Irwin Cohen and Janet Del Benel
Youngstown State University Youngstown, Ohio 44503
Hybrid Orbitals in Molecular Orbital Theory
Hybrid orbitals have been extremely useful in the study of first-row elements but have not yet found much application for second-row and transition elements, although the theoretical basis is the same in all the cases. The chief reason for this difference is that spectral energy levels are a major source of information for the heavier but not the lighter atoms, and hybrid orbitals do not correspond to spectral transitions. Even in first-row chemistry we find the description of double bonds more useful in uv interpretation than the bent-bond description, which is structurally equivalent but which replaces the u and T orbitals by hybrids. But the chemist is concerned with reactions at least as much as with energy levels. We want to know the likelihood of a certain reaction and if possible we want to predict its thermodynamic and kinetic parameters. This sort of inquiry leads to questions like "how strongly is this atom tied to this molecule and what is the charge distribution in the neighborhood of these atoms?" The chemical bond concept has been amazingly successful in answering such questions, so successful indeed that the theoretician has not only not caught up but has not yet even been able to offer clear reasons for that success in terms of orbitals (I). The theoro ticians have orbitals and the chemists have bonds and the question is, what is the connection between them? The best answer to date (2) appears to be expressed in terms of localized hybrid orbitals. However, there is no available satisfactory discussion of these matters on a level open to non-specialists. Our purposes here, then, are to review for nonspecialists the basis of hybrid orbitals in terms of molecular orbital ( M O ) theory, to show how the chemical bond is most closely approximated in orbital theory, and to present some new orbital diagrams. The Slater Determinant and Hybridization
In 1 1 0 theory, one obtains a set of molecular orbitals which must then be combined to form the molecular wave function. In the simplest caseZ the molecular wave function is a Slater determinant constructed as follows. The orbitals including spin, IC.1, IC2, . . ., Gn, are each written in a set of n columns and the electrons 1, 2, . . . , n, are each assigned to all the orbitals in each column as shown
nant produces an antisymmetric wave function since the interchange of any two columns of a determinant changes its sign and this corresponds here to the interchange of two electrons. The exclusion principle also follows since if any two rows of a determinant are identical the value of the determinant is zero and this corresponds here to the inclusion of two identical orbitals (including spin). The Slater determinant is readily illustrated by the H z molecule. We usually think of this in terms of one orbital which has two electrons, of spin a and P, respectively. This one orbital may he referred to as a space orbital, which may be the simple linear combination, N [ ( ~ s ) A ( ~ s ) B where ] N is a normalization constant and A and B refer to the two protons, or it may he a more complicated expression. In any event, at this point we must now combine the space orbital, whatever it is, with the spin, a or P, so that wherever we have one and +% or space orbital we get two spin-orbitals, +a and +P. For H2 we may label the two spin-orbitals simply a and P. Then the normalized Slater determinant is
+
+
; ; ;;
4%
=
4%2
) -
I
(2)
Two important conclusions follow from this. First, electrons do not occupy individual orbitals but are "exchanged" over all possible orbitals. This does not merely refer to the exchange of spin as in the case of HZ, with just one space orbital, but rather as eqn. ( 1 ) above shows, each electron shares each and every orbital. Thus, assignment of individual electrons to individual orbitals is not allowable, only the Slater determinant is, and the old question, "how does an electron cross a node," is meaningless in systems with more than one electron (8). This leads to the second point: any other set of molecular orbitals, hi, say, which leads to the same expansion of the Slater determinant as the original set, +,, is equally valid. There is no one set of orbitals which is uniquely correct but rather there are many sets each of which corresponds to the same total effect and consequently the same physical reality. For example, the two H2 orbitals
x,
.\/%(a (3) hz = 4%- B ) are a valid set since the value of the Slater determinant =
+
'
The molecular wave function then is simply the value of this array taken as a determinant, multiplied by the normalization constant, (n!)-'1%. The Slater determi-
Present address: Theoretical Chemistry Institute, University of Wisconsin, Madison, Wi'iseonsin53706. This is the c1osed;shell ease, in which there are no partly filled energy levels. In the open-shell case, a combination of two or more Slater determinants must be used, but this does not affect our conclusions here.
