Localized and Spectroscopic Orbitals: Squirrel Ears on Water -
R. Bruce Marlin University of Virginia, Charlottesvllle, VA 22903 When comparing aspects of the electronic structure of a molecule as simple as water, teachers and students confront two apparently divergent views. On one hand they learn that in the structure of ice each molecule donates two hvdroeen bonds to other molecules and, in turn, accepts two hidrogen bonds a t two eouivalent lone oair orbitals. But. in discussine the ultraviolet absorption or photoelectron spectra or ionization enerrr of water,. onlv . one lone pair is invoked, the other being a t much lower energy. The latter viewpoint has been emphasized in an article entitled "No Rabbit Ears on Water"(I). However, the article considers only one aspect of the structure of water. In this presentation we show that though the rabbit ears description represents an exaggeration, i t is not only permissible but also useful to consider two equivalent. less nrominent lone oairs-sauirrel ears. Clarification of these ioints is necessaiy because the principles involved apply to almost all molecules, even diatomic molecules. For most molecules a t least two different orbital descriptions are vossible. The molecular orbital enerm level diagram resuits from solution of the ~artree-~ockaell-consistent field (scf~formulation of the electron structure of the molecule. he resulting orthogonal orhitals are symmetry orbitals, beloneine to irreducible representations of the syto which the molecule belongs. ~ h o u g h metry point frequently called canonical orhitals, herein they will be termed spectroscopic molecular orbitals (SMO) because the orbital energy differences yield values for electronic transitions in a molecule. We know that only the spectroscopic orbitals derived from a scf formulation may be used to evaluate electronic transition energies (2). Thus electronic transitions with energy differences giving rise to bands in the visible and ultraviolet regions of the spectrum demand the use of spectroscopic orhitals for their interpretation. Similarly, for the ionization processes measured in photoelectron spectroscopy, Koopman's theorem states that the spectroscopic scf orhital with the highest energy is vacated upon ionization. All observable quantities are determined by the total wave function, which for a closed shell molecule is formed from a single Slater determinant. Such a determinant is invariant with resnect to unitarv transformations amone its molecular orbitals:As a result, there exist for most moiecules an infinite number of different molecular orbital sets that eive the same total wave function. Are any of the orhital sets other than the SMO's useful. and what criteria might be used to select one? Chemical experience shows the d&irahility of a description of molecules in which the mutually orthogonal orbitals are localized to the greatest extent possible. A quantitative procedure for finding those molecular orbitals that maximizes the sum of the orhital self-repulsion energies leads to an orhital set called localized molecular orbitals (LMO) (3). The idea of LMO is to confine each molecular orbital to the smallest nossible soace and to have each confined molecular orhital as remote'as possible from others in the same molecule. The LMO's are concentrated in relatively small regions of space and have undergone minimization of interorhital Coulomhic and exchanee reoulsions. I t is usuallv impossible to reduce the interorbital exchange energy to 668
Journal of Chemical Education
zero, and so, to some extent, electrons are exchanged among the orhitals. The procedure employed for finding LMO's yields orbitals that may most nearly be identified with nonexchaneeahle electrons (4.5). LM& provide the chdmist with the familiar picture of bonds and lone oairs in molecules. The value of the confined and separated LMO's with nearly nonexchangeable electrons is that they are resistant to change when other parts of the molecule are modified. As a result some LMO's are transferable among molecules with related parts of structure, permitting comparison among them. The mutual orthogonality of the LMO's allows formulations of parts of molecules that are additive as well as transferable ( 4 j . ~ ~ ~ l i cations of this kind are frequent in comparina- bond eneraies . and bond dipole moments. The unitary transformation that converts SMO's to LMO's or vice versa transforms individual orhitals with a consequent alteration in their energies and electron densities. However. the total enerev and total electron densitv of the entire molecule is unaf&ted by unitary transforkations. These total rovert ties of a molecule mav he decomposed in nearly an fnfinity of different ways, of which SMO's and LMO's are but two. SMO's are useful