Estude fácil! Tem muito documento disponível na Docsity
Ganhe pontos ajudando outros esrudantes ou compre um plano Premium
Prepare-se para as provas
Estude fácil! Tem muito documento disponível na Docsity
Prepare-se para as provas com trabalhos de outros alunos como você, aqui na Docsity
Encontra documentos específicos para os exames da tua universidade
Prepare-se com as videoaulas e exercícios resolvidos criados a partir da grade da sua Universidade
Responda perguntas de provas passadas e avalie sua preparação.
Ganhe pontos para baixar
Ganhe pontos ajudando outros esrudantes ou compre um plano Premium
Um problema relacionado à teoria do hamiltoniano geral de matéria condensada, especificamente sobre a aproximação de born-oppenheimer para a separação de variáveis electrônicas e iônicas, e as limitações desta abordagem. O texto também discute a teoria de kohn-sham e a definição de um funcional universal (f) com base no hamiltoniano de kohn-sham. Além disso, são discutidos os equações de kohn-sham, o princípio variacional e a definição de um estado de densidade de elétrons estacionário.
Tipologia: Notas de estudo
1 / 22
Esta página não é visível na pré-visualização
Não perca as partes importantes!
P Kratzer
FritzHaberInstitut der MPG
D BerlinDahlem Germany
I From the manyparticle problem to the
KohnSham functional
I I How to p erform a total energy calculation
I I I From the total energy to materials science
e
ion
ee
eion
ionion E
wavefunction r r N
I
M
electronic co ordinates r k k N
ionic co ordintes R I
T
e
N X
k
p
k
m
T
ion
M X
I
P
I
MI
V
ee
N N X
k k
e
jr k r k j
V
ionion
M M X
I I
ZI Z I
jRI R I
j
V
eion rk RI
N X
k
M X
I
v
ion
I
jRI rk j
do esnt account for correlated dynamics of ionic and
electronic co ordinates
Example
supra uid He p olaroninduced sup erconductivity
breakdown of the restriction to a single groundstate
BornOpp enheimer surface
Example chemoluminescence
0 1 2
distance
−
0
1
energy
breakdown of the adiabatic approximation
Example
excitation of surface plasmons during scattering of
an ion from a metal surface
timedep endent theories eg TDDFT
N
k X
r
k
m
v (^)
r
k
N
(^) N
X k
k
(^) W
r
k r
k
r r N E
r r N
W r r
e
(^)
j r r
j
v
(^)
r
M
I X (^)
v
I (^) ion
j R I r j
still
many
fo
r
a
t ypical
solid
degrees
of
freedom
The
manypa
rticle
p
roblem
can
b
(^) e
solved
only
fo
r
rather
small
systems
atoms
molecules
and
clusters
using
established
metho
(^) ds
of
quantum
chemistry
eg
conguration
interaction
KohnSham single particle Hamiltonian
To nd the stationary p oint we do variations at xed
dr n r which leads to
v
n r
Lagrange parameter
If we write the density as a sum over singleparticle
functions
n r
N
j
kBZ
j jk r j
the variational principle E v
r leads
to the KohnSham equations
r
m
e
n r
jk r jk
jk r
with the eective p otential
e
n r v
R I
r
dr
e
n r
jr r
j
X C n
n
r
The
total
energy
fo
r
static
ions
tot
n T n Z d r v
R (^)
i r n r
Z d r Z d r e
n^
r n r
j r
r j^
X
(^) C
n
ion
ion
R
I
tot
n
N
j X (^)
X
k
BZ
j
k
(^) e
e ^ n
X
(^) C
n
ion
ion
R
I
(^) e
e ^ n
Z d r Z d r e
n^
r n r
j r
r j^
(^) e
e ^ n
X
(^) C
n
X
(^) C
n Z d r V X
(^) C
n
r n r
E
tot
is
stationa
ry
with
resp
to
va
riations
of
n
a round
n
(^)
but
the
individual
terms
a re
not
crystal
structure
Bravais
lattice
basis
translational
symmetry
symmetries
of
the
basis
compatible
with
the
Bravais
lattice
group
p
(^) oint
group
Example
t w
o
(^) dimensional
honeycomb
lattice
Bravais
