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台灣大學開放式課程
Pictorial → Symmetry → MO AXn n=3, 4, 5, 6
Chemical Bonding
Lecture 10: MO for AXn system, Multicenter Bonding, VSEPR,
Hybridization
Spring 2012
5/11/2012
A X1
X2
X 3
X4
y
z
r1
r2 r3 x
r4
A set of vectors, r1, r2, r3, and r4, representing the four ⌠ bonds from A to the B atoms in a tetrahedral AB4
molecule.
ΓA-B = A1+T2 A1: s T2: px,py,pz & dxy, dxz, dyz
AX4 system-2 (Td)
Td E 8C3 3C2 6S4 6σd A1 1 1 1 1 1 x2+y2+z2 A2 1 1 1 -1 -1 E 2 -1 2 0 0 (2z2-x2-y2, x2-y2) T1 3 0 -1 1 -1 (Rx, Ry, Rz)
T2 3 0 -1 -1 1 (x, y, z) (xy, xz, yz)
ΓC-H 4 1 0 0 2 A1 + T2
Ene
rgy
Ψb
s + (σ1+ σ2+ σ3+ σ4)
Ψa
s − (σ1+ σ2+ σ3+ σ4)
A atom X atoms MOs
AX4 system-2 (Td)
a1
a1
s a1 σ1+ σ2+ σ3+ σ4 a1
By combining the central s orbital with ligand orbital SALC (symmetry-adapted linear combinations) to give either positive or negative overlap, we get Ψb and Ψa, respectively. The expressions written for these are only meant to express this sign relationship; the actual expression for Ψa and Ψ b contain different coefficients.
+
+
+
−
−
− +
+
+
−
−
−
+ +
+
−
− −
AX4 system-2 (Td)
Consider ligand group orbitals (LGOs) that match with each of the center-atom atomic orbitals Symmetry-adapted linear combination (SALC) Note that s or p along the axis both can form the σ bond
Qualitative representations of the three T2 type σ MOs for an AX4 molicule.
Ene
rgy
Ψ a(T2)
Ψ b(T2) Ψ b(A1)
A atom B atoms MOs
etc.
AX4 system-2 (Td)
Ψ a(A1) p t2 s a1
σ1+ σ2+ σ3+ σ4
σ1− σ2+ σ3− σ4
a1+ t2
An MO energy level diagram for a AX4 molecule showing both the A1 and T2 type interaction.
Experimental evidence for MO
Not compatible with the picture of electrons in four equivalent localized bonds.
A.W. Potts and W. C. Price, Proc. R. Soc. London, A326, 165 (1972).
The T2 ionization (~14eV) is more intense than A1 ionization (~23eV) partly because of the 3:1 ratio of populations. It is much broader because of a Jahn-Teller effect.
CH4
AX4 system-1 (D4h) XeF4
F
F
F x
y F Xe
Γπ’ : out of plane Γπ” : in plane
s+dx2-y2+(px+py) Xe: dsp2
D4h E 2C4 (z) C2 2C'2 2C''2 i 2S4 σh 2σv 2σd A1g 1 1 1 1 1 1 1 1 1 1 x2+y2, z2 A2g 1 1 1 -1 -1 1 1 1 -1 -1 Rz B1g 1 -1 1 1 -1 1 -1 1 1 -1 x2-y2 B2g 1 -1 1 -1 1 1 -1 1 -1 1 xy Eg 2 0 -2 0 0 2 0 -2 0 0 (Rx, Ry) (xz, yz) A1u 1 1 1 1 1 -1 -1 -1 -1 -1 A2u 1 1 1 -1 -1 -1 -1 -1 1 1 z B1u 1 -1 1 1 -1 -1 1 -1 -1 1 B2u 1 -1 1 -1 1 -1 1 -1 1 -1 Eu 2 0 -2 0 0 -2 0 2 0 0 (x, y) Γσ 4 0 0 2 0 0 0 4 2 0 A1g + B1g + Eu
Γπ’ 4 0 0 -2 0 0 0 -4 2 0 A2u + B2u + Eg
Γπ” 4 0 0 -2 0 0 0 4 -2 0 A2g+ B2g + Eu
The MOs for XeF4 and the Xe orbitals and LGOs from which they are obtained.
