Table 2. The Quantum Numbers

Name	Symbol	Values
Principal Quantum Number	n	any integer from 1 to infinity
Azimuthal Quantum Number	l	any integer from 0 to n-1
Magnetic Quantum Number	ml	any integer from -l to +l
Spin Quantum Number	ms	+/- ½

Table 3. Summary of allowed combinations of quantum numbers for values n from 1 to 4.

n	ml	Subshell notation	Number of orbitals in the subshell	Number of electrons needed to fill subshell	Total number of electrons in subshell
1	0	1s	1	2	2
2	0	2s	1	2
2	1,0,-1	2p	3	6	8
3	0	3s	1	2
3	1,0,-1	3p	3	6
3	2,1,0,-1,-2	3d	5	10	18
4	0	4s	1	2
4	1,0,-1	4p	3	6
4	2,1,0,-1,-2	4d	5	10
4	3,2,1,0,-1,-2,-3	4f	7	14	32

Energies of orbitals and building up principles by electrons for many electron atoms

A common feature of all the approximate methods of solving this equation is the so-called single-electron approximation, i.e. the assumption that the wave function of a many-electron system can he represented in the form of the sum of the wave functions of the individual electrons. Even with this simplification, however, the solution of the Schrodinger equation for many-electron atoms and molecules requires a great volume of time-consuming calculations.

The electron configuration of an many electron atom describes the orbitals occupied by electrons on the atom. The basis of this prediction is a rule known as the aufbau principle (meaning “building up principle” in German), which assumes that electrons are added to an atom, one at a time, starting with the lowest energy orbital, until all of the electrons have been placed in an appropriate orbital.

Of great importance for determining the state of an electron in many-electron atom is the principle formulated by Wolfgang Pauli (the Pauli exclusion principle) according to which:

no two electrons in any given atom can have exactly the same set of four quantum numbers.

Two such electrons in the same orbital and having opposite (antiparallel) spins are called paired, in contrast to a single (i.e. unpaired) electron occupying orbital. From the Pauli exclusion principle follows that

each atomic orbital characterized by detinite values of n, l, and m can be occupied by not more than two electrons whose spins have opposite signs.

This can be represented symbolically as follows:

A shell or energy level is a group of orbitals whose radial distribution from the nucleus is approximately equal. The energy of the orbitals in a shell is not necessarily equal.

The three p-orbitals of the second shell (each containing a pair of electrons when full) are all at the same energy level. These p-orbitals therefore represent a separate subshell or sublevel of the second shell. The second shell can contain two subshells altogether, one containing an s-orbital and one containing three p-orbitals: a total of 2² (= 4) orbitals in the second shell. The total number of orbitals and of electrons in a shell and subshell can be defined as follows, Table 3.

Table 3. Expressions for the total number of electrons and orbitals in a shell and a subshell

Shell		Subshell
Number of		Number of
Orbitals	Electrons	Orbitals	Electrons
n2	2n2	2l+1	2(2l+1)

As mentioned above, the energy of an electron in any shell or subshell of a one-electron atom (H) or ion (He⁺ , Li²⁺ , etc.) is completely determined by the quantum number n. This means that all orbitals in the n=2 shell (i.e., 2s and all three 2p) have the same energy (are degenerate) in these species.

In many-electron atoms, however, the subshells within a given shell have different energies. An electron in an atom is not only attracted by the nucleus, but is also repelled by the elec­trons between the given electron and the nucleus. The internal elec­tron layers form, as it were, a peculiar screen that weakens the attrac­tion of the electron to the nucleus or screen the outer electron from the nuclear charge. The order of increasing orbital energies for multielectron atoms see at the figure below.

The Soviet scientist V. Klechkovsky found that the energy of an electron grows as the sum of these two quantum numbers increases, i.e. with increasing n + l. Accordingly, he formulated the following rules:

Klechkovsky's first rule: with an increase in the charge of the nucleus of an atom, the electron orbitals are filled consecutively from orbitals with a smaller value of the sum of the principal and orbital quantum numbers (n + l) to orbitals with a greater value of this sum. Klechkovsky's second rule: at identical values of the sum n+ l the orbitals are filled consecutively in the direction of the growth in the value of the principal quantum number n.

There are several important exceptions to the order above that you are expected to know. It is some d elements (for instance, chromium, molybdenum, the elements of the copper subgroup) whose atoms have only one s electron in the outer electron layer:

₂₄Cr: 1s²2s²2p⁶3s²3p⁶4s¹3d⁵ (instead of 1s²2s²2p⁶3s²3p⁶4s²3d⁴)

₂₉Cu: 1s²2s²2p⁶3s²3p⁶4s¹3d¹⁰ (instead of 1s²2s²2p⁶3s²3p⁶4s²3d⁹)

(Notice that the configurations above can be abbreviated as [Ar]4s¹3d⁵ and [Ar]4s¹3d¹⁰, respectively.)

₄₇Ag: [Kr]5s¹4d¹⁰ (instead of [Kr]5s²4d⁹)

The deviation from Klechkovsky's second rule we encountered in lanthanum and some others lantanides and actinides. Consider the first one as example. According to Klechkovsky's rules, the sublevel 4f (n = 4, l = 3) with the sum n + l equal to 7 and with the smallest possible value of the principal quantum number for this sum ought to be filled. Actually, however, in lanthanum (Z=57), which directly follows barium, a 5d electron appears instead of a 4f one, so that its electron configuration is 1s²2s²2p⁶3s²3p⁶3d¹⁰4s²4p⁶4d¹⁰5s²5p⁶5d¹6s². The building up of the sublevel 4f, to which the single 5d electron, which was present in the lanthanum atom, also transfers, actually does begin, however, in the element cerium (Z = 58) directly following lanthanum. The configuration of the cerium atom is 1s²2s²2p⁶3s²3p⁶3d¹⁰4s²4p⁶4d¹⁰4f²5s²5p⁶6s².

Electrons fill degenerate orbitals according to rules first stated by Friedrich Hund. Hund's rule can be summarized as follows:

the distribution of the electrons within the limits of an energy sublevel at which the absolute value of the total spin of the atom is maximum corresponds to the stable state of an atom.

This rule states that the maximum value of the total spin of an atom corresponds to the stable, i.e. unexcited state in which the atom has the smallest possible energy; at any other distribution of the electrons, the energy of the atom will be higher, so that it will be in an excited, unstable state.

For example, it is exactly the last electronic configuration of carbon atom is correct— it corresponds to the greatest possible value of the total spin of an atom (this signifies the sum of the spins of all the electrons in an atom; for the first two diagrams of the carbon atom this sum is zero, while for the third diagram it is unity).

Using Hund's rule, it is a simple matter to compile a diagram of the electron s

Fig. 11. Possible electronic configuration for carbon and real one for nitrogen

tructure for the atom of the elements carbon and nitrogen:

The configuration 1s²2s²2p² and 1s²2s²2p³ corresponds to this diagram.