10-3 Extended H ̈
Chapter 10 The Extended H ¨uckel Method used here, and he obtained gross populations for hydrogen atoms of around 0.9, giving net positive charges of 0.1. Thus, absolute values are not very useful, but changes
in gross population as we go from one hydrogen to another in the same hydrocarbon molecule or in closely related molecules do appear to be rather insensitive to VSIE choice and are often in accord with results of more accurate calculations.
10-3 Extended H ¨uckel Energies and MullikenPopulations In Chapter 8 it was shown that a simple quantitative relation exists between the energy of a simple H¨uckel MO and the contributions of the MO to bond orders [Eq. (8-56)].
The more bonding such an MO is, the lower is its energy. A similar relationship will now be shown to hold for extended H¨uckel energies and Mulliken populations.
The orbital energy for the real MO φi is (ignoring the spin variable) φ ˆ i H φi dv
Ei =
(10-22)
φ2 dv
i
AOs c
=
j,k j i cki Hj k
(10-23)
AOs c j,k j i cki Sj k
Assume that φi is normalized: The denominator is unity.
The numerator of Eq. (10-23) contains diagonal and off-diagonal terms.
If we separate these, and substitute the relation (10-4) for the off-diagonal terms, we obtain
AOs
AOs
Hjj + Hkk Ei = c2jiHjj + 2
cjickiKSjk
(10-24)
2
j,k j
AOs
AOs
Hjj + Hkk Ei = qi H pi K
(10-25)
j
jj +
j k
2
j
j
This equation indicates that the energy of an extended H¨uckel MO is equal to its net contributions to AO populations times AO energy weighting factors plus its contributions to overlap populations times overlap energy weighting factors.
By summing over all MOs times occupation numbers, we arrive at a relation between total EH energy and net AO and overlap populations:
AOs
AOs
Hjj + Hkk E =
qj Hjj +
pjkK
(10-26)
2
j
j
Section 10-3 Extended H ¨uckel Energies and Mulliken Populations
There is an important difference between the extended H¨uckel formulas (10-25) and (10-26) and their simple H¨uckel counterparts. In a simple H¨uckel MO, the charge density contributions qi always add up to unity, contributing α to the MO energy (omitting heteroatom cases). The deviation of the MO energy from α is thus due only to the bond-order contributions pi . As a result, for hydrocarbons, the simple H¨uckel MO that is the more bonding of a pair is always lower in energy. This is not so, however, in extended H¨uckel MOs. Since the sum of AO and overlap populations must equal the number of electrons present, we can increase the total amount of overlap population only at the expense of net AO populations. Therefore, energy lowering due to the second sum of Eq. (10-25) is purchased at the expense of that due to the first sum. This complicates the situation and requires that we take a more detailed look before issuing any blanket statements.
The following simple example provides a convenient reference point for discussion.
Consider two identical 1s AOs, χa and χb, on identical nuclei separated by a distance R.
These AOs combine to form MOs of σg and σu symmetry. The σg MO has an associated positive overlap population and equal positive net AO populations for χa and χb. If R now is decreased slightly, the overlap population increases slightly (say by 2δ), so the net AO populations must each decrease by δ. The energy change for the MO, then, is Haa + Hbb Eσ = −δ (H
(10-27)
g
aa + Hbb) + 2δK
2
Since the molecule is homonuclear, Haa = Hbb, and Eσ = 2δH
g
aa (K − 1)
(10-28)
If K > 1, Eσ is negative (since H g
aa is negative). This analysis shows that the choice of 1.75 as the value for K has the effect of making the increase in overlap population dominate the energy change. If K were less than unity, the net AO population changes would dominate.
The above example suggests that the EHMO method lowers the energy by maxi mizing weighted overlap populations at the expense of net AO populations. It also suggests that the EH energy should be lowered whenever a molecule is distorted in a way that enables overlap population to increase. These are useful rules of thumb, but some caution must be exercised since the existence of several different kinds of atom in a molecule leads to a more complicated relation than that in the above example.
