Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley Molecular Orbital
Reading: DG 2.11-14, 3.1-5, 4; MT 3.1 Lewis Structure
G. N. Lewis (UC, Berkeley, 1915)
Octet Rule: Closed shell configuration of 8 surrounding e
1 Chem 104A, UC, Berkeley
Resonance
The best structure has the fewest formal charges and has the negative charge on the highest electronegativity atom.
Chem 104A, UC, Berkeley
Valence Bond theory
Heitler, London (1927)
L. Pauling (orbital hybridization) Localized orbital approach
Investigating ground state molecules, molecular geometry Bond dissociation energy
2 Chem 104A, UC, Berkeley
Molecular Orbital theory
Mulliken
Delocalized orbital approach
Unoccupied orbital
Spectroscopic properties (ionization, excited states)
Chem 104A, UC, Berkeley Valence Bond Theory
Valence bond theory (VBT) is a localized quantum mechanical approach to describe the bonding in molecules. VBT provides a mathematical justification for the Lewis interpretation of electron pairs making bonds between atoms. VBT asserts that electron pairs occupy directed orbitals localized on a particular atom. The directionality of the orbitals is determined by the geometry around the atom which is obtained from the predictions of VSEPR theory.
In VBT, a bond will be formed if there is overlap of appropriate orbitals on two atoms and these orbitals are populated by a maximum of two electrons.
bonds: have bonds: a node on the symmetric about inter-nuclear axis the internuclear and the sign of axis the lobes changes across the axis.
3 Valence Bond TheoryChem 104A, UC, Berkeley
Valence bond theory treatment of bonding in H2 and F2
F 2s 2p H 1s1 H 1s1 A B F A B 2s 2p
z axis
2pz 2pz
This gives a 1s-1s bond This gives a 2p-2p bond between between the two H atoms. the two F atoms.
For the VBT treatment of bonding, we ignore the anti-bonding.
Chem 104A, UC, Berkeley
Valence bond theory treatment of bonding in O2
z axis O 2s 2p 2pz 2pz This gives a 2p-2p bond O between the two O atoms. 2s 2p OO z axis Lewis structure
2p 2p (the choice of 2py is arbitrary) y y Double bond: bond + bond Triple bond: bond + 2 bonds This gives a 2p-2p bond between the two O atoms. In VBT, bonds The Lewis approach and VBT are predicted to be weaker than predict that O is diamagnetic – bonds because there is less overlap. 2 this is wrong!
4 Chem 104A, UC, Berkeley
Valence Bond theory
A B
0.74 A
Bond dissociation energy: 103 kcal/mol
Chem 104A, UC, Berkeley
Our Goal
Use atomic orbitals to compose a wavefunction that accurately describe these observations.
5 Chem 104A, UC, Berkeley 1 1 Energy 1s ( )3/ 2 er / a0 a0
1s (1)1s (2) a A B 0
-2
f Experimental curve -4 eV
0.5 1.0 1.5 Internuclear distance
1 1 3/ 2 r / a0 Chem 104A, UC, Berkeley Energy 1s ( ) e a0 g 1sA (1)1sB (2) 1sA (2)1sB (1)
a 0 b
1sA (1)1sB (2) 1sA (2)1sB (1) -2 Heitler-London Wavefunction
f Experimental curve -4 eV
0.5 1.0 1.5 Internuclear distance
6 Chem 104A, UC, Berkeley
Variation Principle:
If an arbitrary wavefunction is used to calculate the energy, then the value obtained is never less than the true value.