AX4 system-1 (D4h)
MO
A1g
B1g
Eu
⌠* ⌠
4
1 x
1
2
3
Xe orbitals LGO (4F) 4
2y
y
x s
y
3 x d x 2 �y2
y y
Γσ → a1g + b1g + eu
x x
y y
x
y
x
y
AX4 system-1 (D4h)
LGO – Symmetry adapted Group orbitals
dsp2
Xe XeF4 4F
2p (12)
5p
a1g* eu*
2s (4) b1g
eu a1g
a2u(pz)
5s a1g
eu a2u
(a1g + b1g + b2g + eg) 5d eg(dxz,dyz) a1g(dz2)
b1g* b2g(dxy)
a2u eu a1g
a2g+b2g+eu+a2u+b2u+eg nonbonding a1g+b1g+eu nonbonding
AX5 system (D3h)
A1’ : s, dz2 A2” : pz E’ : (px,py) & (dxy, dx2-y2)
Γ total = 2A 1'+A2"+E' σ
X
Y F P
F
F F
F
D3h E 2C3 3C2 σh 2S3 3 σv
A1’ 1 1 1 1 1 1 x2 + y2, z2
A2’ 1 1 -1 1 1 -1 Rz
E’ 2 -1 0 2 -1 0 (x, y) (x2-y2, xy) A1” 1 1 1 -1 -1 -1
A2” 1 1 -1 -1 -1 1 z
E” 2 -1 0 -2 1 0 (Rx,Ry)
Γσ total 5 2 1 3 0 3 2A1’ + A2” + E’ Γσ ax 2 2 0 0 0 2 A1’ + A2” Γσ eq 3 0 1 3 0 1 A1’ + E’
Molecular orbitals for σ bonding in PF5
Γσ → 2A1’ + E’ + A2”
a1’
a1’
a2”
e’
e’
⌠ MO (AX5)
2 z
P orbitals (A) s + d
2 z s � d
px
pz
py
LGO (5X) 4 5
2
3
1
2
3
1
4
5
1
3
2
3
1
4
5
2
3
1
1
3
AX5 system (D3h)
AX5 system (D3h)
Determining the equatorial LGO SALC coefficients: • Signs of SALC coefficients from overlapping with
AOs of the center atom • Values of coefficients from normalization &
orthogonal conditions • LGO energy determined by number of nodal planes
AX5 system (D3h)
a1’
a1’
a2”
e’
σ MO (AX5)
2
3
1
4
5
2
3
1 1
e’ 3
A2 (axial) = ± pz 1 2
'' (σ 4 −σ5)
A 1 (axial) = 1 2
' ±dz2 (σ 4 +σ5)
E (equatorial) = ' 1 6 1 2
=
(2σ2 −σ 1 −σ3)
(σ 1 −σ3)
px py
±
SALC of X A
(σ1 +σ 2 +σ3) A 1 (equatorial) = 1 3
' ± s
a2
P PF5 5F
P (15 orbitals) s (5)
1e’ 1a2” 1a1’
2a1 ’
px, py pz s
e’ ” a1’
e’ e” a1’
d dz2
4a1 2a2” 2e’ 3a1’
e’
e”
AX5 system (D3h)
AX6 system (Oh)
A1g : s Eg : dz2, dx2-y2 T1u : px, py, pz
A
Z
Y
X
Γσ = A 1g + Eg +T1u
Oh E 8C3 6C2 6C4 3C2( = C42) i 6S4 8S6 3σh 6 σd
A1g 1 1 1 1 1 1 1 1 1 1 x2 + y2 +z2
A2g 1 1 -1 -1 1 1 -1 1 1 -1
Eg 2 -1 0 0 2 2 0 -1 2 0 (2z2–x2–y2, x2 –y2)
T1g 3 0 -1 1 -1 3 1 0 -1 -1 (Rx, Ry, Rz)
T2g 3 0 1 -1 -1 3 -1 0 -1 1 (xy, xz, yz)
A1u 1 1 1 1 1 -1 -1 -1 -1 -1
A2u 1 1 -1 -1 1 -1 1 -1 -1 1
Eu 2 -1 0 0 2 -2 0 1 -2 0
T1u 3 0 -1 1 -1 -3 -1 0 1 1 (x, y, z)
T2u 3 0 1 -1 -1 -3 1 0 1 -1
Γσ 6 0 0 2 2 0 0 0 4 2 A1g + Eg +T1u
AX6 system (Oh)
1 6
(σ1 +σ 2 +σ3 +σ 4 +σ5 +σ6) A 1g =
1 12
(2σ5 +2σ6 −σ1 −σ 2 −σ3 −σ 4) Eg =
1 2
