chimica inorganica 3. 2018... · chimica inorganica 3 hückel theory in the 1930's a theory...

Chimica Inorganica 3

A common approximation employed in the construction of molecular orbitals (MOs) is the linear combination of atomic orbitals (LCAOs). In the LCAO method, the kth

molecular orbital, ψk, is expanded in an atomic orbital basis,

ψ k = cii=1

n

∑ φi

Hψ k = Eψ k

H -E ψ k = 0

H -E caφa + cbφb ++ cnφn = 0

Left-multiplying by each φi yields a set of i linear homogeneous equations

φi are normalized atomic wavefunctions and ∫φiφidτ= 1. Solving Schrödinger’s equation and substituting for ψk yields,


ca φa H -E φa + cb φa H -E φb ++ cn φa H -E φn = 0

ca φb H -E φa + cb φb H -E φb ++ cn φb H -E φn = 0

ca φn H -E φa + cb φn H -E φb ++ cn φn H -E φn = 0

Solving the secular determinant

H aa -ESaa H ab -ESab H ac -ESac H an -ESanHba -ESba Hbb -ESbb Hbc -ESbc Hbn -ESbnH ca -ESca H cb -EScb H cc -EScc H cn -EScn

Hna -ESna Hnb -ESnb Hnc -ESnc Hnn -ESnn

= 0


Hückel Theory

In the 1930's a theory was devised by Hückel to treat the π electrons of conjugated systems such as aromatic hydrocarbon systems, benzene and naphthalene. Only π electron MO's are included because these determine the general properties of these molecules and the s electrons are ignored. This is referred to as σ-π separability. The extended Hückel Theory introduced by Lipscomb and Hoffmann (1962) applies to all the electrons.

Erich Armand Arthur Joseph Hückel (Berlino, 09/08/1896 – Marburgo, 16/02/1980)


H ii = φi∫ Hφidτ ; H ij = φi∫ Hφ jdτ ;

Sii = φi∫ φidτ ; Sij = φi∫ φ jdτ

In the Hückel approximation,

H ii =αH ij = 0 for φi not adjacent to φ j

H ij = β for φi adjacent to φ j Sii =1Sij = 0

The foregoing approximation is the simplest. Different computational methods treat these integrals differently. Extended Hückel Theory (EHT) includes all valence orbitals in the basis (as opposed to the highest energy atomic orbitals), all Sijs are calculated, the Hiis are estimated from spectroscopic data (as opposed to a constant, α) and Hijs are estimated from a simple function of Sij, Hii and Hjj.


The EHT (and other Hückel methods) are termed semi–empirical because they rely on experimental data for quantification of parameters. Other semi-empirical methods include CNDO, MINDO, INDO, etc. in which more care is taken in evaluating Hij (these methods are based on self-consistent field procedures). Still higher level computational methods calculate the pertinent energies from first principles – ab initio and DFT. Here core potentials must be included and high order basis sets are used for the valence orbitals.

As an example of the Hückel method, we will examine the frontier orbitals (i.e. determine eigenfunctions) and their associated orbital energies (i.e. eigenvalues) of benzene. The highest energy atomic orbitals of benzene are the C 2pπ orbitals. Hence, it is reasonable to begin the analysis by assuming that the frontier MO’s will be composed of LCAO of the C 2pπ orbitals:


φ1

φ2 φ3 φ4

φ5φ6


The matrix representations for this orbital basis in D6h is,

E ⋅

φ1φ2φ3φ4φ5φ6

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

φ1φ2φ3φ4φ5φ6

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

φ1φ2φ3φ4φ5φ6

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

χ E( ) = 6

φ1

φ2 φ3 φ4

φ5φ6


C6 ⋅

φ1φ2φ3φ4φ5φ6

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

0 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 11 0 0 0 0 0

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

φ1φ2φ3φ4φ5φ6

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

φ2φ3φ4φ5φ6φ1

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

χ C6( ) = 0

C2' ⋅

φ1φ2φ3φ4φ5φ6

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

−1 0 0 0 0 00 0 0 0 0 −10 0 0 0 −1 00 0 0 −1 0 00 0 −1 0 0 00 −1 0 0 0 0

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

φ1φ2φ3φ4φ5φ6

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

=

φ1φ6φ5φ4φ3φ2

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

χ C2( ) = −2

The only orbitals that contribute to the trace are those that transform into +1 or –1 themselves (i.e. in phase or with opposite phase, respectively). Thus the trace of the remaining characters of the 2pπ basis may be determined by inspection:


