College 1

Struktuur van Biomolekulen

Wat is ψ(r,t)?

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

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:

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.

39.3 Hydrogen Atom Wave Functions

Moleculen:Born-Oppenheimer Approximation

Nuclei fixed in a frame: use04

1

k

Schrödinger equation: rr

R

ke

Rr

ke

Rr

ke

m eeere

222

22

2/2/2

T Vb Va VR

A B

Solutions, for the case of only one potential:

2/1 RrAs

2/1 RrBs

Max Born Robert Oppenheimer

Symmetric and anti-symmetric wave functionsin H2

+

Bs

Asr 11

2

1 Bs

Asr 11

2

1 andDefine:

Two hydrogenic wave functions for 1s orbital: 2/Rrae

BAMO ss 11 BAMO ss 11

Bonding moleculaire orbitalen

Antibonding moleculaire orbitalen

represents an anti-bonding molecular orbital

BAMO ss 11

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 2p-bonding orbital and (b) the corresponding antibonding MO.

Fig. 2.1.10. (a) The interference of 2px- or 2py-

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

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 - 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 -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

BABA dS *

Heteronuclear diatomairemoleculen: e- verdeling asymmetrisch

BBAAMO cc

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 = cB

2 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

ddHE **

0 AcE 0 BcE and

Hybridization and the Structure of Polyatomic Molecules

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.

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: sHpLisLi 187.0229.0241.0

Hybridization and the Structure of Polyatomic Molecules

sHpLisLi 187.0229.0241.0

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.

Deze coefficientenoptimale compromis

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: sHpLisLi 187.0229.0241.0

Simpele atomic orbital overlap idee weg…?

Zie de AO’s van Li als een hybride AO:

sHLihybride 187.05.0

pLisLiLihybride 258.0281.0

Hybridization 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 vanmet 4 elektronen te verdelen over deze bindingenoverlap elke H1s met een O2p, resulterend in 2 σ-bonds, met elk 2 e,dus:

11222 22221 yxz pppss

BAyx sHsHpOpO 1,1,2,2

22222 221 BAzpss

Maar: hoek van 90o, in werkelijkheid 104o…

Hybridization and the Structure of Polyatomic Molecules

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’.

In the MO description of H2O we aim to construct two O-H bonds that are chemically equivalent, but spatially distinct.

H2O

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% 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’.

2221,'

21,

2 2''1OBA zshsho phs

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

yxiiisp

yxiisp

xisp

pps

pps

ps

22

12

6

123

1

22

12

6

123

1

2223

1

2

2

2

More hybrids

)22(2

1

)22(2

1

xiisp

xisp

ps

ps

BeH2 ground state electronic configuration of the Be-atom is 22 21 ss

Koolstof atoom: 222 221 pss

De buitenste 4 valentie electronen kunnen 4 equivalentesp3 hybride orbitalen vormen:

zyxisp

zyxiiisp

zyxiisp

zyxisp

ppps

ppps

ppps

ppps

22224

1

22224

1

22224

1

22224

1

3

3

3

3

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

1112 2222 zyx ppps

221,'

21,'

21,

2 ''1 hsCBA shshsho

Lone pairs: H-bonding

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.2.1.24 Bonding in ethylene. (a) In the plane of the nuclei: the formation of a bond between the carbon atoms 1 and 2 using sp2 hybrid orbitals and -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 2p-orbitals (that were not involved in the sp2-hybrids.

Voorbeeld ethyleen

Π Electronen houden het molekuul vlak!

Huckel theorieπ 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