Lecture 3: DFT for solid condensed matter and surfaces

Question 1

What are PBC’s and why are they essential in the calculation of macroscopic properties?

Answer 1

• 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.

Question 2

• Give some examples of systems that would require PBC’s

Answer 2

• 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.

Question 3

• How is periodicity described in DFT?

Answer 3

• 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.
• b_1/2 = 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

Question 4

• What is a Wigner-Seitz cell?

Answer 4

• A primitive unit cell where atom defines centre and borders of cell and distance between neighbouring atoms/2

Question 5

• What is the 1^st Brioullin zone of an FFc crystal lattice?

Answer 5

• 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.

Question 6

• 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.

Answer 6

• 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.

Question 7

What is the largest problem in representing periodic systems?

Answer 7

• Describing delocalisation of electrons in a periodic crystal as they are periodically distributed in a metal due to ψ.

Question 8

• How do free electrons differ from those in a periodic crystal?

Answer 8

• 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.

Question 9

• What is Blochs theorem?

Answer 9

• Every wavefunction is ‘quasi-periodic’ and can be represented as a product of a plane wave (e^ikr) and a periodic wave function (u_k[r])

Question 10

• Describe this wavefunction representation using a sketch to explain your answer

Answer 10

• Identical periodic atom in neighbouring cells, as must have same electron function (u_k) across all unit cells as periodic.
• How plane wave (e^ikr) 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.

Question 11

• Describe the consequences of Block theorem

Answer 11

• 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

Question 12

• Describe how k is related to calculation on expectation values

Answer 12

• 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.

Question 13

• 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

Answer 13

• Need to calculate k on a regular grid within the 1^st 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

Question 14

• Why can k points not be summed infinitely in the calculation of k? What is a compromise taken instead?

Answer 14

• 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

Question 15

• Describe how the plane wave modulating the wave function changes as you move away from the centre of the Brillouin zone

Answer 15

• 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.

Question 16

(IMP) Define k-grid convergence. When is it reached?

Answer 16

• 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.

Question 17

• What is a basis set?

Answer 17

• A set of mathematical functions that represent wave functions - u_k(r)

Question 19

• Give examples of two types of basis functions and how they differ

Answer 19

• Modern local basis functions (electrons localised)
• Non-local basis methods (electrons free)
• Generally local are more accurate but take much longer to compute

Question 20

Describe a non-local basis

Answer 20

• 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.

Question 21

• Describe the pseudopotential non-local basis

Answer 21

• 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

Question 22

Describe local atomic orbitals

Answer 22

• Numerical atomic orbitals or gaussian orbitals where all electrons are described locally with their atoms

Question 23

• Why are local basis’ generally more accurate than non-local

Answer 23

• Local: Only specifies functions where atoms sit
• Non-local: must fill all space with wavefunctions, demanding larger unit cells and therefore a larger basis

Question 24

(IMP) Calculations are converged when their result is … of the … approximations used to perform them

Answer 24

Calculations are converged when their result is independent of the numerical approximations used to perform them

Question 25

(IMP) What defines the accuracy of a calculation?

Answer 25

• Errors due to theoretical approximations (e.g. BOA) and levels of theory used

Question 26

(IMP) List 3 factors that affect accuracy in DFT calculations

Answer 26

• Xc-functional choice
• System size (# layers, unit cell)
• PBC (vacuum amount given, type of PBC used)

Question 27

(IMP) What defines the precision of a calculation?

Answer 27

• Errors due to numerical approximations made in calculations

Question 28

(IMP) List 4 factors that affect precision in DFT calculations

Answer 28

• Choice of basis and pseudopotential
• Number of k points and sampling grid used
• Basis cutoff or another specification
• SCF convergence threshold and numerical accuracy.