= (σ1 −σ 2 +σ3 −σ 4)
1 2 1 2 1 2
T1u =
=
=
(σ1 −σ3)
(σ 2 −σ 4)
(σ5 −σ 6)
s
dx2-y2, dz2
px, py, pz
SALC of X A
A orbitals 6X orbitals MOs
σ SALCs of A1g Eg, T1u Symmetry
dz2 dx2�y2
px、py、pz
a1g*
t1u*
eg* s
t1u eg a1g
AX6 system (Oh)
Character table for D3h point group
D3h E 2C3 3C'2 σh 2S3 3σv
A'1 1 1 1 1 1 1 x2+y2, z2 A'2 1 1 -1 1 1 -1 Rz E' 2 -1 0 2 -1 0 (x, y) (x2-y2, xy)
A''1 1 1 1 -1 -1 -1 A''2 1 1 -1 -1 -1 1 z E'' 2 -1 0 -2 1 0 (Rx, Ry) (xz, yz)
ΓS-O(σ) 3 0 1 3 0 1 A1’ + E’
1
2
1
1' [1 3 1 2 0 1 3 1 1 1 3 1 2 0 1 3 1 1] 1121' [1 3 1 2 0 1 3 1 ( 1) 1 3 1 2 0 1 3 1 ( 1)] 0
121' [1 3 2 2 0 ( 1) 3 1 0 1 3 2 2 0 ( 1) 3 1 0 ]1
121'' [1 3 1 2 0 1 3 1 1 1 3 ( 1) 2 0 ( 1) 3 1
12
nA
nA
E
nA
= × × + × × + × × + × × + × × + × × =
= × × + × × + × × − + × × + × × + × × − =
= × × + × × − + × × + × × + × × − + × × =
= × × + × × + × × + × × − + × × − + ×
2
( 1)] 0
1'' [1 3 1 2 0 1 3 1 ( 1) 1 3 ( 1) 2 0 ( 1) 3 1 1] 012
1'' [1 3 2 2 0 ( 1) 3 1 0 1 3 ( 2 ) 2 0 1 3 1 0 ] 012
nA
E
× − =
= × × + × × + × × − + × × − + × × − + × × =
= × × + × × − + × × + × × − + × × + × × =
A’1 + E’ ΓS-O(σ)
σ bonding
Character table for D3h point group
D3h E 2C3 3C'2 σh 2S3 3σv
A'1 1 1 1 1 1 1 x2+y2, z2
A'2 1 1 -1 1 1 -1 Rz
E' 2 -1 0 2 -1 0 (x, y) (x2-y2, xy)
A''1 1 1 1 -1 -1 -1
A''2 1 1 -1 -1 -1 1 z
E'' 2 -1 0 -2 1 0 (Rx, Ry) (xz, yz)
ΓS-O(π⊥) 3 0 -1 -3 0 1 A2” +E”
1
2
1
1' [1 3 1 2 0 1 3 ( 1) 1 1 ( 3) 1 2 0 1 3 1 1] 0121' [1 3 1 2 0 1 3 ( 1) ( 1) 1 ( 3) 1 2 0 1 3 1 ( 1)] 0
121' [1 3 2 2 0 ( 1) 3 ( 1) 0 1 ( 3) 2 2 0 ( 1) 3 1 0 ] 0
121'' [1 3 1 2 0 1 3 ( 1) 1
12
nA
nA
E
nA
= × × + × × + × − × + × − × + × × + × × =
= × × + × × + × − × − + × − × + × × + × × − =
= × × + × × − + × − × + × − × + × × − + × × =
= × × + × × + × − × +
2
1 ( 3) ( 1) 2 0 ( 1) 3 1 ( 1)] 0
1'' [1 3 1 2 0 1 3 ( 1) ( 1) 1 ( 3) ( 1) 2 0 ( 1) 3 1 1] 112
1'' [1 3 2 2 0 ( 1) 3 ( 1) 0 1 ( 3) ( 2 ) 2 0 1 3 1 0 ] 112
nA
E
× − × − + × × − + × × − =
= × × + × × + × − × − + × − × − + × × − + × × =
= × × + × × − + × − × + × − × − + × × + × × =
A’’2 + E’’
ΓS-O(π⊥)
π⊥ bonding
Character table for D3h point group D3h E 2C3 3C'2 σh 2S3 3σv A'1 1 1 1 1 1 1 x2+y2, z2 A'2 1 1 -1 1 1 -1 Rz E' 2 -1 0 2 -1 0 (x, y) (x2-y2, xy)
A''1 1 1 1 -1 -1 -1 A''2 1 1 -1 -1 -1 1 z E'' 2 -1 0 -2 1 0 (Rx, Ry) (xz, yz)
