The following hierarchy of perturbations has been found useful in discussing the spectra of aromatic molecules derived from a (4N + 2)-electron perimeter through the introduction of a series of perturbations.
( I ) 1 Bridging Hydrocarbon (2) Cross-linking 1
Hydrocarbon
(3) 2-Electron or 0-Electron
I Heteroatom Replacement Heterocycle
Heterocycle t
(4)
( 5 ) 1 Substitution Target Molecule
1 -Electron Heteroatom Replacement
Bridging is defined as the insertion of additional n centers within the pe- rimeter. Examples are the production of acenaphthylene (16) from the [ I l]annulenium cation and a CG fragment or the production of pyrene (17) from [14]annulene and the C=C ethylene unit:
88 2 ABSORPTION SPECTRA OF ORGANIC MOLECULES 2.2 CYCLIC CONJUGATED n SYSTEMS 89 Bridging units tend to introduce new orbitals between the HOMOs and
the LUMOs of the perimeter. This is especially true in cases such as in 16 where the bridge contains an odd number of n centers and possesses non- bonding orbitals. Excitations to or from these additional orbitals cannot of course be classified in terms of the unperturbed perimeter transitions.
Cross-linking is defined as the introduction of n bonds between nonadja- cent centers in the perimeter, producing, for example, naphthalene (14) and azulene (15) from [IOIannulene; 2-electron and 0-electron heteroatom re- placement refers to the replacement of C@ by &, -NR-, -S, or a similar group with two n electrons, and the replacement of C@ by -BH- or a similar heteroatom with no n electron in its p, AO, respectively, as il- lustrated in 18 and 19.
Examples of a 1-electron heteroatom replacement are the replacements of =CH- by d B - , 4@-, =NHQ-, =N- or a similar group with one n electron in its p, AO. (Cf. 20 and 21.)
Substitution finally refers to the replacement of an H atom by an element from the second or lower row of the periodic table. (Cf. 22 and 23.) Since all such atoms possess an A 0 of p, symmetry, substitution causes an increase in size of the n system, either by conjugation or by hyperconjugation. For some substituents this n effect may be negligible. (Cf. Section 2.4.1.)
All these structural changes may be discussed by applying the concepts of PMO theory (Dewar and Dougherty, 1975). that is, by means of pertur- bation treatments based on the HMO approximation. From first-order per- turbation theory it is seen that the introduction of a resonance integral /&
between the perimeter atoms Q and cross-linking) or varying the Coulomb
integral &aP of the atom Q (heteroatom replacement) modifies the energy E~
of the i-th perimeter orbital by
The change in the orbital energy is proportional to the product ccoi or to the square <. of the LCAO coefficients, respectively. The effect of introduc- ing a bond between different fragments R and S (bridging or substitution) on the energy E~ of the i-th perimeter orbital is approximated by the second- order expression
[ c;c;/3@,
S E ~ = 2 2
k q - o (E? - E;) I
as long as the orbitals c#f and ~ are not degenerate.
For applications of the PMO method it is convenient to use real combi- nations (1/2)G(fpk + @-,) and (i/2)G(@, - @-,) instead of the complex perimeter MOs given in Equation (2.2). For (4N+ 2)-electron perimeters the real MOs of the degenerate pair of HOMOs are
@a = vz 2 P (sin y)xp
where the sum over p runs from 0 to n- 1; that is, the perimeter atoms p and the AOs X, are numbered 0, 1, . . . , n - 1. The MO @a possesses a nodal plane perpendicular to the molecular plane and passing through the center p = 0; it is antisymmetric with respect to this nodal plane. The MO @, is orthogonal to @a and is symmetric with respect to the plane of symmetry.
The corresponding MOs of the degenerate LUMO pair are*
= d~ 2 (sin
Ic 2n(N n + l)p)~P
* In order to be consistent with the nomenclature introduced in Section 1.2.4 for the MOs o f alternant hydrocarbons and in order to avoid confusion with the angular momentum quantum numbers of the perimeter MOs qiL and qi ,. the original notation C#J ., and C # J . for
the LUMOs (Michl. 1978. 1984) has been changed.
90 2 ABSOKt"f1ON SPECTKA OF ORGANIC MOLECULES 2.2 CYCLIC CONJUGATED n SYSTEMS 9 1 Example 2.7:
The location of the nodal planes and hence the signs of the LCAO coefficients of real perimeter MOs obtained from the complex MOs @A and @_, may be determined quite easily by the following procedure. A polygon with 2k vertices is incribed into the perimeter in such a way that it sits on a vertex in the case of the MOs @a and and that it lies on its side in the case of the MOs GS and
@,.. Lines joining opposite vertices of the polygon identify the location of nodal planes and indicate the regions of the perimeter where the LCAO coefficients have the same sign and the locations where sign changes occur. The absolute magnitude of the coefficient cPi is derived by observing how close the vertex p that represents the atom in question lies to the nearest nodal plane of the MO in question. The smaller the distance, the smaller the coefficient. This is illus- trated in Figure 2.14, where the frontier orbitals of benzene and of the [glannulene dianion are shown together with the HMO coefficients. As usual, the diameter of the circles is proportional to the magnitude of the coefficients and different colors indicate different signs.