Our methane example illustrates the above ideas. We have already seen that the total EH energy for methane goes through a minimum around RC–H = 1 Å. Let us see how the individual MO energies change as a function of RC–H and try to rationalize their behaviors in terms of the overlap population changes. A plot of the MO energies is given in Fig. 10-8. The lowest-energy MO is s type and C–H bonding. The overlap between H 1s AOs and the C 2s STO increases as RC–H decreases. As a result, the energy of this MO decreases as RC–H decreases, favoring formation of the united atom. The second-lowest energy level is triply degenerate and belongs to p-type C–H bonding MOs. A glance at Fig. 10-3 indicates that overlap between AOs will first increase in magnitude as RC–H decreases. But ultimately, at small RC–H this behavior must be reversed because the 1s and 2p AOs are orthogonal when they are isocentric. Therefore, as RC–H decreases, the overlap population first increases, then decreases toward zero. Consequently, the EH energy of this level first decreases, then increases. Since there are six electrons in this
Chapter 10 The Extended H ¨uckel Method
Figure 10-8 Extended H¨uckel MO energies for methane as a function of RC–H.
level and only two in the lowest-energy MO, this one dominates the total energy and is responsible for the eventual rise in total energy at small R that creates the minimum in Fig. 10-4. The two remaining (unoccupied) levels belong to antibonding MOs, and the overlap population becomes more negative as RC–H decreases. The p-type level rises less rapidly and eventually passes through a maximum (not shown), again because the overlap population must ultimately approach zero as RC–H approaches zero.
It is important to remember that all these remarks apply to the EH method only. The relationships between the EH method and other methods or with experimental energies is yet to be discussed.
10-4 Extended H ¨uckel Energies and ExperimentalEnergies
We have seen that a change in EH energy reflects certain changes in calculated Mulliken populations. We now consider the circumstances that should exist in order that such EH energy changes agree roughly with actual total energy changes for various systems.
Section 10-4 Extended H ¨uckel Energies and Experimental Energies
In essence, there are two requirements. First, the population changes calculated by the EH method ought to be in qualitative agreement with the charge shifts that actually occur in the real system. This condition is not always met. Some systems, especially those having unpaired electrons, are not accurately representable by a singleconfiguration wavefunction (that is, by a single product or a single Slater determinant).
An example of this was seen for some 1s2s states of helium (see Chapter 5). Since the EHMO method is based on a single configuration, it is much less reliable for treating such systems. Special methods exist for handling such systems, but they normally are not applied at the H¨uckel level of approximation.
The second requirement is that, as the charge shifts in the real system, the total energy should change in the way postulated by the EH method. For instance, if the charge increases in bond regions, this should tend to lower the total energy. In actuality, this does not always happen. We can see this by returning to our example of two identical 1s AOs separated by R. The associated σg MO increases its overlap population monotonically as R decreases. Therefore, the EH postulate predicts that the total energy should decrease monotonically. It is clear, however, that this does not happen in any real system. If our system is H2, we know that the experimental energy decreases with decreasing R until R = Re, and increases thereafter (Fig. 7-18). If our system is He2+, the experimental total energy increases as R decreases, due to the dominance
2
of internuclear repulsion.8 Consideration of many examples such as these indicates that the postulate is usually qualitatively correct when we are dealing with interactions between neutral, fairly nonpolar systems that are separated by a distance greater than a bond length typical for the atoms involved. These, then, are circumstances under which the EHMO method might be expected to be qualitatively correct.
In brief, we find two kinds of condition limiting the applicability of the EH method.
The first is that the system must be reasonably well represented by a single configuration.
Hence, closed-shell systems are safest. The second condition is that we apply the method to uncharged nonpolar systems where the nuclei that are undergoing relative motion are not too close to each other.
