Lecture 3: DFT

Question 1

How is DFT conceptually different from HF?

Answer 1

It considers electron density as a proxy for the wave function

Question 2

What does DFT state about electron density?

Answer 2

Electron density determines the external potential to within an additive constant.
Therefore the electron density uniquely determines the Hamiltonian operator

Question 3

What is the fundamental statement of DFT?

Answer 3

There is a universal functional that does not depend on the external potential which represents the particular system of interest which gives the exact ground state density and energy

Question 4

What is the problem with the universal functional in DFT?

Answer 4

We don’t know what it is

Question 5

What is energy a functional of?

Answer 5

Electron density

Question 6

What is the energy as a functional the sum of?

Answer 6

Electrons kinetic energy
External potential energy (electron-nuclei)
Electron-electron repulsion
Exchange correlation function

Question 7

How do we find the kinetic energy of the electrons in DFT?

Answer 7

Use the Kohn-Sham equation

Question 8

What is the Kohn-Sham approach?

Answer 8

-imagine non-interacting system where electrons have same density as real system
-we can work out the kinetic energy of the electrons of a non-interacting system

Question 9

Why is using a non-interacting system good in the KS approach in DFT?

Answer 9

-Easy Hamiltonian: a sum of one-electron operators
-eigenfunctions can be Slater determinants if individual one-electron eigenfunctions
-Eigenvalues can be sum of one-electron eigenvalues

Question 10

What is the exchange correlation function in DFT?

Answer 10

-includes corrections from non-interacting system
-includes non-classical correction to electron-electron repulsion

Question 11

What is the electron-electron interaction part in DFT?

Answer 11

-classical electron-electron repulsion
-doesn’t include exchange correlation in a many-body system

Question 12

What are some similarities between HF and DFT?

Answer 12

-both have a variation also principal
-KE and nuclear attraction component are the same
-both use SCF iterative procedure
-there have been modified HF codes to perform DFT

Question 13

What are some differences between HF and DFT?

Answer 13

-DFT is an exact theory that solves the relevant equations approximately
-HF is an approximate theory that solves equations exactly
-when approximations are used for the exchange coefficient, DFT is no longer variational
-DFT is size extensive

Question 14

What is self interaction error?

Answer 14

-artificial interaction with itself due to imperfect cancellation of Exc and self-coulomb interactions
-results in over-delocalisation
-requires expensive methods to overcome

Question 15

Does HF display self interaction error?

Answer 15

No because the coulomb and exchange terms cancel for an electron interacting with itself

Question 16

What are missing long-range correlation and how can they be fixed

Answer 16

-missed London dispersion forces
-can be fixed by including a dispersion correction

Question 17

What should you consider when choosing a functional?

Answer 17

Self interaction error and does it miss long-range correlation effects?

Question 18

What is local spin density approximation? (LSDA)

Answer 18

-assumes value of Exc at position r can be computed exclusively from the value p(r) i.e local value
-depends solely upon the value of electronic density at each point in space

Question 19

Why is LSDA good?

Answer 19

-simplest functional

Question 20

Why is LSDA not so good?

Answer 20

Tends to overestimate correlation energy and underestimate exchange energy

Question 21

What is generalised gradient approximation functional? (GGA)

Answer 21

-accounts for non-homogeneity of the true electron density by allowing for changes in the density away from the coordinate
-based on both density and density gradient

Question 22

Why is GGA good?

Answer 22

-accounts for non-uniform field

Question 23

What is meta-Generalised Gradient Approximation functional?

Answer 23

Like GGA but includes a second derivative of electron density

Question 24

What is hybrid and double hybrid functional?

Answer 24

-incorporates Fock-exchange functional as (from HF)
-Fock exchange is often the computational bottleneck so semi-local functional are often used
-double-hybrid contained both Fock-exchange and perturbation-based theory

Question 25

What is the difference between hybrid and double hybrid functional?

Answer 25

Hybrid scales as N^4, double hybrid scales as N^5

Question 26

What is worse, picking poor basis set or poor functional choice?

Answer 26

Picking a poor Functional as basis sets are less prone to error

Question 27

What are the two types of error basis sets can display?

Answer 27

Basis set incompleteness error (BSIE)
Basis set superposition error (BSSE)

Question 28

What is BSIE

Answer 28

-basis set incompleteness error
-not flexible enough to describe fine details of electron density

Question 29

What is BSSE?

Answer 29

-Basis set superposition error
-close atoms “borrow” basis functions, artificially lowering energy

Question 30

What is the most popular functional/basis choice?
Why is it bad?

Answer 30

B3LYP/6-31G*
Known to perform poorly for even some simple cases