Constructing Symmetry-Adapted Basis Vectors

1. The Projection Operator

The core tool for this lesson is the projection operator, ${\hat{P}}^{(\alpha )}$, which projects an arbitrary state or function onto the subspace that transforms according to the irreducible representation ${\Gamma }^{(\alpha )}$. Its form, derived from the Great Orthogonality Theorem, is a weighted sum over all symmetry operations in the group $G$:

where:

• ${\ell }_{\alpha }$ is the dimension of the irrep ${\Gamma }^{(\alpha )}$.
• $h$ is the order of the group (the total number of symmetry operations).
• ${\chi }^{(\alpha )}(g)$ is the character of the element $g$ in the irrep ${\Gamma }^{(\alpha )}$. The asterisk denotes complex conjugation.
• ${\hat{R}}_g$ is the operator that performs the symmetry operation $g$.

When we apply ${\hat{P}}^{(\alpha )}$ to an arbitrary starting function or "trial vector" $\psi$, the result, ${\hat{P}}^{(\alpha )}\psi$, is a new function that is guaranteed to transform according to the ${\Gamma }^{(\alpha )}$ irrep. If the original function $\psi$ has no component with that symmetry, the result will be zero. Since the pre-factor $\frac{{\ell }_{\alpha }}{h}$ is just a constant, we often work with the un-normalized projector ${\hat{P}}^{(\alpha )}\propto {\sum }_g{\chi }^{(\alpha )}(g{)}^{\ast }{\hat{R}}_g$ and normalize the resulting function at the end.

For a formal introduction to the projection operator, you can consult the following document.

2. A Detailed Example: Molecular Orbitals of Ammonia (C₃ᵥ)

Let's put this operator to work on a concrete physical problem: finding the symmetry-adapted molecular orbitals for the hydrogen atoms in ammonia (NH₃). This molecule has $C_{3v}$ symmetry. The procedure is a direct application of the techniques you've learned in this module.

The video below provides an excellent, step-by-step walkthrough of this entire process. We will follow its structure, and I will add commentary to formalize the concepts.

Step 1: Define a Basis and Find the Reducible Representation

As shown in the video, we choose the 1s orbitals of the three hydrogen atoms as our basis: $\{s_1,s_2,s_3\}$. To find out what symmetries these orbitals can support, we construct a reducible representation, ${\Gamma }_H$, by examining how many orbitals are left unchanged by the symmetry operations of $C_{3v}$.

• E (identity): All 3 orbitals stay put. Character = 3.
• C₃ (rotation by 120°): All 3 orbitals move to a new position. Character = 0.
• σᵥ (reflection): One orbital stays put, the other two swap. Character = 1.

This gives the reducible representation ${\Gamma }_H=(3,0,1)$.

Step 2: Decompose the Representation

Using the reduction formula from lesson 2.5, we decompose ${\Gamma }_H$. The character table for $C_{3v}$ is:

You can verify that the decomposition is ${\Gamma }_H=A_1\oplus E$. This tells us we can form one linear combination of our hydrogen s-orbitals that transforms as the totally symmetric $A_1$ irrep, and a pair of combinations that transform as the two-dimensional $E$ irrep.

Step 3: Apply the Projection Operator

Now we find the explicit form of these combinations.

Let's formalize the projection for the A₁ irrep, starting with the trial orbital $s_1$. The characters for $A_1$ are $(1,1,1)$.

Summing the last column gives $2s_1+2s_2+2s_3$. Since an overall constant does not change the symmetry properties, we can simplify this to $s_1+s_2+s_3$. After normalization (as shown in the video), this becomes:

(By convention, lowercase letters like $a_1$ and $e$ are used to label orbitals, while uppercase letters label the representations themselves.)

Now for the E irrep, whose characters are $(2,-1,0)$.

