339.1 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:
439.3 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.
539.3 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:
639.3 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.
7Born-Oppenheimer Approximation Max BornRobertOppenheimerMoleculen:Born-Oppenheimer ApproximationNuclei fixed in a frame: useSchrödinger equation:TVbVaVRABSolutions, for the case ofonly one potential:
8Symmetric and anti-symmetric wave functions in H2+andDefine:Two hydrogenic wave functions for 1s orbital:
10Antibonding moleculaire orbitalen represents an anti-bonding molecular orbital
11Diatomaire 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.
12Diatomaire 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.
13σ- 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 smallNu p orbitalen, sigma orbitalen, knopen
14Energy level diagramsFig 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 figFig 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)
15S en p overlapFig 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
16Heteronuclear 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 asThe proportion of ψA in the bond is cA2, and the proportion of ψB is cB2. A nonpolar bond has cA2 = cB2pure ionic bond has one coefficient zero (so that A+B- would have cA = 0and cB=1).Polar bond unequal coefficients.Hoe deze coefficienten te vinden??The variation principle, states: If an arbitrary wavefunction isused to calculate the energy, then the value obtained is never lessthan the true energy.
18Hybridization and the Structure of Polyatomic Molecules Welke AO’s te combineren in een MOLi 2p ligt te dicht in energie bij Li 2s en H1s om te negeren,Variatie berekening geeft:Core electronen en valentie elektronenFig 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.
19Hybridization and the Structure of Polyatomic Molecules Hybride is compromis tussen energie nodig om Li naar hogere energie configuratie te promoveren en betere overlapDeze coefficientenoptimale compromisFig (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.
20Hybridization 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.
21Hybridization and the Structure of Polyatomic Molecules Waarom hebben moleculen bepaalde vormen? H2O driehoek, NH3 pyramideCH4 tetrahedral, CO2 linear?H2OO 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…
22Hybridization 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’.
23H2OFig 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 strengthincreases because the overlap improves, but a promotion energy is required because the 2s-electronsnow take part in the bonding. The actual shape of the molecule, which is found by minimizing the totalenergy 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’.
24More hybridsBepaal aan de hand van het aantal equivalente bonds de hybridisatie.Bv drie equivalente bonds met (s,p2) hybridisatie, met (2s, 2px, 2py) als set
25More hybridsBeH2 ground state electronic configuration of the Be-atom isKoolstof atoom:Hybrides vormen sigma bonds met hydrogen of iets andersDe buitenste 4 valentie electronen kunnen 4 equivalente sp3 hybride orbitalen vormen:
26Lone 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 pairZo kun je ook H2O zien, alleen met een H minder en 2 lone pairsLone pairs: H-bonding
27Molecular 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 ethyleenFig 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!
28Huckel 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