1. MODELING MATTER AT NANOSCALES
5. The theory of molecular orbitals for the description of nanosystems (part I)
Hückel Molecular Orbitals

2. Linear combination of Slater pz orbitals

3. The first quantum approach to nanoscales
Erick Hückel stated in 1931 that the a linear combination of atomic orbitals (LCAO), taken as pz hydrogenoid functions allow the calculation of electronic states or molecular electronic wave functions p of planar conjugated hydrocarbons. E. Hückel, Z. Physik, 70, 204 (1931); 76, 628 (1932)

4. The first quantum approach to nanoscales
Such consideration is based on the quasi orthogonality between the s backbone of a planar conjugated molecule and the p electronic cloud.

5. The first quantum approach to nanoscales
Such consideration is based on the quasi orthogonality between the s backbone of a planar conjugated molecule and the p electronic cloud.
The allyl (or isopropyl) radical structure is illustrative:
c1 c2 c3

6. The first quantum approach to nanoscales
The “chemical” approach to bonding in this molecular system is truly challenging.

7. The first quantum approach to nanoscales
The linear combination of atomic monoelectronic states (orbitals) would be expressed as:
where cim coefficients provide the participation of each pz atomic orbital c (as an algebraic basis) in the p molecular orbital corresponding to the ith. monoelectronic state of the whole system.

8. The first quantum approach to nanoscales
The linear combination of atomic monoelectronic states (orbitals) would be expressed as:
where cim coefficients provide the participation of each pz atomic orbital c (as an algebraic basis) in the p molecular orbital corresponding to the ith. monoelectronic state of the whole system.
It is a representation of a molecular state in terms of atomic states.

9. The first quantum approach to nanoscales
The convention for HMO includes that only one electron in a 2pz atomic orbital participates in bonding, even in the case of a multielectronic atom, like carbon.

10. The first quantum approach to nanoscales
The convention for HMO includes that only one electron in a 2pz atomic orbital participates in bonding, even in the case of a multielectronic atom, like carbon.
It is considered sufficient for describing the most labile and chemically active p electron cloud of planar conjugated compounds.

11. The first quantum approach to nanoscales
The convention for HMO includes that only one electron in a 2pz atomic orbital participates in bonding, even in the case of a multielectronic atom, like carbon.
It is considered sufficient for describing the most labile and chemically active p electron cloud of planar conjugated compounds.
It means that the atomic nucleus and the resulting charge of each atom are surrogated as a non polarizable core consisting in the nucleus together with all electrons except that, or those, in the atomic 2pz orbital participating in the p cloud.

12. The system’s matrix in terms of an atomic basis
The allyl radical system’s matrix is built as:
and then the problem is reduced to evaluate Hmn atomic terms or matrix elements of the molecule to be diagonalized and thus obtaining eigenvalues and eigenvectors, being accounted that the matrix is symmetrical.

13. “Simple” Hückel Molecular Orbitals (HMO)
In the so called “simple” molecular orbital theory of Hückel (Hückel Molecular Orbitals or HMO) several new considerations must be taken into account to simplify and evaluate the Hmn matrix elements:
1.- Hmn terms corresponding to non adjacent atoms are taken as zero.

14. “Simple” Hückel Molecular Orbitals (HMO)
2.- Diagonal Hmm terms can be considered as evaluable by pz valence state ionization potentials of a carbon atom. Eventually they are substituted by a a parameter (named by Hückel himself as the “Coulomb integral”) that is related with the corresponding element.

15. “Simple” Hückel Molecular Orbitals (HMO)
3.- Non diagonal Hmn matrix elements between adjacent atoms are treated as fixed parameters corresponding to each interaction between atoms or centers, named as b (named by Hückel himself as “resonance integral”).

16. “Simple” Hückel Molecular Orbitals (HMO)
The equation or determinant to diagonalize would be, then:
|H – E| = 0

17. “Simple” Hückel Molecular Orbitals (HMO)
The equation or determinant to diagonalize would be, then:
|H – E| = 0
It must be observed that diagonalizing this matrix will also give the transformation matrix C, i.e. the LCAO’s coefficient or “eigen vector” matrix.

18. “Simple” Hückel Molecular Orbitals (HMO)
Diagonal matrix E contains all three roots or solutions of the secular equation.

19. “Simple” Hückel Molecular Orbitals (HMO)
Diagonal matrix E contains all three roots or solutions of the secular equation. They are the eigenvalues of the system’s or molecular p monoelectronic states corresponding with molecular orbitals built from pz atomic orbitals.