ΓS-O(π//) 3 0 -1 3 0 -1 A’2 + E’
1
2
1
1' [1 3 1 2 0 1 3 ( 1) 1 1 3 1 2 0 1 3 ( 1) 1] 0121' [1 3 1 2 0 1 3 ( 1) ( 1) 1 3 1 2 0 1 3 ( 1) ( 1)] 1
121' [1 3 2 2 0 ( 1) 3 ( 1) 0 1 3 2 2 0 ( 1) 3 ( 1) 0] 1
121'' [1 3 1 2 0 1 3 ( 1) 1
12
nA
nA
E
nA
= × × + × × + × − × + × × + × × + × − × =
= × × + × × + × − × − + × × + × × + × − × − =
= × × + × × − + × − × + × × + × × − + × − × =
= × × + × × + × − × +
2
1 3 ( 1) 2 0 ( 1) 3 ( 1) ( 1)] 0
1'' [1 3 1 2 0 1 3 ( 1) ( 1) 1 3 ( 1) 2 0 ( 1) 3 ( 1) 1] 012
1'' [1 3 2 2 0 ( 1) 3 ( 1) 0 1 3 ( 2 ) 2 0 1 3 ( 1) 0 ] 012
nA
E
× × − + × × − + × − × − =
= × × + × × + × − × − + × × − + × × − + × − × =
= × × + × × − + × − × + × × − + × × + × − × =
A’2 + E’
ΓS-O(π//)
π// bonding
SO3 The resulting representations for the oxygen group orbitals (three orbitals for each type, one from each oxygen atom) are
The representations for the sulfur orbitals
3O
→ A1’ + E ’ → A2’ + E ’ → A1’ + E ’ → A2” + E ”
Γ2s
Γ2px
Γ2py
Γ2pz
3 3 3 3
0 0 0 0
1 -1 1 -1
3 3 3 -3
0 0 0 0
1 -1 1 1
S
AX3 system (D3h)
D3h E 2C3 3C'2 σh 2S3 3σv A'1 1 1 1 1 1 1 x2+y2, z2 A'2 1 1 -1 1 1 -1 Rz E' 2 -1 0 2 -1 0 (x, y) (x2-y2, xy)
A''1 1 1 1 -1 -1 -1 A''2 1 1 -1 -1 -1 1 z E'' 2 -1 0 -2 1 0 (Rx, Ry) (xz, yz)
3s → A1’ 3px , 3py → E ’ 3pz → A2”
The molecular orbitals resulting from combining these orbitals are shown as figure. z
x
x x
x
y y y
y
z z
z
pz
A2” + E ”
Oxygen orbitals px py
A2’ + E ’ A1’ + E ’ s A1’
pz
A2”
Sulfur orbitals px ; py E ’
AX3 system (D3h)
LGO or SALC s A1’ + E ’
sσ π// pσ π⊥
y direction is along each S=O bond
AX3 system (D3h)
2e’
3e’ 4a1’
1e”
Qualitative MO energy-level diagram for SO3 5e’ 5a1’ 2a2”
1a2’ 1a2” 4e’
3a1’ 2s
2p
3s
3p
S 3O SO3
A1’
A2’’ + E’
A1’+ E’
A1’ + E’ A2’ + E’ A2” + E”
pσ π//
π⊥
sσ
A set of vectors representing the four π-type p orbitals on the four B atoms in a tetrahedral AB4 molecule.
Γπ = E+T1+T2 E: dz2, dx2-y2 T1: None T2: px,py,pz & dxy, dxz, dyz
AX4 system-2 (Td)
Td E 8C3 3C2 6S4 6σd A1 1 1 1 1 1 x2+y2+z2 A2 1 1 1 -1 -1 E 2 -1 2 0 0 (2z2-x2-y2, x2-y2) T1 3 0 -1 1 -1 (Rx, Ry, Rz)
T2 3 0 -1 -1 1 (x, y, z) (xy, xz, yz)
Γπ 8 -1 0 0 0 E + T1 + T2
A set of vectors representing the four π-type p orbitals on the four B atoms in a tetrahedral AB4 molecule.