These conditions are often satisfied by molecules undergoing internal rotation about single bonds. Thus, it is indeed appropriate that the internal rotation barrier in ethane was used to help calibrate the method. A comparison of other EH calculated barriers with experimental values (Table 10-11) gives us some idea of the capabilities and limitations of the method. For all the molecules listed, the most stable conformation predicted by the EH method agrees with experiment. This suggests that one can place a fair amount of reliance on the conformational predictions of the method, at least for threefold symmetric rotors. Also, certain gross quantitative trends, such as the significant reduction as we proceed down the series ethane, methylamine, methanol, are displayed in the EH results, but it is evident that the quantitative predictions of barrier height are not very good. This is not too surprising since the method was calibrated on hydrocarbons. Introduction of halogens, nitrogen, or oxygen produces greater polarity and might be expected to require a different parametrization.
For reasons outlined earlier, we expect even less accuracy in EH calculations of molecular deformations involving bond-angle or bond-length changes. This expectation is generally reinforced by calculation. Molecular shape predictions become poorer 8Actually, He2+ has a minimum at short R in its energy curve due to an avoidance of curve crossing (discussed
2
in Chapter 14), but this minimum is unstable with respect to the separated ions. See Pauling [9].
Chapter 10 The Extended H ¨uckel MethodTABLE 10-11 Energy Barriers for Internal Rotation about Single Bondsa Barrier (kcal/mole)b
Molecule
Calculated
Experiment
CH3–CH3
3.04
2.88
CH3–NH2
1.66
1.98
CH3–OH
0.45
1.07
CH3–CH2F
2.76
3.33
CH3–CHF2
2.39
3.18
CH3–CF3
2.17
3.25
CH3–CH2Cl
4.58
3.68
CH3–CHCH2
1.20
1.99
cis–CH3–CHCHCl
0.11
0.62
CH3–CHO
0.32
1.16
CH3–NCH2
0.44
1.97
a Calculated barriers are for rigid rotation, where no bond length or angle changes occur except for the torsional angle change about the internal axis.
bThe stable form for the first seven molecules has the methyl C–H bonds staggered with respect to bonds across the rotor axis. For the last four molecules, the stable form has a C–H methyl bond eclipsing the double bond.
for more polar molecules (water is calculated to be most stable when it is linear), and bond-length predictions are quite poor too.
Results such as these have tended to restrict use of the EH method to qualitative predictions of conformation in molecules too large to be conveniently treated by more accurate methods. However, just as the simple H¨uckel method underwent various refinements (such as the ω technique) to patch up certain inadequacies, so has the EH method been refined. Such refinements9 have been shown to give marked improvement in numerical predictions of various properties. The EH method has been overtaken in popularity by a host of more sophisticated computational methods. (See Chapter 11.)
However, it is still sometimes used as a first step in such methods as a way to produce a starting set of approximate MOs. The EHMO method also continues to be important as the computational equivalent of qualitative MO theory (Chapter 14), which continues to play an important role in theoretical treatments of inorganic and organic chemistries (as, for example, in Walsh’s Rules and in Woodward-Hoffmann Rules).
10-4.A Problems
The following output is produced by an EH calculation on the formaldehyde molecule, and is referred to in Problems 10-1 to 10-10.
9See Kalman [10], Boyd [11], and the references cited in these papers. Also, see Anderson and Hoffmann [12] and Anderson [13].
Section 10-4 Extended H ¨uckel Energies and Experimental Energies Formaldehyde (Ground State) Orbital Numbering
Orbital Atom
n
l
m
x
y
z
exp
Hii
1
H-1
1
0
0
−0.550000
0.952600
0.0
1.200–13.60
2
H-2
1
0
0
−0.550000 −0.952600
0.0
1.200–13.60
3
C-3
2
0
0
0.0
0.0
0.0
1.625–19.44
4
C-3
2
1
0
0.0
0.0
0.0
1.625–10.67
5
C-3
2
1
1
0.0
0.0
0.0
1.625–10.67
6
C-3
2
1
1
0.0
0.0
0.0
1.625–10.67
7
O-4
2
0
0
1.220000
0.0
0.0
2.275–32.38
8
O-4
2
1
0
1.220000
0.0
0.0
2.275–15.85
9
O-4
2
1
1
1.220000
0.0
0.0
2.275–15.85
10
O-4
2
1
1
1.220000
0.0
0.0
2.275–15.85 Hij = KSij (Hii + Hjj )/2, with K = 1.75.