Decomposition of reducible representations may be accomplished with the following relation:

Returning to the above example,

aA1g =124

6 ⋅1⋅1+ 0 + 0 + 0 + −2( ) ⋅ 1( ) ⋅ 3( )+ 0 + 0 + 0 + 0 + −6( ) ⋅ 1( ) ⋅ 1( )+ 2 ⋅1⋅3+ 0⎡⎣ ⎤⎦ = 0

Continuing the procedure, one finds,

Γ pπ= A2u + B2g + E1g + E2u


With symmetries established, LCAOs may be constructed by “projecting out” the appropriate linear combination. A projection operator, P(i), allows the linear combination of the ith irreducible representation to be determined,

A drawback of projecting out of the D6h point group is the large number of operators. The problem can be simplified by dropping to the pure rotational subgroup, C6. In this point group, the full extent of mixing among φ1 through φ6 is maintained; however, the inversion center, and hence u and g symmetry labels are lost. Thus in the final analysis, the Γis in C6 will have to be correlated to those in D6h. Reformulating in C6,


http://www.staff.ncl.ac.uk/j.p.goss/symmetry/D6h2_cor.html


The projection of the SALC that from φ1 transforms as A is,


Continuing,

The projections contain imaginary components; the real component of the linear combination may be realized by taking ± linear combinations:

For ψ(E1a) SALC’s:


the E1a LCAO’s reduce to (again ignoring the constant prefactor),

ψ 3 E1( ) =ψ 3' E1a( ) +ψ 4

' E1b( ) = 2φ1 +φ2 −φ3 − 2φ4 −φ5 +φ6

ψ 4 E1( ) =ψ 3' E1a( )−ψ 4

' E1b( ) = i 3 φ2 +φ3 −φ5 −φ6( ) ∼ φ2 +φ3 −φ5 −φ6

ε = ei 2π6 = cos 2π

6 + isin 2π6

ε + ε * = cos 2π6 + isin 2π

6 + cos 2π6 − isin 2π

6 = 2cos 2π6 = 1

ε − ε * = cos 2π6 + isin 2π

6 − cos 2π6 + isin 2π

6 = 2isin 2π6 = i 3

ε * − ε = cos 2π6 − isin 2π

6 − cos 2π6 − isin 2π

6 = −2isin 2π6 = −i 3


ψ 1 A( ) = 16

φ1 +φ2 +φ3 +φ4 +φ5 +φ6( ) ψ 2 B( ) = 16

φ1 −φ2 +φ3 −φ4 +φ5 −φ6( )

ψ 3 E1a( ) = 112

2φ1 +φ2 −φ3 − 2φ4 −φ5 +φ6( ) ψ 4 E1b( ) = 12φ2 +φ3 −φ5 −φ6( )

ψ 5 E2a( ) = 112

2φ1 −φ2 −φ3 + 2φ4 −φ5 −φ6( ) ψ 4 E2b( ) = 12φ2 −φ3 +φ5 −φ6( )

φi∫ φidτ = Sii =1; φi∫ φ jdτ = Sij = 0

Within the assumption that

The normalization constant N is

N = 1

ci2

i=1

n

∑


The pictorial representation of the SALC’s are,

The energies (eigenvalues) may be determined by using the Hückel approximation

ψ(B2g)

ψ(E1g)ψ(E1g)

ψ(E2u)ψ(E2u)

ψ(A2u)


