimer. The dashed horizontal line represents D∞ at 2.118 eV.

As we can see from the energetic profiles and from the interpolated minima, reported in Table 3, the inclusion of the higher angular momenta functions does not improve considerably the dissociation energy that is correctly represented already by the F basis set (with this basis set we can reproduce about the 99% of the extrapolated infinite basisset De). The same considerations can be made also on the interpolated equilibrium distance (Table 3) that vary only by about 0.001 bohr passing from the D to I basis set. The long-range part of the potential energy curve has also been analyzed for the G basis set. Figure 5 shows that the

Figure 6. Charge shift (η) and potential energy curve comparison for the G basis set. Figure 5. Long range potential energy curve fitted by a −k/R4 function for the G basis set. See text for more details.

G basis set have been reported. As we can see from the plot, for large value of R the charge is completely localized on the hydrogen atom (η = 2). At R = 10 bohr, the initial stage of the formation of the Be−H bond, the η value starts to decrease considerably, and for geometries close to the equilibrium distance we found that the negative charge in excess is closer to the beryllium than to the hydrogen atom (η = 0.6). When the value of R decreases even more, η passes through zero and then reaches negative values/for R = 1.8 bohr; η assumes a zero value, whereas at 1.0 bohr it is equal to about −0.7. This means that a relevant part of the negative charge in excess is located on the opposite position of Be with respect to the H atom, especially for distances much shorter than the equilibrium distance.

calculated FCI dissociation energies, for R = 40, 60, 80, and 100 bohr of separation of the nuclei, are perfectly fitted by a function −k/R4. This form is, in fact, the expression expected for the interaction between two isotropic closed shells, a neutral and a charged ones. It can be shown that the value of the constant k is related to the polarizability of the neutral fragment, and we have U(R) = q2α/2R4, where U is the interaction energy of the two atoms, q is the charge of the ion (1 in au) and α the polarizability of the neutral atom.29,30 In our case, the polarizability of the Be atom, computed with the G basis set, is 37.662 (au), whereas the same quantity, obtained as 195

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1s orbital on the H atom at long distance of the nuclei giving the largest contribute to the total electronic dipole moment that, for large values of R, has a linear behavior as a function of R. The σ2 orbital, on the contrary, shows a contribution that is practically zero when the two atoms are well separated. Its contribution, however, becomes very important at small value of R. This gives a negative contribution to the charge average position, which is the principal cause of the negative value reached by the CS. This orbital can be associated to the 2s orbital on the beryllium which at large R gives a vanishing contribution to the total dipole, but when the interatomic distance becomes small, it gives the formation of the sp hybrid which, following our interpretation, host the negative charge. We also plotted the two-dimensional contour maps of the lowest NO’s for various values of the atomic separation. Figure 8 shows the σ1 and σ2 orbitals, calculated at the FCI level with the F basis set, for R = 1.0, 2.7, 10.0, and 15.0 bohr.a As we can see from these plots, the σ1 orbital is well localized on the H atom for large R and at short distance combines with one sp hybrid to form the σ bonding Be−H orbital. The σ2 orbital, on the other hand, is localized on the Be when the two atoms are well separated but, with the shortening of R, changes gradually its shape and moves to negative z. When the interatomic distance is small, also the influence of other higher orbitals becomes important and the contributions of these orbitals become relevant to understand the movement of the charge. In Figure 9 we have reported the same information of Figure 7 but for the π (because of degeneracy πy and πx have the same behavior) and σ3 orbitals. As can be seen from the plot, for large separation of the nuclei, these orbitals show a contribution to the total dipole moment equal to zero (despite an occupation different from zero) that suggests that both are localized on the Be atom. Their contributions, however, change drastically when the two atoms approach each other. The π orbital, indeed, gives a small negative contribution to the total dipole moment that starts to slightly decrease for values of R shorter than 13 bohr and reach a minimum (−0.055) for R = 2. More interesting is, instead, the behavior of the σ3 orbital. At long distances this orbital corresponds to the pz atomic orbital, because it has an occupation equal to that of the px and py. Its contribution to the total dipole moment shows a maximum (0.4) at around 11 bohr that suggests a movement of the orbital toward the H atom for that distances. When the atoms become closer, however, the contribution strongly decreases and assumes also negative value for R ≤ 4 bohr. Unlike the π orbitals, the contribution of σ3 does not tend rapidly to its asymptotic value, confirming that this orbital has

