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Fundamentals of
Quantum Mechanics

   Contents



© The scientific sentence. 2010

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 En = - 13.6 Z2/n2 (eV).


Simplest approximation:
Interactions between electrons neglected:

En = - 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 Veff(r). This Veff(r) contains the attractive interaction between the electron and the nucleus Vn, and the repulsive interaction between the electron and the (N - 1) other electrons Ve.
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 ith electron, from the group, located at the radius ri 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:
Vn(ri) = - Z e2/ri, and

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

Therefore:
Veff(ri) = - Z e2/ri + Σ(j) e2/rij
ij denotes the different electron pairs.


CFA approximation:

Veff(ri) = - Z e2/ri + Σj e2/rij

ij denotes the different electron pairs.



4. The Hamiltonian of one electron

The Hamiltonian of the ith electron is:

Hi = - (ħ2/2m) ∇i2 +Veff(ri)

i is the Laplacian operator:
i = Δi2 = ∂2 /∂r2 = ∂2 /∂x2 + ∂2 /∂y2 + ∂2 /∂z2



5. The Hamiltonian of the atom

The Hamiltonian of the atom is:

H = Σ(i) Hi
i goes from 1 to N

H = - (ħ2/2m) Σ(i) ∇i2 - Z Σ(i) e2/ri + Σ(i) Σ(j) e2/rij
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)Veff(ri)
i goes from 1 to N



We do not have an exact and complete form of express the effective central potential Veff(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
Veff(r) = - Z e2/r.

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

For intermediate distances, Veff(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, Veff(r) = + e2/r.



6. The related energies

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

The eigenstate of the system is written as:
ψ = φα1(r1) φα2(r2) φα3(r3) ... φαN(rN)

ψ = Π φαi(ri)
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:
Hi φαi(ri) = Enl φαi(ri)
Or:
[- (ħ2/2m) ∇i2 + Veff(ri)] φαi(ri) = Enl φαi(ri)

It is important to note that Veff(r) is not a Coulomb-like potential energy; that is not of 1/r form. the energy Enl 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 Eni li
i goes from 1 to N


Atom with N electrons
H ψ = E ψ     ψ = Πφαi(ri)
Hi φαi(ri) = Eni li φαi(ri)       αi = {n,l,ml}
E = Σ Eni li      i from 1 to N



7. The expressions of energies


Energy levels of hydrogen atom:

En = - 13.6 /n2 (eV)

Energy levels of hydrogen-like atom:

En = - 13.6 Z2/n2 (eV)

Energy levels of many-electrons atom:

En = - 13.6 Zeff2/n2 (eV)


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

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


Zeff Calculator
Enter an atom's symbol
Get its effective nuclear charge
Results:


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




  


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