Transcript Slide 1

College 1
Struktuur van Biomolekulen
Wat is ψ(r,t)?
39.1 Quantum-Mechanical View
of Atoms
Since we cannot say exactly where an electron
is, the Bohr picture of the atom, with electrons
in neat orbits, cannot be correct.
Quantum theory
describes an electron
probability distribution:
39.3 Hydrogen Atom Wave
Functions
The wave function of the ground state
of hydrogen has the form:
The probability of finding the electron
in a volume dV around a given point
is then |ψ|2 dV.
39.3 Hydrogen Atom Wave
Functions
The ground state is spherically symmetric;
the probability of finding the electron at a
distance between r and r + dr from the
nucleus is:
39.3 Hydrogen Atom Wave
Functions
This figure shows the three probability
distributions for n = 2 and = 1 (the distributions
for m = +1 and m = -1 are the same), as well as
the radial distribution for all n = 2 states.
Moleculen:
Born-Oppenheimer Approximation
Max Born
Nuclei fixed in a frame: use
Schrödinger equation:
A
k
Robert
Oppenheimer
1
4 0
2

ke2
ke2
ke2 
 

2








e r    e e r 
r
2
m
r

R
/
2
r

R
/
2
R


e


T
Vb
Va
VR
B
Solutions, for the case of
only one potential:
1As r  R / 2 
1Bs r  R / 2 
Symmetric and anti-symmetric wave functions
in H2+
Define:
 r  

1 A
1s  1Bs
2

and
 r  
MO  1s A  1s B
Two hydrogenic wave functions for 1s orbital:

1 A
1s  1Bs
2

MO  1s A  1s B
  e ar  R / 2
Bonding moleculaire orbitalen
Antibonding moleculaire orbitalen
MO  1s A  1s B
represents an anti-bonding molecular orbital
Diatomaire moleculen, H2
Fig. 2.1.5. A molecular orbital energy
level diagram for orbitals constructed
from (1s, 1s)-overlap, the separation of
the levels corresponding to the
equilibrium bond length.
Fig. 2.1.6. The ground electronic
configuration of H2 is obtained by
accomodating the two electrons in the
lowest available orbital (the bonding
orbital.
Diatomaire moleculen, He2
Fig.2.1.7. The ground electronic
configuration of the four-electron molecule
He2 has two bonding electrons and two
antibonding electrons. It has a higher energy
than the separated atoms, and so He2 is
unstable.
σ- and π-molecular orbitals
Fig. 2.1.9. (a) the constructive
interference leading to the formation of a
2ps-bonding orbital and (b) the
corresponding antibonding MO.
Fig. 2.1.10. (a) The interference of 2px- or 2pyAO’s leading to the formation of a 2p-bonding
orbital and (b) the corresponding antibonding
orbital. Note that for the -orbitals the
contribution to the binding energy of a molecule
is relatively small
Energy level diagrams
Fig. 2.1.11. The molecular
orbital energy level
diagram for (2p, 2p)overlap. While simple
overlap considerations
suggest the order in (a),
the order often found in
practice is that shown in
fig 2.1.12
Fig. 2.1.12 Variation of the s- and -orbital energies of Period 2
homonuclear diatomics.
S en p overlap
Fig. 2.1.13. Overlapping s- and p-orbitals. (a)
End-on overlap leads to non-zero overlap and
to the formation of an axially symmetric sbond. (b) Broad-side overlap leads to no net
accumulation of electron density in the
internuclear region.
A measure of the extent to which two orbitals overlap is the overlap integral S
S    * A B d   A B
Heteronuclear diatomaire
moleculen: e- verdeling asymmetrisch
The range of bond types, from nonpolar through polar to ionic is
captured in MO theory by writing the LCAO as
MO  c A A  cB B
The proportion of ψA in the bond is cA2, and the proportion of ψB
is cB2.
A nonpolar bond has cA2 = cB2
pure ionic bond has one coefficient zero (so that A+B- would have cA = 0
and cB=1).
Polar bond unequal coefficients.
Hoe deze coefficienten te vinden??
The variation principle, states: If an arbitrary wavefunction is
used to calculate the energy, then the value obtained is never less
than the true energy.
Variatieprincipe
E    * Hd
E c A  0
and
*

