1. Lecture 5. Many-Electron Atoms. Pt
Lecture 5. Many-Electron Atoms. Pt.3 Hartree-Fock Self-Consistent-Field Method
References
Ratner Ch. 9, Engel Ch. 10.5, Pilar Ch. 10
Modern Quantum Chemistry, Ostlund & Szabo (1982) Ch
Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch.7
Computational Chemistry, Lewars (2003), Ch.4
A Brief Review of Elementary Quantum Chemistry

2. Hartree (single-particle) self-consistent-field method based on Hartree products (D. R. Hartree, 1928)
Proc. Cambridge Phil. Soc. 24, 89
Nobel lecture (Walter Kohn; 1998)
Electronic structure of matter
Impossible to search through
all acceptable N-electron
wavefunctions.
Let’s define a suitable subset.
N-electron wavefunction
is approximated by
a product of N one-electron
wavefunctions. (Hartree product)

3. Constrained minimization with the Hartree product

4. (J. C. Slater; V. Fock, 1930) Z. Physik, 61, 126; Phys. Rev. 35, 210
Hartree-Fock Self-Consistent-Field Method based on Slater determinants (Hartree+Pauli)
(J. C. Slater; V. Fock, 1930) Z. Physik, 61, 126; Phys. Rev. 35, 210
Restrict the search for the minimum E[] to a subset of , a Slater determinant.
To build many-electron wave functions, assume that electrons are uncorrelated. (Hartree products of one-electron orbitals)
To build many-electron wave functions, use Slater determinants, which is all antisymmetric products of N spin orbitals, to satisfy the Pauli principle.
Use the variational principle to find the best Slater determinant (which yields the lowest energy) by varying the spatial orbitals {i}.

5. Beyond Hartree: the ground state of He (singlet)
notation
 |1s>  1s 1s
Slater determinant
 1s2
Total spin quantum number S = Ms = 0 (singlet)
S2 (1,2) = (s1 + s2)2 (1,2) = 0, Sz (1,2) = (sz1 + sz2) (1,2) = 0

6. Energy of the Slater determinant of the He atom:
the “singlet” ground state
spatial-symmetric
spin-antisymmetric
no spin in the Hamiltonian
Coulombic repulsion between
two charge distributions
|1s(1)|2 and |1s(2)|2
Coulomb
integral
<1s|h|1s>
=<1s|T+VNe|1s>=Tss+Vs

7. Coulombic repulsion between two charge distributions
|1s(1)|2 and |1s(2)|2

8. Excited state of He (singlet and triplet states)
antisymmetric
spatial-symmetric
spin-
symmetric
spatial-antisymmetric
spatial-symmetric
spatial-antisymmetric

9. Energy of the Slater determinant of the He atom:
a “triplet” first excited state
triplet
singlet
? (quiz)
Coulomb integral > 0
includes in it
wave function
(final solution)!
where
Exchange integral (>0)

10. Energy of the Slater determinant of the He atom:
a “triplet” first excited state
triplet
singlet
Coulomb integral > 0
includes in it
wave function
(final solution)!
where
Exchange integral (>0)

11. Two-electron interactions (Vee)
Coulomb integral Jij (local)
Coulombic repulsion between electron 1 in orbital i and electron 2 in orbital j
Exchange integral Kij (non-local) only for electrons of like spins
No immediate classical interpretation; entirely due to antisymmetry of fermions
> 0, i.e., a destabilization

12. Each term includes the wave function (the final solution) in it!

13. Hartree-Fock Self-Consistent-Field Method based on Slater determinants (Hartree+Pauli)
(J. C. Slater & V. Fock, 1930)
Each  has variational parameters (to be changed to minimize E) including the effective nuclear charge  (instead of the formal nuclear charge Z)
Variational condition
Variation with respect to the one-electron orbitals {i}, which are orthonormal
or its combination for lower E
= ij

14. Constrained (due to the orthonormality of {i}) minimization of EHF[SD] leads to the HF equation.
vergil.chemistry.gatech.edu/notes/hf-intro/node7.html
; Pilar Ch.10.1, Ostlund/Szabo Ch.1.3

15. Constrained minimization with the Slater determinant

16. After constrained minimization with the Slater determinant

19. Hartree-Fock equation (one-electron equation)
Fock operator: “effective” one-electron operator
Two-electron repulsion operator (1/rij) is replaced by one-electron operator VHF(i), which takes it into account in an “average” way
&
Two-electron repulsion cannot be separated exactly into one-electron terms. By imposing the separability, the orbital approximation inevitably involves
an incorrect treatment of the way in which the electrons interact with each other.

20. Hartree-Fock Self-Consistent Field (HF-SCF) Method
Problem:
Fock operator (V) depends on the solution.
The answer (solution) must be known in order to solve the problem!
HF is not a regular eigenvalue problem that can be solved in a closed form.
Solution (iterative approach):
Start with a guessed set of orbitals.
Solve the Hartree-Fock equation.
Use the resulting new set of orbitals in the next iteration and so on
Until the input and output orbitals differ by less than a preset threshold (i.e. converged to a self-consistent field).

21. Hartree-Fock equation (One-electron equation) spherically symmetric
Veff includes
spherically symmetric
&
- Two-electron repulsion operator (1/rij) is replaced by one-electron operator VHF(i), which takes it into account in an “average” way.
- Any one electron sees only the spatially averaged
position of all other electrons.
- VHF(i) is spherically symmetric.
- (Instantaneous) electron correlation
is ignored.
Spherical harmonics (s, p, d, …) are valid
angular-part eigenfunction (as for H-like atoms).
- Radial-part eigenfunction of H-like atoms are not valid any more.
optimized

22. EHF > E0 (the exact ground state energy)
Electron Correlation
Ref) F. Jensen, Introduction to Computational Chemistry, 2nd ed., Ch. 4
A single Slater determinant never corresponds to the exact wave function.
EHF > E0 (the exact ground state energy)
Correlation energy: a measure of error introduced through the HF scheme
EC = E0 - EHF (< 0)
Dynamical correlation
Non-dynamical (static) correlation
Post-Hartree-Fock method
Møller-Plesset perturbation: MP2, MP4, …
Configuration interaction: CISD, QCISD, CCSD, QCISD(T), MCSCF, CAFSCF, …

23. Solution of HF-SCF equation gives

24. Solution of HF-SCF equation: Effective nuclear charge
(Z- is a measure of shielding.)

25. Aufbau (Building-up) principle