# Characterizing the E⊗e Jahn–Teller Potential Energy Surfaces by Differential Geometry Tools

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{3}H

_{3}.

## 1. Introduction

## 2. Methods

^{TM}computer algebra software [26,27]. More concretely, the data discussed in Section 3.2 were obtained with the function NDSolve for numerically solving differential equations, the outcome being represented with the Plot3D and ParametricPlot3D graphic routines. The data in Section 3.3 were handled in the Matlab–Octave environments [28,29], using fminsearch for fitting and surf for graphical rendering.

_{3}moiety, while the positions of hydrogen atoms were the object of gradient optimization. The computed energy surface consisted of 275 points, each one the object of an input file running the above-mentioned constrained geometry optimization. The GAMESS input files were automatically generated with a Matlab–Octave script performing the conversion of polar coordinates of the e-type Jahn–Teller modes into molecular geometry, as defined specifically in Section 3.3, and writing out the necessary input keywords. The presented molecular geometry images were realized with the Molden software [34].

## 3. Results and Discussion

#### 3.1. The E⊗e Potential Energy Surfaces in the Linear Coupling Model

_{−}(ρ) or e(ρ) ≡ e

_{+}(ρ) if we conventionally allow the radial coordinate to take negative values. In this system, we have axial symmetry, which allows for the use of the idea of a rotational surface. The points of the surface can be described as the following set of coordinates:

^{−1})

_{ij}elements. It follows that:

#### 3.2. Illustrating Geodesic Patterns on the “Mexican Hat” and Conical Intersection Surfaces

_{+}(ρ) solution in Equation (2), being depicted with darker (violet) coloring in Figure 1, while the lowest sheet e

_{−}(ρ) is drawn in a lighter color (yellow). The e

_{−}(ρ) surface has a circular trough at the the ρ

_{min}= V/K radius. With a certain scaling and cropping in the graphical representation of the surfaces, a certain resemblance to a Mexican hat can be observed, the E⊗e profile also being called this name. In our vertically elongated scaling, the “Mexican hat” name, as figure of speech, is not so obvious, but this is an irrelevant aspect.

_{P}= 0, other arbitrary values being obtainable by the rotation with φ

_{P}of the presented figures around the vertical axis.

_{−}(ρ) surface, produced with P at the respective smaller, equal, and larger radial coordinates, in comparison to the radius ρ

_{min}of the minimum trough. Strictly speaking, the non-null initial condition due to the first-order derivative of the angular coordinate determines only one half of the geodesic line evolving in the clockwise or anti-clockwise direction with respect to P’s origin depending on the ${\dot{\phi}}_{P}>0$ or ${\dot{\phi}}_{P}$ < 0 case. We represented these halves in different colors, their meeting marking the starting point. In the discussed ${\dot{\rho}}_{P}=0$ situation, the geodesic halves modulated by the sign of ${\dot{\phi}}_{P}$ are symmetrical with respect to a mirror plane passing through the point P and the vertical axis at ρ = 0. The two branches of the geodesic spiral on the surface, crossing each other at polar coordinates equal to those of the initial point φ = φ

_{P}and, on the diametrically opposed side, φ = φ

_{P}+ π. For the starting points on the upper surface (see panel d), e

_{+}(ρ), and for the ones placed on the lower surface, e

_{−}(ρ), after the minimum radius (see panels b and c), ρ

_{P}≥ ρ

_{min}, the curves evolve upwards at elevations higher than the initial point e

_{±}(ρ) > e

_{±}(ρ

_{P}).