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Figure 1. CHI orbital energies wim rnrrelationS indicmted for the cononi. col orbitals. (Schematic.)
is identical with set (2), as may be seen by substitution and expansion. More generally
is a valid set where a and b are any arbitrary numbers limited by the normalization condition, la l2 /b 1% = 1. Thus, once we have a set of MO's, we can generate an infinity of other, equally correct sets. I n the case of Hz, the original set, a and 0, may he preferred for simplicity and set (3) may be preferred because it leads more directly to the physical interpretation that each electron shares each orbital, whereas set (4) may be preferred for some other reason. Of the infinity of correct sets, then, some may he more useful than others in certain ways but not more correct. One particular set of molecular orbitals is especially important. The best way to obtain a set of orbitals is the Hartree-Fock self-consistent field method, in which each electron is treated as travelling in the combined potential field of the nuclei and the averaged out charge distribution of all the other electrons. Now it is possible to formulate this method so that the resultant orbitals are stationary (that is, have invariant energy) with respect to a mathematical device called the Hartree-Fock operator. This stationary or invariant energy is called the "orbital energy"; it completely neglects the actual electron interdependence or correlation, due to collisions, which makes the orbital concept and the orbital energy mathematical tools rather than physical reality. Nevertheless, the orbital energies for these orbitals do provide very useful approximations to real properties such as ionization potentials and spectral transition energies. Orbitals obtained this way have been called "HartreeFock orbitals" (4), %anonical orbitals" (5), and "spectroscopic orbitals" (6). We will refer to them here as canonical since that term has gained the greatest currency. Now when we combine a set of canonical orbitals of different orbital energies (in the atomic case this might be s p or spd hybridization) the hybrid orbitals are no
+
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Figure 2. CHI orbitol energies with correlations of hybrid M O with sononicd AO. ISchernotic.)
longer useful for approximating ionization potentials and spectral transitions. We may calculate the energy of a hybrid orbital as the weighted average of the energies of the constituent canonical orbitals, hut if we represent these hybrids on orbital energy diagrams we should not infer anything about electronic transitions to or from such "energy levels," for such transitions are not observed. Figure 1 shows schematically an orbital energy level diagram for CHa with correlations shown for the canonical orbitals. I n Figure 2 the correlations are shown for the hybrid, or equivalent, A t 0 . Figure 3 correlates the hybrid MO with the hybrid AO, which is the straightforward chemical interpretation that the individual sp3 atomic hybrids each combine with one H orbital to make the four individual, equivalent, localized C-H bonds. This would appear to be the simplest picture but correlations with atomic hybrids can easily be misleading, as shown below. Also possible is a still different type of diagram, with canonical A t 0 correlated
figure 3. C H ~ energies with c bed no. ISchematicl
~
~
~of ~hybrid ~ ~MOt with i hy~ n
~
with hybrid A 0 (7). All of these diagrams should be considered as different modes of expression of the same situation and not as different situations. The "formation" of atomic (or molecular) hybrids by a linear combination of atomic (or molecular) orbitals is sometimes confused with the fornation of molecular orbitals by a linear combination of atomic orhitals. These situations are fundamentally quite different: in atomic (or molecular) hybridization there is no physical process involved and we can generate no change in orbital energies or any obsemable property, whereas in molecular orbital formation we have a drastic physical change-the close approach of the atoms-and we must generate new orbital energies and new properties.
z
they may be. Each MO is the sum of the products of the coefficients in the indicated row and the A 0 listed above the coefficients. For example +n = c ( 2 p d d ( L f Ia - 4 - I,,)
+
+
+
where e2 4dZ overlap = 1. MO theory now has two problems to solve: the exact forms of the A 0 that give the best MO must be determined, and the values of the coefficients must be calculated. These problems can be solved and we then have a set of canonical molecular orbitals with welldefined orbital energies, but they each extend over the whole molecule and therefore do not a t all correspond to the chemists' four valence bonds. However, tetrahedral symmetry allows these four MO's to be transformed into four equivalent MO's, At, aa follows (8)
The results of this transformation are given in Table 2, in which the coefficients have been multiplied by two for simplicity. If we include spin and expand the Slater determinant, the result for Table 2 is the same as for Table 1. The four equivalent MO's in Table 2 turn out to he considerably localized since each includes principally 3d) together with only a one ligand (coefficient b small contribution (b-d) from each of the other three. Localization like this is a general result of symmetry transformations such as (5), so that such a set of equivalent orbitals may be called a localized set (9). Now if in Table 2 we set a = e and b = d and neglect overlap, then the three small contributions in each MO completely disappear. Also, in this case, all the centralatom coefficientsare equal so that the central-atom part of each equivalent MO is exactly an sp3hybrid which is bonded with just one ligand. This is the precise connection between MO theory and the idealized chemical bond. However, there is no real justification for setting a = c and b = d. Indeed, since the 2s orbital is more stable than the 2p, we must have a > c and b < d, so that the MO are not perfectly localized after all (i.e., b - d # 0 and the simple, isolated chemical bond turns out to be merely an approximation. Furthermore, the s character in each of the four A, is greater than 25%. This variation in s character is perhaps more surprising than the blurring of the chemical bond, for .we know that bonds are not really perfectly isolated from each other but we may have thought that the total s character must always add up to just 100%. However, we distinguish between the central-atom electron distribution as it is in the molecule (the left half of Table 2) and as it may be imagined in the hypothetical valence state . . . "'* > -"*". ' + " , . ~. '. .-*.-..- .. $,li"':'r.,-." . .'- . ,. : r
+
Figure 4. Coordinote system for four ligondr tetrohedrolly disposed obovt o central atom a t the origin.