To reach a better understanding of this peculiar feature of the system, we analyzed in detail the five most relevant FCI natural orbitals (NO) (the 1s orbital of beryllium being kept frozen at the SCF level, it was not considered in the FCI expansion), and the center of charge associated to each orbital, from R = 1 to R = 20 bohr. These orbitals, when the interatomic distance becomes small, give rise to the formation of three σ orbitals and two degenerate π orbitals, which are labeled herein after as σ1, σ2, σ3, πx, and πy. Figure 7 shows the occupation and the

Figure 7. Occupations and contributions to the total dipole moment for the σ1 (A) and σ2 (B) natural orbitals for the G basis-set.

contribution to the total dipole moment of the orbitals σ1 (panel A) and σ2 (panel B) at various values of R. As can be seen from the reported curves, the σ1 orbital corresponds to the

Figure 8. Two dimensional plotsa of the σ1 (top) and σ2 (bottom) natural orbitals, for different interatomic distances, calculated with the F basis-set. 196

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Figure 10. Occupations and contributions to the total dipole moment, in the avoided crossing region, for the σ3 and σ4 natural orbitals.

in R, because the H nucleus is far from this orbital. The bond orbital σ1, on the other hand, tends to reduce its size as the two nuclei become closer. This explains why, in the region between 1.0 and 2.0 bohr, the charge contribution of σ1 strongly decreases as R is reduced, whereas the corresponding value of σ2 is almost constant (see Figure 7, where the different scales should be noticed). MultiReference CI Calculations. It is not the aim of the present work to systematically study the effect of different basis sets on the description of the BeH− bond, nor to see how different active spaces and multireference configuration interaction (MRCI) calculations are able to reproduce the FCI results. However, because BeH− is an unconventional system, which is risky to be presented to first year students as a typical case of “impossible molecule”, it is wise to be cautious before drawing conclusions. For instance, one can wonder about the size of core-correlation effects, by definition not taken into account in our valence FCI calculations. Moreover, although the v6z basis set we used at the FCI level is extremely large, one could wonder whether the use of diffuse atomic functions could alter the FCI picture we obtained. To answer these and related questions, a set of MRCI calculations on BeH− were performed by using the correlation consistent augcc-pVQZ basis set. After a preliminary HF step, we performed CAS-SCF calculations with a valence CAS(4/4) space, obtained by distributing the four valence electrons into four valence sigma orbitals. Then MRCI calculations were done, by both freezing and correlating the two beryllium core electrons. The first Σ singlet and triplet excited states were also obtained, after the appropriate averaged CAS-SCF calculations (again with ia CAS(4/4) active space). In Figure 11A, the HF, valence MRCI (v-MRCI) and corecorrelated MRCI (c-MRCI) calculations are presented. The well is already present at the HF level, which means that its origin is not in a correlation effect. The C-MRCI curve is very much parallel to the v-MRCI one, showing that core correlation effects, although in absolute not negligible, are very little distance dependent. The equilibrium distances (Req) and the dissociation energies (De) for these potential energy curves have been interpolated with a cubic function and the results reported in Table 4.