Γπ = E+T1+T2
E: dz2, dx2-y2 T1: None T2: px,py,pz & dxy, dxz, dyz
AX4 system-2 (Td)
• It is impossible to form a complete set of π bonds (i.e. two to each X atom)
• The π bonding will not be entirely independent of the σ bonding
A atom orbitals
Batom orbitals
Molecular Orbitals
pσ SALC’s: A1, T2
pπ SALC’s: E, T1, T2 σs SALC’s: A1, T2
T2(σ*, π*) E(π*)
T1(π) A1(σ), T2(σ), E(π), T2(π) A1(σ), T2(σ)
T2(σ*, π*) 4p: T2 A1(σ*)
4s: A1 3d: E, T2
AX4 system-2 (Td) Complete qualitative MO diagram applicable to an MCl4
2- complex
An approximate MO diagram for a tetrahedral AX4 molecule or complex, where A is a +2 ion from the first transition and X’s are O or Cl atoms.
AX6 system (Oh)
T1g : None T2g : dxz, dyz, dxy
T1u : px, py, pz
T2u : None
Γπ = T1g+T2g+T1u+T2u
Oh E 8C3 6C2 6C4 3C2( = C42) i 6S4 8S6 3σh 6 σd
A1g 1 1 1 1 1 1 1 1 1 1 x2 + y2 +z2
A2g 1 1 -1 -1 1 1 -1 1 1 -1
Eg 2 -1 0 0 2 2 0 -1 2 0 (2z2–x2–y2, x2 –y2)
T1g 3 0 -1 1 -1 3 1 0 -1 -1 (Rx, Ry, Rz)
T2g 3 0 1 -1 -1 3 -1 0 -1 1 (xy, xz, yz)
A1u 1 1 1 1 1 -1 -1 -1 -1 -1
A2u 1 1 -1 -1 1 -1 1 -1 -1 1
Eu 2 -1 0 0 2 -2 0 1 -2 0
T1u 3 0 -1 1 -1 -3 -1 0 1 1 (x, y, z)
T2u 3 0 1 -1 -1 -3 1 0 1 -1
Γπ 12 0 0 0 -4 0 0 0 0 0 T1g + T2g +T1u + T2u
AX6 system (Oh)
px1
px4
py2
py3
px3
px5
py6
py1
py5
px2
px6
py4
T2g SALC of X atom πorbitals are derived by matching to T2g orbitals (dxy, dxz, dyz) on the A atom.
AX6 system (Oh)
An approximate MO diagram for an actahedral AX6 molecule or ion, where A is a +2 or +3 ion from the transition series and the X’s are F, O, or Cl atoms.
A X
MOs
cyclo-butadiene (C4H4) in D4h
D4h E 2C4 C2 2C'2 2C''2 i 2S4 σh 2σv 2σd
Γσ
Γπ
4 4
0 0
0 0
0 -2
2 0
0 0
0 0
4 -4
0 2
2 0
C
C C
C
Reducible Representations of C-C σ- and π-bonds of cyclo-butadiene
C4H4 (D4h)
Γσ = A1g + B2g + Eu
Γπ = Eg + A2u + B2u
cyclo-butadiene (C4H4) in D4h C4H4 (D4h)
D4h E 2C4 (z) C2 2C'2 2C''2 i 2S4 σh 2σv 2σd
A1g 1 1 1 1 1 1 1 1 1 1 x2+y2, z2
A2g 1 1 1 -1 -1 1 1 1 -1 -1 Rz
B1g 1 -1 1 1 -1 1 -1 1 1 -1 x2-y2
B2g 1 -1 1 -1 1 1 -1 1 -1 1 xy
Eg 2 0 -2 0 0 2 0 -2 0 0 (Rx, Ry) (xz, yz)
A1u 1 1 1 1 1 -1 -1 -1 -1 -1
A2u 1 1 1 -1 -1 -1 -1 -1 1 1 z
B1u 1 -1 1 1 -1 -1 1 -1 -1 1
B2u 1 -1 1 -1 1 -1 1 -1 1 -1
Eu 2 0 -2 0 0 -2 0 2 0 0 (x, y)
Γσ 4 0 0 0 2 0 0 4 0 2 A1g + B1g + Eu
Γπ 4 0 0 -2 0 0 0 -4 2 0 Eg + A2u+ B2u
C4H4 (D4h) Trick: determining MOs by matching to central atomic orbital with proper symmetries
Γπ = Eg + A2u + B2u
Eg xz, yz
A2u z B2u z(x2-y2)
http://www.chemistry.ucsc.edu/~soliver/151A/Handouts/
C4H4 (D4h)
Sketches of the πmolecular orbitals for planar C4H4
(eg)a (eg)b
a2u b2u
a2u
b2u
eg
Ener
gy
Diborane (D2h)
Diborane (B2H6) has a planar ethylene-like framework (D2h)
Total 3+3+6 = 12 electrons 2x4 = 8 electrons forming 4 edge B-H bonds 12-8 = 4 electrons in the center square electron deficient
Bridge bond: three atoms sharing 2 electrons Three-center two-electron bond
Diborane (D2h) σ bridge bonding in diborane
Γboron = Ag + B2g + B1u + B3u
ΓH = Ag + B3u
