Nelson F. Phelan ond Milton Orchin University of Cincinnati Cincinnati, Ohio
I I
Symmetry Simplifications in Calculation of HMO Wave Functions
The Hiickel molecular orbital (HMO) method (1) of obtaining approximate wave functions for conjugated a systems involves four basic steps: 1. 2. 3. 4.
Setting up the secular determinant Expansion of the secular determinant Solution of the resulting equation Determination of the normalized coefficients
Carrying through the second and third steps of the procednre, without the aid of a computer, can be prohibitively tedious if the a system contains a large number of atoms. For example: The a system of phenanthrene has fourteen carbon atoms, and accordingly a 14 X 14 secular determinant is required. The expansion of the determinant by cofactors (8) or by other means (3) is laborious and the resultant equation, once obtained, is a 14th power equation. Swain and Thorson (4) have described a procedure for reducing a large n X n secular determinant to a number of smaller ones by employing symmetry considerations, and this treatment appears in recent texts (5-7). According to this method we can reduce an n X n secular determinant to smaller determinants if the molecule possesses a two or k-fold axis of rotation. The basic procednre involves choosing trial wave fnnctions belonging to d i e r e n t symmetry species appropriate to themolecule. Since only trial functions of the same symmetry can interact, the large n X n secular determinant can thus be separated into smaller determinants, each containing trial functions of only a single symmetry species. To find the number of each symmetry species in the molecule, we first obtain a reducible representation and then determine the irreducible representations )'I( contained in it. We then separate each set of irreducible representations into a smaller determinant. For example: in a ten atom r system such as naphthalene with D z symmetry ~ we obtain the irreducible representations: 2% Zb,, Sbau 3bZg. The original 10 X 10 secular determinant may thus be separated into two sets of 2 X 2 and two sets of 3 X 3 determinants. However, the procedure as it commonly appears (4-6) neglects the antisymmetric or ungerade character of the 2 p orbitals in the a system by ignoring the nodal plane of the molecule as an element of the point group on the ground that all a systems obviously have such a plane. In effect the atoms are treated as spheres undergoing the symmetry operations of the point group (or sub-group if a reduced character table is used); this procednre, for purposes of the present paper, may be abbreviated as the atom method (AM). When the p r orbitals rather than the atoms are transformed under the symmetry operations, a reducible
+
+
+
representation diierent from that obtained by the more common AM is obtained. The procednre involving orbital transformation may be called the orbital method (OM) (7). Including the nodal plane as a symmetry element has some positive advantages without complicating the procedure. An example will help illustrate the two approaches. Table 1.
Chorocter Table for C2v
The allyl system belongs to point group C2. (Table 1). The preferred orientation of this molecule in a coordinate system is shown in Figure 1. Now in order to obtain the reducible representation by the AM, the symmetry operations of C2,are performed on the atoms.
Fig. 1.
Orientation of the ally1 system
If an atom is transformed into another atom by the operation, it is given the character zero, and if transformed into itself, it is given the character +I. On the other hand, in order to obtain the reducible representation by the OM, the character of -1 is given to the transformation when the operation converts the orbital into minus itself, and as before, zero and plus one when transformed into something else or itself, respectively. Tables 2 and 3 show the results of applying the AM and OM procedures to the allyl system. To obtain the number of each irreducible representation r contained in these reducible representations we use the equation (8): nr = ' / g ~ ~ n r r ~ Y ~ , ~ Volume 43, Number 1 1 , November 1966
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571
Table 2.
I
Gnu
Cn"
+1 +1
Atom 1 Atom 2 Atom 3
*w
0 1
+1
+0
+3
+l
Table 3.
To obtain the results shown in Table 4 we have used the complete character table for the point group Dm, appropriate to naphthalene. For linear stilbene we have used D2, a reduced character table or subgroup of D m which is the actual point group of the molecule (no errors are made if a molecule is assumed to have lesser symmetry than it actually has). A study of Table 4 shows that in all cases we obtain the same number of reduced n X n secular determinants, regardless of the AM or OM approach; however, the symmetry species are always different.
