A tri-atomic Renner-Teller system entangled with Jahn-Teller conical intersections.

TL;DR: The present study is characterized by planar contours that intersect the collinear axis, thus, forming a unique type of RT-non-adiabatic coupling terms (NACT) expressed in terms of Dirac-δ functions, and reveals an unexpected result of the following kind.

Abstract: The present study concentrates on a situation where a Renner-Teller (RT) system is entangled with Jahn-Teller (JT) conical intersections. Studies of this type were performed in the past for contours that surround the RT seam located along the collinear axis [see, for instance, G. J. Halasz, A. Vibok, R. Baer, and M. Baer, J. Chem. Phys. 125, 094102 (2006)]. The present study is characterized by planar contours that intersect the collinear axis, thus, forming a unique type of RT-non-adiabatic coupling terms (NACT) expressed in terms of Dirac-δ functions. Consequently, to calculate the required adiabatic-to-diabatic (mixing) angles, a new approach is developed. During this study we revealed the existence of a novel molecular parameter, η, which yields the coupling between the RT and the JT NACTs. This parameter was found to be a pure number η = 22/π (and therefore independent of any particular molecular system) and is designated as Renner-Jahn coupling parameter. The present study also reveals an unexpected result of the following kind: It is well known that each (complete) group of states, responsible for either the JT-effect or the RT-effect, forms a Hilbert space of its own. However, the entanglement between these two effects forms a third effect, namely, the RT/JT effect and the states that take part in it form a different Hilbert space.

Summary

I. INTRODUCTION

• This article is one additional link in a series of articles [1] [2] [3] [4] devoted to the problem of revealing rigorous, efficient, and accurate methods to construct diabatic potential energy surfaces (PES) for multi-state, poly-atomic molecular systems.
• The severity of this issue increases significantly if one is interested in studying chemical exchange processes that require two or more arrangement channels.
• In what follows, the authors suggest calculating the ADT matrices for the tri-atomic grid directly.
• The only problem encountered here is that these calculated NACTs are extremely spiky-reminiscent of the Dirac δ-function (see, e.g., Fig. 2 in Ref. 21(a))-and therefore their correct shape is frequently missed.

A. Introductory remarks

• The authors approach is based on solving the following multidimensional first order differential equation: 5, 12 ∇A(s) + τ (s)A(s) =0, where A(s), as previously mentioned, is the ADT matrix, τ (s) is an anti-symmetric matrix that contains the above mentioned vectorial NACTs and s is a variable that presents the collection of internal nuclear coordinates.
• 13, 14 One way to avoid these singularities is to eliminate the τ (s)-matrix and form a modified SE free of all singularities but governed by V(s), a full potential matrix, which replaces the original, diagonal matrix, u(s).
• The matrix V(s) is known as the diabatic PES-its diagonal elements are the corresponding diabatic potentials and its off diagonal elements form the diabatic coupling terms (reminiscing of the NACTs).
• Since V(s) has to be presented at a given grid of points the authors must guarantee that the chosen contours (along which A(s) and V(s), are calculated) cover efficiently the full corresponding CS.
• 18(a), 23 (b) So far the authors referred to CSs in general but the above mentioned planar CSs are the ones to be considered for their purposes.

1. Tri-state NACT-matrix

• The authors remind the reader that the two states 1A and 1A form a RT degeneracy line along the collinear HHF axis.
• The NACT-matrix takes the form EQUATION where the authors assumed that τ 21 ≡0.
• This form is not well suited for the numerical treatment as will be elaborated next.
• To achieve this arrangement, the authors permute between the last two rows and then between the last two columns so that τ (s) becomes EQUATION which is the NACT-matrix of the desired form.

2. Tetra-state NACT matrix

• As in the previous case, this matrix is not well suited for their numerical treatment.
• To achieve this, the authors permute between the second and the third rows and then between the second and the third columns, so that τ (s) becomes EQUATION ).
• Next, the authors permute between the two last rows and then between the two last columns so that τ (s) becomes EQUATION which is the NACT-matrix of the required form.