First-Row Molecules and Localization
Let us now consider the molecular orbitals for the combination of a first-row atom with four equivalent ligand atoms, as in CH,, C(CH&, NH4+, BFI-, etc. We use the coordinate system shown in Figure 4, with the central atom a t the origin and the ligand atoms a t the numbered corners. Molecular orbital theory permits the symmetry combinations listed in Table 1, where the AIO's are listed at the left and the constituent atomic orbitals (AO) are indicated a t the tops of the columns. I n the A 0 list, s, x, y, z, and the L, refer respectively to the central atom 2s, 2p,, 2p,, and 2p, orbitals and the four ligand valence orbitals, whatever
Table 1. Molecular Orbitols for a First-Row Central Atom with Four Tetrahedrally Disposed Ligonds
.... Table 2.
'
"..
, . %
f'"
Equivalent MO far a First-Row Central Atom with Four Tetrahedrally Disposed Equivalent Ligands 8
Z
21
Z
L.
L
I,.
I,.
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-the free atom ready for combination. I n the tetrahedral valence state the only way to have four equivalent hybrids is for each to have 25% s character. But in the molecule this hypothetical valence state must suffer electron flow towards or away from the ligands, and this will in general he relatively away from the less stable 2p and towards the more stable 2s. This change in sp character that occurs in molecule formation is the reason for the suggestion above that i t is best not to use tielines for the hybrid orbitals as in Figure 3, for to do so would seem to imply that the atomic hybrids combine as such to form the MO whereas they really combine with sp redistribution. The redistribution of sp character in some in-bond hybrids can be estimated by an approximate, semiempirical calculation using the set of orbital electronegativities published by Hinze, Whitehead, and Jaff6 (10) together with the relation they propose between electronegativity and ionic character. For example, their figures predict an ionic character of 31.2% for a C(2s)-H(1s) bond (C negative) and 5.7y0 (C positive) for C(2p)-H(1s). Therefore, the C(2s) population in Table 1 is 1.312 and each (2p) is 0.943, for a total for the shell of 4.141 (indicating a net negative charge at the carbon). If we now describe this molecule in terms of hybrids (Table 2) we distribute the s population evenly among the four bonds for an s character in each of 1.312/4.141 or 31.7'%. I n a similar way the s characters in Table 3 can be calculated.
Table 3.
Tables like 1and 2 can be constructed for these cases (8, 12) and the results are entirely analogous. If the orbitals are assumed to be of equal electronegativity and if overlap is neglected, then the central-atom part reduces to a pure octahedral (or square-planar, etc.) hybrid which is bonded with just one ligand. But again, in the actual molecule there will be some spd redistribution, and the h 4 0 will not be perfectly localized. Spatial diagrams of d-hybrid orbitals appear not to have been published (IS), even though they are as valid as the more familiar sp hybrids of the first row (14). Therefore, we show in Figures 5 and 6, respectively, the d2sp3 hybrid for iron and the spad2 hybrid for sulfur. The reason for the great difference between these two
Calculated s Character in Carbon sp Hybrid Bonds
Bond C(spa)-C(spa) C(spi)-H(s) C(sp8)-Li(s) C(.vp3)-Li(p) C(sp3)-O(spa) C(sp3)-0(p) C(spi)-CKs) C(sp"-Cl(p) C(sp9-F(s1 C(spJl-F(p) C(spz)-L(spz) C(sp)-C(spP) C(SP)-~(SP~)
Si~bstanee diamond methane CLi, CLi, C(OR)r C(Olt), CCl, CCl, CFI CF, graphite allene CO:.