Figure 9. Occupations and contributions to the total dipole moment for the π (A) and σ3 (B) natural orbitals for the G basis set.

an active role in the description of the movement of the charge from H to Be which mainly take place in the σ space. By looking at the occupation of the σ3 orbital, one can notice the presence of an extremely sharp minimum close to R = 3 bohr. The corresponding charge contribution has a sudden change, although a careful analysis shows that both quantities are continuous. By analyzing the occupation of the other natural orbitals, we found the presence of an avoided crossing in the occupation number between the σ3 and σ4 (the next in occupation after σ3). This has been reported in Figure 10, where one can see a weakly avoided crossing between the two orbital occupations. Correspondingly, we can notice a switch in the charge contributions, indicating the presence of two diabatic states (in the occupation number). Finally, we give here an interpretation of these rather unexpected results in term of the sp hybrid orbitals of beryllium. At long internuclear distance, the valence orbitals of Be are the atomic 2s, doubly occupied, and the three 2p orbitals, empty. For short internuclear distances, there is the formation of two sp hybrids. One of them combines with the hydrogen 1s orbital to give rise to the bond orbital. The other one, which points in the opposite direction with respect to the H atom, is a lone pair. When these two orbitals have roughly the same size, the total charge contribution is close to zero. It is the case for the region close to equilibrium (actually, a vanishing η is obtained for slightly shorter distances). However, the size of these orbitals has a very different evolution as a function of R. The lone pair σ2 is not very sensitive to variations 197

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Figure 11. MRCI potential energy curves obtained with the aug-ccpVQZ basis set. In panel A a comparison of valence-MRCI, core correlated-MRCI and HF results has been reported. Panel B shows the results for the ground state (X1Σ) and first excited state (A1Σ) for the ion compared to ground state (2Σ) of the neutral dimer.

system, with an estimated internuclear distance of 2.7021 bohr, and a bond energy of about 2.12 eV. These values are close to the corresponding quantities of the linear BeH2 system (2.5179 bohr and 2.55 eV, respectively) and the BeH radical, whose bond length is 2.5354 bohr. In particular, in the present work we studied the charge distribution of the anion, which is a very effective tool to understand the structure of the molecule. Our results show that for large value of interatomic distance the excess of negative charge is completely localized on the hydrogen atom, as confirmed by the value of η = 2. At the initial stage of formation of the Be−H bond (R = 10 bohr), the value of η starts to decrease considerably, down to a negative value for R < 1.8 bohr, which means that the relevant part of the negative charge is located on the opposite position of Be with respect to the H atom, giving rise to a rather strongly bonded system (2.1 eV) having a (:Be−H)− Lewis structure. These unexpected results are analyzed in details in term of the sp hybrid orbital of beryllium. Indeed, at long internuclear distance, the valence orbitals of Be are the atomic 2s, doubly occupied, and the three empty 2p orbitals. When the formation of the two sp hybrids occurs, one of them combines with the hydrogen 1s orbital, to give rise to the bond orbital, and the other one is a lone pair oriented on the opposite direction with respect to the H atom. This orbital is not very sensitive to variations in R, whereas the orbital σ1 tends to reduce its size as the two nuclei become closer. This explains the negative value of η in the region between 1 and 2 bohr, which is strongly affected by the size of these two orbitals. In synthesis, we can say that the mechanism of bond formation in the BeH− ion is the result of two combined effects: the quasi-degeneracy of 2s and 2p energies in beryllium atom, and the very weak electron affinity of hydrogen (that can be seen also as the ionization potential of H−). The first effect implies an easy formation of two sp hybrid linear orbitals, whereas the second effect results in a low energy cost to move the H− extra electron into one of these hybrid orbitals. The first aspect of the bond mechanism is confirmed by the fact that no bond is formed (i.e., there is a very tiny minimum) if only s orbitals are included in the basis set, thus preventing the formation of sp hybrid orbitals. The second aspect is confirmed by the fact that also the isoelectronic BeHe species has a purely repulsive energy curve (apart from a tiny van der Waals well at very large distance), because of the impossibility of mobilizing

Table 4. Interpolated Dissociation Energies and Equilibrium Internuclear Distances for the HF, v-MRCI, and c-MRCI Obtained with the aug-cc-pVQZ Basis Set basis set

