Chao-Yang Hsu and Milton Orchin University o f Cincinnati Cincinnati, Ohio 45221
II
A Simple Method for Generating Sets of Orthonormal Hybrid Atomic Orbitals
The introduction of essentially nonmathematical descriptions of molecular orhital theory into the chemical curriculum a t a very early stage makes it desirable to give a comparable description of hyhrid atomic orhitals a t about the same time and a t about the same level. Excellent elementary quantum mechanical treatments of hybridization are available (1-6) and several authors have given quite satisfactory treatments which require relatively little knowledge of quantum mechanical theory (7-9) However, these treatments require some group theory hackground, which although relatively easy to master, is not a t the disposal of the student when he usually first encounters hybridization. Students are introduced to descriptions of orhitals very early in their careers. The angular distribution of the 2p atomic orhital in the form of a figure 8 is already familiar to a second year student. What perhaps is not appreciated is that an orhital is in fact a description of a wave function. What the student should now he taught is the form of the wave function which corresponds to the orbital displayed. In the case of the pure p atomic orbital this presents no special problem' hut the correspondence between a hyhrid orhital and the complete analytic form of the wave function it represents may present some difficulty. In the first portion of this aiticle we wish to point out that equivalent sets of hybrid atomic orbitals may he generated by a very simple geometric construction method. This construction method will lead by inspection to the correct assignment of algebraic signs and when combined with the simple arithmetic requirements of orthonormal sets, the method leads immediately to a correct set of hybrid orbitals or, more exactly, to the correct wave functions which these orbitals represent. The arithmetic depends on the application of three basic requirements of wave functions. ( 1 ) Normalization. Assume that we are dealing with one of the three familiarly shaped sp2 hyhrid atomic wave functions consisting of one part s orbital and two parts p orhital. Assume the particular hybrid orhital, 4,, has the form
The normalization condition requires that the sum of the squares of the coefficients, c, of each atomic wave function, j, in the ith hybrid wave function must he equal to unity, i.e.
The above sp2 hybrid wave function is normalized since
(2) Orthogonality. Each hyhrid wave function (orbital) i? the set of hyhrids must be orthogonal to each of the other hybrids, i.e.
114 /Journal
of Chemical Education
Thus a set of three sp2 hyhrid orhitals of the form
are each orthogonal to the other, for example
(3) Unit Orbital Contribution. The squares of the coefficients of any one component atomic wave function summed over all the hybrids in which it participates must be equal to unity, i.e.
Thus in the three sp2 hyhrid orhitals shown ahove, all of the s orhital contributions must sum to a total of one complete s orbital
In the first portion of this paper we will consider equivalent hybrid wave functions made up of va~iousmixtures of s and p orhitals all of which have the form d, = c,l(s) + c&J
+ C ~ P +Jc , r ( p , )
In a set of equivalent hyhrid wave functions each wave function, 4,, of the set must necessarily possess the same fraction of each kind of (s, p, or d ) atomic orbital. Procedure for Generating Equivalent Hybrid Orbitals
Step I . Specify a cartesian coordinate system, and place one of the i hyhrid orbitals along one of the coordinate axes and as many of the other orhitals as possible in the plane containing this axis and any other coordinate axis. Step 2. Since the s orbital has spherical symmetry each equivalent hyhrid orbital of a set contains l l n of the total of the unit s orbital distributed in the n hyhrid orbitals. Therefore, the coefficient in each hyhrid orhital is 11IX. Step 3. Apply the three quantum mechanical conditions mentioned ahove to generate all the hyhrid orhitals.
However, since the orbital resembling the number 8 actually represents the square of the angular part of the wave function, both lobes should he positive. For discussion, see ref. (7). p. 21.
