D. A. Sweigartl Oxford University Oxford, England
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Lone Pair Orbital Energies in Group V1 and % Hydrides
I find that many chemists are surprised when told that water does not contain two lone pair orhitals of the same energy. The purpose of this communication is to clarify this point. The conceot of hvbrid orhitals has enioved . " considerable attention over the years. However, chemists frequently forzet that hvbridization is simnlv a useful mathematical de;ice and siould not he considered as representing physical reality. For example, it has been pointed out numerous times that hybrid orbitals lead to totally incorrect conclusions regarding orhital energies, ionization potentials, and spectroscopic properties of molecules (1,2). The best computation method available that retains the concept of orbitals is the Hartree-Fock self-consistentfield method (HF-SCF). Each of the molecular orbitals (MO's) generated by this procedure belongs to an irreducible representation of the point group of the molecule in question. Koopmans' theorem (3) states that for closed shell systems the ionization potential (IP) of an electron in an orhital is just the orbital energy as calculated by the HF-SCF method. One can apply transformations to these delocalized symmetry MO's to produce localized MO's closely corresponding to hyhrid orbitals. (I, 2, 4-6). We then have the orhitals localized along the bonds, and therehy obtain a picture that conforms to the conventional ideas of directed valence. In eeneral these localized orbitals do not individually belong to the irreducible representations of the molecule's noint " moun. . Now when talking about energy one must not use hyhrid orbitals because Koopmans' theorem applies only to the delocalized symmetry MO's. Thus an observed ionization potential refers to the energy of a Hartree-Fock orbital. Any energy parameter associated with a hyhrid orbital is not an ohservable. Another way of stating the matter is to observe that the delocalized symmetry MO's are eigenfunctions of the Fock operator with associated eigenvalues ("orhital energies"). Localized MO's or hyhrid orhitals are not in general eigenfunctions of the Fock operator and hence do not represent stationary states. We should note that when considering the total energy or total electron density of a system it matters not which type of orhital is used because the Slater determinant is invariant to an unitary transformation. A very close analogy to the above statement can he seen
'Address correspondence to the author at: Department of Chemistry, Swarthmore College, Swarthmore, Pa. 19081.
322 /Journal of Chemical Education
in vibrational spectroscopy. Thus only wavefunctions corresponding to delocalized normal coordinates are eigenfinctions of the Hamiltonian while wavefunctions of individual internal coordinates in general are not. In the Group VI and VII hydrides, H,X, there are symmetry MO's sufficiently localized on the X atom to warrent terming them "lone pair" orbitals. For Group VI (H2X) group theory tells us that there are no required degeneracies. Calculated IP's as well as ones measured by the powerful technique of ultraviolet photoelectron spectroscopy (UPS) (7-9) clearly show that the higher energy (lower IP) lone pair is essentially a pure p-type orbital while the second lone pair is roughly an s-type orhital. UPS shows that these are separated by at least 10 eV-a large amount of energy. For example, the first lone pair in water has an IP of 12.6 eV and the second has an IP meater than 30 eV. In the Group VII hydrides (HF, HCI, HBr, HI) two of the lone pairs are required by group theory to he degenerate. These degenerate lone pairs are p orhitals and have the lowest IP. The third lone pair is a s type orhital and has a much higher IP. Hence water (or R20) does not have two lone pairs of equal energy and HC1 (or RCl) does not have three lone pairs of the same energy. Many chemists seem puzzled by this. This may he because many take the hybridization process too literally. Used within reason hybrid orbitals can aid the interpretation of bond polarities, dipole moments, and molecular structure. As regards molecular structure we should remember that several quite successful theories exist that do not explicitly invoke hqbridization (10, 11).
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Acknowledgment The author acknowledges fruitful discussions with N. J. Hoyle and Professor R. B. Martin. Literature Cited I11 Hall. G. G., and Lenard-Jones. J.. Fmc. Roy. Soe. ( l a n d o n ) , A202. 155 11950); Pop1e.J.A.. Quart. Reu(LondonJ, 11,278 119571. 121 Peters. D.. J Amer Cham. S o c. 94.. 7W 119721: . . Cahen. I.. and Del Bone.. J... J. CHEM.EOUC..d6.487(19691. (31 Kooprnans. T.,Phyrira. I. I04 (1933). (41 Edmiiton,C..andRuedenberg, K..RP". M o d . P ~ Y s 35.457(,963). ., (61 Lennard-Ions, J., andPoplo, J. A , Proc Rqv. Soc. (London), AM2, 166(19501. 161 Pilar. F. L.. "Elementary Quantum Chemistry..' MeCrsw-Hill Book Company. Ns","",k l I. F.. . l,i r-. h.s n-, . 1, , , . .... ....., .
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