A chemist's view of bonding - IJS
Transcript of A chemist's view of bonding - IJS
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A chemist's view of bonding &
a very informal intro to Bloch theorem( from molecules to solids and back )
Anton KokaljDepartment of Physical and Organic Chemistry
Jožef Stefan Institute
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Basics ...
time-independent Schrödinger equation
Born-Oppenheimer approximation
one-electron approximation Hartee-Fock DFT (Kohn-Sham)
molecular orbitals
formally exact, but in practice approximative
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Molecular orbitals
molecular orbital = linear combination of atomic orbitals (MO) (LCAO)
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Molecular orbitals
molecular orbital = linear combination of atomic orbitals (MO) (LCAO)
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Molecular orbitals
molecular orbital = linear combination of atomic orbitals (MO) (LCAO)
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A chemist's view of bonding
molecular orbital (MO) diagram
unstable !
HAB
= interaction between sA & s
B
SAB
= overlap between sA & s
B
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A chemist's view of bonding
molecular orbital (MO) diagram
unstable !
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A chemist's view of bonding
molecular orbital (MO) diagram
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A chemist's view of bonding
molecular orbital (MO) diagram
unstable !
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A chemist's view of bonding
molecular orbital (MO) diagram
unstable !
keep antibonding states emptykeep antibonding states empty
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A chemist's view of bonding
from H2 to solid
HOMO – highest occupied MOLUMO – lowest unoccupied MO
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A chemist's view of bonding
from H2 to solid
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A chemist's view of bonding
from H2 to solid
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Density of states (DOS)
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Density of states (DOS)
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Electronic structure of metals
two types of bands:delocalized (broad) sp-bandslocalized (narrow) d-bands
Transition metals (TM):s-band is half filled for all TMs (similar bonding for all)
variation in bonding comes from d-band
Fermi energy (EF)
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From H2 to solids … only 1D for simplicity
lets’ consider a simple 1D system: an infinite chain of H atoms
due to translational symmetry:ρ(x) = ρ(x + R)
V(x) = V(x + R)
what about wave-function? Let’s try:ψ(x) ?=? ψ(x + R)
, where R = na
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From H2 to solids … only 1D for simplicity
what about wave-function? Let’s try:ψ(x) ?=? ψ(x + R)
I can only put 2 electrons in this state …I need to form linear combinations!
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From H2 to H
∞ ...
H2
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From H2 to H
∞ ...
H3
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From H2 to H
∞ ...
H4
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From H2 to H
∞ ...
H9
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H9
sin(kx)
more general: exp(ikx)
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From H2 to H
∞ ...
HN
(suppose N is very large)
ψ(0) = ψ(L)ψ(x) = ψ(x + L)
Born−von Karman boundary conditions
0
L
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cos(2π/L∙x)
cos(2π/L∙2x)
cos(2π/L∙3x)
cos(2π/L∙4x)
cos(2π/L∙Nx/2) = cos(π/a∙x)
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cos(2π/L∙x)
cos(2π/L∙2x)
cos(2π/L∙3x)
cos(2π/L∙4x)
cos(2π/L∙Nx/2) = cos(π/a∙x)
more general:
exp(ikx), where k = 2πκ/(Na) and κ ∈ (−N/2, N/2](if κ is outside this range it folds back due to periodicity)
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cos(2π/L∙1x)
cos(2π/L∙2x)
cos(2π/L∙3x)
cos(2π/L∙4x)
cos(2π/L∙Nx/2) = cos(π/a∙x)
cos(2π/L∙0x) = 1
ψk (x) = exp(ikx) u(x)
more general:
exp(ikx), where k = 2πκ/(Na) and κ ∈ (−N/2, N/2]
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Bloch theorem (still 1D chain of H atoms)
ψk(x) = exp(ikx) u
k(x) notice:
ψk(x+R) = exp(ikR) ψ
k(x)
where R is any lattice translationcell functionhas periodicity of lattice u
k(x+R) = u
k(x)
sum over lattice vectors(R = na, n ∈ (−∞,∞))
sum over „reciprocal“ lattice vectors(G = n/a, n ∈ (−∞,∞))
sum „all“ 1s orbitals plane-waves
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K-points
ψk(x) = exp(ikx) u
k(x) where k = 2πκ/(Na) and κ ∈ (−N/2, N/2]
band structure
k = 0
k = π/a
note that ε(k) = ε(−k)
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Supercell
Let’s double the unit cell: anew
= 2a
but: k 1/∝ a, hence knew
∈ [0, π/anew
] while k ∈ [0, π/a]
Remember !
when supercell increases:
# of k-points decreases# of bands increases
π/2a
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Supercell & symmetry lowering
uniform chain of H-atoms is not stable !
unstable
stable
Jahn-Teller or Peierls distortion
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Supercell & symmetry lowering
uniform chain of H2 pairs
LOMO
HOMO
LUMO
HUMO
combinations of σ-bonding orbitals
combinations of σ*-antibonding orbitals Unoccupied
Occupied
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Supercell & symmetry lowering
uniform chain of H2 pairs
LOMO
HOMO
LUMO
HUMO
combinations of σ-bonding orbitals
combinations of σ*-antibonding orbitals Unoccupied
Occupied
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Supercell & symmetry lowering
uniform chain of H2 pairs: band-gap opening
band structure
occupied band
empty band
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What determines the band-width ?
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What determines the band-width ?
overlap !
Effect of strain on band-width
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What determines the band-width ?
overlap !
Effect of coordination on band-width
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From molecules to solids: interaction of a molecule with a surface
concept ofcharge donation and back-donation
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Electronic structure of transition metals (TM)
Hammer-Nørskov chemisorption model → d-band center (ed)
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Molecule – TM-surface interaction
d-band center
d-band
TM ≡ transition metal
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Molecule – TM-surface interaction
TM ≡ transition metal
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Molecule – TM-surface interaction
TM ≡ transition metal
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Molecule – TM-surface interaction
integrates totwo electrons
TM ≡ transition metal
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Molecule – TM-surface interaction
integrates totwo electrons
TM ≡ transition metal
projwfc.xmolecularpdos.x hands-on excercise on Tue: CO@Rh(100)
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Molecule – TM-surface interaction
TM ≡ transition metal
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Tuning the molecule – TM-surface interaction
TM ≡ transition metal
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Tuning the molecule – TM-surface interaction
TM ≡ transition metal
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Tuning the molecule – TM-surface interaction
TM ≡ transition metal
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Position of d-band
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TM: bonding decreases from left to right
1Kokalj, Chem. Phys. 393 (2012) 1–12
Chemisorption of N, O, Cl @ 3d TMs1
weak bonding
strong bonding
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Take home message
recipe for bonding: keep antibonding states empty
ψk(x) = exp(ikx) u
k(x) where k = 2πκ/(Na) and κ ∈ (−N/2, N/2]
or k ∈ 2π(−1/(2a), 1/(2a)] = 2π(−a*/2, a*/2] (for 1D)
when supercell increases: # of k-points decreases # of bands increases
Brillouin zone
a* = reciprocal lattice vector
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Literature
Roald Hoffmann, “SOLIDS and SURFACES: A Chemist’s View of Bonding in Extended Structures” (Wiley-VCH)
Roald Hoffmann, “A chemical and theoretical way to look at bonding on surfaces”, Rev. Mod. Phys. 60, 601–628
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Acknowledgment
to my daughter Katarina for calculating and plotting all pictures of orbitals
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Thank you very much for your attention