D2h E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz) Ag 1 1 1 1 1 1 1 1 B1g 1 1 -1 -1 1 1 -1 -1 B2g 1 -1 1 -1 1 -1 1 -1 B3g 1 -1 -1 1 1 -1 -1 1 Au 1 1 1 1 -1 -1 -1 -1 B1u 1 1 -1 -1 -1 -1 1 1 B2u 1 -1 1 -1 -1 1 -1 1 B3u 1 -1 -1 1 -1 1 1 -1
Γboron
4 0 0 0 0 0 4 0
ΓH 2 0 0 2 0 2 2 6
φ1
φ2
φ3
φ4
H1
H2
x
z y
Diborane (D2h) Qualitative energy-level diagram for the bridge bonding in diborane
Four electrons in two bonding orbitals two electron three-center bridge bonds
Γboron = Ag + B2g + B1u + B3u
ΓH= Ag + B3u
Diborane (D2h) Molecular orbitals for bridge bonding in diborane
Γboron = Ag + B2g + B1u+B3u
ΓH = Ag + B3u
Lewis Theory G.N. Lewis, who introduced it in his 1916 article The Molecule and the Atom
Lewis stuctures, also called Electron-dot Structures or Electron-dot Diagrams, are diagrams that show the bonding between atoms of a molecule, and the lone pairs of electrons that may exist in the molecule.
The Octet Rule
Eight electrons in their valence shells, similar to the electronic configuration of a noble gas.
H Cl [Na]+ [ Cl ]−
Lewis Structures:NaCl , HCl and H2O •• H:O:H
•• Sorce from: http://en.wikipedia.org/wiki/Category:Chemical_bonding
Sorce from: http://en.wikipedia.org/wiki/Category:Chemical_bonding
VSEPR Theory The Valence Shell Electron Pair Repulsion Theory or VSEPR is a model in chemistry that aims to generally represent the shapes of individual molecules. 1. Construct a valid Lewis structure that shows all of the bond pairs and the lone pairs of electrons. 2. To predict the molecular geometry based on the total number of pairs around the central atom.
Three types of repulsion :
☆ The lone pair-lone pair ☆ The lone pair-bonding pair ☆ The bonding pair-bonding pair
(lp-lp) > (lp-bp) > (bp-bp) repulsion.
AXnEm A: central atom; X: bonded ligand; E; lone pairs
Type Shape Examples
AX1E* Linear HF
AX2E0 Linear BeCl2, HgCl2
AX2E1 Bent SO2, O3
AX2E2 Bent H2O
AX2E3 Linear XeF2
AX3E0 AX3E1
Trigonal planar Trigonal Pyramidal
BF3 NH3
AX3E2 T-shaped ClF3, BrF3
AX4E0 Tetrahedral CH4
AX4E1 Seesaw SF4
AX4E2 Square Planar XeF4
AX5E0 AX5E1 AX6E0 AX6E1 AX7E0
Trigonal Bipyramidal Square Pyramidal Octahedral Pentagonal pyramidal Pentagonal bipyramidal
PCl5 BrF5 SF6 XeF6 IF7
Hybridization
Hybridization is the mixing of atomic orbitals belonging to a same electron shell to form new orbitals suitable for the qualitative description of atomic bonding properties. Hybridized orbitals are very useful in explaining the shape of molecular orbitals for molecules. Hybridisation is an integral part of the valence shell electron-pair repulsion (VSEPR) theory
--- LCAO from the same atom
φ1sp3; φ2
sp3 ; φ3sp3 ; φ4
sp3
Hybridization --- LCAO from the same atom A ∑=j
jiji χcHBφ
φ1sp φ2
sp
φ1sp2; φ2
sp2 ; φ3sp2
Gives Triangular geometry Another approach:
Hybridization the AOs first
http://www.mhhe.com/physsci/chemistry/essentialchemistry/flash/hybrv18.swf
The Hybrid orbitals動畫網站(英文說明)
The hybrid orbitals for various electron pair arrangments of A
s px py
s px py pz dz2
s p3 dz2 dx2-y2
O
H H
x
z
Recall H2O MOs: Hybridization of the two a1 AOs further stabilized the 1a1 bonding MO.
