1. Struktuur van Biomolekulen
College 1
Struktuur van Biomolekulen

2. Wat is ψ(r,t)?

3. 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:

4. 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.

5. 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:

6. 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.

7. Born-Oppenheimer Approximation
Max Born
Robert
Oppenheimer
Moleculen:
Born-Oppenheimer Approximation
Nuclei fixed in a frame: use
Schrödinger equation: T Vb Va VR A B
Solutions, for the case of
only one potential:

8. Symmetric and anti-symmetric wave functions
in H2+
and
Define:
Two hydrogenic wave functions for 1s orbital:

9. Bonding moleculaire orbitalen

10. Antibonding moleculaire orbitalen
represents an anti-bonding molecular orbital

11. Diatomaire moleculen, H2
Fig 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 The ground electronic configuration of H2 is obtained by accomodating the two electrons in the lowest available orbital (the bonding orbital.

12. Diatomaire moleculen, He2
Fig 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.

13. σ- and π-molecular orbitals
Fig (a) the constructive interference leading to the formation of a 2ps-bonding orbital and (b) the corresponding antibonding MO.
Fig (a) The interference of 2px- or 2py-AO’s leading to the formation of a 2pp-bonding orbital and (b) the corresponding antibonding orbital. Note that for the p-orbitals the contribution to the binding energy of a molecule is relatively small
Nu p orbitalen, sigma orbitalen, knopen

14. Energy level diagrams
Fig 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
Fig Variation of the s- and p-orbital energies of Period 2 homonuclear diatomics.
2 s banen worden 1 sigma, li atoomgetal is 3, bondorder = ½ (n-n*), voor N2: ½ (8-2)

15. S en p overlap
Fig Overlapping s- and p-orbitals. (a) End-on overlap leads to non-zero overlap and to the formation of an axially symmetric s-bond. (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

16. 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
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.

17. Variatieprincipe
and

18. 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:
Core electronen en valentie elektronen
Fig 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.

19. Hybridization and the Structure of Polyatomic Molecules
Hybride is compromis tussen energie nodig om Li naar hogere energie configuratie te promoveren en betere overlap
Deze coefficienten
optimale compromis
Fig (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.

20. 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:
Simpele atomic orbital overlap idee weg…?
Zie de AO’s van Li als een hybride AO:
Geen echte fysische of mathematische reden, wel aantrekkelijk, plus kan helpen in het bepalen van verschillende bijdragen aan de energieen van bonden.

21. Hybridization and the Structure of Polyatomic Molecules
Waarom hebben moleculen bepaalde vormen? H2O driehoek, NH3 pyramide
CH4 tetrahedral, CO2 linear?
H2O
O elektron configuratie:
Dus een basis set van met 4 elektronen te verdelen over deze bindingen overlap elke H1s met een O2p, resulterend in 2 σ-bonds, met elk 2 e, dus:
En dus hybridisatie…
Maar: hoek van 90o, in werkelijkheid 104o…

22. 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 (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’.

23. H2O
Fig 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% 2s-character.
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.
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’.

24. 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

25. More hybrids
BeH2 ground state electronic configuration of the Be-atom is
Koolstof atoom:
Hybrides vormen sigma bonds met hydrogen of iets anders
De buitenste 4 valentie electronen kunnen 4 equivalente sp3 hybride orbitalen vormen:

26. Lone e- pairs NH3 N: atoomgetal 7, dus: 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
Zo kun je ook H2O zien, alleen met een H minder en 2 lone pairs
Lone pairs: H-bonding

27. 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 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 p-bond between the two 2p-orbitals (that were not involved in the sp2-hybrids.
Π Electronen houden het molekuul vlak!

28. Huckel theorie Excitation energy is 2β (resonance integral)
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 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