Lecture 3: DFT for solid condensed matter and surfaces Flashcards
What are PBC’s and why are they essential in the calculation of macroscopic properties?
- A central box is embedded in an infinitely-repeating set of identical images.
- Translational symmetry of this small part of the system is utilized to capture the extended nature of complex systems that could otherwise not be simulated feasibly.
- Give some examples of systems that would require PBC’s
- Metal bulk/surface: electrons are delocalised, making it hard to cut out a meaningful chunk to analyse. Must use PBCs to estimate entire system
- Metal organic frameworks: largely conjugated systems, therefore chemistry likely to be similar throughout.
- How is periodicity described in DFT?
- Huge systems of unit cells of length a are converted in to reciprocal lattice vectors
- An infinitely long set of unit cells, each of length a, can be defined in a reciprocal lattice vector of length b.
- b1/2 = 2a
- 4-unit cells each of length a would be 1/4 b each
- a larger real space vector corresponds to a smaller reciprocal unit cell and vice versa
- What is a Wigner-Seitz cell?
- A primitive unit cell where atom defines centre and borders of cell and distance between neighbouring atoms/2

- What is the 1st Brioullin zone of an FFc crystal lattice?
- The Wigner-Seitz cell in reciprocal space
- All points are unique in symmetry and describe how electrons are transported across a material
- Γ defines the centre atom of the unit cell.

- The … … makes up all the unit cells and their electrons we define in our system.
- Unit cells are identified with a … vector defined by multiples of the reciprocal unit cell vectors, which describe how real space unit cells … … … … … in the crystal
- This quasi-continuous description of cells is called the …-…-…(…) boundary condition.
- The crystal volume makes up all the unit cells and their electrons we define in our system.
- Unit cells are identified with a k vector defined by multiples of the reciprocal unit cell vectors, which describe how real space unit cells interact with far away neighbours in the crystal
- This quasi-continuous description of cells is called the Born-von-Karman (BvK) boundary condition.

What is the largest problem in representing periodic systems?
- Describing delocalisation of electrons in a periodic crystal as they are periodically distributed in a metal due to ψ.
- How do free electrons differ from those in a periodic crystal?
- Ψ of electron is the same for both (an infinitely oscillating plane wave), however the periodic crystal modulating the ψ of an electron associated with a crystal.
- What is Blochs theorem?
- Every wavefunction is ‘quasi-periodic’ and can be represented as a product of a plane wave (eikr) and a periodic wave function (uk[r])

- Describe this wavefunction representation using a sketch to explain your answer
- Identical periodic atom in neighbouring cells, as must have same electron function (uk) across all unit cells as periodic.
- How plane wave (eikr) modulates each cell can be different
- Interference shows how electrons are spread over space instead of being local.
- The simplest state if one where all unit cells are in phase due to infinitely large wave.
- As the wave increases in frequency, more complex linear combinations of cells in/out on phase form.
- Allows creation of linear combinations of different functions of atoms that describe periodicity that is larger than a unit cell.

- Describe the consequences of Block theorem
- Electronic state can be defined by 2 quantum numbers
- K – a discrete quasi-crystal momentum
- N – tells us energy of state
- These give us many combinations of states in which to define different parts of our periodic system
- Describe how k is related to calculation on expectation values
- When calculating observables must integrate of over all possible k values, as tells us how we mix different orbitals to describe collective waves in the crystal.

- Need to calculate k on a … … within the … … zone in reciprocal space.
- This grid must be … enough so that the observable is converged
- The grid is called the … -… grid
- Need to calculate k on a regular grid within the 1st Brillouin zone in reciprocal space.
- This grid must be dense enough so that the observable is converged
- The grid is called the Monkhorst-Pack grid
- Why can k points not be summed infinitely in the calculation of k? What is a compromise taken instead?
- Would be too computationally expensive
- Must instead calculate as many k points as possible on a regular grid
- The denser the grid, the more k points and therefore the more interactions between neighbouring unit cells can be included, giving a more precise sampling of collective periodic structure
- Describe how the plane wave modulating the wave function changes as you move away from the centre of the Brillouin zone
- At the gamma central gamma point, all electronic state are perfectly coherent (lowest energy combination)
- As you move further out in to reciprocal unit cell space more interactions between neighbouring unit cells are included, leading to anticoherence, and a more complex wave function to describe.
(IMP) Define k-grid convergence. When is it reached?
- The saturation points at which increasing the value of k grid (increasing density) is outweighed by the computational expense associated with that increase in precision.
- This is to ensure the k point sampling is sufficient for all periodic states of the system.
- K gird converged when increase in density does no change observable significantly.
- What is a basis set?
- A set of mathematical functions that represent wave functions - uk(r)
- Give examples of two types of basis functions and how they differ
- Modern local basis functions (electrons localised)
- Non-local basis methods (electrons free)
- Generally local are more accurate but take much longer to compute
Describe a non-local basis
- Traditional plane waves: describes any function in phase space through linear combinations of different oscillations on top of modulation.
- Sums oscillations up a certain energy cut off, defining accuracy and quality of basis set.
- Describe the pseudopotential non-local basis
- As valence electrons are of interest, removes core electrons from calculation and instead describes pseudopotential that already includes them as in frozen states.
- Valence electrons see atom that includes core electrons
- Smooths function as opposed to the result of many core electrons having to be described
Describe local atomic orbitals
- Numerical atomic orbitals or gaussian orbitals where all electrons are described locally with their atoms
- Why are local basis’ generally more accurate than non-local
- Local: Only specifies functions where atoms sit
- Non-local: must fill all space with wavefunctions, demanding larger unit cells and therefore a larger basis
(IMP) Calculations are converged when their result is … of the … approximations used to perform them
Calculations are converged when their result is independent of the numerical approximations used to perform them
(IMP) What defines the accuracy of a calculation?
- Errors due to theoretical approximations (e.g. BOA) and levels of theory used
(IMP) List 3 factors that affect accuracy in DFT calculations
- Xc-functional choice
- System size (# layers, unit cell)
- PBC (vacuum amount given, type of PBC used)
(IMP) What defines the precision of a calculation?
- Errors due to numerical approximations made in calculations
(IMP) List 4 factors that affect precision in DFT calculations
- Choice of basis and pseudopotential
- Number of k points and sampling grid used
- Basis cutoff or another specification
- SCF convergence threshold and numerical accuracy.