20. “Simple” Hückel Molecular Orbitals (HMO)
If we consider a certain reference value b0 for all conjugate interactions between carbon atoms, this value can normalize all terms in the secular determinant creating a new g variable and a kmn new bonding parameter:

21. “Simple” Hückel Molecular Orbitals (HMO)
If we consider a certain reference value b0 for all conjugate interactions between carbon atoms, this value can normalize all terms in the secular determinant creating a new g variable and a kmn new bonding parameter:
Then, secular determinant becomes:

22. “Simple” Hückel Molecular Orbitals (HMO)
In the case of hydrocarbons and certain other molecules bmn = b0 and then:

23. The case of propyl structure
Then, the case of propyl radical illustrates an HMO secular determinant becoming a topological image of a molecular structure.

24. The case of propyl structure
Then, the case of propyl radical illustrates an HMO secular determinant becoming a topological image of a molecular structure.
Diagonal terms are the unknown variables and non – diagonal terms remain as 1 if there are adjacent bonding or 0 if not.

25. The case of propyl structure
Diagonalizing the case of propyl, i.e. finding the solutions for variables and coefficients, results:
i state gi Ei c1 c2 c3 1
-1.414
a b0
0.5
0.707 2 a
-0.707 3
1.414
a b0
where:

26. Energy calculations
Total electronic energy
ni is the ccupation number of the i molecular orbital.
N is the total number of states or molecular orbitals.

27. Energy calculations
Total electronic energy
ni is the ccupation number of the i molecular orbital.
N is the total number of states or molecular orbitals.
m is the number of double bonds in the molecule.
is ethene energy as calculated with the same parameters.
Resonance Energy

28. Energy calculations
Total electronic energy
ni is the ccupation number of the i molecular orbital.
N is the total number of states or molecular orbitals.
m is the number of double bonds in the molecule.
is ethene energy as calculated with the same parameters.
l is the index of an empty orbital and i the index of the occupied.
Resonance Energy
Excitation energy

29. Energy calculations
Allyl system brings three filling possibilities at energies available in a chemical environment: an anion 1[C2H5]- with 4 electrons, a radical 2[C2H5]• with 3 electrons and a cation 1[C2H5]+ with 2 electrons.

30. Energy calculations Energies of each corresponding filling level are:
C3H5-
C3H5•
C3H5+
Eelect
4a+2.828b0
3a+2.828b0
2a+2.828b0

31. Energy calculations Energies of each corresponding filling level are:
C3H5-
C3H5•
C3H5+
Eelect
4a+2.828b0
3a+2.828b0
2a+2.828b0
It must be observed that:
Either the intrinsic value of a as that of b0 are negative, and therefore their corresponding energies are also essentially negative.

32. Energy calculations Energies of each corresponding filling level are:
C3H5-
C3H5•
C3H5+
Eelect
4a+2.828b0
3a+2.828b0
2a+2.828b0
It must be observed that:
Either the intrinsic value of a as that of b0 are negative, and therefore their corresponding energies are also essentially negative.
As the eigenvalue of the i = 2 MO is zero, this crude approximation ignores the contribution of that orbital to the total energy of the system, either occupied or not.

33. Energy calculations Energies of each corresponding filling level are:
C3H5-
C3H5•
C3H5+
Eelect
4a+2.828b0
3a+2.828b0
2a+2.828b0
It must be observed that:
Either the intrinsic value of a as that of b0 are negative, and therefore their corresponding energies are also essentially negative.
As the eigenvalue of the i = 2 MO is zero, this crude approximation ignores the contribution of that orbital to the total energy of the system, either occupied or not.
Occupation only influences multiples of a.

34. Charges and Chemical Bond
If a molecular orbital fi has a ni occupation number, the part of those electrons which are present in the neighborhood of the cm basis orbital must be a certain integration of the electronic cloud around that center:

35. Charges and Chemical Bond
If such integral is developed in terms of the LCAO and it is accounted that the basis is orthogonal, then:

36. Charges and Chemical Bond
If such integral is developed in terms of the LCAO and it is accounted that the basis is orthogonal, then:
Also, for the case of any m center:

37. Charges and Chemical Bond
If such integral is developed in terms of the LCAO and it is accounted that the basis is orthogonal, then:
Also, for the case of any m center:
Then:

38. Charges and Chemical Bond
If such integral is developed in terms of the LCAO and it is accounted that the basis is orthogonal, then:
Also, for the case of any m center:
Then:
Therefore, the sum of all molecular orbital contributions on center m, is the p charge density and remains as:

39. Charges and Chemical Bond
We can define a P density matrix as depending from molecular wave functions expressed by the coefficients, which terms are given by:

40. Charges and Chemical Bond
We can define a P density matrix as depending from molecular wave functions expressed by the coefficients, which terms are given by:
Diagonal terms of this expression are coincident with the previous p charge density:

41. Charges and Chemical Bond
We can define a P density matrix as depending from molecular wave functions expressed by the coefficients, which terms are given by:
Diagonal terms of this expression are coincident with the previous p charge density:
The remaining pmn (where m  n) off diagonal terms of the density matrix expression in the simple Hückel theory are known as p bond orders.

