Dr M. Mehrdad University of Guilan, Department of Chemistry, Rasht, Iran

MO theory considers the electrons in molecules to occupy MOs that are formed by linear combinations (addition and subtraction) of all the atomic orbitals on all the atoms in the structure. In MOT, electrons are not confined to an individual atom plus the bonding region with another atom. Instead, electrons are contained in MOs that are highly delocalized spread across the entire molecule. MOT is based on the Schrödinger equation. Ψ = EΨ : amiltonian operator Ψ: wavefunction describing an orbital E: the energy of an electron in a particular orbital obtain Ψ and this equation Ψ = Σc i φ i (linear combinations of all the atomic orbitals) c i = coefficient φ i = atomic orbital To construct group MO s, one needs to understand how to combine AO s properly. 2 This procedure is called qualitative molecular orbital theory(qmot)

Rules of QMOT 1. Consider valence orbitals only (e.g., for Carbon, 2s, 2p x, 2p y and 2p z ) 2. Form completely delocalized MO s as linear combinations of s and p AO s. Remember, combination of n AO s gives n MO s 3. MO s must be either symmetric or antisymmetric with respect to the symmetry operations of the molecule. 4. Compose MO s for structures of higher symmetry and then produce MOs for related, but less symmetric systems by systematic distortion of the MOs for higher symmetry. For example, for the C 2 system, start with linear C (D h ) then bend the system (C 2v ). 5. Molecules with similar molecular structures, e.g., C 3 and N 3, have qualitatively similar MO s, the major difference being the number of valence electrons that occupy the common MO system. 6. The total energy of the system is the sum of the MO energies of the individual valence electrons. ( occupied MO s) 7. If the two highest energy MO s of a given symmetry derive primarily from different kinds of AO s (e.g., s and p), then mix the two MO s to form hybrid orbitals. For example, for the A 2 system (p.3), mix C and E orbitals to form hybrid C and E. 3

8. When two orbitals interact, the lower energy orbital is stabilized and the higher energy orbital is destabilized. The out-of-phase or antibonding interaction between the two starting orbitals always raises the energy more than the corresponding in-phase or bonding interaction lowers the energy.. (energy of stabilization, e stab, is always smaller that energy of destabilization, e destab. Thus, 4electron-2center interaction is always repulsive. ) 4

9. When two orbitals interact, the lower energy orbital mixes into itself the higher energy one in a bonding way, whereas the higher energy orbital mixes into itself the lower energy one in an antibonding way. (If orbitals of different energy interact (b), the one of lower energy, B, will contribute more in binding orbital; the one of higher energy, A, will contribute more in antibonding orbital.) b) 5

10. The smaller the initial energy gap between the two interacting orbitals, the stronger the mixing interaction. 11. The larger the overlap between interacting orbitals, the larger the interaction. (σ-bonds are stronger than π-bonds.) 12. The more electronegative elements have lower energy AO s. 13. A change in the geometry of a molecule will produce a large change in the energy of a particular MO if the geometry change results in changes of AO overlap that are large. 14. The AO coefficients are large in high energy MO s with many nodes or complicated nodal surfaces. 15. Energies of orbitals of the same symmetry classification cannot cross each other. Instead such orbitals mix and diverge. 6

example 7

Symmetry Elements E: Identity operation عنصر یکسانی C n : Proper rotation محور چرخشی 3 C C 3 C 2 3 C C 3 C 6 C 2 9

Symmetry Elements i: Inversion عمل وارونگی Cl Cl Cl W Cl Cl Cl i Cl Cl Cl W Cl Cl Cl C 4 s h : orizontal Mirror Plane صفحه آیینه ای افقی s h C 2 s v : Vertical Mirror Plane صفحه آیینه ای عمودی Br Br s v Br Br 10

Symmetry Elements S n : Improper rotation: combination C n and s h عمل چرخشی انعکاسی S 2 is equivalent to inversion (i) S 4 3 4 C 4 4 3 s h 2 1 1 2 1 2 3 4 S 2 Me center of symmetry Me Me Me Me C 2 Me Me s h Me Me Me Me Me 11

