In computational physics and Quantum chemistry, the Hartree–Fock (HF) method also known as self consistent method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system or many electron system in a stationary state

2. CONTENTS
ALGORITM
OF
HATREE
METHOD
HATREE
APPROXI
MATION
HATREE-
FOCK
METHO
D
Appro
ximati
ons
Observa
tion
OVER
VIEW

3. PERTURBATION THEORY
VARIATIONAL PRINCIPLE
WKB APPROXIMATION
Judge the quality of the wave functions by the energy
Any approximation to the ground state wave function will yield an
expectation value of the Hamiltonian that is greater than or equal to the
ground state energy i.e.

4.  The Born-Oppenheimer Approximation
The motion of atomic nuclei and electrons in a molecule
can be separated.
ψ 𝑇=ψ 𝑒×ψ 𝑛

5. HARTREE APPROXIMATION
• The simplest approximation.
Major simplifications

6. Electronic Schrödinger Equation for a System of
Many Electrons
1
2
𝑖=1
𝑁
𝛻2
KINETIC ENERGY
OF ELECTRONS
INTERACTION
BETWEEN ALL
NUCLEI AND
ELECTRONS
𝒊=𝟏
𝑵
𝒋>𝒊
𝑵
𝟏
𝒓𝒊𝒋
COULOMBIC
REPULSION
BETWEEN
ELECTRONS
𝒊=𝟏
𝑵
𝑨=𝟏
𝑴
𝒁 𝑨
𝒓 𝒂𝒊

7. Hartree’s Self-Consistent Field (SCF)
Approach.
For a trial wave function ,It permits the many-particle problem
to be reduced to problem of single particle.
This method invokes orbital approximation i.e. Complex wave
function of many electron system
ψ (r1, r2,…..,𝑟𝑛) = ψ𝑖(𝑟𝑖).
WHATIS SCF
METHOD?
An iterative
method

9.  𝑯 𝒆𝒍𝒆𝒄= 𝟏
𝑵
𝒉 𝒆𝒍𝒆𝒄
𝒊
Ideal method for a computer as easily written as an algorithm.
Gaussian
Wave
function
Calculate
Charge
Density
Calculate
Potential
Solve
Schrodinger
equation
Calculate
Charge
density
Is Charge
Density
Same as
before?
YES
NO
ST
O
P

10. Comparison with other approximation:
Perturbation
theory with
hydrogen-
like wave
functions
Variational
theory with
effective Z
Numerical
Hartree(-
Fock)
result
Experi
mental
result
E=-5.5𝑅 𝑦 E=-5.695Ry E=-5.724Ry E=-5.807Ry

11. Hatree-Fock Method
• In 1930,Fock pointed out that Hatree
wave function violates Pauli exclusion principle.
Two identical fermions cannot
occupy the same quantum state simultaneously
Wave function has to be anti-symmetric.

12. • Antisymmetric Function is defined as ψ (r1,r2)=-ψ (r2,r1).
• Main simplification :
In this model we need to use simple possible antisymmetric space one
can imagine : Antisymmetric Tensor product.
• For N=2, we have
where α is the electron spin Eigen function.

13. Slater Determinant
• A function which is determinant of atomic orbitals Slater determinant.
• For many electrons
 Wave function of a multi-electron system that
satisfies
1.Anti-symmetric requirements
2.The Pauli exclusion principle.

14. Schrodinger Energy in ground state for a single slater determinant
Hatree-Fock energy.
The quantity we can measure is Electronic density ρ.
Answer is
Energy
 Determine energy directly from
the minimizing orbitals BUT………

15. Remedy
 𝐹(𝑖) = ℎ(𝑖) + 𝑗=1
𝑛/2
[2𝐽𝑗 𝑖 − 𝐾𝑖 𝑖 ]

16. CRUCIAL OBSERVATION
• In theory HF limit is achieved by an infinite basis set.
• In practice finite basis sets that can approach HF limit as
efficiently as possible.
• Hatree-Fock energy is invariant under Unitary transformation on
ψ 𝟏to ψ 𝑵.
• Non-linear problem , so HF ground state energy will be
smallest Eigen value and Eigen operator of Fock operator.
“Hartree-Fock method, give us approximate wave functions
for the atoms which have errors in total enery of only a
fraction of 1%” - J. C. SLATER