Solid state physics

Lecture 3: chemical bonding

Prof. Dr. U. Pietsch

Electron charge density distribution fromx-ray diffraction data

Electronic charge densityof silicon

Valence charge density difference charge density(bonding charges)

U.Pietschphys.stat.sol.(b) 137, 441,(1986)

dkikrkFr )exp()()(

h

h

k

k

l

l

ii HrHFV

r );2exp()(2

)( )(1

hkla

H

Electron density distribution of GaAs fromx-ray diffraction data

Valence charge densitydifference charge density(bonding charges)

Ga As

Ga

As

Charge density determines the type of bond

Depending on the type of interaction bonding can be classified as :Ionic bond (e.g. NaCl): 1…10eV / bondCovalent bond (e.g. Diamond, Si): 1…10eV / bondMetallic bond (e.g. Alkali-metals Li,Na…): ~1eV / bondVan der Waals bonding (nobel-gas crystals): ~0.1eV / bondHydrogen bond (water /ice): ~0.1eV / bond

Classification of bonding in crystals

Covalent bond in H2 molecule

)111111

(4

²)(

2

²

121221

21rRrrrr

e

mH

baba

nnn WH

Linear combination of atomic orbitals (LCAO approach)

1221

1221

baba

baba

BA

BA

Ansatz:

1_

0_

Sspinsparallel

Sspinselantiparall

Zero approximation: R∞; separated atoms

)11

(4

²)(

2

²

21

210

ba rr

e

mH

Difference H1=H-H0

)1111

(4

²

1212

1rRrr

eH

ba

²12

S

WWWW AC

H

By perturbationtheory :

Bonding energies

²1 S

WWW AC

dVHW babaC 211

*

2

*

1

Coulomb integral

Exchange integral

dVHW babaA 121

*

2

*

1

dVS baba 12

*

2

*

1²

overlap integral

Bonding state

Antibonding state

Model using 1s obitals only : R=0.88 A, Wmin=-2.9 eVExact model (experiment) : R=0.74 A, Wmin=-4.5 eV

1. Ionic contribution(5%)

21211221 bbaababa DCBA

Further improvements:

H2+ H2

-

...)exp()exp( 21 21 raAraA2. Considering morewave functions

3. Considering highershell wave functions

)......2()2()1( pCsBsA

Note, wave functions are not physical quantities : one can use anyfunction, the use of „atomic like“ wave function is an approximation

||2 electron density distribution can be observed

s ss

ssp

s pd

pp

dp dd

s and bonds

Carbon bonds

Solution for solidsIn 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 electronsis independent from movement of these other electrons. Individual interaction of thiselectron with each of the other electrons is neglected. That mean that Coulomb termin Hamiltonian is replaced by mean potential Vi=∑1/rjj which depends solely byposition 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* V i+1

Poisson equ. Cyclic repetition up to E becomes minimum, vary V, HARTREE (1920) described the total wave function of the multi –electron system by theproduct of orbitals (+ spin)of the contributing electrons (Hartree-product).

)()()...2()1(1

21 inn

inel

iii EVKH )(

0

)()(

rrV

V. Fock (1930) improved the approach considering the permutability ofelectrons. This is described by the Slater determinate where each electroncan occupy each orbital. By exchange of two electrons (two lines of thedeterminate) changes the sign of wave function (anti symmetry condition). Hartee approach is meaning the diagnal elements of this matrix only

Solution of Hartree-Fock equation

with Fock operator Fi=∑Hi+∑(Ji−Ki)

Containing the Hamiltonian of the electron in the mean potential of otherelectrons and nuclei (Fock-operator ); it contains a single-electron kinetic oprator and a two-electron operatorreplacing (Coulomb- and exchange-operator).

)(...)2()1(

........

)(...)2()1(

)(...)2()1(

!

1 222

111

n

n

n

N

nnn

iiii E=F

Hartree-Fock approach

Ionic bond

Charge densityof NaCl

Cohesive energy of ionic crystal

Attractive interaction:

Repulsive interaction: or

Sum or

Total energy: per ion pair

Madelung constant

Finding bonding energy andequilibrium distance

4

1k

)²

)exp((r

eZu r

nmtot

0

r

utot

Bonding distance and cohesive energy

Cohesive energyper ion pair

nm

r

Zr

er

0

2

0

²)exp(

Equilibrium distance

Madelung constant

74756.1....5

24

4

6

3

8

2

126

For NaCl

Linear chain

badly converging problem

(CsCl)= 1.76267(ZnS) = 1.6381(NaCl)= 1.74756

Metallic bond

In metals at least one electron/atoms isexcited into the conduction band. Overlap ofrespective wave functions over a length ofmany atomic distances gives rise to decreaseof mean potential energy .This term isattractive and is partially cancelled by therepulsive mean kinetic energy . In this scheme bonding energy is

potE

kinE

kinpotpot EEE

Conducton band

valence band

Free electrons

Valence electrons

Fermienergy

E0=Epot

More detailed picture

correxchageCoulPauliAB EEEEEEE )(

AEE energy difference between lower band edge of conduction band and energy of free atom - bonding energy of ions

pauliE Due to Pauli principle only 2 electrons occupy energy E, all other haveto 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 importantfor metals.

exchangeE

CoulE

corrE

Example for Na: EB Exp) = -1.13 eV/atom

correxchageCoulPauliAB EEEEEEE )(

- 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 atomicdipoles while oscillation of negative chagedelectrons against the positive nucleus.

2

22

12

22

12

12

12

12

10 CxpCxpH

mm

Hamiltonian of decoupled oscillators

Oscillator frequency mC /0

H1 is Coulomb interaction of both oscillators

³

))((2

³

²2²²²² 2121

2121

1R

exex

R

xxe

xR

e

xR

e

xxR

e

R

eH

)();(

)();(

212

1212

1

212

1212

1

pppppp

xxxxxx

as

as

2

2

12

2

12

2

12

2

110 )

³

²2()

³

²2( aamssm

xR

eCpx

R

eCpHH

...)³

²2()

³

²2(1[ 2

8

1

2

10

³²2

CR

e

CR

e

m

CRe

Second term cancels out first valuable change comes from 3rd term

6

2

81

0 )³

²2(

R

A

CR

eU Attractive

Quantum effect, because U 0 if h=0, A scales with electron polarisability

Repulsive part (guess) 12R

BU rep

Both together ])()[(4)( 612

612 RRR

A

R

BRU

ss

Van der Waals term

Lennard-Jones potential

Hydrogen bonds

Hydrogen bridging in molecules and polymers

Polar hydrogen bond in water

hexagonal structure of ice

Often described as dipole-dipole interaction with covalent features direction of bond

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

Hydrogen bonds in DNA