Dr M. Mehrdad - دانشگاه...
Transcript of Dr M. Mehrdad - دانشگاه...
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. HΨ = EΨ H: Hamiltonian operator Ψ: wavefunction describing an orbital E: the energy of an electron in a particular orbital
obtain Ψ and this equation Ψ = Σciφi (linear combinations of all the atomic orbitals) ci = coefficient φi = atomic orbital
To construct group MO’s, one needs to understand how to combine AO’s properly.
This procedure is called qualitative molecular orbital theory(QMOT) 2
Rules of QMOT
1. Consider valence orbitals only (e.g., for Carbon, 2s, 2px, 2py and 2pz) 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 CH2
system, start with linear HCH (D∞h) then bend the system (C2v). 5. Molecules with similar molecular structures, e.g., CH3 and NH3, 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 AH2
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, estab, is always smaller
that energy of destabilization, edestab. 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
Symmetry Elements E: Identity operation
عنصر یکسانیCn: Proper rotation
محور چرخشی
CH3
H
H
H3C
C2
H3C
H
H
CH3
H
H
H
H
H
H
H
H
H
H
H
HC6
H
H
H
H
H
H
C2
9
Symmetry Elements
i: Inversion
عمل وارونگی W
Cl
Cl Cl
Cl
Cl
Cl
i
WCl
Cl Cl
Cl
Cl
Cl
sh: Horizontal Mirror Plane
صفحه آیینه ای افقی
sv: Vertical Mirror Plane
صفحه آیینه ای عمودی
C4
H
H
H
H
H
H
H
H
sh H
H
H
H
H
H
H
H
Br Br
HH
C2
sv
Br Br
HH
10
Symmetry Elements Sn: Improper rotation: combination Cn and sh
عمل چرخشی انعکاسی S2 is equivalent to inversion (i)
Me
Me
Me
MeC2
Me
Me
Me
Mesh
Me
Me
Me
Me
S2
center of symmetry
H1 H2
H4H3C4
H4H3
H1 H2
sh
H3 H4
H1H2
S4
11
Symmetry Groups
• Groups with no proper rotation axis
– C1: Only E (i.e. no symmetry elements)
– Cs: E and s
– Ci: E and i
– Sn: E, Sn (S1 = Cs; S2 = Ci)
• Groups with one proper rotation axis
– Cn: E, Cn only
– Cnv: E, Cn, and n sv (linear unsymmetrical molecules are C∞v)
– Cnh: E, Cn, and sh
• Dihedral Groups: Groups with n C2 axes to Cn
– Dn: E, Cn, and n C2 axes to Cn
– Dnh: E, Cn, n C2 axes, and sh (linear symmetrical molecules are D∞h)
– Dnd: E, Cn, n C2 axes, and n sv
• Cubic Groups: Groups with more than one Cn (n ≥ 3)
– Td: symmetry of a regular tetrahedron: 4 C3
– Oh: symmetry of a regular octagon: 6 C4
– Ih: symmetry of a regular icosahedron: 12 C5
12
F
CCl Br
H
C1
H
Br BrBr
HH
Cs
HH
FCl
FCl
Ci
CH3
CH3
C2
NH H
H
C3v
H F
F H
C2h
D6h
H
CH H
H
Td
WC
C C
C
C
C
O
O
O
O
O
O
Oh
Ih
D2
S4
Br
Br Br
Br
HH
H
H
H
H HH
HH
D3d
13
O C S
C•v∞
O C O
D•h∞
Symmetry Decision Tree
Linear?
Find
principal
axes
C∞v or
D ∞h
More than one
Cn (n ≥ 3)
Cubic
T, O, I
Cs, Ci
or C1
Cn is the
principal
axis?
nC2
to Cn?
S2n colinear
w/ Cn?
n vertical
mirror
planes
S2n
sh? sv? Dnd
Dnh Dn
sh? sv? Cn
Cnv
Cnh
Yes
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes None
No
No
No
No No
No
No
Physical Chemistry, Joseph H.
Noggle, 2nd ed., Scott Foresman
& Co, Glenview, IL, 1996, pg 840.
Yes
14
To illustrate the procedures of qualitative MO theory, we will “build” the MO’s of planar CH3. We choose planar CH3 because it is more symmetrical.
We will be using: - three H 1s orbitals, - one C 2s orbital, - and three C 2p orbitals.
15
1. First, mix the carbon 2s orbital with the three H AO’s in-phase to
produce orbital A
which is symmetric with respect to the C3-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 pz AO cannot mix with any of the H 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 px and py AO’s can mix with the 1s orbitals of the H atoms to give
favorable interaction patterns, as seen in MO’s B and C.
B and C are degenerate – of the same energy. 17
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 H-H
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-H bonding, whether planar or pyramidal.
In a VB model, we would want three C-H bonds, each with two electrons, for
a total of six C-H bonding electrons.
The two models agree. With QMOT, we still have three C-H bonds, described
by three occupied MOs that are strongly C-H 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 spi 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-H bonding orbitals derived from carbon 2s orbitals and of s-symmetry are termed s(CH3) orbitals. 2. The C-H bonding orbitals that are derived from carbon 2p orbitals, of -symmetry, are termed (CH3) orbitals. They are a degenerate pair. 3. The other orbital of s-symmetry, that derived from the carbon pz AO, points away from the H’s and is termed the s(out) orbital.
http://www.chem.ox.ac.uk/vrchemistry/orbitals/html/page11.htm 25