Even Function
Definition and meaning of Even Function in chemistry.
An even function is a mathematical function that graphs perfectly symmetrically across the vertical y-axis. In strict mathematical terms, it satisfies the specific condition where the value f(−x) exactly equals f(x). In chemistry, this concept describes quantum wavefunctions and molecular vibrations that remain unchanged when spatially inverted.
In more detail
Symmetry plays a massive role in understanding quantum chemistry and modern molecular spectroscopy. When describing the shapes of orbitals or molecular vibrations, chemists classify these mathematical functions by parity. Even functions are officially labeled with the lowercase letter g, which stands for the German word gerade.
Odd functions change their mathematical sign upon spatial inversion and get the label u for ungerade. This simple parity classification completely governs which electronic transitions and vibrational movements are physically allowed to happen. For symmetrical molecules with a clear center, this leads directly to the famous mutual exclusion rule.
This rule states that molecular vibrations with even character are easily seen in Raman spectroscopy experiments. However, these exact same even vibrations remain completely invisible during standard infrared absorption spectroscopy. Conversely, the odd vibrations will show up strongly in infrared scans but remain invisible in Raman tests.
This happens because the mathematical dipole moment operator used in infrared physics is an odd function. Multiplying an even molecular vibration by this odd dipole operator produces a forbidden odd mathematical result. Understanding these even functions helps chemists interpret confusing spectral graphs and identify unknown molecular geometries quickly.
Key facts
| Field | Physical Chemistry |
|---|---|
| Mathematical condition | f(−x) = f(x) |
| Symmetry label | gerade (g) |
| Spectroscopic rule | Mutual exclusion principle |
| Raman activity | Gerade (even) vibrations are Raman active |
| Example orbitals | Spherical s orbitals and four-lobed d orbitals |
The classic 1s atomic orbital wavefunction for a hydrogen atom is a perfect example of an even function. If you invert every coordinate point through the central atomic nucleus, the spherical shape remains completely unchanged. Because the wavefunction looks mathematically identical after this spatial inversion, the 1s orbital possesses gerade symmetry. This same even symmetry rule holds true for all spherical s orbitals and all four-lobed d orbitals.
Frequently asked questions
What is the difference between an even and an odd function in chemistry?
An even function remains completely unchanged when inverted through a molecule's center of symmetry. An odd function will flip its mathematical sign when subjected to that exact same inversion.
Which common atomic orbitals are considered even functions?
All spherical s orbitals and all d orbitals are classified as even or gerade functions. The p orbitals and f orbitals are odd functions because they change sign upon inversion.
How does this relate to the mutual exclusion rule?
In highly symmetrical molecules, the mutual exclusion rule separates the visible vibrational patterns. Even vibrations only appear in Raman spectra, while odd vibrations only appear in infrared spectra.