Even Function
Definition and meaning of Even Function in chemistry.
An even function is a mathematical function f(x) that satisfies f(−x) = f(x), meaning its graph is symmetric about the y-axis. In chemistry, this property describes wavefunctions and vibrational modes that remain unchanged upon inversion through a molecule's center of symmetry.
In more detail
In quantum chemistry and spectroscopy, even functions are classified as gerade ("g," German for even), while odd functions (f(−x) = −f(x)) are ungerade ("u," odd). This parity governs which electronic transitions and vibrational modes are allowed. For centrosymmetric molecules, the mutual exclusion rule follows directly from this symmetry: vibrations with even (gerade) character are Raman active but IR inactive, while odd (ungerade) vibrations are IR active but Raman inactive, because the dipole moment operator is itself an odd function.
Key facts
| Field | Physical Chemistry |
|---|---|
| Mathematical condition | f(−x) = f(x) |
| Symmetry label | gerade (g) |
| Example orbitals | s and d orbitals |
The 1s atomic orbital wavefunction is an even function: inverting every point through the nucleus (r → −r) leaves ψ unchanged, so the 1s orbital has gerade (g) symmetry. The same holds for all s orbitals and for d orbitals.
Frequently asked questions
What is the difference between an even and an odd function in this context?
An even (gerade) function is unchanged by inversion through a center of symmetry, f(−x) = f(x), while an odd (ungerade) function changes sign, f(−x) = −f(x). Orbitals and vibrations are classified this way to predict spectroscopic selection rules.
Which atomic orbitals are even functions?
All s orbitals and d orbitals are even (gerade); p and f orbitals are odd (ungerade), changing sign upon inversion through the nucleus.