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Chemistry 1000 Lecture 7: Hydrogenic orbitals
Marc R. Roussel
September 10, 2018
Marc R. Roussel Hydrogenic orbitals September 10, 2018 1 / 24
Uncertainty principle
Heisenberg uncertainty principle
Fundamental limitation to simultaneous measurements of position andmomentum:
∆x∆px ≥1
2~
with ~ =h
2π.
Uncertainty is, roughly, the experimental precision of themeasurement.
Position and momentum can’t simultaneously both be known toarbitrary accuracy.
Marc R. Roussel Hydrogenic orbitals September 10, 2018 2 / 24
Uncertainty principle
Why not?
Suppose that we want to locate an object in a microscope.
Photons reflect (or refract) from the sample.Photons have momentum so they give the object a “kick” (i.e. changethe momentum) during interaction with an object.
Resolution ∆x ∼ λKick ∆px ∼ h/λ
}∆x∆px ∼ h > h
4π
Marc R. Roussel Hydrogenic orbitals September 10, 2018 3 / 24
Uncertainty principle
Example: Suppose that we use X-rays to determine the position of anelectron to within 10−10 m (diameter of a hydrogen atom). Since
∆x∆px ≥1
2~, we have
∆px ≥~
2∆x= 5× 10−25 kg m s−1,
or
∆v ≥ ∆pxme
= 6× 105 m/s.
Marc R. Roussel Hydrogenic orbitals September 10, 2018 4 / 24
Uncertainty principle
Important consequence:
Bohr theory has orbits of fixed r , i.e. ∆r = 0.
The radial momentum component would then have to have infiniteuncertainty.
(∆pr =~
2∆r)
Infinite uncertainty in momentum not possible(sorry, Douglas Adams)∴ Bohr orbits not possible
Marc R. Roussel Hydrogenic orbitals September 10, 2018 5 / 24
Hydrogenic orbitals
Wavefunctions in modern quantum mechanics
Quantum systems are described by a wavefunction ψ.
Square of wavefunction = probability density
ψ2 dV = probability of finding the particle in a small volume dV .
dV
ψ
Volume
wavefunction at this point =
Orbital: one-electron wavefunction
Marc R. Roussel Hydrogenic orbitals September 10, 2018 6 / 24
Hydrogenic orbitals
Hydrogenic orbitals
Depend on three quantum numbers
n: principal quantum number
Total energy of atom depends on n (as in Bohr theory):
En = −Z 2
n2RH
`: orbital angular momentum quantum number
Size of orbital angular momentum vector (L) dependson `:
L2 = `(`+ 1)~2
m`: magnetic quantum number
z component of L depends on m`:
Lz = m`~
Marc R. Roussel Hydrogenic orbitals September 10, 2018 7 / 24
Hydrogenic orbitals
Rules for hydrogenic quantum numbers
n is a positive integer (1,2,3,. . . )
` can only take values between 0 and n − 1` 0 1 2 3 4 5 . . .
code s p d f g h . . .m` can only take values between −` and `
The orbitals are therefore the following:
n ` subshell m` number of orbitals
1 0 1s 0 12 0 2s 0 12 1 2p −1, 0 or 1 33 0 3s 0 13 1 3p −1, 0 or 1 33 2 3d −2, −1, 0, 1 or 2 5...
Marc R. Roussel Hydrogenic orbitals September 10, 2018 8 / 24
Hydrogenic orbitals
Degeneracy
All orbitals corresponding to the same value of n have the sameenergy.Different orbitals with the same energy are said to be degenerate.
Example: The 2s, 2p−1, 2p0 and 2p1 orbitals all correspond to n = 2and are degenerate in hydrogenic atoms.
The degeneracy between orbitals can be lifted by external fields.Example: A magnetic field removes the degeneracy between orbitalswith different values of m` (Zeeman effect).
Marc R. Roussel Hydrogenic orbitals September 10, 2018 9 / 24
Hydrogenic orbitals
Real-valued orbitals
The orbitals corresponding to the quantum numbers (n, `,m`) arecomplex-valued, i.e. they involve i =
√−1.
In many cases, there is no distinguished z axis, and therefore noparticular meaning to the quantum number m`.