Example 1. Trigonal Planar sp7 Hybrid Orbitals Step I. Assume that the p, and p, orbitals are the two p orbitals involved. Choose the @l(sp2)hybrid orbital oriented in the positive direction of the x axis as shown in Figure 1. Step 2. Since there are three equivalent hybrid orbitals sharing one s orbital, each of the three sp2 hybrid orbitals receives a contribution of one-third of the s orbital, i.e., c,l = \'1/3. Step 3. Since the hybrid orbital &(spa) only contains the s and p, orbitals, (the p, orbital makes no contribution because its nodal plane includes the x axis), we may calculate the coefficient of the p, orbital by using the normalization condition, i.e., cl12 + c1z2 = 1,thus
Figure 1 . (left) Schematic diagram showing three si12 hybrid orbitals in thexy plane. Figure 2. (right) Schematic diagram showing four so3 hybrid orbitals is in the xz plane)
(ma
The three other sp3 hybrid orbitals 42, 43, and b4 are all equivalent with respect to the z axis. The coefficients of p, for these three hybrid orbitals can then be obtained from the unit orbital contribution requirement
and hence, we obtain one of the hybrid orbitals
Figure 1 shows that the two other sp2 hybrid orbitals @2 and @3 are equivalent, with respect to both the x and y axis. Accordingly, we may calculate the coefficients of the p, and p , orbitals from the condition of unit orbital contribution. The contribution of the p, orbital is thus: clzZ + c2z2 + ~ 3 2 ' = 1 where c2z2 = c~~~and clz2 = ( w 3 l 2 . Hence Figure 2 shows that 62, From Figure 1 it is seen that both 42 and $3 point in a direction such that the p, contribution is along the -x axis. 2 have a value of - d r 6 . Since the Hence c22 and ~ 3 both total contribution of the p, orbital is also equal to 1, each of the two hybrid orbitals receiving contributions from the p, orbital contains one-half of the unit orbital contribution, and the coefficients are f \/I/Z. The two complete hybrid orbitals are thus
m3 and $4
all point in the negative
z direction and hence each has the coefficient -dT/12.
The $2 orbital lies in the xz-plane, the nodal plane of the p, orbital, and therefore the p, orbital makes no contribution to 42. The coefficient of the p, orbital in 42, which according to Figure 2 will have a positive sign, is calculated from the normalization condition
whence
The positive and negative signs used in the component atoinic p, orbitals are apparent from Figure 1, where the p, orbital in @2 has a positive sign and in $3 a negative sign. Example 2. Tetrahedral sp3 Hybrid Orbitals Step I. Orient the four spa hybrid orbitals as shown in Figure 2. The $I orbital points along the positive direction of the z axis, while 42 orbital has its plane of symmetry in the xz plane and therefore has its large positive lobe below the xy plane as required by the 10Y28' bond angle of the tetrahedral molecule. Step 2. Each of the four spa hybrid orbitals contains one-fourth of the s orbital and hence each 41 receives a contribution of = 112 s. Step 3. The qh(sp3) orbital is generated from the s and p, orbitals only and the coefficient of the p, orbital is obcl12 = tained from the normalization condition: ell2 (1/2)2 - c14 = 1.
and
According to the orientation specified in Figure 2, the coefficients on p, in the hybrid orbitals and b4 must be equal and negative. From the unit orbital contribution condition
The calculation of the coefficients of p, in the hybrid orbitals $3 and $4 is straightforward. Since both the hybrid orbitals are equivalent with respect to t h e y axis and according to Figure 2 have opposite signs, ca3 = and c43 = The resulting hybrid orbitals are
v'w
+
and
A more common orthonormal set of tetrahedral sp3 hybrid orbitals is obtained if the orientation of the hybrid orbitals and the coordinate system are specified as shown in Figure 3. This figure shows that all four hybrid orbitals Volume 50, Number 2. February 1973 / 115
Figure 3. schematic diagram showing the four hybrid sp3 orbitals equivalent with respect to three coordinate axes.