_{−}(ρ), comprised of the origin and radius of the minimum trough, 0< ρ

_{P}< ρ

_{min}(see panel a), the geodesic goes first toward the minimum and then starts spiraling upward after crossing the valley. The rotation of the geodesics on the surfaces around the ρ = 0 vertical axis can be assigned to the positive curvature of the e

_{±}(ρ) surfaces with respect to the ρ coordinate. This situation tells us that, in the long term, the geodesics cannot be interpreted with the intuition of the shortest path between two points because the cycling imposes lengthier routes than when drawing a curve with a monotonous variation in the angular parameter between the φ

_{P}and φ

_{Q}values of the generally distant P and Q references placed on the same geodesic. In turn, the minimum path interpretation holds in the infinitesimal sense of Q’s arrival occurring in the vicinity of P’s starting point.

_{P}. This is the trivial case of the geodesic and is defined as the intersection of the e

_{±}(ρ) surfaces with the vertical plane containing the ρ = 0 axis and the initial P point. These are the analogs of meridians, which are geodesics on the sphere. In turn the circles defined at a constant radial parameter ρ = ρ

_{P}, are not allowed to be geodesics in similar manner to a globe in which parallels other than the equator do not fulfill the underlying equations.

_{_}(ρ) sheet, as in panel (a), it crosses the degeneracy at ρ = 0 and goes onto the e

_{+}(ρ) surface. Vice versa, as shown in panel (b), it can pass from e

_{+}(ρ) to e

_{−}(ρ), the different colors of the curve branches helping us to visually locate the initial point at their confluence. Actually, in this situation, the geodesics are parabolas with the minimum located at the ρ

_{min}contained in the plane defined by the vertical ρ = 0 axis and the point P.

_{−}(ρ), the lower surface resulting then for ρ > 0 and the upper one being accounted for with an enforced negative ρ having e

_{+}(ρ) = e(−ρ) = e

_{_}(−ρ). Accepting such a non-standard convention is not essential to the discussion, but it helps us to deal with the inter-surface evolution of the particular meridian-like geodesics. In turn, in the above-mentioned case, as well as in the cases discussed below, with a non-null initial angular derivative ${\dot{\phi}}_{P}\ne 0$ the geodesics are not allowed to either touch the ρ conical intersection or to pass through onto the companion surface. This is determined by the avoidance of a singularity in the last term of Equation (13), or, in other words, by the physical denial of infinite angular momentum in the conservation law from (12).

_{−}(ρ), the geodesic has a part that approaches an orbit near the conical intersection. The larger the initial radial derivative, the closer the loop comes to the conical intersection, as seen in frames (a) and (b) in Figure 3, where the blue line climbs to a smaller radius and performs a complete cycle around the conical intersection, while the green one escapes before completing a cycle. Both branches evolve, in the long term, towards the upper part of e

_{_}(ρ) at ρ > ρ

_{min}. In the frames (c) and (d), both the blue and green lines perform a tour of the conical intersection, although, in the (d) panel, corresponding to the point P on the e

_{+}(ρ) sheet, this is not easily visible because of the small radius near the conical point.

_{+}(ρ) or e

_{−}(ρ), this fact being possible only in the particular condition shown in Figure 2.

_{±}(ρ) = ±V ρ. Note that a cone is obtained if we take the half-difference of the eigenvalues: V ρ = (e

_{+}(ρ) − e

_{−}(ρ))/2.

_{0}= r

_{0}, φ

_{0}= 0, with the initial gradients $\dot{\rho}={r}_{0}{p}_{0}\mathrm{sin}\left({u}_{0}\right)/\sqrt{{V}^{2}+1}$ and $\dot{\phi}={p}_{0}\mathrm{cos}\left({u}_{0}\right)$, can be ascribed as follows:

_{0}to define a ratio between the linear tangential and radial velocities if the evolution parameter t is formally regarded as time.