A M Transformation
++11 +1
0 +1 0 1
+
+3
OM Transformation
l(1 1 (2 1 (2 2 (2 2 (2
ally1 toluene benzyl naphthalene linear stilbene
where g is the order of the group (the total number of symmetry operations in the point group--in Cz. this is four), g, is the order of the class of the symmetry operation R (each vertical column in the point group is a class and the coefficient before the symmetry operatom in the character table is the order of the classin Cz. this is one for all operations), rrRis the character of the symmetry operation in the irreducible representation r, and ropR is the character of the symmetry opers, tor in the reducible representation (or direct product as it is often called). Thus we obtain the irreducible representations for the AM : n,,
-
'/,(1.1.3
+ 1.1.1 + 1.1.1 + 1.1.3) = 2
+ 1.1.1 - 1.1.1 - 1.1.3) = 0 ns, = '/,(1.1.3 - 1 . 1 . 1 + 1 . 1 . 1 - 1 . 1 . 3 ) = 0 na. = '/4(1.1.3 - 1 . 1 . 1 - 1 . 1 . 1 + 1 . 1 . 3 ) = 1 n., = 1/,(1.1.3
X 1 ) a n d l (2 X 2) X 2 ) and 1 (4 X 4)
X 2 ) and 1 (5 X 5 ) X 2) and 2 (3 X 3) X 2) and 2 (5 X 5 )
In the allyl example we see that we have reduced the original 3 X 3 determinant by either method to two smaller determinants; one 1 X 1 and one 2 X 2. The next step involves obtaining trial wave functions. To do this we now repeat the procedure for obtaining the reducible representation except that instead of giving the character of the transformation, we substitute the atoms or orbitals that are generated by the transformation. The results are found in Tables 5 (AM) and 6 (OM). To obtain the appropriate combination of these atomic orbitals which correspond to the trial wave functions, we multiply the results of the transformation by the characters of the appropriate symmetry species. For this purpose we may employ matrix multiplication (9). Multiplication of a column matrix by a row matrix can be symbolized:
and for the OM: n . ,
=
'/,(1.1.3
- 1.1.1 + 1.1.1 - 1.1.3) = 0 - 1.1.1 + 1.1.3) = 1
If we treat the representations in Tables 5 and 6 for allyl as the row vector, and the irreducible representations we derived for allyl (Table4) as the column vector,
n,, = 1/,(1.1.3 - 1 . 1 . 1 ns = '/&.1.3
ns,
=
'/r(1,1.3
+ 1.1.1 + 1.1.1 + 1.1.3) = 2 + 1.1.1 - 1.1.1 - 1.1.3) = 0
Table 5.
These results and, as further examples, those for toluene, benzyl, naphthalene, and stilbene are given in Table 4. Toluene is an even alternant hydrocarbon, abbreviated AH (a conjugated system with an even number of atoms in the a system such that if alternating atoms in the a system are starred, the starred atoms are attached to only unstarred atoms and vice versa), as are naphthalene and stilbene. Benzyl, however, is an odd alternant hydrocarbon (a conjugated system with an odd number of atoms in the rr system such that the starred atoms are bonded only to unstarred atoms).
o
Naphthalenea (Dab)
AM OM AM OM AM OM AM OM
.. .. .. .. .. .. .. ..
Linear stilbeneb (DB)
AM OM
5 2
Ally1 (CaO) Toluene (CZ,) Benzyl (CI.)
a
++I
*98 +I
+Q8
+& +h
+d. +9* Table 6.
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Journal o f Chemical Mucafion
+Q1
+Q3
+9z +9r
++I
Orbital Transformations
al 2 0
a
a.
0
4 0 5 0
0 2 0
.. .. .. .. .. ..
.. .. .. ..
.. .. .. ..
1
2
03
.. ..
a. .. .. .. ..
br
.. ..
bl 0 2 0 4 0 5
02
.. ..
.. ..
..
5
2 5
..
2
1 0 2 0 2 0
br .. ..
..
.. .. .. . . 2 5
big
blY
bo
bl,
bs
.. .. .. .. .. ..
.. .. .. .. .. ..
