Atoms and the Periodic Table VI: Many Electrons System and Periodicity

If we observe closely to the periodic table, the system or the pattern that is not created by single electron system such as hydrogen. Therefore, we need a different approach from what we used in hydrogen atom. In this section we will discuss about this matter and how the electron configuration for many electrons system, and its implication, the periodicity.


If we see the Schrodinger's equation that we have used to describe the electron in H atom, it depends on 3 different variable.

However, this equation cannot solve the problem precisely for many electron system. In the other sides, if we observe the electron closely, it is similar with a small magnetic field. Basically, the electron is a charge particle and it spins. The spinning of electron produces a magnetic field which the direction of the magnetic field depends on the direction of spinning (clockwise or anticlockwise). This approach was proved by Stern-Gerlach experiment.

The experiment used silver atom (Ag) that shot trough a homogenous magnetic field and as the result, there was a trajectory path from the electron that deviated by the magnetic field (either upward or downward direction). Therefore, an electron has another quantum number that describes this behaviour and it is called spin quantum number (m_s). To conclude, an electron has 4 degrees of freedom which are describes the spin movement and the value is + 1/2; n, l, m_l, that describes about the orbital.

In H atom, the interaction is only the attraction between electron an nucleus. However, in many electron system because there are another electrons in the system, so there will be a repulsion between electron. Therefore, it requires another Schrodinger's equation to solve this problem and the implication is the energies will be different. Moreover, the energy of each electron in many electron system depends on the position of the other electron.

The energy in this system depends on the n and l, so the distribution of electron will be different. Since it base on n and l, the energy between s-, p-, and d-orbital will be different. However, the shape of the orbital is still the same as the orbitals in H atom, but the different only at the size. Therefore, to arrange the electron in many electron is based on 3 rules.

1. Aufbau Principle
The electron will start to fill the orbital from the available orbital with the lowest energy. Therefore, the energy order of the orbital is:

2. Pauli Exclusion Principle
This principle is basically about the availability of the orbital. Pauli exclusion principle states that there is no electron has the same set of quantum number. Therefore based on the first 2 principles we can generate the electron configuration from H to B.
H: 1s¹
He: 1s²
Li: 1s² 2s²
Be: 1s² 2s²
B: 1s² 2s² 2p¹

If we move to C, the problem is where the second electron of p-orbital should be placed. Should we pair the electron with the first electron or put in another p-orbital. Therefore, the 3rd rule regulates this problem.

3. Hund's Rule
Hund's rule is applied for degeneracy orbital (n and l is the same). It states that in the degenerate orbital, the electron will occupy separate orbital first with parallel spin, the it will be paired until the orbital is full.

As we already stated briefly, the energy of electron depends on the other electron, and the distribution of the electron itself determines the energy as well. If we see the repulsion force between 2 electrons which strong, one electron will be pushed away and its energy will increase. The consequence of this matter is the electron that located further will be less bound than the inner electron. Therefore, it causes a self consistency problem, which means to find an electron, another electron should be found.

If we see from the point of view of the outer electron, the interactions are the repulsion from inner electron and the attraction from the nucleus. The repulsion from the inner electron push away the electron, so it will be less bound with the proton. If we move to an electron in the inner part, it still has the attraction from nucleus which stronger, but the repulsion from the outer electron can be neglected. The reason is in one position the outer electron will repel the inner electron closer to nucleus, but when the outer electron at the other sides, it will pushed away the inner electron. Therefore, the effect can be neglected.

From those description, it rises 2 main question, for given electron. Firstly, how many electron or how close the electron to the nucleus. Since electron behaves in such similar way as electron cloud, so the classical-based question will be changed into what fraction of electron cloud is close to the nucleus.
Therefore, based on our wave function, the question will be how does ψ² behave close to the nucleus and this phenomenon is the penetration (local maximum at radial distribution function).

Secondly, the question is how much repulsion is experienced by given electron. Based on our wave function, the actual question is what fraction of electron cloud lie closer to nucleus than the given electron. This phenomenon is related to screening or shielding of the nuclear charge. If the given electron has shielded electron it will experience lower nuclear charge. The effective nuclear charge (Z_eff) can be calculated as:

Z_eff = Z - S

where Z is the nuclear charge and S is the screening or shielding constant. The value of S can be calculated based on Slater's rule.

