Density Functional Theory Flashcards
Why are ab-initio MO-based methods (HF, MCSCF, CISD, MP2, CCSD(T),…) are called wavefunction-based methods?
They compute energy as a functional of a wavefunction
E = f(Y(r))
Functional
Something that changes a function into a value
Function and operator
Function: value into a value
Operator: function into a function
Density functional theory
Incorporation of quantum mechanics into density-functional ideas
Why was it hoped that density based methods may be faster than wavefunction based methods?
Wavefunction base methods = function of 3N dimensions (N = number of electrons) so involve a lot of integrals. But for a stationary state of moving charged particles, E could be expressed as a functional of electron density which is only 3 dimensions (less integrals)
Hohnberg-Kohn existence theorem
For electronic systems, there exists a functional Ehk such that the Schrodinger equation becomes E = Ehk[p] where p(r) is the true electron density of stationary state
Hohnberg-Kohn variational theorem
For every approximate or trial density, if p(trial) is not equal to p(true), Ehk[p(trial)] is greater than or equal to than Etrue
Challenges of DFT theory
(1) How to generate good electron densities
(2) Ehk is unknown - how to approximate
How is problem (1) solved?
Kohn Sham MO’s are obtained with an SCF procedure but are variationally optimized to suit a different energy expression
Kohn-Sham SCF technique
p(trial) is built from Slater determinants of ‘N’ orbitals
The KS MO’s are obtained through an SCF procedure, but are variationally optimized to suit a different energy
Which term in Ehk is varied in DFT
The fourth term (exchange correlation functional)
Exchange correlation functionals
Usually taken as the sum of Ex[p] (exchange functional - Fermi correlation) and Ec[p] (correlation functional - Coulomb)
Require numerical integration
GGA DFT calculation comparison to HF
Slower than HF, but faster than MP2 and almost as accurate, thus cost effective for molecules 8-50 atoms, as well as solid state calculations
What forces to DFT methods do poorly for
London-dispersion forces and complexes
Geometry errors (DFT)
(mean average, Angstroms)
0.008 B2LYP
0.011 B3PW91
