# Exact Rovibronic Equivalence of the Adiabatic and Diabatic Representations of N‐Coupled State Diatomic Systems

## TL;DR

This paper shows that adiabatic and diabatic representations in diatomic systems are numerically equivalent, enabling accurate assessment of non-adiabatic effects in spectroscopy.

## Contribution

A novel method demonstrates numerical equivalence between adiabatic and diabatic representations for N-state diatomic systems.

## Key findings

- Numerically exact equivalence is achieved for adiabatic and diabatic representations using ab initio data for N2, CH, and a 10-state model.
- The method efficiently assesses the importance of non-adiabatic effects in rovibronic energy calculations.
- The equivalence is implemented in the diatomic code duo for spectroscopic modeling.

## Abstract

The Born–Oppenheimer approximation assumes nuclear motion evolves on single, uncoupled potential energy surfaces, widely used to solve the time‐independent Schrödinger equation for atomistic systems. However, for near‐degenerate same‐symmetry electronic states, avoided crossings in the potential energy curves occur and non‐adiabatic couplings (NACs) become significant. In such cases, the adiabatic approximation is unsuitable for high‐resolution spectroscopy. A unitary transformation to the diabatic representation can eliminate NACs, resulting in smooth molecular property curves that may cross. Computing this adiabatic‐to‐diabatic transformation (AtDT) is desirable but non‐analytic for multi‐state coupled systems, necessitating numerical solutions. It remains unclear if current methods yield numerically exact AtDTs ensuring rovibronic energy level equivalence between adiabatic and diabatic pictures. We demonstrate (for the first time) numerically exact equivalence of adiabatic and diabatic representations for N‐state diatomic molecules using ab initio data for N2, CH, and a model 10‐state system. We show how the equivalence can be efficiently used to assess the importance of non‐adiabatic effects and the impact of omitting them when computing rovibronic energies of diatomic molecules. The adiabatic and diabatic representations of the spectroscopic model, including all coupling terms, have been implemented in the diatomic code duo.

A novel method demonstrates the numerical equivalence of adiabatic and diabatic representations in N‐state coupled diatomic systems. This crucial finding provides an efficient and accurate way to assess the importance of non‐adiabatic effects, which are often disregarded without rigorous quantitative checks.