First step of Slater's rule is the electron is placed in groups; the ascending order of n, s and p orbital in the same group; d,f,... are separate group. Therefore the group is:

[1s] [2s2p] [3s3p] [3d] [4s4p] [4d] [4f] [5s5p] [5d] ....

All electron in a group have equal S. Then, the rules are:

1. Ignore electrons in groups to the right

2. Every other electron in the same group contributes 0.35 to S (0.3 if the group is 1s)

3. For an electron in [nsnp]: electrons with n-1 contribute 0.85 each to S; and all electrons further left contribute 1.0 each to S.

4. For electrons in [nd] and [nf]: electrons in all groups to the left contribute 1.0 each to S.

Example:

The value for electron in 2s and 3s orbital at Na atom.

The electron configuration atom of Na = 1s² 2s² 2p⁶3s¹ and Z for Na is 11.

Now, the electron group for Na atom based on Slater's rule =  [1s² ] [2s² 2p⁶] [3s¹]

For 3s orbital the S value is:
S = (0 x 0.35) + (8 x 0.85) + (2 x 1.0) = 8.8,

so, the Z_eff = 11 - 8.8 = 2.2

For 2s orbital the S value is:

Ignore the electron at 3s orbital

S = (7 x 0.35) + (2 x 1.0)

S = 2.45

So, the Z_eff = 11 - 2.45 = 8.55

The value of for 2p is the same as 2s, since they are in the same group.

The weakness of this method is if you try to calculate for 4s orbital of K, it will provide the same number as Na (2.2). This means the electrons at 3s of Na and 4s at K feel the same nuclear charge, therefore the ionisation energy should be the same. However in reality the 1st ionisation energy of Na is lower than K (495.8 and 418.0 kJ mol^-1 respectively).

The value of Z_eff affects the periodicity, which creates a not-smooth pattern across the periodic table. The pattern is caused by the shell structure of the atom itself. As across the table, the nuclear charge increases, so the Z_eff also increases, which means the electron is held more tightly. In this section, we will discussed about the electronegativity.

Before we discuss about electronegativity, it might be better to have a look about the bonding first. As we know, the electron behave as wave function of electron distribution or electron cloud. A bonding is basically a combining between 2 electron cloud or wave function. The electron bonding is in the between 2 nuclei of atoms, so it can act as a shield between 2 positive nuclei. If we describe the bond formation in term of energy we can get a graph between energy and radius or distance between 2 nuclei.

Therefore, the the radius where the energy is at the lowest point is called the covalent bond radius. For covalent compound, if it is a pure covalent the bonding electron will be equally distributed, but if the atom is different it will not be equally distributed. The atom with high electronegativity will have higher density of electron as shown in figure below.

Therefore, we can define electronegativity as the tendency for an atom in a molecule to accept or donate electron. In this section we only discuss about the Pauling scale (χ^P) and briefly about Mulliken (χ^M).

Pauling scale is based on his observation about dissociation energy of molecule. He observed that bond energy of XY [D(XY)] are not simply the average bond energy of X₂ [D(XX)] and Y₂ [D(YY)].

Then, Pauling formulated the difference between experimental D(XY) and the average of dissociation energy of X₂ and Y₂ as ΔD. Therefore, the formula is:

The value of bond energy are measured in electron-volts (eV; 1 eV = 96.5 kJ mol^-1). Then, Pauling proposed that ΔD is a measure of the ionicity (ionic nature) of the bond between X and Y, and this is itself a consequence of the different abilities of atoms X and Y to attract electrons to themselves.

He proposed that ΔD should be the square of the difference in electronegativity between X and Y, so:

Moreover, he fixed the value for fluorine atom as the most electronegative atom with the value of 4 and fitted the rest, giving a scale that attaches to every element a number between 4 and 0. The strength of Pauling's scale is simple, empirical (based on experiment), and intuitive. However, it cannot describe why the square of root is used, so it is not relate-able to. In the other sides, the Mulliken's scale is based on the avearge of the electron affinity and the ionisation energy of the atom.

In addition, the value of Mulliken's scale is close to the Pauling's scale, and the Mulliken's scale can be empirical or calculated.