Description of the lowest-energy surface of the $\mathrm{C}\mathrm{H}+\mathrm{O}$ system: Interpolation of ab initio configuration-interaction total energies by a tight-binding Hamiltonian

It is demonstrated that the potential-energy surface and the lowest-energy path for a polyatomic molecule (applied to the $\mathrm{C}\mathrm{H}+\mathrm{O}$ system) is accurately calculated via bond-length-dependent tight-binding Hamiltonian, fitted to ab initio configuration-interaction (CI) total energies. This Hamiltonian not only reproduces the CI energies accurately and efficiently, but also effectively recognizes and identifies CI energy values that may erroneously converge to excited states. The resulting normal mode frequencies are in very good agreement with experiment.

Figure 1

Geometric structures indicated by open stars were included in the fits of the first and second CI roots. The horizontal axis refers to the ordinal number of the geometric structure in the CI database. The open squares and triangles are the original CI points. The solid circles and squares are predictions (not included in the fit). The TB Hamiltonian predicts the diabatic states: The squares of structures 77 and 82 belong to the lower-energy set, but clearly are diabatic continuations of the open triangle curves, which, before the crossing, belong to the higher-energy set and vice versa. The unbiased TB Hamiltonian fit predicts the diabatic continuation. A conventional fitting method could not recognize such crossings.

Figure 2

TB Hamiltonian fit to the 1st CI root for $\mathrm{H}-\mathrm{C}$ distance $2.1295\mathrm{a.u.}$, $\theta =180°$. The CI points $A$ and $B$ are predicted to belong to the second root, as the TB Hamiltonian fit of the first CI root suggests to discard them. The TB Hamiltonian fit prompted our attention to look for the missing correct CI values. If the CI points were from a third party database, the $A$ and $B$ points should have to be disregarded, a recognition impossible by conventional fitting methods. The meaning of the horizontal axis is as in Fig. 1. It is remarkable that the seven leftmost TB points “in the well,” although not fitted, coincide with the CI values.

Figure 3

TB Hamiltonian fit to the second CI root. The first three points of each curve were fitted. Nonfitted open circles are predictions. As in Fig. 2, in the second curve the fit predicts that the CI points $A$ and $B$ of structure 659 actually belonged to the second and third roots, respectively, to which the MCSCF iterative procedure had originally converged, missing the first root. The TB Hamiltonian fit prompted attention to look for the missing CI root. A conventional fitting method could not recognize that point $A$ of the database belongs to the second root and that point $B$ should be entirely aborted. The meaning of the horizontal axis is as in Fig. 1.

Figure 4

Comparison of the TB predicted to the corresponding CI not fitted total energy $E$ in a.u. for $\mathrm{C}-\mathrm{H}$ $\text{distance}=3.01\mathrm{a.u.}$ Several (connected) $E$ vs $\mathrm{C}-\mathrm{O}$ distance curves are shown, each for a different HCO angle between 50° and 180°. The fitted CI points are indicated by asterisks, the predicted by open circles, and the differences by upper triangles. The black squares represent the original CI energies of the lowest state (first root) and the down triangles of the second root. In both fitted and predicted points the differences between the ab initio CI energies and the TB Hamiltonian total energies (upper triangles) are less than $0.6\mathrm{kcal}∕\mathrm{mol}$. The meaning of the horizontal axis is as in Fig. 1.

Figure 5

Comparison of the TB predicted to the corresponding CI not fitted total energy $E$ in a.u. for HCO $\text{angle}=100°$. Four (connected) $E$ vs $\mathrm{C}-\mathrm{H}$ distance curves are shown, each for a different $\mathrm{C}-\mathrm{O}$ distance. The first, third, and fourth sets (curves) from the left, for $\mathrm{C}-\mathrm{O}$ distances 2.5, a.u., $3.8\mathrm{a.u.}$, and $4.8\mathrm{a.u.}$, respectively, were fitted; the second $(\mathrm{C}-\mathrm{O}=3.2\mathrm{a.u.})$ is entirely predicted and coincides with the corresponding ab initio CI, but not fitted, set. The black squares represent the original CI energies of the lowest state (first root). In both fitted and predicted points the differences between the ab initio CI energies and the TB Hamiltonian total energies (upper triangles) are less than $0.6\mathrm{kcal}∕\mathrm{mol}$. The meaning of the horizontal axis is as in Fig. 1.