Groups with no proper rotation axis C 1 : Only E (i.e. no symmetry elements) C s : E and s C i : E and i S n : E, S n (S 1 = C s ; S 2 = C i ) Groups with one proper rotation axis C n : E, C n only Symmetry Groups C nv : E, C n, and n s v (linear unsymmetrical molecules are C v ) C nh : E, C n, and s h Dihedral Groups: Groups with n C 2 axes to C n D n : E, C n, and n C 2 axes to C n D nh : E, C n, n C 2 axes, and s h (linear symmetrical molecules are D h ) D nd : E, C n, n C 2 axes, and n s v Cubic Groups: Groups with more than one C n (n 3) T d : symmetry of a regular tetrahedron: 4 C 3 O h : symmetry of a regular octagon: 6 C 4 I h : symmetry of a regular icosahedron: 12 C 5 12

Cl F C C 1 Br Br Br C s Br F Cl F Cl C i Br Br S 4 Br Br C 3 N O C S F C 3 C 3v C v F C 2h C 2 D 2 D 6h O C O D h D 3d O C O O C C C W C C C O O T d O O h I h 13

Symmetry Decision Tree C nh Yes No No s h? s v? C n C v or D h More than one C n (n 3) Cubic T, O, I No S 2n colinear w/ C n? Yes C nv Yes Yes Linear? No Yes Find principal axes C n is the principal axis? No nc 2 to C n? n vertical mirror planes No S 2n None Yes Yes C s, C i or C 1 s h? No s v? Yes D nd Yes No D nh D n Physical Chemistry, Joseph. Noggle, 2nd ed., Scott Foresman & Co, Glenview, IL, 1996, pg 840. 14

To illustrate the procedures of qualitative MO theory, we will build the MO s of planar C 3. We choose planar C 3 because it is more symmetrical. We will be using: - three 1s orbitals, - one C 2s orbital, - and three C 2p orbitals. 15

1. First, mix the carbon 2s orbital with the three AO s in-phase to produce orbital A which is symmetric with respect to the C 3 -axis of symmetry of the molecule. [Focus on the low-lying, bonding MO s, the orbitals of which are mixed in-phase (bonding).] Using out-of-phase mixing, one gets the high energy, antibonding orbital E. 16

2. Next, use the carbon p-orbitals. The p z AO cannot mix with any of the orbitals, since they all lie on the nodal plane of this orbital. We thus have an MO that is simply an atomic p-orbital, D. The p x and p y AO s can mix with the 1s orbitals of the atoms to give favorable interaction patterns, as seen in MO s B and C. B and C are degenerate of the same energy. 17

Orbitals A, B, C and D are the group orbitals for planar C 3. 18

MOs for Planar A 3 19

A diagram that follows orbital energies as a function of angular distortions is termed a Walsh diagram. Pyramidalization lowers the energy of orbital A slightly (slight - interaction); it raises the energies of B and C more because of the loss of overlap between the p orbitals and the hydrogen orbitals. 21

The biggest impact, however, is on orbital D. This orbital is non-bonding when planar, but becomes increasingly bonding upon pyramidalization. Orbitals A-C are strongly C- bonding, whether planar or pyramidal. In a VB model, we would want three C- bonds, each with two electrons, for a total of six C- bonding electrons. The two models agree. With QMOT, we still have three C- bonds, described by three occupied MOs that are strongly C- bonding. 22

Consistent with Rule 7, 7. If the two highest energy MO s of a given symmetry derive primarily from different kinds of AO s then mix the two MO s to form hybrid orbitals. MO s D and E, having the same symmetry, but one based on a carbon p orbital and the other using a 2s orbital, leads to mixing of these two orbitals to form D and E. D now looks more like a lone pair orbital, and it resembles an sp i hybrid at carbon. 23

Group orbitals a collection of partially delocalized orbitals that is consistently associated with a functional group or similar collection of atoms in a molecule. A, B, C, D (or D ) are the Group orbitals of the methyl group, and we can use these orbitals to model the bonding in any molecules that contain the methyl group. 1. Low-lying C- bonding orbitals derived from carbon 2s orbitals and of s-symmetry are termed s(c 3 ) orbitals. 2. The C- bonding orbitals that are derived from carbon 2p orbitals, of -symmetry, are termed (C3) orbitals. They are a degenerate pair. 3. The other orbital of s-symmetry, that derived from the carbon p z AO, points away from the s and is termed the s(out) orbital. http://www.chem.ox.ac.uk/vrchemistry/ orbitals/html/page11.htm 25