Types of Symmetry in Molecules
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Types of Symmetry in Molecules
1. axis of symmetry (Cn)2. plane of symmetry (s)
3. center of symmetry (i)
4. improper axis of symmetry (Sn)
“Wavefunctions have symmetry and their symmetry can be used to understand their properties and to define and describe molecular wavefunctions more easily.”
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Symmetry Operations Cn ― rotation by 2p/n radians gives an indistinguishable view of molecule
.. NH H 1 X C3
H
6 X C2
1 X C6prinicpal axis
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Symmetry Operations
s ― reflection through molecular plane gives an
indistinguishable view of the molecule
1 x sh
.. NH H 3 X Cv
H 6 X sv
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Symmetry Operations
i ― inversion through center of mass gives an indistinguishable view of the molecule
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Symmetry Operations Sn ― Rotation by 2p/n & reflection through a plane ┴ to axis of rotation gives an indistinguishable view of the molecule
S2
Ball – table 13.1 – p423Each symmetry element can be defined by a 3x3 matrix.
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Ball – p421Molecules do not have random sets of symmetry elements – only certain specific sets of symmetry elements are possible. Such sets of symmetry always intersect at a single point. Therefore the groups of symmetry elements are referred to as point groups.
Character Tables are lists for a specific point group that indicates all of the symmetry elements necessary for that point group. These can be found in Ball - appendix 3, p797.The number of individual symmetry operations in the point group is the order (h) of the group. The character tables are in the form of an hxh matrix.
Point Groups
E/C1 Cs (C1h) Ci (S2) Cn Cnv Cnh
Dn Dnh Dnd Sn
Td Oh Ih Rh
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Linear?
no
yesyes
yes
no
no
no
no
no i?yes D∞h
no C∞v
≥ 2Cn, n > 2? yes yesi? C5?
no
yes Ih
Oh
Cn? Select highest Cn
yess?
i?no
Cs
no C1
yesCi
nC2 to Cn?yes sh?
Dnh
nsd?
sh?yesCnh
nonsv?
yes CnvS2n?
yes S2nnoCn
no Td
no Dn
yesDnd
Point Group Flow chart
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Polyatomic Molecules: BeH2
Linear? yesno i?yes D∞h
no C∞v
Point Group Flow chart
D∞h
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Be1s, Be2s, Be2pz HA1s, HB1s(HA1s + HB1s), (HA1s - HB1s)(HA1s + HB1s), (HA1s - HB1s)Be1s, Be2s, Be2pz
Polyatomic Molecules: BeH2
Minimum Basis Set?
s
p
Be H
Be2px, Be2py
How can you keep from telling H atoms apart?
Separate into symmetric and antisymmetric functions?
Minimum Basis Setsg = (HA1s + HB1s), Be1s, Be2s, su = Be2pz & (HA1s - HB1s)pu = Be2px & Be2py
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BeH2 – Minimum Basis Set Be HA HB
1s -115 eV -13.6 eV -13.6 eV
2s -6.7 eV
2p -3.7 eV
AO energy levels
Does LCAO with HA and HB change energy?
-13.6 eV = (HA1s + HB1s) and (HA1s - HB1s)
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Polyatomic Molecules — BeH2
Spartan – MNDO semi-empirical
7.2eV su* = 0.84 Be2pz + 0.38 (HA1s - HB1s)
3.0eV sg* = 0.74Be2s - 0.48(HA1s + HB1s)
2.5eV pu = (0.95Be2px - 0.30Be2py)& (0.30Be2px + 0.95Be2py)
-12.3 eV su = -0.51Be2pz + 0.59 (HA1s - HB1s)-13.8 eV sg = -0.67Be2s - 0.52(HA1s + HB1s)-115 eV sg = Be1s
pu = Be2px & Be2py su = Be2pz | (HA1s - HB1s) sg = Be1s | Be2s | (HA1s + HB1s)
su = 0.44(Be2pz) + 0.44 (HA1s - HB1s)sg = -0.09(Be1s) + 0.40(Be2s) + 0.45 (HA1s + HB1s) sg = 1.00(Be1s) + 0.016(Be2s) -0.002 (HA1s + HB1s)
HF SCF calculation : J. Chem. Phys. 1971
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Be HA & HB
1sg
2sg
1su
1pu
3sg*2su*
-13.6 eV
-115 eV
-6.7 eV-3.7 eV
-13.8 eV
-12.3 eV
3.0 eV2.5 eV
7.2 eV
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Average Bond Dipole Moments in Debyes (1 D = 3.335641 Cm)
H - O 1.5 C - Cl 1.5 C = O 2.5H - N 1.3 C - Br 1.4 C - N 0.5H - C 0.4 C - O 0.8 C º N 3.5
e = 1.6022 x 10-19
Dipole Moments & Electronegativity
In MO theory the charge on each atom is related to the probability of finding the electron near that nucleus, which is related to the coefficient of the AO in the MO
CH2O Geometry
Use VSEPR and SOHCAHTOA to find dipole moment in debyes.