zA
x
Figure 4. Schematic diagram showing formation of dspZ hybrid orbitals from 4s. 4p,. 4p,and 3d,2 . orbitalr.
are equivalent with respect to the three coordinate axes and hence the coefficients for e w h component atomic orbital is q 4 , It only remains to assign the positive or negative sign of the coefficients and this is easily done by inspection of Figure 3. The coefficients of the p component of all four atomic orbitals are positive in the $1' hybrid orbital since it points in the positive direction of the three axes. In the $z', $a', and 4 4 ' , the respective p,, p,, and p, component atomic orbitals are positive and the others are negative. The hybrid sp3orbitals are thus
@)- f%"-:~l") @$ 0.255 -0.437 -0.163 0.952 -0.45% 0.952 0.544 -0.43i
Example 3. Square Planar dsp2 Hybrid Orbitals
.
Steo 1. Snecifv . the four atomic orbitals in the cartesian coordinate system as shown in Figure 4. Step 2. Each of the four equivalent dsp2 orhitals receives a contribution of one-fourth of the s orbital, thus the coefficient of the s orbital in each hvbrid orbital is 112.
Step 3. From Figure 4 it is clear that $1 is made up of one-half of the p, orbital and one-fourth of the dx2.,2 orbital and hence coefficients are vm and 112, respectively, for these orhitals
Both sets of tetrahedral hybrid orbitals, 6, and A', are equivalent and equally acceptable. A linear transformation can change one set into the other. (Actually, for any molecule we can generate an infinite number of equivalent sets depending on the choice of the coordinate system.) The transformation matrix which has elements that we will call a,, may be found as follows
and $2 is that the sign of The only difference between the coefficient of the p, component orbital is negative in $2
The $3 orbital consists of 114 the s orbital, 112 of the p, orbital, and 114 of thed,z_.:!orbital
Substituting the appropriate components making up and 61we obtain
a2, ( ~ + p . - p ,
- p ~ + %2 ? ( s - p ~ + p , - p J
=
1
fi
For $n it is only necessary to change the sign of the coefficient of thep, component in $3
3 s + - p2l
Because s, p,, p,, and p, are independent wave functions, the coefficients of each of these functions on both sides of the equation are equal 8:
a,,
+ a,, + a , ,
+
g,(dspq
+ a,,
-
0
We now have four equations containing four unknowns. From the second and third equations we get all = am and a13 = ala. Substituting these results into the first and fourth equations, we obtain
116 /Journal
of Chemical Education
1
d
+
+
+ fl
I
1 a,, = a,, = --- = 0.633 4 1- fi als = a , , = ----- = -0.183 4
The other elements of the transformation matrix can be obtained in an a n a b 3 m manner, giving the equation
-
Step I. The orientation of the six d2sp3 hybrid orbitals are specified as shown in Figure 5 where four of the orhitals, $1, 62, ma, and $4, are shown in the xy plane, and the other two, and $8, point along the z axis.
+
a,, - a,, =
-
Example 4. Octahedral d2sp3Hybrid Orbitals
+ n,, = I
P,: a,, - n,, a,:, - a,, = 0 Pi: a,) - n,? - an ai, = 0 P,: a,,
Ls
= 2
S
4.b% xL
dr2
dx7,7
d%p3
- Y
F i u r e 5. Schematic diagram showing the six d2sp3 hybrid orbital9 from 4s. 40,. 3d,2 and 3d,2 .z component orbitals.
4p,, 4p,.
