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College 1 Struktuur van Biomolekulen. Wat is ψ(r,t)?

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Presentation on theme: "College 1 Struktuur van Biomolekulen. Wat is ψ(r,t)?"— Presentation transcript:

1 College 1 Struktuur van Biomolekulen

2 Wat is ψ(r,t)?

3 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: 39.1 Quantum-Mechanical View of Atoms

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 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 Hydrogen Atom Wave Functions

7 Moleculen: Born-Oppenheimer Approximation Nuclei fixed in a frame: use Schrödinger equation: T VbVb VaVa VRVR AB Solutions, for the case of only one potential: Max Born Robert Oppenheimer

8 Symmetric and anti-symmetric wave functions in H 2 + 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, H 2 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 H 2 is obtained by accomodating the two electrons in the lowest available orbital (the bonding orbital.

12 Diatomaire moleculen, He 2 Fig The ground electronic configuration of the four-electron molecule He 2 has two bonding electrons and two antibonding electrons. It has a higher energy than the separated atoms, and so He 2 is unstable.

13 σ- and π-molecular orbitals Fig (a) the constructive interference leading to the formation of a 2p  -bonding orbital and (b) the corresponding antibonding MO. Fig (a) The interference of 2p x - or 2p y - AO’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

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  - and  -orbital energies of Period 2 homonuclear diatomics.

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  - 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 ψ A in the bond is c A 2, and the proportion of ψ B is c B 2. A nonpolar bond has c A 2 = c B 2 pure ionic bond has one coefficient zero (so that A + B - would have c A = 0 and c B =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 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. 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:

19 Hybridization and the Structure of Polyatomic Molecules 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. Deze coefficienten optimale compromis

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:

21 Hybridization and the Structure of Polyatomic Molecules Waarom hebben moleculen bepaalde vormen? H 2 O driehoek, NH 3 pyramide CH 4 tetrahedral, CO 2 linear? H 2 O 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: Maar: hoek van 90 o, in werkelijkheid 104 o …

22 Hybridization and the Structure of Polyatomic Molecules Fig (a) p and p’ can be expressed as linear combinations of p x and p y but the combinations are not orthogonal. (b) the orthogonal hybrids, h and h’, obtained by mixing 2s-character into p and p’. In the MO description of H2O we aim to construct two O-H bonds that are chemically equivalent, but spatially distinct.

23 H2OH2O 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 H 2 O 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 H 2 O molecule is given by. The two electron pairs that are put in the third hybrid and 2p z0 are called ‘lone pairs’.

24 More hybrids Bepaal aan de hand van het aantal equivalente bonds de hybridisatie. Bv drie equivalente bonds met (s,p 2 ) hybridisatie, met (2s, 2p x, 2p y ) als set

25 More hybrids BeH 2 ground state electronic configuration of the Be-atom is Koolstof atoom: De buitenste 4 valentie electronen kunnen 4 equivalente sp 3 hybride orbitalen vormen:

26 Lone e - pairs NH 3 N: atoomgetal 7, dus: Maak vier sp 3 hybrides 3 maken een σ bond met H, met elk 1 e van N, 2 e - over in 4e hybrid = lone electron pair 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. Fig Bonding in ethylene. (a) In the plane of the nuclei: the formation of a  bond between the carbon atoms 1 and 2 using sp 2 hybrid orbitals and  -bonds between the 4 H 1s electrons and the remaining sp 2 orbitals on each C-atom. (b) Perpendicular to the plane of the 6 nuclei: formation of a  -bond between the two 2p- orbitals (that were not involved in the sp 2 - hybrids. Voorbeeld ethyleen Π Electronen houden het molekuul vlak!

28 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 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


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