Figure 2.14. Frontier orbitals of benzene and [Ilannulene dianion. The nodal planes
(--.) are obtained from the polygons with 2k vertices inscribed into the perimeter, and the magnitude of the LCAO coefficients (indicated by the size of the circles) may be estimated from the location of the nodal planes.
Uncharged perimeters with 4N + 2 n electrons are alternant hydrocarbons and the pairing theorem (cf. Section 1.2.4) applies; @s and @,, are paired, as are @a and their coefficients are equal on all odd atoms p, c,,,. = c,, and c,,,. = cPar respectively, and they are equal in magnitude but opposite In sign
on all even atoms y, c,,. = - c,,, and c,,, = - c,,. Due to this relation, those perturbations that are proportional to the product c,c, of LCAO coefficients according to Equation (2.13) can be divided in two classes: even perturba- tions, for which the sum g + a is even (this includes the case Q = a), and odd perturbations, for which g + a is odd (Moffitt, 1954a).
The following regularities are useful in applying PMO arguments:
When a perturbation is even, as for heteroatom replacements or cross- links that introduce odd-membered rings, c,c, = ceS,cmt and c,c, = c,.c,,. Hence, @, and @,. are either both stabilized or both destabilized by the same amount, and the same is true for @a and It follows that for an even perturbation AHOMO = ALUMO to the first order and that the orbital order must be either @,, @a, @,,, @a, or @,, @,, @,., @,.. The relative phase angle q, = a - /3 has the value q, = a. (Cf. Example 2.5.)
However, when the perturbation is odd, as for cross-links that introduce
pa m.. Thus, even-membered rings, then cesc, = -c,.c,, and c,c, = - c ,c
if @, is stabilized, @,. is destabilized by the same amount, and if @, is de- stabilized, @st is stabilized by the same amount. A similar relation holds between @, and It follows that for an odd perturbation AHOMO = ALUMO, too. This is fulfilled exactly within the HMO and PPP models since an odd perturbation does not remove the property of alternancy. The orbital order is either @,, @a, @,., @,, or @,, @,., @,., and the relative phase angle can only have the value g, = 0.
An example is shown in Figure 2.15 where an even perturbation (B,,) pro- duces the orbitals of azulene and an odd perturbation (&,) those of naphtha- lene from the perimeter orbitals of cyclodecapentaene. (Cf. Example 2.5.) The energies of @a and @at are not affected to the first order by the odd perturbation producing naphthalene, whereas @, is stabilized and 9,. desta- bilized. Therefore, becomes the HOMO and @a. the LUMO, and the HOMO-LUMO splitting AEHo,o~,,,o is the same as for cyclodecapentaene.
The HOMO-+LUMO transition is referred to as 'L, according to Platt. In azulene, on the other hand, @a is destabilized and becomes the HOMO, whereas @,, is stabilized and becomes the LUMO. The HOMO-LUMO splitting is markedly smaller than for cyclodecapentaene, and the HOM-LUMO transition is of the 'L, type.
For a mixed even-odd perturbation one has in general AHOMO # ALUMO, with either AHOMO > ALUMO or AHOMO < ALUMO. The effects of n substituents and charged bridges with an odd number of centers (e.g., C@ or C@) can be viewed as a superposition of an even and an odd perturbation, so AHOMO f ALUMO in these cases.
The fact that due to symmetry th'ere is no mixing between the L states in charged perimeters with 4N+ 2 electrons also has consequences for systems derived from these charged perimeters. Even after introducing the pertur- bation the polarization directions of the 'L, and 'L, transitions as well as those of the 'Bl and 'B, transitions are mutually perpendicular, and the split- ting of the 'L bands is only small. Important examples are indole derivatives
2 ABSORPTION SPECTRA OF ORGANIC MOLECULES 2.2 CYCLIC CONJUGATED n SYSTEMS 93
Figure 2.15. The real form of the four frontier MOs of [lO]annulene. The double- headed arrows indicate the cross-links that produce naphthalene (black arrows) and azulene (white arrows), respectively. First-order shifts are shown as vertical arrows on the orbital energy levels in the center (by permission from Michl, 1984).
(cf. 18), which can be produced formally from the [9]annulene anion by one cross-link and one 2-electron heteroatom replacement.