* (x)H(x)dx H *(x)(x)
Chem 104A, UC, Berkeley
Variation Principle:
If an arbitrary wavefunction is used to calculate the energy, then the value obtained is never less than the true value. (x)
Can always minimize E (x) (x)
7 Chem 104A, UC, Berkeley 1 1 1s ( )3/ 2 er / a0 a0
+ +
1 Z 3/ 2 Zr / a0 1s 100 ( ) e a0
Chem 104A, UC, Berkeley 1 Z Energy 1s ( )3/ 2 eZr / a0 a0 g
a b 0 c 1sA (1)1sB (2) 1sA (2)1sB (1) -2 Zeff=1.17
f Experimental curve -4 eV
0.5 1.0 1.5 Internuclear distance
8 Chem 104A, UC, Berkeley Promoting electron Orbital Hybridization
- + + A
2pzA A N(1sA 2 pzA ) 1sA N(1s 2 p ) + + - B B zB B
2pzB 1sB 1 s contribution 1 2 2 p contribution 1 2
Chem 104A, UC, Berkeley
- + + A N(1sA 2 pzA ) A za 1sa 2p + + - B B N(1sB 2 pzB ) zb 1sb 2p
=0.1
9 Chem 104A, UC, Berkeley Energy N(1s 2 p ) g A A zA B N(1sB 2 pzB )
a b 0 c
-2 d A (1) B (2) A (2) B (1) f Experimental curve -4 eV
0.5 1.0 1.5 Internuclear distance
Chem 104A, UC, Berkeley Energy
g
a b 0 c
A (1) B (2) A (2) B (1) -2 d [ (1) (2) (2) (1)] e A A B B f Experimental curve -4 eV =6%
0.5 1.0 1.5 Internuclear distance
10 Chem 104A, UC, Berkeley Energy Valence bond theory is all about promoting e and hybridizing orbitals into the shape required by experiments.! g
a b 0 c
-2 d e f -4 eV
0.5 1.0 1.5 Internuclear distance
Chem 104A, UC, Berkeley
Covalent Bond - + + F N(2 pzF 2sF ) + + H1s
zF 2sF 2p N(2 p 2s ) - + + lp zF F + + H
zF 2sF 2p =0.5 Lone pair
11 Chem 104A, UC, Berkeley
promotion 2 2 2 2 1 2 1 2 2 2 1s 2s 2 px 2 py 2 pz 1s 2s 2 px 2 py 2 pz
E 500kcal / mol 1 eV=96.4 kJ/mol=23.1 kCal/mol Benefit of hybridization 1. Increase in e density between nuclei, enhanced bonding 2. Decrease in e-e repulsion by formation of a lone-pair orbital, LP Orbital more directed away from BP than the Original s LP Ionic Contribution: 50%
1sH (1)F (2) 1sH (2)F (1) F (2)F (1)
Chem 104A, UC, Berkeley
12 Chem 104A, UC, Berkeley
[1sHa (1)2 pyo (2) 1sHa (2)2 pyo (1)][1sHb (3)2 pzo (4) 1sHb (4)2 pzo (3)]
Valence bond theory is all about promotingChem 104A, electron UC, Berkeley and hybridizing orbitals into the shape required by experiments!
- y py s + O - 104.5 + pz HH dv 0 a b a b 2 a dv 1 z
13 Chem 104A, UC, Berkeley - y py s + O - + pz HH a b z N[cos 2 p sin 2 p 2s ] a 2 zo 2 yo o N[cos 2 p sin 2 p 2s ] b 2 zo 2 yo o
Orthogonality andChem Normalization 104A, UC, Berkeley Two properties of acceptable orbitals (wavefunctions) that we have not yet considered are that they must be orthogonal to every other orbital and they must be normalized. These conditions are related to the probability of finding an electron in a given space. Orthogonal means that the integral of the product of an orbital with any other orbital is equal to 0, i.e.: 0 nm where n m and means that the integral is taken over “all of space” (everywhere).
Normal means that the integral of the product of an orbital with itself is equal to 1, i.e.: 1 nn
This means that we must find normalization coefficients that satisfy these conditions. Note that the atomic orbitals () we use can be considered to be both orthogonal and normal or “orthonormal”.
14 N[cos 2 p sin 2 p 2s ] Chem 104A, UC, Berkeley a 2 zo 2 yo o N[cos 2 p sin 2 p 2s ] b 2 zo 2 yo o
104.5 dv 0 Orthogonality a b 2 0.5 dv 1 Normalization a N 1/ 1.25
a 0.552 pzo 0.712 pyo 0.4522so
b 0.552 pzo 0.7122 pyo 0.4522so
[1sHa (1)2 pyo (2) 1sHa (2)2 pyo (1)][1sHb (3)2 pzo (4) 1sHb (4)2 pzo (3)] New Wavefunction:
[1sHa (1)a (2) 1sHa (2)a (1)][1sHb (3)b (4) 1sHb (4)b (3)]
Chem 104A, UC, Berkeley
Unit Orbital Contribution:
The contribution of any atomic orbital to all hybrid Orbitals must add up to 1.0.
15 a 0.55Chem2 pzo 104A, 0.71 UC,2 Berkeleypyo 0.4522so Orbital contribution b 0.552 pzo 0.7122 pyo 0.4522so
a b 30% 30% 60% 2 pzo 0.552 30% 2 p 50% 50% 100% yo 0.712 50% 20% 20% 40% 2so 0.4522 20%
Bonding: Average 80% p character, 20% s character What’s left? 40% p +60% s
lp 0.632 pzo 0.772so
Chem 104A, UC, Berkeley - y py s - + O lp 0.632 pzo 0.772so + pz HH a b z
a 0.552 pzo 0.712 pyo 0.4522so
b 0.552 pzo 0.7122 pyo 0.4522so
16 Chem 104A, UC, Berkeley
p s 1 (2 p ) lp1 2 x lp character character 1 Bonding 80% 20% lp1 (2 px lp ) 2 orbital lp 70% 30% orbital
DirectionalityChem 104A, UC, Berkeley The bonding in diatomic molecules is adequately described by combinations of “pure” atomic orbitals on each atom. The only direction that exists in such molecules is the inter-nuclear axis and the geometry of each atom is undefined in terms of VSEPR theory (both atoms are terminal). This is not the case with polyatomic molecules and the orientation of orbitals is important for an accurate description of the bonding and the molecular geometry.
