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Ionic and multicenter bonds

eFF can describe compounds containing the elements hydrogen, helium, and carbon; we now consider the elements that lie in between -- lithium, beryllium, and boron. These early elements present us with an opportunity to observe ionic bonding, since they are electronegative, as well as electron deficient multicenter bonding, since they lack enough electrons to complete a full octet. A collection of compounds containing lithium, beryllium, boron, and carbon are shown in Figure 4.14, and their dissociation energies are given in Table 4.5.

Figure 4.14: eFF can account for ionic and multicenter bonding.

Table 4.5: Dissociation energies.
    $ \Delta E$ (kcal/mol)
relative to energy of eFF exact
LiH Li + H 58.1 56.6
$ \mathrm{BeH_{2}}$ BeH + H 113.0 98.9
BeH Be + H 109.6 52.8
$ \mathrm{B_{2}H_{6}}$ 2 $ \mathrm{BH_{3}}$ 27.6 41.2
$ \mathrm{CH_{5}^{+}}$ $ \mathrm{CH_{3}^{+}}$ + $ \mathrm {H_{2}}$ 20.8 45.5

Lithium atom adopts a clear $ \mathrm{1s^{2} 2s^{1}}$ configuration, with a valence electron much larger ( $ \mathrm{s_{e}}$ = 7.45 bohr) than the spin-paired core electrons ( $ \mathrm{s_{e}}$ = 0.71 bohr). We can form lithium hydride by combining lithium with a hydrogen atom of opposite spin. The resulting ionic compound has a bond length slightly longer than the exact value (1.689 A versus 1.596 A exact), and because of the greater bond length, a slightly higher dipole moment as well (6.51 D versus 5.88 D). The dissociation energy is very near the exact value (58.1 kcal/mol versus 56.6 kcal/mol exact), which is not surprising since the electrons in the species Li, H, and LiH are all well represented by single-peaked functions.

In a similar manner, we form beryllium dihydride by adding two hydrogens to a beryllium atom. The bond length of $ \mathrm{BeH_{2}}$ is shorter than the bond length of LiH, due to the greater nuclear charge of Be; the difference in eFF bond lengths parallels that found in the exact values (-0.29 $ \mathrm{\AA}$ shrinkage versus -0.25 $ \mathrm{\AA}$ exact). The energy of breaking $ \mathrm{BeH_{2}}$ into BeH and H is near the exact value (113.0 kcal/mol versus 98.9 kcal/mol exact); however, the energy of breaking BeH into Be and H atoms is too high (109.6 kcal/mol versus 52.8 kcal/mol exact). This too-high energy is a consequence of a well-known difficulty in describing the valence electrons of beryllium atom as a single configuration wavefunction [34]

In beryllium, the $ \mathrm{{1s}^{2}{2s}^{2}}$ configuration is nearly degenerate to the $ \mathrm{{1s}^{2}2p_{x}^{2}}$, $ \mathrm{{1s}^{2}2p_{y}^{2}}$, $ \mathrm{{1s}^{2}2p_{z}^{2}}$ configurations; hence the wavefunction should be a resonance combination of these configurations. This static correlation is not well-described by Hartree-Fock or other single determinant methods, but the ``floating'' nature of the eFF electrons can account for these other configurations to some extent by shifting themselves to an average position between configurations. In the case of beryllium, eFF recognizes the $ 2s-2p$ degeneracy and separates the two valence electrons along an arbitrary axis to relieve electron-electron repulsion. However, it cannot shift electrons along the other two axes simultaneously, hence Be atom cannot gain its full measure of resonance stabilization. Static correlation in less symmetric cases, such as the breaking of a linear bond, should be better handled by eFF.

Ionic compounds can include as participants not only hydrides but carbanions as well. Tert-butyl lithium contains a very polar carbon-lithium bond, and an eFF model shows why this is the case. Imagine the compound $ \mathrm{(CH_{3})_{3}CH}$, but with the terminal $ \mathrm{H^{+}}$ replaced by $ \mathrm{Li^{+}}$. The $ \mathrm{Li^{+}}$ contains a $ \mathrm{1s^{2}}$ core of electrons, and so unlike the proton, moves far away from the center of electron density in what was formerly the C-H bond. Thus the carbon-lithium bond is polar, and the species acts as a carbanion; the tert-butyl group makes it too hindered to be nucleophilic, but the overall species can act as an active base. eFF makes the carbon-lithium bond too long (4.215 $ \mathrm{\AA}$ versus 2.026 $ \mathrm{\AA}$ B3LYP) because of the previously discussed too-diffuse nature of carbanions in eFF. We can describe as well the agglomeration of $ \mathrm{^{t}BuLi}$ into tetramers based upon tetraheral $ \mathrm{Li_{4}}$, where the lithium-lithium bonds optimize to the correct length (2.378 $ \mathrm{\AA}$ versus 2.43 $ \mathrm{\AA}$ Li-Li distance in the crystal structure [35]).

The boron compound $ \mathrm{BH_{3}}$ dimerizes into the borane $ \mathrm{B_{2}H_{6}}$ via a resonance combination of covalent and donor-acceptor bonds, which can also be viewed as two three-center two-electron bonds [36]. eFF describes nearly all aspects of the $ \mathrm{BH_{3}}$ and $ \mathrm{B_{2}H_{6}}$ geometries correctly ( $ \mathrm{BH_{3}}$ bond length 1.252 $ \mathrm{\AA}$ versus 1.190 $ \mathrm{\AA}$ exact; $ \mathrm{B_{2}H_{6}}$ B-B 1.347 $ \mathrm{\AA}$ versus 1.331 $ \mathrm{\AA}$ exact; B-H covalent = 1.243 $ \mathrm{\AA}$ versus 1.207 $ \mathrm{\AA}$ exact; B-H bridging = 1.347 $ \mathrm{\AA}$ versus 1.331 $ \mathrm{\AA}$ exact). However, the dimerization energy is too low (27.6 kcal/mol versus 41.2 kcal/mol exact).

Another example of resonance between covalent and donor-acceptor bonds is found in $ \mathrm{CH_{5}^{+}}$, a fluxional molecule [37] that can be viewed as an interaction between $ \mathrm{CH_{3}^{+}}$ and $ \mathrm {H_{2}}$ where the hydrogens are similar and rapidly interconvert. However, no such resonance appears in the eFF description, where the hydrogens remain clearly distinguishable -- the H atoms that were originally apart of $ H_{2}$ remain close together (0.798 $ \mathrm{\AA}$ versus 0.869 $ \mathrm{\AA}$ exact [38]) and far away from the carbon (1.744 $ \mathrm{\AA}$ versus 1.231 $ \mathrm{\AA}$ exact). Like borane, the association energy is also too low (20.8 kcal/mol versus 45.5 kcal/mol exact), suggesting a future need for explicit resonance/electron delocalization terms in eFF.

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Next: Application to matter at Up: Validation against ground state Previous: Allowed versus forbidden reactions   Contents
Julius 2008-04-29