Probability of Symmetry Species occurring in Reducible Representation Calculator

Formula Used:

\[ n_i = \frac{1}{h} \times \sum (\chi_r \times \chi_i \times g_c) \]

Probability (n_i):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Probability of Symmetry Species occurring in Reducible Representation?

Definition: This calculator determines how many times an irreducible representation appears in a reducible representation using group theory in chemistry.

Purpose: It helps chemists and physicists analyze molecular symmetry and vibrational modes by decomposing reducible representations into irreducible ones.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ n_i = \frac{1}{h} \times \sum (\chi_r \times \chi_i \times g_c) \]

Where:

\( n_i \) — Number of times the irreducible representation appears
\( h \) — Order of the group (total number of symmetry operations)
\( \chi_r \) — Character of the reducible representation
\( \chi_i \) — Character of the irreducible representation
\( g_c \) — Number of symmetry operations in the class

Explanation: The formula calculates the projection of the reducible representation onto each irreducible representation of the group.

3. Importance of This Calculation

Details: This calculation is fundamental in molecular spectroscopy, helping predict vibrational modes, orbital symmetries, and selection rules in quantum chemistry.

4. Using the Calculator

Tips: Enter the order of the group, characters of reducible and irreducible representations, and number of symmetry operations. Order and symmetry operations must be > 0.

5. Frequently Asked Questions (FAQ)

Q1: What is the order of a group?
A: The order is the total number of symmetry operations in the point group (e.g., 6 for C_3v).

Q2: Where do I find character values?
A: Character tables for common point groups are available in chemistry textbooks or online resources.

Q3: What if my result isn't an integer?
A: Check your inputs. The result should be a whole number as it counts occurrences. Non-integer results suggest incorrect inputs.

Q4: How is this used in spectroscopy?
A: It helps determine which vibrational modes are IR or Raman active by analyzing how they transform under symmetry operations.

Q5: Can this be used for any point group?
A: Yes, the formula works for all finite point groups in molecular symmetry analysis.