c - Quantum Chemistry Group

Transcription

Murcia 2008
Electronic structure of solids
Vı́ctor Luaña
Departamento de Quı́mica Fı́sica y Analı́tica, Universidad de Oviedo
http://web.uniovi.es/qcg/
Programa Interuniversitario de Doctorado en Quı́mica Teórica y Computacional
Murcia, 21-01-2008 al 16-02-2008
c V. Luaña, QTC Murcia 2008
(1)
Lgm1: Geometry and symmetry of crystals
Geometry and symmetry of crystals
(2)
Crystal lattice
Periodic crystals: The crystal lattice and crystal coordinates
A periodic crystal can be described as a convolution of two elements: a periodic network of points
(the lattice) and a molecular motif.
+
Molecular
motif
b
a
Cristal
Lattice
Z
Convolution of two 1D functions:
h(x) = f ◦ g(x) =
f (t) g(x − t) dt.
Ω
Lattice: infinite but discrete network of points that exhibit translational symmetry. There exist a
(non-unique) set of independent lattice vectors, {~a, ~b, ~c}, such that all lattice points are connected
by primitive translations:
T ≡ {~t = n1~a + n2~b + n3~c | (n1 , n2 , n3 ) ∈ Z3 }
(1)
(3)
Crystal lattice
Some basic definitions
Unit cell: The {~a, ~b, ~c} vectors define a parallelepipedon that reproduces the whole lattice network by means of primitive translations. There is an inifinite number of ways for selecting a unit
cell.
Primitive cell: Any primitive cell contains exactly one lattice
point and encloses the minimal volume that reproduces the whole
crystal by translations alone.
Centered cell: n primitive cells can be glued together to form a
centered cell containing n lattice points.
The use of a primitive cell means that the lattice is formed by primitive translations alone.
The use of a centered cell means that the lattice contains fractional translations in addition to the
primitive ones.
Centered cells are used for convenience: showing better the symmetry of the crystal, evidencing the
relationship of two structures, etc.
(4)
Crystal lattice
Crystal geometry
c
Cell parameters: (a, b, c, α, β, γ).
α
b
γ
a
cos α = (~b · ~c)/bc,
β
Director cosines:
cos β = (~c · ~a)/ca,
cos γ = (~a · ~b)/ab.
Cell volume: V = ~a · (~b × ~c) = ~b · (~c × ~a) = ~c · (~a × ~b).
Crystallographic coordinates: Any arbitrary point in the crystal is described by a vector:
 
x i 
 ax ,
~
ri = xi~a + yi~b + zi~c = ~a ~b ~c 
xi ∈ R3 .
i
 yi  = ˜
zi
(2)
Main cell: Points in the main cell are such that 0 ≤ xi , yi , zi < 1. Any point in the crystal has an
equivalent in the main cell, and it can be converted to it by a primitive translation.
The crystal is an euclidean space described in terms of a non-orthonormal basis set. Most
of the vector operations must be learned again.
(5)
Crystal lattice
The scalar product (aka inner product) of two vectors is a measurement of the length of one of the vectors multiplied by the length of the
perpendicular projection of the other on it.
~
u · ~v = ~v · ~
u = uv cos θ.
√
Vector’s norm: k~
uk = u = ~
u·~
u.
For two perpendicular vectors ~
u · ~v = 0.
The vector product of two vectors, ~
u × ~v , is another vector such that:
• it is perpendicular to both vector factors;
• the length is equal to the area of the parallelogram defined by the
vector factors, k~
u × ~v k = uv sin θ; • the orientation is given by the right
hand rule (the thumb points as the vector product when the rest of
fingers move from the first to the second factor).
For any vector: ~
u×~
u = ~0.
(3)
Mixed triple product (aka box product): ~
u · (~v × w)
~ = ~v · (w
~ ×~
u) = w
~ · (~
u × ~v ) gives the volume
of the parallelepipedon defined by the three vectors.
(6)
Crystal lattice
Vector operations using the crystallographic coordinates
Vector addition:
Scalar product:


xi ± xj 
~
ri ± ~
rj = a(xi ± xj ) = ~a ~b ~c 
(4)
yi ± y j 

.
˜
 
zi ± zj
xj


T
T T
T

~
ri · ~
rj = (a xi ) (a xj ) = xi a a xj = xi G xj = xi yi zi G  yj 
 , (5)
˜
˜
˜ ˜
zj
where
 
~a 

T

~
G = a a = b
 ~a
˜ ˜
~c
~b

~a · ~a
~
~c = 
 b · ~a
~c · ~a
~a · ~b
~b · ~b
~c · ~b
 
~a · ~c
a2
 
~b · ~c  = ab cos γ
 
~c · ~c
ac cos β
ab cos γ
b2
bc cos α
ac cos β


bc cos α 

c2
(6)
is the metrical matrix (aka metrical tensor) of the lattice.
For the orthonormal {~i, ~j, ~k} basis G = 1.
The determinant of the metrical tensor is the square of the cell volume:
p
2
V = det G = G
=⇒ V = abc 1 − cos2 α − cos2 β − cos2 γ + 2 cos α cos β cos γ
(7)
Vector products can be simplified by the introduction of the reciprocal lattice construction.
(7)
Crystal lattice
The reciprocal cell and the vector product
The reciprocal of the (~a, ~b, ~c) cell is formed by the (~a ? , ~b ? , ~c ? ) vectors, defined is such a way that
the reciprocal of a cell vector has an inverse length and it is perpendicular to the other two vectors
 


of the cell. In other words:
?
~a
1 0 0




? T
?



~
~
(a ) a = s1
=⇒  b  ~a b ~c = s 0 1 0
(8)

˜
˜
~c ?
0 0 1
where s is an adimensional factor. In solid state physics is customary to choose s = 2π, whereas
the crystallographical and mathematical convention is s = 1. We will use s = 1 for reasons that will
be later explained.
The reciprocal lattice definition is satisfied by choosing:
~a
?
~b × ~c
=
,
V
~b ? = ~c × ~a ,
V
~a × ~b
~c =
.
V
?
(9)
Direct and reciprocal cell have inverse volumes: V ? V = 1 (V ? = ~a ? · (~b ? × ~c ? )). Additionally
?
?
?
~b ? × ~c ?
~b ?
~
c
×
~
a
~
a
×
~b =
~a =
,
,
~c =
.
(10)
?
V
V?
V?
Therefore, our original cell (aka direct cell) is the reciprocal of the reciprocal cell: (a? )? = a.
˜
˜
(8)
Crystal lattice
Reciprocal cell parameters: (a? , b? , c? , α? , β ? , γ ? ). These parameters can be obtained from the
direct cell lenghts and angles as
ca sin β
ab sin γ
bc sin α
,
b? =
,
c? =
,
V
V
V
cos β cos γ− cos α
cos γ cos α− cos β
cos α cos β− cos γ
cos α? =
, cos β ? =
, cos γ ? =
.
sen β sen γ
sen γ sen α
sen α sen β
a? =
Notice that if a, b, c [=] Å then a? , b? , c? [=] Å
−1
(11)
(12)
.
Vector product (of two direct cell vectors):
~
ri × ~
rj = (xi~a + yi~b + zi~c) × (xj ~a + yj~b + zj ~c)
yi
= yj
x
zi ~
(b × ~c) − i
zj | {z } xj
V~
a?
x
zi (~c × ~a) + i
zj | {z } xj
V~
b?
yi (~a × ~b) = V
yj | {z }
V~
c?
~a ?
xi
xj
~b ?
yi
yj
zi zj ~c ? (13)
Direct and reciprocal vectors. The next convention will be used to distinguish among them:
~
r = a x,
˜
~h = a? h,
˜
~κ = 2π~h = a? κ.
˜
(14)
Thus ~κ recovers the solid state convention of incorporating the 2π factor.
(9)
Crystal lattice
Scalar product on either the direct or reciprocal cell requires a metrical matrix, but the mixed
product of a vector from one space and a vector from the other does not require it:
~
ri · ~
rj = xTi G xj ,
~hi · ~hj = hT G? x ,
i
j
~h · ~
r = hT x,
~κ · ~
r = κT x,
(15)
where

(a? )2
a? b?

G? = G−1 = (a? )T a? = 
a? b? cos γ ?

˜ ˜
a? c? cos β ?
Vector product
~a ?
~
ri × ~
rj = V xi
xj
~b ? ~c ? yi zi ,
yj zj cos γ ?
(b? )2
b? c? cos α?
~a ~b ~c ~hi × ~hj = V ? h k l ,
i i i
h j k j lj 
a ? c?
cos β ?
b? c?
cos α?  .
(c? )2


(16)
~a
~
b
~c ?
~κi × ~κj = V κxi κyi κzi , (17)
κxj κyj κzj Mixed triple product
x y z i
i
i
~
ri · (~
rj × ~
rk ) = V xj yj zj ,
xk yk z k c V. Luaña, QTC Murcia 2008
h k l i
i
i
~hi · (~hj × ~hk ) = V ? h k l .
j j j
h k kk lk (18)
(10)
Crystal lattice
Transformations of the lattice vectors
Linear transformations: They imply a change in the orientation, length, or both of the basis
vectors. The position vectors do not change with the axis transformation.


p11 p12 p13 

r 0 = a0 x0 = a P x0 = a x = ~
r
(19)
~a 0 ~b 0 ~c 0 = ~a ~b ~c 
p21 p22 p23  =⇒ ~
˜
˜
˜
p31 p32 p33
The transformation can be interpreted as operating on the axes or on the coordinates:
a0 = a P
˜
˜
⇐⇒
x = P x0
⇐⇒
x0 = P−1 x
⇐⇒
a = a0 P−1 .
˜ ˜
(20)
Special case: an orthogonal transformation, P−1 = PT , represents a proper (if Det P > 0) or
improper (if Det P < 0) rotation of the cell axes.
Origin shift: The initial origin, O, is moved to O0 by a translation p
~ = ap. This modifies all
˜
0
0
0
position vectors: ~
r −→ ~
r =~
r−p
~, or a x = a(x − p).
˜
˜
Affine transformation: General operations have both, linear and shift terms: (P|p)x = P x + p.
(11)
Crystal lattice
Properties of the reciprocal space
The reciprocal lattice. Translational symmetry in the direct space also implies translational
symmetry in the reciprocal space. If ~t is a primitive translation:
r + ~t = ~
r
∀ ~
=⇒
~
~
~
~
~ ~
eih·~r = eih·(~r+t) = eih·~r eih·t
=⇒
~ ~
eih·t = 1.
(21)
~
r ∈R3
The network of lattice points in the direct space is transformed into a network of reciprocal lattice
~
points through the condition eih·~t = 1.
Crystalographic planes. In Miller’s notation, indices (h, k, l) define a plane intersecting the ~a, ~b
and ~c axes at a distance a/h, b/k and c/l, respectively, from the origin.
(0,0,1)
(1,1,1)
(1,0,2)
(1,−1,2)
(0,2,0)
Common notations: [uvw] a direction in the crystal; <uvw> a set of directions equivalent by
symmetry; (hkl) Miller indices of a plane; {hkl} a set of planes equivalent by symmetry.
(12)
Normal vector to a crystallographic plane: Vectors ~v1 = ~b/k − ~a/h and
~v2 = ~c/l − ~b/k lie in (h, k, l) plane. Their vectorial product:
c
c/l
Crystal lattice
v2
H
b/k
O
~h = f (~v1 × ~v2 ) = ... = f V (h~a ? + k~b ? + l~c ? )
hkl
b
v1
a/h
a
(22)
is perpendicular to the plane. Using f = hkl/V as a normalization factor, we
see that the reciprocal lattice vector ~h = a?~h = h~a ? + k~b ? + l~c ? is the normal
˜
to all the planes (λh, λk, λl), where λ ∈ R is a scale factor.
Planes in direct space are associated to directions in the reciprocal space, and viceversa. Both
spaces form a dual.
c
Special hexagonal plane indexing:
• Plane (hkil), where i is the reciprocal of the fractional intercept of the
~a3 axis. The three first indices are not independent, but h + k = −i.
−
(1100)
a3
• Direction [uvtw], related to the usual [U V W ] notation by: U = u−t,
V = v − t, W = w.
a2
a1
−
(1011)
(13)
Crystal lattice
Internal geometry
The crystallographic coordinates of the atoms within the unit cell gives the best description of the
crystal structure. The internal geometry (distances and angles) is easily computed from here. The
inverse is not so simple.
Distance between two atoms: dij
q
= k~
rj − ~
ri k = k~
rij k = + (xij )T G xij ..
Angle between three atoms:
~
rji · ~
rjk = rji rjk cos θijk ,
~
rji × ~
rjk = rji rjk sin θijk ,
tan θijk
~
rji × ~
rjk =
.
~
rji · ~
rjk
(23)
Dihedral angle between four atoms:
~
rijk · ~
rjkl = rijk rjkl cos θijkl ,
where
~
rijk = ~
rij × ~
rjk
and
~
rjkl = ~
rjk × ~
rkl .
(24)
(14)
Crystal systems
The seven crystal systems and the fourteen Bravais lattices
The 7 crystal systems appear as a consequence of imposing symmetry conditions upon the lattice
vectors. The consideration of primitive or centered cells on the crystal systems produces 14 different
cell types known as Bravais lattices.
System
Symmetry
Cell parameters
Bravais latt.
Cubic
Four C3 (3) axes
a = b = c, α = β = γ = 90◦
P, I, F
Hexagonal
a C6 (6) or S3 (6̄) axis
a = b, α = β = 90◦ , γ = 120◦
P
Trigonal (H)
a C3 (3) or S6 (3̄) axis
a = b, α = β = 90◦ , γ = 120◦
R
Trigonal (R)
a C3 (3) or S6 (3̄) axis
a = b = c, α = β = γ
R
Tetragonal
a C4 (4) or S4 (4̄) axis
a = b, α = β = γ = 90◦
P, I
Orthorhombic
three ⊥ C2 (2) or S2 (2̄) axes
α = β = γ = 90◦
P, I, F. C
Monoclinic
a C2 (2) or S2 (2̄) axis
α = β = 90◦ or α = γ = 90◦
P, B or C
Triclinic
E(1) or i(1̄)
none
P
Trigonal or rhombohedral: The true R cell is primitive, but the R cell with hexagonal axes contains
three lattice points and it is centered on (2/3, 1/3, 1/3) (obverse) or (1/3, 2/3, 1/3) (reverse).
(15)
Crystal systems
Bravais lattices
P: primitive; I: centered in the middle point (Inner); F: centered on each cell face (Faces); A (B
or C): centered of the faces perpendicular to the ~a lattice vector (A) or to the ~b (B) or ~c (C); R:
rhombohedral (primitive only when rhombohedral axes are used).
Cubic P
(sc)
Cubic I
(bcc)
Cubic F
(fcc)
Hex. P
Ortho. P
Ortho. I
Ortho. F
Trigonal R
Ortho. C
Tetrag. I
Tric. P
Mono. P
Tetrag. P
Mono. B
(16)
Crystal systems
Every Bravais cell can be converted to a primitive form:
~aP
~bP
~cP
F:
(~a + ~b)/2
(~b + ~c)/2
I:
(~a + ~b − ~c)/2
(−~a + ~b + ~c)/2
(~a − ~b + ~c)/2
(~c + ~a)/2
C:
(~a − ~b)/2
(~a + ~b)/2
~c
c
bP
cP
b
a
aP
c
cP
c = cP
bP
a
aP
b
bP
aP
b
a
(17)
Crystal systems
The particular case of the R cell:
Using hexagonal axes, the cell can be centered at:
• (0, 0, 0), (2/3, 1/3, 1/3), (1/3, 2/3, 2/3) (Obverse setting, modern, ITC-A).
• (0, 0, 0), (1/3, 2/3, 1/3), (2/3, 1/3, 2/3) (Reverse setting, classical, IT1).
A rhombohedral cell for the obverse setting:
• 3~aR = 2~aH + ~bH + ~cH ,
• 3~bR = −~aH + ~bH + ~cH ,
• 3~cR = −~aH − 2~bH + ~cH .
Up to six different fully different hexagonal-rhombohedral
arrangements can be described.
The cell described with hexagonal axes contains 3 primitive R cells.
(18)
Crystal systems
Bravais lattices properties
Orthohedral (cubic, tetragonal, orthorhombic):
V = abc,
kxk = [(ax)2 + (by)2 + (cz)2 ]
(1/a, 1/b, 1/c, 90, 90, 90),
1/2
,
(25)
khk = [(h/a)2 + (k/b)2 + (l/c)2 ]
1/2
.
Hexagonal and rhombohedral (H):
√
3 2
1
a c,
kxk = [(ax)2 − (ax)(by) + (by)2 + (cz)2 ] /2 ,
V =
2
1/2
√
√
4(h2 + hk + k2 )
2
khk =
+
(l/c)
.
(2/a 3, 2/a 3, 1/c, 90, 90, 60),
3a2
(26)
(27)
(28)
Rhombohedral (R):
V = a3 R(α),
R(α) = [1 − 3 cos2 α + 2 cos3 α]
kxk = a[x2 + y 2 + z 2 + 2(xy + yz + zx) cos α]
(a? , a? , a? , α? , α? , α? ),
a? = sin α/aR(α),
1/2
1/2
,
1/2
(29)
(30)
cos α? = − cos α/(1 + cos α),
khk = a? [h2 + k2 + l2 + 2(hk + kl + lh) cos α? ]
,
.
(31)
(32)
(19)
Crystal systems
Monoclinic: (a, b, c, 90, β, 90)
V = abc sin β,
kxk = [(ax)2 + (by)2 + (cz)2 + 2(ax)(cz) cos β]
1/2
,
(1/a sin β, 1/b, 1/c sin β, 90, β + π, 90),
khk = [(ha? )2 + (kb? )2 − 2hla? c? cos β + (lc? )2 ]
(33)
(34)
1/2
.
(35)
Triclinic:
V = abc[1 − cos2 α − cos2 β − cos2 γ + 2 cos α cos β cos γ] /2 ,
V ? = V −1 ,
1/2
kxk = (ax)2 + (by)2 + (cz)2 + 2(ax)(by) cos γ + 2(by)(cz) cos α + 2(cz)(ax) cos β
,
1
(a? , b? , c? , α? , β ? , γ ? ),
bc
sin α,
a =
V
ca
b? =
sin β,
V
ab
sin γ,
c? =
V
?
V ?a
sin α = ? ? ,
b c
V ?b
?
sin β = ? ? ,
c a
V ?c
?
sin γ = ? ? ,
a b
?
cos β cos γ − cos α
,
sin β sin γ
cos γ cos α − cos β
cos β ? =
,
sin γ sin α
cos α cos β − cos γ
,
cos γ ? =
sin α sin β
cos α? =
(36)
(37)
(38)
(39)
(40)
(41)
1/2
khk = (a? h)2 + (b? k)2 + (c? l)2 + 2(a? h)(b? k) cos γ ? + 2(b? k)(c? l) cos α? + 2(c? l)(a? h) cos β ?
.
(42)
(20)
Wigner-Seitz cells
The Wigner-Seitz (WS) cell
Crystalographic cells are parallelepipedic by convenience, but they could have arbitrary forms and
still reproduce the crystal through the symmetry translations.
Wigner-Seitz lattices are conceptually quite important. A WS cell about a lattice point is the crystal
region closer to that point than to any other lattice point. To form the WS cell, a lattice point is
connected by lines to its lattice neighbors, and each line is divided into two halves by mean of a
perpendicular plane. The WS cell is the smallest polyhedron that is interior to all the planes.
bcc
fcc
(21)
Crystal symmetry
Crystal symmetry
Molecular symmetry is usually described on Schönflies’ scheme, but the Hermann-Mauguin (aka
IUCr) scheme is preferred on crystals. The main difference lies in the selection of improper
rotations: Schönflies uses rotation-reflection operations, whereas Hermann-Mauguin is based on
rotation-inversion operations. Both schemes are equivalent but not the same.
Hermann-Mauguin: n (= 1, 2, ...) is a rotation axis of order n (1 is the identity); n̄ is a rotationinversion axis of order n (1̄ is the inversion itself); m is a reflecting plane; n|m means a mirror plane
perpendicular to a n axis; and nm indicates n mirror planes that contain a n rotation axis.
H-M (IUCr)
1
n
m
n|m
nm
1̄
2̄
3̄
4̄
5̄
6̄
...
Schönflies
E
Cn
σ
Cn y σh
Cn y nσv
i
σ
S6
S4
S10
S3
...
3
3
5
1
1
5
1
3
3
6
2
1
5
3
S6
6
2
4
6
S3
5
6
4
4
4
6
2
2
(22)
Crystal symmetry
Crystallographic point groups
Molecules can have symmetry axes of any order. Translational invariance, however, excludes proper
axes of order 5 and ≥ 7. Only 32 point groups are thus compatible with the 3D crystal lattices.
Essential
N◦
4 × 3 (C3 )
4
T (23), Td (4̄3m), Th (m3), O(432), Oh (m3m).
Tetrag.
4 (C4 )
7
S4 (4̄), C4 (4), C4v (4mm), C4h (4|m), D2d (4̄2m), D4 (422),
D4h (4|mmm).
Hexag.
6 (C6 )
6
C6 (6), C6v (6mm), C6h (6|m), D6 (622), D6h (6|mmm), S6 (3̄).
3 (C3 )
6
C3 (3), C3v (3m), C3h (6̄), D3 (32), D3h (6̄m2), D3d (3̄m).
Ortho.
222 or mm
3
D2 (222), D2h (mmm), C2v (2mm).
Monoc.
2 or m
3
C2 (2), Cs (m), C2h (2|m).
–
2
C1 (1), Ci (1̄).
System
Cubic
Tric.
Notation: Schönflies (HM-IUCr)
The molecular motif can include C5 axes (for instance C60 or some bacteriophage virii) but the
forbidden axes do not belong to the crystal. Quasicrystals, lacking true translational symmetry, use
to show pentagonal and heptagonal local symmetry.
(23)
Crystal symmetry
Space symmetry: Seitz operators
Crystal symmetry operations are affine transformations, including rotational and translational parts:

