Allowed values for quantum numbers

n             1,2,3....

l                0, 1, 2, ...(n-1)

m_l             +l, +(l -1),... -l

Special labels for quantum number l

l = 0     s

     1     p

     2     d

     3     f

Orbitals in atoms with more than one electron.

We assume that they look like hydrogen orbitals, but are distorted somewhat by the electrostatic repulsions of the other electrons and by the higher-charged nucleus.

Click here  for some pictures of hydrogen-like orbitals: note the different numbers and types of nodes in these.

The nuclear charge experienced by a particular electron depends upon how many electrons lie between it and the nucleus.

Electrons experience an "effective nuclear charge"

Z_eff = Z - S

where S represents the "screening effect" of the inner electrons.

A fourth quantum number

m_s = +½ or -½

Meaning?

Electrons have the property of possessing an intrinsic angular momentum ("spin"). This is in addition to orbital angular momenta that they may have when l is not zero. The spin generates a magnetic moment for the electron.

In the presence of an external magnetic field, the electron's own magnetism can be aligned with or against that field, and this corresponds to the two values of m_s.
[Review description of the Stern-Gerlach experiment, Fig. 6.26]

Occupation of orbitals in many-electron atoms is controlled by the

Pauli Exclusion Principle = No two electrons in an atom can have the same set of four quantum numbers.

"Meaning" of an electron's quantum numbers.

n        "distance from the nucleus"

l        "shape of orbital" (1,2,4...lobes). Electron property: orbital angular momentum

m_l     "direction of orbital" (with respect to a defined axis) : component of orbital angular momentum wrt axis

m_s    "electron spin"

The Pauli Exclusion Principle implies that a single orbital can "hold" only two electrons with opposite spins (m_s = +½ and m_s = -½)

This can be represented by

"spin-up" and "spin-down" arrows in a box.

We use a different box for each set of n, l, and m_l values

The electron configuration of an atom summarizes the quantum numbers for all the electrons.

We can represent this information in two ways.

E.g. for the carbon atom, ₆C the six electrons occupy 1s, 2s, and 2p orbitals
The electron configuration is written as 1s²2s²2p²

or in "box" form
 

1s  (2e)

2s  (2e)

2px  (1 e)

2py  (1 e)

2pz  (0 e)

Note: the three boxes for "p" orbitals correspond to the three values of m_l (+1, 0, -1). Because of electron-electron repulsions, electrons "prefer" to avoid one another when possible, and to have their spins parallel. (Hund's Rule of maximum multiplicity)

Electron configurations of atoms can be deduced from the position of the atom in the Periodic Table. The configurations are based on the "Aufbau" (Build-up) Principle.

Orbitals are arranged in the order of increasing energy and the necessary number of electrons are added to the lowest-energy orbitals using the Pauli Exclusion Principle and Hund's Rule.

In the hydrogen atom the orbital energies are 1s<2s=2p<3s=3p=3d<4s ....

But the ground state of Be is 1s²2s² and not 1s²2s¹2p¹

This shows that the energy of 2s < 2p in a Be atom.

Why?

The effective nuclear charge, Z_eff, felt by a 2s electron in Be is greater than that felt by a 2p electron.
We say that the 2p electron is more "screened" from the nucleus by the inner core of 1s electrons.
We also say that the 2s electron "penetrates" the core electrons and experiences a greater nuclear charge.

The arrangement of the atoms in the Periodic Table shows that all "ns" electrons are more tightly bound than are the "np" electrons.After every inert gas atom (which has a complete shell of s and p electrons) is an alkali metal atom with an ns¹ electron configuration.

"d" and "f" electrons are even less penetrating (more screened) than "p" and "s" electrons.

After the 3p subshell is filled at Argon, the next electron occupies a 4s orbital (K). Only when the 4s is full, do electrons begin to occupy the 3d orbitals.

Another look at the Periodic Table (a link to loads of periodic table websites)

Early 19th century:
groups of similar elements were recognized:
Ca,Sr,Ba     Cl,Br,I      Li,Na,K ("Döbereiner's triads")

Later: the known elements were arranged in the order of their atomic weights. Newlands, Mendeleev, and Lothar Meyer noted periodic similarities

Mendeleev published one form of the Periodic Table in 1869. Blank spaces for elements yet to be discovered.
Properties predicted for six of these (three were discovered in his lifetime).

"eka-boron" (scandium)
"eka-aluminum" (gallium)
"eka-silicon" (germanium)
"eka-manganese" (technetium)
"dvi-manganese" (rhenium)
"eka-tantalum" (polonium)

After the Mendeleev Table, the rare gases were discovered ("group 0"), and the concept of atomic number was developed.