#### 3.3. Quantum Chemical Calculation of the Jahn–Teller Effect in Triangular Molecular Systems

_{3}. However, although it presents a conical intersection, the potential energy surface of H

_{3}does not show bonded minima [39,40], the system being prone to rapid dissociation in the molecular and atomic hydrogen H

_{2}+ H, or to ionization, the H

_{3}

^{+}cation being a well-known molecule in the interstellar space [41]. Then, the next molecule on the simplicity scale would be the cyclo-propenyl radical C

_{3}H

_{3}. This molecule can be described as a triangle composed of carbon atoms bordered by a triangle composed of hydrogens bonded to carbons. According to chemical intuition, the Jahn–Teller effect is due to the so-called π electrons on the C

_{3}unit, the H

_{3}moiety just following the distortion of the carbon frame.

_{3}H

_{3}radical [51] used highly rated computational methods, the so-called coupled cluster routines [52], which, although not suited for multi-configurational and degenerate wave functions, are good at tackling the optimal distorted molecular geometries at the energy minima of the Jahn–Teller problem at hand.

_{3}H

_{3}have not been presented previously.

^{2}E″ term in the case of the π-type electronic system of the cyclo-propenyl in the D

_{3h}symmetry point group. The molecular term has the same symmetry as those of the singly occupied degenerate orbital ascribed conventionally the non-capitalized label e″ (see the scheme in the left-bottom part of Figure 5). The double degeneracy of the

^{2}E″ state comes from the two equivalent possibilities to place the unpaired electron in the two lodges of the e″ orbital set. One may see in the qualitative scheme shown in the lower part of Figure 5 that the distortion leads to the removal of orbital degeneracy (energy equivalence) in the e″ orbitals. The stable molecular geometry is an isosceles triangle with one elongated edge, the frontier orbital having the a

_{2}representation that gives rise to the corresponding

^{2}A

_{2}ground state in the C

_{2v}point group.

_{3}H

_{3}system. In this view, we did a relaxed geometry scan, where the structure of the C

_{3}moiety is generated by tuning the distortion coordinates having the e′ symmetry, while the hydrogen atoms, i.e., the C-H bond lengths and CCH bond angles, were the subject of gradient refinement to optimized values.

_{1}, R

_{2}, and R

_{3}, along axes fixed at mutual 120° angles. In spite of the constrained directions, the degrees of freedom are sufficient for defining any triangular configuration. In principle, even a negative R

_{i}can be admitted, but such an extreme evolution is not needed in our problem since relatively small displacements from the equilateral reference are expected when ${R}_{1}={R}_{2}={R}_{3}={l}_{CC}\left({D}_{3h}\right)/\sqrt{3}$ as a function of the carbon–carbon bond-length in the system at the conical intersection. Then, we chose the symmetrized coordinates as a linear transformation over the above-defined radii:

_{−}(ρ) surface, the realistic system is warped in a trigonal pattern, showing three minima and three saddle points in the quasi-circular valley. To account for this, the model should be updated with second-order vibronic coupling terms [2,3], namely the W parameter in the new eigenvalue equations:

_{3}axis of the molecule. The angular dependence modulates three minima and three maxima around the former circular trough of the minimum from the previous simplified level of the model (with W = 0). Since on the bottom of the valley the curvature with respect to the radial variable remains positive (corresponding to the minimum), the maxima with respect to the angular parameter (negative curvature) are actually saddle points. The absolute minima (the three equivalent positions at φ = 0 and ±120°) show a positive curvature (the second derivative of the above-defined eigenvalues) with respect to both the ρ and φ variables.

^{−1}Å

^{−2}, V = 102.996 kcal·mol

^{−1}Å

^{−1}, and W = 34.349 kcal

^{2}·mol

^{−2}Å

^{−4}. The goodness of the fit can be measured by the linearity in the representation of the e

_{calc}ab initio calculated energies vs. the e

_{fit}values fitted by Equation (18). The least-square line e

_{fit}= a·e

_{calc}+ b has a slope a = 1.001 and an R

^{2}= 0.994, close to the ideal unity values, while the intercept is small (b = 0.202 kcal/mol).

## 4. Conclusions

_{3}H

_{3}undergoing spontaneous distortion from the conceivably higher geometry of the equilateral triangle. We performed a CASSCF-based complete mapping of the two potential surfaces originating from the E term in D

_{3h}symmetry as a function of e-type distortion coordinates. The realistic surfaces show a trigonal warping fitted by the intervention of the second-order vibronic coupling parameter. Here, we did not explore the case of the second-order coupling in the geodesics analysis; however, we aim to devote further systematic attention to this class of problems and plan to extend the analysis to other prototypic Jahn–Teller and related effects.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Jahn, H.A.; Teller, E. Stability of polyatomic molecules in degenerate electronic states-I—Orbital degeneracy. Proc. R. Soc. Lond. A
**1937**, 161, 220–235. [Google Scholar] - Bersuker, I.B. The Jahn–Teller Effect and Vibronic Interactions in Modern Chemistry; Plenum Press: New York, NY, USA, 1984. [Google Scholar]
- Bersuker, I.B.; Polinger, V.Z. Vibronic Interactions in Molecules and Crystals; Springer: Berling, Germany, 1989. [Google Scholar]
- Born, M.; Oppenheimer, R. Zur Quantentheorie der Molekeln. Ann. Physik
**1927**, 389, 457–484. [Google Scholar] - Jensen, F. Introduction to Computational Chemistry; Wiley: Chichester, UK, 2007. [Google Scholar]
- Putz, M.V.; Cimpoesu, F.; Ferbinteanu, M. Structural Chemistry, Principles, Methods, and Case Studies; Springer: Cham, Switzerland, 2018. [Google Scholar]
- Bersuker, I.B. Pseudo-Jahn–Teller effect: A two-state paradigm in formation, deformation, and transformation of molecular systems and solids. Chem. Rev.
**2013**, 113, 1351–1390. [Google Scholar] [CrossRef] [PubMed] - Martin, R.M. Electronic Structure: Basic Theory and Practical Methods; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Cotton, F.A. Chemical Applications of Group Theory, 3rd ed.; Wiley: New York, NY, USA, 1990. [Google Scholar]
- Bunker, P.R.; Jensen, P. Fundamentals of Molecular Symmetry; CRC Press: Boca Raton, FL, USA, 2005. [Google Scholar]
- Hoffmann, M.R.; Laidig, W.D.; Kim, K.S.; Fox, D.J.; Schaefer, H.F., III. Electronic symmetry breaking in polyatomic molecules. Multiconfiguration self-consistent field study of the cyclopropenyl radical C
_{3}H_{3}. J. Chem. Phys.**1984**, 80, 338–343. [Google Scholar] [CrossRef][Green Version] - Glukhovtsev, M.N.; Laiter, S.; Pross, A. Thermochemical assessment of the aromatic and antiaromatic characters of the cyclopropenyl cation, cyclopropenyl anion, and cyclopropenyl radical: A high-level computational study. J. Phys. Chem.
**1996**, 100, 17801–17806. [Google Scholar] [CrossRef] - Moffitt, W.; Thorson, W. Vibronic states of octahedral complexes. Phys. Rev.
**1957**, 108, 1251–1255. [Google Scholar] [CrossRef] - Deeth, R.J.; Hitchman, M.A. Factors influencing Jahn–Teller distortions in six-coordinate copper (II) and low-spin nickel (II) complexes. Inorg. Chem.
**1985**, 25, 1225–1233. [Google Scholar] [CrossRef] - O’Brien, M.C.M. Dynamic Jahn–Teller effect in octahedrally Co-ordinated d
^{9}ions. Proc. R. Soc. Lond. A**1964**, 281, 323–339. [Google Scholar] - Lee, J.H.; Delaney, K.T.; Bousquet, E.; Spaldin, N.A.; Rabe, K.M. Strong coupling of Jahn–Teller distortion to oxygen-octahedron rotation and functional properties in epitaxially strained orthorhombic LaMnO
_{3}. Phys. Rev. B**2013**, 88, 174426. [Google Scholar] [CrossRef][Green Version] - Berry, M.V. Quantal phase factors accompanying adiabatic changes. Proc. Roy. Soc. Lond. A