 d
E cB  0
Hybridization and the Structure of
Polyatomic Molecules
Welke AO’s te combineren in een MO
Li 2p ligt te dicht in energie bij Li 2s en H1s om te negeren,
Variatie berekening geeft:
  0.41Li 2s  0.29Li 2 p  0.87H1s
Fig. 2.1.16. Hydrogen and lithium
atomic energy levels: H1s overlaps
with both Li2s and Li2p, and the
resulting MO can be viewed as arising
from the overlap of H1s with a
(Li2s,Li2p)-hybrid orbital. Li1s is a
core orbital and plays only a minor role
in the bonding.
Hybridization and the Structure of
Polyatomic Molecules
  0.41Li 2s  0.29Li 2 p  0.87H1s
Deze coefficienten
optimale compromis
Fig. 2.1.17. (a) A cross-section through the (Li2s, Li2p)-hybrid showing the accumulation of
amplitude on one side of the nucleus. (b) The H1s-orbital overlaps the hybrid strongly, and a
stronger bond is formed than with Li2s alone.
Hybridization and the Structure of
Polyatomic Molecules
Li 2p ligt te dicht in energie bij Li 2s en H1s om te negeren,
Variatie berekening geeft:   0.41Li 2s  0.29Li 2 p  0.87H1s
Simpele atomic orbital overlap idee weg…?
Zie de AO’s van Li als een hybride AO:
  0.5Lihybride 0.87H1s
Lihybride 0.81Li 2s  0.58Li 2 p
Hybridization and the Structure of
Polyatomic Molecules
Waarom hebben moleculen bepaalde vormen?
H2O driehoek, NH3 pyramide
CH4 tetrahedral, CO2 linear?
H2O
2
2
2
1
1
O elektron configuratie: 1s 2s 2 p z 2 px 2 p y
Dus een basis set van O2 px , O2 p y , H1s A , H1sB 
met 4 elektronen te verdelen over deze bindingen
overlap elke H1s met een O2p, resulterend in 2 σ-bonds, met elk 2 e,
dus: 1s 2 2s 2 2 p 2s 2s 2
z
A
B
Maar: hoek van 90o, in werkelijkheid 104o…
Hybridization and the Structure of
Polyatomic Molecules
In the MO description of H2O we aim to construct two O-H bonds that are
chemically equivalent, but spatially distinct.
Fig. 2.1.19 (a) p and p’ can be
expressed as linear combinations of
px and py but the combinations are
not orthogonal. (b) the orthogonal
hybrids, h and h’, obtained by
mixing 2s-character into p and p’.
H2 O
Fig. 2.1.20 Three orthogonal AO’s
hybridize to give three orthogonal
hybrids. While the first two are
chemically equivalent and each bind an
H-atom, the third (dark) is different and
in the case of H2O contains 60% 2scharacter.
Ignoring the O2s contribution, there is no promotion energy, and moderately good (s,p)-overlap.
When O2s-hybridization is allowed, forming the two equivalent hybrids h and h’, the bond strength
increases because the overlap improves, but a promotion energy is required because the 2s-electrons
now take part in the bonding. The actual shape of the molecule, which is found by minimizing the total
energy is a compromise between strong bonding and promotion energy.
2 2
2
2
2
1
s
s
s
h
'
'
2
p
zO
. o h,1s A h',1sB
Finally, the electronic configuration of the H2O molecule is given by
The two electron pairs that are put in the third hybrid and 2pz0 are called ‘lone pairs’.
More hybrids
Bepaal aan de hand van het aantal equivalente bonds de hybridisatie.
Bv drie equivalente bonds met (s,p2) hybridisatie, met (2s, 2px, 2py) als set
sp 2 i  
sp 2 ii  
sp 2 iii 
1
3
1
2s  2 p
3
1
3
2s 
2s 
x
1
6
1
6
2

2px
2px
1
2
1
2
2 py
2 py
More hybrids
BeH2 ground state electronic configuration of the
Be-atom is 1s 2 2s 2
sp i  
sp ii  
2
2
Koolstof atoom: 1s 2s 2 p
1
2
1
2
(2s  2 p x )
(2s  2 p x )
2
De buitenste 4 valentie electronen kunnen 4 equivalente
sp3 hybride orbitalen vormen:     1 2s  2 p  2 p  2 p 
sp 3 i
sp 3 ii  
sp 3 iii 
sp 3 i  
x
4
1
2s  2 p
4
1
4
1
4
y
z
x
 2 p y  2 pz 
2s  2 p
 2 p y  2 pz 
2s  2 p
x
x
 2 p y  2 pz 
Lone e- pairs
NH3
N: atoomgetal 7, dus:
2s 2 2 p1x 2 p1y 2 p1z
Maak vier sp3 hybrides
3 maken een σ bond met H, met elk 1 e van N,
2 e- over in 4e hybrid = lone electron pair
1s s
2
o
2
h,1s A
Lone pairs: H-bonding
s
2
h',1sB
s
2
h',1sC
2
h' '
Molecular orbital theory
Molecules with extensive π-bonding systems, like benzene or for instance
the photosynthetic pigments chlorophyll a, b-carotene, are not described
very well by valence bond theory, because the π-electrons are often not
localized in a single bond, but instead are delocalized over the whole
molecule.
Voorbeeld ethyleen
Fig.2.1.24 Bonding in ethylene. (a) In the
plane of the nuclei: the formation of a s
bond between the carbon atoms 1 and 2
using sp2 hybrid orbitals and s-bonds
between the 4 H 1s electrons and the
remaining sp2 orbitals on each C-atom. (b)
Perpendicular to the plane of the 6 nuclei:
formation of a -bond between the two 2porbitals (that were not involved in the sp2hybrids.
Π Electronen houden het molekuul vlak!
Huckel theorie
π electrons do not interact with one another, and so the many-electron wavefunction is just a product
of one-electron molecular orbitals. Furthermore it assumes that the structure of the molecule is given
by the σ-framework, plus some simplifications gives….
Fig.2.1.25. Hückel molecular
orbitals for ethylene. The
carbon nuclei are represented
by dots, and the nodal planes
for the MO’s are represented
by the dashed lines.
Excitation energy is 2β (resonance integral)
Highest occupied molecular orbital is called HOMO, the lowest unoccupied
molecular orbital is called LUMO