r11

~
r 0i = {R̂|t̂} ~
ri = R xi + t = 
r21
r31
r12
r22
r32
   
xi
r13
tx
   
   
r23 
  yi  + ty  .
r33
zi
tz
(43)
• Pure rotation: {R̂|0̂}.
• Pure translation: {1̂|t̂}.
• Product of operations: {R̂|t̂R }{Ŝ|t̂S } = {R̂Ŝ|t̂R + R̂t̂S }.
• Unit operation (neutral): {1̂|0̂} =⇒ {1̂|0̂}{R̂|t̂R } = {R̂|t̂R }{1̂|0̂} = {R̂|t̂R }.
• Inverse operation: {R̂|t̂R }−1 = {R̂−1 | − R̂−1 t̂R }.
• Null operation: {0̂|0̂} =⇒ {0̂|0̂}{R̂|t̂R } = {R̂|t̂R }{0̂|0̂} = {0̂|0̂}.
The set of symmetry operations in a crystal forms a mathematical group: the product of any two
operations is another symmetry operation; there exists a single unit operation; and any operation
has an inverse that also belongs to the group. This space group of the crystal contains inifinite
operations, because the rotational part can be arbitrarily combined with any primitive translation
(symmorphic operations).
(24)
Crystal symmetry
Non-symmorphic operations: The space group can also contain operations in which a fractionary
translation is combined with a non null rotational part.
• Screw axes (aka rotation-translation axes). A nk operation is a rotation of 2π/n followed by a
translation of k/n parts of the cell lenght along the rotation axis.
Ex: 32 [0, 0, 1] = {3[0, 0, 1]|τ (0, 0, 2/3)}.
Possible cases: 21 , 31 , 32 , 41 , 42 , 43 , 61 , 62 , 63 , 64 y 65 .
• Glide planes: Composed of the reflection in a plane plus a fractional translation. The translation
can be:
axial (a, b, or c): the translation (~a/2, ~b/2 or ~c/2) is contained within the plane (ab, bc or
ac). Ex: {m[100]|τ (0, 1/2, 0)}.
diagonal (n): glides along a diagonal face ((~a + ~b)/2, (~b + ~c)/2 or (~c + ~a)/2). Ex:
{m[001]|τ (1/2, 1/2, 0)}.
diamond (d): translation along a (~a ± ~b)/4, (~b ± ~c)/4, (~c ± ~a)/4, or (~a + ~b + ~c)/4 cubic
direction.
(25)
Crystal symmetry
Space groups
Space group (G): Formed by the infinite symmetry operations of the crystal.
Translation group (T ): The infinite set of primitive translations also forms a group.
Factor group (G/T ): This quotient group removes from G all operations containing translations
that involve displacements of one cell length or more on any direction. Centering translations
are left. The result is a finite group. Most references to the space group in the literature
correspond to G/T rather than to G.
Point group (Gp ): Further collapse of all translations to 0 produces the associated point group,
that describes exclusively the rotational symmetry of the crystal.
Laue class: In the absence of anomalous scattering the diffraction pattern of a crystal shows
inversion invariance even if the crystal lacks it (Friedel’s law). The Laue class, which adds the
inversion to the point group, is the symmetry of the diffraction pattern. The Bravais centering
added to the Laue class forms the Patterson symmetry.
Gp and G/T dimension:
Size
Gp
P
I,A,B,C
R (R axes)
R (H axes)
F
h
h
2h
h
3h
4h
Example: The cubic m3̄m (Oh , h = 48) point group gives rise to three different space groups:
P m3̄m, Im3̄m and F m3̄m, of dimensions 48, 96 and 192, respectively,
(26)
Crystal symmetry
The 230 space groups
By 1891 E. S. Fedorov and A. M. Schönflies, working independently, completed the list of the
230 crystallographic space groups. 73 of these are symmorphic groups that come from the
combination of the 14 Bravais lattices and the 32 crystallographic point groups. The 157 remaining
non-symmorphic groups include glide translations and screw rotations.
System
#SG
SG
Bravais
Point groups
Laue class
Triclin.
2
1–2
P
1, 1̄
1̄
Monoc.
13
3–15
P, C
2, m, 2|m
2|m
Ortho.
59
16–74
P, C, I, F
222, mm2, mmm
mmm
Tetrag.
68
75–142
P, I
4, 4̄, 4|m, 422, 4mm, 4̄2m, 4|mmm
4|m, 4|mmm
Trig.
25
143–167
P, R
3, 3̄, 32, 3m, 3̄m
3̄, 3̄m
Hexag.
27
168–194
P
6, 6̄, 6|m, 622, 6mm, 6̄2m, 6|mmm
6|m, 6|mmm
Cubic
36
195–230
P, I, F
23, m3̄, 432, 4̄3m, m3̄m
m3̄, m3̄m
The space groups and their main properties are described in the International Tables for Crystallography. The Bilbao Crystallographic Server (http://www.cryst.ehu.es) provides a high quality
tool on Internet.
(27)
Crystal symmetry
Grupo puntual
Sistema
cristalino
Grupo Espacial
Vectores
de centrado
The International
Tables of Crystallography
Posiciones
de Wyckoff
Extinciones
sistematicas
(28)
Crystal symmetry
Wyckoff’s positions
The application of the factor group symmetry operations on the coordinates of an arbitrary point
produces all the equivalent positions of that point. For instance, a generic (x, y, z) has 8 equivalent
positions in the P 42 /m group (num. 84):
4
4
4
2
2
2
(x, y, z) −−−−
−→ (ȳ, x, z + 1/2) −−−−
−→ (x̄, ȳ, z) −−−−
−→ (y, x̄, z + 1/2)








my
my
my
my
4
4
(44)
4
2
2
2
−→ (y, x̄, 1/2 − z).
−→ (x̄, ȳ, z̄) −−−−
−→ (ȳ, x, 1/2 − z) −−−−
(x, y, z̄) −−−−
Position (x, y, z) is said to have a multiplicity of 8 in this group.
The multiplicity of a general position is equal to the order nh of the G/T factor group. However,
if a point is contained within one or more of the symmetry elements, several of the symmetry
operations will produce the same image when applied to the point coordinates. The real multiplicity
of the position will be reduced to a divisor of nh. For instance, in the same P 42 /m group we find:
4
4
4
2
2
2
(x, y, 0) −−−−
−→ (ȳ, x, 1/2) −−−−
−→ (x̄, ȳ, 0) −−−−
−→ (y, x̄, 1/2)








my
my
my
my
4
4
(45)
4
2
2
2
(x, y, 0) −−−−
−→ (ȳ, x, 1/2) −−−−
−→ (x̄, ȳ, 0) −−−−
−→ (y, x̄, 1/2),
and the (x, y, 0) has a multiplicity of only 4, half of the order of the factor group.
(29)
Crystal symmetry
Important: The list of special symmetry positions within a group (Wyckoff’s positions) is fully
exploited for the description of crystal structures. In the case of P 42 /m group:
Wyckoff
8k
Local symm.
1
4j
4i
4h
4g
2f
2e
2d
2c
2b
2a
m..
2..
2..
2..
4̄..
4̄..
2|m..
2|m..
2|m..
2|m..
Equivalent positions
(x, y, z), (ȳ, x, z + 1/2), (x̄, ȳ, z), (y, x̄, z + 1/2), (x, y, z̄),
(ȳ, x, 1/2 − z), (x̄, ȳ, z̄), (y, x̄, 1/2 − z).
(x, y, 0), (x̄, ȳ, 0), (ȳ, x, 1/2), (y, x̄, 1/2).
(0, 1/2, z), (1/2, 0, z + 1/2), (0, 1/2, z̄), (1/2, 0, 1/2 − z).
(1/2, 1/2, z), (1/2, 1/2, z + 1/2), (1/2, 1/2, z̄), (1/2, 1/2, 1/2 − z).
(0, 0, z), (0, 0, z + 1/2), (0, 0, z̄), (0, 0, 1/2 − z).
(1/2, 1/2, 1/4), (1/2, 1/2, 3/4).
(0, 0, 1/4), (0, 0, 3/4).
(0, 1/2, 1/2), (1/2, 0, 0).
(0, 1/2, 0), (1/2, 0, 1/2).
(1/2, 1/2, 0), (1/2, 1/2, 1/2).
(0, 0, 0), (0, 0, 1/2).
Notice: when the coordinates of a transformed point are outside of the main cell we can put them
back within 0 ≤ x, y, z < 1 by adding up integers. Example: x = 0.2 and y = 0.3 produce the
next 4j positions: (0.2, 0.3, 0), (−0.2, −0.3, 0) = (0.8, 0.7, 0), (−0.3, 0.2, 0.5) = (0.7, 0.2, 0.5), and
(0.3, −0.2, 0.5) = (0.3, 0.8, 0.5).
(30)
Crystal symmetry
The general position and the Seitz operators
A list of the general position equivalents
operators that create them.


1 0 0 0


,
(x, y, z) : 
0
1
0
0


0 0 1 0


0 −1 0
0


(ȳ, x, z + 1/2) : 
0 0
0
1
,
0
0 1 1/2


−1
0 0 0



(x̄, ȳ, z) : 
 0 −1 0 0 ,
0
0 1 0


0 1 0
0


(y, x̄, z + 1/2) : 
0
−1 0 0
,
0 0 1 1/2
contains the same information than a list of the Seitz