**1984**, 392, 45–57. [Google Scholar] - Cimpoesu, F.; Ferbinteanu, M.; Humelnicu, I.; Mihai, A. The symmetry blueprints of the molecular edifices. Symmetry Cult. Sci.
**2008**, 19, 397–414. [Google Scholar] - Toader, A.M.; Buta, C.M.; Frecus, B.; Mischie, A.; Cimpoesu, F. Valence bond account of triangular polyaromatic hydrocarbons with spin: Combining ab initio and phenomenological approaches. J. Phys. Chem. C
**2019**, 123, 6869–6880. [Google Scholar] [CrossRef] - Buta, M.C.; Frecus, B.; Enache, M.; Humelnicu, I.; Toader, A.M.; Cimpoesu, F. Intra- and inter-molecular spin coupling in phenalenyl dimeric systems. J. Phys. Chem. A
**2021**, 125, 6893–6901. [Google Scholar] [CrossRef] - Toader, A.M.; Buta, M.C.; Mischie, A.; Putz, M.V.; Cimpoesu, F. The density functional theory account of interplaying long-range exchange and dispersion effects in supramolecular assemblies of aromatic hydrocarbons with spin. Molecules
**2022**, 27, 45. [Google Scholar] [CrossRef] - Toader, A.M.; Buta, M.C.; Maftei, D.; Putz, M.V.; Cimpoesu, F. Atoms in generalized orbital configurations: Towards atom-dedicated density functionals. Int. J. Mol. Sci.
**2019**, 20, 5943. [Google Scholar] [CrossRef][Green Version] - Jianu, M.; Achimescu, S.; Daus, L.; Mihai, A.; Roman, O.A.; Tudor, D. Characterization of rectifying curves by their involutes and evolutes. Mathematics
**2021**, 9, 3077. [Google Scholar] [CrossRef] - Aydin, M.E.; Mihai, A. A note on surfaces in space forms with Pythagorean fundamental forms. Mathematics
**2020**, 8, 444. [Google Scholar] [CrossRef][Green Version] - Aydin, M.E.; Mihai, A. Ruled surfaces generated by elliptic cylindrical curves in the isotropic space. Georgian Math. J.
**2019**, 26, 331–340. [Google Scholar] [CrossRef] - Wolfram, S. The Mathematica Book, 5th ed.; Wolfram-Media: Champaign, IL, USA, 2003. [Google Scholar]
- Mathematica Software, Version 13.0; Wolfram Research Inc.: Champaign, IL, USA, 2014.
- MATLAB, Version 6; The MathWorks Inc.: Natick, MA, USA, 2000.
- Eaton, J.W.; Bateman, D.; Hauberg, S.; Wehbring, R. GNU Octave, Version 3.8.1; 2014. Available online: https://www.gnu.org/software/octave/ (accessed on 20 January 2022).
- Schmidt, M.W.; Baldridge, K.K.; Boatz, J.A.; Elbert, S.T.; Gordon, M.S.; Jensen, J.H.; Koseki, S.; Matsunaga, N.; Nguyen, K.A.; Su, S.; et al. General atomic and molecular electronic structure system. J. Comput. Chem.
**1993**, 14, 1347–1363. [Google Scholar] [CrossRef] - Gordon, M.S.; Schmidt, M.W. Advances In electronic structure theory: GAMESS a decade later. In Theory and Applications of Computational Chemistry, the First Forty Years; Dykstra, C.E., Frenking, G., Kim, K.S., Scuseria, G.E., Eds.; Elsevier: Amsterdam, The Netherlands, 2005; Chapter 41; pp. 1167–1189. [Google Scholar]
- Dunning, T.H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys.
**1989**, 90, 1007–1023. [Google Scholar] [CrossRef] - Hilderbrandt, R.L. Cartesian Coordinates of Molecular Models. J. Chem. Phys.
**1969**, 51, 1654–1659. [Google Scholar] [CrossRef] - Schaftenaar, G.; Noordik, J.H. Molden: A pre- and post-processing program for molecular and electronic structures. J. Comput.-Aided Mol. Design.
**2000**, 14, 123–134. [Google Scholar] [CrossRef] - do Carmo, M.P. Differential Geometry of Curves and Surfaces; Prentice-Hall: Hoboken, NJ, USA, 1976. [Google Scholar]
- Applegate, B.E.; Barckholtz, T.A.; Miller, T.A. Explorations of conical intersections and their ramifications for chemistry through the Jahn—Teller effect. Chem. Soc. Revs.
**2003**, 32, 38–49. [Google Scholar] [CrossRef] - Domcke, W.; Yarkony, D.R.; Koppel, H. (Eds.) Conical Intersections: Electronic Structure, Dynamics and Spectroscopy; World Science: Singapore, 2004. [Google Scholar]
- Lipschutz, M. Schaum’s Outline of Theory and Problems of Differential Geometry; McGraw-Hill Book Company: New York, NY, USA, 1969. [Google Scholar]
- Mahapatra, S.; Köppel, G. Quantum mechanical study of optical emission spectra of Rydberg-excited H
_{3}and its isotopomers. Phys. Rev. Lett.**1998**, 81, 3116–3120. [Google Scholar] [CrossRef] - Mistrík, I.; Reichle, R.; Helm, H.; Müller, U. Predissociation of H
_{3}Rydberg states. Phys. Rev. A**2001**, 63, 042711. [Google Scholar] [CrossRef] - Oka, T. Interstellar H
_{3}^{+}. Chem. Rev.**2013**, 113, 8738–8761. [Google Scholar] [CrossRef] - Wheeler, S.E.; Robertson, K.A.; Allen, W.D.; Schaefer, H.F.; Bomble, Y.J.; Stanton, J.F. Thermochemistry of key soot formation intermediates: C
_{3}H_{3}isomers. J. Phys. Chem. A**2007**, 111, 3819–3830. [Google Scholar] [CrossRef] - Cirelli, G.; Graf, F.; Günthard, H.H. ESR spectrum of the cyclopropenyl radical. Chem. Phys. Lett.
**1974**, 28, 494–496. [Google Scholar] [CrossRef] - Closs, G.L.; Redwine, O.D. Characterization of matrix- isolated cyclopropen-3-yl by EPR Spectroscopy. J. Am. Chem. Soc.
**1986**, 108, 506–507. [Google Scholar] [CrossRef] - Closs, G.L.; Evanochko, W.T.; Norris, J.R. Structure and dynamics of the trimethylcyclopropenyl radical as determined by electron and nuclear magnetic resonance. J. Am. Chem. Soc.
**1982**, 104, 350–352. [Google Scholar] [CrossRef] - Schreiner, K.; Ahrens, W.; Berndt, A. ESR proof of the antiaromaticity of a cyclopropenyl radical. Angew. Chem. Int. Ed. Engl.
**1975**, 14, 550–551. [Google Scholar] [CrossRef] - Davidson, E.R.; Borden, W.T. The potential surface for planar cyclopropenyl radical and anion. J. Chem. Phys.
**1977**, 67, 2191–2196. [Google Scholar] [CrossRef] - Poppinger, D.; Radom, L.; Vincent, M.A. On the Jahn-Teller distortion in the cyclopropenyl radical. Chem. Phys.
**1977**, 23, 437–442. [Google Scholar] [CrossRef] - Chipman, D.M.; Miller, K.E. Theoretical study of the cyclopropenyl radical. J. Am. Chem. Soc.
**1984**, 106, 6236–6242. [Google Scholar] [CrossRef] - Tachibana, A.; Asai, Y.; Ikeuchi, S.; Ishikawa, S.; Yamabe, T. Isomorphic electron orbitals for vibronic flexibility in a cyclopropenyl radical molecular device. Theoret. Chim. Acta
**1990**, 78, 1–9. [Google Scholar] [CrossRef] - Guo, M.; Wang, Z.; Wang, F. Stationary points on potential energy surface of cyclic C
_{3}H_{3}with coupled-cluster approaches and density functional theory. J. Phys. Chem. A**2021**, 125, 4079–4088. [Google Scholar] [CrossRef] - Bartlett, R.J.; Musiał, M. Coupled-cluster theory in quantum chemistry. Rev. Mod. Phys.
**2007**, 79, 291–352. [Google Scholar] [CrossRef][Green Version] - Knowles, P.J.; Werner, H.-J. An efficient second-order MCSCF method for long configuration expansions. Chem. Phys. Lett.
**1985**, 115, 259–267. [Google Scholar] [CrossRef] - Schmidt, M.W.; Gordon, M.S. The construction and interpretation of MCSCF wavefunctions. Annu. Rev. Phys. Chem.
**1998**, 49, 233–266. [Google Scholar] [CrossRef][Green Version] - Kayi, H.; Garcia-Fernandez, P.; Bersuker, I.B.; Boggs, J.E. Deviations from Born-Oppenheimer Theory in Structural Chemistry: Jahn-Teller, Pseudo Jahn-Teller, and Hidden Pseudo Jahn-Teller Effects in C
_{3}H_{3}and C_{3}H_{3}^{−}. J. Phys. Chem. A**2013**, 117, 8671–8679. [Google Scholar] [CrossRef]

**Figure 1.**Geodesics with the ${\dot{\rho}}_{P}=0$ initial condition, qualitatively distinct as a function of the placement of the starting point ρ

_{P}. The panels (

**a**–

**c**) correspond to an initial point on the lower surface (ground-state), while (

**d**) corresponds to the point on the higher solution (excited state). The situations (

**a**–

**c**) for ρ

_{P}< ρ

_{min}, ρ

_{P}= ρ

_{min}, and ρ

_{P}> ρ

_{min}, respectively, where ρ

_{min}is the radius of the circular trough defining the minimum of the ground surface. In the case of (

**d**), the qualitative pattern is similar, irrespective of the position of the starting point. The initial point (ρ

_{P}, φ

_{P}= 0) occurs where the red and blue colors come into contact. The two halves of the geodesic, with different colors, result from the opposite signs of the non-null ${\dot{\phi}}_{P}\ne 0$ initial condition.

**Figure 2.**Geodesics with the ${\dot{\phi}}_{P}=0$ initial condition: (

**a**) the starting point is placed on the lower energy surface; (

**b**) the starting point is situated in the higher state. The two halves of the geodesic result from opposite signs of the non-null ${\dot{\rho}}_{P}\ne 0$ initial condition.

**Figure 3.**Upper view (along the vertical axis) of general geodesics, illustrated for a distinct starting point P: (

**a**) 0 < ρ

_{P}< ρ

_{min}on the e

_{−}(ρ) surface; (

**b**) ρ

_{P}= ρ

_{min}on the e

_{−}(ρ) surface; (

**c**) ρ

_{P}> ρ

_{min}on the e

_{−}(ρ) surface; (

**d**) any ρ

_{P}vs. ρ

_{min}’s relative placement on the e

_{+}(ρ) surface. The coloring of the curves discriminates qualitatively different initial conditions of the subsequent differential equations. The red curves represent the ${\dot{\rho}}_{P}=0$ initial derivative, being just a different view of the geodesics represented in Figure 1 (the merged red and blue halves from Figure 1 being shown here all in red). The green and blue geodesics stand for, respectively, in the order of increasing absolute value of the $\left|{\dot{\rho}}_{P}\right|$ radial velocity. The red, green, and blue curves start from the same point, their crossing in P being marked by a circle.

