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General aspects of electronic ground states�: toward HF / DFT

Young Min Rhee

Department of Chemistry

KAIST

Young Min Rhee, 2023

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Please recall what you learned as a freshman

  • You learned about
    • H atom

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      • 1-electron system: Exactly solvable
      • s, p, d, f, … orbitals
    • He, Li, …
      • Ignore e-e interactions and take 1-electron solutions
      • Recover some e-e interactions

by lifting 2s/2p; 3s/3p/3d degeneracies

      • Fill up (“Aufbau”): electron configuration

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Young Min Rhee, 2023

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The idea is: Just ignore the scary thing…

  • H versus He

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H

He

Young Min Rhee, 2023

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Please recall what you learned as a freshman

  • You learned about
    • H atom

​

​

      • 1-electron system: Exactly solvable
      • s, p, d, f, … orbitals
    • He, Li, …
      • Ignore e-e interactions and take 1-electron solutions
      • Recover some e-e interactions

by lifting 2s/2p; 3s/3p/3d degeneracies

      • Fill up (“Aufbau”): electron configuration

​

http://singlet.kaist.ac.kr

Young Min Rhee, 2023

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How to jump from H atom to a molecule: Think simplest!

  • You learned about
    • H2+ molecule
      • 1-electron system: Exactly solvable with Born-Oppenheimer approximation
      • σ, π, δ, φ, … orbitals

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    • H2, He2; O2, N2, F2; CO…
      • Ignore e-e interactions and

take 1-electron solutions

      • Somehow “understand” the

orbital energy orders

      • Fill up (“Aufbau”):

electron configuration

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http://singlet.kaist.ac.kr

Young Min Rhee, 2023

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How can we BETTER consider e-e interaction?

  • Self consistent field (SCF) approximation
    • Don’t delete the scary term!

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    • But still assume that the solution is a production function

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    • And somehow get the optimal solution within the product space
      • How do we optimize?

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Young Min Rhee, 2023

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With some super-difficult (?) math

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Coulomb integral

Things can be computed for a given ψ

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Self-consistent field method

  • Reduction scheme

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    • Integrate out for r2 with

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Self-consistent field method

  • Scheme

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  • Can be extended for cases with more than 2 electrons

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    • “Hartree method”

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Electrons are fermions

  • Let’s revisit:

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Electrons are fermions

  • Use Slater determinant

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Hartree-Fock approximation

  • Self consistent field (SCF) approximation
    • Don’t delete the scary term!

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    • But still assume that the solution is a production function

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    • And somehow get the optimal solution within the “product” space
      • How do we optimize?

http://singlet.kaist.ac.kr

Young Min Rhee, 2023

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We need E

  • What E looks like is:

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Is n! (or, n! times n2 times n!) scary?

  • What E looks like is:

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n! (or, n! times n2 times n!) is NOT scary…

  • when MO’s are orthogonal

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Caution: i can point to an electron (dummy)

or a function (non-dummy)

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Reduce duplicates by dummies

  • when MO’s are orthogonal

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Coulomb integral

exchange integral

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RHF

  • Same spatial orbitals for two spins

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Some useful terminologies

  • Physicist’s versus chemist’s notations

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    • For real orbitals:

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How to do the optimization: Fock operator

  • Adopt the variational principle

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How to do the optimization: Fock operator

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How to do the optimization: Fock operator

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    • Constrained minimization with

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http://singlet.kaist.ac.kr

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Utilize the symmetry of ε

  • Consider a unitary transform U that diagonalizes ε

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    • Thus, we can take the diagonal eigen-solutions of the Fock operator (“canonical” orbitals)
    • Any unitary transform is still a solution

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How to do the optimization: Fock operator

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    • Constrained minimization with

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http://singlet.kaist.ac.kr

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How to do the optimization: Fock matrix

  • Atomic orbital (AO) basis set expansion
    • Optimizing functions is tricky
    • With basis functions, it becomes easy

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    • SCF: Construct f matrix and find the eigenvectors as MO’s

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      • Finding MO coefficients is equivalent to

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Eigenvalue / eigenvector of F

  • C is the generalized eigenvectors of F with S

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    • Only the eigenvectors with the n (= # of electrons) lowest eigenvalues are used for the Slater determinant
    • The remaining (N – n) eigenvectors form virtual orbitals

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Brillouin condition for an error vector

  • Occ-Virt block of the Fock matrix must be zero

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  • This condition is met only when SCF has converged
    • We can use as a metric of measuring SCF error

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What is the cost of SCF?

  • Fock build

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    • Integrals are obtained in AO basis:
    • And then contracted as

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  • Diagonalization
    • To get eigenfunctions as MO’s

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Some practical issues

  • Shell pairing
    • Chemist’s notation is really convenient for this concept
    • AO’s are sparse: and scale not as ~N2 but as ~N
    • Integrals constructed in batches
      • Algorithm designs are important for efficiency
  • Direct SCF
    • is not stored but re-calculated when needed
  • Summation orders matter a lot
    • E.g.

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1-particle density matrix (1PDM)

  • Definition

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    • For a Slater determinant

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    • 1-electron density

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Utility of 1PDM

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  • MO / AO basis transform

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2-particle density matrix (2PDM)

  • Definition

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    • For a Slater determinant

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    • pair density

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      • Should reflect both Coulomb-like and exchange-like stuffs
      • Should be zero when

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These topics lead…

  • Directly to
    • HF theory (Tue)

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    • MP2 and correlation (not covered during our eChem)
      • What aspect should we fix for HF?
      • Full CI (FCI) concept: complete function space for a perfect solution

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    • DFT (Tue)

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    • Optimization: gradient-based method (Wed)
      • Case with

http://singlet.kaist.ac.kr

Young Min Rhee, 2023