Req (bohr)

De (eV)

HF v-MRCI c-MRCI

2.6664 2.6921 2.6737

2.0087 1.9158 1.9636

One could also wonder if the well in the 1Σ ground-state curve could be the effect of an avoided crossing with a higher state. To investigate this point, we computed the first singlet 1Σ excited state, as well as the first triplet, 3Σ (of course, for symmetry reason, the triplet cannot have any avoided crossing with the singlet ground state). We investigated also, at the same MRCI level, the behavior of the 2Σ ground state of the neutral species BeH, which is found to lie, for any value of the internuclear distance, well above the anion ground-state curve. In Figure 11B, the energies of the ground state and first excited states of the anion, and the neutral ground state, are shown as a function of the distance. It appears that the neutral 2Σ state is above the ion ground state but below the first singlet excited state of the ion. This means that the ion excited state is an autoionizing state, having a disposition to spontaneously lose its extra electron. This fact is confirmed by the position of the first triplet excited state (not shown in the figure for graphical reasons), whose curve is placed almost perfectly on the top of the singlet state. In fact, these two states, having almost identical energies, and being at every distance very much parallel to the 2Σ state, are described by an electron at large distance from the neutral molecule (as far as this is permitted by the finite basis set). The coupling between the unpaired electron of the BeH radical and the distant electron can give either a singlet or a triplet spin function, with virtually no effect on the energy. It should also be stressed that the fact that the two excited-state curves do not superpose to the neutral 2Σ one is simply due to the incompleteness of the basis set. (In principle, the convergence of an autoionizing-state energy to the neutral one as a function of the basis set size is very slow, because extremely diffuse basis functions are needed to describe an electron at very large (“infinite”) distance.)

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CONCLUSIONS We presented a valence FCI study of the electronic structure of the BeH− anion. This molecule turned out to be a fairly bonded 198

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L. Bendazzoli, S. Evangelisti, with contributions from L. Gagliardi, E. Giner, A. Monari, and M. Verdicchio. (13) Bendazzoli, G. L.; Evangelisti, S. J. Chem. Phys. 1993, 98, 3141− 3150. (14) Bendazzoli, G. L.; Evangelisti, S. Int. J. Quantum Chem. Symp. 1993, 27, 287−301. (15) Gagliardi, L.; Bendazzoli, G. L.; Evangelisti, S. J. Comput. Chem. 1997, 18, 1329−1343. (16) Rossi, E.; Bendazzoli, G. L.; Evangelisti, S. J. Comput. Chem. 1998, 19, 658−672. (17) DALTON, a molecular electronic structure program, Release 2.0 (2005), see http://www.kjemi.uio.no/software/dalton/dalton.html. (18) Angeli, C.; Cimiraglia, R. Private communication. (19) Rossi, E.; Emerson, A.; Evangelisti, S. Lecture Notes in Computer Science (LNCS2658); Springer: Berlin, 2003; Vol. II, pp 316−323. (20) Borini, S.; Monari, A.; Rossi, E.; Tajti, A.; Angeli, C.; Bendazzoli, G. L.; Cimiraglia, R.; Emerson, A.; Evangelisti, S.; Maynau, D.; Sanchez-Marn, J.; Szalay, P. G. J. Chem. Inf. Model. 2007, 47, 1271− 1277. (21) Scemama, A.; Monari, A.; Angeli, C.; Borini, S.; Evangelisti, S.; Rossi, E. Lect. Notes Comput. Sci. 2008, 5072, 1094−1107. (22) Feller, D. J. Comput. Chem. 1996, 17, 1571−1586. (23) Schuchardt, K. L.; Didier, B. T.; Elsethagen, T.; Sun, L.; Gurumoorthi, V.; Chase, J.; Li, J.; Windus, J. T. L. Chem. Inf. Model 2007, 47, 1045−1052. (24) Dunning, J. J. Chem. Phys. 1989, 90, 1007−1023. (25) Evangelisti, S.; Bendazzoli, G. L.; Gagliardi, L. Chem. Phys. 1994, 185, 47−56. (26) Helal, W.; Evangelisti, S.; Leininger, T.; Monari, A. Chem. Phys. Lett., submitted for publication. (27) Helgaker, T.; Klopper, W.; Koch, H.; Noga, J. J. Chem. Phys. 1997, 106, 9639−9646. (28) Halkier, A.; Helgaker, T.; Jørgensen, W. P.; Klopper; Koch, H.; Olsen, J.; Wilson, A. K. Chem. Phys. Lett. 1998, 286, 243−252. (29) Stone, A. J. The theory of intermolecular forces; Claredon Press: Oxford, U.K., 1996. (30) Bottcher, C. J.; Bordewijk, P. Theory of electric polarization; Elsvier: New York, 1980; Vol. II.

