Density Functional Theory (DFT) Flashcards
What does the DFT do?
Takes the complex many electron problem and focuses all the effort into dealing with electronic density n(r).
What is the equation for the electronic density n(r)?
n(r) = Ψ(r1=r, r2=r, …)Ψ(r1,r, r2=r,…) i.i. probability of finding an electron at position r.
What does DFT show?
That n(r) can be found from solving the Schrodingers equation for a single electron moving in an effective potential set up by the nuclei and other electrons.
What is the Schrodinger equation including the electron density n(r)?
(-ћ^2/2m ∇^2 + Veff[n(r)] + Veff(r))ф(λ)(r) = ε(λ)ф(r), where n(r) = sum from λ=1 to Nelec of ф(λ)(r)*ф(λ)(r)
What can we approximate the Veff to in the Schrodinger equation including electron density n(r)?
Veff(r) = e^2 integral of n(r’)/(4πε0|r-r’|) dr’ (Hartree approximation) i.e. repulsive coulomb potential from a cloud of charge set up by the other electrons.
What do crystal structures possess?
Translational symmetry.
How can a perfect crystal be created?
By decorating each point in a lattice with the same basis (atom or group of atoms).
What is a lattice?
A lattice is an infinite array of mathematical points in space with translational periodicity.
What does the diagram of a lattice look like?
Load of dots forming squares.
What does the diagram of a crystal structure look like?
Same as lattice but with an x in the middle of each square, with the basis being the dot and the cross below and to the left together.
How can we define the lattice in 3D?
3 fundamental translation vectors: a, b, c, so R(n1n2n3) = n1a+n2b+n3c, where the n number are integers.
When choosing a,b and c, what does it mean if they are primitive?
They satisfy the equation for R.
What is the equation for the volume of the parallelepiped for a lattice?
v(c) = a.(bXc)
How many crystal structures are there and what are these called?
14 3D crystal structures with different symmetries: the Bravais lattices.
What are the 7 groups in the Bravais lattices?
Cubic, tetragonal, orthorhombic, hexagonal, triagonal, monoclini and triclinic.