C. ADT matrices and privileged angles 1. Two-state case and the ADT angle

• The only exceptional case is the two-state case where the A-matrix which can be expressed in terms of one angle γ (s) (to be termed as the ADT or mixing angle) and this leads to the following simple line integral: 12 EQUATION.
• Here, τ 12 was introduced earlier, designates the contour along which is carried out the integration and the dot presents the scalar product.
• Since the authors intend to consider circular contours only the integration can be simplified to become over an angle, ϕ EQUATION where (ϕ, q) are polar coordinates: q is the radius, ϕ is the angle associated with the rotation.
• About two decades ago it was suggested 24 to identify the topological phase with the Berry phase for a twostate system, also known as Comment.
• 25 This connection was found to be valid for all reported numerical studies of molecular systems with two quasi-isolated states.

2. Tri-state privileged angle

• Substituting this product in Eq. ( 1) yields three coupled first-order differential equations for the three corresponding quasi-Euler angles, γ ij .
• The final set of equations as well as their solution depends on the order of the Q-matrices.
• These two equations are solved with the aim of calculating the privileged ADT angle γ 12 (ϕ, q).
• The introduction of the privileged angle enables the extension of the earlier defined two-state topological phase, α 12 , to three-state systems.

2. The tri-state JT/RT coupled equations

• In order to derive the corresponding differential equations for the tri-state RT/JT coupled system, the authors consider the NACT-matrix given in Eq. ( 3) and substitute the relevant matrix elements in Eqs. (9a) and (9b).
• Thus, EQUATION ) where the integration is done along a circular contour, and therefore the first-order differentiation operator ∇, is replaced by the angular derivative: (∂/∂ϕ).
• Parts of Eqs. ( 12) can be integrated analytically taking advantage of Eq. (11b).

3. The tetra-state JT/RT coupled equations

• In what follows are derived the differential equations for the tetra RT/JT coupled system.
• These equations are solved for the initial conditions γ.

4. Derivation of the Renner-Jahn coupling parameter η

• The authors start with the single-valuedness for the diabatic potentials which is fulfilled when sin(π.
• Substituting this outcome in ( 16), the authors find that the quantization is fulfilled whenever 2ε−χ = 0 thus, as before, this equality yields for η the result given in Eq. ( 23) and consequently also Eq. (24).
• Indeed, for all considered cases (even for vertical shifts up to ∼ 2 Rad.), the authors find the differences between the theoretical shifts and the required numerical ones to be negligibly small (see a comparison along the two last columns of Table 1 given in Ref. 3 ).
• So far this derivation was carried out for the tri-state case.
• In other words, the transition from a tri-state system to a tetra-state system leaves RJCP unaffected.

A. Introductory comments

• Employing MOLPRO, 33(b) only JT-NACTs and the corresponding ADT angles.
• The corresponding RT-NACTs required for the present study are not calculated but assumed to be quasi-Dirac-δ functions as discussed in Sec. II D.
• The calculations and the theoretical study are done (as frequently mentioned) along closed circular contours.
• This common center is chosen in such a way as to guarantee that the various circles (with the varying radii) cover the whole planar CS of interest.

B. JT-NACTs along closed circles

• The feature that characterizes the NACTs for q = 0.4 a.u. is that the circle does not sur- round the point of ci which is located at a distance of 0.5 a.u. from the center of the circle.
• In all other cases, the circles surround the ci-point and therefore the various τ 12 (ϕ|q >0.5 a.u.)'s exhibit a slightly more complicated structure.
• As for the two NACTs that couple the third state the following can be said: (a) τ 13 (ϕ|q) hardly changes as q increases; (b) τ 23 (ϕ|q) changes significantly and becomes spikier.
• These spiky NACTs may lead to inaccuracies in calculating the corresponding ADT angles.