(46)
(47)
(48)
(49)
1

(x, y, z̄) : 
0
0

0

(ȳ, x, 1/2 − z) : 
1
0

−1

(x̄, ȳ, z̄) : 
 0
0

0

(y, x̄, 1/2 − z) : 
−1
0

0
0
0
1
0

0
,
0
−1
−1
(50)
0
0
0


0
0
0
,
0 −1 1/2

0
0 0

−1
0 0
,
0 −1 0

1
0
0

0
0
0
.
0 −1 1/2
(51)
(52)
(53)
(31)
Crystal symmetry
Centering vectors
The International Tables use a simple trick to reduce the list of equivalent positions on centered
cells. Only the positions generated by the point group Gp are explicitely given, and the centering
vectors must be added to them to produce the whole set of the factor group.
P or R†
I
F
A
B
C
R‡+ (H)
R‡− (H)
(0, 0, 0)
(0, 0, 0)
(0, 0, 0)
(0, 0, 0)
(0, 0, 0)
(0, 0, 0)
(0, 0, 0)
(0, 0, 0)
(1/2, 1/2, 1/2) (1/2, 1/2, 0) (0, 1/2, 1/2) (1/2, 0, 1/2) (1/2, 1/2, 0) (2/3, 1/3, 1/3) (2/3, 1/3, 2/3)
(1/2, 0, 1/2)
(1/3, 2/3, 2/3) (1/3, 2/3, 1/3)
(0, 1/2, 1/2)
R† : trigonal cell with rhombohedral axes. R‡ (H): trigonal cell with hexagonal axes.
Ex: In the case of the R32 group (Num. 155) the ITA gives:
Wyckoff
18f
9e
9d
6c
3b
3a
Sim. local
1
.2
.2
3.
32
32
Posiciones equivalentes
(x, y, z), (ȳ, x − y, z), (y − x, x̄, z), (y, x, z̄), (x − y, ȳ, z̄), (x̄, y − x, z̄)
(x, 0, 1/2), (0, x, 1/2), (x̄, x̄, 1/2)
(x, 0, 0), (0, x, 0), (x̄, x̄, 0)
(0, 0, z), (0, 0, z̄)
(0, 0, 1/2)
(0, 0, 0)
and the three R(H) centering vectors must be added to complete the list.
(32)
Crystal symmetry
An example: CaF2, fluorite type structure (C1)
Cubic, F m3̄m (225), a = 5.4626 Å, Z = 4.
Ca
4a
(0, 0, 0)
F
8c
(1/4, 1/4, 1/4)
From the International Tables of
Crystalography:
Wyckoff Sim. Equiv. positions
4a
m3̄m (0, 0, 0)
8c
4̄3m (1/4, 1/4, 1/4), (1/4, 1/4, 3/4)
Centering vectors
(0, 0, 0), (1/2, 1/2, 0), (1/2, 0, 1/2), (0, 1/2, 1/2)
Simple tasks: Cell volume? Density? Neighbor distances? Atoms in the main cell? Cell
plot?
(33)
Crystal symmetry
Some simple crystal structures
Grouped by structure symbols. (other alternatives: http://cst-www.nrl.navy.mil/lattice/)
A1
Cu, Cubic, a = 3.609 Å, F m3̄m, Z = 4.
fcc
Cu (4a) (0, 0, 0).
A2
Li, Cubic, a = 3.46 Å, Im3̄m, Z = 2.
bcc Li (2a) (0, 0, 0).
A3
Be, Hexag., a = 2.2860, c = 3.5843 Å, (hcp ideal: c/a ≈ 1.63) P 63 /mmc, Z = 2.
Be (2c) (1/3, 2/3, 1/4).
A4
C (diamante), Cubic, a = 3.5667 Å, F d3m, Z = 8.
C (8a) (1/8, 1/8, 1/8).
A9
C (grafito), Hexag., a = 2.456, c = 6.696 Å, P 63 /mmc, Z = 4.
C (2b) (0, 0, 1/4); C (2c) (1/3, 2/3, 1/4).
B1
B2
B3
B4
NaCl, Cubic, a = 5.6402 Å, F m3̄m, Z = 4.
Na (4a) (0, 0, 0); Cl (4b) (1/2, 1/2, 1/2).
CsCl, Cubic, a = 4.123 Å, F m3̄m, Z = 1.
Cs (1a) (0, 0, 0); Cl (1b) (1/2, 1/2, 1/2).
β-ZnS (blenda), Cubic, a = 5.4060 Å, F 4̄3m, Z = 4.
Zn (4a) (0, 0, 0); S (4c) (1/4, 1/4, 1/4).
ZnO (zincita, wurtzita), Hexag., a = 3.2495, c = 5.2069 Å, P 63 mc, Z = 1.
Zn (2b) (1/3, 2/3, z ≈ 0); O (2b) (1/3, 2/3, z ≈ 0.345).
(34)
A1: Cu
B1: NaCl
Crystal symmetry
A2: Li
B2: CsCl
A3: Be
A4: diamante
B3: ZnS (blenda)
A9: grafito
B4: ZnO (wurtzita)
Plots made with tessel (http://web.uniovi.es/qcg/tessel/tessel.html) and POVRay (http:
//www.povray.org).
(35)
Crystal symmetry
Crystal Data Bases
Cambridge Structural Database (CSD) (http://www.ccdc.cam.ac.uk/products/csd/): The
main source of crystal structures for organic and organometallic compounds. Commercial, access by
subscription. 436436 structures as of 2008-01-01.
Inorganic Crystal Structure Database (ICSD) (http://www.fiz-karlsruhe.de/icsd.html?&L=
0): Natural and synthetic inorganic compounds. Commercial, access by subscription. > 100000
entries (feb 2007). See a fully functional subset at http://icsd.ill.fr/icsd/ (3592 samples).
CRYSTMET (http://www.tothcanada.com/databases.htm): Metals, including alloys, intermetallics and minerals. Commercial, access by subscription. 109877 entries (2006-11-10).
American Mineralogist Crystal Structure Database (AMCSD) (http://rruff.geo.arizona.
edu/AMS/amcsd.php): Minerals. Free access (financed by NSF). 3156 different minerals (many
more entries, as a mineral may appear at different pressures and temperatures).
Reciprocal Net (http://www.reciprocalnet.org/): Small but well choosen set of quite common
molecules and materials. Data from CSD and other sources. Free access (financed by NSF).
Protein Data Bank (PDB) (http://www.rcsb.org/pdb/): Structures of large biological
molecules, including proteins and nucleic acids. Free access (international support). 48638
structures (2008-01-29). Indispensable portal for anyone working on biomolecules.
(36)
Crystal symmetry
Structural Classification of Proteins (SCOP) (http://scop.mrc-lmb.cam.ac.uk/scop/): Further analysis of the proteins contained in PDB: folds, superfamilies, evolutionary relationship, etc.
Free access.
Nucleic Acids Data Bank (NADB) (http://ndbserver.rutgers.edu/): Similar to PDB, but
specialized on oligonucleotides. Free access (international support). 3745 structures (2008-01-17).
Mineralogy Database (http://webmineral.com/): Good database on minerals and gems
maintained by commercial dealers. Free access.
Crystal Lattice Structures (http://cst-www.nrl.navy.mil/lattice/): Very good description
of the common crystal lattice structures of the elements and simple compounds. Free access.
Crystallography Open Database (COD) (http://www.crystallography.net/): Voluntary effort
to provide an open alternative to CSD and ICSD. Absolutely free access. Some 48000 entries (dec
2006) and growing fast. A sister PCOD database specializes on theoretically predicted structures.
Powder diffraction file (PDF) (http://www.icdd.com/): Largest DB on single phase powder
diffraction pattern. Widely used to identify compounds based on their fingerprint spectra.
Commercial.
Database of Macromolecular Movements (http://molmovdb.mbb.yale.edu/molmovdb/): Analysis and prediction of the dynamical behaviour of macromolecules. Movies, morphings, etc. Free
access.
(37)
Exercises
Do you know any other good structures database? Please, e-mail me the address and details
(mailto:[email protected])
References
Ashcroft and Mermin, 1976 [1]; Bhadeshia, 2006 [2]; Bradley and Cracknell, 1972 [6]; Burns and
Glazer, 1990 [3]; Giacovazzo et al., 2002 [4]; International Tables, vol A [5].
Exercises
1. The polymorph I of benzene has been measured again recently by A. Budzianowski and A.
Katrusiak, Acta Cryst. B 62 (2006) 94. The observed structure is orthorhombic, space group
P bca (Num. 61), a = 7.287(6), b = 9.20(2), c = 6.688(9) Å, four molecules per cell (Z = 4),
and the atoms occupy the non-equivalent positions shown in the table. Using this data: (a)
determine the crystal density; (b) obtain the average C-C and C-H distances, the average
C-C-C, H-C-H, and H-C-C angles, and the average deviation from the planarity of the benzene
units; (c) plot the benzene stacking using any molecular modeller.
C −0,053 7 (8) +0,142 5 (9)
+0,009 7 (12) H −0,085 (6) +0,246 (7) +0,034 (8)
C +0,084 0 (7) +0,092 4 (10) +0,137 3 (10) H +0,140 (6) +0,156 (6) +0,219 (8)
C −0,134 3 (7) +0,052 1 (9)
−0,123 5 (12) H −0,220 (6) +0,080 (6) −0,204 (9)
(38)
Exercises
2. The general position of the P bca group is: (1) (x, y, z), (2) (1/2−x, −y, 1/2+z), (3)
(−x, 1/2+y, 1/2−z), (4) (1/2+x, 1/2−y, −z), (5) (−x, −y, −z), (6) (−1/2+x, y, −1/2−z), (7)
(x, −1/2−y, −1/2+z), (8) (−1/2−x, −1/2+y, z). (A) Get the Seitz matrices of the group. (B)
Obtain the multiplication matrix of the factor group and use it to determine the equivalence
classes of the Seitz operators.
Remember: two operations {R̂|t̂R } and {Ŝ|t̂S } are equivalent if there exists another operation
{X̂|t̂X } in the group such that {X̂|t̂X }−1 {R̂|t̂R }{X̂|t̂X } = {Ŝ|t̂S }.
3. Transform the F and I cells into their primitives. Obtain then the reciprocal cells of both
primitives and use this to explain a typical sentence reproduced in most solid state textbooks:
the reciprocal of a F cell is an I cell, and viceversa.
4. The non-orthogonal parallelepipedic cells are quite useful for most crystallographic tasks, but
there are occasions in which a cartesian frame is far more convenient. Let us define one of
the possible cartesian axes in the following way: First, the origin remains unchanged; the
cartesian system is described by the three orthonormal vectors {~i, ~j, ~k}; ~i is collinear with
the crystallographic ~a vector; both, ~i and ~j lie in the ab plane; and, finally, ~i × ~j = ~k. Find
the transformation from crystallographic to cartesian coordinates and viceversa, for a general
crystal (a, b, c, α, β, γ).
(39)
Ltm1: Thermodynamic properties of a pure crystal I
Thermodynamic properties of a pure crystal
(40)
Ltm1: Thermodynamic properties I
The Standard Model
The Standard Model of Material Science
The macroscopic behavior of matter (gases, liquids, solids, ...) is ruled by a
thermodynamical potential. For an hydrostatic system under fixed pressure and
temperature conditions, the ruling potential is the general Gibbs free energy:
G? (p, T ; ~x) = E(V, T ; ~x) + pV (~x) − T S(V, T ; ~x) = A(V, T ; ~x) + pV (~x),
(54)
where
• p: pressure; V : volume; T : absolute (Kelvin) temperature;
• ~x: variables describing the internal geometry of the system;
• E: internal energy; S: entropy; A: Helmhotz free energy.
Thermodynamic equilibrium conditions are found by minimizing G? with respect to
the internal variables:
min G? (p, T ; ~x)
~
x
=⇒
G(p, T ) = G? (p, T ; ~xmin ).
(55)
(41)
The Standard Model
Thermodynamic equilibrium embraces two parts:
The gradient of G? with respect to the ~
x coordinates must be zero:
Equilibrium:
(necessary cond.)
~ ~x G? (p, T ; ~
∇
x) = 0
(56)
The points fulfilling this equation are critical points of the G? surface.
Critical points can be classified according to the sign of the eigenvalues of the
Hessian matrix:
~ ⊗ ∇G
~ ?
H=∇
=⇒
U−1 H U = Λ = diag(λ1 , λ2 , · · · ).
(57)
Minima: all the eigenvalues are positive;
• Stability:
(sufficient cond.)
n-order saddle points: n of the eigenvalues are negative;
maxima: all the eigenvalues are negative.
The absolute minimum represents the most stable conformation.
The other minima correspond to metastable conformations.
First order saddle points are the transition states in a reaction pathway: the
path being the gradient line going down from the saddle to two different
minima.
(42)
Calculation of G?
Calculation of G? = A(V, T ; ~x) + pV (~x)
A(V, T ; ~
x) = Emec (V ; ~
x) + Atrasl (V, T ; ~
x) + Arot (V, T ; ~
x) + Avib (V, T ; ~
x) + · · ·
(58)
• Quantum mechanics (QM): Emec is the eigenvalue of the system hamiltonian: ĤΨ = EΨ.
Solving the time independent Schrödinger equation is difficult and time consuming.
• Molecular mechanics (MM): If quantum mechanics is too slow, we can resort to obtain Emec by
adding up interaction potentials.
• QM/MM: A combination of techniques (multiscale methods) is the basis to tackle very large
systems.
• Statistical mechanics: Provides the equation for the remaining terms. For a crystal under small
harmonic vibrations, for instance:
Z ∞
n
o
−hν/kB T
g(ν) hν/2 + RT ln(1 − e
) dν.
(59)
Avib ≈
0
The vibration frequencies and degeneration, ν and g(ν) respectively, are obtained from a quantum
or molecular mechanics calculation.
(43)
Equilibrium properties
Equilibrium properties of a crystal phase
The problem: Crystal space is too large to perform blind searches looking for stable structures.
Strategy: Impose some symmetry on the cell parameters and the atomic occupancy. The
restrictions define a structure subspace in which minimizations and derivatives are performed. Care
must be taken to ensure that the resulting phases are truly stable (this may involve breaking the
assumed symmetry).
Example: ZnO, wurtzite phase, hexagonal, P 63 mc, Zn (2b) (1/3, 2/3, z ≈ 0), O (2b) (1/3, 2/3, zO )
with zO ≈ 0,345. In this case ~
x = (a, c, zO ), and obtaining the equlibrium geometry requires a
3-parameter minimization.
Pressure and temperature effects: It is simple to include pressure effects on the theoretical
simulations but it is rather dificult to deal accurately with the temperature. Fortunately: (1)
pressure can induce larger and more significant changes than temperature; and (2) experimentation
on high pressure conditions is difficult and theoretical help is usually much welcomed.
The static model: assume T = 0 K and ignore completely T effects, including zero-point
contributions to Emec . Under this approximation:
A(V, T ; ~
x) ≈ Ecell (V ; ~
x),
G? (p, T ; ~
x) ≈ Ecell (V ; ~
x) + pV (~
x).
(60)
Significant equilibrium properties: geometry (~
xeq ); equation of state (EOS, p(V ), V (p), or
f (p, V, T ) = 0); bulk modulus (B0 = −V (∂p/∂V )T , and B(p)); elastic constants; ...
(44)
EOS (method 1): p = −(∂A/∂V )T . Data: A(V ) = min~x A(V ; ~
x) vs V (~
x).
EOS (method 2): Define a pressure, p, and compute G(p, T ) = min~x G? (p, t; ~
x). The minimum
of G determines V for this pressure. Repeat for other pressures.
EOS (empirical equations) Let’s define: x = (V /V0 )1/3 ; ` = ln x; f = (x−2 − 1)/2. Then:
Murnaghan:
B0
p(V ) = 0
B0
"
V0
V
B 0
0
#
−1 ,
V (p) = V0
P
1 + B00
B0
−1/B 0
0
Birch-Murnaghan (4th order):
3
35
3 0
5/2
B0 B000 + (B00 −4)(B00 −3) +
f2 .
p = 3B0 f (1 + 2f )
1 + (B0 −4)f +
2
2
9
.
(61)
(62)
Poirier-Tarantola:
3
3
p = 3B0 x−3 ` 1 + (B00 −2)` +
1 + B0 B000 + (B00 −2) + (B00 −2)2 `2 .
2
2
1−x
3 0
Vinet: p = 3B0 2 exp
(B0 −1)(1−x) .
x
2
(63)
(64)
Least squares fitting of the (p, V ) values to the above analytical equations can be used to determine
B0 = B(p = 0), B00 = (dB/dp)p=0 , B000 , etc.
(45)
Elastic equation of state
For small deformations of the crystal cell around an equilibrium configuration:
E − E0
1 X
1 X
φ=
=
cijkl ij kl =
sijkl τij τkl .
V
2 ij,kl
2 ij,kl
(65)
Strain tensor : Let ij be a symmetric deformation component
ij
1
=
2
∂δxi
∂δxj
+
,
∂xj
∂xi
(66)
where xi is one of the cell parameters (a, b, c), and δxi = x0i − xi . The ij form a 3 × 3 symmetric
matrix that can be diagonalized to determine the main deformation directions.
Stress tensor τ : The crystal strain is the consequence of stress forces
acting on the crystal faces. Let τij = τji be the component of force
acting on the face normal to the xi -axis and directed along the xj -axis.
Stress and strain components are related as
τij
∂φ
=
,
∂ij
τij =
X
kl
cijkl kl ,
ij =
X
sijkl τkl .
(67)
kl
(46)
Elastic stiffness constants (aka elastic moduli). The elastic moduli are the curvatures of the
elastic energy with respect to the strain components:
cijkl
∂2φ
=
,
∂ij ∂kl
invariant under i ↔ j, k ↔ l, ij ↔ kl.
(68)
Strain components are adimensional. The magnitude of c’s is then energy/volume=pressure. The
symmetry of the τ , and c tensors is made explicit by defining a new suffix notation:
τ11 τ22 τ33 τ23 τ31 τ12 11 22 33 223 231 212
τ1 τ2 τ3 τ4 τ5 τ6 1 2 3
4
5
6
Strain non-diagonal terms are duplicated because they appear twice as many times than diagonal
P
terms in the 65 equation. In this way, a cmn is exactly one of the cijkl and the φ = (1/2) i,j cij i j
expression is maintained. Stress terms, however, are not duplicated, so the 2 factor is included into
the s’s definition: s11 = s1111 , but s16 = 2s1112 , and s66 = 4s1212 .
The index change transforms and τ into six-element vectors, and the fourth rank tensors c and
s into 6 × 6 symmetric matrices. In the worst case (a triclinic crystal), there are 21 independent
elastic moduli. The unit cell symmetry can reduce much this number.
~ = ~0 and
Intrinsic stability of a phase. A crystal structure is said to be mecanically stable if ∇φ
the elastic energy is positive definite with respect to all strains, i.e. all the eigenvalues of the c
matrix are positive. This would ensure the phase to be a true minimum on the energy surface at
zero p and T . Were an eigenvalue negative or close to zero, its eigenvector would show how the
(47)
crystal shape should deform to attain a lower energy minimum. The stability condition can be
generalized to any p and T by using φ = G? /V to compute the effective ceff
ij .
The stability condition impose restrictions on the elastic moduli. For instance, on a cubic crystal
c11 > |c12 | ,
c11 + 2c12 > 0,
c44 > 0,
(69)
whereas for an hexagonal cell
c11 > |c12 | ,
(c11 + c12 )c33 > 2c313 ,
c44 > 0,
c66 > 0.
(70)
Elastic moduli and the crystal systems (See Bhagavantam [7] or Nye [8])
Cubic
Hexagonal
Orthorhombic
c11 c12 c12
0
0
0
c11 c12 c13
0
0
0
c11 c12 c13
0
0
0
c11 c12
0
0
0
c11 c13
0
0
0
c22 c23
0
0
0
c11
0
0
0
c33
0
0
0
c33
0
0
0
c44
0
0
c44
0
0
c44
0
0
c44
0
c44
0
c55
0
c44
(c11 −c12 )/2
c66
For a isotropic medium c44 = (c11 −c12 )/2 (Cauchy relationship).
(48)
Trigonal (3, 3̄)
c11 c12 c13 c14
Trigonal (3m, 32, 3̄m)
c15
0
c11 c13 −c14 −c15
c33
c11 c12 c13 c14
0
0
c11 c12 c13
0
0
0
0
c11 c13
0
0 −c16
0
0
0
c33
0
0
0
c44
0
0
c44
0
0
c44
0
c11 c13 −c14
0
c33
Tetrag. (4, 4̄, 4|m)
0
0
0
c44
0
−c15
c44
c14
c44 c14
c‡66
c‡66
c16
c66
c‡66 = (c11 −c12 )/2.
Tetrag. (4mm, 4̄2m, 422, 4|mmm)
Monoclinic
Triclinic
c11 c12 c13
0
0
0
c11 c12 c13
0
0 c16
c11 c12 c13 c14 c15 c16
c11 c13
0
0
0
c22 c23
0
0 c26
c22 c23 c24 c25 c26
c33
0
0
0
c33
0
0 c36
c33 c34 c35 c36
c44
0
0
c44 c45
0
c44 c45 c46
c44
0
c55
0
c55 c56
c66
c66
c66
(49)
Calculation of elastic moduli using the crystal static deformation method pioneered by Catti
[9, 10]. The only requirement is a computational method able to provide the crystal energy for
arbitrary crystal geometries. Starting from an equilibrium configuration, the technique works by
performing controlled deformations of the crystal. Each deformation is designed to produce a
simplified form of the elastic energy 2φ = T c.
For instance, the deformation T = [η, 0, 0, 0, 0, 0] in a cubic crystal has an elastic energy 2φ = c11 η 2
and it is clear that c11 can be obtained as (∂ 2 φ/∂η 2 ). A short collection of calculations for some
small values of η plus a least square fitting is enough to complete the task.
This particular deformation transforms the unit cell from a cube to a square-based prism of
dimensions [a(1+η), a, a, 90, 90, 90], thus lowering the symmetry from cubic to tetragonal. The
symmetry reduction can have the added consequence that some of the atoms within the cell gain
degrees of freedom and their position within the cell should be reoptimized for each new value of
η. Failing to take into account this inner strain can produce a significant impact on the calculated
value of the elastic moduli.
Designing a set of sensible deformations can be a creative task with a large influence on the
computational effort. The best routes will elude creating inner strain and lowering the unit cell
symmetry as much as possible.
The bulk modulus, B, is related to the elastic moduli. In a cubic system, for instance, B =
(c11 + 2c12 )/3.
(50)
Example [11]: Cubic CaF2 [fluorite; F m3̄m (Num. 225); Ca (4a) (0, 0, 0); F (8c) (1/4, 1/4, 1/4)].
Strain
Space G. Inner strain 2φ
Cell
[η, η, 0, 0, 0, 0]
4|mmm
No
2(c11 +c12 )η 2
[a(1+η), a(1+η), a, 90, 90, 90]
[η, η, −2η, 0, 0, 0] 4|mmm
No
6(c11 −c12 )η 2
[a(1+η), a(1+η), a(1−2η), 90, 90, 90]
[η, η, η, 0, 0, 0]
F m3̄m
No
3(c11 +2c12 )η 2 [a(1+η), a(1+η), a(1+η), 90, 90, 90]
[0, 0, 0, η, η, η]
R3̄m
F (x, x, x)
3c44 η 2
[a, a, a, α, α, α], 2η = cos(90−α)
x ≈ 1/4
Ca and F remain on a symmetry fixed position for the first three proposed deformations, and F
has only one degree of freedom in the fourth one. [η, η, η, 0, 0, 0] is a breathing dilatation of the
cubic unit cell and, consequently, the elastic energy is proportional to the bulk modulus. The c44
modulus could also be obtained through a [0, 0, 0, 0, 0, η] deformation, but this would reduce the
cell to moclinic symmetry, instead of rhombohedral as the proposed [0, 0, 0, η, η, η].
The space group that results from each deformation is usually the highest subgroup of the original
group cell that belongs to the final crystal system.
The importance of the elastic characterization. Bulk modulus and elastic moduli are very
important properties of the crystal beyond their significant role on establishing the stability of the
crystal phase and helping to predict the possible deformation of the unit cell shape in a potential
phase transition. Elastic data is required in crystallography to calculate the thermal diffuse scattering
(51)
Exercises
correction to the Bragg diffraction intensities. The interpretation of the observed seismic waves,
that constitute our best probe on Earth innards, depend on the assumed elastic properties of the
solid phases present in the earth mantle. Material scientist are engaged in a long quest for superhard
materials that maintain their hardness on a high pressure, high temperature, or high radiation
working ambient.
References
Bhagavantam, 1966 [7]; Nye, 1985 [8]; Catti, 1985 [9], 1989 [10]; Catti et al., 1991 [11]; Jona and
Marcus, 2001 [12]; Ashcroft and Mermin, 1976 [1].
Exercises
1. Use the data below and the equations of state described previously to determine E0 , V0 , B0
and the appropriate number of derivatives of B for the bcc phase of lithium. Compare the
values predicted by the different EOS models. Examine the stability of the Birch-Murnaghan
results if the EOS is truncated to second and third order. Check the suitability of using five or
six points centered around the minimum and a least squares fit to a parabola, as it is commonly
done by some authors. Compare in a plot the original data and the diferent EOS.
Caution and hints: Take care to convert appropriately the energy and lenght units. Report pressures and bulk moduli in GPa. The open source code gnuplot (http://www.gnuplot.info/)
provides a robust an easy to use approach to perform non-linear least squares fitting for
(52)
Exercises
arbitrary functions.
Data: WIEN97 fp-LAPW calculations on bcc Li [See details on J. Chem. Phys. 119 (2003)
6341]. Lattice parameters are given in bohr and energies in Rydberg (2 Ry = 1 hartree).
a
Ecell
a
Ecell
a
Ecell
a
Ecell
4,10
−14,810 791
6,10
−15,042 100
6,73
−15,043 745
7,30
−15,037 646
4,30
−14,869 500
6,20
−15,043 233
6,75
−15,043 627
7,40
−15,036 125
4,50
−14,914 924
6,30
−15,043 977
6,77
−15,043 497
7,50
−15,034 511
4,80
−14,964 410
6,40
−15,044 393
6,80
−15,043 291
7,60
−15,032 719
5,00
−14,988 021
6,45
−15,044 489
6,83
−15,043 068
7,62
−15,032 367
5,20
−15,005 972
6,50
−15,044 509
6,85
−15,042 907
7,64
−15,032 013
5,40
−15,019 404
6,55
−15,044 464
6,90
−15,042 478
7,66
−15,031 657
5,60
−15,029 240
6,60
−15,044 353
7,00
−15,041 487
7,68
−15,031 296
5,80
−15,036 004
6,65
−15,044 184
7,10
−15,040 340
7,69
−15,031 116
6,00
−15,040 551
6,70
−15,043 954
7,20
−15,039 056
7,70
−15,030 934
2. Design a collection of static deformations that could be used to obtain the bulk modulus and
the elastic moduli of the A1 (fcc), A2 (bcc), A3 (hcp), A4 (diamond), and A9 (graphite)
structures (See table in page 34).
(53)
Ltm3: Thermodynamic properties of a pure crystal II
Example: Bulk properties and phase stability
(54)
Ltm3: Thermodynamic properties II
Equilibrium properties of MgO
Equilibrium properties of MgO (aiPI calculations)
Cubic, F m3̄m, a = 4.210 Å
Mg
4a
0
0
0
O
4b
1/2
1/2
1/2
aiPI input:
uchf
crystal
title MgO. Experimental geometry.
spg f m -3 m
cell 7.9557 7.9557 7.9557 90.0 90.0 90.0
neq 0.0 0.0 0.0 mg.ion mg.int mg.cint mg.lint
neq 0.5 0.5 0.5 o2.ion o2.int o2.cint o2.lint
endcrystal
end
The *.ion files contain a description of the basis set, electron occupancy, starting orbitals, etc, for
each ion type in the crystal. CPU: ≈44 s per 100 geometries (Intel PIV/1.1 GHz).
(55)
Equilibrium properties of MgO
The quantum aiPI calculation provides Emec (V ). Under the static approximation (T = 0 K and the
zero point energy is neglected) G? ≈ Emec (V ) + pV and
Elatt = Ee − Evac (Mg+2 ) − Evac (O−1 ),
2 ∂A
∂p
∂ E
p=−
, Be = −V
≈V
.
∂V T
∂V T
∂V 2
min E → {ae , Ee },
a
A = E + T S ≈ E,
(71)
(72)
−560
Prop.
Unit
aiPI
Exptal.
ae
Å
4.212
4.210
Elatt
kcal/mol
708
725
Be
GPa
156
163
−680
(dB/dp)e
—
3.28
4.13
−700
Cp
J/mol K
38.16
37.89
α
10−6 K−1
24.8
13.5
−580
Elatt (kcal/mol)
−600
−620
−640
−660
−720
3.5
4.0
4.5
a (Å)
5.0
5.5
Thermal properties (Cp and α = V −1 (∂V /∂T )p ) have been obtained using a quasiharmonic Debye
model[14], in which the volume dependent bulk modulus, B(V ), is used to determine Debye’s
temperature, θD = hνD /kB , as a function of volume.
(56)
B1-B2 phase transition in alkali halides
B1 phase
B2 phase
8
7
rhombohedral intermediate
7
(a, a, a, α, α, α)
Cl
5
4
60
3
4
R3̄m
6
Cl
3
2
A
(1a)
0 0 0
X
(1b)
1/2 1/2 1/2
α ∈ [60◦ (B1), 90◦ (B2)]
1
8
90
6
5
1
2
P m3̄m
F m3̄m
(a, a, a, 90◦ , 90◦ , 90◦ )
A
(4a)
0 0 0
X
(4b)
1/2 1/2 1/2
(Buerger’s 2D
mecanism)
(a, a, a, 90◦ , 90◦ , 90◦ )
A
(1a)
0 0 0
X
(1b)
1/2 1/2 1/2
(57)
Some results on LiCl:
E (hartree)
-0.22
-0.24
-0.26
-0.28
-0.30
-0.32
100
90
5.5
6.0 6.5
a (bohr)
70
7.0
7.5
60
80
α (deg)
8.050
B1 and B2 phases are metastable as far as G? (x; p, T )
is a minimum at their corresponding positions in the
phase space. Of particular importance is the curvature
represented by the ceff
44 elastic constant. Remember:
ceff
ij =
∂G?
1
V ∂εi ∂εj
(73)
where εi are the Voigt-Lagrange deformation parameters. In particular ε4 = 2ε23 , and c44 is the curvature
with respect to the Buerger’s R3̄m opening.
700
600
500
c44, c44
eff (GPa)
5.0
The quantum-mechanical energy, E(a, α), is the potential energy surface that determines stability under
0 K. The effect of pressure is obtained by adding the
work term , pV . Temperature effects have been estimated using a simple quasiharmonic Debye model.
B2
B2(eff)
400
300
B1
200
100
B1(eff)
0
-100
0
50
100
150
P (GPa)
200
250
300
(58)
8.0
0 GPa
40 GPa
80 GPa
300 GPa
7.5
7.0
A reaction path, (a, α) within Buerger’s mechanism,
has been obtained by minimizing G? (x; p, T ) for fixed
values of α. Pathways obtained for different pressures
are almost parallel. This is a consequence of the sym~ ⊗ ∇G
~ ? , at both
metry of the hessian matrix, H = ∇
metastable phases.
a (bohr)
6.5
6.0
5.5
5.0
4.5
4.0
3.5
50
60
70
80
α (grados)
90
100
(59)
0.10
0.00 GPa
40.00 GPa
80.00 GPa
300.00 GPa
0.08
0.06
G (hartrees)
0.04
The free energy profile along the reaction pathway,
G(α), determines the most stable phase and the kinetical aspects of the phase transition. The B1-B2
transition pressure is Ptr ≈ 80 GPa in the equilibrium
(LiCl), but the transition barrier is large and different
for the compression-decompression steps: significant
hysteresis should be expected.
0.02
0.00
-0.02
-0.04
-0.06
50
60
70 80 90 100 110
α (grados)
(60)
A qualitative model for the Hysteresis can be expressed
in simple kinetical terms:
2500
T(K); ∆G‡ (K)
2000
‡
dnB1
= ωB1 e−∆GB1→B2 /kB T nB1
dP
1500
‡
B2 → B1
1000
− ωB2 e−∆GB2→B1 /kB T nB2 ,
B1 → B2
500
0
0
50
100
P (GPa)
150
200
(74)
where nB1 and nB2 are the concentrations of both
phases, ∆G‡ are the energy barriers (different for the
B1 → B2 and B2 → B1 processes except at equilibrium), and the ωi are unknown kinetical constants.
B1-B2 equilibrium pressures at 300 K.
Ptr
Li
F
252
Cl
I
79
94
113
16
16
Na
12 (23–24)
21 (26–30)
K
5.6 (1.73)
2.0 (1.9–2.1)
Rb
−0.2 (0.9–3.3)
Cs
−2.6 (2)
Br
0.3 (0.5)
−1.3 (B2)
1.64 (1.77)
2.7 (1.8–1.9)
0.1 (0.45)
0.8 (0.3–0.4)
−1.5 (B2)
−0.7 (B2)
(61)
Exercises
References
Gale and Rohl, 2003 [16]; Luaña and Pueyo, 1990 [17]; Luaña and Flórez, 1992 [18]; Luaña, Flórez,
and Pueyo, 1993 [19]; Pendás et al., 1994 [20]; Blanco, 1997 [21]; Schlick, 2002 [22].
Exercises
1. The contiguous figure represents the energy of the
CaO crystal at the B1 and B2 phases calculated
by Mota et al. [23]. Determine the equilibrium
transition pressure.
2. Search the recent literature (e.g. 2000 to current
date) on the B1-B2 phase transition and write a
short resume of the most relevant experimental
and theoretical advances on the subject.
(62)
Ltm4: Thermodynamic properties of a pure crystal II
Atomistic simulation of matter
(63)
Atomistic simulation
Atomistic simulation of matter
Let assume the system to be made of N interacting bodies (neutral atoms or charged ions, usually:
atoms, from now on, to simplify). The mechanical energy can be obtained as:
Emec =
N
X
v1 (~
ri ) +
i=1
|
where
P
i>j
N
X
v2 (~
ri , ~
rj ) +
i>j
{z
one body
}
|
N
X
v3 (~
ri , ~
rj , ~
rk ) +...
(75)
i>j>k
{z
two body
sums over all different pairs of bodies,
}
|
P
i>j>k
{z
three body
}
over all different triplets, and so on.
The properties of ordinary matter are controlled by the kinetic energy and the electrostatic interaction
of the fundamental particles (electrons and nuclei). The two-body nature of the Coulomb law makes
only natural that the v2 (~
ri , ~
rj ) ≈ v2 (rij ) terms dominate the mechanical energy.
The required self-consistency between the quantum mechanical wavefunction and the potencial
acting on the particles can be viewed as the origin of the (N 6= 2)-body terms. In other words,
the local environment of a body is affected by its neighbors and this modifies the way in which
they interact. In any way, the many-body terms are responsible for many of the fine details that
determine phase stability and thermodynamical properties.
(64)
Additive and effective energies of atoms
Dealing with the energy of atoms and groups of atoms will be simpler by defining the next three
quantities:
• The effective energy of an atom i includes all the contributions to the total energy of the whole
system in which the atom i participates:
i
Eeff
= v1 (~
ri ) +
| {z }
V1i
N
X
v2 (~
ri , ~
rj ) +
j(6=i)
|
N
X
v3 (~
ri , ~
rj , ~
rk ) + . . .
(76)
j>k(6=i)
{z
V2i
}
|
{z
}
V3i
where Vni describes the sum of all the n-body terms that contain i. As a consequence of this
definition,
i with respect to the position and internal structure of i is
the minimization of Eeff
equivalent to the minimization, respect to the same variables, of the total energy of
the system, Emec [PRB 39 (1989) 11093].
This idea comes from McWeeny’s and Huzinaga’s Quantum Electronic Separability Theory,
where it has been formulated as a Restricted Variational Principle.
(65)
• The additive energy of i contains an even part of all the energy terms in which the atom is
included. Accordingly, the Vni n-body term is multiplied by an 1/n factor:
i
Eadd
= V1i +
1 i
1
1 i
V2 + V3i + · · · +
V + ...
2
3
n n
(77)
The additive energies can be added together to give the total energy of the system:
Emec =
X
i
Eadd
.
(78)
i
The additive energy of an arbitrary subset of atoms can be also formed easily:
C
B
A
+ ...
+ zEadd
+ yEadd
Eadd (Ax By Cz . . . ) = xEadd
(79)
• The local energy of i within G is a quantity created to help in the calculation of the effective
energy of a group of atoms. We want to do
G
Eeff
=
X
i(G)
Eloc .
(80)
i∈G
i(G)
Therefore, Eloc has to be additive within G but effective with respect to the rest of the
system. The interaction of i with jkl . . . must be shared evenly only between the atoms that
(66)
belong to G [JCP 97 (1992) 6544]. In other words:
i(G)
Eloc
X
= v1 (~
ri ) +
j(6=i)
1
v2 (~
ri , ~
rj ) +
nG
ij
|
{z
}
i(G)
X
1
j>k(6=i)
nG
ijk
|
v3 (~
ri , ~
rj , ~
rk ) + . . .
{z
(81)
}
i(G)
V2
V3
where nG
ijk... is the number of atoms among i, j, k, ... that belong in G. The calculation of
i(G)
Eloc
is rather cumbersome, as it depends on both, the atom i and the group G. When G is
i(G)
terms as the difference between the general
small it may be more convenient to get the Vn
Vni minus a local correction. For the two and three-body terms we can use:
i(G)
V2
i(G)
V3
(6=i)
X
1
i
= V2 −
v2 (~
ri , ~
rj ),
2 j∈G
=
V3i
2
−
3
(6=i)
X
(j>k)∈G
(82)
(6=i)
1 X X
v3 (~
ri , ~
rj , ~
rk ) −
v3 (~
ri , ~
rj , ~
rk ).
2 j∈G
(83)
j ∈G
/
The formulas of four and further body terms become progressively more complicated.
(67)
Additive energies are particularly useful on the treatment of periodic structures. The cell energy of a crystal, for instance, is just the additive energy of the cell
group:
Ecell
X
Ecrystal
i
cell
Eadd
.
≡ Eadd =
=
ncell
i∈cell
(84)
Notice that for an infinitely periodic model of a perfect
crystal Ecrystal and ncell are both inifinite but Ecell is a
finite and well defined quantity.
The effective energy of a group, on the other hand, is the
quantity to determine on embedded cluster model calculations, i.e. on the calculation and optimization of a cluster
of atoms embedded into a lattice that is assumed to remain
frozen:
X
i(cluster)
cluster
Ecluster ≡ Eeff
=
Eloc
.
(85)
Frozen lattice
Cluster
i∈cluster
(68)
Typical two-body potentials
Coulomb potential: When the atoms carry net charges, the electrostatic charge-charge interaction
provides a leading contribution to the energy of the system. The term ionic matter alludes to the
leading role of this term.
q i qj
v2 (rij ) =
.
(86)
4πε0 rij
The summation of the Coulomb terms on an infinite system, like a crystal, gives rise to a conditionally
convergent term that needs special techniques.
U(r) / ε
Lennard-Jones potential:
0.0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
−0.8
−0.9
−1.0
[John Edward Lennard-Jones, Trans. Faraday Soc. 25 (1929) 668]
De
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
r/σ
A −r −6 term describes the long-range London dispersion
attraction. The +r−12 term emulates the short-range repulsion between atoms. The LJ6-12 term is widely used in
modelling the van der Waals interactions:
σ 12 σ 6
−
, (, σ ∈ R > 0). (87)
U (r) = 4
r
r
The equation can be transformed into a universal form by
using y = U/ and x = r/σ. Equilibrium properties: re =
√
σ 6 2, Ue = −, U∞ = 0, ke = 57,146 437 . . . /σ 2 .
(69)
Morse potential:
[Philip M. Morse, Phys. Rev. 34 (1929) 57]
1.00
Proposed as a reasonable fit of the nuclear potential curves
of diatomic molecules:
2
ρ(1−r/σ)
U (r) = 1 − e
, (, ρ, σ ∈ R > 0). (88)
0.60
0.40
0.20
0.00
0.6
Conversion to a universal form: y = U/ and x = r/σ, with
ρ left as a free parameter. Equilibrium properties: re = σ,
Ue = 0, U∞ = , De = , ke = 2ρ2 /σ 2 .
ρ=3
ρ=6
ρ=9
0.8
1.0
1.2
1.4
1.6
r/σ
1.8
2.0
2.2
2.4
Common transformations: U (r) − , (ρ/σ) = α.
Generalized Morse potential:
Several generalizations have
been proposed. For instance:
qρ(1−r/σ)
ρ(1−r/σ)
U (r) = e
− qe
, (q > 1).
(89)
Conversion to a universal form: y = U/ and x = r/σ, with ρ
and q left as free parameters: ρ influences the curvature at the
minimum and q the well depth.
Equilibrium properties: re = σ, Ue = (1−q), U∞ = 0, De =
(q−1), ke = q(q−1)ρ2 /σ 2 .
0.00
−0.20
−0.40
−0.60
U(r) / ε
U(r) / ε
0.80
−0.80
−1.00
−1.20
−1.40
ρ=6, q=1.5
ρ=6, q=2.0
ρ=6, q=2.5
−1.60
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
r/σ
(70)
Buckingham potential:
rmin / σ
Designed as an improvement of LJ6-12 potential, it conserves the −r −6 London attraction, but changes the +r −12
respulsive wall into an exponential term, thus the frequent
exp-6 nickname:

σ 6 6
α


eα(1−r/σ) −
for r>rmin ,
r
(90)
U (r) = α−6 α


∞
for r≤rmin .
(2/π) atan[U(r) (α−6)/ αε]
1.0
0.5
0.0
rmin
1.0
0.8
0.6
0.4
0.2
0.0
6
re
8 10 12 14
α
Buckingham: α=9.3
−0.5
U(r)
U’(r)
−1.0
0.5
1.0
r/σ
1.5
2.0
The internal cutoff rmin is defined as the smallest positive value for which U 0 (r) = 0 (a maximum
of the potential) and it helps to prevent the U → −∞ collapse at very short distances.
Transformation into a universal form: y = U (α−6)/α and x = r/σ. The free parameter must be
α > 7 or the potential would have no minimum.
Equilibrium properties: re = σ, Ue = −, U∞ = 0, De = , ke = α(α−7)/(α−6)σ 2 .
Internal cutoff: The iterative equation x = [eα(x−1) ]1/7 converges to xmin starting from x ≈ 0.9.
Then rmin = xmin σ.
Generalization: Sometimes the London term is seen as −C(σ/r)6 .
(71)
2.5
Lorentz-Berthelot combination rules: The homogeneous parameters can be used to estimate the
potential parameters for the heterogeneous combinations. Distances are combined as arithmetic
means: σij = (σii + σjj )/2. Energies and energy-like parameters are combined as geometric means:
√
√
ij = ii jj , αij = αii αjj .
Cutoffs and switching functions
where rcut is the cutoff radius and rsw ≈ 0.95rcut is the
switching radius.
1.0
0.8
0.6
S(r)
Most simulations define a cutoff radius beyond which the
atom-atom interactions are assumed to be null. To avoid
discontinuities in the energy, that can bring havoc to the
simulation, the cutoff can be introduced by means of a
switching function that ensures the analyticity of the product
U (r)S(r). A standard cubic switch is:


1
if r ≤ rsw ,


 (r +2r−3r )(r −r)2
cut
sw
cut
S(r) =
if rsw <r≤rcut , (91)
3

(r
−
r
)
cut
sw



0
if r > rcut ,
0.4
rsw
rcut
0.2
0.0
2.80
2.85
2.90
2.95
r/σ
3.00
3.05
(72)
3.10
Three and four-body potentials
There are many ways to introduce these terms. Bond angle potentials, for instance, can introduce
a penalty when an angle θijk , between bonded atoms, deviates from some expected value:
v3 (i, j, k) = Kh (θijk − θ¯j )2 ,
or
v3 (i, j, k) = Kt (cos θijk − cos θ̄j )2 ,
(92)
where the expected value θ̄j depends on the assumed properties of the middle atom j. The bond
angle arrangement of an atom is assumed to be governed by its Lewis pairs. For instance, we should
expect 180 ◦ around a sp-C atom, 120 ◦ around a sp2 -C, 109,47 ◦ around a sp3 -C, and so on.
This scheme can be augmented to include penalties for the deviation of a torsional angle from some
expected value.
A different strategy is including potentials between non-bonded next nearest neighbors (nnn), nnnn,
and so for.
The development of such a molecular mechanics force field is a quite specialized tasks that involves,
first, developing the force field constants and, second, assesing its suitability for a given collection
of compounds.
(73)
The shell model
−q
This is a quite successful approximation introduced for the dynamical
treatment of pure and defective ionic crystals. Going beyond the point
charge approximation by introducing dipole terms produces an intrinsic
instability (polarization catastrophe) unless the polarizability decreases
with the increased ionic overlap. This is automatically accounted for
in the shell model by using two different entities to represent every
atom/ion: a positively charged core that contains all atom mass, and
negatively charged massless shell. Core and shell from the same atom
interact through a strong harmonic potential. Shell-shell interactions
are described by an appropriate two body potential. Core-core interactions are usualy neglected.
+q
m
kx2
Vij
Julian Gale’s Gulp code [16] (www.ivec.org/GULP) implements a huge collection of potential
schemes, including the shell model, and a variety of dynamical methods to determine the properties
of pure and defective crystals. GULP is the last of a large family of codes developed under the
influence of the english Atomic Energy Authority at Harwell. The code is available upon request for
academic usage.
(74)
Gulp example: Shell model optimisation and property calculation for alumina.
1
1
2
3
4
5
6
7
8
9
10
11
12
13
optimise property conp
title
alumina test file
end
cell
4.7602 4.7602 12.9933 90 90 120
fractional
Al core 0.00000 0.00000 0.35216
Al shel 0.00000 0.00000 0.35216
O core 0.30624 0.00000 0.25000
O shel 0.30624 0.00000 0.25000
spacegroup
167
2
3
4
5
6
7
8
9
10
11
12
13
species
Al core 0.043
Al shel 2.957
O core 0.513
O shel -2.513
buckingham
Al shel O shel 2409.505 0.2649
0.00 0.0 10.0
O shel O shel
25.410 0.6937 32.32 0.0 12.0
spring
Al 403.98
O
20.53
output xr example1
output marvin example1 . mvn
(75)
Exercises
References
Gale and Rohl, 2003 [16]; Luaña and Pueyo, 1990 [17]; Luaña and Flórez, 1992 [18]; Luaña, Flórez,
and Pueyo, 1993 [19]; Schlick, 2002 [22].
Exercises
1. The typical LJ6-12 potential is sometimes assumed to have a too strong short range repulsion
and it is therefore changed into LJ6-11 or LJ6-10. Compare the behaviour of the three
potentials. Determine, in general, the equilibrium properties of the LJn-m potential if n < m.
2. Do a quantum mechanical calculation of the binding energy of the Ne2 molecule. Check the
influence of the correlation energy and the basis set. Fit the LJ and Morse potentials to your
best results.
3. The “Embedded Atom Method” (EAM) [24, 25] is a popular and successful technique in
developing interaction potentials for metallic systems. Write a short resume on the foundation
of the method and some recent applications of the technique.
(76)
Ltm2: Thermodynamic properties of a Lennard-Jones solid
Thermodynamic properties of a Lennard-Jones solid
(77)
The Lennard-Jones potential
The Lennard-Jones LJ6-12 internuclear potential
Proposed in 1929 by John Edward Lennard-Jones [Trans. Faraday Soc. 25 (1929) 668]:
σ 12 σ 6
y=U/
−
=⇒
y(x) = 4(x−12 − x−6 )
U (r) = 4
x=r/σ
r
r
(93)
U(r) / ε
The scaling of U and r transforms the potential into a universal function y(x). LJ6-12 has been
a favorite for the simulation of gases and liquids. The potential works well on closed shell atoms,
particularly noble gases.
0.0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.7
−0.8
−0.9
−1.0
De
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
r/σ
/kB (K)
σ (Å)
He
10,2
2,28
Ne
47,0
2,72
Ar
119,8
3,41
Kr
164,0
3,83
H
8,6
2,81
C
51,2
3,35
N
37,3
3,31
O
61,6
2,95
F
52,8
2,83
(Allen & Tildesley, 1989)
(78)
The Lennard-Jones potential
The potential can be arbitrarily derived:
n n
nU
(11+n)!
(5+n)!
d
dx d y
dn y
dn y
n
= (−1) 4
−
⇒
=
= n
.
dxn
11! x12+n
5! x6+n
drn
dr dxn
σ dxn
(94)
The equilibrium properties can be easily derived from here:
xe =
√
6
2 = 1,122 462 048 . . . ⇒ re = σ
√
6
2,
(95)
ye = −1 ⇒ De = ,
ye00
h
= 4 156 · 2
−7/3
(n)
ye
− 42 · 2
−4/3
i
= 57,146 437 84 . . . ⇒ ke = ye00
(11+n)!
(5+n)!
= (−1)n 4
−
11! 22+n/6
5! 21+n/6
(n)
⇒ Ue
(96)
,
σ2
(n)
= ye
(97)
.
σn
(98)
Anharmonic factors:
n
2
(n)
ye
5,714 643 787 086 · 10+01
5 −2,602 316 012 800 · 10+05
8
1,168 201 198 101 · 10+09
n
(n)
ye
n
3 −1,069 145 453 154 · 10+03
4
6
4,122 720 000 000 · 10+06
(n)
ye
1,682 750 554 240 · 10+04
7 −6,789 075 863 614 · 10+07
9 −2,103 513 743 008 · 10+10 10
3,962 896 605 241 · 10+11
(79)
The fcc structure
Simulation of the face centered cubic (fcc) structure
The fcc structure will serve to illustrate the calculation of a crystal phase:
Cell: (a, a, a, 90◦ , 90◦ , 90◦ ).
Space group: cubic, F m3̄m (No. 225).
1 (0, 0, 0), 2 (1/2, 1/2, 0), 3 (1/2, 0, 1/2), 4 (0, 1/2, 1/2).
Atoms in the main cell: Z = 4, 

2
a
0
0
q 

G = a 3 .
Metric matrix: G = 
a2
0 
 0
 . Cell volume: V =
0
0
a2
2
2
2
+ zij
Interatomic distance: rij
) where ~
rij = ~
rj − ~
ri .
= rTij G rij = a2 (x2ij + yij
Scaling: Length: r ? = r/σ; Energy: E ? = E/; Pressure: p? = pσ 3 /; . . .
Cutoff and switch function: The r −6 term dominates the long range tail of the potential. To cut
contributions < 10−n the cutoff radius must be rcut = 10n/6 σ. No switching off will be used.
Cell energy: Ecell =
Z
X
i=1
i
i
Eadd
, where Eadd
=
1 X
ULJ (rij ) and j runs over all atoms with rij ≤ rcut .
2
j(6=i)
(80)
The fcc structure
Strategy: How to run over shells of neighbor cells in order of increasing distance to the main cell?
j
3
+3
2
+2
1
+1
0
0
−1
−2
−3
−3 −2 −1
0
+1 +2 +3
i
Caution: Many quite small contributions
are added to larger values, producing
large truncation errors. How to avoid
the problem?
Ecell = 0
do iat = 1, nat
Eadd (iat) = 0; Econtri=0; L=0; doagain=.TRUE.
while (doagain .and. L.le.100)
do i = -L, L
do j = -L, L
if (abs(i).eq.L .or. abs(j).eq.L) then
do jat = 1, nat
xij = x(jat) + i - x(iat)
yij = y(jat) + i - y(iat)
rij = distance(xij, yij)
if (rij.gt.0d0 .and. rij.le.rcut ) then
Econtri = Econtri + 0.5d0 * ULJ(rij)
endif
enddo
endif
enddo
enddo
Eadd (iat) = Eadd (iat)+Econtri
L = L+1; doagain = (abs(Econtri) .ge. 1d-6)
enddo
Ecell = Ecell + Eadd (iat)
enddo
(81)
The fcc structure
Strategy: Many tasks involve obtaining the energy for a collection of geometries. Much effort can
be saved by keeping intermediate quantities between successive calculations.
"
12 6 #
Z X
Z X
Z
σ 12 X
σ 6 X
X
X
σ
σ
4
−12
−
=
2
hij
−
2
h−6
Ecell =
ij ,
2
ah
ah
a
a
ij
ij
i=1
i=1
i=1
j(6=i)
j(6=i)
|
{z
A12
j(6=i)
}
|
{z
}
A6
(99)
q
2 + z 2 ). The index i runs over the atoms in the main cell, but j sums all
where hij = (x2ij + yij
ij
atoms in the whole crystal until convergence is achieved. However, once A12 and A6 are obtained,
the optimization of a is a very simple analytical calculation:

−12
−6

Y = A12 X
− A6 X

1/6

a
2A
(α)
12
min
=
,
Minimun: Y 0 =0 ⇒ Xmin =
Y 0 = 6X −7 [−2A12 X −6 + A6 ]

