Lecture 2 - Current developments in DFT

Question 1

• E_XC is … and needs to be …

Answer 1

• E_XC is unknown and needs to be approximated

Question 2

• What is the basic assumption of Local-density approximations (LDA)?

Answer 2

• E_XC only depends on the value of the local electron density at one point

Question 3

• How does an ensemble generalized LDA approximate XC energy as a functional of electron density?

Answer 3

• Exactly calculates ϵ^XC_HEG­ energy density for a homogenous electron gas (HEG)

Question 4

• What is a homogeneous electron gas?

Answer 4

• Box of homogeneous positive charge background filled with electrons (no atoms)
• These electrons spread out homogeneously allowing exact calculation depending on the density at that separation.

Question 5

• Discuss the structural, elastic and vibrational properties as a result of the use of LDAs
• what approximation yields these results

Answer 5

• Generally, give satisfactory results
• Crystal bulk lattice (distance of unit cell vector) more accurate as usually underestimated
• Bulk moduli (E to compress/expand solid) too large but 10% error not uncommon
• Vibrational frequencies too high/stiff
• These are due to local approximation which compacts the entire system.

Question 6

• Discuss some other problems with LDA’s

Answer 6

• Binding energies are too negative i.e.overbinding
• Activation energies are unreliable
• Band gaps, ionization energies and electron affinities are strongly underestimated.

Question 7

• What are Generalized-gradient approximation (GGA’s) and how do they improve upon LDAs

Answer 7

• Similar form but a local gradient of electron density is included as well as the value of density
• This is to gain information on the local variation in neighbouring electron density at other positions

Question 8

• What strategies can be taken in designing functionals

Answer 8

• Non-empirical functionals: satisfy constraints via certain known mathematical/physical boundary conditions (less accurate)
• Empirical functionals: satisfy a property by empirically fitting to yield prediction of a molecule/material property (more accurate – regularly used)

Question 9

• What is the disadvantage of empirical design of functionals?

Answer 9

• Are derived from QM with good accuracy, however, are not truly ab initio.

Question 10

• How does the bulk lattice constants and cohesive energies with GGA compare with LDA?

Answer 10

• Bulk lattice constants (unit cell vecotr between atoms): GGA increase due to more repulsive core-valence XC
• Cohesive energies (E released by binding atoms in to a solid): GGA reduction mostly due to valence effect, giving better description.

Question 11

• How do Energy barriers of GGAs compare to LDAs

Answer 11

• Free energy of molecule better described with GGAs, reducing the degree to which the barrier is underestimated
• LDAs underestimate to a large extent.

Question 12

Name an improvement GGAs make on LDAs

Answer 12

• GGA correct LDA overbinding, with less stiffness of tightly packed system

Question 13

• Where do GGAs still fall short

Answer 13

• Still no long-range description of vdW forces as local approximation (same as LDA)
• As GGA favours low coordination (large gradient), can now interpret different E sites on a surface (LDA could not distinguish). However can do so incorrectly.

Question 14

• For their simplicity, LDA/GGAs perform well for a large range of materials, marking ‘semi-…’ DFT as an important improvement in how we describe the …
• However major failures in the … of chemical reduction barriers and … … as well as the overestimation of … mark a need for further improvement.

Answer 14

• For their simplicity, LDA/GGAs perform well for a large range of materials, marking ‘semi-local’ DFT as an important improvement in how we describe the E_XC
• However major failures in the underestimation of chemical reduction barriers and band gaps as well as the overestimation of polarizabilities mark a need for further improvement.

Question 15

• What is an example of a GGA functional?

Answer 15

• Perdew-Burke-Ernzerhof (PBE)

Question 16

• What is the general idea of meta-GGAs?

Answer 16

• Expand the local function dependence to occupied KS orbitals.
• 2^nd derivative of KE added via T[ρ] of non-interacting electrons i.e. the product of the derivative of KS ψ of single particle space
• This leads to the function becoming explicitly orbital dependent.
• Still technically local derivative however accounts for many variations in different orbitals.