%S
.
32 32 33 33 33 32 47 33 100 47 41 57 58
"+"
Figure 5. Iron d'rpa hybn'd. The marks tho nucleus. Reading from the outennost ports inward, the Rlled-in bands indicote densities of 10~b10-2.6, 10-el O-'J, and l V L 1 0-OJ electrons per cubic bohr. Solid liner ore noder. Distance from nucleus to outermost contour along the principal oxis is 4.6 bohr. This is a direct photograph of line printer ovtpvtllBM 360/40),except h o t the noder were drawn in.
Apparently the carbon sp population in a tetrahedral hybrid in the molecule is generally about 32Y0 s (the C-F bond is a singular exception). Similarly, the trigonal hybrid is about 41% s and the digonal about 58% s. This contrasts with the familiar valence-state figures of 25y0, 33%, and 50y0, respectively, which would obtain in the absence of sp redistribution. A number of correlations have been made with valence state s character (11) which perhaps would be better expressed if sp redistribution were considered, although exact calculations of this sort are not yet feasible. Transition Metals
The most common complexes of the transition metals are octahedral, tetrahedral, and square-planar. Pauling first showed how these structures can he interpreted in terms of d2sp3(octahedral), dsp2 (square), and other atomic hybrids. 490
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Figure 6. Sulfur w a d ' hybrid. Same manner as Figure 5. Distance fmm nucleus to outermost contour dong the principal axis is 4.5 bohr.
octahedral hybrids is that the iron 4s and 4p orbitals have an extra node. The sulfur hybrid is very strongly directional, as predicted by Pauling, but the iron hybrid, although also strongly directional, has a pronounced density in the back region. Since this is cut by a node, there can be no good back side bonding here, but there can be significant electronic repulsion. The sulfur hybrid is plotted from accurate wavefunctions calculated for the appropriate excited state (15); the iron orbital is calculated from accurate ground state 4s and 3d orbitals (16) together with a (Schmidt orthogonalized) 4p orbital constructed by extrapolation from the occupied s and p orbitals. The use of ground state and extrapolated orbitals renders the diagram less than exact although experience with these functions indicates that an accurate plot would not be greatly different..
Figure 7. Polar groph of the octohedrol onguior function, Y(d'rpal. This is obtained by ignoring or factoring out the different radio1 functions,
+ +
leaving Yld'rpal = 1 d T ~ d Y,
V'~YJ/&.
Figure 7 is sometimes used to represent an octahedral hybrid. This is a polar graph of the so-called angular portion of the hybrid, derived as follows. Each simple A 0 can be written as a product of a radial function R(r), dependent on 1. alone, and an angular function Y(B,4), dependent on spherical coordinates B and 4 but not on 1. (-40)= R(r).Y(@,+)
I n a hybrid (Hy) composed of AO,, A02, AOa, or RIYl, R2Y2,etc., with the hybridization coefficients el, cz, c3, etc., we have (Hy) = C X R ~ Y X~1RsYa crRaYa .. . We now assume that Rl = Rz = R3 = . . . either in general or at least over significant regions of the orbitals. Then for those regions we can factor and obtain
+
+
+
+ C ~ Y+I . .. )
(Hy) = R (CIYI
(6)
where the expression in parenthesis is the "angular part" of the hybrid. Obviously, eqn. (6) is valid only when the radial parts of the A 0 constituents are alikeotherwise there is no such equation and there is no "angular part." Now, in sp hybridization, the radial parts are indeed alike for the outer regions of the orbitals. But d orbitals have very different radial functions, so that eqn. (6) does not fit dsp hybrids and it should not be surprising that Figure 7 is so different from Figures 5 and 6. Energy Localizalion
The localization referred t.o up to this point has been accomplished by symmetry considerations. The problem of localization in the absence of symmetry is not
so readily solved. We cannot apply symmetry to CHClBrF or even to HF, for example, but we feel-that here, too, there should be a relation between A 1 0 theory and the chemical bond concept. Several methods of localization in the absence of symmetry have indeed been suggested (17, IS), culminating in the definitive method of C. Edmiston and I
