Abstract
The quantum mechanical description of isomerization is based on bound eigenstates of the molecular potential energy surface. For the near-minimum regions there is a textbook-based relationship between the potential and eigenenergies. Here we show how the saddle point region that connects the two minima is encoded in the eigenstates of the model quartic potential and in the energy levels of the [H, C, N] potential energy surface. We model the spacing of the eigenenergies with the energy dependent classical oscillation frequency decreasing to zero at the saddle point. The eigenstates with the smallest spacing are localized at the saddle point. The analysis of the HCN ↔ HNC isomerization states shows that the eigenstates with small energy spacing relative to the effective (v1, v3, ℓ) bending potentials are highly localized in the bending coordinate at the transition state. These spectroscopically detectable states represent a chemical marker of the transition state in the eigenenergy spectrum. The method developed here provides a basis for modeling characteristic patterns in the eigenenergy spectrum of bound states.
Highlights
Correspondence principle and molecular dynamics in the frequency domain The correspondence principle[10] of quantum mechanics is a rule that any quantum mechanical theory we set up must satisfy, but has fundamental importance in interpreting any quantum mechanical result we obtain. The consequence of this principle is the possibility to connect the quantum description to the corresponding classical one in some region of system parameters. One such classical to quantum mechanical correspondence is the correlation between the spacing of the discrete eigenenergies and the classical oscillation frequency in a bound system
The correspondence principle allows us to study systems where there is no analytical solution for the quantum problem but where we can find an analytical solution for the classical problem
To obtain a visualization of the quantum dynamics in the frequency domain we can represent the spacing of a set of eigenenergies ΔEn(En) and compare it with the result obtained in the classical limit
Summary
Has a saddle point between two symmetric minima; for this potential an analytic solution of the classical motion[12] exists. For these series of states the corresponding classical oscillations take place due to the effective adiabatic-bend potential. As the pure bending eigenstates pass through the energy of the isomerization barrier, the hydrogen atom no longer “belongs” to one of the ends of the CN fragment These vibrational states are extremely important for chemistry as they correspond to a third type of molecule that we can form from the H, C and N atoms: H0.5CNH0.53.
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