Question 17

• How do meta-GGAs compare to LDA/GGA

Answer 17

• Better molecular binding E’s
• Better cohesive/structural description
• Band gaps still underestimated
• Long range vdW still not accounted for as still a local functional

Question 18

(IMP) Describe the band-gap problem in GGA’s

Answer 18

• Underestimation of difference in ionisation energy/electron affinity due to self-interaction error
• Contrary to HF, V_X and V_H + V_C don’t cancel for an H-atom
• Missing V_X causes half electrons to move apart from each other instead of contracting in to a positive and negative ion in dissociation.

Question 19

(IMP) What is the over localisation error also associated with the band gap problem?

Answer 19

• Electronic states are too localised as electrons do not sufficiently repel each other (Pauli repulsion missing)
• Electrons do not come close in contact unless they must (run away from themselves as orbitals more diffuse.

Question 20

(IMP) Describe the problem of missing long-range correlation in local functionals

Answer 20

• Electron correlation V_C treated incorrectly (due to local approximation)
• Leading to no long range interaction between non overlapping densities
• As a result, LDA/GGA/meta-GGA do not capture dispersion/vdW effects

Question 21

(IMP) How does the error in delocalization affect how the ionisation energy and electron affinity are related to the energies of the KS states at this stage? Use a sketch to support your answer

Answer 21

• Energies of the KS auxiliary states do not equal the electron affinity and ionisation energy of the system as they should
• This is due to the presence of half electrons in local description causing a bowing of the curve instead of clear definition
  
  • means half an electron in LUMO and HOMO instead of 1 in LUMO (I underest, A overest, E underest)
• Gradient represents energy levels and are incorrect

Question 22

(IMP) What is the solution to the long-range correlation error?

Answer 22

Include explicit non-local correlation or long-range dispersion terms, describing sum of all pairwise interactions as a function as 1/r^6 e.g. Many body dispersion/ vdWsurf

Question 23

• What is an example of a meta-GGA

Answer 23

• TPSS

Question 24

(IMP) How do 4th-rung hybrid functionals solve the localization problem in meta-GGAs? Use a sketch to support your answer

Answer 24

• Includes HF exchange, which strongly over localizes/stabilizes electrons, making graph more convex.

Question 25

* Give 2 examples of hybrid functionals

Answer 25

* B3LYP: most used xc, empirical mix of 3 functionals and HF exchange * HSE06: PBE with 25% HF exchange at short range and PBE in the long range (range separated functional).

Question 26

* How do hybrid functionals compare to local?

Answer 26

* Large improvements in barriers, band gaps and dissociation energies (≈0.1 eV expt) * More accurate and consistent description of CO site adsorption.

Question 27

* What is the idea of 5^th rung RPA (Random Phase Approximation) functionals?

Answer 27

* Xc functionals become explicitly dependent on unoccupied orbitals as well as occupied orbitals previously * Similar to CCSD in HF (mixing excitation to unoccupied states)

Question 28

* How is the use of RPAs carried out?

Answer 28

* E_c is described via non-local dynamic correlation response functions allowing electrons to interact more * Calculated via many body perturbation theory

Question 29

* What is the Jacobs ladder of DFT? commment on its use in functional selection

Answer 29

* Form of finding a systematic relationship between functional approximations * However not a truly systematic improvement process like we do in improving a wavefunction * Therefore, can worse by moving up ladder, must be careful

Question 31

One important numerical factor in PBC material simulations is Brillouin zone/k gird sampling, why do we do k grid sampling, and to what degree to ensure high enough accuracy(?)

Answer 31

Periodic systems are **translational invariant**, Blocks theorem tells us that **translation symmetry operator commutes with the Hamiltonian**, thereby every single wavefunction must be an eigenfunction of the translation operator, this is done with a **plane wave product**