Draw pictures of the two d orbitals in the x y-plane as flat...

Draw pictures of the two d orbitals in the $x y$-plane as flat projections in the plane of the paper. Label each drawing with the appropriate mathematical function, and include a labelled pair of Cartesian coordinate axes. Label the orbital lobes correctly with $+$ and - signs.

Key Concepts

Atomic Orbitals

Atomic orbitals are mathematical functions that describe the probability distribution of an electron in an atom. They represent regions in space where there is a high likelihood of finding an electron, and are derived from solutions of the Schrödinger equation.

d Orbitals

d orbitals are a set of five atomic orbitals corresponding to the angular momentum quantum number l = 2. Their complex shapes arise from the angular part of the wavefunctions, and they have distinct orientations and nodal patterns that determine their spatial distributions.

Real Spherical Harmonics

Real spherical harmonics are the mathematical functions used to express the angular portion of atomic orbitals in a real form. For d orbitals, these functions define the shapes and orientations of the orbitals, such as identifying lobes and nodal planes in specific directions.

Phase and Sign of Wavefunctions

The phase (often indicated by plus and minus signs) of a wavefunction in different regions is crucial as it determines the constructive or destructive interference when orbitals combine. In orbital drawings, this phase information is represented by labeling the lobes with positive and negative signs.

Cartesian Coordinate System

The Cartesian coordinate system is used to depict the spatial orientation of atomic orbitals by providing a set of axes (typically x, y, and z). This system allows for accurate representation of the orbital shapes and the positioning of their lobes relative to each axis.

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