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
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
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
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
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