ψ A2u= 1

6φ1 +φ2 +φ3 +φ4 +φ5 +φ6( )

E A2u( ) = ψ A2uH ψ A2u

=

16

φ1 +φ2 +φ3 +φ4 +φ5 +φ6( ) H 16

φ1 +φ2 +φ3 +φ4 +φ5 +φ6( ) =

16

H11α

+H12β

+H13 +H14 +H15 +H16 +β

H21β

+H22α

+H23β

+H24 +H25 +H26 +

H3i i =1− 6( )α+2β

+H4i i =1− 6( )α+2β

+H5i i =1− 6( )α+2β

+H6i i =1− 6( )α+2β

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

16⋅6 ⋅ α + 2β( ) =α + 2β

1 2

5

3

4

6


ψ B2 g= 1

6φ1 −φ2 +φ3 −φ4 +φ5 −φ6( )

E B2g( ) = ψ B2 gH ψ B2 g

=

16

φ1 −φ2 +φ3 −φ4 +φ5 −φ6( ) H 16

φ1 −φ2 +φ3 −φ4 +φ5 −φ6( ) =

16

H11α

−H12β

+H13 +H14 +H15 −H16 +β

H2i i =1− 6( )α−2β

+

H3i i =1− 6( )α−2β

+H4i i =1− 6( )α−2β

+H5i i =1− 6( )α−2β

+H6i i =1− 6( )α−2β

+

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

16⋅6 ⋅ α − 2β( ) =α − 2β

E ψE1ga( ) = E ψ

E1gb( ) =α + β

E ψE2ua( ) = E ψ

E2ub( ) =α − β

Note the energies of the E orbitals are degenerate. Constructing the energy level diagram, we set α = 0 and β as the energy parameter (a negative quantity, so an MO whose energy is positive in units of β has an absolute energy that is negative),


The energy of benzene based on the Hückel approximation is

ETot = 2 2β( )+ 4β = 8β


What is the delocalization energy (i.e. π resonance energy)?

To determine this, we consider cyclohexatriene, which is a six-membered cyclic ring with 3 localized π bonds; in other terms, cyclohexatriene is the product of three condensed ethylene molecules. For ethylene,

ψ 1 A( ) = 12

φ1 +φ2( )

ψ 2 B( ) = 12

φ1 −φ2( )

E ψ 1( ) = 12

φ1 +φ2( ) H 12

φ1 +φ2( ) = 122α + 2β( ) =α + β

E ψ 2( ) = 12

φ1 −φ2( ) H 12

φ1 −φ2( ) = 122α − 2β( ) =α − β

The above was determined in the C2 point group. Correlating to D2h point group gives A in C2 → B1u in D2h and B in C2 → B2g in D2h:


The Hückel energy of ethylene is,

ETot = 2β

The energy of cyclohexatriene is then 3(2β) = 6β. The resonance energy is therefore,

ERes C6H6( ) = 8βETot (benzene )

− 6βETot (cyclohexatriene )


The bond order is given by,

Consider the B.O. between the C1 and C2 carbons of benzene

ψ 1 A2u( )⎡⎣ ⎤⎦ = 2 ⋅16

⎛⎝⎜

⎞⎠⎟⋅ 1

6⎛⎝⎜

⎞⎠⎟= 13

ψ 3 E1g

a( )⎡⎣

⎤⎦ = 2 ⋅

212

⎛⎝⎜

⎞⎠⎟⋅ 1

12⎛⎝⎜

⎞⎠⎟= 13

ψ 4 E1g

b( )⎡⎣

⎤⎦ = 2 ⋅ 0( ) ⋅ 1

2⎛⎝⎜

⎞⎠⎟ = 0

Electronic Population Analysis on LCAO-MO Molecular Wave Functions. I Mulliken, R. S. J. Chem. Phys. 1955, 23, 1833–1840

chimica inorganica 3. 2018... · chimica inorganica 3 hückel theory in the 1930's a theory...

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