σ
A
(α)
6


00
−14
−8
Y = 156A12 X
− 42A6 X
(100)
00 = 156A X −14 − 42A X −8 .
where Y = Ecell / and X = a/σ. Then: Ymin = −A26 /4A12 , Ymin
12 min
6 min
In the case of a Lennard-Jones fcc crystal: A6 = 925,050 946 665 605 and A12 = 6 211,522 660 630 82.
The equilibrium properties are then: amin = 1,541 737 σ, Vmin /Z = 0,916 159 770 σ 3 , Emin /Z =
−8,610 200 154 , and Bmin = −V (∂p/∂V ) = 75,185 141 /σ 3 .
(82)
The fcc structure
Pressure effects: There are two basic ways of determining the effects of hydrostatic pressure (i.e.
pressure applied isotropically) on the crystal properties:
Method 1: the equilibrium pressure, as a function of the volume, can be determined as the slope
of the A(V, T ) function:
∂A
.
p(V ) = −
(101)
∂V T
Method 2: the equilibrium volume, as a function of the pressure, can be determined by minimizing
the general Gibbs free energy:
min G? (p, T ; V ) = G(p, T )
V
⇒
V (p),
(102)
where G(p, T ) = A(V, T ) + pV .
Both ways provide the static Equation of State (EOS), V (p) or p(V ), and the crystalBulk Modulus:
BT (p) = −V
∂p
∂V
=V
T
∂2A
∂V 2
.
(103)
T
The static approximation consists on the complete neglect of thermal effects (i.e. T = 0 K), even
of the zero point vibrational energy. Using this approximation:
A(V, T ) ≈ Ecell (V ),
G(p, T ) ≈ Ecell (V ) + pV.
(104)
(83)
The fcc structure
The pressure equation, 101, can be evaluated analytically for the Lennard-Jones fcc crystal:
Y0
σ −1
(3a2 )−1
z }| {
z }| { z}|{ ∂X ∂V −1
∂(Ecell /)
=−
=−
Y0
2
∂X
∂a
∂a
σ3a
∂Ecell
∂V
h i
−15
−9
=
(4A
X
−2A
X
).
12
6
σ3
p≈−
(105)
This equation provides the pressure as a function of the reduced length, X = a/σ. It is easy to
transform it into a p(V ) relationship:
p? =
p
? −5
? −3
=
4A
(V
)
−
2A
(V
)
12
6
σ −3
(106)
where V ? = V /σ 3 is the reduced volume.
A similar treatment can be applied to eq. 103. This gives rise to the next form of the reduced bulk
modulus
B
52A12
16A6
B? =
=
−
.
(107)
σ −3
3(V ? )5
3(V ? )3
Both relationships, p? (V ? ) and B ? (V ? ), can be used together to produce an implicit description of
the more interesting B ? versus p? relationship.
(84)
The fcc structure
100
V / V0
p / (εσ−3)
80
60
40
1.00
0.95
0.90
0.85
0.80
0.75
0.70
Pressure contraction: The cell volume diminishes as the applied pressure increases. The effect is highly non-linear, being dominated, for
small volumes, by the p? ∝ (V ? )−5 term.
0
20
40
60
80 100
−3
p / (εσ )
20
3
V0 / (Zσ )
0
0.6
0.7
0.8
0.9
1.0
1.1
3
Vcell / (Zσ )
1.2
1.3
1.4
A small negative pressure could produce large
cell expansions.
A pressure of 100 /σ 3 contracts the cell to
some 70 % of its V0 = V (p=0) value.
2
The cohesive energy, i.e. the energy needed to
separate the crystal into non-interacting atoms,
is −8,610 200 154 per atom.
0
Ecell / (Zε)
The cell energy versus volume follows a curve
similar to the Lennard-Jones potential. Notice,
however, that Ecell is a summation of many interaction energies.
V0/(Zσ3)
-2
Ecoh/(Zε)
-4
-6
-8
0.5
1.0
1.5
2.0
2.5
Vcell / (Zσ3)
3.0
3.5
4.0
(85)
50
The fcc structure
p* = 0
5
10
15
20
40
The enthalpy is the static approximation to the
Gibbs free energy.
H* / Z
30
The minimum of each H ? (V ? ) curve determines the equilibrium properties for that particular pressure.
20
10
0
0.5
1
1.5
2
2.5
Whereas the p=0 curve has a null slope asymptote, the asymptotes for the other curves corresponds to the pV ≡p? V ? term.
V* / Z
140
Bulk modulus is loosely related to the crystal
hardness.
The B(p) relationship is almost linear:
B(p)/σ −3 = 75,37 + 7,160p − 0,056 1p2 + ...
130
120
B / (εσ-3)
Bulk modulus increases with pressure, showing
that compressing the crystal becomes progressively harder as the crystal shrinks.
110
100
90
80
70
0
2
4
6
8
10
-3
p / (εσ )
(86)
Generalization for arbitrary crystal structures
The equations developed for the fcc crystal can be generalized to any crystal structure.
Cell energy:
Ecell
A12
A6
E? =
= 12 − 6 ,
X
X
a
X= ,
σ
A12 = 2
Z X
X
h−12
ij ,
A6 = 2
i=1 j(6=i)
Z X
X
h−6
ij .
(108)
i=1 j(6=i)
2 + z 2 ]1/2 .
Cubic cells: hij = [x2ij + yij
ij
2 + r 2 z 2 ]1/2 , with r = c/a.
Tetragonal cells: hij = [x2ij + yij
c
c ij
2 − x y + r 2 z 2 ]1/2 , with r = c/a.
Hexagonal cells: hij = [x2ij + yij
c
ij ij
c ij
2 + z 2 ) + 2 cos α (x y + y z + z x )]1/2 .
Rhombohedric cells: hij = [(x2ij + yij
ij ij
ij ij
ij ij
ij
2 + r 2 z 2 ]1/2 , with r = b/a, and r = c/a.
Orthorhombic cells: hij = [x2ij + rb2 yij
c
b
c ij
2 + r 2 z 2 + 2r x z cos β]1/2 .
Monoclinic cells (β 6= 90◦ ): hij = [x2ij + rb2 yij
c ij ij
c ij
2 +r 2 z 2 +2r x y cos γ +2r r y z cos α+2r z x cos β]1/2 .
Triclinic cells: hij = [x2ij +rb2 yij
c ij ij
b ij ij
b c ij ij
c ij
(87)
Cell volume:
V? =
Vcell
= A3 X 3 ,
3
σ
X = a/σ.
(109)
Cubic cells: A3 = 1.
Tetragonal cells: A3 = rc = c/a.
√
√
Hexagonal cells: A3 = rc 3/2 = c 3/2a.
√
Rhombohedric cells: A3 = 1 − 3 cos2 α + 2 cos3 α.
Orthorhombic cells: A3 = rb rc = (b/a)(c/a).
Monoclinic cells (β 6= 90◦ ): A3 = rb rc sin β.
p
Triclinic cells: A3 = rb rc 1 − cos2 α − cos2 β − cos2 γ + 2 cos α cos β cos γ.
Equilibrium pressure:
p
A12
A6
4A12
p? =
=
4
−
2
=
σ −3
A3 X 15
A3 X 9
A3
A3
V?
5
2A6
−
A3
A3
V?
3
.
X = a/σ.
(110)
(88)
Equilibrium volume: The p? (V ? ) relationship is highly non-linear and it cannot be directly
inverted. However, the following iterative equation can be used to produce a series of values,
{V0? ≈1, V1? , V2? , ...}, that converges to the equilibrium volume for a given pressure:
?
Vi+1
4A12 A43
=
p? + 2A6 A23 (Vi? )−3
1/5
.
(111)
General Gibbs free energy:
G? ≈ H ? (V ? ; p? ) =
H
A12 X −12 − A6 X −6 + p? A3 X 3
A3 4
A3 2
?V ?.
A12
−
A
+
p
6
V?
V?
=
=
(112)
It must be noticed that this is not the equilibrium value, that would be a funcion of just the pressure
and temperature. The equilibrium value for G? is easily obtained by using the equilibrium volume
or the equilibrium pressure on the above equation. For instance:
?
Heq
(V ? ) = 5A12
A3
V?
4
− 3A6
A3
V?
2
.
(113)
(89)
Bulk modulus:
B
1
52 A12
−15
−9
B? =
=
(156A
X
−
42A
X
)
=
12
6
σ −3
9A3
3 A3
A3
V?
5
14 A6
−
3 A3
A3
V?
5
.
(114)
Parameters for some prototypical structures:
Structure
Z
nn nnn
A3
A6
A12
simple cubic (sc)
1
6
12
1
1,680 384 783 87 · 10+01
1,240 429 809 01 · 10+01
bcc
2
8
6
1
1,161 829 239 24 · 10+02
2,048 378 308 15 · 10+02
fcc
4
12
6
1
9,250 509 336 92 · 10+02
6,211 522 660 62 · 10+03
diamond
8
4
12
1,241 973 100 22 · 10+04
1,487 222 535 45 · 10+06
ideal hcp
2
12
6
1
√
2
5,781 958 858 92 · 10+01
4,852 917 507 64 · 10+01
The ideal hcp structure corresponds to the densest hcp packing, with rc = c/a =
p
8/3.
The number of nearest neighbors (nn), next nearest neighbors (nnn), etc. shows the atomic
arrangement from a local perspective. Comparison of the neighborhhod arrangement can be used
to show that different cell descriptions can correspond to the very same crystal.
(90)
2
E* / Z
0
The stability of the structures follows the order
of density packaging: hcp ≈ fcc > bcc > sc >
diamond.
-2
-4
-6
-8
0.5
The fcc and the ideal hcp structures have almost identical properties. The hcp is slightly
more stable (8,696 · 10−4 ).
fcc
bcc
sc
ideal hcp
diamond
1.0
1.5
2.0
2.5
V* / Z
3.0
3.5
4.0
100
?
Vmin
80
sc
1,067 084
1,215 057
60
bcc
1,233 719
0,938 899
fcc
1,541 737
0,916 160
Diamond
2,492 005
1,934 445
Ideal hcp
1,090 167
0,916 144
p*
Xmin
Structure
40
20
0
0.6
fcc
bcc
sc
ideal hcp
diamond
0.7
0.8
0.9
1.0
V* / Z
1.1
1.2
1.3
1.4
(91)
70
-4
-5
-6
-7
-8
-9
60
50
H* / Z
40
0 1 2 3 4 5
30
20
fcc
bcc
sc
ideal hcp
diamond
10
0
-10
0
Fcc and the ideal hcp structure have
the same equilibrium enthalphy under
any pressure. They are the most stable
phases under any p? >0 pressure.
The sequence of stability follows an
order that do not changes with pres?
? < H?
?
sure: Hhcp
≈ Hfcc
bcc < Hsc <
?
Hdiamond
.
5 10 15 20 25 30 35 40 45 50
p*
(92)
140
120
100
Fcc and the ideal hcp structure have
also the same bulk modulus under any
hydrostatic pressure.
B*
80
The bulk modulus of every structure
increases almost linearly with pressure. The sequence of hardness do not
changes with pressure: Bhcp ≈ Bfcc >
Bbcc > Bsc > Bdiamond .
60
40
fcc
bcc
sc
ideal hcp
diamond
20
0
0
2
4
6
8
10
p*
(93)
0
1
2
3
4
5
72
sc
48
24
Radial neighbor density: g(r)
72
The nearest neighbors
(nn) shell appears at
almost the same distance for all the structures.
bcc
48
24
72
fcc
48
Fcc and hcp appear
identical for the first
two shells (nn and
nnn), but start to differ on the third shell
(nnnn) of neighbors.
24
72
ideal hcp
48
24
72
diamond
48
24
0
0
1
2
r/σ
3
4
5
(94)
512
256
fcc
bcc
sc
ideal hcp
diamond
The accumulated number of neighbors,
R
G(R) = 0R g(r)dr (notice the log2 G
scale), shows the close relationship between the fcc, bcc and ideal hcp structures. Fcc and hcp are identical on the
nn and nnn shells. Fcc, hcp and bcc
tend to be equivalent, on average, at
large distances. The sc and diamond
structures are far less dense.
128
G(r)
64
32
The stability of the crystal structures
follows the order of the nn coordination
index: 12(fcc and hcp) > 8(bcc) >
6(sc) > 6(diamond).
16
8
4
1.0
1.5
2.0
2.5
3.0 3.5
r/σ
4.0
4.5
5.0
The Lennard-Jones solid shows a clear
preference for the dense structures. Is
this a necessary consequence of the
radial dependence of the two-body LJ
potential?
(95)
Treatment of the hcp crystal
The geometry of all the crystal structures examined up to now depended on a single internal variable,
a, that we converted into X after scaling. Determining the equilibrium geometry and properties
involved minimizing the general Gibbs energy, G? (p, T, X), with respect to this single variable.
For most crystal structures, however, the geometry depends on a number of cell parameters and
atomic cell positions, and the minimization of G? (p, T, ~
x) with respect to all of them requires some
techniques beyond those used up to now.
Let us start considering the general hcp structure:

a2

2
Cell: (a, a, c, 90, 90, 120). Metric matrix: G = 
 −a /2
0
−a2 /2
a2
0
0

√

3 3
.
Cell
volume:
V
=
a rc .
0 

2
c2
Space group: P 63 /mmc (194). Atoms in the main cell: Z = 2; (2c) : (1/3, 2/3, 1/4), (2/3, 1/3, 3/4).
2
2
2
Interatomic distance: rij
= rTij G rij = a2 x2ij + yij
− xij yij + rc2 zij
where rc = c/a and
~
rij = ~
rj − ~
ri .
p
We have already examined this structure, but only when the rc = 8/3 ratio was enforced. We
will examine now the dependence of the energy and crystal properties on the two geometrical
parameters: a and c or rc .
(96)
3.0
c/σ
2.5
2.0
3.0
−2
−3
−4
−5
−6
−7
−8
−9
2.5
rc = c/a
1.5
The ideal hcp ratio (rc = c/a = 8/3
represented by the straight line in
the plot) truly corresponds to the
minimum energy configuration of the
LJ crystal.
2.0
1.5
1.0
1.0
0.5
0.5
1.0
1.5
a/σ
2.0
1.0
1.5
2.0
2.5
3.0
V/(Zσ3)
(97)
E/(Zε)
−4
−5
−6
−7
−8
−9
E/(Zε)
p
From clusters to the crystal
0
−1
The properties of
LJ clusters should
converge to the
crystal
properties
as
the
cluster
size
increases.
The convergence is,
however, quite slow.
Nanomatter properties significantly
differ from bulk
range properties.
−2
Eopt/n (ε)
−3
−4
−5
−6
−7
−8
−9
0
100
200
300
400
500
600
700
800
900 1000
n (LJ atoms)
(98)
The significant contribution of the surface and the accusated tendency to form an icosahedral
arrangement are the main cause for the slow convergence of the clusters towards the bulk.
LJ135
LJ147
LJ549
LJ557
LJ900
LJ906
LJ907
LJ910
(99)
Exercises
References
Calvo et al. [26]; Doye & Calvo, 2002 [27]; Noya & Doye, 2006 [28]; Stillinger, 2001 [29]; Somasi
et al., 2000 [30].
Exercises
1. Analyze the energy of the LJ crystal in the graphite phase as a function of the a and c lattice
parameters. Determine the best rc = c/a ratio as a function of the external pressure. Compare
the relative stability and hardness of the graphite and diamond phases. Determine the neighbor
shell structure for graphite and compare it with the phases represented in the figure of page
94.
2. Determine the value of the independent elastic moduli for a Lennard-Jones solid in the sc,
bcc, fcc, hcp, graphite and diamond phases. Compare the prediction of the model with the
experimental values for noble gases, alkaline and alkaline-earth metals, the diamond and
graphite phases of C and Si, etc. Cite the sources of your data and discuss the successes and
failures of the LJ model.
3. Check very carefully the relative stability of the fcc and ideal hcp compact phases. Which is
most stable? Is this result modified if the (σ/r)12 repulsion term is changed into (σ/r)m with
m = 8, 9, 10, · · ·
(100)
Exercises
4. The fcc and the ideal hcp are just two of the infinite compact structures formed by stacking
layers of close contact hard spheres. A detailed description of the close-packed structures
can be found in Krishna & Pandey [31] (http://www.iucr.org/iucr-top/comm/cteach/
pamphlets/5/5.pdf). Check the relative stability of A, AB (hcp), ABC (fcc), ABA, ABAC,
etc LJ6-12 crystal.
5. Use a Morse potential to examine the relative stability of the compact phase as a function of
the internal ρ parameter.
(101)
Les1: The electronic structure problem
The electronic structure problem (overview)
(102)
Overview
The equation to solve
We want to solve the Hartree-Fock or Kohn-Sham equation for an electron in the average
field of all other (infinite) crystal charges:
~2 2
−
∇~r + G(~r, [ρ]) ψλ (~r) = λ ψλ (~r)
2me
(115)
where G(~r, [ρ]) = V̂n + V̂H + V̂xc :
• V̂n : electrostatic potential of the nuclei (or sum of the pseudopotentials of the
chemical kernels);
Z
ρ(~r 0 )d~r 0
• V̂H (~r) =
: Coulomb potential due to the (posibly valence) electron
0|
|~
r
−
~
r
R3
density. Hartree term.
• V̂xc : Exchange and correlation term. Common approaches: Hartree-Fock (HF), LDA
or LSDA (Local [Spin] Density Approach), GGA (Generalized Gradient Approach), . . .
(103)
Overview
Implicit assumptions
• Non-relativistic hamiltonian. However, many (most) solid state calculations
do need and use relativistic corrections. This, in most cases, involves using a
relativistic pseudopotential (an atomic necromancy problem) plus a spin-orbit
operator.
• Born-Oppenheimer approximation: the dynamic of electrons and nuclei can
be approximately decoupled. Electrons move on a fixed nuclear frame. Nuclei
move adiabatically over the electronic ground state potential energy surface.
However, electron-phonon interaction is quite important: use perturbation theory
afterwards.
• Independent electron approximation: the many electron wave function
is implicitely described as a Slater determinant over some spinorbital set.
Correlation is treated through the V̂xc functional.
(104)
Overview
Specific solid state issues
• Properties of the ψλ : translational symmetry, direct and reciprocal space, Bloch
theorem, . . .
• Choice/generation of pseudopotentials (if used).
• Type of basis set: planewaves (PW), orthogonal PWs, augmented PWs, crystal
GTOs, . . .
• Simplified models: free electron gas, weakly interacting electron gas, tightbinding models, . . .
• Properties: band diagrams, density of states, electron density, transport
properties, optical properties, response functions, . . .
(105)
Timeline
A solid state methods rough timeline
1928 Bloch theorem.
1931 Wilson: band theory shows the difference between insulators and metals.
1934 Slater: calculation of the bands of Na.
1935 Wigner and Seitz: first quantitative calculation of band structure in Na.
1935 Bardeen: Fermi surface of a metal.
1937 Slater: formulation of the APW method.
1940 Herring: formulation of the OPW method.
1947 Shockley-Bardden-Brattain: build a point contact transistor.
1947 Korringa (completed by Kohn and Rostoker in 1954): KKR method.
1951 Philips-Kleinman on pseudopotentials.
1953 Herrman-Callaway: first realistic band structure of a semiconductor (Ge).
1965 Hohenberg-Kohn theorem, and Kohn-Sham method.
1975 Anderson: LMTO method. Opens the door to the full potential ”L” methods.
1985 Car-Parrinello method.
(106)
Les2: The free electron model and beyond
−1.0 −0.5 0.0
0.5
1.0
1.0
T = 0K
βµ=50
βµ=10
βµ=5
βµ=1
Parte real
0.8
A
ψ(x)
fFD
0.6
−1
3π
0.4
2π
π
−π
0
π
kx
0.2
ωt
0
2π
0.0
0.0
−π
3π
0.5
1.0
ε/µ
1.5
2.0
The free electron model and beyond
Parte real
A−1 ψ(x)
3π
2π
π
−π
0
π
kx
ωt
0
2π
−π
3π
(107)
Free electron (1D)
Free electron (1D)
The Hamiltonian and the Schrödinger equation:
d2 ψ
2m
2
=
−
ψ
=
−k
ψ
dx2
~2
=⇒
~ d2
p̂2x
ĥ = −
=
,
2m dx2
2m
ikx
ψk (x) = e
,
k2 ~2
=
,
2m
p̂x = −i~
k ∈ R.
d
.
dx
(116)
The solution is a 1D planewave (PW). Some properties:
• PW(1D) are eigenfunctions of p̂x : p̂x ψk (x) = k~ψk (x).
• ψk (x) and ψ−k (x) have the same energy and move on opposite directions with the same
speed.
• A PW has an uniform density in R: ψk? ψk = e−ikx eikx = e0 = 1
R
R
2
• A PW is not normalizable on R: R |ψk | dx = R dx −→ ∞. This should be evident from
the fact that a PW is an eigenfunction of momentum: the position uncertainty should go to
infinity through Heisenberg principle.
• Spacial periodicity: Let x and x + λ be the closests different points in phase, i.e. ψk (x + λ) =
ψk (x) for arbitrary x. Then eikx eikλ = eikx and eikλ = 1, so kλ = 2π. A PW has a
wavelength λ = 2π/k .
(108)
Free electron (1D)
Periodic boundary conditions
If the PW is forced to be periodic with period A, i.e. if for any x ∈ R
ψk (x + a) = ψk (x)
=⇒
eika = 1
=⇒
ka = n2π,
n ∈ N.
(117)
Only those PW with k being an integer multiple of 2π/a will have the desired periodicity. There
still are an infinite number of PW, but we have passed from k ∈ R to n ∈ N.
R
The PW can be normalized to a single cell of the 1D lattice: |N |2 0a |ψk |2 dx = 1. The set of
normalized an periodic planewaves is
1
|ki = ψk (x) = √ eikx ,
a
2π
k=n
,
a
k2 ~2
h2 n2
k =
=
,
2m
2ma2
Any two different PWs from this set are orthogonal:
Z a
i2π(n−n0 ) − 1
i(k−k0 )a − 1
0
1
1
e
e
hk0 |ki =
ei(k−k )x dx =
=
=0
a 0
a
k − k0
2π(n−n0 )
n ∈ N.
(118)
iff n 6= n0 .
(119)
The periodic PWs form, in fact, a complete set and, for any function
Z ∞
X
1 X ik(x−x0 )
e
= δ(x−x0 ) =⇒ f (x) =
fk eikx ⇐⇒ fk =
f (x)e−ikx dx. (120)
a k
−∞
k
f (x) is the Fourier transform of fk , and fk is the inverse Fourier transform of f (x).
(109)
Free electron (3D)
Free electron (3D)
~ and the solutions are planewaves (PW) again:
The hamiltonian is ĥ = p̂2 /2m, p
~ˆ = −i~∇,
~
1
|~ki = ψ~k (~
r ) = V − /2 eik·~r ,
~2 k2
~k =
,
2m
~
r , ~k ∈ R3 .
(121)
with V being the volume of the normalization box. Some properties
• The PWs are eigenfunctions of the momentum operator: p
~ˆ |~ki = ~~k |~ki.
• The particle velocity is proportional to ~k, the wavevector : ~v = ~~k/m.
• The |~ki and |−~ki PWs are degenerated.
• The wavelength of a PW is λ = 2π/k.
To enforce periodic boundary conditions in a general way we define the parallelepipedic cell a and
˜
3
3
an arbitrary primitive translation ~t = a n, n ∈ N . For any ~
r∈R :
˜
~ ~
ψ~ (~
r + ~t ) = ψ~ (~
r ) =⇒ eik·t = 1.
(122)
k
k
If ~k is a vector in the reciprocal cell, ~k = a? k = 2πa? h, the periodicity condition is
˜
˜
~ ~
T
1 = eik·t = ei2π(h
n)
=⇒
h1 nx + h2 ny + h3 nz ∈ N
for all n ∈ N3 .
(123)
(110)
~k = 2π(h1~a ? +h2~b ? +h3~c ? ) = 2πa? h = a? k, (h1 , h2 , h3 ) ∈ N3 . (124)
˜
˜
The wavevector of the periodic PWs is then
Free electron (3D)
This discrete mesh of allowed ~k-points is distributed uniformly in the reciprocal space. Each ~k point can be associated with a small parallelepiped
of volume v~k = (2π)3 V ? = (2π)3 /V , where V is the volume of the main
cell. It is implicitely assumed that a primitive cell is used to describe the
reciprocal space.
The periodic PWs form an orthonormal set
1
h~k|~k 0 i =
V
Z
V
~
~ 0 )~
r
ei(k−k
d~
r = δ~k,~k 0 ,
(125)
where the integral is done on a unit cell. The set is also complete, which means that any 3D
function can be expanded as
Z
X
~
~
f (~
r) =
f~k eik·~r ⇐⇒ f~k =
f (~
r )e−ik·~r d~
r.
(126)
~
k
V
(111)
Free electron gas
Model: A gas of free and independent electrons
Let’s assume a very simple model for a metal. Ne electrons are confined within the metal obeying
the crystal lattice periodicity and the Pauli principle but independent and suffering no interaction
otherwise. Each electron can be described using a PW plus a spin function: |~kσi, where σ = ±1/2
is the spin-azimuthal quantum number. Let n~k,σ (either 0 or 1) be the occupation number of
spin-PW |~kσi. A microscopic description of the state of the whole gas of electrons is obtained by
giving the occupancies of all spin-PWs: {n~k,σ }. The total number of electrons and the total energy
of the gas is
X
X
Ne =
n~kσ ,
E=
n~kσ ~k ,
(127)
~
kσ
~
kσ
where ~k = ~2 k2 /2m is the energy of PW |~ki.
Fermi energy/sphere/radius: In the ground state of the gas, all energy levels from zero up to
some energy F are occupied without gaps. The occupied states form a sphere in ~k-space. The
volume of the sphere must be enough to accomodate the Ne /2 electron pairs, taking into account
that each state occupies a volume of (2π)3 /V :
4 3
Ne (2π)3
πk =
3 F
2
V
=⇒
kF =
Ne
3π 2
V
1/3
=⇒
F
~2
=
2m
Ne
3π 2
V
2/3
.
(128)
(112)
Free electron gas
Example: Li bcc (a = 3,46 Å for the I cell). Ne /V = 1/(a3 /2) = 4,828 m−3 (each Li
transfers one valence electron to the electron gas). Then: kF = 1,13 · 1010 m−1 = 1,13 Å−1 ,
F = 7,75 · 10−19 J = 4,84 eV. Those values are typical of many metals.
Density of states: Allowed states form a discrete mesh in ~k-space. Their number is, however,
so huge that no significant error is made by assuming that they form a continuous distribution. If
n = N/V = k3 /3π 2 is the density of electrons with wavevector modulus ≤ k:
3
k2
1
2m /2 √
dn = 2 dk = g(k)dk =
d = g()d
(129)
π
2π 2 ~2
where k and belong in the [0, ∞) interval. If we assume that all levels up to the Fermi energy are
fully occupied, the total number of electrons and the total energy of the electron gas would be
Z F
Z F
N
E
=
g()d,
=
g()d.
(130)
V
V
0
0
Statistical treatment: Fermi-Dirac distribution. We can introduce the above energy distribution
into an statistical treatment of the electron gas. The grand canonical partition function for a
system of independent and identical fermions is
Y
Y
−βi
Ξ(µ, V, T ) =
(1 + λe
)=
[1 + e−β(i −µ) ]
(131)
i
i
where µ (chemical potential), V (volume), and T (absolute temperature) determine the thermodyc V. Luaña, QTC Murcia 2008
(113)
Free electron gas
namical state of the Fermi gas; β = 1/kB T ; λ = eµβ = eµ/kB T is the Lewis-Randall activity of
Q
the gas; and the i runs over the one-particle quantum states, i.e. the |~kσi states in the case of
the electron gas.
The average properties of the grand canonical ensamble are obtained from its partition function:



