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 and momentum: ∆ x ∆ p x ≥ 1 2 � with � = h 2 π . 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
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 h ∆ x ∆ p x ∼ h > 4 π Kick ∆ p x ∼ h /λ Marc R. Roussel Hydrogenic orbitals September 10, 2018 3 / 24
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 ∆ p x ≥ 1 2 � , we have � 2∆ x = 5 × 10 − 25 kg m s − 1 , ∆ p x ≥ or ∆ v ≥ ∆ p x = 6 × 10 5 m / s . m e 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 infinite uncertainty. � (∆ p r = 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 ψ 2 dV = probability of finding the particle in a small volume dV . ψ wavefunction at this point = Volume dV 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): E n = − Z 2 n 2 R H ℓ : orbital angular momentum quantum number Size of orbital angular momentum vector ( L ) depends on ℓ : L 2 = ℓ ( ℓ + 1) � 2 m ℓ : magnetic quantum number z component of L depends on m ℓ : L z = m ℓ � 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,. . . ) ℓ can only take values between 0 and n − 1 ℓ 0 1 2 3 4 5 . . . code s p d f g h . . . m ℓ can only take values between − ℓ and ℓ The orbitals are therefore the following: ℓ subshell number of orbitals n m ℓ 1 0 1s 0 1 2 0 2s 0 1 2 1 2p − 1, 0 or 1 3 3 0 3s 0 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
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 , 2p 0 and 2p 1 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 m ℓ (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 , ℓ, m ℓ ) 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 m ℓ . 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 m ℓ , and that are real-valued. Marc R. Roussel Hydrogenic orbitals September 10, 2018 10 / 24
Hydrogenic orbitals Electron density maps n = 1 1.5 1s 1 0.5 z z 0 -0.5 -1 -1-0.5 0 0.5 1 1.5 y -1.5 -1 -0.5 0 0.5 1 x 1.5 Marc R. Roussel Hydrogenic orbitals September 10, 2018 11 / 24
Hydrogenic orbitals Electron density maps n = 2, ℓ = 0 2 s 8 6 4 2 z z 0 -2 -4 -6 8 -8 4 0 -8 -6 -4 -2 0 2 4 6 8 -8 y -4 x Marc R. Roussel Hydrogenic orbitals September 10, 2018 12 / 24
Hydrogenic orbitals Electron density maps n = 2, ℓ = 1 2 p x 8 2 p y 6 8 4 6 2 4 z z 0 2 z z -2 0 -4 -2 -6 -4 -8 -8 -6 -4 4 0 0 -8 y 4 -4 x -8 8 6 -6 4 -4 2 8 -8 0 -2 0 -2 2 -4 4 -6 8 6 -8 x y 2 p z 8 6 4 2 z z 0 -2 -4 -8 -6 -4 0 -8 y 4 8 6 4 2 8 0 -2 -4 -6 -8 x Marc R. Roussel Hydrogenic orbitals September 10, 2018 13 / 24
Hydrogenic orbitals Electron density maps n = 3, ℓ = 0 3 s 20 10 z z 0 -10 -20 -20 -10 0 10 20 -20 y -10 0 10 20 x Marc R. Roussel Hydrogenic orbitals September 10, 2018 14 / 24
Hydrogenic orbitals Electron density maps n = 3, ℓ = 1 3 p x 3 p y 20 20 10 z z 10 0 z z 0 -10 -10 -20 20 20 10 -20 0 y 0 -10 -20 x -20 -20 -10 -20 -10 0 0 10 20 10 20 x y 3 p z 20 10 z z 0 -10 20 -20 0 -20 y -20 -10 0 10 20 x Marc R. Roussel Hydrogenic orbitals September 10, 2018 15 / 24
Hydrogenic orbitals Electron density maps n = 3, ℓ = 2 20 20 20 3 d xy 3 d xz 3 d yz 10 10 10 0 0 0 y z z -10 -10 -10 -20 -20 -20 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 x x y 3 d z 2 20 3 d x 2 -y 2 20 10 10 0 y z z 0 -10 -10 20 -20 -20 0 y -20 -20 -10 0 10 20 -20 -10 0 10 20 x x 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 illustrations 1 s orbital Marc R. Roussel Hydrogenic orbitals September 10, 2018 18 / 24
Hydrogenic orbitals Hydrogenic orbital illustrations 2 s orbital Marc R. Roussel Hydrogenic orbitals September 10, 2018 19 / 24
Hydrogenic orbitals Hydrogenic orbital illustrations 2 p z orbital Marc R. Roussel Hydrogenic orbitals September 10, 2018 20 / 24
Hydrogenic orbitals Hydrogenic orbital illustrations 3 s orbital Marc R. Roussel Hydrogenic orbitals September 10, 2018 21 / 24
Hydrogenic orbitals Hydrogenic orbital illustrations 3 p z orbital Marc R. Roussel Hydrogenic orbitals September 10, 2018 22 / 24
Hydrogenic orbitals Hydrogenic orbital illustrations 3 d x 2 − y 2 orbital Marc R. Roussel Hydrogenic orbitals September 10, 2018 23 / 24
Hydrogenic orbitals Hydrogenic orbital illustrations 3 d z 2 orbital Marc R. Roussel Hydrogenic orbitals September 10, 2018 24 / 24
Recommend
More recommend
