Lecture 3: DFT Flashcards
How is DFT conceptually different from HF?
It considers electron density as a proxy for the wave function
What does DFT state about electron density?
Electron density determines the external potential to within an additive constant.
Therefore the electron density uniquely determines the Hamiltonian operator
What is the fundamental statement of DFT?
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
What is the problem with the universal functional in DFT?
We don’t know what it is
What is energy a functional of?
Electron density
What is the energy as a functional the sum of?
Electrons kinetic energy
External potential energy (electron-nuclei)
Electron-electron repulsion
Exchange correlation function
How do we find the kinetic energy of the electrons in DFT?
Use the Kohn-Sham equation
What is the Kohn-Sham approach?
-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
Why is using a non-interacting system good in the KS approach in DFT?
-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
What is the exchange correlation function in DFT?
-includes corrections from non-interacting system
-includes non-classical correction to electron-electron repulsion
What is the electron-electron interaction part in DFT?
-classical electron-electron repulsion
-doesn’t include exchange correlation in a many-body system
What are some similarities between HF and DFT?
-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
What are some differences between HF and DFT?
-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
What is self interaction error?
-artificial interaction with itself due to imperfect cancellation of Exc and self-coulomb interactions
-results in over-delocalisation
-requires expensive methods to overcome
Does HF display self interaction error?
No because the coulomb and exchange terms cancel for an electron interacting with itself
What are missing long-range correlation and how can they be fixed
-missed London dispersion forces
-can be fixed by including a dispersion correction
What should you consider when choosing a functional?
Self interaction error and does it miss long-range correlation effects?
What is local spin density approximation? (LSDA)
-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
Why is LSDA good?
-simplest functional
Why is LSDA not so good?
Tends to overestimate correlation energy and underestimate exchange energy
What is generalised gradient approximation functional? (GGA)
-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
Why is GGA good?
-accounts for non-uniform field
What is meta-Generalised Gradient Approximation functional?
Like GGA but includes a second derivative of electron density
What is hybrid and double hybrid functional?
-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
What is the difference between hybrid and double hybrid functional?
Hybrid scales as N^4, double hybrid scales as N^5
What is worse, picking poor basis set or poor functional choice?
Picking a poor Functional as basis sets are less prone to error
What are the two types of error basis sets can display?
Basis set incompleteness error (BSIE)
Basis set superposition error (BSSE)
What is BSIE
-basis set incompleteness error
-not flexible enough to describe fine details of electron density
What is BSSE?
-Basis set superposition error
-close atoms “borrow” basis functions, artificially lowering energy
What is the most popular functional/basis choice?
Why is it bad?
B3LYP/6-31G*
Known to perform poorly for even some simple cases