C. (1,2) ADT angles along closed circles 1. Tri-state results

• The authors distinguish between three types of curves: (1) The curve for q = 0.4 a.u. is not shifted (in other words, the shift is zero) and its topological phase, α 12 (q), is zero which results from the fact that the circle does not surround the (1,2)ci.
• Figure 4 reveals one interesting (and important) feature:.
• The ϕ-dependence of the various curves become similar and the curves are converging to each other as the radius, q, of the circles increases.
• As it happens this is the region where the chemical reaction takes place.

2. Tetra-state results

• These ADT angles are calculated along circles with the following radii: q = 3.0, 3.7, 4.0 a.u. presented in the relevant panels.
• As is noticed the results are well converged for all three cases.
• The encouraged fact from this comparison is that although the tetra-state ADT angles are calculated using two additional NACTs, namely, τ 13 (ϕ|q) and τ 23 (ϕ|q) (and therefore are based on more involved expressions to calculate the shifts at ϕ = π -see Eqs. ( 20) and (21) -still, the ADT angles, are reasonably well converged.

IV. DISCUSSION AND CONCLUSIONS

• ADT angle γ 12 (ϕ|q) due to two entangled NACTs-the JT-NACT and the RT-NACTs.
• This was not the situation when the authors started studying the FHH system.
• In other words, increasing the JT sub-Hilbert space from two states to three did not yield the expected quantization (unlike in numerous other cases 27, 29 This fact led to the conclusion that the sharp increase of γ 12 (ϕ|q) in the region of π.
• To test this possibility, the authors developed a new methodology which enabled the numerical study of the entangled RT/JT system.
• But this is not always sufficient as convergence might be attained for a non-physical situation (see, for example, in the first paragraph of this section and Fig. 7 ).

Figures

FIG. 6. A comparison between tri-state and tetra-state ADT (mixing) angles γ 12(ϕ|q), for circular contours at R = Rce = 6 a.u. along the interval 0 ≤ ϕ ≤ 2π . The tri-state angles were calculated employing the RT/JT Eqs. (15a) and (16) and the tetra-state angles were calculated employing the RT/JT Eqs. (19a)–(21b). The comparison is presented in panel (a) for q = 3.0 a.u., in panel (b) for q = 3.7 a.u. and in panel (c) for q = 4.0 a.u.

FIG. 7. ADT (mixing) angles, γ 12(ϕ|q), calculated for a circular contour at R = Rce = 6 a.u. with radius q = 4.0 a.u., along the interval 0 ≤ ϕ ≤ 2π : Two JT curves are shown: a two-state curve calculated with Eq. (6) and a three-state curve calculated with Eqs. (19a) and (19b).

FIG. 1. A schematic picture of the system of coordinates: (a) Positions of atoms, point of (1,2) ci and the center of all circular contours. (b) System of coordinates: (R,θ |r) vs. (ϕ,q|r). In this figure, atoms F, H1, and H2 stand for atoms A, B, and C, respectively, mentioned in the text.

FIG. 3. Angular NACTs, τϕ (ϕ|q), for circular contours at R = Rce = 6 a.u. along the interval 0 ≤ ϕ ≤ 2π : (a) τ 12ϕ (ϕ|q), for q = 0.4 a.u.; (b) τ 12ϕ (ϕ|q), τ 13ϕ (ϕ|q) and τ 23ϕ(ϕ|q), for q = 3 a.u.; (c) the same as in (b) but for q = 3.7 a.u.; (d) the same as in (b) but for q = 4 a.u.

FIG. 2. Schematic figures describing the two RT/JT models: (a) the tri-state RT/JT model; (b) the tetra-state RT/JT model.

FIG. 4. The tri-state ADT (mixing) angle, γ 12(ϕ|q), for circular contours at R = Rce = 6 a.u. along the interval 0 ≤ ϕ ≤ 2π as calculated employing the RT/JT Eqs. (15a) and (16). Results are shown for q = 0.4, 1.0, 2.0, 3.0, 3.7, 4.0, 4.6, 5.0 a.u.

FIG. 5. The transition region, presented as a square, from reagents channel to products channel.