Post on 25-Jan-2015
description
Mihir
iiT,BhuBaneswar
Quantum Mechanical Wave Function gives all information about a given system.
For a Hydrogen Atom, We can solve Schrodinger Equation exactly and determine the allowed energy state of the System.
It is difficult to solve Schrodinger Equation for N- body System. We must involve some approximation for the solution .
. It is a method of obtaining an approximate solution to Schrodinger Equation for many body system. . It is Primarily a theory of Electronic ground state structure in terms of electronic density distribution n(r).
The Time independent Schrodinger Equation Hππ π₯π₯1,π₯π₯2, β¦ . . π₯π₯ππ.π π 1.π π 2 β¦ . .π π ππ = πΈπΈπΈπΈ (π₯π₯1,π₯π₯2, . π₯π₯ππ,π π 1 β¦ β¦π π ππ)
Where H is the Hamiltonian for a system consisting of M nuclei and N electrons.
π»π» = β1
2πππποΏ½π©π©ππ2ππ
ππ=1
β1
2πππποΏ½π©π©ππ2ππ
ππ=1
+12οΏ½ππ
ππ=1
οΏ½ππ2
ππππ β ππππ
ππ
ππ=1,ππβ ππ
βοΏ½οΏ½ππππππ2
|ππππ β ππππ|+
12οΏ½ππ
ππ=1
οΏ½πππππππ€π€ππ2
|ππππ β πππ€π€|
ππ
π€π€=1,π€π€β ππ
ππ
ππ=1
ππ
ππ=1
π»π» = ππππ + ππππ + ππππππ + ππππππ + ππππππ
First two terms of the above equation are Kinetic Energy of Electrons and nuclei respectively. Third term is electron- electron Potential Energy. Fourth term is the Potential energy between electrons and nuclei. Last term is nuclei-nuclei potential energy.
Born-oppenheiMer approxiMaTion
ππππ β« ππππ Nuclei are much slower than electron. Hence Nuclei Kinetic Energy is zero. ππππ= 0 Nuclei-Nuclei Interaction is Constant. ππππππ= πΆπΆπΆπΆπΆπΆπΆπΆπΆπΆπΆπΆπΆπΆπΆπΆ Hence we can eliminate the term ππππ πΆπΆπΆπΆππ ππππππ from the Hamiltonian. Hence New Hamiltonian for many body system
π»π» = ππππ + ππππππ + ππππππ
All ground state properties are determined by ground state density n(r).
The total energy of a many body system is a unique functional of electron density.
Density πΆπΆ ππ = β«ππ3ππ2 β«ππ3ππ3 β¦ β¦β«ππ3ππππ |ππ ππ, ππ2, β¦ ππππ |2 n(r) uniquely determines ππππππππ. ππππππππ = β« π£π£ππππππ ππ πΆπΆ ππ ππππ The total energy functional can be written as πΈπΈ πΆπΆ = πΉπΉ πΆπΆ + β«π£π£ππππππ ππ πΆπΆ ππ ππππ
Universal function F(n) is Independent of External potential but it is unknown.
INPUT OUTPUT
INCAR POSCAR POTCAR KPOINTS
OUTCAR OSCZICAR CONTCAR CHGCAR EIGENVAL WAVECAR PROCAR XDATCAR DOSCAR
INCAR: Central Input file. :Different parameters for different properties. . POSCAR: Position of Ions : Lattice Constant .POTCAR : Psuedopotential from VASP .KPOINTS: Kpoint meshes
For Binding Energy calculation, I have to calculate single point Energy calculation.
I take Oxygen as an example. Here I do spin Polarised calculation. For single O atom, Input files are INCAR,
POSCAR, POTCAR, KPOINTS. For O2 dimer, only POSCAR file is modified. B.E =E(O2)-2*E(O)
INCAR POSCAR
KPOINTS
POTCAR
FREE Energy value for O Atom
Here all the Input files are same as those for Oxygen Atom.
Only POSCAR File is slightly changed. Here No. of atoms is 2 Hence two co-ordinate points are required for
describing the position of atoms in the molecule.
Energy of Oxygen Molecule
Binding Energy = E (O2)-2*E(O) = -9.83466139+3.96809746 = -5.86656393eV
Here I choose Cl2 dimer. Input files : INCAR,POSCAR, POTCAR, KPOINTS Open POSCAR file
Cl2 Molecule ! Comment line 1.00 !Length Unit in Angstrom 10.0 0.00 0.00 0.00 10.0 0.00 0.00 0.00 10.0 2 Cartesian 0.00 0.00 0.00 1.97 0.00 0.00
Lattice Vectors
No. of Atoms
Position of Atoms
Now edit the POSCAR file and Change X-Co-ordinate of second Cl atom to 1.99A
Save the file and run VASP Record Energy value. Repeat the Calculation for successive increase in bond length up
to 2.07A and record corresponding Energy values. Plot the graph between Energy and bond Length. The Length corresponds to minimum energy is the equilibrium
bond length.
Here I choose Cu atom in FCC phase. Keep all INPUT files (INCAR,POSCAR,POTCAR,KPOINTS) POSCAR File
Cu 3.55 0.5 0.5 0.0 0.0 0.5 0.5 0.5 0.0 0.5 1 Cartesian 0 0 0
Lattice vector in FCC System
Edit POSCAR file and Change Lattice Constant to 3.60 A Save the file and run the job and record energy Repeat the calculation by successive increase in length up to 3.70A
and record all corresponding Energy values. Plot the graph between energy with lattice constant. The lattice constant corresponds to minimum energy is actual
lattice constant.
The silicon Crystal structure is FCC . FCC primitive unit cell with 2 atoms in the
unit cell. For band structure calculation first run self-
consistency calculation to get the charge density.
Then fix the charge density and run a non self consistency calculation for desired K points to get band structure.
INCAR for SCF Calculation
For SCF calculation ICHARG =2
For GGA psuedopotential
POSCAR
For FCC, no. of atoms per unit cell
KPOINTS
For Automatic Mesh Generation
POTCAR file is provided by VASP Then run VASP After job is completed, we get charge density in
CHGCAR file To get band structure, run non-self consistence
calculation for each desired k points.
Input files are INCAR, POSCAR, POTCAR, KPOINTS, CHGCAR.
INCAR and KPOINTS files will be modified. In INCAR file, we change ICHARG=11 We modify KPOINTS file, to specify along
some high symmetry direction to calculate energy.
Reciprocal space
Along each line, 10 kpoints are calculated.
Run VASP The above KPOINTS file instructs vasp to
calculate the Energy at each k point between L point and Gamma point, gamma point and K point.
Along each line 10 Kpoints are calculated. In output, we find EIGENVAL file. From this we find all information to plot band
structure.
EIGENVAL file
Kpoints
Energy of 8 bands at that particular energy point
Since energy of bands at each k points, we get information about band structure.
I Plot band structure using MATLAB.
Band Structure of Silicon
Graphene is a 2-dimensional crystalline allotrope of carbon.
In graphene, Carbon atoms are densely packed in a regular hexagonal pattern.
It is one atom thick layer of graphite. Input files INCAR, POSCAR, POTCAR,
KPOINTS
INCAR POSCAR KPOINTS
INPUT files for SCF Calculation for graphene
POTCAR
KPOINTS for Graphene Along high symmetry line
Band Structure of Graphene
W.Kohn. L.J.Sham, physical rev. 140,A1133(1965)
H.J. Monkhrost and J.Pack, Phys. Rev.B 13,5188(1976)
W.Kohn, A.D.Becke,R.G.Parr,J.Phys.Chem.1996, 100, 12974-12980