1. Solid state physics
Lecture 3: chemical bonding
Prof. Dr. U. Pietsch

2. Electron charge density distribution from x-ray diffraction data
Electronic charge density
of silicon
U.Pietsch
phys.stat.sol.(b)
137, 441,(1986)
Valence charge density
difference charge density
(bonding charges)

3. Electron density distribution of GaAs from x-ray diffraction data
difference charge density
(bonding charges)
Valence charge density
Charge density determines the type of bond

4. Classification of bonding in crystals
Depending on the type of interaction bonding can be classified as :
Ionic bond (e.g. NaCl): 1…10eV / bond
Covalent bond (e.g. Diamond, Si): 1…10eV / bond
Metallic bond (e.g. Alkali-metals Li,Na…): ~1eV / bond
Van der Waals bonding (nobel-gas crystals): ~0.1eV / bond
Hydrogen bond (water /ice): ~0.1eV / bond

5. Covalent bond in H2 molecule

6. Linear combination of atomic orbitals (LCAO approach)
Ansatz:
Zero approximation: R∞; separated atoms
Difference H1=H-H0
By perturbation
theory :

7. Bonding energies Model using 1s obitals only : R=0.88 A, Wmin=-2.9 eV
Coulomb integral
Antibonding state
Exchange integral
overlap integral
Bonding state
Model using 1s obitals only : R=0.88 A, Wmin=-2.9 eV
Exact model (experiment) : R=0.74 A, Wmin=-4.5 eV

8. Further improvements:
1. Ionic contribution (5%)
H2+  H2-
2. Considering more wave functions
3. Considering higher shell wave functions
Note, wave functions are not physical quantities : one can use any function, the use of „atomic like“ wave function is an approximation
 |y|2  electron density distribution can be observed

9. s and p bonds
ssp
s ss
ppp
s pd
p dp
p dd

10. Carbon bonds

11. sp3- hybrid-Orbital
sp2- hybrid-Orbital
sp- hybrid-Orbital

12. Solution for solids
In solids quantum mechanical calculations is based on Born–Oppenheimer (BO) approximation making the assumption that the motion of atomic nuclei and the electrons in a solid can be separated. Schrödinger equation of electrons movement is solved for fixed positions of nuclei.
For electron system single particle movement in mean potential of all other electrons is independent from movement of these other electrons. Individual interaction of this electron with each of the other electrons is neglected. That mean that Coulomb term in Hamiltonian is replaced by mean potential Vi=∑1/rjj which depends solely by position i,j of the selected electron.
Solution of Schrödiger equation for one electron
(K- kinetic enery, V- potential) in self-constistent field of the other electrons
Viyyy*=r  V i+1
Poisson equ.
Cyclic repetition to minimase E, varies V, y
HARTREE (1920) described the total wave function of multi –electron system by the product of orbitals (+ spin) of all contributing electrons (Hartree-product).

13. Hartree-Fock approach
V. Fock (1930) improved the approach considering the permutability of electrons. This is described by the Slater determinate where each electron can occupy each orbital. By exchange of two electrons (two lines of the determinate) the sign of wave functions is changing (anti symmetry condition). Hartee approach is realized by using the diagnal elements of this matrix only
Solution of Hartree-Fock equation
with Fock operator Fi=∑Hi+∑(Ji−Ki)
contains the Hamiltonian of the electron in the mean potential of all other electrons and nuclei (Fock-operator );
it contains a single-electron kinetic operator and a two-electron operator replacing (Coulomb- and exchange-operator).

14. Ionic bond
Charge density
of NaCl

15. Cohesive energy of ionic crystal
Attractive interaction:
Repulsive interaction: or or
Sum
Total energy:
per ion pair
Madelung constant
Finding bonding energy and equilibrium distance

16. Bonding distance and cohesive energy
Equilibrium distance
Cohesive energy
per ion pair

17. Madelung constant Linear chain a badly converging problem For NaCl
a(CsCl)=
a(ZnS) =
a(NaCl)=

18. Metallic bond
In metals at least one electron/atoms is excited into the conduction band. Overlap of respective wave functions over a length of many atomic distances gives rise to decrease of mean potential energy This term is attractive and is partially cancelled by the repulsive mean kinetic energy
In this scheme bonding energy is
Fermi
energy
Free electrons
Conducton band
Valence electrons
valence band
E0=EB

19. More detailed picture
energy difference between lower band edge of conduction band
and energy of free atom - bonding energy of ions
Due to Pauli principle only 2 electrons occupy energy E, all other have to occupy higher states.
Coulomb interaction of cunducting electrons - repulsive
Exchange interaction between electrons of opposite spin - attractive
Correlation energy - Impact of many body interaction - very important for metals.
Example for Na: EB Exp) = eV/atom
= eV
= eV
Problem : difference of large numbers

20. Chemical bond in Noble gases
Key assumption: formation of atomic dipoles while oscillation of negative chaged electrons against the positive nucleus.
Hamiltonian of decoupled oscillators
Oscillator frequency
H1 is Coulomb interaction of both oscillators

21. Lennard-Jones potential
Second term cancels out  first valuable change comes from 3rd term
Attractive
Van der Waals term
Quantum effect, because DU 0 if h=0, A scales with electron polarisability a
Repulsive part (guess)
Both together
Lennard-Jones potential

23. Hydrogen bonds
Often described as dipole-dipole interaction with covalent features  direction of bond
hexagonal structure of ice
Polar hydrogen bond in water
Hydrogen brides in molecules and polymers
In the X−H···Y system, the dots represent the hydrogen bond: the X−H distance is ≈110 pm, whereas the H···Y distance is ≈160 …200 pm.

24. Hydrogen bridging bonds
Quelle : Wikipedia, Chemgapedia

25. Hydrogen bonds in DNA
Hydrogen bonding between guanine and cytosine, one of two types of base pairs in DNA.