Examine the predicted bonding in ammonia using “pure” atomic orbitals:
2s 2p N
The 2p orbitals on N are oriented along the X, Y, 3 H and Z axes so we would predict that the angles
1s 1s 1s between the 2p-1s bonds in NH3 would be 90°. We know that this is not the case. HNH H
17 Chem 104A, UC, Berkeley
H N H 107.3 p s character character Bonding 77% 23% orbital lp 69% 31% orbital
Chem 104A, UC, Berkeley
2 2 0 2 1 1 1s 2s 2 p Promotion 1s 2s 2 p
1 Chem 104A, UC, Berkeley
1 (2s 2 p ) a 2 x
1 (2s 2 p ) b 2 x p s character character Bonding 50% 50% orbital
Chem 104A, UC, Berkeley
2 Chem 104A, UC, Berkeley
HybridizationChem 104A, UC, Berkeley The problem of accounting for the true geometry of molecules and the directionality of orbitals is handled using the concept of hybrid orbitals. Hybrid orbitals are mixtures of atomic orbitals and are treated mathematically as linear combinations of the appropriate s, p and d atomic orbitals.
Linear sp hybrid orbitals
A 2s orbital superimposed 1 1 on a 2p orbital 1 sp The two resultant sp x 2 2 hybrid orbitals that are 1 1 directed along the x-axis 2 sp (in this case) 2 2 The 1/2 are normalization coefficients.
3 Chem 104A, UC, Berkeley Example of the orthogonality of 1 and 2
1 1 1 1 1 2 sp2 2 2 sp2
1 1 1 1 12 2 spsp2 2 2
1 1 1 1 12 ss sp sp pp 2 2 2 2
1 1 1 1 12 1 0 0 1 2 2 2 2
1 1 0 12 2 2
Thus our hybrid sp orbitals are orthogonal to each other, as required.
HybridizationChem 104A, UC, Berkeley
Valence bond theory treatment of a linear molecule: the bonding in BeH2
BeH2 The promotion energy can 2s 2p be considered a part of the Be energy required to form hybrid orbitals. Be*
sp 2p Be* (sp)
2 H 1s 1s H Be H The overlap of the hybrid orbitals on Be with the 1s orbitals on the H atoms gives two Be-H (sp)-1s bonds oriented 180° from each other. This agrees with the VSEPR theory prediction.
4 Chem 104A, UC, Berkeley
p s character character Bonding 67% 33% orbital
Chem 104A, UC, Berkeley
Valence bond theory treatment of a trigonal planar molecule: the bonding in BH3
2s 2p B
B*
sp2 2p B* (sp2)
1 1 1 2 1 spxy p y This gives three sp orbitals 3 6 2 that are oriented 120° 1 1 1 apart in the xy plane – be 2 3 sp6 xy2 p x careful: the choice of axes 1 2 in this example determines 3 3 sp6 x the set of coefficients.
5 Chem 104A, UC, Berkeley
The coefficients in front of each atomic wavefunction indicate the amount of each atomic orbital that is used in the hybrid orbital. The sign indicates the orientation (direction) of the atomic orbitals. Remember that you have to use each atomic orbital completely (columns) and that each hybrid must be normal (rows). Check this by summing the squares of the coefficients.
The signs in front 1 1 1 of the coefficients 1/3 + 1/6 + 1/2 = 1 1 spxy pSo this hybrid is normal indicate the 3 6 2 direction of the hybrid: 1 1 1 1/3 + 1/6 + 1/2 = 1 1: -x, +y 2 3 sp6 xy2 pSo this hybrid is normal 2: -x, -y
1 2 3: +x, 0y 1/3 + 4/6 = 1 3 sp 3 6 x So this hybrid is normal y
1/3 + 1/3 + 1/3 = 1 1/6 + 1/6 + 4/6 = 1 1/2 + 1/2 = 1 x
So the entire s orbital So the entire px orbital So the entire py orbital has been used has been used has been used
Chem 104A, UC, Berkeley
Valence bond theory treatment of a trigonal planar molecule: the bonding in BH3
sp2 2p B*
3 H 1s 1s 1s
H HB H
The overlap of the sp2 hybrid orbitals on B with the 1s orbitals on the H atoms gives three B-H (sp2)-1s bonds oriented 120° from each other. This agrees with the VSEPR theory prediction.