42. Charges and Chemical Bond
Working with the total energy expression in terms of P:
where charge densities and bond orders become explicit.

43. Charges and Chemical Bond
As the working condition has been that the molecular orbital basis set (the atomic orbitals) is orthonormal, as well as themselves, then non diagonal pmn terms mean nothing regarding charge, i.e. electron densities, because ALL charge is expressed in the diagonal of matrix P ≡[ pmn]:

44. Charges and Chemical Bond
As the working condition has been that the molecular orbital basis set (the atomic orbitals) is orthonormal, as well as themselves, then non diagonal pmn terms mean nothing regarding charge, i.e. electron densities, because ALL charge is expressed in the diagonal of matrix P ≡[ pmn]:
However, departing from the previous energy expression:

45. Charges and Chemical Bond
As the working condition has been that the molecular orbital basis set (the atomic orbitals) is orthonormal, as well as themselves, then non diagonal pmn terms mean nothing regarding charge, i.e. electron densities, because ALL charge is expressed in the diagonal of matrix P ≡[ pmn]:
However, departing from the previous energy expression:
It can be understood that bond orders (cases when m ≠n) are related with contributions of the cm and cn atomic orbital interaction probability to the total energy of the system.

46. Charges and Chemical Bond
Summarizing:
p Bond order

47. Charges and Chemical Bond
Summarizing:
p Bond order
p Charge density

48. Charges and Chemical Bond
Summarizing:
p Bond order
p Charge density
p Formal charge
p formal charge expresses the remaining electron charge on a given center and serves to appreciate how positive or negative it becomes after the molecular wave function calculation.

49. Charges and Chemical Bond1 2 3 gi
-1.414
1.414
pmn Ni
n=1
n=2
n=3
ci1
0.5
0.7071
m =1
1.500
0.707
-0.500
ci2
m =2
1.000
ci3
m =3

50. Charges and Chemical Bond1 2 3 gi
-1.414
1.414
pmn Ni
n=1
n=2
n=3
ci1
0.5
0.7071
m =1
1.500
0.707
-0.500
ci2
m =2
1.000
ci3
m =3
C3H5· i 1 2 3 gi
-1.414
1.414
pmn ni
n=1
n=2
n=3
ci1
0.5
0.7071
m =1
1.000
0.707
0.000
ci2
m =2
ci3
m =3

51. Charges and Chemical Bond1 2 3 gi
-1.414
1.414
pmn Ni
n=1
n=2
n=3
ci1
0.5
0.7071
m =1
1.500
0.707
-0.500
ci2
m =2
1.000
ci3
m =3
C3H5· i 1 2 3 gi
-1.414
1.414
pmn ni
n=1
n=2
n=3
ci1
0.5
0.7071
m =1
1.000
0.707
0.000
ci2
m =2
ci3
m =3
C3H5+ i 1 2 3 gi
-1.414
1.414
pmn ni
n=1
n=2
n=3
ci1
0.5
0.7071
m =1
0.500
0.707
ci2
m =2
1.000
ci3
m =3

52. Bond distances
An interesting empirical relationship exists between interatomic distances of conjugated carbon atoms and bond orders. The interpolated result is:
R. Daudel, R. Lefebvre, and C. Moser, Quantum Chemistry. Methods and Applications. (Interscience Publishers, Inc., New York, 1959).

53. Bond distances
An interesting empirical relationship exists between interatomic distances of conjugated carbon atoms and bond orders. The interpolated result is:
There are certain analytical expressions, too:
For trigonal C atoms: rmn = 1.52 – 0.19 pmn
For digonal C atoms: rmn = 1.45 – 0.12 pmn
R. Daudel, R. Lefebvre, and C. Moser, Quantum Chemistry. Methods and Applications. (Interscience Publishers, Inc., New York, 1959).
A. Julg, Chimie Quantique. (Dunod, Paris, 1967).

54. Bond distances
By applying these relationships, the sole bond order found for all three bonds in propyl testing molecule give an equal bond distance of Å, independently of the valence forms.

