1. Electronic Structure of Atoms
Glenn V. Lo
Department of Physical Sciences
Nicholls State University

2. Modern Quantum Theory
Bohr’s concept of a well-defined orbit is no longer acceptable; it is inconsistent with Heisenberg’s uncertainty principle
Modern Quantum Theory is consistent with
“energy quantization” as proposed by Bohr
The Heisenberg Uncertainty Principle
Modern Quantum Theory proposes a probabilistic model for atoms.
Erwin Schrodinger: use “wave functions”
Orbital = mathematical function that allows us to describe the probabilities of locating electrons around nucleus; represented by  (psi).
Probability per unit volume = ||2

3. Atomic Orbitals
Atomic orbitals are conveniently expressed as a function of location in spherical polar coordinates: distance from the nucleus (r) and direction (theta and phi).  =  (r, , ) Relation to Cartesian Coordinates: z = r cos() x = r sin() cos() y = r sin() sin()

4. Radial Distribution Functions
Figure shows probability vs. distance for “1s” and “2s” orbitals of H.
Most probable distance of electron in 1s orbital corresponds to the radius of Bohr’s first orbit: 52.9 pm
Bohr’s theory says: H is most stable with electron restricted to orbit 52.9 pm from nucleus. Modern quantum theory says: electron can be anywhere, but 52.9 pm is the most probable distance from the nucleus.

5. 3-D Representation
Electron clouds are used to visually represent orbitals. Regions with thicker cloud --- higher probability of finding electron. Follow this link:
Examples: 1s and 2s orbitals
spherically symmetric (independent of direction)
The 2s orbital has a radial node --- a spherical surface where probability goes to zero; ns orbital has (n-1) radial nodes.

6. 3-D Representation
A boundary surface is often drawn such that the probability of finding the electron (being described by the orbital) is low outside the surface. Algebraic sign + or -) of  is oftentimes indicated, or color coded.
Shape depends on how  depends on  and .
Size depends on how  depends on r.
Examples: 1s, 2s, 2px, 2py, and 2pz orbitals

7. Web Resource
Website with visual and mathematical representations for orbitals

8. Orbital names
Each atom has a set of orbitals, which can be grouped into “shells” or “levels.” The shell or level number is called “n” or the “principal quantum number”
One orbital in n=1: 1s
For n=2, there are 2 types of orbitals, 4 in all s, 2p (set of 3)
For n=3, there are 3 types of orbitals, 9 in all s, 3p (set of 3), 3d (set of 5)
In general, there are n types of orbitals.
Subshell or sublevel = orbitals of a given type in a given level. There is one subshell in n=1: 1s There are two subshells in n=2: 2s and 2p

9. Principal Quantum Number
Each atom has a set of orbitals, which can be grouped into “shells” or “levels.” The shell or level number is called “n” or the “principal quantum number”
One orbital in first shell (n=1): 1s
For n=2, there are 2 types of orbitals, 4 in all:
2s, 2p (set of 3)
For n=3, there are 3 types of orbitals, 9 in all:
3s, 3p (set of 3), 3d (set of 5)
In general there are n types of orbitals and n2 orbitals in level n.
In general, orbital with larger n describes an electron that is, on average, farther away from the nucleus and has more energy.

10. Orbital Quantum Number
Orbital or azimuthal quantum number, represented by the letter ell in script font (l), is an alternative way of specifying the type of orbital or subshell. For s orbitals: l = 0 For p orbitals: l = 1 For d orbitals: l = 2 For f orbitals: l = 3 For g orbitals: l = 4
Highest allowed value of l for a given level is n-1. Example: for n=4, possible l values are 0, 1, 2, 3 “4s” orbital corresponds to n=4, l =0 “4p” orbital corresponds to n=4, l =1

11. Magnetic Quantum Number
Magnetic quantum number, represented by the letter m or “em sub ell” (ml), distinguishes orbitals within a subshell. Example: consider a p subshell
The p subshell consists of 3 orbitals
One orbital each corresponds to m = -1, 0, and +1
In general, for a given subshell (l), there are (2l+1) possible m values: - l to +l. Example: for a d subshell,
There are 5 orbitals. Possible m values = -2, -1, 0, 1, 2

12. Magnetic Quantum Number
Why is m called the “magnetic quantum number”?
Motion of electron around the nucleus generates a magnetic field. Any type of curved motion of a charged particle generates a magnetic field. In the animation, imagine sphere is an electron: the green vertical arrow is called the “orbital angular momentum vector” and is pointing to the south pole of magnetic field generated by its motion.

