Solid State Computing. Peter Ballo. Models. Classical: Quantum mechanical: H = E Semi-empirical methods Ab-initio methods. Molecular Mechanics. atoms = spheres bonds = springs
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H = E
Energy = E(str) + E(bend) + E(tor) + E(NBI)
all-electron or pseudo- potential ..
Two body , three body…, multi-body potential
no self consistency loop and no eigenvalue computation
Very Complex many body Problem !!
(Because everything interacts)
From density, work out Effective potential
Solve the independents K.S. =>wave functions
Deduce new density from w.f.
New density ‘=‘ input density ??
Finita la musica
Electrons are fermions (antisymmetric wave function)
At this stage, the only thing we need is:
Still a functional (way too many variables)
Generalised Gradient Approximation:
LDA or GGA
M. I. Baskes, Matter. Chem. Phys. 50, 152 (1997)
k-vector (or k-point)
See X-Ray diffraction for instance
Also, Fourier transform and Bloch theorem
Energy levels (eigenvalues of SE)
Corresponds to a supercell 36 time bigger than the primitive cell
Which require a finer mesh, Metals or Insulators ??
Sum of plane waves of increasing frequency (or energy)
One has to stop: Ecut
kx=kx-step ! Move to the Gamma point (0,0,0)
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)
else if (i.EQ.2*n) then
kx=1.0 ! Jump to the U,K point
else if (i.GT.2*n) then
kx=kx-2.0*step ! Now go back to Gamma
#Definition of the unit cell
acell 3*10.18 # This is equivalent to 10.18 10.18 10.18
rprim 0.0 0.5 0.5 # In lessons 1 and 2, these primitive vectors
0.5 0.0 0.5 # (to be scaled by acell) were 1 0 0 0 1 0 0 0 1
0.5 0.5 0.0 # that is, the default.
#Definition of the atom types
ntypat 1 # There is only one type of atom
znucl 14 # 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 2 # There are two atoms
typat 1 1 # 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
0.0 0.0 0.0 # 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)
#Definition of the planewave basis set
ecut 8.0 # Maximal kinetic energy cut-off, in Hartree
#Definition of the k-point grid
kptopt 1 # Option for the automatic generation of k points, taking
# into account the symmetry
ngkpt 2 2 2 # This is a 2x2x2 grid based on the primitive vectors
nshiftk 4 # of the reciprocal space (that form a BCC lattice !),
# repeated four times, with different shifts :
shiftk 0.5 0.5 0.5
0.5 0.0 0.0
0.0 0.5 0.0
0.0 0.0 0.5
# 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 10 # Maximal number of SCF cycles
toldfe 1.0d-6 # Will stop when, twice in a row, the difference
# between two consecutive evaluations of total energy
# differ by less than toldfe (in Hartree)
iter Etot(hartree) deltaE(h) residm vres2 diffor maxfor
ETOT 1 -8.8611673348431 -8.861E+00 1.404E-03 6.305E+00 0.000E+00 0.000E+00
ETOT 2 -8.8661434670768 -4.976E-03 8.033E-07 1.677E-01 1.240E-30 1.240E-30
ETOT 3 -8.8662089742580 -6.551E-05 9.733E-07 4.402E-02 5.373E-30 4.959E-30
ETOT 4 -8.8662223695368 -1.340E-05 2.122E-08 4.605E-03 5.476E-30 5.166E-31
ETOT 5 -8.8662237078866 -1.338E-06 1.671E-08 4.634E-04 1.137E-30 6.199E-31
ETOT 6 -8.8662238907703 -1.829E-07 1.067E-09 1.326E-05 5.166E-31 5.166E-31
ETOT 7 -8.8662238959860 -5.216E-09 1.249E-10 3.283E-08 5.166E-31 0.000E+00
At SCF step 7, etot is converged :
for the second time, diff in etot= 5.216E-09 < toldfe= 1.000E-06
cartesian forces (eV/Angstrom) at end:
1 0.00000000000000 0.00000000000000 0.00000000000000
2 0.00000000000000 0.00000000000000 0.00000000000000
| eig [eV]: -5.8984 1.7993 1.9147 1.9147 2.8058 2.8058 141.3489 313.9870 313.9870
| focc: 2.0000 1.9999 1.9998 1.9998 1.9979 1.9979 0.0000 0.0000 0.0000