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.”

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

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

Symmetry Operations

i ― inversion through center of mass gives an indistinguishable view of the molecule

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.

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

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

Polyatomic Molecules: BeH2

Linear? yesno i?yes D∞h

no C∞v

Point Group Flow chart

D∞h

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

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)

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

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

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.

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)

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

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)

Polyatomic Molecules: BeH2

Minimum Basis Set

sg = (HA1s + HB1s), Be1s, Be2s,

su = Be2pz & (HA1s - HB1s)

pu = Be2px & Be2py

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

Be HA & HB

1sg

2sg

1su

1pu

3sg*2su*

+-

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

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)