Renner-Teller Intersections Research Articles

Spectroscopy and heats of formation of CXI (X = Br, Cl, F) iodocarbenes: quantum chemical characterisation of the X̃(1 A′), ã(3 A″) and Ã(1 A″) states

The equilibrium energies of the iodocarbenes CXI (X = Br, Cl, F) in their , and states and their atomisation and dissociation energies in the complete basis limit were determined by extrapolating valence correlated (R/U)CCSD(T) and Davidson corrected multi-reference configuration interaction (MRCI) energies calculated with the aug-cc-pVxZ (x = T,Q,5) basis sets and the ECP28MDF pseudopotential of iodine plus corrections for core and core–valence correlation, scalar relativity, spin–orbit coupling and zero-point energies. Spin–orbit energies were computed in a large basis of configurations chosen so as to accurately describe dissociation to the 3P and 2P states of C and of the halogens X and I, respectively. The computed singlet–triplet splittings are 13.6, 14.4 and 27.3 kcal mol−1 for X = Br, Cl and F, respectively. The enthalpies of formation at 0 K are predicted to be 97.4, 82.6 and 38.1 kcal mol−1 with estimated errors of ±1.0 kcal mol−1. The excitation energies (T00) in CBrI and CClI are calculated to be 41.1 and 41.7 kcal mol−1, respectively. The Renner–Teller intersections in both molecules are predicted to be substantially higher than the dissociation barriers on the surfaces. By contrast, in CFI the state is found to be unbound with respect to dissociation.

Quantum chemical characterization of the $${\tilde{X}}(^{1} A_{1} )$$ X ~ ( 1 A 1 ) , $${\tilde{a}}(^{3} B_{1} )$$ a ~ ( 3 B 1 ) , $${\tilde{A}}(^{1} B_{1} )$$ A ~ ( 1 B 1 ) and $${\tilde{B}}(2{}^{1}A_{1} )$$ B ~ ( 2 1 A 1 ) states of diiodomethylene and the enthalpies of formation of diiodomethylene, iodomethylene and

The equilibrium energies of the \(\tilde{X}(^{1}A_{1}),\tilde{a}(^{3}B_{1}),\tilde{A}(^{1}B_{1})\; \mathrm{and}\;\tilde{B}(2^{1}A_{1})\) states of diiodomethylene (CI2) and its atomization and dissociation energies in the complete basis limit were determined by extrapolating valence correlated (R/U)CCSD(T) and Davidson corrected multi-reference configuration interaction energies calculated with the aug–cc–pVxZ (x = T, Q, 5) basis sets and the ECP28MDF pseudopotential of iodine plus corrections for core and core-valence correlation, scalar relativity, spin–orbit coupling and zero-point energies. The geometries and vibrational frequencies were obtained at the CCSD and complete active space second-order perturbation theory levels of theory with the cc–pVTZ basis. Spin– orbit energies were computed in a large basis of configurations chosen so as to accurately describe dissociation to the 3P and 2P states of C and I, respectively. These computations were extended to iodomethylene (CHI) and iodomethylidyne (CI), resulting in small corrections to the thermochemistry and the singlet–triplet gap of CHI computed previously. The onset (T00) of the \(\tilde{A}\leftarrow \tilde{X}\) excitations in CI2 is predicted to be 12,680 cm-1. The Renner–Teller intersection is computed to have a substantially lower energy (6.5 kcal mol-1) than the dissociation barrier on the \(\tilde{A}\) surface, thus internal relaxation via Renner–Teller coupling is expected to be the dominant photochemical channel. The predicted enthalpies of formation of CI2, CHI and CI in their ground states at 0 K are 109.1 ± 1, 102.8 ± 1 and 132.9 ± 1 kcal mol-1, respectively. The computed singlet–triplet gaps in CI2 and CHI are 11.1 and 4.4 kcal mol-1, respectively.

Performance of Tamm-Dancoff approximation on nonadiabatic couplings by time-dependent density functional theory

The Tamm-Dancoff approximation (TDA), widely used in physics to decouple excitations and de-excitations, is well known to be good for the calculation of excitation energies but not for oscillator strengths. In particular, the sum rule is violated in the latter case. The same concern arises within the TDA in the calculation of nonadiabatic couplings (NACs) by time-dependent density functional theory (TDDFT), due to the similarities in the TDDFT formulations of NACs and oscillator strengths [C. Hu, H. Hirai, and O. Sugino, J. Chem. Phys. 127, 064103 (2007)]. In this study, we present a systematic evaluation of the performance of TDDFT/TDA for the calculation of NACs. In the cases we considered, including a variety of systems possessing Jahn-Teller and Renner-Teller intersections, as well as an example with accidental conical intersections, it is found that the TDDFT/TDA performs better than the full TDDFT, contrary to the conjecture that the TDA might cause the NAC results to deteriorate and violate the sum rule. The surprisingly good performance of the TDA for NACs is probably because the TDA can partially compensate for the local-density-approximation error and give better excitation energies in the vicinity of intersections of potential energy surfaces. Our study also shows that it is important to use the TDA based on the rigorous full-TDDFT formulation of NACs, instead of using it based on an alternative approximate formulation.

Nonadiabatic couplings from time-dependent density functional theory: Formulation by the Kohn-Sham derivative matrix within density functional perturbation theory

We present an improved method for accurately calculating nonadiabatic couplings (NACs) between the ground and excited states of electronic systems, using time-dependent density functional theory (TDDFT) within the plane-wave pseudopotential framework. Based on our previous work that the TDDFT formulation of NACs by the Kohn-Sham derivative matrix elements can avoid the accuracy problem caused by the nonlocal pseudopotentials, we extend the evaluation of these matrix elements to the analytical scheme, achieved by density functional perturbation theory (DFPT). As DFPT has been a standard implementation in many ab initio codes, our scheme requires little effort in the coding. Extensive calculations of NACs near various Jahn-Teller or Renner-Teller intersections show the good accuracy of our results. A comparison of our present and previous work has also clarified an important fact in the TDDFT computation of NACs, regarding the evaluation of the nuclear derivative of the many-body Hamiltonian after mapping to the Kohn-Sham system.

