Draft:Projection Operator Method

  • Comment: Please read WP:NOTATEXTBOOK. Currently this is a howto guide to the method, not an encyclopedic article. In addition, it is not clear that this merits an article of its own as it is a variant of the general projection operator, not really something different. Ldm1954 (talk) 12:39, 26 April 2026 (UTC)


The projection operator method is a mathematical technique used to generate symmetry-adapted linear combinations (SALCs) from either the full matrices of a set of symmetry operations or from their corresponding characters. This method is commonly used in chemistry, particularly for the generation of molecular orbitals from atomic orbitals.[1] [2]

Process

The application of the projection operator method to generate MO diagrams from atomic orbitals involves several steps.

  1. Identify the point group of the molecule.
  2. Identify the central atom and label the valence orbitals of outer atoms which will be used as the basis functions for a representation.
  3. Generate a reducible representation using the basis functions and the appropriate character table.
  4. Reduce the reducible representation to its irreducible components.
  5. Determine how each set of valence orbitals transforms in response to the symmetry operations present in the point group. Multiply the result by the irreducible representation determined in the last step. The resulting combinations of orbitals are the SALCs for these atomic orbitals.
  6. Match the symmetry and energy of the SALCs with the atomic orbitals of the central atom to generate an MO diagram.[1]

Example

An example of the use of the projection operator method to construct the MO diagram of ammonia is shown below.

  1. Identify point group. Ammonia has a C3 axis and three mirror planes, meaning its point group is C3v.
    Symmetry elements present in ammonia
  2. Label the orbitals which will be used as the basis functions. In this case, the central atom is nitrogen and the σ orbitals of the hydrogen atoms will be used as the basis functions for the representation.
    Ammonia with labelled hydrogen atom σ orbitals
  3. Generate a reducible representation using the basis functions and the appropriate character table. Remembering that any basis function that moves during a symmetry operation does not contribute to the character for that function, the following reducible representation is obtained.
C3v E C3 C32 σv σ'v σ"v
Γ 3 0 0 1 1 1
  1. Reduce the reducible representation to its irreducible components. For ammonia, the following set of irreducible components is obtained: Γ = A1 + E. In this case, there should be a total of 3 MOs for this molecule, one which has A1 symmetry and two which have E symmetry (E is doubly degenerate).
  2. Determine how each set of valence orbitals transforms in response to the symmetry operations present in the point group. Multiply the result by the each of the irreducible representations determined in the last step.
    C3v E C3 C32 σv σ'v σ"v
    Γ S1 S2 S3 S1 S2 S3
    A1 1 1 1 1 1 1
    E 2 −1 −1 0 0 0

    The resulting SALCs are S1+S2+S3 for A1 and S1−S2−S3 and S1−S2 for E. The second SALC for E was obtained by determining how a different valence orbital (S2, in this case) transforms in response to the symmetry operations in the point group and subtracting the result from the other SALC for the E representation.

  3. Match the symmetry and energy of the SALCs with the atomic orbitals of the central atom to generate an MO diagram. In this case, the best orbital overlaps are from combining the 2S orbital of nitrogen with the A1 representation and the 2px and 2py of nitrogen with the E representation.
    SALCs of orbitals for ammonia

    These orbitals can then be used to generate an MO diagram for ammonia.

    MO diagram for ammonia

References

  1. ^ a b Cotton, F. Albert. Chemical Applications of Group Theory (3rd ed.). Wiley. ISBN 0-471-51094-7.
  2. ^ Davidson, G. (1991). Group Theory for Chemists. Palgrave Macmillan, London. pp. 81–90. ISBN 978-0-333-49298-7.

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