pV = kB T ln Ξ,

∂
ln
Ξ





,
 hSi = −kB T

dE = −pdV + T dS + µdN,
∂T
µ,V
=⇒
(132)
∂
ln
Ξ




E = −pV + T S + N µ,
.
 hN i = kB T



∂µ
T,V
d(pV ) = pdV − SdT + N dµ, 
The average number of electrons is not a constant in the grand canonical esamble, but a function
of the thermodynamic state. The average number of electrons per state follows the Fermi-Dirac
distribution:
hN i =
X
i
1
∂ X
ln(1+e−β(i −µ) ) ⇒ n() = β(−µ)
= fF D (/µ; βµ). (133)
hni i = kB T
∂µ i
e
+1
The Fermi-Dirac distribution is a universal function of two adimensional variables: /µ, the energy
relative to the chemical potential; and βµ = µ/kB T , the reduced temperature. At low temperature
µ ≈ F is a very good approximation. The Fermi temperature TF = F /kB is usually 1·103 –1·104 K
for most metals. Under room temperature T ≪ TF and the distribution is almost a step function.
(114)
Free electron gas
1.0
T = 0K
βµ=50
βµ=10
βµ=5
βµ=1
0.8
fFD
0.6
0.4
0.2
0.0
0.0
0.5
1.0
ε/µ
1.5
2.0
(115)
Free electron gas
0.00
βµ=50
βµ=10
βµ=5
β
−1 ’
f FD
−0.05
−0.10
−0.15
−0.20
−0.25
0.0
0.5
1.0
ε/µ
1.5
2.0
(116)
Free electron gas
Derivative of the Fermi-Dirac function and integrals involving it. The function
βeβ(−µ)
∂f
= −
2
∂
eβ(−µ) + 1
(134)
gets involved in many integrals related to the FD distribution. Most of the integrals lack a closed
analytical form, but we can exploit the fact that ∂f ∂ is close to a Dirac’s delta function to expand
the integrals into a rapidly convergent series.
The prototype integral is
Z
∞
A=
0




∂f
F ()
d =

∂


η = β(−µ)
dη = βd
η ∈ [−βµ, ∞)







Z
∞
=−
F
−βµ
η
µ+
β
eη
dη ≈ −
(eη + 1)2
Z
∞
(135)
−∞
We have seen that µ ≈ F kB T , in other words µ η/β. We can expand the F () function in
a Taylor’s series:
2
η
1
η
η
= F (µ) + F 0 (µ) + F 00 (µ)
+ ...
(136)
F µ+
β
β
2
β
and using the auxiliary integrals
Z ∞
1
η m eη dη
Am =
(m ≥ 0) : A0 = 1, A1 = 0, A2 = π 2 /6, . . .
(137)
η
2
m! −∞ (e + 1)
(117)
Free electron gas
we get
Z
∞
A=
0
π 2 00
∂f
d = −F (µ) −
F (µ) + . . .
F ()
∂
6β
(138)
Using this technique we can obtain:
3
Ne
1
2m /2 3/2
π2
The average number of electrons: n =
=
µ
1+
+ ...
V
3π 2 ~2
8µ2 β
#
"
2
2
π
kB T
+ ...
The chemical potential: µ = F 1 −
12
F
3
2m /2 5/2
5π 2
F 1 +
+ ...
~2
122F β 2
2 2m 3/2
k
√
B
The cv heat capacity: cel
F T
v =
6
~2
1
Eel
The internal energy:
=
V
5π 2
If we add up the vibrational contribution, the heat capacity at low temperatures for a nonmagnetic
metal is cv = α1 T + α2 T 3 . The lineal term comes from the conducting electrons. The cubic term
is due to the elastic vibrations.
(118)
Brillouin zones
The Brillouin zones
The first Brillouin zone (BZ-1) is the common name for the Wigner-Seitz primitive cell of the
reciprocal or ~k-space lattice. Whereas primitive and centered cells are both used for the direct
lattice, centering is avoiding in the manipulation of the ~k lattice.
An alternative definition for BZ-1 is the set of points in ~k space that can be reached from the origin
(~k = ~0) without crossing any Bragg plane. A Bragg plane for any two points in the lattice being
the plane which is perpendicular to the line between the two points and passes through the bisector
of that line.
The concept of BZ-1 can be generalized. The second BZ (BZ-2) is the set of points that can be
reached from the first zone by crossing only one Bragg plane. BZ-(n+1) is formed is the set of
points not in {BZ-1, BZ-2, ... BZ-(n−1)} that can be reached from BZ-n by crossing only one
Bragg plane. Alternatively, the n Brillouin zone can be reached from the origin by crossing n−1
Bragg planes, but not fewer.
The construction of the BZ is illustrated in the next slides for a simple square lattice. An important
point to check is that every Brillouin zone has the same volume, namely the volume of a primitive
reciprocal lattice. In addition, any BZ can be mapped back to the first zone by using just primitive
translations, i.e. all the BZ’s are equivalent by translation symetry.
(119)
Brillouin zones
4th nn
nn
2nd nn
3rd nn
(120)
Brillouin zones
4th nn
nn
2nd nn
3rd nn
(121)
Brillouin zones
4th nn
nn
2nd nn
3rd nn
(122)
Brillouin zones
4th nn
nn
2nd nn
3rd nn
(123)
Brillouin zones
Al BZs are equivalent due to the translational part of the crystal space group. Furthermore,
the rotational symmetry determines the equivalence between different positions within a Brillouin
zone. The special symmetry points receive particular names. Although there are different naming
conventions, Γ is typically used to designate the origin of the reciprocal cell (~k = ~0).
Let’s examine the BZ for the cubic P, I and F Bravais lattices, directly from the Kvec plots of the
excellent Bilbao Crystallographic Server. The plot and the list of special ~k positions is available for
all the space groups.
Im3̄m
F m3̄m
P m3̄m
kz
kz
H3
X3
L
X1
kx
Γ
Σ
Q
K
R
S
P
U
∆
X3
R
F1
F2
Λ
kz
N1
Λ
S
W
D
Γ
X
V ky
S1 M
Λ
kx
Γ
∆
Σ
T
F
N
G
H
∆
kx
ky
X ky
M
Σ
Z
(124)
Fermi surface and Brillouin zones
Mapping the Fermi surface onto the Brillouin zones
At 0 K the electrons occupy spin states up to filling up some region of ~k-space (a sphere in the case
of a free electron gas). The most important part is the boundary between the populated and the
empty regions, i.e. the Fermi surface. This boundary will tipically extend along several BZs, but it
can be mapped back to the first BZ by means of primitive translations.
The next two examples correspond to a free electron gas on a square lattice with a density, Ne /V ,
of 1 and 2 electron pairs per primitive lattice, respectively. Even this simple system can show a
rather involved surface.
For temperatures above 0 K the electron population is governed by the Fermi-Dirac statistics, so
the boundary between populated and unpopulated states is broken by the thermal excitation of the
electrons. The Fermi surface remains a useful concept, however, because TF = F /kB is quite
large in real materials. Thermal effects can be described as a broadening of the surface boundary.
The Fermi surface, is useful for predicting the thermal, electrical, magnetic, and optical properties
of metals, semimetals, and doped semiconductors. As we shall see, F for an insulator lies on an
energy gap, and only conducting materials have a Fermi Surface.
Fermi surfaces can be measured through observation of the oscillation of transport properties
in magnetic fields. The most direct technique is the angle resolved photoemission spectroscopy
(ARPES, see [32]), that resolves the electronic structure in the momentum-energy space.
(125)
4th nn
nn
2nd nn
3rd nn
(126)
4th nn
nn
2nd nn
3rd nn
(127)
Periodic Table of the Fermi Surfaces of Elemental Solids
http://www.phys.ufl.edu/fermisurface
Tat-Sang Choy, Jeffery Naset , Selman Hershfield, and Christopher Stanton
Physics Department, University of Florida
Jian Chen
Seagate Technology
(15 March, 2000)
Ferromagnets:
Co_fcc
Co_fcc
Alternate Structures :
Source of tight binding parameters (except for fcc Co ferromagnet): D.A. Papaconstantopoulos, Handbook of the band structure of elemental solids, Plenum 1986.
This work is supported by NSF, AFOSR, Research Corporation, and a Sun Microsystems Academic Equipment Grant.
(128)
Independent electrons in a periodic potential
Independent electrons in a periodic potential. Bloch theorem.
The Scrödinger equation for an independent electron in a potential V (~
r ) is
~2
r ) ψ(~
r ) = ψ(~
r)
∇~r2 + V (~
ĥ(~
r )ψ(~
r) = −
2me
(139)
~ where L
~ is any direct lattice vector. As
and the potential has the lattice periodicity, V (~
r + L),
~ ~r = ∇
~ ~ , because L
~ is constant, it follows that ĥ(~
~ = ĥ(~
∇
r + L)
r ).
~
r +L
In the case of a non-degenerate energy level: Let’s assume
~
~ = ĥ(~
~ = ψ(~
~
ĥ(~
r + L)ψ(~
r + L)
r )ψ(~
r + L)
r + L)
and
ĥ(~
r )ψ(~
r ) = ψ(~
r)
(140)
~ and ψ(~
As ψ(~
r + L)
r) share the same non-degenerate energy, they must coincide up to a multiplier:
~ = ζ(L)ψ(~
~
~ must be such that
ψ(~
r + L)
r). The multiplier ζ(L)
~ 2
~ and ψ(~
• ζ(L) = 1, if we ask that both ψ(~
r + L)
r ) are normalized;
~ +L
~ 0 ) = ζ(L)ζ(
~
~ 0 ), because two succesive translations by L
~ and L
~ 0 are equivalent to a
• ζ(L
L
~ +L
~ 0.
single translation by L
→
−
~ = ei~q L , where q~ can be any vector in ~k-space. Then:
To achieve this: ζ(L)
→
−
~ = ei~q L ψ(~
ψ(~
r + L)
r)
(Bloch theorem).
(141)
(129)
~ will be, in general, a linear combination of
In the case of a degenerate energy level: ψi (~
r + L)
the degenerate states in ~
r:
X
~
~ j (~
ψi (~
r + L) =
ζij (L)ψ
r ).
(142)
j
We can always ask that the degenerate functions form an orthonormal set: hψi |ψj i = δij . The
unitary character of the ζ matrix follows from here:
δik
~
~ =
= hψi (~
r + L)|ψ
r + L)i
k (~
X
j,l
∗ ~
~ hψj (~
ζij
(L)ζkl (L)
r )|ψl (~
r )i
=
X
∗
ζij
ζkj ⇒ ζ −1 = ζ† . (143)
j
~
This means that we can diagonalize ζ and the eigenvectors will not mix under the translation L.
From here we continue as in the non-degenerate case.
~ is a reciprocal lattice vector
Restrictions on q~ due to the lattice network in ~k-space: If G
~~
~ ~
~~
~
~
eiGL = 1. Then, for any q~ in ~k-space ei(~q+G)L = eiGL ei~qL = ei~qL . In other words, q~ can be
chosen to be within the first BZ.
Periodic Boundary Conditions (Born-von Karman): Let’s assume the crystal to be formed by
N = N1 × N2 × N3 parallelepipedic cells (along the ~a, ~b, ~c directions, respectively). By impossing
equivalence of the one particle wavefunctions on the opposite faces we introduce quantization of
(130)
the q~ vector. For instance, along the ~a direction
ψq~ (~
r ) ≡ ψq~ (~
r + N1~a) = ei~qN1~a ψq~ (~
r)
⇒
ei~qN1~a = 1
⇒
q1 N1 = i2π
(144)
The q1 component can get N1 independent values: 2π/N1 , 4π/N1 , . . . , 2N1 π/N1 . Equivalence
at the boundaries along ~b and ~c introduce the quantization of the components q2 and q3 . The
N1 × N2 × N3 independent values for the q~ vector reside in the first BZ and have equivalent copies
on each and every BZ.
As N is usually quite large (≈ NA ), q~ can be treated as a continuous variable for most purposes.
Each independent q~ labels a one particle orbital that can be occupied by a pair of electrons of
different spin. The ground state of a crystal with n electron pairs per primitive unit cell will consist
of n different sets of ψn~q (~
r ), with energies n~q , populated by the electron pairs.
Thermal effects will introduce some smearing of the electron population along the states close to
the Fermi surface.
Bloch functions: Felix Bloch demonstrated in 1928 that one electron wavefunctions in a periodic
potential can be described as the product of an arbitrary planewave and a function, called envelope,
that satisfies the periodicity of the crystal. In other words
ψn~q (~
r ) = ei~q~r un~q (~
r)
with
~ = un~q (~
un~q (~
r +L)
r)
(145)
~ is a lattice vector in direct space, and ~
where q~ ∈ R3 is any vector in ~k-space, L
r is an arbitrary
(131)
vector, also in direct space. The ψq~ (~
r ) wavefunctions are called Bloch functions or Bloch states.
This equation is different but equivalent to the Bloch theorem seen before. For a Bloch function:
~
~ i~
~
~ = ei~q(~r+L)
~ = ei~qL
ψn~q (~
r +L)
un~q (~
r +L)
e q~r un~q (~
r ) = ei~qL ψn~q (~
r)
(146)
which is the first form of the theorem (eq. 141). Inversely, for any function φq~ (~
r ) satisfying 141 we
can build a Bloch state using uq~ (~
r ) = φq~ (~
r )/ei~q~r .
Bloch states are delocalized on the whole crystal, showing identical shape on every single unit cell.
On the contrary, they are localized in ~k-space, as q~ has a definite value (and we need only to
consider q~ in the first BZ).
Wannier functions: An equivalent but different way of describing the electronic states is using
one particle wavefunctions that are localized in ~
r -space but delocalized in ~k-space. This Wannier
functions are related to the Bloch states as:
1 X i~qL
~
~
ψn~q (~
r) = √
e Wn (~
r −L)
N ~
L
or
X
1
~
~
Wn (~
r −L) = √
e−i~qL ψn~q (~
r)
N q~∈BZ1
(147)
(132)
Energy bands: zone schemes
Mapping the energy states onto the Brillouin zones. Energy
bands.
The representation of the energy levels n~q ≡ n (~
q ) as a function of q~ gives rise to the band structure
diagram, perhaps the most typical outcome of band theory. To avoid dealing with a 4D map, it is
customary to define a trajectory in ~k-space that follows the directions of maximal symmetry on the
reciprocal cell. The n (~
q ) values are then ploted versus the length of the trajectory from ~0 to q~.
The band structure diagram can be limited to the first Brillouin Zone (reduced sone scheme),
because all BZ are equivalent by the translational symmetry. The n band can also be represented
in the n BZ (extended zone scheme) and this is the origin of a very simple recipe to use the free
electron model as a method of drawing a first approximation to the band structure diagram:
~ is considered the source of a set of planewaves. Each G
~ will
• Each reciprocal lattice point, G,
~
produce a band within the first BZ, i.e. n ≡ G.
2
2
~
~
• The energy at q~ of the planewave originated at G is n≡G
q ) = ~ q~ − G /2m.
~ (~
~ runs over the whole
• In the reduced zone scheme we only need to sample q~ ∈ BZ1 and G
~ = ~0 but q~ samples all BZ’s.
lattice. In the extended zone scheme we only use G
• Bands will show degeneracy at the boundaries of the BZ’s. This can be avoided with a simple
perturbation theory treatment. (PW’s interfere for the Bragg planes)
(133)
3.0
3a
2a
BZ−1
2b
3b
ε / (h2/2mea2)
2.5
2.0
1.5
1.0
0.5
0.0
−1
0
n = k / (2π/a)
1
(134)
3.0
3a
2a
BZ−1
2b
3b
ε / (h2/2mea2)
2.5
2.0
1.5
1.0
0.5
0.0
−1
0
n = k / (2π/a)
1
(135)
3.0
3a
2a
BZ−1
2b
3b
ε / (h2/2mea2)
2.5
2.0
1.5
1.0
0.5
0.0
−1
0
n = k / (2π/a)
1
(136)
5.0
Μ
Σ
∆
Χ
The red band originates at the (0, 0)
~k-point,
green
bands come from
(1, 0) and equivalents, blue bands
from (1, 1), and
magenta ones from
(2, 0).
ε / (h2/2mea2)
Γ
4.0
Ζ
3.0
2.0
1.0
0.0
X
∆
Γ
Σ
M
Z
X
∆
(137)
Γ
Nearly free electron model
The Nearly Free Electron Model (NFEM)
A simple way to improve on the free electron bands is by treating the collective effect of the rest of
the crystal as a little perturbation on top of the free electron hamiltonian. Orthonormal PWs are
just the solution for the unperturbed problem:
Ĥ
(0)
~2 2
=−
∇~r ,
2m
(0)
ψ~
k
~
1
(~
r ) = |~ki = V − /2 eik~r ,
(0)
~
k
~2 k2
=
,
2m
~k = 2πa? h, h ∈ Z3 .
˜
(148)
Let V̂ (~
r ), the lattice potential on an electron, be treated as a first order perturbation. Wave
functions and eigenvalues expanded in a perturbation series,
(0)
(1)
(2)
k
k
k
ψ~k (~
r ) = ψ~ + ψ~ + ψ~ + . . . ,
(0)
(1)
(2)
k
k
k
~k = ~ + ~ + ~ + . . . ,
(149)
can be substituted into the one-particle Schrödinger equation,
(Ĥ (0) + V̂ )ψ~k = ~k ψ~k .
(150)
Grouping together the terms with the same order in the perturbation we get
(0)
(0)
k
k
(0)
(1)
(1)
(0)
k
k
k
k
(0)
(2)
(1)
(1)
(2)
(0)
k
k
k
k
k
k
zero order:
0 = (Ĥ (0) − ~ )ψ~ ;
first order:
0 = (Ĥ (0) − ~ )ψ~ + (V̂ − ~ )ψ~ ;
second order:
(151)
(152)
0 = (Ĥ (0) − ~ )ψ~ + (V̂ − ~ )ψ~ + (−~ )ψ~ ; . . .
(153)
(138)
The zero order equation is just the unperturbed problem. To solve the first order equation we
(1)
expand ψ~ in PW series
k
X (1) (0)
(1)
ψ~ (~
r) =
c~ ~ 0 ψ~ 0 (~
r ).
(154)
k
kk
k
~
k0 6=~
k
This expansion is always possible because PWs for a complete set. Substituting 154 into 152 and
(0)
multiplying by hψq~ |
X (1)
(0)
(0)
(0)
(0)
(1)
(0)
0=
c~ ~ 0 hψq~ |(Ĥ (0) − ~ )|ψ~ 0 i + hψq~ |(V̂ − ~ )|ψ~ i
kk
k
k
k
k
~
k0 6=~
k
=
X
(1)
(0)
(0)
kk
k
k
(1)
(0)
(0)
kk
k
k
(0)
(0)
(0)
(0)
(1)
k
k
(0)
(0)
c~ ~ 0 (~ 0 − ~ ) hψq~ |ψ~ 0 i + hψq~ |V̂ |ψ~ i − ~ hψq~ |ψ~ i
k
k
~
k0 6=~
k
=
X
(0)
(0)
(1)
k
k
c~ ~ 0 (~ 0 − ~ )δq~,~k0 + hψq~ |V̂ |ψ~ i − ~ δq~,~k
~
k0 6=~
k
(1)
(0)
(0)
(0)
(0)
(1)
k
k
= c~ (q~ − ~ ) + hψq~ |V̂ |ψ~ i − ~ δq~,~k
k~
q
k
where we have used the orthonormality of the lattice PWs. In fact, we have used
Z
1
~
~ = 2πa? n, n ∈ Z3
eiG~r d~
r=0
iff G
V V
˜
(155)
(156)
where V is the direct lattice cell.
(139)
The integral of the perturbation is very important. Using a Fourier series expansion of the potential:
V (~
r) =
X
V~g ei~g~r
(157)
~
g
h~k|V̂ |~k0 i =
X
V~g h~k|ei~g~r |~k0 i =
~
g
X
V~g h~k|~k0 +~g i =
~
g
X
V~g δ~k,~k0 +~g = V~k−~k0
(158)
~
g
Notice: (1) The Fourier series of V (~
r ) sums only over reciprocal lattice vectors, due to the periodic
nature of the potential; (2) The integral holds true whenever ~k−~k0 is a reciprocal lattice vector,
even if ~k and ~k0 are not.
Using 158 in 155 we arrive to
(1)
(0)
(0)
(1)
k
k
0 = c~ (q~ − ~ ) + Vq~−~k − ~ δq~,~k
k~
q
(159)
and
Case q~ = ~k:
(1)
~
k
= V~0 ;
Case q~ 6= ~k:
(1)
c~
k~
q
=−
Vq~−~k
(0)
(0)
q~ − ~
.
(160)
k
All energy levels are increased by the value of the lattice potential at the Γ ~k-point. The zero of
the energy scale is arbitrary, however, and we can set V~0 = 0. Up to first order, then, the lattice
potential adds nothing to the free electron band structure and we need to proceed to second order,
at least.
(140)
(2)
Obtaining the second order correction is similar to the first order already done. The ψ~
k
function can also be expanded in PW series
X (2) (0)
(2)
ψ~ (~
r) =
c~ ~ 0 ψ~ 0 (~
r ).
k
kk
k
wave
(161)
~
k0 6=~
k
(0)
Using 161 and 157 in 153 and multiplying by hψq~ |:
X (2) (0)
X (1)
(0)
(0)
(0)
(0)
(0)
(2)
(0)
(0)
0=
c~ ~ 0 (~ 0 − ~ ) hψq~ |ψ~ 0 i +
c~ ~ 0 hψq~ |V̂ |ψ~ 0 i − ~ hψq~ |ψ~ i
kk
k
k
k
kk
~
k0 6=~
k
=
X
(2)
~
k~
k0
c
(0)
~
k0
(
(0)
− ~ )δq~,~k0 +
X
k
k
k
(1)
(2)
kk
k
c~ ~ 0 Vq~−~k0 − ~ δq~,~k
~
k0 6=~
k
~
k0 6=~
k
(2)
k
~
k0 6=~
k
(0)
(0)
= c~ (q~ − ~ ) −
k~
q
X V~k0 −~k V~k−~k0
k
~
k0 6=~
k
(0)
~
k0
(0)
− ~
(2)
− ~ δq~,~k .
(162)
k
k
The second order corrections to energy and wave functions are then
For q~ = ~k: ~
(2)
k
=−
X |V~k−~k0 |2
~
k0 6=~
k
X
V~k0 −~k Vq~−~k0
~k: c(2) =
;
For
q
~
=
6
;
~
(0)
(0)
(0)
(0)
(0)
(0)
k~
q
~ 0 − ~
(~ 0 − ~ )(q~ − ~ )
~0 ~
k
k
k 6=k
k
k
(163)
k
where we have used that V−~g = V~g? because V (~
r) is a real function.
(141)
All the derivation above fails when we try to get the perturbative corrections to a degenerated
state. This is, however, a most important situation, as degeneracy occurs systematically at the BZ
boundaries.
(0)
(0)
k
k+~
q
We can treat the problem using perturbation theory for degenerate states. Let |ψ~ i and |ψ~
i
be two almost degenerated zero order states. The first order correction to the energy is obtained by
solving the secular equation:
(0)
(0)
(0)
(0)
?
hψ~ |V̂ |ψ~ i ~ − ~ − V
q
~
k+~
q
k
k
k
0=
=
(0)
(0)
(0)
(0)
− Vq~
~
~
−
hψ~ |V̂ |ψ~ i
k+~
q
k+~
q
k
k+~
q
h
i
(0)
(0)
(0) (0)
− |Vq~ |2
(164)
= 2 − (~ − ~ ) + ~ ~
k
k+~
q
k
k+~
q
and
=
(0)
In the degenerated limit ~
k
(0)
(0)
k
k+~
q
~ + ~
2
(0)
= ~
k+~
q
 (0)
1/2
2
(0)
 ~ − ~