Summing the last column gives $2s_1-s_2-s_3$. After normalization, we get our first SALC for the E representation:

3. Finding the Partner Functions

We have found one of the two basis functions for the 2D irrep $E$. How do we find the second, orthogonal partner, ${\psi }_2(e)$? There are a few systematic methods.

Method 1: Project from a different starting orbital.
We can simply apply the same projection operator ${\hat{P}}^{(E)}$ to a different orbital in our original basis, for example $s_2$. This gives $2s_2-s_1-s_3$. This new function is linearly independent of ${\psi }_1(e)$, but it is not orthogonal to it.

Method 2: Gram-Schmidt Orthogonalization.
This is the most robust and practical approach. Once you have two linearly independent functions, you can construct an orthogonal set. The video you just watched uses this exact procedure.

Following the video's calculation, the process yields the second, orthogonal partner function:

Together, ${\psi }_1(e)$ and ${\psi }_2(e)$ form a complete, orthonormal basis for the $E$ symmetry subspace.

Method 3: The "Full" Projection Operator.
A more advanced method uses a projection operator that targets a specific partner function (i.e., a specific row of the representation matrices).

This operator projects onto the $i$-th partner function, starting from a trial function that transforms like the $j$-th partner. This is powerful but requires knowledge of the full representation matrices, not just the characters. Marianetti's notes (Section titled "More on the projection Operator") and Winkler's notes (LINK, Section "Decomposition into Irreducible Invariant Subspaces") discuss this generalized operator.

4. Application: Block-Diagonalization of the Hamiltonian

We've invested significant effort to find this symmetry-adapted basis. The payoff is immense. Let the Hamiltonian of our system be $\hat{H}$. Since $\hat{H}$ is invariant under the symmetry operations of the group, it must commute with all the group operators: $[\hat{H},{\hat{R}}_g]=0$ for all $g\in G$.

An important consequence of this, which follows from Schur's lemmas, is that the Hamiltonian cannot connect states of different symmetries. That is, the matrix element $⟨{\psi }_i^{(\alpha )}|\hat{H}|{\psi }_j^{(\beta )}⟩$ is zero if $\alpha \ne \beta$ (different irreps) or if $i\ne j$ (different orthogonal partner functions of the same irrep).

This means that if you write the Hamiltonian matrix in the SALC basis, it will be block diagonal.

Each block corresponds to a single irreducible representation. To find the eigenvalues of the full Hamiltonian, you only need to diagonalize these much smaller blocks independently. This simplifies a quantum problem from potentially intractable to straightforward.

The following reading provides concrete examples using the tight-binding model, which is highly relevant to your interests in condensed matter physics.

5. Consolidation Exercise: $\sigma$-orbitals of BF₃

To solidify your understanding, I encourage you to watch the following video, which applies the same projection operator method to a more complex molecule, boron trifluoride (BF₃), which belongs to the $D_{3h}$ point group.

This concludes our module on the fundamental tools of representation theory. You have progressed from basic group axioms to constructing character tables, decomposing representations, and now, to building the explicit symmetry-adapted basis functions that are so vital for solving physical problems.

Conclusion

Key Takeaways:

• The projection operator, ${\hat{P}}^{(\alpha )}\propto {\sum }_g{\chi }^{(\alpha )}(g{)}^{\ast }{\hat{R}}_g$, is a systematic tool for constructing Symmetry-Adapted Linear Combinations (SALCs) from an arbitrary set of basis functions.
• For multi-dimensional irreps, the character projector yields one linear combination. Orthogonal partner functions can then be found by applying group operations or by using the Gram-Schmidt orthogonalization procedure.
• The primary physical motivation for constructing SALCs is that they form a basis in which the system's Hamiltonian becomes block diagonal.
• This block diagonalization vastly simplifies quantum mechanical calculations, separating a large problem into a set of smaller, independent problems, one for each irrep.

In our next module, we will begin to apply this powerful machinery to the concrete symmetries of molecules and crystals, starting with the classification of point groups. You now have the complete theoretical toolkit to analyze the consequences of these symmetries.