**Figure 4.**Different patterns for a geodesic on the cone. Left side: the case with the ${\dot{\rho}}_{P}=0$ initial condition. Right side: a situation with ${\dot{\rho}}_{P}>0.$ The red and blue halves starting from the initial point correspond to the ${\dot{\phi}}_{P}>0$ and ${\dot{\phi}}_{P}<0$ situations, respectively.

**Figure 5.**Optimized molecular geometries and qualitative π-type orbital schemes for the C

_{3}H

_{3}radical in a high-symmetry structure (left side, D

_{3h}point group) and at a distorted stationary point (right side, C

_{2v}point group). The bond lengths marked on molecular geometries are all in Ångstrom units (Å).

**Figure 6.**Left-side snippet: convened system for defining the vertices of general triangles by the lengths of three radii fixed at mutual 120° angles. The following frames mark the displacements along the $\overrightarrow{{R}_{1}}$, $\overrightarrow{{R}_{2}}$, and $\overrightarrow{{R}_{3}}$ vectors corresponding to the total symmetry at the a

_{1}coordinate and the two degenerate elements of the e′ representation, respectively.

**Figure 7.**Computed relaxed potential energy surfaces emulating the E⊗e Jahn–Teller effect in the cyclo-propenyl radical C

_{3}H

_{3}. The triangles annotated on the extrema points at the bottom of the lowest surface qualitatively suggest the distortion trend. The green triangles with open angles correspond to the minima (their absolute geometry being realistically represented on the right side of Figure 5). The acute red triangles stand for saddle points around the circular trough. The geometry at the conical intersection is the equilateral triangle (realistically shown on the left side of Figure 5). The relative energy is given in kcal/mol, with the zero value fixed at the conical intersection. The distortion coordinates are presented in Ångstrom units.

**Figure 8.**Example of a geodesic on the model potential energy adapted from the parameters fitted for the cyclo-propenyl radical. (

**a**) The surface represented by the mesh corresponds to the lowest eigenvalue from the model with the fitted K, V, and W parameters, the solid transparent surface results taking the fitted K and V parameters and imposing W = 0. The red and blue curves evolving on the solid surface correspond to the symmetrical branches of a geodesic starting at ρ

_{P}= 0.05 Å and φ

_{P}= 0 with the ${\dot{\rho}}_{P}=0$ and ${\dot{\phi}}_{P}=1.0$ initial conditions. (

**b**) The lengths of the edges in the C

_{3}triangle along the geodesic sequence presented in the red line from the left-side panel as a function of the conventional evolution parameter t. The blue half of the geodesic has the same evolution if we permute the C1-C3 vs. C1-C2 bond length values. The atom labels are given in Figure 5. The vertical dashed line passes through the minimum of the surface. The small differences between the values of bond lengths at the minimum inferred from this graph vs. those from quantum optimization, described in Figure 5, come from the W = 0 approximation enforced in the actual case.

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Cimpoesu, F.; Mihai, A.
Characterizing the *E*⊗*e* Jahn–Teller Potential Energy Surfaces by Differential Geometry Tools. *Symmetry* **2022**, *14*, 436.
https://doi.org/10.3390/sym14030436

**AMA Style**

Cimpoesu F, Mihai A.
Characterizing the *E*⊗*e* Jahn–Teller Potential Energy Surfaces by Differential Geometry Tools. *Symmetry*. 2022; 14(3):436.
https://doi.org/10.3390/sym14030436

**Chicago/Turabian Style**

Cimpoesu, Fanica, and Adela Mihai.
2022. "Characterizing the *E*⊗*e* Jahn–Teller Potential Energy Surfaces by Differential Geometry Tools" *Symmetry* 14, no. 3: 436.
https://doi.org/10.3390/sym14030436