Introduction to Computational Quantum Chemistry
Lesson 12: Periodic calculations
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Periodic systems
“1D”-3D objects
Closely connected to crystals
Periodic systems (explicit-solvent MD)
Boundary conditions:
System is enclosed in an object which upon replication fills the
whole space
Commonly used space-filling objects:
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Input structure
Structure of the molecule
Information about the periodic box
In crystallography usually you get:
3 crystallographic axes: a, b, c
3 principal angles: α, β, γ
These define the translational operations in three directions
For planar or chain systems you do not need 3 vectors
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Wigner-Seitz cell and Brillouin zone
W-S cell is a real-world object which contains unique information
about the system
Brillouin zone has the same property, however it is defined in a
reciprocal space
Wigner-Seitz cell has dimension units in Å, Brillouin zone in Å−1
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Plane wave basis functions
The periodicity of the system must be satisfied
Either we can replicate STOs/GTOs
Define a new type of basis functions: Plane waves
φ(x) = Aeikx
+ Be−ikx
(1)
φ(x) = Acos(kx) + Bsin(kx) (2)
E =
1
2
k2
(3)
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Plane wave basis functions II
χk(r) = eik·r
(4)
where k is similar to the exponent of the GTO
k values must satisfy:
k · t = 2πm (5)
t is the translational vector of the system
m is a positive integer
Typical spacing between k vectors is 0.01 eV
Size of basis set is defined with a threshold (200 eV)
Use of pseudopotentials for core regions is mandatory
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Bands
As in molecules, the electrons are filled into the bands
Each band is represented by a function in k-space
HOMO is called the Fermi level
3 different states occur:
Insulator: large band gap
Semimetal: small band gap
Metal: no band gap
Insulators and semimetals can be viewed as closed-shell systems
Metals as open-shell (fractional occupations of bands)
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Band Diagram and Density Of States
Band diagram shows the evolution of bands along key-symmetry
paths
DOS shows how are the levels in the crystal populated
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Visualization of crystallographic data
CIF files - Crystallographic Information Files
Contain the structure in a unit cell along with the unit cell
Detailed description of the structure
Mercury - Free visualization software
Gaussview - can read and prepare input for gaussian
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How to obtain structure
From a crystallographer
From a database:
PDB - biological structures (proteins, NA)
CSD - organic crystals
COD - inorganic crystals
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QM packages
Most of the commercially available packages have some
implementation of PBC
Gaussian, Turbomole, ADF...
Still work with GTOs or STOs
Packages developed specially for solids:
VASP
SIESTA
CP2K
ONESTEP
GPAW
Unfortunately we don’t have a licences for native solid-state QM :(
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Task
Calculate the band gap for:
Lithium fluoride
Silicon
Aluminium
Use PBC implemented in Gaussian09 and Turbomole
Use Gaussview for generation of the structure (reuse in TM)
PBEPBE/STO-3G; PBE/def2-SV(P) method
Increase the memory to max. of your machine
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Solution
Go to COD and download appropriate structures
Prepare standard G09 job and include the translational vectors
(Symbol TV)
Converge the SCF and look on the bottom of the output for the
band gaps
Silicon and Lithium fluoride should be treated as singlet states
Aluminium (metal) should be treated as open shell system
Increase the multiplicity to get minimum energy
Gap should increase Al < Si < LiF
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