.. .. .. .. .. ..
.. .. .. .. .. ..
.. .. .. .. .. ..
4. .. .. .. .. .. ..
2 0
0 3
. .
0 2
.. ..
3 0
.. ..
The y plane is the plane of the molecule and the z axis passes through the bridging atoms 9 and 10. The yz plane is the plane of the molecule with the z axis the long axis.
572
+C
Svmmetw. Species for Various Molecules .
Table 4. System
Atom Transformations
0 3
.. ..
.. ..
2 0
.. ..
.. ..
we obtain the desired dot products. Thus for the AM case we have the 2a, and lb2irreducible representations. T a k i their corresponding characters from Table 1 and the representations for atom transformation from Table 5 we have:
where H u
=
JJ.1HJ.ldr which for J.swould be
=
and where Hz2 = Jfi2HJ.zdr which for J . Gwould be J+zH+zdr and where HI? = Hz1 = SJ.lHJ.2dr which for J.6 and J.6would be Jl/v'2(qh +dH+zd.r. The 1 X 1 determinant of species a* would involve a 1 X 1determinant of theform:
+
However, these trial wave functions are not normalied. If we normalize these wave functions (lo), we obtain the AM trial wave functions:
Now for the OM case we found la, and 2bl irreducible representations. Taking their correspondmg characters from Table 1 and the representations for atom transformation from Table 6 and repeating the multiplications as above, we obtain, after normalization:
We could now convert these trial wave functions into secular determinants (1) such that each secular determinant corresponds to only a single symmetry species: for the AM case: .,*I
-+, -
and .3$z a&
( 2 X 2 ) determinant of a, symmetry
(1 X 1) determinant of bn symmetry
for OM case:
md h
( 2 X 2) determinant of b, symmetry
(1 X 1) determinant of m symmetry
However, an inspection of the form of the trial wave functions shows that:
so that the determinants obtained in both AM and OM cases would be identical and therefore the HMO's resulting from them will be identical. Solution by the HMO method first requires setting up the appropriate secular determinant. I n the OM method, the 2 X 2 determinant of species bl would involve a secular determinant of the form:
where Hll = JJ.IHJ.1d7 which for J.& would be = Jl/-\/Z(61 - +3)H1/v'z(+1 - +3)dr. The actual procedures for solving these determinants and evaluating the wave functions by the HMO method are not the topic of this paper and are well illustrated elsewhere (I,.$). We are interested here in the expressions for the molecular orbitals of the allyl system that come from the HIMO method. The allyl MO's are (11):
Thus both AM and OM result in the same wave equations for the allyl system. Since the electron densities, charge densities, bond orders, free valence, etc., are all derived from the molecular wave equations, they will be the same for both AM and OM, which is, of course, one of the justifications for neglecting a consideration of the nodal plane of the p r systems. The point of difference in the two approaches is that the OM does hut the AM does not give the correct symmetry species of the final wave functions. This can be seen by drawing the MO's in the usual way as shown in Figure 2. The C2. character table, (Table 1) shows that the correct species for the above MO's are b,, a,and b, respectively. Because the allyl system is a simple ?r system, it is perhaps instructive to consider more complicated examples in order to illustrate the two procedures. Although ordinarily we must carry out the complete HMO treatment in order to obtain the complete wave equations, we can by judicious choice of our r system, avoid the necessity of doing so. Specifically, we can select examples from odd AH systems and focus on the nonbonding molecular orbital (NBMO) which every such system possesses. There is a very simple method, developed by Longuet-Higgins (Is),for determining the coefficients of the atomic wave functions in the NBMO without solving the secular determinant. This method depends on the fact that in the NBJIO, the coefficients of each unstarred atom must be zero (there is a node through these atoms) which can be assured by having the coefficients of the starred atoms surrounding each unstarred atom, sum to zero. Thus in the allyl system, which is the simplest odd AH, CI-C&, if the coefficient of C1 is arbitrarily assigned the value a, in order for the coefficient of Cz to he zero, the coefficient of the atomic orbital on C3 must be -a. According to the normalization condition, the sum of the squares of the coefficients c, of the AO's making up the J., MO's must be unity, i.e., W,, = 1. I n the allyl system NBMO Volume 43, Number 1 1, November 1966
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573
+ +
Thus none of the seven MO's according to the AM will be of species bl, whereas the OM predicts bl species for five of the seven MO's. If a complete HMO treatment is carried out, the NBMO will be found to belong to the correct (bl) symmetry species.