6 Chem 104A, UC, Berkeley
p s character character Bonding 75% 25% orbital
Chem 104A, UC, Berkeley
Valence bond theory treatment of a tetrahedral molecule: the bonding in CH4
2s 2p C
C*
sp3 C* (sp3)
1 1 1 1 This gives four sp3 orbitals 1 4 sp4 xyz4 p4 p that are oriented in a 1 1 1 1 H tetrahedral fashion. 2 4 sp4 xyz4 p4 p 1 1 1 1 HCH 3 4 sp4 xyz4 p4 p H 1 1 1 1 4 4 sp4 xyz4 p4 p
1 Chem 104A, UC, Berkeley
Valence bond theory treatment of a tetrahedral molecule: the bonding in CH4
2s 2p C
C* sp3 C* (sp3)
4 H H 1s 1s 1s 1s C H H H
The overlap of the sp3 hybrid orbitals on C with the 1s orbitals on the H atoms gives four C-H (sp3)-1s bonds oriented 109.47° from each other. This provides the tetrahedral geometry predicted by VSEPR theory.
2 Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
1 Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
Trigonal bipyramidal
2 Chem 104A, UC, Berkeley Valence bond theory treatment of a trigonal bipyramidal molecule:
the bonding in PF5 3s 3p PF5 has a VSEPR theory AX5 P geometry so we need hybrid 3d orbitals suitable for bonds to P* 5 atoms. ns and np combinations can only P* (sp3d) 3d provide four, so we need to use nd orbitals (if they are available).
2 3s 3pz 3py 3px 3dz 3 sp dz2
The appropriate mixture to form a trigonal bipyramidal arrangement
of hybrids involves all the ns and np orbitals as well as the ndz2 orbital.
Chem 104A, UC, Berkeley Valence bond theory treatment of a trigonal bipyramidal molecule The orbitals are treated in two different sets.
1 1 1 These coefficients are 1 3 sp6 xy2 pexactly the same as the 1 1 1 result for the trigonal 2 sp pplanar molecules 3 6 xy2 because they are 1 2 derived from the same 3 3 sp6 x orbitals (sp2)
1 1 These coefficients are 4 pdsimilar to those for the sp 2 z 2 z2 hybrids because they are 1 1 formed from a combination 5 pd 2 z 2 z2 of two orbitals (pd).
Remember that d orbitals are more diffuse than s or p orbitals so VBT predicts that the bonds formed by hybrids involving d orbitals will be longer than those formed by s and p hybrids.
3 Chem 104A, UC, Berkeley Valence bond theory treatment of a trigonal bipyramidal molecule:
the bonding in PF5
P* (sp3d) 3d
F 2s 2p F F 2s 2p 2s 2p F 2s 2p F 2s 2p The overlap of the sp3d hybrid orbitals on P with the 2p orbitals on the F atoms gives five P-F (sp3d)-2p bonds in two sets: the two axial bonds along the z-axis (180° from each other) and three equatorial bonds in the xy plane (120° from each other and 90° from each axial bond). This means that the 5 bonds are not equivalent!
Chem 104A, UC, Berkeley
octahedral
4 Chem 104A, UC, Berkeley Valence bond theory treatment of an octahedral molecule:
the bonding in SF6 3s 3p S 3d S*
S* (sp3d2) 3d
F FFFFF
3 2 3s3pz 3py 3px 3dz2 3dx2-y2 sp d
The overlap of the sp3d2 hybrid orbitals on S with the 2p orbitals on the F atoms gives six S-F (sp3d2)-2p bonds 90° from each other that are equivalent. You can figure out the normalization coefficients.
5 Chem 104A, UC, Berkeley
Valence bond theory treatment of bonding: a hypervalent molecule, ClF3 3s 3p F Cl 3d FCl Cl* F There are five “objects” around Cl* (sp3d) 3d Cl so the geometry is trigonal bipyramidal and the shape is FFF given by AX3E2 (T-shaped).
The overlap of the sp3d hybrid orbitals on Cl with the 2p orbitals on the F atoms gives three P-F (sp3d)-2p bonds in two sets: the two axial bonds along the z-axis (less than 180° from each other because of the repulsion from the lone pairs) and the one equatorial bond halfway between the other Cl bonds. Again, the bond lengths will not be the same because there is more d contribution to the axial hybrid orbitals.
Chem 104A, UC, Berkeley
1 Chem 104A, UC, Berkeley
VBT does not adequately explain:
1. Excited state properties OO 2. Paramagnetism of oxygen: Lewis structure presence of unpaired electrons
3. Bond in hypervalent molecules, energy cost too high
2