Applications of Hybridization in MO Theory
2A1+B1+B2 A1+B1 sp-hybridized orbitals
Hybridization --- LCAO from the same atom A ∑=j
jiji χcHBφ
φ1sp φ2
sp
AH2
A 2H
sp
pz
s
σ
σ*
py pz py
A
s
p
sp hybrid
Linear AH2 g2σu2σ
u1σ
g1σ
uπ
AH2 A.O. Hybrid A.O.
Hybridization
∑=j
jiji χcHBφ
φ1sp2; φ2
sp2 ; φ3sp2
--- LCAO from the same atom A
AH2
A 2H
sp2
pz
s σ
σ*
pz Bent AH2
A
s
p
sp2 hybrid
Hybridization
∑=j
jiji χcHBφ
φ1sp2; φ2
sp2 ; φ3sp2
--- LCAO from the same atom A σ*
planar AH3
AH3
A 3H
sp2
pz
s σ
pz
A
s
p
sp2 hybrid
Gives a Triangular geometry
Hybrid orbitals of A AL2
linear
bent
BeH2
sp
OH2
D2h E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz)
Ag 1 1 1 1 1 1 1 1 x2, y2, z2
B1g 1 1 -1 -1 1 1 -1 -1 Rz xy
B2g 1 -1 1 -1 1 -1 1 -1 Ry xz
B3g 1 -1 -1 1 1 -1 -1 1 Rx yz
Au 1 1 1 1 -1 -1 -1 -1
B1u 1 1 -1 -1 -1 -1 1 1 z
B2u 1 -1 1 -1 -1 1 -1 1 y
B3u 1 -1 -1 1 -1 1 1 -1 x
Γ(sp) 2 2 0 0 0 0 2 2 Ag + B1u
C2v E C2 (z) σv(xz) σ’v (yz)
A1 1 1 1 1 z x2, y2, z2
A2 1 1 -1 -1 Rz xy
B1 1 -1 1 -1 x, Ry xz
B2 1 -1 -1 1 y, Rx yz
Γσ 2 0 2 0 A1 + B1 sp; p2
Γt sp3
Γlp 2 0 0 2 A1 + B2 sp; p2
Trick: C∞v (D∞h) symmetry use the C2v (D2h) table
AL3 planar
bent
BH3
sp2; sd2 ; dp2; d3 (s; dz2) ((x,y); (dxy, d x2-y2))
NH3 sd2,, p3; sp2; pd2; d3 (s; ps; dz2) ((x,y); (dxy, d x2-y2); (dxz, dyz))
D3h E 2C3 3C2 σh 2S3 3 σv
A1’ 1 1 1 1 1 1 x2 + y2, z2
A2’ 1 1 -1 1 1 -1 Rz
E’ 2 -1 0 2 -1 0 (x, y) (x2-y2, xy)
A1” 1 1 1 -1 -1 -1
A2” 1 1 -1 -1 -1 1 z
E” 2 -1 0 -2 1 0 (Rx,Ry)
Γ 3 0 1 3 0 1 A1’ + E’
C3v E 2C3 3 σv
A1 1 1 1 z x2 + y2, z2
A2 1 1 -1 Rz
E 2 -1 0 (x,y), (Rx, Ry) (x2-y2, xy), (xz, yz)
Γ 3 0 1 A1 + E
AL4 Planar
Pt Cl42-
sp2d ; p2d2 (s, dz2) (d x2
–y2 ) (px
py)
D4h E 2C4 (z) C2 2C'2 2C''2 i 2S4 σh 2σv 2σd
A1g 1 1 1 1 1 1 1 1 1 1 x2+y2, z2
A2g 1 1 1 -1 -1 1 1 1 -1 -1 Rz
B1g 1 -1 1 1 -1 1 -1 1 1 -1 x2-y2
B2g 1 -1 1 -1 1 1 -1 1 -1 1 xy