the helium electrons. It is the combination of these two properties that lead to the formation of an unexpectedly strong bond in this surprising molecular system.

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ASSOCIATED CONTENT

S Supporting Information *

Plot of the potential curve of the ground state of BeH− using the s-only basis set at the MRCI level as well as the table of the energy for the singlet and triplet states using the same basis set at the HF, CASSCF, and MRCI levels. This material is available free of charge via the Internet at http://pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We would like to thank Prof. Kirk Peterson (Washington State University) for kindly providing the cc-pV6Z basis set for the beryllium atom, and Prof. Celestino Angeli and Prof. Renzo Cimiraglia, (University of Ferrara) who wrote the four-index transformation program (unpublished) needed to produce the FCI integrals. This work was partly supported by the University of Toulouse and the French “Centre National de la Recherche Scientifique” (CNRS, also under the PICS action 4263), and the Italian Ministry of University and Research (MIUR). The authors thank CINECA (Bologna) for the generous PhD grant accorded to one of the authors (M.V.). GLB thanks the University of Bologna. This work was also granted access to the HPC resources of CALMIP (Toulouse) under the allocation 2011-p1048.

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ADDITIONAL NOTE 2D plots details: maximum contour value = 1.00. Number of contours = 50. a

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REFERENCES

(1) Salomonson, S.; Wartson, H.; Lindgren, I. Phys. Rev. Lett. 1996, 76, 3092−3095. (2) Kramida, A.; Martin, W. C. J. Phys. Chem. Ref. Data 1997, 26, 1185−1194. (3) Bernath, P. F.; Shayesteh, A.; Tereszchuk, K.; Colin, R. Science 2002, 297, 1323−1324. (4) Monari, A.; Vetere, V.; Bendazzoli, G. L.; Evangelisti, S. Chem. Phys. Lett. 2008, 465, 102−105. (5) Vetere, V.; Monari, A.; Scemama, A.; Bendazzoli, G. L.; Evangelisti, S. J. Chem. Phys. 2009, 130, 24301−24309. (6) Monari, A.; Bendazzoli, G. L. J. Chem. Theory Comput. 2009, 5, 1266−1273. (7) Pastore, M. C.; Monari, A.; Angeli, C.; Bendazzoli, G. L.; Cimiraglia, R.; Evangelisti, S. J. Chem. Phys. 2009, 131, 34309−34320. (8) Evangelisti, S.; Monari, A.; Leininger, T.; Bendazzoli, G. L. Chem. Phys. Lett. 2010, 496, 306−309. (9) Rienstra-Kiracofe, J. C.; Tschumper, G. S.; Schaefer, H. F., III. Chem. Rev. 2002, 102, 231−282. (10) Bendazzoli, G. L.; S., E.; Passarini, F. Chem. Phys. 1997, 215, 217−225. (11) Pauling, L. The Chemical Bond; Cornell University Press, Ithaca, NY, 1967. (12) NEPTUNUS is a Quantum-Chemistry FORTRAN code for the calculation of FCI and CAS-CI energies and properties written by G. 199

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