We can replace the original set of orbitals with ones corresponding tothe same values of n and ` (so same energy and angular momentumsize), but that don’t correspond to any particular value of m`, andthat are real-valued.
Marc R. Roussel Hydrogenic orbitals September 10, 2018 10 / 24
Hydrogenic orbitals
Electron density mapsn = 1
-1.5 -1 -0.5 0 0.5 1 1.5-1-0.5 0 0.5 1 1.5
-1
-0.5
0
0.5
1
1.5
z
1s
x
y
z
Marc R. Roussel Hydrogenic orbitals September 10, 2018 11 / 24
Hydrogenic orbitals
Electron density mapsn = 2, ` = 0
-8 -6 -4 -2 0 2 4 6 8-8
-4 0
4 8-8
-6-4-2 0 2 4 6 8
z
2s
x
y
z
Marc R. Roussel Hydrogenic orbitals September 10, 2018 12 / 24
Hydrogenic orbitals
Electron density mapsn = 2, ` = 1
-8-6-4-2 0 2 4 6 8
-8-4
0 4
8-8-6-4-2 0 2 4 6 8
z
2px
x
y
z
-8-4
0 4
-8-6-4-2 0 2 4 6 8
-8-6-4-2 0 2 4 6 8
z
2py
x
y
z
-8-6-4-2 0 2 4 6 8
-8-4
0 4
8-8-6-4-2 0 2 4 6 8
z
2pz
x
y
z
Marc R. Roussel Hydrogenic orbitals September 10, 2018 13 / 24
Hydrogenic orbitals
Electron density mapsn = 3, ` = 0
-20 -10 0 10 20-20 -10 0 10 20
-20
-10
0
10
20
z
3s
x
y
z
Marc R. Roussel Hydrogenic orbitals September 10, 2018 14 / 24
Hydrogenic orbitals
Electron density mapsn = 3, ` = 1
-20 -10 0 10 20-20
0 20
-20
-10
0
10
20
z
3px
x
y
z
-20-10 0 10 20
-20-10 0 10 20
-20
-10
0
10
20
z
3py
x
y
z
-20 -10 0 10 20-20
0 20
-20
-10
0
10
20
z
3pz
x
y
z
Marc R. Roussel Hydrogenic orbitals September 10, 2018 15 / 24
Hydrogenic orbitals
Electron density mapsn = 3, ` = 2
-20
-10
0
10
20
-20 -10 0 10 20
y
x
3dxy
-20
-10
0
10
20
-20 -10 0 10 20z
x
3dxz
-20
-10
0
10
20
-20 -10 0 10 20
z
y
3dyz
-20
-10
0
10
20
-20 -10 0 10 20
y
x
3dx2-y2
-20 -10 0 10 20-20
0 20
-20
-10
0
10
20
z
3dz2
x
y
z
Marc R. Roussel Hydrogenic orbitals September 10, 2018 16 / 24
Hydrogenic orbitals
Wavefunctions have a phase
The wavefunction has a phase, i.e. a sign.
The sign changes at nodal surfaces.
Diagrammatically, we represent the phase using color.
Marc R. Roussel Hydrogenic orbitals September 10, 2018 17 / 24
Hydrogenic orbitals
Hydrogenic orbital illustrations1s orbital
Marc R. Roussel Hydrogenic orbitals September 10, 2018 18 / 24
Hydrogenic orbitals
Hydrogenic orbital illustrations2s orbital
Marc R. Roussel Hydrogenic orbitals September 10, 2018 19 / 24
Hydrogenic orbitals
Hydrogenic orbital illustrations2pz orbital
Marc R. Roussel Hydrogenic orbitals September 10, 2018 20 / 24
Hydrogenic orbitals
Hydrogenic orbital illustrations3s orbital
Marc R. Roussel Hydrogenic orbitals September 10, 2018 21 / 24
Hydrogenic orbitals
Hydrogenic orbital illustrations3pz orbital
Marc R. Roussel Hydrogenic orbitals September 10, 2018 22 / 24
Hydrogenic orbitals
Hydrogenic orbital illustrations3dx2−y2 orbital
Marc R. Roussel Hydrogenic orbitals September 10, 2018 23 / 24
Hydrogenic orbitals
Hydrogenic orbital illustrations3dz2 orbital
Marc R. Roussel Hydrogenic orbitals September 10, 2018 24 / 24