Chemistry 1000 Lecture 7: Hydrogenic orbitals Marc R. Roussel - - PowerPoint PPT Presentation

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Chemistry 1000 Lecture 7: Hydrogenic orbitals Marc R. Roussel - - PowerPoint PPT Presentation

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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

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SLIDE 1

Chemistry 1000 Lecture 7: Hydrogenic orbitals

Marc R. Roussel September 10, 2018

Marc R. Roussel Hydrogenic orbitals September 10, 2018 1 / 24

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SLIDE 2

Uncertainty principle

Heisenberg uncertainty principle

Fundamental limitation to simultaneous measurements of position and momentum: ∆x∆px ≥ 1 2 with = h 2π. Uncertainty is, roughly, the experimental precision of the measurement. Position and momentum can’t simultaneously both be known to arbitrary accuracy.

Marc R. Roussel Hydrogenic orbitals September 10, 2018 2 / 24

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SLIDE 3

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. change the momentum) during interaction with an object. Resolution ∆x ∼ λ Kick ∆px ∼ h/λ

  • ∆x∆px ∼ h >

h 4π

Marc R. Roussel Hydrogenic orbitals September 10, 2018 3 / 24

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SLIDE 4

Uncertainty principle

Example: Suppose that we use X-rays to determine the position of an electron 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,
  • r

∆v ≥ ∆px me = 6 × 105 m/s.

Marc R. Roussel Hydrogenic orbitals September 10, 2018 4 / 24

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SLIDE 5

Uncertainty principle

Important consequence: Bohr theory has orbits of fixed r, i.e. ∆r = 0. The radial momentum component would then have to have infinite uncertainty. (∆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

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SLIDE 6

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

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SLIDE 7

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 n2 RH

ℓ: orbital angular momentum quantum number

Size of orbital angular momentum vector (L) depends

  • n ℓ:

L2 = ℓ(ℓ + 1)2

mℓ: magnetic quantum number

z component of L depends on mℓ: Lz = mℓ

Marc R. Roussel Hydrogenic orbitals September 10, 2018 7 / 24

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SLIDE 8

Hydrogenic orbitals

Rules for hydrogenic quantum numbers

n is a positive integer (1,2,3,. . . ) ℓ can only take values between 0 and n − 1 ℓ 1 2 3 4 5 . . . code s p d f g h . . . mℓ can only take values between −ℓ and ℓ The orbitals are therefore the following: n ℓ subshell mℓ number of orbitals 1 1s 1 2 2s 1 2 1 2p −1, 0 or 1 3 3 3s 1 3 1 3p −1, 0 or 1 3 3 2 3d −2, −1, 0, 1 or 2 5 . . .

Marc R. Roussel Hydrogenic orbitals September 10, 2018 8 / 24

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SLIDE 9

Hydrogenic orbitals

Degeneracy

All orbitals corresponding to the same value of n have the same energy. Different orbitals with the same energy are said to be degenerate. Example: The 2s, 2p−1, 2p0 and 2p1 orbitals all correspond to n = 2 and are degenerate in hydrogenic atoms. The degeneracy between orbitals can be lifted by external fields. Example: A magnetic field removes the degeneracy between orbitals with different values of mℓ (Zeeman effect).

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SLIDE 10

Hydrogenic orbitals

Real-valued orbitals

The orbitals corresponding to the quantum numbers (n, ℓ, mℓ) are complex-valued, i.e. they involve i = √−1. In many cases, there is no distinguished z axis, and therefore no particular meaning to the quantum number mℓ. We can replace the original set of orbitals with ones corresponding to the same values of n and ℓ (so same energy and angular momentum size), but that don’t correspond to any particular value of mℓ, and that are real-valued.

Marc R. Roussel Hydrogenic orbitals September 10, 2018 10 / 24

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SLIDE 11

Hydrogenic orbitals

Electron density maps

n = 1

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

  • 1-0.5 0 0.5 1 1.5
  • 1
  • 0.5

0.5 1 1.5 z 1s x y z

Marc R. Roussel Hydrogenic orbitals September 10, 2018 11 / 24

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SLIDE 12

Hydrogenic orbitals

Electron density maps

n = 2, ℓ = 0

  • 8 -6 -4 -2 0 2 4 6 8 -8
  • 4

4 8

  • 8
  • 6
  • 4
  • 2

2 4 6 8 z 2s x y z

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SLIDE 13

Hydrogenic orbitals

Electron density maps

n = 2, ℓ = 1

  • 8
  • 6
  • 4
  • 2

2 4 6 8

  • 8
  • 4

4 8

  • 8
  • 6
  • 4
  • 2

2 4 6 8 z 2px x y z

  • 8
  • 4

4

  • 8
  • 6
  • 4
  • 2

2 4 6 8

  • 8
  • 6
  • 4
  • 2

2 4 6 8 z 2py x y z

  • 8
  • 6
  • 4
  • 2

2 4 6 8

  • 8
  • 4

4 8

  • 8
  • 6
  • 4
  • 2

2 4 6 8 z 2pz x y z

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SLIDE 14

Hydrogenic orbitals

Electron density maps

n = 3, ℓ = 0

  • 20
  • 10

10 20

  • 20 -10 0 10 20
  • 20
  • 10

10 20 z 3s x y z

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SLIDE 15

Hydrogenic orbitals

Electron density maps

n = 3, ℓ = 1

  • 20
  • 10

10 20

  • 20

20

  • 20
  • 10

10 20 z 3px x y z

  • 20
  • 10

10 20

  • 20
  • 10

10 20

  • 20
  • 10

10 20 z 3py x y z

  • 20
  • 10

10 20

  • 20

20

  • 20
  • 10

10 20 z 3pz x y z

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SLIDE 16

Hydrogenic orbitals

Electron density maps

n = 3, ℓ = 2

  • 20
  • 10

10 20

  • 20
  • 10

10 20 y x 3dxy

  • 20
  • 10

10 20

  • 20
  • 10

10 20 z x 3dxz

  • 20
  • 10

10 20

  • 20
  • 10

10 20 z y 3dyz

  • 20
  • 10

10 20

  • 20
  • 10

10 20 y x 3dx2-y2

  • 20
  • 10

10 20

  • 20

20

  • 20
  • 10

10 20 z 3dz2 x y z

Marc R. Roussel Hydrogenic orbitals September 10, 2018 16 / 24

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SLIDE 17

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.

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SLIDE 18

Hydrogenic orbitals

Hydrogenic orbital illustrations

1s orbital

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SLIDE 19

Hydrogenic orbitals

Hydrogenic orbital illustrations

2s orbital

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SLIDE 20

Hydrogenic orbitals

Hydrogenic orbital illustrations

2pz orbital

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SLIDE 21

Hydrogenic orbitals

Hydrogenic orbital illustrations

3s orbital

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SLIDE 22

Hydrogenic orbitals

Hydrogenic orbital illustrations

3pz orbital

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SLIDE 23

Hydrogenic orbitals

Hydrogenic orbital illustrations

3dx2−y2 orbital

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SLIDE 24

Hydrogenic orbitals

Hydrogenic orbital illustrations

3dz2 orbital

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