Distance Matrix (a.u.)
1
2
3
4
1
0.0
3.6004
2.0787
3.7985
2
3.6004
0.0
2.0787
3.7985
3
2.0787
2.0787
0.0
2.3055
4
3.7985
3.7985
2.3055
0.0
Total effective nuclear repulsion = 17.69537317 a.u.
Formaldehyde Eigenvalues Eigenvalues (a.u.)
Occ. no.
Eigenvalues (a.u.)
Occ. no.
E(1) = 1.039011
0
E(6) = −0.587488
2
E(2) = 0.472053
0
E(7) = −0.597185
2
E(3) = 0.314551
0
E(8) = −0.611577
2
E(4) = −0.342162
0
E(9) = −0.755816
2
E(5) = −0.517925
2
E(10) = −1.242836
2
Sum = −8.625654 a.u.
0
0
1
.0392
.0392
.0
.0
.0
.2146
.0
.0
.0
.0000
1
.0011
.0011
.2550
.0
.0685
.0000
.8481
.0
.0252
.0000
0
0
0
0
0
0
0
0
0
0
0
0
0
−
−
.0729
.0729
.3070
.00
.3056
.00
.00
.00
.0000
.01
.2721
.2721
.4875
.00
.2245
.0000
.3066
.00
.3327
.0000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−
−
−
−
−
−
−
−
89
89
.0
.0
.0
.2146
.0
.00
.00
.0000
.01
.00
.2141
.2141
.0000
.00
.0000
.3179
.0000
.00
.0000
.7600
0
0
0
0
0
0
0
0
0
0
0
0
0
−
−
−
−
−
−
.0813
.0813
.3734
.00
.4580
.00
.0000
.01
.00
.00
.0
.00
.0
.2456
.00
.0
.0
.9181
.0
.0
0
0
0
0
0
0
0
0
0
0
0
0
67
67
.4204
.4204
.00
.00
.00
.0000
.01
.00
.00
.2146
.2016
.2016
.0460
.00
.2768
.0000
.0884
.00
.8317
.0000
0
0
0
0
0
0
0
0
0
0
0
0
0
−
−
−
−
−
−
.2428
.2428
.00
.00
.0000
.01
.4580
.00
.3056
.00
.4281
.4281
.0000
.00
.0000
.3813
.0000
.00
.0000
.6475
0
0
0
0
0
0
0
0
0
0
−
−
−
−
−
−
.0
.0
.00
.0000
.01
.00
.00
.2146
.0
.00
.0
.00
.0
.9940
.00
.0
.00
.4532
.00
.00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−
345
345
.5133
.5133
.0000
.01
.00
.00
.3734
.00
.3070
.00
.8924
.8924
.0000
.00
.0000
.2519
.0000
.0
.0000
.2511
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
−
−
−
−
−
2
2
.1534
.0000
.5133
.00
.2428
.4204
.0813
.00
.0729
.0392
.7683
.7683
.5553
.00
.1727
.0000
.4799
.00
.3412
.0000
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
−
−
−
−
−
−
−
Matrix
1
1
.0000
.1534
.5133
.00
.2428
.4204
.0813
.00
.0729
.0392
.5279
.5279
.3964
.00
.6043
.0000
.8367
.00
.6960
.0000
erlap
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
ectors
−
−
−
−
Ov
−
−
v
otal
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
T
10
Eigen
10
344
0
0
1
.0180
.0180
.0
.0
.0
.0046
.0
.0
.0
.9939
.0004
.0004
.2882
.0
.0614
.0000
.6534
.0
.0037
.0000
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
−
−
−
−
.0225
.0225
.1443
.00
.1880
.0
.00
.00
.6061
.01
.3363
.3363
.7358
.00
.1775
.0000
.1123
.00
.3017
.0000
0
0
0
0
0
0
0
0
0
0
0
0
−
−
,
89
89
.0
.0
.00
.1936
.00
.00
.00
.6857
.01
.00
.1476
.1476
.0000
.00
.0000
.4203
.0000
.00
.0000
.2845
0
0
0
0
0
0
0
.0210
.0210
.1058
.00
.1876
.00
.6420
.01
.00
.00
.00
.00
.00
.2175
.00
.00
.00
.7825
.00
.01
0
0
0
0
0
0
0
0
0
0
0
0
−
−
67
.3890
.3890
.00
.00
.00
.4929
.01
.00
.00
.0046
67
.1088
.1088
.0028
.00
.3257
.0000
.0021
.01
.4518
.0000
0
0
0
0
0
0
0
0
0
0
0
0
1
0
−
.1134
.1134
.00
.00
.2634
.00