Figure 6

The fitting error shows a systematic (not random) smooth distribution along the PES, due to the differences between two smooth surfaces—i.e., that of of TB and CI—as shown, for example, in the same to Fig. 2 subset of the CI and TB values, but as a function of the actual $\mathrm{C}-\mathrm{O}$ distance (not the structure ordinal number). In this scale points $A$ and $B$ are close to nearby database points, therefore, not clearly distinguishable.

Figure 7

The TB minimum energy path for the system of $\mathrm{H}\mathrm{C}+\mathrm{O}$. For each $\mathrm{O}-\mathrm{C}$ distance the TB energy was minimized (i.e., searched for minimum) with respect to the $\mathrm{C}-\mathrm{H}$ distance and $\mathrm{O}-\mathrm{C}-\mathrm{H}$ angle. The minimum energy and the corresponding $\mathrm{C}-\mathrm{H}$ distance and $\mathrm{O}-\mathrm{C}-\mathrm{H}$ angle are displayed as follows: If C is considered on the energy curve and O on the energy axis keeping the $\overrightarrow{\mathit{C}\mathit{O}}$ arrow horizontal, then the $\overrightarrow{\mathit{C}\mathit{H}}$ arrow indicates the position of H relative to the C and O. At large $\mathrm{O}-\mathrm{C}$ distances, CH is bent toward O, the $\mathrm{O}-\mathrm{C}-\mathrm{H}$ angle gradually becoming 121.417°. The displayed data are entirely interpolated predictions based on the TB parameters. There were no ab initio CI points to be fitted on this curve, since the optimal values (minima), used to produce the curve, were unknown before the optimization.

Figure 8

TB Hamiltonian prediction of the $\mathrm{H}\mathrm{C}-\mathrm{O}$ PES at $\mathrm{C}-\mathrm{H}$ distance of $2.1295\mathrm{a.u.}$ for various $\mathrm{C}-\mathrm{O}$ distances and $\mathrm{H}-\mathrm{C}-\mathrm{O}$ angles around the $\mathrm{H}\mathrm{C}-\mathrm{O}$ equilibrium geometry, for which the fit was performed. The corresponding ab initio CI points (either fitted or not) are shown on the surface. Angles smaller than 50° corresponding to $\mathrm{H}-\mathrm{C}\mathrm{O}$ were not fitted.

Figure 9

TB Hamiltonian prediction of the $\mathrm{H}\mathrm{C}-\mathrm{O}$ PES, as in Fig. 8, but with optimized (lowest energy) $\mathrm{C}-\mathrm{H}$ distance. On a regular mesh of $\mathrm{C}-\mathrm{O}$ distances and $\mathrm{H}-\mathrm{C}-\mathrm{O}$ angles, a minimization of the TB total energy were performed with respect to the $\mathrm{C}-\mathrm{H}$ distance, and the optimal energy is displayed (vertical axis). There are no ab initio CI points on this surface, since the optimal $\mathrm{C}-\mathrm{H}$ distance, used to produce the surface, was unknown in advance. Therefore, this PES is an interpolating prediction based on the TB parameters fitting the existing CI database. The TB Hamiltonian clearly predicts the existence of three energy minima, the main at 121.417°, and two higher lying, corresponding to the linear $\mathrm{H}-\mathrm{C}\mathrm{O}$ or $\mathrm{C}-\mathrm{O}\mathrm{H}$, as explained in the text.