Many electrons atoms

1. Introduction

Hydrogen atom is the simplest atom at to apply Bohr model ans Schrodinger equation. many electrons atoms require other theories and more complex analysis.

An atom, in its normal state has Z protons and Z electrons in order to be neutral. Z is called the atomic number. The more Z is high, the more complex to study the atom becomes. The electrons interact both with each other and with protons of nucleus. If the problem of finding solutions for the hydrogen atom is completely solved, It is not the case for the atom with many electrons starting with the neutral helium, which has only two electrons. The only tools available to study Z-atoms are the approximations.

2. First approximation

The first and simplest one is to ignore the interactions between electrons with each other and assume each electron s moving under the action of the nucleus as a point charge +Ze. In this approximation, each electrons has an independent potential energy and wave function. We can represent this situation as a hydrogen atom with Ze proton, and replace in all the formula for hydrogen atom e by Ze, and e² by Ze²; therefore, rewrite the related potential energy as -Ze²/r. Hence, its energy levels become E_n = - 13.6 Z²/n² (eV).

Let's note that this approximation is not useful.

3. The central-field approximation: CFA

The central field approximation for many-electron assume that the global electric field is radial of any electron within an atom considered as a sphere . This global field depends only on the distance r between the electron and the nucleus.

The corresponding potential energy for this global field E(r) is V(r), called effective potential and denoted by V_eff(r). This V_eff(r) contains the attractive interaction between the electron and the nucleus V_n, and the repulsive interaction between the electron and the (N - 1) other electrons V_e.
The overall effect of these (N - 1) other electrons is to screen the Coulomb attraction between this electron and the nucleus.

For an atom with Z protons and N electrons, the i^th electron, from the group, located at the radius r_i from the nucleus has the two potential energy:

1. The one corresponding to the attractive interaction between this electron and the (N - 1) other electrons:
V_n(r_i) = - Z e²/r_i, and

2. The one corresponding to the repulsive interaction:
V_e(r_i) = + Σ(j) e²/r_ij,
j goes from 1 to N - 1.

Therefore:
V_eff(r_i) = - Z e²/r_i + Σ(j) e²/r_ij
ij denotes the different electron pairs.

4. The Hamiltonian of one electron

The Hamiltonian of the i^th electron is:

H_i = - (ħ²/2m) ∇_i² +V_eff(r_i)

∇_i is the Laplacian operator:
∇_i = Δ_i² = ∂² /∂r² = ∂² /∂x² + ∂² /∂y² + ∂² /∂z²

5. The Hamiltonian of the atom

The Hamiltonian of the atom is:

H = Σ(i) H_i
i goes from 1 to N

H = - (ħ²/2m) Σ(i) ∇_i² - Z Σ(i) e²/r_i + Σ(i) Σ(j) e²/r_ij
i goes from 1 to N, and j goes from 1 to N - 1.

Using the CFA, the Hamiltonian of the atom is:

H = - (ħ²/2m) Σ(i) ∇_i² + Σ(i)V_eff(r_i)
i goes from 1 to N



We do not have an exact and complete form of express the effective central potential V_eff(r). Nevertheless, we can obtain a qualitative expression by boundary conditions at large and small distances:

At small distances r → 0 , we have no screening, so the electron sees the entire nucleus, then
V_eff(r) = - Z e²/r.

At large distances r → ∞, we have a complete screening, so the electron sees Z - (N - 1) "free" protons, then
V_eff(r) = - (Z - (N - 1)) e²/r.

For intermediate distances, V_eff(r) is between the two limits.

In general, an atom in its normal state (electrically neutral), has Z protons and Z neutrons. Therefore: Z - (N - 1) = 1, then at large distances r → ∞, where we have a complete screening, V_eff(r) = + e²/r.

6. The related energies

The Hamiltonian of the atom is:
H = Σ(i) H_i
i goes from 1 to N
That is a system of N independent electrons.

The eigenstate of the system is written as:
ψ = φ_α1(r₁) φ_α2(r₂) φ_α3(r₃) ... φ_αN(r_N)

ψ = Π φ_αi(r_i)
i goes from 1 to N

αi = {n,l,ml} is the set of the three quantum numbers:
principal, orbital, and magnetic for each electron.

The Schrodinger equation for the atom is:
H ψ = E ψ

The Schrodinger equation for a single electron is:
H_i φ_αi(r_i) = E_nl φ_αi(r_i)
Or:
[- (ħ²/2m) ∇_i² + V_eff(r_i)] φ_αi(r_i) = E_nl φ_αi(r_i)

It is important to note that V_eff(r) is not a Coulomb-like potential energy; that is not of 1/r form. the energy E_nl of each electron depends on both n and l; not on n only as for the Hydrogen or Hydrogen-like atoms.

The total energy of the N-electron system is the sum of the energies for each electron i ; that is:

E = Σ E_{ni li}
i goes from 1 to N

7. The expressions of energies

Z_eff is the effective charge obtained by the difference between the nuclear charge Z and the screening factor. We can use the Z_eff calculated from Slater Type Orbitals (STO) method.

In the following calculator, Just enter the atomic symbol of an atom to obtain its Z_eff.

Recall that for Slater method, we have:
E = - (Z^*/n^*)² (13.6 eV),
and the principal quantum number n is corrected as follows:
n = 1, 2, 3, 4, 5, 6
n^* = 1, 2, 3, 3.7, 4.0, 4.2