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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 ar 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 * Hd 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