k
k+~
q
2
 + |Vq~ |
± 


2
(165)
(0)
and = ~ ± |Vq~ |.
k
The figure in page 144 shows the corrections to the free electron bands of a one dimensional lattice
calculated with eq. 163. It is clear that the formulas diverge in the neighborhood of a degeneracy.
In fact, a 0/0 indeterminacy happens in that case.
(142)
Eq. 165, on the other hand, works like a charm as the figures in page 145 and 146 clearly show.
The result is easier to describe in the extended or repeated zone schemes. The energy parabola
typical of the free electrons curves up or down in the nearby of a BZ boundary. This generates gaps
of forbidden energy, i.e. energy ranges for which no state occurs. The width of the gap is, in this
model, twice the absolute value of the lattice potential at the BZ boundary.
The reason of the success of the eq. 165 can be questioned. The derivation is purported for the
degenerated case but the formula works well quite far from this situation. In fact, it works for any
~k-point and it does not suffer the problems of the second order perturbation formula, eq. 163.
Even though solid state physics books systematically describe the nearly free electron model in
terms of perturbation theory, it is better to analyze the problem as a variational linear calculation.
The complete, albeit approximated, hamiltonian is Ĥ = T̂ + V̂ (~
r ), and the PWs are the basis set
functions. The Schrödinger equation is transformed into a set of secular equations:
det |H − ~k 1| = 0
and
(H − ~k 1)c~k = 0
(166)
where the H matrix elements are
(0)
(0)
k
k
H~k,~k0 = hψ~ |T̂ + V̂ |ψ~ 0 i = T~k,~k0 + V~k,~k0 ,
(0)
T~k,~k0 = ~ δ~k,~k0 ,
k
V~k,~k0 = V~k−~k0 .
(167)
A basis set formed by the two PWs |~ki and |~k+~
q i will give rise to the eq. 165. The variational
method, however, can be used with a basis of any size, the only trouble being the use of an efficient
method for diagonalizing the H matrix.
(143)
NFEM: Perturbation Vq = 0.05 (h2/2ma2)
3.0
3a
2a
BZ−1
2b
3b
ε / (h2/2mea2)
2.5
2.0
1.5
1.0
0.5
0.0
−1
0
n = k / (2π/a)
1
(144)
3.0
3a
2a
BZ−1
2b
3b
ε / (h2/2mea2)
2.5
2.0
1.5
1.0
0.5
0.0
−1
0
n = k / (2π/a)
1
(145)
3.0
3a
2a
BZ−1
2b
3b
ε / (h2/2mea2)
2.5
2.0
1.5
1.0
0.5
0.0
−1
0
n = k / (2π/a)
1
(146)
Les2: Codes
Solid state codes
Some important solid state codes
Abinit: GPL, www.abinit.org, X. Gonze et al. (Université Catholique de Louvain; Corning Inc.;
etc).
Pwscf: GPL, www.pwscf.org, S. Baroni et al. (Trieste, . . . ).
Wien2k: distributed as source for a small subscription, www.wien2k.at, P. Blaha (K. Schwartz) et
al. (Technical University Wien, . . . )
Siesta (Spanish Initiative for Electronic Simulations with Thousands of Atoms): distributed as
source for a small subscription, www.uam.es/siesta/, E. Artacho (U. Cambridge), P. Ordejón (ICM,
Barcelona), J. Junquera (U. Cantabria), D. Sánchez Portal (UPV), J. M. Soler (UAM), . . . et al.
crystal06: only binary is distributed (06) or source could not be modified (03 and 98 versions),
www.crystal.unito.it, R. Dovesi (U. Torino), V.R. Saunders (Daresbury Lab., U. Cambridge) et
al.
References
Kantorovich, 2004 [33]; Kaxiras, 2003 [34]; Kittel, 2005 [35]; Marder, 2000 [36]; Martin, 2004 [37];
McQuarrie, 1976 [38]; Pueyo, 2005 [39];
See Damascelli et al. [32] for a description of the Fermi Surface measurement of high-Tc cuprate
(147)
Exercises
superconductors.
Exercises
1. Demonstrate the next properties of planewaves: (a) the wavelength of the ψ~k (~
r ) planewave is
λ = 2π/k; (b) periodic PWs form an orthonormal set;
2. The image in page 110 sketches a direct and its corresponding reciprocal lattice. The image
is, however, wrong. Can you point the mistakes included and plot schematically the correct
reciprocal lattice?
3. Determine F for a free electron gas in a simple square lattice if there are three electron pairs
per unit cell. Use the corresponding Fermi radius to map the Fermi sphere over the Brillouin
zone scheme, and plot the shape of the Fermi surface once translated to the first Brillouin
zone.
4. Determine and plot the band structure diagram for a free electron in a simple square lattice.
The special points in the first Brillouin zone are (k-coordinates in 2π/a units): Γ(0, 0),
X(1/2, 0), and M (1/2, 1/2). The special lines are ∆ (from Γ to X), Σ (Γ-M ) and Z (X–M ).
5.
(148)
Les3: Tight-binding methods
Tight-binding methods
(149)
The tight-binding method
The tight-binding (TB) method
The free electron model totally ignores the nature of the material apart from the crystal geometry
and the assumption on the number of electrons donated, per unit cell, to the electron gas. The
nearly free electron model introduces a dependency on the lattice potential while retaining the
concept that the valence electrons belong to the whole crystal. The tight-binding model, on the
other hand, is based under the assumption that the electronic structure of the solid very much
resembles that of their atomic components. Crystalline orbitals are thus derived from the atomic
orbitals by imposing on them the periodicity required by Bloch’s theorem. Albeit the TB scheme can
be applied ab initio, it is common to introduce great simplifications to produce final TB formulas
depending upon a short collection of parameters, empirically fitted to some key experimental data.
Many TB models can be linked back to the influential article by Slater and Koster [40]. A good
detailed description of the TB calculations is found in chapter 4 of Kaxiras [34]. A recent review of
TB techniques and applications can be found in Goringe, Bowler and Hernández, 1997 [41].
Let φl (~
r − ~ti ) be the l-th atomic orbital attached to atom i, whose position is ~ti with respect to
the origin of the unit cell in which it is placed. Bloch states can be obtained as
1 X i~kR
~
~
χ~kil (~
r) = √
e
φl (~
r − ~ti − R)
(168)
N ~
R
~ sums over the N primitive unit cells that from the crystal.
where R
(150)
The crystal orbitals can be expanded in the basis of the TB Bloch functions:
X
ψn~k (~
r) =
cn~kil χ~kil (~
r)
(169)
il
where the sum runs on the atoms in the primitive unit cell (i) and the atomic orbitals (l).
Substitution of this expansion in the Schrödinger equation gives
X
ĥψn~k (~
r) = n~k ψn~k (~
r ) =⇒
cn~kil ĥ − n~k χ~kil (~
r ) = 0.
(170)
il
Upon multiplication by hχ~kmj |
X
cn~kil hχ~kmj |ĥ|χ~kil i − n~k hχ~kmj |χ~kil i = 0
(171)
il
the differential equation is converted into a secular equation, on which the band energies, n~k , and
the band coefficients, cn~kil , are the unknowns to solve. We can write the secular equations in terms
of atomic orbitals. The overlap between two Bloch states is given by
1 X i~k(R
~ 0 −R
~ 00 )
~ 00 )|φl (~
~ 0 )i (†)
e
hφm (~
r − ~tj − R
r − ~ti − R
(172)
hχ~kmj |χ~kil i =
N 0 00
~ ,R
~
R
X ~~
1 X i~kR
~
~
~
~
~
e
=
hφm (~
r − tj )|φl (~
r − ti − R)i = (‡)
eikR hφm (~
r − ~tj )|φl (~
r − ~ti − R)i
N
0
~ R
~
R,
(173)
~
R
(151)
where we have taken into account that (†) the atomic overlap integral should not change if we
~ =R
~0 − R
~ 00 , and (‡) the term added up in 173 do not
translate equally both atomic functions, R
~ 0 , so (1/N ) P ~ 0 → 1. The hamiltonian integrals can be treated is a similar way
longer depend on R
R
to give
X ~~
~ .
hχ~kmj |ĥ|χ~kil i =
eikR hφm (~
r − ~tj )|ĥ|φl (~
r − ~ti − R)i
(174)
~
R
The secular equation is then
i
X
X ~~ h
ikR
~
~
~
~
~
~
cn~kil
e
hφm (~
r −tj )|ĥ|φl (~
r −ti −R)i − n~k hφm (~
r −tj )|φl (~
r −ti −R)i = 0
∀
jm
il
(175)
~
R
and it has to be solved on each ~k point within the first BZ.
So far, the TB equations could be solved ab initio. However, most of the TB calculations introduce
severe simplifications to let the equations be solved ”by hand” and even produce analytical
expressions for the band energies.
Let us examine one of the most common schemes. First, atomic orbitals on different centers and
different orbitals on the same center are assumed to be orthogonal
~ = δlm δij δ(R).
~
r −~ti −R)i
hφm (~
r−~tj )|φl (~
(176)
All hamiltonian elements are asumed to be negligible except the orbital energy of an atomic orbital
on a single center (on site energies, el ) and the integrals between atomic orbitals on nearest
(152)
neighbor centers (hopping integrals, Vlm.ij ):

 = e δ δ δ(R),
~
l lm ij
~
hφm (~
r −~tj )|ĥ|φl (~
r −~ti −R)i
 = Vlm,ij δ((~tj −~ti − R)
~ − d~nn ),
(177)
where d~nn is a vector between the i-th atom and its nearest neighbors.
Example: 1D linear chain
Let x be the direction aligned with the chain. The Bloch funcions are
1 X ikna
χkl (x) = √
e
φl (x − na)
N n∈Z
(178)
where l is the type of atomic orbital (s, pσ = px and pπ = {py , pz }); N is the number of primitive
unit cells, i.e. the number of n values. In the simplified TB scheme:
Overlap: hφm (x−ia)|φl (x−[i+n]a)i = hφm (x)|φl (x−na)i = δlm δn0 ;

 =eδ δ ;
l lm n0
Hamilton: hφm (x−ia)|ĥ|φl (x−[i+n]a)i = ... =
 = Vlm δn,±1 .
(179)
(180)
(153)
To complete the calculation we need the next hopping integrals
(181)
Vssσ
Vspσ
Vppσ
Vppπ
whereas the next cases are trivially null due to the symmetry
(182)
The Schrödinger equation takes the form of a secular equation
el δlm δn0 +Vlm δn,±1
∀ :
m
0=
X
=
X
l
l
ckl
δ
δ
lm n0
}|
{
z
z
}|
{
X
ikna
hφm (x)|ĥ|φl (x−na)i −k hφm (x)|φl (x−na)i
e
n∈Z
ika
ckl (el − k )δlm + [e
|
−ika
+e
{z
2 cos(ka)
] Vlm
}
(183)
(154)
We can simplify the notation defining: Vklm = 2Vlm cos(ka) and Ekl


