Lecture 1: Intro to DFT

Question 1

• When a system is an … of observable A (when ψ is an eigenfunction of operator Â) then the expectation value of A is an … of ψ
• Âψ(r) = aψ(r)

Answer 1

• When a system is an eigenstate of observable A (when ψ is an eigenfunction of operator Â) then the expectation value of A is an eigenvalue of ψ
• Âψ(r) = aψ(r)

Question 2

• The electronic Schrodinger equation is Ĥ_eψ = E_eψ, which is a result of what approximation of the Hamiltonian?

Answer 2

Internuclear term (Vnn­) removed via Born Oppenheimer approximation which assumed this value is fixed as nuclei are very slows with respect to electrons

Question 3

• How is the ESE approximated further?

Answer 3

• ESE still too complex to solve e.g. due to 2^nd differential in KE term
• Separate into a universal (F_e) and an external potential (V_ext)

Question 4

• A function is the mapping from a … to a number e.g. f(x) = y
• A functional is a mapping from a … to a number e.g. the area A under the function f(x) is a single … dependent on the whole … .
• The total energy is a functional of the … … (… – functional)
• ESE takes us from … which is known, to expectation value through …

Answer 4

• A function is the mapping from a number to a number e.g. f(x) = y
• A functional is a mapping from a function to a number e.g. the area A under the function f(x) is a single number dependent on the whole function.
• The total energy is a functional of the wave function (ψ – functional)
• ESE takes us from V_ext which is known, to expectation value through ψ

Question 5

• (IMP) Outline briefly how the ESE is solved ab-intio via wavefunction theory
• (DFT is also ab initio just different appraoch)

Answer 5

• Exact ESE is solved through approximation of the ψ through variational or perturbation theory
• Uses HF of non-interacting electrons in a mean field
• Electron correlation captured through post HF methods
• E.g. CI, MP2 only differing in how they approximate ψ

Question 6

• (IMP) Why might it be more appropriate to use electron density, ρ(r) to solve the ESE

Answer 6

• Ψ depends on 3n coordinates for n electrons
• Storing this info for >100 electrons is unfeasible
• ρ(r) depends only on 3 coordinate (x,y,z), describing the probability distribution of electrons in space at that point
• much simpler as only depends on position you measure it.

Question 7

• Hohenberg-Hohn (HK) theorem proves that there is a one-to-one invertible mapping between the … potential, … and the electron …, …
• This means the energy can formulated as an … density …

Answer 7

• Hohenberg-Hohn (HK) theorem proves that there is a one-to-one invertible mapping between the external potential V_ext and the electron density ρ
• This means the energy can formulated as an electron density functional

Question 8

Through HK theory, how can we now approximate our Energy operator? (dntk)

Answer 8

• Energy can be expressed as a density functional where KE/V_ee is a universal functional F[ρ], independent of the external potential and atomic positions (also determined by electron density and dictate energy)
• Energy as a functional of electron density is minimised through variational

Question 9

• What is the problem with expressing the universal function F in terms of density?

Answer 9

• Expansion of the original KE term contains a 2^nd derivative of ψ which cannot be formulated as a simple function of ρ
• We know that F[ρ] must exist but can’t write it down.

Question 10

• Every external potential can have only … density
• But the same density can connect to … … external potentials
• Therefore, we can calculate the GS density of …-… e­^-s (via Hartree theory) in an … … v_s that is known
• Use this … … potentials equivalent … to calculate the energy of the system described by …

Answer 10

• Every external potential can have only one density
• But the same density can connect to two different external potentials
• Therefore, we can calculate the GS density of non-interacting e­^-s (via Hartree theory) in an effective potential v_s that is known
• Use this made up potentials equivalent ρ to calculate the energy of the system described by v_ext

Question 11

• (IMP) Describe the details of this figure

Answer 11

• Adiabatic connection of an artificial auxiliary system, v_s and a real many-body system, v_ext connected by the same electron density, ρ
• Real system defined by coulomb interaction of all particles (complex)
• Mapped into a non-interacting system encoded by overall potential describing indirect interaction between atoms.

Question 12

• Write the energy as a functional of electron density in terms of this new auxiliary potential (dntk)

Answer 12

• The kinetic energy is now described as a functional of some artificial wavefunctions

Question 13

Where is the exchange correlation functional, E_XC[ρ] derived from?

Answer 13

• The scaling parameter λ describing difference in the adiabatic connection between the artificial KE and true KE as well as the correlation and exchange energy of the non-interacting system.

Question 14

• What is our KS potential, v_s made up of

Answer 14

• v_s = v_H + v_ext + v_xc
• v_H = Hartree potential
• v_ext = external potential
• v_XC = exchange-correlation

–> the last describes the interaction of the electrons in our non interacting system

Question 15

• Describe the Kohn-Sham equations that form (dntk)

Answer 15

• Set of 1-particle equations form, each desrbing KE/v_s of that particle
• Iteratively change {ψ_i} and v_s until they are consistent.
• KS eigen-energies reminiscent of MOs but for effective non interacting auxiliary states

Question 16

(IMP) Describe the key differences between Wave function and KS theory

Answer 16

• Wave function theory
  
  • Complexity in high dimensional ψ via exact ESE
  • Finite sum over slater determinants
  • Finite approximation of ψ àMP2,CCSD etc
  • Very slow convergence to get exact E­_corr but results in exact path via systematic improvements through variational theory
  • Very computationally expensive (limited to ~30 atoms)
• DFT
  
  • Complexity in v_XC (less as based on density alone)
  • FInite sum over KS equations describing orbital like states
  • Non-local potential in space (and time) connects all possible interactions
  • Local approximations to v_XC makes systematic path to true answer difficult
  • Good error calculations
  • Computationally very efficient