1. Solid State Computing
Peter Ballo

2. Models Classical: Quantum mechanical: H = E Semi-empirical methods
Ab-initio methods

3. Molecular Mechanics atoms = spheres bonds = springs
math of spring deformation describes bond stretching, bending, twisting
Energy = E(str) + E(bend) + E(tor) + E(NBI)

4. From ab initio to (semi) empirical
Quantum calculation
First principles
Reliability proven within the approximations
Basis sets,
functional,
all-electron or pseudo- potential ..
Computationally expensive
Based on fitting parameters
Two body , three body…, multi-body potential
No theoretical background empirical
Applicability to large system
no self consistency loop and no eigenvalue computation
Reliability ?

5. Climbing Mt. Psi

6. The Framework of DFT DFT: the theory Elements of Solid State Physics
Schroedinger’s equation
Hohenberg-Kohn Theorem
Kohn-Sham Theorem
Simplifying Schroedinger’s
LDA, GGA
Elements of Solid State Physics
Reciprocal space
Band structure
Plane waves
And then ?
Forces (Hellmann-Feynman theorem)
E.O., M.D., M.C. …

7. Using DFT Practical Issues Applications Input File(s) Output files
Configuration
K-points mesh
Pseudopotentials
Control Parameters
LDA/GGA
‘Diagonalisation’
Applications
Isolated molecule
Bulk
Surface

8. The Basic Problem
Dangerously
classical
representation
Cores
Electrons

9. Schroedinger’s Equation
Wave function
Potential Energy
Kinetic Energy
Coulombic interaction
External Fields
Energy levels
Hamiltonian operator
Very Complex many body Problem !!
(Because everything interacts)

10. First approximations Adiabatic (or Born-Openheimer)
Electrons are much lighter, and faster
Decoupling in the wave function
Nuclei are treated classically
They go in the external potential

11. New density ‘=‘ input density ??
Self consistent loop
Initial density
From density, work out Effective potential
Solve the independents K.S. =>wave functions
Deduce new density from w.f.
New density ‘=‘ input density ?? NO
YES
Finita la musica

12. Electrons are fermions (antisymmetric wave function)
DFT energy functional
Exchange correlation funtional
Contains:
Exchange
Correlation
Interacting part of K.E.
Electrons are fermions (antisymmetric wave function)

13. Exchange correlation functional
At this stage, the only thing we need is:
Still a functional (way too many variables)
#1 approximation, Local Density Approximation:
Homogeneous electron gas
Functional becomes function !! (see KS3)
Very good parameterisation for
LDA
Generalised Gradient Approximation:
GGA

14. DFT: Summary
The ground state energy depends only on the electronic density (H.K.)
One can formally replace the SE for the system by a set of SE for non-interacting electrons (K.S.)
Everything hard is dumped into Exc
Simplistic approximations of Exc work !
LDA or GGA

15. Bulk properties
zero temperature equations of state (bulk modulus, lattice constant, cohesive energy)
structural energy difference (FCC,HCP,BCC)
two shear elastic constants in FCC structure

17. M. I. Baskes, Phys. Rev. B 46, 2727 (1992)
M. I. Baskes, Matter. Chem. Phys. 50, 152 (1997)

18. And now, for something completely different: A little bit of Solid State Physics
Crystal structure
Periodicity

19. Reciprocal space sin(k.r) Reciprocal Space bi Real Space ai
(Inverting effect)
sin(k.r)
Reciprocal Space bi
Real Space ai
Brillouin Zone
k-vector (or k-point)
See X-Ray diffraction for instance
Also, Fourier transform and Bloch theorem

20. Band structure E
Energy levels (eigenvalues of SE)
Crystal
Molecule

21. The k-point mesh
Corresponds to a supercell 36 time bigger than the primitive cell
Brillouin Zone
Question:
Which require a finer mesh, Metals or Insulators ??
(6x6) mesh

22. Sum of plane waves of increasing frequency (or energy)
Project the wave functions on a basis set
Tricky integrals become linear algebra
Plane Wave for Solid State
Could be localised (ex: Gaussians) + + =
Sum of plane waves of increasing frequency (or energy)
One has to stop: Ecut

23. Solid State: Summary
Quantities can be calculated in the direct or reciprocal space
k-point Mesh
Plane wave basis set, Ecut

25. if (i.LE.n) then
kx=kx-step ! Move to the Gamma point (0,0,0)
ky=ky-step
kz=kz-step
xk=xk+step
else if ((i.GT.n).AND.(i.LT.2*n)) then
kx=kx+2.0*step ! Now go to the X point (1,0,0)
ky=0.0
kz=0.0
else if (i.EQ.2*n) then
kx= ! Jump to the U,K point
ky=1.0
else if (i.GT.2*n) then
kx=kx-2.0*step ! Now go back to Gamma
ky=ky-2.0*step
end if

31. # Crystalline silicon : computation of the total energy
#Definition of the unit cell
acell 3* # This is equivalent to
rprim # In lessons 1 and 2, these primitive vectors
# (to be scaled by acell) were
# that is, the default.
#Definition of the atom types
ntypat # There is only one type of atom
znucl # The keyword "znucl" refers to the atomic number of the
# possible type(s) of atom. The pseudopotential(s)
# mentioned in the "files" file must correspond
# to the type(s) of atom. Here, the only type is Silicon.
#Definition of the atoms
natom # There are two atoms
typat # They both are of type 1, that is, Silicon.
xred # This keyword indicate that the location of the atoms
# will follow, one triplet of number for each atom
# Triplet giving the REDUCED coordinate of atom 1.
1/4 1/4 1/4 # Triplet giving the REDUCED coordinate of atom 2.
# Note the use of fractions (remember the limited
# interpreter capabilities of ABINIT)

32. + + = #Definition of the planewave basis set
ecut # Maximal kinetic energy cut-off, in Hartree
#Definition of the k-point grid
kptopt # Option for the automatic generation of k points, taking
# into account the symmetry
ngkpt # This is a 2x2x2 grid based on the primitive vectors
nshiftk # of the reciprocal space (that form a BCC lattice !),
# repeated four times, with different shifts :
shiftk
# In cartesian coordinates, this grid is simple cubic, and
# actually corresponds to the
# so-called 4x4x4 Monkhorst-Pack grid
#Definition of the SCF procedure
nstep # Maximal number of SCF cycles
toldfe 1.0d # Will stop when, twice in a row, the difference
# between two consecutive evaluations of total energy
# differ by less than toldfe (in Hartree)

33. iter Etot(hartree) deltaE(h) residm vres2 diffor maxfor
ETOT E E E E E+00
ETOT E E E E E-30
ETOT E E E E E-30
ETOT E E E E E-31
ETOT E E E E E-31
ETOT E E E E E-31
ETOT E E E E E+00
At SCF step 7, etot is converged :
for the second time, diff in etot= E-09 < toldfe= E-06
cartesian forces (eV/Angstrom) at end:
Metals (T=0.25eV)
ik=1
| eig [eV]:
| focc: