Quantum states for quantum processes: A toy model for ammonia inversion spectra

Gustavo A. Arteca and O. Tapia

Phys. Rev. A 84, 012115 – Published 25 July 2011

Abstract

Chemical transformations are viewed here as quantum processes modulated by external fields, that is, as shifts in reactant to product amplitudes within a quantum state represented by a linear (coherent) superposition of electronuclear basis functions; their electronic quantum numbers identify the “chemical species.” This basis set can be mapped from attractors built from a unique electronic configurational space that is invariant with respect to the nuclear geometry. In turn, the quantum numbers that label these basis functions and the semiclassical potentials for the electronic attractors may be used to derive reaction coordinates to monitor progress as a function of the applied field. A generalization of Feynman's three-state model for the ammonia inversion process illustrates the scheme; to enforce symmetry for the entire inversion process model and ensure invariance with respect to nuclear configurations, the three attractors and their basis functions are computed with a grid of fixed floating Gaussian functions. The external-field modulation of the effective inversion barrier is discussed within this conceptual approach. This analysis brings the descriptions of chemical processes near modern technologies that employ molecules to encode information by means of confinement and external fields.

Received 23 February 2011

DOI:https://doi.org/10.1103/PhysRevA.84.012115

Authors & Affiliations

Gustavo A. Arteca^1,2 and O. Tapia^2,*

¹Département de Chimie et Biochimie & Biomolecular Sciences Programme, Laurentian University, Ramsey Lake Road, Sudbury, Ontario, Canada P3E 2C6
²Department of Physical Chemistry, Uppsala University, Ångströmlaboratoriet, Box 259, S-751 05 Uppsala, Sweden

^*orlando.tapia@fki.uu.se

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Vol. 84, Iss. 1 — July 2011

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Figure 1
Semiclassical potential energy curves $U_{1}$ (top) and $U_{3}$ (bottom) built with a fixed 10-center grid of 6-311G $^{* *}$ $Bq$ orbitals (“Banquo” or “ghost” atoms). A set of nitrogenlike 6-311G $^{* *}$ functions is placed at the center of the grid. The remaining nine centers correspond to H-like 6-311G $^{* *}$ $Bq$ orbitals placed at the equilibrium geometries for the classical charges in the attractors. The diagrams show the fixed $Bq$ atoms (blank circles) and the variable test charges (solid circles; $+$ 1 for H and $+$ 7 for N). $X$ (Å) plane of $+$ 1 charges (perpendicular to the $C_{3}$ axis). The full grid sustains the equilibrium geometry of the $C_{3 v}$ -singlet structures and the $D_{3 h}$ open-shell singlet. The corresponding (diabatic) $U_{i}$ values for these highlighted test-charge configurations are indicated by arrows and dashed circles.Reuse & Permissions
Figure 2
Effective energy for the total quantum states arising from coupling the model basis functions for NH $_{3}$ . The results are derived with the fixed 10-center grid of 6-311G $^{* *}$ FGOs. The top diagram shows the semiclassical potential energy curves ${$ $U_{i}$ ( $x$ )} (thin full and dashed lines), and the total energy $E_{full}$ (thick full line) with the coupling $V_{13}$ $=$ $V_{23}$ $=$ 0.1 a.u. The $x$ coordinate refers to the position of the plane of three $H$ nuclei ( $+$ 1 test charges) along the $C_{3}$ -symmetry axis. The stationary points are indicated with arrows. The bottom diagram shows the coefficients ${$ $|$ $c_{i}$ $|$ $}$ for the linear combination of ${$ $Ψ$ $_{i}$ $}$ functions in terms of the reaction coordinate for the $|$ R $〉$ → $|$ P $〉$ process. There is a rapid increase in $|$ P $〉$ -state amplitude as soon as one enters the regions dominated by the $|$ TS $〉$ state, indicating that all three basis states contribute significantly to the effective barriers.Reuse & Permissions
Figure 3
Effective potential energy $E_{full}$ ( $x$ ) for the total quantum states arising from coupling the model electronic basis functions, as a function of the matrix elements $V_{13}$ $=$ $V_{23}$ . As the field intensity increases, the effective barriers disappear and the system exhibits a single minimum, that is, the NH $_{3}$ will become a stable planar ( $D_{3 h}$ ) structure at high fields. $V_{13}$ couplings are as follows: (a) 0.2, (b) 0.3, (c) 0.4, (d) 0.5, (e) 0.7. The critical value for the disappearance of the barriers is estimated to be $V_{13}$ $^{(c)}$ $=$ 0.605 ± 0.005 a.u. The results use the 10-center grid discussed in the appendix.Reuse & Permissions
Figure 4
Change in $E_{full}$ ( $x$ ) value at its stationary points ( $x$ $_{a}$ $^{*}$ , $x$ $_{b}$ $^{*}$ , $x$ $_{c}$ $^{*}$ ) as a function of the external coupling matrix elements. The minima dominated by the $|$ R $〉$ and $|$ TS $〉$ states are $E_{full}$ ( $x$ $_{a}$ $^{*}$ ) and $E_{full}$ ( $x$ $_{c}$ $^{*}$ ), respectively. Note that the barrier maximum [ $E_{full}$ ( $x$ $_{b}$ $^{*}$ )] decreases faster than the minima as the field increases. Moreover, the energy difference between the barrier maximum and the intermediate minimum decreases if the barrier is endothermic ( $V_{13}$ < 0.47 a.u.), yet increases when the barrier becomes exothermic until it disappears (0.47 ⩽ $V_{13}$ < 0.61, in a.u.). The results use the 10-center grid discussed in the Appendix.Reuse & Permissions
Figure 5
Change in location of $E_{full}$ extrema $x$ $_{a}$ $^{*}$ , $x$ $_{b}$ $^{*}$ , $x$ $_{c}$ $^{*}$ as a function of the external coupling (top diagram) in the three-state ammonia model. Note the $|$ R $〉$ -like minimum ( $x$ $_{a}$ $^{*}$ ) and the barrier maximum ( $x$ $_{b}$ $^{*}$ ) merge eventually but are affected differently by the external field. At low $V_{13}$ values, both $x$ $_{a}$ $^{*}$ and $x$ $_{b}$ $^{*}$ are displaced toward the central minimum ( $x$ $_{c}$ $^{*}$ ). After reaching a maximum approach to $x$ $_{c}$ $^{*}$ at $V_{13}$ ≈ 0.2 a.u., the barrier maximum begins to move away from the $|$ TS $〉$ -like region and toward the regions with larger reactant amplitude. The bottom diagram characterizes the composition of the quantum state at $x$ $_{b}$ $^{*}$ , the top of the effective barrier. Note that even when the barrier becomes exothermic ( $V_{13}$ ≈ 0.47 a.u.), still the $|$ TS $〉$ state makes the largest contribution to the barrier maximum. The $|$ R $〉$ state makes the largest contribution to the barrier maximum for $V_{13}$ > 0.55 a.u., when the two extrema are very close. The results use the 10-center grid discussed in the Appendix.Reuse & Permissions

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