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Heteronuclear Diatomic Molecules
MO = LCAO same type (sp) — similar energyAll same type AO’s = basis set
minimum basis set (no empty AO’s)
Resulting MO’s are delocalized
Coefficients = weighting contribution
HF minimum basis sets = H(1s), F(1s), F(2s), F(2pz)p = F(2px), F(2py)
without lower E AO’s s = H(1s), F(2s), F(2pz)p = F(2px), F(2py)
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HF
12.9eV
19.3eV
H1s
F2s
F2pz
13.6eV
18.6eV4s*
s
s
px py
0.19 H1s + 0.98 F2pz
0.98 H1s - 0.19 F2pz
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Delocalized HF Molecule
1px & 1py = F(2px) & F(2py)
3s = -0.023F(1s) - 0.411F(2s) + 0.711F(2pz) + 0.516H(1s)
2s = -0.018F(1s) + 0.914F(2s) + .090F(2pz) + .154H(1s)
1s = 1.000F(1s) + 0.012F(2s) + 0.002F(2pz) - 0.003H(1s)
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Polyatomic Molecules: BeH2
Minimum Basis Set
sg = (HA1s + HB1s), Be1s, Be2s,
su = Be2pz & (HA1s - HB1s)
pu = Be2px & Be2py
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Polyatomic Molecules — BeH2
su* = C7 Be2pz - C8
(HA1s - HB1s)
sg* = C5
Be2s + C6 (HA1s + HB1s)
pu = Be2px & Be2py
su = C3Be2pz + C4 (HA1s - HB1s)sg = C1Be2s + C2 (HA1s + HB1s)sg = Be1s
What is point group?What are the basis set AOs for determining MOs ?
pu = Be2px & Be2py su = Be2pz | (HA1s - HB1s) sg = Be1s | Be2s | (HA1s + HB1s)
su = 0.44(Be2pz ) + 0.44 (HA1s - HB1s)sg = -0.09(Be1s) + 0.40(Be2s) + 0.45 (HA1s + HB1s) sg = 1.00(Be1s) + 0.016(Be2s) -0.002 (HA1s + HB1s)
HF SCF calculation : J. Chem. Phys. 1971
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Be HA & HB
1sg
2sg
1su
1pu
3sg*2su*
+-
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MO: 1 2 3 4 5Eigenvalues:-1.82822 -0.63290 -0.51768 -0.51768 0.24632 (ev): -49.74849 -17.22198 -14.08688 -14.08688 6.70261 A1 A1 ??? ??? A1 1 H2 S 0.37583 -0.46288 0.00000 0.00000 0.80281 2 F1 S 0.91940 0.29466 0.00000 0.00000 -0.260523 F1 PX 0.00000 0.00000 -0.78600 0.61823 0.000004 F1 PY 0.00000 0.00000 0.61823 0.78600 0.000005 F1 PZ -0.11597 0.83601 0.00000 0.00000 0.53631
HF 1s2 2s2 3s2 (1p22p2) 4s0 Semi-empirical treatment of HF from Spartan (AM1)
One simpler treatment of HF is given in Atkins on page 428 gives the following results....4s = 0.98 (H1s) - 0.19(F2pz) -13.4 eVpx = py = F2px and F2py -18.6 eV3s = 0.19(H1s) + 0.98(F2pz) -18.8 eV2s = F2s ~ -40.2 eV1s = F1s << -40.2 eV
H1s
F2s
F2p
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Localized MO’s
6e- = 6 x 6 determinant
adding cst • column to another column leaves determinant value unchanged
adjust so resultant determinant represents localized MO’s
CH4 - localized bonding MO J. Chem. Phys. 1967 C - HA MO = ....
0.02(C1s) + 0.292(C2s) + 0.277(C2px + C2py + C2pz) + 0.57(HA1s) - 0.07(HB1s + HC1s + HD1s)