13. Magnetic Quantum Number
Each m value corresponds to one allowed orientation of the magnetic field. For p orbital, imagine an invisible bar magnet inside the atom that is oriented in one of three ways shown here.

14. Spin Quantum Number
Magnetic behavior of atoms suggest that electrons have an “intrinsic” magnetism, in addition to the magnetism that is generated by their motion around the nucleus.
Intrinsic magnetism is attributed to “spin angular momentum” of electron. Theory becomes consistent with experimental results if we assume it can be oriented in one of only two ways. Imagine the two possibilities by imagining an invisible bar magnet which can be oriented as shown on the right.
Two possible orientations of spin are called “spin up” or “spin down”, or have spin quantum number ms=+1/2 or –1/2.

15. Pauli’s Exclusion Principle
An electron in an atom is completely described by specifying
the orbital that describes the probability of finding it around the atom,
its spin
Or by specifying four quantum numbers (n, l, m, ms)
Pauli’s Exclusion Principle: no two electrons in an atom can have the same set of quantum numbers.
For a given orbital (n, l, and m), there can be only two possible ms values… +1/2 and –1/2
Maximum number of electrons assigned to an orbital is, therefore, two. One “spin up,” the other “spin down.”

16. Orbital energies
In a hydrogen atom: energy of electron depends only on n, the “principal quantum number, which is the number in the name of the orbital. Higher n, higher energy.
Ground state refers to the lowest energy state. Ground state of H is described by the 1s orbital (n=1)
Excited states. Higher energy states. A hydrogen atom with its electron in a 2s orbital (n=2) is in an “excited state”
Excited states are unstable. Eventually, atoms find their way back to ground state --- by emitting light, or losing energy through collisions with other atoms.
For Hydrogen, modern quantum theory predicts: E = eV / n2; essentially the Bohr’s result

17. Orbital energies
In atoms with more than one electron: energy of electron depends on n+l,
the higher the n+l value, the higher the energy;
for orbitals with same n+l value, higher n has higher energy
Example: 3d vs. 4s vs. 4p For 3d, n+l =3+2 = 5 For 4s, n+l =4+0 = 4 For 4p, n+l =4+1 = 5
Orbital energies: 4s < 3d < 4p

18. Orbital energies
In atoms with more than one electron, the typical relative orbital energies are
1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < etc.

19. Orbital energies * 1s
Relate the sequence below to the memory aid shown on the right, which resembles the layout of the periodic table.
1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < ? < ? < ? < ? 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 5d 6p
*4f

20. Electron Configuration
Electron configuration: a listing of subshells assigned to electrons in an atom. Example: He (two electrons)
Ground state configuration: 1s2
Excited state configuration: 1s1 2s1
Another excited state configuration: 1s1 2p1
Unless otherwise specified, “electron configuration” refers to ground state.

21. Pauli’s Exclusion Principle
Pauli’s Exclusion Principle: (postulate)
no more than two electrons per orbital
two electrons in an orbital must have opposite spins.
How many electrons can be assigned to
A p orbital? 2
A p subshell? 6 (two per orbital x 3 orbitals)

22. Orbital Diagram
Ground state electron configuration can be obtained by:
Filling low-energy orbitals first (Aufbau principle)
Filling orbitals in a sublevel singly, with parallel spins, before pairing up (Hund’s rule)
Orbital diagram of Oxygen (8 electrons) is shown here. Ground state electron configuration is
1s2 2s2 2p4

23. Ground Configuration Patterns
Valence shell = outermost shell
Valence Configurations of Main Groups
Group IA: s1
Group IIA: s2
Group IIIA: s2p1
Group IVA: s2p2
Group VA: s2p3
Group VIA: s2p4
Group VIIA: s2p5
Noble gases: completely filled s2p6 (or 1s2 for He)

24. Patterns
Transition Metals: if n is the valence shell, the ground state electron configuration is
[Noble gas core] ns2 (n-1)dx
Where x=1 to 10
There are exceptions: half-filled subshells seem to be preferred. Ex. Cr

25. Web Resources