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UPSC Civil Services (Mains)

Chemistry optional

  • 2 papers
  • 250 marks each
  • 3 h per paper
  • 5 of 8 questions, Q1 and Q5 compulsory

Paper I covers physical and inorganic chemistry.

UPSC Civil Services (Mains), Chemistry optional · 2016 · Paper I

Q3(c). Angular wave functions of the d orbitals and their signs

  • 10 marks
  • Answered
Question

What are the angular wave functions of d-orbitals? Explain them, including their geometric signs, with the help of neat and labelled diagrams.

Answer in brief

For l=2 the angular part Y(θ,φ) of ψ=Rₙ₂(r)Y(θ,φ) takes five real forms: d_(z²)∝3²θ-1, d_(xz)∝θθφ, d_(yz)∝θθφ, d_(xy)∝²θ2φ, d_(x²-y²)∝²θ2φ. Their sign changes across two angular nodes, so lobes alternate + and −.

Key points

  • The hydrogen-like wave function separates as ψₙₗₘ=Rₙₗ(r) Yₗₘ(θ,φ). The angular part Y fixes the shape and orientation of the orbital, the same for 3d, 4d and 5d. The complex functions with mₗ=±1,±2 are combined into real ones…
  • The five real functions (with normalisation constants; x=rθφ, y=rθφ, z=rθ):
  • Signs. The sign is the sign of Y, read from the Cartesian form: d_(xy)∝ xy is + where x and y have the same sign (first and third quadrants) and − in the other two. d_(x²-y²) is + along x, − along y. d_(z²)∝2z²-x²-y² is +…
  • Nodes. Each d orbital has l=2 angular nodes: two planes for the four-lobed orbitals, two cones (²θ=13) for d_(z²). The sign changes on crossing every node; opposite lobes carry the same sign (gerade).

The full model answer

Written out in full, as it should go in the answer booklet. The solution page with its diagrams follows it.

  1. The hydrogen-like wave function separates as ψₙₗₘ=Rₙₗ(r) Yₗₘ(θ,φ). The angular part Y fixes the shape and orientation of the orbital, the same for 3d, 4d and 5d. The complex functions with mₗ=±1,±2 are combined into real ones that point along or between the axes. The panel shows polar plots of |Y| in one plane, + shaded and − hatched.
  2. The five real functions (with normalisation constants; x=rθφ, y=rθφ, z=rθ):
OrbitalY(θ,φ)LobesAngular nodes
d_(z²)0.3154 (3²θ-1)+ along ± z; − torus in xy planetwo cones, θ=54.74° and 125.26°
d_(xz)1.0925 θθφbetween x and zxy and yz planes
d_(yz)1.0925 θθφbetween y and zxy and xz planes
d_(xy)0.5463 ²θ2φbetween x and yxz and yz planes
d_(x²-y²)0.5463 ²θ2φalong ± x (+) and ± y (−)planes x=± y
  1. Signs. The sign is the sign of Y, read from the Cartesian form: d_(xy)∝ xy is + where x and y have the same sign (first and third quadrants) and − in the other two. d_(x²-y²) is + along x, − along y. d_(z²)∝2z²-x²-y² is + along z and − in the ring.
  2. Nodes. Each d orbital has l=2 angular nodes: two planes for the four-lobed orbitals, two cones (²θ=13) for d_(z²). The sign changes on crossing every node; opposite lobes carry the same sign (gerade).

Application. the signs decide overlap: in an octahedral complex d_(x²-y²) and d_(z²) point at the ligands, form σ bonds and become e_g; d_(xy), d_(xz), d_(yz) can only π-bond (t_(2g)).

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Model answer to Q3(c), 2016 Paper I: solution page 18
Solution page 18 of 46
Read the whole 2016 solution2016 question paper

All parts of Question 3