Single surface beyond Born–Oppenheimer equation for a three-state model Hamiltonian of $Na3$ cluster

When a set of three states is coupled with each other but shows negligibly weak interaction with other states of the Hilbert space, these states form a sub-Hilbert space. In case of such subspace [J. Chem. Phys. 124, 074101 (2006)], (a) the adiabatic-diabatic transformation (ADT) condition, $∇⃗A+τ⃗A=0$ [Chem. Phys. Lett. 35, 112 (1975)], provides the explicit forms of the nonadiabatic coupling (NAC) elements in terms of electronic basis function angles, namely, the ADT angles, and (b) those NAC terms satisfy the so-called curl conditions [Chem. Phys. Lett. 35, 112 (1975)], which ensure the removal of the NAC elements [could be singular also at specific point(s) or along a seam in the configuration space] during the ADT to bring the diabatic representation of the nuclear Schrödinger equation with a smooth functional form of coupling elements among the electronic states. Since the diabatic to adiabatic representation of the Hamiltonian is related through the same unitary transformation $(∇A+τA=0)$⁠, it could be quite interesting to explore the nature of the nonadiabatic coupling terms starting from a diabatic Hamiltonian and, thereafter, to formulate the extended Born–Oppenheimer (EBO) equation for those adiabatic states transformed from diabatic ones. We consider a three-state diabatic potential matrix constructed for the excited states of $Na3$ cluster [J. Chem. Phys. 88, 6068 (1988)] at the pseudo-Jahn–Teller model situation, which can reproduce experimentally measured vibrationally resolved absorption lines [Surf. Sci. 156, 770 (1985)] with appropriate choice of coupling parameters, analytically calculate the nonadiabatic coupling elements along with their curls, and numerically evaluate the ADT angles to explore the nature of its nonadiabaticity. While formulating the single surface beyond the BO equation, our theoretical derivation demonstrates that the existence of zero curls of the NAC terms is a necessity. Indeed, when the energy gap between the third state $(12A1′/22A1′)$ and the doubly degenerate states $(22E′/32E′)$ of the model Hamiltonian for $Na3$ cluster is considered to be either identically or approximately zero, the curl for each NAC element naturally approaches zero, leading to a theoretically valid EBO equation. We demonstrate the numerical validity of the EBO equation by calculating the nonadiabatic effects on the photoabsorption spectrum starting with the initial wave function located on the ground electronic state and compare with the corresponding diabatic spectrum when the three states are either degenerate at a point or approaching to form three-state degeneracy at the same point. Finally, we calculate the vibrational eigenspectrum of the ground adiabatic state by using (so to say) theoretically and numerically valid EBO equation to compare with those experimentally measured and BO/geometric phase calculated spectra (Tables I-III).