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Principles of Chemical Science_Hydrogen Atom Wave functions (Orbitals) - Lec6

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Chemistry

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Chemistry

Lecture 6: Hydrogen Atom Wave functions (Orbitals) 1. Wavefunctions (Orbitals) for the Hydrogen Atom 2. Shape and Size of S and P Orbitals 3. Electron Spin and the Pauli Exclusion Principle

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5.111 Lecture Summary #6 Monday, September 15, 2014

Readings for today: Section 1.9 – Atomic Orbitals. Section 1.10 – Electron Spin, Section 1.11 –
The Electronic Structure of Hydrogen. (Same sections in 4th ed.)
Read for Lecture #7: Section 1.12 – Orbital Energies (of many-electron atoms), Section 1.13 –
The Building-Up Principle. (Same sections in 4th and 5th ed.)

Topics: I. Wavefunctions (Orbitals) for the Hydrogen Atom
II. Shape and Size of S and P Orbitals
III. Electron Spin and the Pauli Exclusion Principle

I. WAVEFUNCTIONS (ORBITALS) FOR THE HYDROGEN ATOM
Solving the Schrödinger Equation provides values for En and Ψ(r,θ,φ).

A total of 3 quantum numbers are needed to describe a wavefunction in 3D.

1. n ≡ principal quantum number
n =
determines binding energy (energy level or shell)

2. l ≡ angular momentum quantum number
l =
l is related to n, determines angular momentum, describes subshell, shape of orbital
largest value of l = n – 1

3. m ≡ magnetic quantum number
m=
m is related to l, determines behavior in magnetic field, describes the specific orbital

To describe an orbital, we need to use all three quantum numbers:
Ψnlm(r,θ,φ)
The wavefunction describing the ground state is .

Using the terminology of chemists:

The Ψ100 orbital is instead called the orbital.
n designates the shell or energy level (1,2,3…)
l designates the subshell (shape of orbital) (s, p, d, f…)
m designates orbital orientation (specific orbital) (px, py, pz…)

 =0⇒ orbital  = 1 ⇒ orbital  = 2 ⇒ orbital  = 3 ⇒ orbital

for  = 1: m = 0 is pz orbital, m = ±1 are the px and py orbitals

1

, State label wavefunction orbital H atom En H atom En[J]
n=1
 =0 ψ100 –2.18 × 10–18J
m=0
n=2
 =0 ψ200 -5.45 × 10–19J
m=0
n=2
 =1 ψ211 -5.45 × 10–19J
m = +1
n=2
 =1 210 ψ210 –RH/22 -5.45 × 10–19J
m=0
n= 2
 =1 21-1 ψ21-1 –RH/22 -5.45 × 10–19J
m = -1
What is the corresponding orbital for a 5,1,0 state?

For a hydrogen atom, orbitals with the same n
value have the same energy: E = -RH/n2.

≡ having the same energy

For any principle quantum number, n, there are
degenerate orbitals in hydrogen (or any other 1 electron atom).

IN THEIR OWN WORDS
MIT graduate student Benjamin Ofori-Okai discusses how energy levels relate to
research in nanoscale MRI (magnetic resonance imaging), a technique that allows
3-D imaging of biological molecules, such as proteins, and viruses.

Image from "Behind the Scenes at MIT”. The Drennan Education Laboratory. Licensed
under a Creative Commons Attribution-NonCommercial-ShareAlike License. 2

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Exam (elaborations)

Principles of Chemical Science_Hydrogen Atom Wave functions (Orbitals) - Lec6

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