Gallup G.A. - Valence Bond Methods, Theory and applications (CUP, 2002)

196

14 Rings of hydrogen atoms

The real reactions that most

resemble these

are

the production of

cyclobu-

tane from two ethylene molecules ((H

2 )2 ) and the Diels–Alder reaction between

butadiene and ethylene ((H

2 )3).

Even these cannot be made to react in the bare

forms, but fairly simple activation by substituents

will

allow the

(H

2 )3 analog to

proceed. Apparently, no form of the (H

2 )2

analog

has ever been observed. Our

MO theory.

Table 15.1.

The four most important categories of HLSP functions in

the wave function for benzene.

1

2

3

4

Num.

a

2

3

12

12

2 p a

2 p b

2 p a

2 p a

2 p b

2 p c

2 p a

2 p a

HLSP

b

2 p c

2 p d

2 p a

2 p d

2 p c

2 p d

2 p c

2 p c

C i c

2 p e

2 p f

R

2 p e

2 p f

R

2 p e

2 p f

R

2 p e

2 p f

R

0.160 88

−0.057 63

0.051 62

0.027 44

a The

number of terms in the symmetry

function

that

is generated from the tableau

shown. (See text.)

15.1 STO3G calculation

1 A 1g

symmetry functions. Table 15.1 shows the important HLSP functions for a

π-only

σ

orbitals are all treated in the “core”, as described in Chapter 9, and the

π electrons

characteristics.

1. The two functions

of this type are the classical Kekul´

e structures for

benzene. One might

expect the coefficient to be larger, but we will see below why it is not.

15.1

STO3G calculation

199

Kekul´structures and denote them by

K

1 and K 2. As discussed in Chapter 5, when

these HLSP functions are projected from the appropriate product of

p orbitals

K

1 = θ PNPρ1,

K

2 = θ PNPρ2,

they are not normalized to 1. The “raw” 2

× 2 overlap matrix is

0.799 515 5

2 .246 364 9

,

2.246 364 9

and, hence, the true overlap between a normalized

K

1 and

K

2 is 0.355 915 2. Thus,

if we consider the

1 A 1g symmetry

function

involving

K

1 and

K 2, we obtain in its

normalized form

1 A

1g = 0.607 251 7(K 1 + K 2),

individual

Kekul´structures. In these terms, the Kekul´structures appear to have a

larger coefficient. A similar analysis for the Dewar structures leads to an apparent

enhancement of the coefficient magnitude to

−0.133 825 9.

15.1 STO3G calculation

201

Table 15.2.

Comparison of different calculations

of the π system of benzene. All energies

are in hartrees.

E

a

− E core

K

1

−6.711 44

K 1 & K 2

−6.755 53

Full cov.

−6.760 79

SCF

−6.834 10

SCVB

−6.904 88

Full π

−6.911 87

a

E core

= −222.142 48.

Orbital amplitude

0.4

0.3

0.2

0.1

0.0

3

4

−4

0

1

2

−3

−2

−1

0

−2

−1 x-direction (Å)

1

2

y-direction (Å)

3

−3

4 −4

Figure 15.1. Altitude plot of the SCVB orbital for the

π system of benzene. There are six

symmetrically equivalent versions of this around the ring. The amplitude is given in a plane

z -direction from the plane of the nuclei.

0.5 A in the positive

predominantly a single 2

p z

with smaller “satellites” in the ortho positions, essen-

tially nothing at the meta positions, and a negative contribution at the para position.

The coefficients in

the two covalent-only wave

functions,

(pure

2 p z ) and

(SCVB 2 p z ) are not very different:

(pure 2

p z ) = 0.402 88( K

1 + K

2) − 0.150 26(D

1 + D

2 + D

3),

(SCVB 2

p z ) = 0.403 53( K

1 + K

2) − 0.122 03(D

1 + D

2 + D

3).

We may, however, examine the 2

× 2 Kekul´-only matrices for these two cases. For

the pure 2 p z orbitals we have, in hartrees, the secular equation

= 0,

−2.448 49

−

0.355 92 E

6.711 44

E

−

Table 15.3.

Lowering of the energy from resonant

mixing for pure and SCVB

2 p z

orbitals.

Pure eV

SCVB eV

K

− only

−1.199

−0.870

K

+ D

−1.343

−0.878

Pure

SCVB

T a

1.2299

1.0891

V b

c

−2.9613

−2.8750

[11|11]d

0.5795

0.5125

[11|22]

and for the SCVB orbitals,

−

= 0.

−4 .616 57

0.664 00E

6.872 56

E

−

1

1

We see that the

K

–K

diagonal

element for the SCVB

orbitals is already

about

4.4 eV below that for the pure 2

p z orbitals. This is the most immediate explanation

for the lower energy of the SCVB result. In fact, this is the larger effect. As seen in

Eq. (14.2), the amount of energy lowering in 2

× 2 systems like these is

H 12 − S 12 H

11

.

1 + S 12

p z and SCVB orbitals,

In this case we have 1.20 eV and 0.87 eV for the pure 2

respectively, and the resonance appears somewhat more beneficial for the localized

orbital. These results are included in Table 15.3.

We may obtain

more information

from

a comparison of some of

the one-

and two-electron integrals for the individual orbitals. The values are shown

in

Table 15.4. It is seen that the changes in the potential energy terms nearly cancel

with the repulsive self-energy and the nuclear and core potential energies chang-

ing in opposite directions. The change in the adjacent repulsion energy is also not

15.1 STO3G calculation

203

Table 15.5. Comparison of different calculations

of the

π system of linear 1,3,5-hexatriene. All

energies are in hartrees.

K 1

−6.129 74

Full cov.

−6.133 78

SCF

−6.164 40

SCVB

−6.251 49

Full π

−6.252 41

a E core

= −223.91056.

the delocalization in that orbital, and it thus makes the same sort

of contribution

as do ionic structures in MCVB wave functions. The values of the

n -electron ma-

trix elements are the result of an interplay of considerable complexity among the

simpler oneand two-electron matrix elements, and it is not really possible to say

much more about the effects of these latter quantities upon the total energies.

We show another aspect of these numbers in Table 15.3, where we detail the

effects of resonance between the two Kekul´ structures and among all of the co-

valent structures for the two sorts of 2

p z orbitals. The results suggest

that within

one structure the SCVB orbitals duplicate, to some extent, the effect of multiple

structures, and the configurational mixing produces less energy lowering with them.

Those familiar with the long history of the attacks on the question of the resonance

energy of benzene may be somewhat surprised at the small numbers in Table 15.3.

The energy differences that are given there are for just the sort of process that might

be expected to yield a theoretical value for the resonance energy, but experimental

determinations yield numbers in the range 1.7–2.3 eV. This is an important question,

which we will take up in Section 15.3, where it will turn out that some subtleties

must be dealt with.

15.1.2 Comparison with linear 1,3,5-hexatriene

In order to put the structure of benzene into better perspective, we give a similar

calculation of 1,3,5-hexatriene for comparison. Table 15.5 shows the energies for

a set of calculations parallel to those in Table 15.2 for benzene. The most obvious

difference is the smaller total spread in the energies, about 3.3 eV rather than the

5.5 eV for benzene, and the SCVB energy is closer to the full

π than in benzene.

Standard arguments say that there is only

a rather small amount of resonance