in aovlications involving electronic transitions such as in visible and ultraviolet absorption spectroscopy and ionization energies. Indeed, SMO's are the only orhitals whose energies may be used directly in describing electronic transitions. LMO's correspond closely to the chemist's ideas of directed valence where the electron density is considered to he predominantly localized between two atoms. The LMO's are useful in considering ground state properties of a molecule such as structure. divole moments. and hond enereies. I t is generally not useful-to consider the orhital energes of all LMO's in a molecule. Other avvlications of LMO's are to Ytructures of boron hydrides (&.to internal rotations about single bonds (71,and to nuclear motions in large molecules of lo&symmetry not easily handled by the ~offmann-woodward rules (8). In simple diatomic molecules the SMO's belong to one of four categories, namely a and a bonding and antibonding orhitals. They are frequently used in describing electronic structures. Only three categories result from transformation of the above orbitals to yield LMO's: inner shells of atoms and two classes of valence orbitals, atomic lone pair and diatomic bond orbitals. The valence orbitals of unstable Be? are described by SMO's in terms of a bonding and antibond: ine orbitals delocalized over both atoms and bv LMO's bv distorted atomic 2s orbitals on each atom. In Nntwo a bonding and one a antibonding valence SMO's and two a honding SMO's are transformed to LMO's of three bent (banana) bonds between atoms and one lone pair on each nitroeen atom. The addition of two antibonding a orbitals to t h e SMO formulation for F2 results in one sigma hond and three lone pairs on each fluorine atom in the LMO description. The shapes of the filled LMO's are more dependent upon the number of electrons involved than are the shapes of the SMO's, which mainly only contract as the nuclear charge increases (4). The LMO description of molecules is akin to the localized
structures in the valence bond description due to Pauling (9).However, unlike the latter, LMO's are always transformable to SMO's. The simolest valence bond descrintion of F? joins two tetrahedra at'a point t o form a hond and places lone nairs a t the remainine corners of the tetrahedra leadine to a ione pair-lone pair angle of 109'. In the LMO f o r m u l ~ tion the lone pairs of F? are nearly. sn2 with an angle . hybrids . of nearly 120' (4). In methane the eight valence electrons are in four bonding SMO's of two distinctly different shapes, three in one irreducible representation of the point group and one in another. All four SMO's are delocalized over all five atoms in the molecule and are not the most useful ones for describing the electron density distribution in the molecule. After transformation four equivalent LMO's that helong to a reducible representation are obtained. They may he permuted by operations of the symmetry group of the molecule and hence are called equivalent orbitals (10). In this case the four equivalent LMO's are identical to the tetrahedral sp3 hybrids of valence hond theory. For water, four of the eight valence electrons are distrihuted over two bonding a-type SMO's of very different energies. Both of these a orbitals are delocalized over all three atoms of the molecule, one symmetric and one antisymmetric with respect to the plane bisecting the molecule. The other four valence electrons occur in nearly nonbonding SMO's, one a and one a.Ionization occurs from the higher energy a SMO which is nearly pure p, while the a SMO of appreciably lower energy is roughly an s-type orbital (I, 11). These last two SMO's are sufficiently localized so that one might refer to them as "lone pairs". The LMO description of water is the more familiar one with two equivalent 0-H bond orhitals, each concentrated around one hond, and two equivalent lone pairs each with appreciable s and p character. Eauivalent LMO's appear in methane and water, and each set df equivalent LM& may he considered to have equal energy. As aresult we speak of the equal hond energies of the four honds in methane and the two in water. These bond energies refer to ground state properties of the molecules and are transferable to other molecules containing C-H and 0-H bonds such as alcohols. However, according to the ahove nrincinle it is not nermissible to derive anv sort of electronic transition energies from LMO's; SMO's only must In the SMO formulation the water "lone pairs" be emoloved. . are not of equal energy, and hence ionization occurs most easily from the highest energy orhital. Despite localization yielding three bent bonds in Nz, the localization procedure yields separated a and a bonds in ethylene. Evidently the prior existence of planar symmetry in