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

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)

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

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

Covalent bond in H2 molecule

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

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

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

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

Carbon bonds

sp3- hybrid-Orbital sp2- hybrid-Orbital sp- hybrid-Orbital

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).

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).

Ionic bond Charge density of NaCl

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

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

Madelung constant Linear chain a badly converging problem For NaCl a(CsCl)= 1.76267 a(ZnS) = 1.6381 a(NaCl)= 1.74756

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

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) = -1.13 eV/atom - 3.09 + 1.95 + 4.12 - 3.14 - 0.9 = -1.06 eV - 3.09 + 1.95 = -1.14 eV Problem : difference of large numbers

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

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

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.

Hydrogen bridging bonds Quelle : Wikipedia, Chemgapedia

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