Eks − k
Vkspσ
0
0
cks


 V
 c
Ekpσ − k
0
0
kspσ

  kpσ



  ckp
0
0
Ekpπ − k
0
y


0
0
0
Ekpπ − k
ckpz
= el + Vkll . Then




 = Mk c k = 0


(184)
and the secular equation is clearly separated into a 2 × 2 and two 1 × 1 independent blocks. The
eigenvalues for the σ bands are
s
2
Eks − k
Vkspσ
E
−E
E
+E
ks
kpσ
ks
kpσ
2
⇒ k =
0 = ±
+ Vkspσ
(185)
2
2
Ekpσ − k Vkspσ
and the energies of the two degenerated π bands are
k = Ekpπ = ep + 2Vppπ cos(ka).
(186)
There is a secular system for each k-point in BZ1, i.e. k ∈ [− π
,+π
]. The s and pσ functions
a
a
mix together to form two σ orbitals. The σ and π functions do not mix in this system due to the
symmetry.
(155)
pσ band
εk
Case: 1D lattice with λ-type orbitals only. There is a single band, with energy k = Ekλ = eλ + 2Vλλ cos(ka).
The band center is eλ , the atomic energy of the λ level,
and its width is 2Vλλ , i.e. the witdh is determined by the
hopping integral between two λ functions in nearest neighbor atoms. The atomic levels follow the aufbau principle,
so we can expect a similar ordering for the crystal bands:
1s < 2s < 2p < 3s.... Assuming that Vssσ < 0, the s band
shape is similar to the nearly free electron fundamental band.
The ssσ hopping integral can be seen as h+| − |+i, where
the signs show the dominant component on each orbital lobe
and on the operator, but then, for the ppσ hopping integral
we have h±| − |∓i, and h±| − |±i for the ppπ. We expect,
then, Vppσ > 0 and Vppπ < 0. The figure on the side shows
the close relationship between the ssσ, ppσ and ppπ bands
with the first three bands in the free electron model. This
striking relationship has been exploited by Harrison and others to determine the TB parameters that best fit the free
electron bands [42].
pπ band
s band
-π/2
0
ka
π/2
(156)
Example: Band structure of group IV diamond structures
Harrison and coworkers [42, 43] have studied systematically the application of TB models to
covalent, ionic and metallic solids. The next table contains all the data required to do a simple TB
calculation on the diamond structure of C, Si, and Ge, prototypes of covalent solids with tetrahedral
coordination. The calculation uses a minimal sp3 atomic basis set to describe the four valence
electrons per atom, i.e. the eight valence electrons per primitive unit cell.
C
Si
Ge
a
(Å)
3.57
5.43
5.66
s
(eV)
-17.52
-13.55
-14.38
p
(eV)
-8.97
-6.52
-6.36
Vssσ (eV)
-4.50
-1.93
-1.79
Vspσ (eV)
5.91
2.54
2.36
Vppσ (eV)
10.41
4.47
4.15
Vppπ (eV)
-2.60
-1.12
-1.04
Hopping integrals strongly depend on the interatomic
distance. Harrison used a bond orbital model to obtain
~2 ηlm ~2
Vlm =
,
= 1 at.u. = 7,62 eV Å2 ,
2
me d
me
(187)
where ηlm is a dimensionless constant and d is the
nearest neighbors distance. This equation works extremely well on tetrahedral semiconductors, but it is
rather poor on, e.g., close-packed structures.
Obtaining the TB equations and developing a simple code for this case is not difficult. We can also
resort to the TBPW provided by R. M. Martin and cols. at the www.electronicstructure.org
website.
(157)
25
20
5
15
5
εk (eV)
10
Si
0
5
Ge
0
C
0
-5
-5
-10
-10
-5
-10
-15
-20
L
Γ
XU
Γ
L
Γ
XU
Γ
L
Γ
XU
Band energies have been displaced to make zero the top of
the valence band. The three
solids are predicted to have a
direct band gap at Γ: 13,89
(C), 3,66 (Si), and 1,90 eV
(Ge). C is predicted to be
an insulator, and Si and Ge
are predicted to be semiconductors. Experimental values
for the direct band gap are
7,3 (C), 3,48 (Si) and 0,81 eV
[44]. The three solids, however, have a significantly lower
indirect band gap: 5,48 (C),
1,11 (Si), and 0,66 eV (Ge).
The sp3 minimal basis is able
to fit well the valence bands,
Γ
but it fails to describe the conduction band structure.
(158)
5
Vogl et al. [45] used a sp3 s∗
basis to produce a very successful model of the tetrahedral semiconductors. The
s∗ excited state very much
improves the fitting of the
conduction band that now
predicts correctly the indirect band gap: 1,17 eV on
Si (Exptal.: 1,11 eV). Vogl
paper describes a general
method for fitting to highsymmetry points on the
band structure.
5
Si
Harrison
Vogl
0
εk (eV)
0
-5
-5
-10
-10
L
Γ
X
U
Γ
L
Γ
X
U
Γ
(159)
Improvements on the TB recipe
The TB ideas can be carried up to a self-consistent ab initio procedure. Failing to reach that goal,
let us present some common improvements over the basic TB recipe.
Arbitrary orientation of the orbitals: The best strategy to deal with an arbitrary crystal is to
~ = xi+yj +zk =
assume a global set of cartesian axes. Given two atoms and a separation vector d/d
sen θ cos φi+sen θ sen φj +cos θk, some of the non-null hopping integrals are (notice the illustration
in next page):
Vss =Vssσ ;
Vs,px =x2 Vspσ ;
Vpx ,px =x2 Vppσ +(1−x2 )Vppπ ;
Vpx ,pz =xz(Vppσ −Vppπ ). (188)
Slater and Koster [40] provide all the formulas required to work with s, p and d orbitals.
Nonorthogonality of the orbitals: The simplified TB equations tend to assume that atomic
orbitals centered of different centers are orthogonal but they are not, in general. The overlap
depends strongly on the internuclear distance and on the space extension of the particular orbitals:
the average radius is proportional to the main quantum number, so orbitals that belong to the same
shell have a similar space range. Taking into account the lack of orthogonality becomes unavoidable
when the TB models go beyond the minimal basis formulation.
Multicenter integrals: Three and four center integrals between exponential type orbitals are
usually neglected, mostly because they are difficult and expensive to compute. Nowadays, there
exist methods to calculate all of them with any required precision.
(160)
Excited state orbitals: The addition of low lying excited state orbitals to the basis set can
improve greatly the behavior of the TB equations. Vogl et al. [45] provide a classical example for
the important role of the s∗ on the description of the conduction band of the tetrahedral C–Ge
semiconductors. As a matter of fact, the TB method tends towards an ab initio technique as the
number and flexibility of basis set increases well beyond the minimal basis range of the basical
scheme.
(161)
Lessons from the TB band structure
• The atomic orbitals give rise to the bands in the solid. The band width is mainly consequence
of the mixing of equivalent orbitals from nearby atoms, henceforth increasing as the hopping
integrals increase.
• Bands are filled by paired electrons in order of increasing energy. The highest occupied band
is called valence band, whereas the conduction band is the lowest unnoccupied one.
• Bands can be separated by gaps of forbidden energy. The most important one being the gap
between the top of the valence band and the bottom of the conduction band. The gap is called
direct when it involves valence and conduction states with the same ~k value, and indirect
otherwise.
• In absence of vibrations, the only spectroscopic transitions allowed by the main electric dipole
mechanism must conserve the ~k value, i.e. they are vertical transitions in the band diagram.
However, ~k → ~k 0 transitions become allowed if a crystal vibration (phonon) of appropriate
symmetry carries away the momentum excess ~(~k 0 −~k).
• Systems are classified according to the band gap energy Eg into: metals and semimetals
(Eg = 0), semiconductors (0 < Eg < 3 ∼ 4 eV), and insulators (Eg 3 eV). Metals are
good electrical conductors, semiconductors have an electrical resistivity lying in the range of
1 · 10−2 –1 · 109 Ω cm, and the resistivity of insulators is much higher.
• The electrical conductivity is related to the number of free carriers: electrons and holes in
(162)
a conduction band and therefore able to pass to a electronic state of different momentum
with a negligible energy cost. A semiconductor lacks free carriers at the absolute zero, but an
increasing number of electrons can jump the energy gap as the temperature increases. The
conductivity of a semiconductor, thus, increases with T .
• Contrarily, the probability of carriers-nuclei collisions grow with temperature and this effect
dominates the conductivity of good metals, that become less conductive when heated.
(163)
Density of states
Density of states (DOS): sampling and van Hove singularities
A quite useful concept in analyzing the electronic band structure of solids is the density of states
(DOS) as a function of the energy. The g(ε)dε distribution represents the number of electronic
states available in the energy range from ε to ε+dε. In the case of a free electron gas, we have
already seen (eq. 129 in page 113) that
3
2me /2 √
1
ε
(189)
g(ε) =
2π 2
~2
√
and the DOS follows a smooth g(ε) ∝ ε curve. Real solids, however, show a far more complex
DOS. Taking into account the spin degeneracy and normalizing g(ε) to the volume of the solid:
Z
XZ
dS~k
V X
2V X
2V
~
(190)
2δ(ε
−
)
=
2δ(ε
−
)d
k
=
g(ε) =
n~
k
n~
k
(2π)3
(2π)3 n
(2π)3 n ~ =ε ∇
~
~
~
~
n
k
k nk nk
The last integral (see, e.g. Kaxiras [34], chap. 5) is over surfaces in ~k-space on which the band
energy n~k is constant, and dS~k is the differential element normal to the surface. No matter how
complex it looks, the last integral makes clear a very important feature: the DOS has sharp spikes
~ ~ ~ = ~0, i.e. on the critical points of
called van Hove singularities [46] at the points such that ∇
k nk
the energy bands in ~k-space.
(164)
Density of states
The different types of van Hove singularities can be
classified according to the rank (r, number of non-zero
eigenvalues) and signature (s, number of positive minus
number of negative eigenvalues) of the hessian matrix:
~~ ⊗∇
~ ~ ~.
∇
k
k nk
sing.
(r, s)
M0
(3, +3)
minimum
pit
M1
(3, +1)
2-saddle
pale
M2
(3, −1)
1-saddle
pass
M3
(3, −3)
maximum
peak
type
The singularities can also be recognized according to the shape of the DOS, but this requires a high
resolution map:
Obtaining the DOS is a delicate problem of integration by sampling within the BZ. Symmetry is used
to reduce the number of independent sampling points, and several techniques have been developed.
Monkhorst and Pack [47] special point method is the most common technique on insulators and
semiconductors. Metals, particularly those with a complex Fermi surface, require a higher precision
sampling [48, 49].
(165)
Density of states
5
(ε-εF) (eV)
0
-5
-10
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Γ
g(ε) (e/eV)
M
K
Γ
A
DOS of hcp Be, according to a FPLAPW GGA calculation [50].
(166)
Exercises
References
Ashcroft-Mermin, 1976 [1]; Harrison, 1989 [42]; Harrison, 2005 [43]; Kantorovich, 2004 [33];
Kaxiras, 2003 [34]; Kittel, 2005 [35]; Marder, 2000 [36]; Martin, 2004 [37]; Yu and Cardona, 2001
[44].
Exercises
1. Show that the χ~kil (~
r ) defined in eq. 168 satisfy Bloch’s theorem.
2. Consider a 2D square lattice with a single type of atoms located at the lattice grid points.
Derive the TB equations for a system of orthogonal s and p orbitals.
3. Extend the above model to a 3D cube lattice.
4. Use the TBPW by R. M. Martin et al., or any other similar TB code, to determine the band
structure of C, Si and Ge using the sp3 s? parametrization of Vogl et al. [45].
5. Derive the DOS for the free electron model in one and two dimensions.
6. Calculate the DOS for the TB model of a 2D square lattice with one atom per unit cell and
s and p orbitals. For this calculation it is not sufficient to obtain the bands along some high
symmetry lines, but the entire BZ must be sampled with enough resolution to be able to show
the characteristic patterns around the critical points.
(167)
Les4: General electronic structure methods
General electronic structure methods
(168)
Independent electron Schrödinger equation
The independent electron Schrödinger equation
We want to solve the Hartree-Fock or Kohn-Sham equation for an electron in the average field of
all other (infinite) crystal charges:
~2
r , [ρ]) ψn~q (~
r ) = n (~
q )ψn~q (~
r)
∇~r2 + V̂ (~
Ĥ(~
r)ψn~q (~
r) = −
(191)
2me
where V̂ (~
r, [ρ]) = V̂n + V̂H + V̂xc :
•
•
•
V̂n : electrostatic potential of the nuclei (or pseudopotentials of the chemical kernels);
Z
ρ(~
r 0 )d~
r0
V̂H (~
r) =
: Coulomb potential due to the (possibly valence) electron density.
0|
3
|~
r
−
~
r
R
V̂xc : Exchange and correlation term. Common approaches: Hartree-Fock (HF), LDA or
LSDA (Local [Spin] Density Approach), GGA (Generalized Gradient Approach), . . .
Notation: vectors on the direct and reciprocal spaces.
~
r -space = {~
r = a x | x ∈ R3 };
˜
~k-space = {~
q = 2πa? q | q ∈ R3 };
˜
~ = a n | n ∈ Z3 };
~
r -lattice = {L
˜
~k-lattice, = {G
~ = 2πa? n | n ∈ Z3 }.
˜
(192)
(193)
(169)
Electronic band structure via plane waves
We will transform Schrödinger equation to a secular form by expanding the crystal orbitals in a basis
of planewaves. We start by describing a crystal orbital as a Bloch function:
i~
q~
r
ψn~q (~
r) = e
i~
q~
r
un~q (~
r) = e
X
~
G
X
X
1 iG~
1 i(G+~
~r
~ q )~
r
~ qi
=
cn~q √ e
=
cn~q |G+~
cn~q √ e
V
V
~
~
G
(194)
G
~
where we have used the fact that the periodic component of the Bloch function, un~q (~
r ) = un~q (~
r +L),
can be expanded in series of orthonormal planewaves. The result is that the planewaves that form
~ where G
~ is a vector in the ~k-lattice and q~ is
the basis of the crystal orbitals have the form |~
q +Gi,
the same ~k-space vector that labels the crystal wavefunction that we want to expand.
Using 194 in 191:
n
o
n
o
X
~
~ 1i
cn~q (G1 ) Ĥ − n (~
q ) |~
q +G
0 = Ĥ − n (~
q ) ψn~q (~
r) =
(195)
~1
G
~ 2|
and multiplying by h~
q +G
X
~ 1 ) h~
~ 2 |Ĥ|~
~ 1 i − n (~
~ 2 |~
~ 1i
0=
cn~q (G
q +G
q +G
q ) h~
q +G
q +G
(196)
~1
G
(170)
but
~ 2 |~
~ 1 i = hG
~ 2 |G
~ 1 i = δ(G
~1 − G
~ 2)
h~
q +G
q +G
(197)
~ 1 and G
~ 2 belong to the ~k-lattice, so |G
~ 1 i and |G
~ 2 i belong to the orthonormal PW
because both G
set.
The hamiltonian element matrix is a sum of the kinetic and potential energy components. The
kinetic energy is trivial, as PWs are eigenfunctions of p
~ˆ and T̂ = p̂2 /2m:
~ 1 )2
~2 (~
q +G
~2 2
~
~1 − G
~ 2 ).
~
∇~r |~
q +G1 i = −
δ(G
TG
q ) = h~
q +G2 | −
~ 1 ,G
~ 2 (~
2m
2m
(198)
~ = V̂ (~
The potential energy integral is greatly simplified if, taking into account the V̂ (~
r +L)
r ) lattice
periodicity, we expand V̂ (~
r ) into a Fourier series:
X
X
~r
iG~
~
~
~
~ ~
VG
q) =
VG
q +G2 |e
|~
q +G1 i =
VG
(199)
~ ,G
~ (~
~ h~
~ δ(G1 +G−G2 ) = VG
~ −G
~ .
1
2
2
~
G
1
~
G
The potential energy is thus reduced to the calculation of a single coefficient in the Fourier series:
Z
V~g =
V̂ (~
r )e−i~g~r d~
r
(200)
V
where the integral is done within the volume, V , of a single unit cell.
(171)
Using 198 and 199 in 196 we obtain the secular equation
("
#
)
2
2
X
~
~ (~
q +G1 )
~ 1)
~ 1 −G
~ 2) + V ~ ~
cn~q (G
0=
− n (~
q ) δ(G
G2 −G1
2m
~1
G
∀
(201)
~2
G
that must be solved independently for each q~ that belongs to the first Brillouin Zone (BZ1). The
normalized crystal orbitals are then
2
X
X ~
~
~
ψn~q (~
r) =
cn~q (G) |~
q +Gi with 1 = hψn~q (~
r )|ψn~q (~
r )i =
(202)
cn~q (G) .
~
G
~
G
The electron density of the crystal is obtained as
X
2
ρ(~
r) =
fn~q ψn~q (~
r )
(203)
n~
q
where fn~q is the electron population on band n~
q . In the case of a metal, this requires a delicate
calculation of the Fermi energy. The electron density is a required component in the calculation
of the V~g Fourier coefficients, because the V (~
r ) lattice potential containts the Coulomb and
the exchange-correlation contributions. Henceforth, the secular equations 201 must be solved
self-consistently on each q~-point in the BZ1. Obtaining the total energy, like the Density of States
(DOS) and related properties, is a last step in the calculation, and it requires sampling efficiently
the BZ1.
(172)
The main difficulty for the scheme described is the number of planewaves required to describe any
material. The electron density is heavily concentrated around the nuclei and, in fact, this is the
region that contributes most to the total energy. Planewaves are quite inefficient to represent a
tightly localized distribution in real space. Let us assume a target resolution of some δr ≈ 0,1 Å
in a cubic lattice of a ≈ 5 Å. This corresponds in ~k-space to δk = 2π/δr ≈ 63 Å−1 and
~ = 2πa n, we need
2πa? = 2π/a ≈ 1,3 Å−1 . Thus, in the sum over reciprocal lattice vectors, G
˜
6
n ∈ [−50, +50] or about 10 planewaves. Solving the secular equations means obtaining the lowest
eigenvalues and eigenvectors of a 106 × 106 matrix plus iterating to self-consistency, and that work
has to be repeated on each q~-point.
There are two main ways, however, to turn practical the use of planewaves. The first method
involves the use of pseudopotentials that simulate the effect of the core electrons, thus limiting
the crystal calculation to the valence electrons. This reduces drastically the spacial resolution and
the number of planewaves required. A different strategy consist in modifying the basis set, adding
core tailored components to the planewaves like in the OPW (Orthogonalized PWs) and APW
(Augmented PWs) methods, or substituting them with exponential (STO, Slater Type functions)
or gaussian functions (GTO, Gaussian Type functions) like in the LCAO (Linear Combination of
Atomic like functions) method.
Treatment of the lattice potential. Atomic form factors.
A great ecomomy is
achieved in the treatment of the potential terms if they can be expressed as a sum of atomic
(173)
contributions:
V (~
r) =
XX
~ i − L)
~
Vi (~
r−R
(204)
i
~
L
~ sums over unit cells, i sums over the atoms in one unit cell, and R
~ i is the nuclear position
where L
of i relative to the origin of its unit cell.
The Fourier coefficients for this potential are

Z
V~g =
−i~
g~
r
V (~
r )e
d~
r=
V
=
X
~
L
XXZ
i
~
L
~
−i~
gL
e| {z }
=1
X
i
~i
−i~
gR
Z
e
R3
−i~
g~
r
~ i − L)e
~
Vi (~
r−R
d~
r=
V
0
−i~
g~
r0
Vi (~
r )e
0
d~
r = Ncell
X
~ i − L;
~
~
r0 =~
r−R
~
r ∈V;
~
vi (~g )e−i~gRi
~
r0
∈
d~
r = d~
r 0;
R3 .


(205)
i
~
~ belong in the ~k-lattice and ~
where we have taken into account e−i~gL = 1 because ~g and L
r -lattice,
respectively. Equation 205 makes clear that V~g is an extensive quantity, proportional to the number
of unit cells in the crystal. On most cases we really want the intensive V~g per cell. The vi (~g )
0
function is called the atomic form factor, whereas e−i~g~r is the geometric term, as it only depends
on the position of the nuclei within the cell. If Vi (~
r 0 ) = Vi (|~
r 0 |) then vi (~g ) = vi (|~g |) and the
computational effort is accordingly reduced.
(174)
Orthogonalized Plane Waves
Orthogonalized Plane Waves (OPW)
OPWs, introduced by Herring in 1940 [51], were the basis for the first quantitative calculations, in
the 40s and early 50s, of bands in general materials. Today OPWs have been surpassed by other
methods, but the technique is still regarded as the natural introduction to pseudopotentials in solid
state physics.
As we have discussed before, plane waves are inefficient to represent the well localized electron
density around nuclei. Not only the core orbitals, but even the valence wavefunctions require very
high ~k components due to their internal nodal structure caused by the required orthogonality to the
core levels. The idea behind OPWs is fulfilling the valence-to-core orthogonality from the beginning,
so that the basis functions do only need to expand the valence density out of the near-core regions.
Let |~
q i be a planewave and |ψc i one of the core states that we want to remove from the solution
space of the crystal orbital Schrödinger equation. The OPW associated to |~
q i is
X
|OPW, q~i = |~
qi −
|ψc i hψc |~
qi .
(206)
c
P
If the |ψc i were a complete set of states c |ψc i hψc | = 1̂ and the OPW would vanish. This is
clearly false in our case, but we expect that the core states show a strong localization around the
nuclear positions and that OPWs remain small there.
Working with OPWs and the normal one particle hamiltonian is equivalent to working with PWs
(175)
Orthogonalized Plane Waves
but a modified hamiltonian:
!
2
X
~
(ĥ − q~ ) |OPW, q~i = −
∇~r2 + V̂ (~
r ) − q~
|~
qi −
|ψc i hψc |~
qi
2m
c
≈ (c − q~ ) |ψc i
z
}|
{
X
~2 2
∇ |~
q i + (V̂ − q~ ) |~
qi −
hψc |~
q i (ĥ − q~ ) |ψc i
=−
2m ~r
c
(
)
2
X
~
= −
∇~r2 + V̂ −
(c − q~ ) |ψc i hψc | − q~ |~
q i = (ĥps − q~ ) |~
qi
2m
c
(207)
where
ĥps
~2 2
=−
∇ + V̂ps ;
2m ~r
V̂ps = V̂ −
X
(c − q~ ) |ψc i hψc | .
(208)
c
In words, the lattice potential V̂ (~
r ) is replaced by a lattice pseudopotential that contains the
orthogonality condition of the PWs to the core sates. This pseudopotential tends to be smoother
on the core regions that the original lattice potential, thus allowing for a significant reduction in the
number of PWs required. On the other hand, V̂ps is a non-local operator that, in addition, depends
on the unknown eigenvalue of the crystal state upon which it is applied. This introduces a further
complication on the one particle equation, but this is more than compensated by the reduction in
the PW basis set.
(176)
Pseudopotentials
Ab initio pseudopotentials
Ab initio pseudopotentials have a very long tradition in many areas of atomic, molecular, nuclear
and condensed matter quantum problems. Chemists will stress its origin on the core-valence and
sigma-pi separability problems, and the names of Lykos, Parr, McWeeny and Huzinaga, fathers
of the Electron Separability Theory, are unavoidable. Solid state physicists will prefer pointing to
the very influential 1959 article by Phillips and Kleinman and quickly step to the modern soft and
ultrasoft potentials by Vanderbilt, Hamann et al., or Troullier-Martins schools.
Common to all the traditions and schools, core pseudopotentials are built to reproduce the valence
AE
levels of a reference all electron atomic calculation. Let {AE
r )} be the reference orbitals,
nlm , ψnlm (~
solution to the atomic one electron equation
~2 2
AE
AE
−
∇~r + V (r) ψnlm
(~
r ) = AE
r ).
(209)
nlm ψnlm (~
2m
The pseudopotential equation
~2 2
ps
ps
ps
∇~r + Vl
ψ̃nlm
(~
r ) = ps
ψ̃
r)
−
nlm nlm (~
2m
(210)
differs from the all electron counterpart in several important aspects:
1. Core states are projected out from the solution space of eq. 210. Valence states are kept
untouched, i.e. ps = AE .
(177)
Pseudopotentials
2. The radial part of the pseudofunction ψ ps is modified by removing out the inner nodal structure
of the true valence function. ψ ps is kept identical to ψ AE beyond a given cutoff radius Rc , but
the inner wiggles are changed into a smooth, nodeless, function for r < Rc .
3. In the norm-conserving pseudopotentials ψ ps and ψ AE integrate identically for r < Rc . This
condition is removed for the soft and ultrasoft potentials. Continuity conditions must be
applied at Rc .
(178)
Pseudopotentials
4. The pseudopotential depends on l. This can be achieved in several ways:
Local ps: V̂ps = U (r) (local in ~
r ).
P
Semilocal ps: V̂ps = l U l (r)P̂l (local in r, nonlocal in θ, φ).
L
X
Nonlocal separable ps: V̂ps = UL+1 (r) +
|Ylm i Ul (r) hYlm |.
l=0
General ps: V̂ps = UL+1 (r) +
L
XX
Dtt0 l |βtlm i hβt0 lm |.
t,t0 l=0
5. Pseudopotentials provide a cheap way to introduce relativistic effects.
Obtaining a good pseudopotential is a delicated handicraft that involves:
1. Deciding which atomic orbitals are kept in the valence. For instance, 3spd4s are typically the
valence in the first transition metal group.
2. Creating the pseudofunctions (cutoff radius, smooth inner zone, continuity conditions, norm
conserving, . . . ).
3. Determining the pseudopotentials by inverting eq. 210.
4. Checking for the occurrence of anomalies: e.g. ghost states.
5. Analyzing the transferability of the potentials for electronic states of the atom different from
the states used as reference. This is a good test for transferability to molecules and crystals.
6. Testing the potentials in the solid.
(179)
Pseudopotentials
The norm-conserving condition makes sure that the pseudopotential scattering properties remain
correct for nearby nl (Hamann et al., 1979 [52]). Norm-conserving potentials are accepted by most
solid state codes.
Soft (Troullier-Martins [53] are currently the most popular) and ultrasoft (i.e. Vanderbilt [54])
have a clear computational advantage by requiring a reduced number of PW’s. In other words, the
energy cutoff for the PW included in the solid state calculation can be considerably reduced. Some
extra coding is required in the solid-state program, however, and not all packages accept ultrasoft
potentials.
There is a close relationship between the ultrasoft potentials and the Projector Augmented-Wave
method [55]. In fact, the PAW technique provides a way for coding the ultrasoft potentials [56].
Some web resources:
• Martins site: bohr.inesc-mn.pt/~jlm/.
• Vanderbilt site: www.physics.rutgers.edu/~dhv/.
• The Octopus code: www.tddft.org/programs/octopus/wiki/index.php/Main_Page.
• A nice Vanderbilt’s report on pseudopotentials: www.physics.rutgers.edu/~dhv/talks/
bangalore-july06.pdf.
(180)
Other basis functions
Other basis functions
(181)
Solid state codes
Some important solid state codes
Abinit: GPL, www.abinit.org, X. Gonze et al. (Université Catholique de Louvain; Corning Inc.;
etc).
Pwscf: GPL, www.pwscf.org, S. Baroni et al. (Trieste, . . . ).
Wien2k: distributed as source for a small subscription, www.wien2k.at, P. Blaha (K. Schwartz) et
al. (Technical University Wien, . . . )
Siesta (Spanish Initiative for Electronic Simulations with Thousands of Atoms): distributed as
source for a small subscription, www.uam.es/siesta/, E. Artacho (U. Cambridge), P. Ordejón (ICM,
Barcelona), J. Junquera (U. Cantabria), D. Sánchez Portal (UPV), J. M. Soler (UAM), . . . et al.
crystal06: only binary is distributed (06) or source could not be modified (03 and 98 versions),
www.crystal.unito.it, R. Dovesi (U. Torino), V.R. Saunders (Daresbury Lab., U. Cambridge) et
al.
References
Kantorovich, 2004 [33]; Kaxiras, 2003 [34]; Kittel, 2005 [35]; Marder, 2000 [36]; Martin, 2004 [37];
McQuarrie, 1976 [38]; Pueyo, 2005 [39];
(182)
Ltm4: General electronic structure methods
Exercises
Exercises
1.
(183)
References
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(186)

c - Quantum Chemistry Group

Transcription

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