ethylene renders both the SMO's and LMO's nearly identical to the familiar a and a description (12). The term banana honds for the bent bonds that occur in LMO's should probably he avoided because more electron density occurs along the internuclear axis than the appellation banana seems to imolv. The node in a bent hond is on the other side of the interrkkear axis from the greatest electron density. It is also ~rofitahleto transform the delocalized a orhitals of aromatic molecules such as benzene into localized orbitals. Structures similnr hut not identical to the Kekule type result. It is not possible to localize the molecular orhitalsto the same extent as in ethylene, and a positive lobe of the localized orbitals extends over a t least three carbon atoms. These "delocalized" localized orhitals provide additional insights into the resonance stabilization of aromatic molecules (4,131. The basic problem in any description of the total electron density distrihution is the small, almost miniscule effect that chemical hondine has. As two nrotons move into ~ o s i tion near an originally sphericaloxide ion, only asmall transfer of electronic charee densitv into the internuclear reeion accounts for the honiing in water. The two protons scarcely ~~~~
~
~
-
-
change the distrihution of the 10 oxide electrons. I t is this subtlety that makes the quantitative description of chemical bonding so difficult. An accurate difference electron density map between that for water and that for the oxide ion would show a small build-up of electron density in the internuclear reeion. - . a necessitv for bondine. and smaller nrotuherances for the two equivalent lone pairs-squirrel ears. Difference electron densitv -mans. from crvstal structure determinations reveal twolone pairs in water (14,15) and two 120' lone pairs in the oxygens of carboxylate groups (14-17). The specific case of the a electron distribution in 1,3butadiene (H2C=CH-CH=CHz) serves to illustrate the above points. Though we have already written two double honds, this depiction does not conform to the spectroscopic molecular orbital description. For purposes of illustration, we simply model butadiene as a one-dimensional box. For the two pairs of a electrons in a box of length L, the solution to the Schrodinger wave equation yields, for the two lowest energy spectroscopic orhitals v.
and where A = (2/L)''2 and the hox length, L, is taken as about 213 longer than the distance between the two terminal carbon nuclei. The corresnondine enereies are such that Eq = 4E,. In the figure the wave fuictionH $1 and $2 are drawn in thk second column with $9 denicted a t hieher enerev than h. In considering electronic transitions thk $1 andu& fun&ons and their corresponding energies El and Ez are employed. The corresponding orbital densities and $2 are plotted in the first column of the figure. I t is apparent that considered separately the orbital-densities ark delocalized along the length of the box. (The orhital densities are analogous to the two different spectroscopic "lone pair" orbita1s;n water.) Because the total (determinantal) wave function is invariant to a unitary transformation, it'is permissible to take a linear combination of the b, . .and b* , - functions. which when normalized are given by and
*+ = ($1
+ *z)lzl"
*-
- *J/21'Z
= (*,
The (average) energies of these hybridized functions are equal, and they have no utility in spectroscopy. The total energy of the system is the same as before. These hybridized wave functions are plotted a t the same level in the third column of the fieure and their corresnondine orhital densities in the last cilumn. The orbital densitiesUforeach of the two orhitals bc2 and L2are seen to be much more localized than $12 and'&z2. he maximum in $+2 occurs about 30% of the distance alone the box and is 55%higher than the maximum in the unhybridized orhitals. ~ h u s - t h eorhital density for $+ is localized mainly in the first half of the hox and that for $- in the second half. This kind of density concentration corresponds to the alternate double bonds in H2C=CH--CH=CH2 (and is analogous to the equivalent lone pairs in water). Simnle maninulation of the ahove eauations shows that $12 $z2 = $+;+ $-2. Therefore, the t o h electron density over both orhitals along the box is identical on the right and left sides of the figure, only the resolution of the total electron densitv into different kinds of airs of orhital densities differs. H O we~ resolve the total eiectron density depends upon the molecular property we wish to discuss. For spectroscopic transitions the and $2 (and higher) solutions to the Schrodinger equation are the proper functions. For the relative positions of electrons the localized orhital description in
+
Volume 65 Number 8 August 1988
669