Eg 2 0 -2 0 0 2 0 -2 0 0 (Rx, Ry) (xz, yz)
A1u 1 1 1 1 1 -1 -1 -1 -1 -1
A2u 1 1 1 -1 -1 -1 -1 -1 1 1 z
B1u 1 -1 1 1 -1 -1 1 -1 -1 1
B2u 1 -1 1 -1 1 -1 1 -1 1 -1
Eu 2 0 -2 0 0 -2 0 2 0 0 (x, y)
Γ 4 0 0 2 0 0 0 4 2 0 A1g + B1g + Eu
CH4
sp3; sd3 (s) (px py pz); (dxy dxz dyz)
Td E 8C3 3C2 6S4 6σd
A1 1 1 1 1 1 x2 + y2 + z2
A2 1 1 1 -1 -1
E 2 -1 2 0 0 (2z2 - x2 - y2, x2 - y2)
T1 3 0 -1 1 -1 (Rx, Ry, Rz)
T2 3 0 -1 -1 1 (x, y, z) (xy, xz, yz)
ΓC-H 4 1 0 0 2 A1 + T2
AL4
AL5
PF5
sp3d ; spd3 ;
XeBr5
sp3d ; sp2d2 ; sd4
D3h E 2C3 3C2 σh 2S3 3 σv
A1’ 1 1 1 1 1 1 x2 + y2, z2
A2’ 1 1 -1 1 1 -1 Rz
E’ 2 -1 0 2 -1 0 (x, y) (x2-y2, xy)
A1” 1 1 1 -1 -1 -1
A2” 1 1 -1 -1 -1 1 z
E” 2 -1 0 -2 1 0 (Rx,Ry)
Γ 5 2 1 3 0 3 2A1’ + E’ +A2”
C4v E 2C4 C2 2σv 2σd
A1 1 1 1 1 1 z x2 + y2, z2
A2 1 1 1 -1 -1 Rz
B1 1 -1 1 0 -1 x2 - y2
B2 1 -1 1 -1 1 xy
E 2 0 -2 0 0 (x, y), (Rx, Ry) (xz, yz)
Γ 5 1 1 3 1 2A1 + B1 + E
AL6
D4h E 2C4 (z) C2 2C'2 2C''2 i 2S4 σh 2σv 2σd
A1g 1 1 1 1 1 1 1 1 1 1 x2+y2, z2
A2g 1 1 1 -1 -1 1 1 1 -1 -1 Rz
B1g 1 -1 1 1 -1 1 -1 1 1 -1 x2-y2
B2g 1 -1 1 -1 1 1 -1 1 -1 1 xy
Eg 2 0 -2 0 0 2 0 -2 0 0 (Rx, Ry) (xz, yz)
A1u 1 1 1 1 1 -1 -1 -1 -1 -1
A2u 1 1 1 -1 -1 -1 -1 -1 1 1 z
B1u 1 -1 1 1 -1 -1 1 -1 -1 1
B2u 1 -1 1 -1 1 -1 1 -1 1 -1
Eu 2 0 -2 0 0 -2 0 2 0 0 (x, y)
Γeq 4 0 0 2 0 0 0 4 2 0 A1g + B1g + Eu
Γax 2 2 2 0 0 0 0 0 2 0 A1g + A2u
sp3d2
sp2d
pd
AL6
Oh E 8C3 6C2 6C4 3C2( = C42) i 6S4 8S6 3σh 6 σd
A1g 1 1 1 1 1 1 1 1 1 1 x2 + y2 +z2
A2g 1 1 -1 -1 1 1 -1 1 1 -1
Eg 2 -1 0 0 2 2 0 -1 2 0 (2z2–x2–y2, x2 –y2)
T1g 3 0 -1 1 -1 3 1 0 -1 -1 (Rx, Ry, Rz)
T2g 3 0 1 -1 -1 3 -1 0 -1 1 (xy, xz, yz)
A1u 1 1 1 1 1 -1 -1 -1 -1 -1
A2u 1 1 -1 -1 1 -1 1 -1 -1 1
Eu 2 -1 0 0 2 -2 0 1 -2 0
T1u 3 0 -1 1 -1 -3 -1 0 1 1 (x, y, z)
T2u 3 0 1 -1 -1 -3 1 0 1 -1
Γσ 6 0 0 2 2 0 0 0 4 2 A1g + Eg +T1u
sp3d2
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