this means that (a=) 02 (-a)% = 1 and a = 1 / 4 2 The same procedure whence J. = 1 / 4 2 qil - 1 / 6 2 may he applied to the benzyl system (see Fig. 3). If the coefficient on Cs is arbitrarily assigned +a, C3, and C7 are -a in order for C, and Ce to be zero. The coef-
--
--
*,I, anti-bonding M O AM or OM b,
*11
non-bonding M O lNBMOl AM b, OM 0s The MO's of the d y l syrtsm
Fig. 2.
ficient a t C1 must he 2a since then the sum of the starred coefficients around the unstarred atom Cz is 2a - a - a = 0. Using the normaliation condition we obtain the coefficients:
+ 2(-a)%+ (a)* =
@a)=
1
7a4 = 1 a = c, =
2/*
cct
=
c:
=
-I/&
oa =
I/&
and thus the complete wave function is:
+ NBMO = 2/dT+1 - l/.\/T+i + I/-/% which can be drawn as in Figure 4. point group Cz., and application of (Table 1) to the NBMO pictured transform as symmetry species b,. tion of Table 4 shows that
-
- 114W
Benzyl belongs to the character table above shows it to However, inspec-
The AIM can be modified to give the correct species. Our original AM assumption was neglect of the nodal plane of the molecule as an element of symmetry in the point group. To correct AM symmetry species for this assumption, we simply consider the antisymmetric character of the 2 p s orbitals with respect to appropriate C2 rotation and with respect to reflection in the molecular plane. To correct the species obtained by the AM, we therefore multiply the characters of Cz and a,, by -1 as shown for al b, conversion in Table 7 and a2 -t a2 conversion as shown in Table 8. However, this correction is frequently not obvious to a beginning student. Because the OM gives directly the correct symmetry species and because its use does not complicate the procedure, the authors recommend that it be taught in basic introductions to HMO theory and application.
-
AM -. 5a, and 2bn OM
Table 7.
2ar and 5bl
a3
Table 8.
* -a Fig.>.
Conversion of AM to O M
I
cxz @z=
cvi
b~
Conversion of AM to O M a2
The benryl system
Correction
++11 -1 -1
Correction
be
...
+1 -1 -1 +l
xi-1)
x(-i)
Literature Cited (1) HOCKEL,E.,Z. Physik, 70, 204 (1931); 72, 310 (1931); 76, 628 (1932); 83, 632 (1933); International Canference on Physics, London, 1934, Vol. 11, The Physical Society, ~ o i d o n ;1935, p. 9. (2) ROBERTS, J. D., "Notes on Molecular Orbital Calculatiom," W. A. Beniamin. Inc.. New York. N. Y.. 1962. DD. 4346. Fig. 4.
574
The +NBMO for bonzyl
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Journal of Chemical Education
.
.
(195'9). ( 5 ) ROBERTS, J. D., op. Cit., pp. 61-72. A., JR., l'Mole~~Iar Orbital Theory for (6) STREITWEISER, Organic Chemists," John Wiley & Sons, Ino., New York, N. Y., 1961, pp. 79-95. F. A., "Chemical Applications of Gmup Theory," (7) COTTON, John Wiley & Sons, Inc., New York, N. Y., 1963, pp. 142
(8)
JAW$, H. H. AND ORCHIN, MILTOX, "Symmetry in Chemis-
try," John Wiley & Sons, Inc., Sew York, S. Y., 1965, p. 125. (9) Ibid., pp. 74-75. (10) ROBERTS, J. D., o p . eit., p. 35. A., JR., o p . tit., p. 50. (11) STREITWEISER, (12) LONGUET-HIGGINS, H. C., J. Chem. P h ~ s .18, , 275 (1950).
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