The Microscopic Mechanism of Spectral Symmetry

Subject: Response regarding the "Analytical Approach" and "Method of Moments"

I agree with Tom Barbara that the "Method of Moments" (Van Vleck moments) offers a powerful perspective on this problem. Indeed, proving that all odd moments of the spectral distribution are strictly zero (\(M_1 = M_3 = \dots = 0\)) would constitute a valid statistical proof of symmetry. As Tom correctly noted, since moments are related to the traces of multiple commutators (e.g., the skewness is proportional to \(M_3 \propto \text{Tr}\{[\hat{H}, [\hat{H}, \hat{I}_x]][\hat{H}, \hat{I}_x]\}\)), this approach has the distinct advantage of being basis-independent.

However, in our work, we have developed a structural analytical approach that goes beyond the statistical description of the lineshape and reveals the fundamental quantum mechanical cause of the observed symmetry.

The Operator Algebra Approach

Instead of analyzing the consequences (i.e., the shape of the spectrum via moments), we identified the specific symmetry operator that generates this phenomenon. We found that for the specific class of spin systems described, the Hamiltonian \(\hat{H}\) commutes with a combined symmetry operator \(\hat{Q}\):

[\hat{H}, \hat{Q}] = 0, \quad \text{where } \hat{Q} = \hat{P} \times \hat{\Pi}

Here, the operator \(\hat{Q}\) is a superposition of two distinct actions:

\(\hat{P}\) (Spin Inversion): The operator that flips the spin projections (\(I_z \to -I_z\)). In the matrix representation, this swaps the "up" and "down" spin blocks.
\(\hat{\Pi}\) (Spatial Permutation): The operator that reflects the order of nuclei (\(i \leftrightarrow N+1-i\)), effectively swapping the chemical shift parameters.

The Mechanism of Symmetry Restoration

The key insight is that separately, neither \(\hat{P}\) nor \(\hat{\Pi}\) commutes with the Zeeman term of the Hamiltonian (\(\hat{H}_Z\)):

Spin inversion flips the sign of the spin operators: \(\hat{P}\hat{H}_Z\hat{P}^{-1} = -\hat{H}_Z\).
Spatial permutation flips the sign of the frequencies (under the condition of antisymmetric shifts): \(\hat{\Pi}\hat{H}_Z\hat{\Pi}^{-1} = -\hat{H}_Z\).

    However, when applied simultaneously, these two "minus signs" cancel each other out. As a result, the combined operation \(\hat{Q}\) leaves the total energy invariant.

Conclusion

This commutation relation \([\hat{H}, \hat{Q}] = 0\) implies that the system possesses a conserved quantum number (a generalized parity). This conservation law imposes strict selection rules on the allowed transitions, enforcing the spectrum to be a perfect palindrome.

While the Method of Moments would analytically describe "how" the spectrum looks (by proving \(M_3=0\)), our operator approach explains "why" it happens: the symmetry of the spectrum is a direct manifestation of a hidden commutational symmetry of the Hamiltonian.