.1876
.00
.1880
.0
.4258
.4258
.0000
.00
.0000
.4593
.0000
.00
.0000
.6891
0
0
0
0
.00
.00
.00
.1207
.00
.00
.00
.1936
.00
.00
.00
.00
.00
.7825
.00
.00
.00
.2175
.00
.00
0
0
0
0
0
Each
0
0
0
0
0
0
0
0
in
Electrons
345
.2920
.2920
.6097
.00
.00
.00
.1058
.00
.1443
.00
345
.4266
.4266
.0000
.01
.0000
.1204
.0000
.00
.0000
.0264
12
0
0
0
0
0
0
0
0
Electrons
0
0
0
0
1
0
0
0
o for
w
T
2
with
.0702
.6876
.2920
.00
.1134
.3890
.0210
.00
.0225
.0180
2
.3881
.3881
.0561
.00
.1155
.0000
.0241
.00
.0282
.0000
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
Populations
−
−
−
−
MOs
for
erlap
1
1
Ov
.6876
.0702
.2920
.00
.1134
.3890
.0210
.00
.0225
.0180
.1664
.1664
.9171
.00
.3198
.0000
.2082
.00
.2220
.0000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
en
−
−
−
−
Matrix
ge
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
Mullik
10
Char
10
345
Chapter 10 The Extended H ¨uckel Method Reduced Overlap Population Matrix Atom by Atom
1
2
3
4
1
0.6876 −0.0702
0.7945 −0.0615
2
−0.0702
0.6876
0.7945 −0.0615
3
0.7945
0.7945
1.4866
0.8148
4
−0.0615 −0.0615 0.8148
6.9277
Orbital Charges
Net Charges
1
1.018973
6
0.879574
1 −0.018973
3
1.311555
2
1.018973
7
1.767672
2 −0.018973
4 −1.273609
3
1.026764
8
1.782529
Total charge = 0.000000.
4
0.217471
9
1.749778
5
0.564637
10
1.973630
10-1. Use the output to determine the orientation of the molecule with respect to Carte sian coordinates. Sketch the molecule in relation to these axes and number the atoms in accord with their numbering in the output.
10-2. Use the orbital numbering data together with the overlap matrix to figure out which of the labels 1s, 2s, 2px, 2py, 2pz goes with each of the ten AOs (i.e., it is obvious that AO 1 is a 1s AO, but it is not so obvious what AO 5 is).
10-3. Use your conclusions from above, together with coefficients in the eigenvector matrix, to sketch the MOs having energies of −0.756, −0.611, and −0.597 a.u.
Which of these are π MOs? Which are σ MOs?
10-4. Label each of the ten MOs “π ” or “σ ” by inspecting the coefficient matrix.
10-5. What is the Mulliken overlap population between C and O 2pπ AOs in this molecule? Should removal of an electron from MO 7 cause the C=O bond to shorten or to lengthen?
10-6. Demonstrate that MO 7 satisfies Eq. (10-24). (Note that Hii values in the first table are in units of electron volts, while orbital energies are in atomic units.)
10-7. Use the reduced overlap population matrix to verify that the sum of AO and overlap populations is equal to the number of valence electrons.
10-8. Using MO 7 as your example, verify that the charge matrix table is a tabulation of the contributions of each MO to gross atom populations.
10-9. In the list of “orbital charges,” are the “orbitals” MOs or AOs? Demonstrate how these numbers are derived from those in the charge matrix.