Step 2. From the diagram, it is seen that each of the six hybrid orbitals receives a contribution of 116 of the s orbital. The orbitals $1 and $2 receive a contribution of 112 of the p, orbital while 63 and qb each contain 112 of the p, orbital and @E, and & each contain 112 of the p, orbital. The four hybrid orbitals in the ny plane each also contains 114 of the d,z,z orbital. Putting this information together gives
(either s, p, or d ) in any hybrid, di, can be calculated from the sum of the square(s) of the coefficient, ci,, of each of the atomic orbitals with magnetic quantum number m thus Contribution from q to hybrid 4, =
c,.,~
In the hybrid orbital 62(sp2) above, for example, the contribution of the s component orbital is (d/3)2= 113, while the total contribution of the p component orbital is
leading to an orbital consisting of 113 s and 213 p character respectively. The other two orbitals likewise have this character and all three sp2 hybrids may be written
As a matter of fact when we write spa we mean s1.0p2.0or an orbital with (111 2) X 100 = 33.3% s character and 66.7% p character. The superscript in p is sometimes called the hybridization index: in general the exact hybrid can be written as sp". In sp, sp2, and spa hybrids X = v'i, 8 2 , and 4;and X2 = 1, 2, and 3, respectively. The per cent of s character in each orbital is
+
Step 3. The positive lobe of the component d,z orbital is along the z axis and thus the coefficient for this component orbital in 4~ and 46 can be calculated from the normalization condition
% s Character Accordingly, 4~ and 46 are given by
The sign of the d,z component orbital in the xy plane is negative. The coefficients of the d,? component orbital in the four hybrid orbitals 41, @2, $3, and 44 can be calculated from the unit orbital contribution condition
=
1
X' X 100
+
It should be obvious that the amount of p character in a set of orbitals will determine the angle, 0, between the hybrid orbitals. Conversely, if the angle between hybrid orbitals is known, we should be able to calculate the amount of p character. The equation relating bond angle and p character can he seen by inspection of the sp case where A2 = l,B = 180°, to be
Solving this equation for .?pip, sp2, and sp3
Accordingly, the complete hybrid orbitals are
Nonequivalent Hybrid Orbitals (10)
The Contribution of Atomic Orbitals to a Set of Equivalent Hybrid Orbitals The hybrid orbitals that we have discussed thus far consist of sets in which the contribution of the component s, p, or d atomic orbitals is the same in each of the hybrid orbitals of the set. The three sp2 hybrid orbitals we discussed earlier are such a set
Each of these orbitals is made up of exactly 113 s and 213 p. The contribution of the component atomic orbital p
Thus far we have considered only equivalent hybrid orbitals; such orbitals are present when all the atoms bonded to the central element in question are equivalent such as the three .sp2 hybrid orbitals of the boron atom in BFs and the four sp3 orbitals of the carbon atom in CH4 or CClr. If the element is involved in bonding with nonequivalent atoms, the orbitals involved need no longer he equivalent. We saw above, that the amount of p character may be calculated from a bond angle. For two equivalent hybrid orbitals, i, eqn. (1) is
Figure 6. Diagram showing band angles and types of hybrid orbitals of formaldehyde.
Volume 50, Number 2. February 1973
/
117
unhybridired P
hybridized
........-
,
~ ->
.
N
P
& 1
=
= 0.60
The wave function for the carbon hybrid orhital involved in honding to oxygen is
Figure 7. Schematic diagram showing t h e energy levels of the orbitals of carbon atom in formaldehyde.