lattice
if
a
basis
of
t w
o
atoms
is
used
a
a
b
b^
Brillouin
zone
lattice
vecto
rs
a
(^)
a
a
recip
ro
(^) cal
lattice
vecto
rs
b
b
b
b^ i
a
j
a
k (^)
f
ijk
g
f
g
cyclic
p
(^) ermutations
of
indices
Since H commutes with the elements of T the wave
functions must b e of the form
jk r R e
ikR jk r Blo chs theorem
for any R n a n a n a
The wavevector k is an index sp ecied by a p oint in the
rst Brillouin zone elementary cell of the recipro cal
lattice and the numb er of such p oints is equal to the
numb er of lattice sites in the crystal
Due to Blo chs theorem it suces to calculate in
just one elementary cell
jk r e
ikr u jk r
where u is a latticep erio dic function
In practice sums over k are evaluated by summing
over a discrete set of sp ecial kp oints
Ha
rris
functional
tot
n i N
j X (^)
X
k
BZ
j
k
(^) e
e ^ n
i
X
(^) C
n
i
ion
ion
R
I
up
date
cha
rge
densit
y
n
i ^ r
N
k
X
k
n
B
X
j
f
j
k (^) j
i
j
k r j
with
f
j
k
exp
j
k
F
k
In
p
rinciple
w
e
a
re
interested
in
but
nite
helps
to
stabilize
the
SC
lo
(^) op
tot
n
i
T n i
Z
d
r
v
R (^)
I r n i r
(^) e
e ^ n
i r
X
(^) C
n
i ^
ion
ion
R
I
KohnSham
functional
tot
n E
tot
n
(^)
with
selfconsistent
densit
y
n
A
t
the
stationa
ry
p
(^) oint
the
KohnSham
equations
lead
us
to
(^) n
N
j X (^)
X
k
BZ
j
k Z d r n r V
e
n
r
F
o
r
the
nonconsistent
case
w
e
ma
y
intro
duce
the
generalization
(^) n
e
N
j X (^)
X
k
BZ
j
k V
e
Z d r n r V
e
r
This
can
b
(^) e
used
to
dene
the
double
functional
tot (^) D
n
e
(^) n
e
(^) e e n E X
(^) C
n
ion
ion
R
I
with
the
va
riational
p
rop
erties
tot (^) D
n
(^) n
e
n
(^)
tot (^) D
n
(^)
V
e
n
(^)
c (^) (^) n
tot (^) D
n
V
e
n
(^)
(^) v
(^)
tot (^) D
n
(^)
V
e
n
(^)
c (^) (^) v
No
rmally
one
has
c c
The
Ha
rris
functional
in
this
notation
is
tot (^) D
n
i
e
n
i
GaAs Ga metal
As
GaAs bulk
Ga
p T G GaAs
As p T
As p T from ideal twoatomic gas S
ion
GaAs
and S
ion
Ga
from Debye mo del of the solid
500 600 700 800 900 1000
temperature [K]
−3.
−3.
−2.
g
Ga
[eV/particle]
μGa(GaAs, p(As 2 )=
− Pa)
μ Ga (GaAs, p(As 2 )=
− Pa)
μ Ga (bulk Ga)
Free enthalpy p er Ga atom in bulk GaAs in
thermo dynamic equilibrium with As vap or at two
pressures compared to the free enthalpy p er Ga atom
in elemental Ga
P Kratzer et al Phys Rev B
o ccupation numb ers from Fermi distribution
f jk exp jk
F k T
The program actually computes the free energy F
entropy of the electronic system
e k B
n B
j
kBZ
f jk lnf jk f jk ln f jk
For a freeelectronlike metal we have
O T
O T
extrap olation to zero electronic temp erature
e T
see M J Gillan J Phys Cond Mat
C second derivatives
Examples
force constant matrix
tot
i
j
calculation of phonon sp ectrum vibrational
entropy
particle numb er uctuations
chemical softness and hardness
sr
nr
v T
E tot
nr nr
dr
hr
nr
v T
E tot
nr nr
nr
dr
Note When calculating second derivatives the
resp onse of the density must b e taken into account
Density Functional Perturbation Theory
structural prop erties
Examples structural phase transitions surface
reconstructions yes
elastic prop erties
Examples bulk mo dulus C
yes
chemical prop erties
Examples thermo chemical stability of comp ounds
reactivity of surfaces yes
transp ort prop erties
Examples transp ort eective mass magneto
resistence developing eld
opticalsp ectroscopic prop erties
Examples photo emission sp ectra cross sections for
light absorption
topic b eyond KohnSham theory encouraging
progress recently
many other applications