55. Bond distances
By applying these relationships, the sole bond order found for all three bonds in propyl testing molecule give an equal bond distance of Å, independently of the valence forms.
Electron diffraction experimental unique bond distance in gas phase is ± Å for the radical.
- DE: E. Vajda, J. Tremmel, B. Rozsondai, I. Hargittai, A. K. Maltsev, N. D. Kagramanov, and O. M. Nefedov, J. Am. Chem. Soc. 108 (15), 4352 (1986).

56. Bond distances
By applying these relationships, the sole bond order found for all three bonds in propyl testing molecule give an equal bond distance of Å, independently of the valence forms.
Electron diffraction experimental unique bond distance in gas phase is ± Å for the radical.
Reliable theoretical results give Å for the radical, Å for the anion and Å for the cation.
- DE: E. Vajda, J. Tremmel, B. Rozsondai, I. Hargittai, A. K. Maltsev, N. D. Kagramanov, and O. M. Nefedov, J. Am. Chem. Soc. 108 (15), 4352 (1986).
- MP2 | 6-31G* Geometry optimization calculations

57. Case of conjugated heteroatoms
In the case of a certain X heteroatom (non carbon):
aX = a0 + hXb0 where a0 is a reference monocentric parameter, as well as b0 and the new hx parameter can be obtained by several ways:
F. L. Pilar, Elemmentary Quantum Chemistry. (McGraw-Hill Book Company, New York, 1968).

58. Case of conjugated heteroatoms
In the case of a certain X heteroatom (non carbon):
aX = a0 + hXb0 where a0 is a reference monocentric parameter, as well as b0 and the new hx parameter can be obtained by several ways:
Where I are valence state ionization potentials of the reference (I0) and the heteroatom (IX).
F. L. Pilar, Elemmentary Quantum Chemistry. (McGraw-Hill Book Company, New York, 1968).

59. Case of conjugated heteroatoms
In the case of a certain X heteroatom (non carbon):
aX = a0 + hXb0 where a0 is a reference monocentric parameter, as well as b0 and the new hx parameter can be obtained by several ways:
Where I are valence state ionization potentials of the reference (I0) and the heteroatom (IX).
Where c are electronegativities of the reference (c0) and the heteroatom (cX).
F. L. Pilar, Elemmentary Quantum Chemistry. (McGraw-Hill Book Company, New York, 1968).

60. Case of conjugated heteroatoms
It is easy to arrive to:
F. L. Pilar, Elemmentary Quantum Chemistry. (McGraw-Hill Book Company, New York, 1968).

61. Case of conjugated heteroatoms
It is easy to arrive to:
It means that the new diagonal term for the heteroatom is g0 + hX and g0 remains as the variable containing the energy term.
F. L. Pilar, Elemmentary Quantum Chemistry. (McGraw-Hill Book Company, New York, 1968).

62. Case of conjugated heteroatoms
Thus, the most general form for the secular equation (determinant) for a planar conjugated systems of N terms (including conjugated heteroatoms) according to the simple Hückel theory would be:

63. Case of conjugated heteroatoms
Thus, the most general form for the secular equation (determinant) for a planar conjugated systems of N terms (including conjugated heteroatoms) according to the simple Hückel theory would be:
Where:
hm is zero for a reference atom, as carbon is, and takes a value different than zero in case of most heteroatoms that behave significantly different to carbon.

64. Case of conjugated heteroatoms
Thus, the most general form for the secular equation (determinant) for a planar conjugated systems of N terms (including conjugated heteroatoms) according to the simple Hückel theory would be:
Where:
hm is zero for a reference atom, as carbon is, and takes a value different than zero in case of most heteroatoms that behave significantly different to carbon.
kmn is one when two reference atoms, as carbons, are bonded and zero when there are no bond between them. It can take other values for heteroatoms.

65. Case of conjugated heteroatoms
Some possible ways to evaluate kmn are:
for carbon – carbon bonds where rmn is in Å
F. L. Pilar, Elementary Quantum Chemistry. (McGraw-Hill Book Company, New York, 1968); A. Julg, Chimie Quantique. (Dunod, Paris, 1967).

66. Case of conjugated heteroatoms
Some possible ways to evaluate kmn are:
for carbon – carbon bonds where rmn is in Å
kC=O = 1.1
kC-O = 0.6
kC-N: = 1.3
kC-F = 1
kC-Cl = 0.6
kC-Br = 0.4
F. L. Pilar, Elementary Quantum Chemistry. (McGraw-Hill Book Company, New York, 1968); A. Julg, Chimie Quantique. (Dunod, Paris, 1967).

67. Procedures
The great majority of simple HMO method input matrices for computer programs only require values for hm because g0 is considered as obviously present.

68. Examples
Molecules to try a HMO matrix: 1 2 3 4 5 6 7 8 9

69. Examples 10 11 12 13 14 15