Since 42 and 43(sp2.'3) are equivalent with respect to both the y and z axes (Fig. 6). the unit orbital contribution requirement can he readily applied for the calculation of the coefficients of the s andp, atomic orbitals mds) = m,(r) =
Jfi0.57 =
hihi cos O,, = -1
whereas for two different hyhrid orbitals, i, and, j, separated by an angle mij, the equation becomes
A particularly instructive example is the formaldehyde molecule (Fig. 6). In this molecule, the carbon atom has three hybrid orbitals involved in the three o bonds, and a bond pure p orbital involved in a n bond. The H-C-H angle is known to be about 118" (Oii), thus the two hybrid orbitals of the carbon atom involved in the C-H o bonds, calculated from eqn. (11, are sp2.13. The H-C-0 bond angle is known to he about 121" (Bij); substituting the values Oij = 121" and Ai = m 3 into eqn. ( 2 ) . we find that the hybrid orbital involved in the C-0 o bond is sp1.77. The p orbital contribution in the two types of the hybrid orbitals of carbon atom in the formaldehyde molecule is different; in the hybrid orbitals ~ ~ 2 . 1the 3 contribution of the p orbital is 2.13/(1 2.13) = 0.68, whereas the carbon hyhrid orbital used for bonding to oxygen is sp'-77 and the contribution of the p orhital is 1.77/(1 1.77) = 0.64. As a consequence, the energies of the bonding orbitals of the carhon atom i n the formaldehyde molecule are in the order p > ~ p ~ . ~ ' > s p as l . ~shown ~ , in Figure 7. The carbon hyhrid atomic orbital used in honding with oxygen has more s character than the orbitals used for honding with the hydrogen atoms. Finally, as an exercise in closing the circle, we might ask what is the equation for the complete wave function corresponding to the spz-13 and the spl.77 orbitals of carbon in CH20 that we have just discussed. We find the answer by reversing the processes we have just gone through. Consider formaldehyde in the orientation shown in Figure 6. The wave function corresponding to the s ~ hy- ~ d 7 7 p , ) where N is the norbrid orbital is 61 = N(s malization factor, calculated in the usual way
+
+
+
The p, orbital makes contributions to only the @zand 63 (sp2-13)orbitals, hence the coefficients on p, are fl = +0.71. Accordingly, the full wave functions are
*
m 2 ( s ~ ~ -=' ~ ) + 0.71py - 0 . 4 2 ~ ~ 0.57s ~ 0.42pn ~ $ ~ ( . F P ~=. '0.57s ~) - 0 . 7 1 Summary The analytic form of sets of equivalent hyhrid orbitals can be readily obtained by drawing these orbitals in a conveniently chosen coordinate system and then applying the arithmetical requirements for correct sets of atomic hybrid wave functions, i.e., normalization, orthogonalization, and unit orbital contribution. The method is illustrated for sp2, sp3, dsp2, and d2sp3 hybrid orbitals. This procedure is also applicable to sets of nonequivalent hyhrid orbitals; the hybrid atomic orbitals of carbon in formaldehyde are derived as an example. Finally, given the hybridization index of an atomic orbital, its analytic form is obtained by use of the same principles. Thus, for example, the sp1.77 atomic hybrid orbital of carbon in formaldehyde is shown to correspond to 0.60s 0.80pZ.
+
Acknowledgment
We wish to thank Drs. Ellis and Macomber for helpful discussions. Literature Cited i l l Kimball,G.Z.,J. Chem.Phys.,R. 188i19401. (21 Psuling, L.. "The Nature of the Chemical Bonding," (3rd E d l , Cornell U n i v ~ r r i t y Press.. New Ymk. 1960. (31 Kuhn. H..J. Chrm. Phyl., l6,727(19481. (41 Aseriesofpapers b y 0 u f f e y . G H.. J. Chem Phys.. I8 119,WI and is, M l l r r ~ l l.I N . I Chsm PhW 12.767 11!,601
-
~
~
(81 Cotton. F. A,, "Chemical Application of Group Theory." (2nd Ed.1. lnterseknreJohn Wiley, NewYork.
1971.
191 Murrcl. .J. N., Kettle. S. F. A,. and Tedder. .J. M.. "Valence Theory," John Wiiey XI snn. MewYnrk I~ S. 6~.. ~~~, (10) Mislow. K.. "Intmduetion to Stercochemiatry," W. A. Benjamin. Inc.. New York. 1965; Hendricluan, J. 8.. Cram. D. J., and Hamrnond. G. S.. "Oieanic Chemir~
whereupon
118 /Journal
I9i1951l.
~
try."(3dEd.l, McCraw-HillCo., NewYurk. 1970.p. 60.
of Chemical Education
