1) The Born-Oppenheimer approximation separates the motions of electrons and nuclei in molecules by treating the nuclei as fixed due to their large mass compared to electrons. 2) The Hamiltonian operator for a molecule is divided into electronic and nuclear terms. 3) A Slater determinant describes the wavefunction of a multi-electron system by incorporating the Pauli exclusion principle through its sign change upon exchange of electrons.

1. Born-Oppenheimer
approximation & Slater
determinant
By
- Gokila.N

2. The Born-Oppenheimer Approximation is the
assumption that the electronic motion and the nuclear
motion in molecules can be separated .
This involves the following assumptions:
• The electronic wave function depends upon the nuclear
positions but not upon their velocities, i.e., the nuclear
motion is so much slower than electron motion that they
can be considered to be fixed.

3. • Electrons are much lighter than the nuclei . Hence
electrons in a molecule would much more faster
than the nuclei
• The nuclear motion (e.g., rotation, vibration) sees a
smeared out potential from the speedy electrons

4. Wave function ψ of a molecule is the product of the
electronic wave function and the nuclear wave function .
ψ = ψe ψN
Hamiltonian operator of a molecule
Ĥ = TN + Te + VeN + Vee + VNN ------------------ (1)
Ĥe = Te + VeN + VNN ------------------ (2)
Te + VeN + VNN = Hamiltonian for electron motion
VNN = Hamiltonian for nu-nu repulsion
^ ^ ^ ^
^ ^ ^
^
^
^
^
^

5. Ĥ = Ĥe + VNN ---------------- (3)
Consider hydrogen molecule
^

6. Operator for the kinetic energy of the electron
T = - ½ ∇1
2
- ½ ∇2
2
Operator for the potential energy due to the electron
nuclear attraction
VeN = −1
𝑟𝑎1
−1
𝑟b1
−1
𝑟𝑎2
−1
𝑟𝑏2
Operator for the potential energy due to the electron-
electron repulsion
Vee = 1
𝑟12
^
^
^

7. Operator for the potential energy due to the nu-nu
repulsion
VNN = 1
𝑅
Hamiltonian of the molecule
Ĥ = −1
𝑟𝑎1
−1
𝑟b1
−1
𝑟𝑎2
−1
𝑟𝑏2 + 1
𝑟12 + 1
𝑅
Ĥ = Ĥe + 1
𝑅
^

8. Slater determinant
In quantum mechanics, a Slater determinant is an
expression that describes the wave function of a multi-
fermionic system. It satisfies anti-symmetry requirements,
and consequently the Pauli principle, by changing sign upon
exchange of two electrons

9. • The Slater determinant arises from the consideration of
a wave function for a collection of electrons, each with
a wave function known as the spin-orbital, where
denotes the position and spin of a single electron.
• A Slater determinant containing two electrons with the
same spin orbital would correspond to a wave function
that is zero

10. For Helium atom
=
=1/ 2! 1𝑠(1)α(1) 1𝑠(2)α(2)
1𝑠(1)β(1) 1𝑠(2)β(2)
ψ(1,2)
=1/ 2!
1𝑠(1)α(1) 1𝑠(1)β(1)
1𝑠(2)α(2) 1𝑠(2)β(2)
-ψ(1,2)

11. For Lithium atom
=1/ 3!
1𝑠(1)α(1) 1𝑠 2 α(2) 1𝑠(3)α(3)
1𝑠(1)β(1) 1𝑠(2)β(2) 1𝑠(3)β(3)
2𝑠(1)α(1) 2𝑠(2)α(2) 2𝑠(3)α(3)
ψ(1,2,3)