TDDFT study on quantization behaviors of nonadiabatic couplings in polyatomic systems

AbstractWe present extensive calculations on the quantization behaviors of first‐order nonadiabatic couplings (NAC) in polyatomic systems, using the method based on time‐dependent density functional theory (TDDFT). Along circular contours around Jahn–Teller conical intersections at various symmetries, such as D4h and D6h, the angular NAC are found to show apparent dependence on the contour angle, no matter how close the conical intersections are approached. This is in contrast to that the angular NAC in triatomic Jahn–Teller systems (D3h) converge to the quantized value of 1/2. The oscillatory feature of our calculated curves qualitatively resembles the analytic result of angular NAC around elliptic conical intersections in the small‐displacement limit. Furthermore, by performing the integral along the whole contour, the quantization rule on the integral of the angular NAC holds for all cases, which gives the geometric phase with a value of π. On the other hand, the angular NAC in the immediate vicinity of Renner–Teller intersections are shown to be always 1. This observation can be understood from the fact that the molecular symmetry property of the Renner–Teller intersections is always determined by the linear atomic chain. Further analysis of our results on Renner–Teller systems shows that although the quantization behavior of angular NAC can determine all NAC vectors on each atom for the triatomic systems, it cannot for the systems with more than three atoms. The NAC vectors of the latter systems are found to show both common and distinct features with respect to the atomic number and species by our TDDFT calculations. © 2012 Wiley Periodicals, Inc.

Second-order nonadiabatic couplings from time-dependent density functional theory: Evaluation in the immediate vicinity of Jahn-Teller/Renner-Teller intersections

For a rigorous quantum simulation of nonadiabatic dynamics of electrons and nuclei, knowledge of not only the first-order but also the second-order nonadiabatic couplings (NACs) is required. Here, we propose a method to efficiently calculate the second-order NAC from time-dependent density functional theory (TDDFT), on the basis of the Casida ansatz adapted for the computation of first-order NAC, which has been justified in our previous work and can be shown to be valid for calculating second-order NAC between ground state and singly excited states within the Tamm-Dancoff approximation. Test calculations of the second-order NAC in the immediate vicinity of Jahn-Teller and Renner-Teller intersections show that calculation results from TDDFT, combined with modified linear response theory, agree well with the prediction from the Jahn-Teller/Renner-Teller models. Contrary to the diverging behavior of the first-order NAC near all types of intersection points, the Cartesian components of the second-order NAC are shown to be negligibly small near Renner-Teller glancing intersections, while they are significantly large near the Jahn-Teller conical intersections. Nevertheless, the components of the second-order NAC can cancel each other to a large extent in Jahn-Teller systems, indicating the background of neglecting the second-order NAC in practical dynamics simulations. On the other hand, it is shown that such a cancellation becomes less effective in an elliptic Jahn-Teller system and thus the role of second-order NAC needs to be evaluated in the rigorous framework. Our study shows that TDDFT is promising to provide accurate data of NAC for full quantum mechanical simulation of nonadiabatic processes.

Nonadiabatic couplings from the Kohn-Sham derivative matrix: Formulation by time-dependent density-functional theory and evaluation in the pseudopotential framework

We study the time-dependent density-functional theory formulation of nonadiabatic couplings (NAC's) to settle problems regarding practical calculations. NAC's have so far been rigorously formulated on the basis of the density response scheme and expressed using the nuclear derivative of the Hamiltonian, $\ensuremath{\partial}H/\ensuremath{\partial}R$, whereby causing the pseudopotential problem. When rewritten using the nuclear derivative operator, $\ensuremath{\partial}/\ensuremath{\partial}R$, or the $d$ operator, the formula is found free of the problem and thus provides a working numerical scheme. The $d$-operator-based formulation also allows us to lay a foundation on the empirical Slater transition-state method and to show an improved way of using the auxiliary excited-state wave-function ansatz, both of which have been utilized in previous works. Evaluation of NAC near either the Jahn-Teller or the Renner-Teller intersection in various molecular systems shows that the values of NAC are much improved over previous calculations when the $d$-operator formula is implemented in the pseudopotential framework.

Renner–Teller intersections along the collinear axes of polyatomic molecules: H2CN as a case study

The tetra-atomic C(2)H(2)(+) cation is known to form Renner-Teller-type intersections along its collinear axis. Not too long ago, we studied the nonadiabatic coupling terms (NACTs) of this molecule [G. J. Halász et al., J. Chem. Phys. 126, 154309 (2007)] and revealed two kinds of intersections. (i) By employing one of the hydrogens as a test particle, we revealed the fact that indeed the corresponding (angular) NACTs produce topological (Berry) phases that are equal to 2pi, which is a result anticipated in the case of Renner-Teller intersections. (ii) However, to our big surprise, repeating this study when one of the atoms (in this case a hydrogen) is shifted from the collinear arrangement yields for the corresponding topological phase a value that equals pi (and not 2pi). In other words, shifting (even slightly) one of the atoms from the collinear arrangement causes the intersection to change its character and become a Jahn-Teller intersection. This somewhat unexpected novel result was later further analyzed and confirmed by other groups [e.g., T. Vertesi and R. Englman, J. Phys. B 41, 025102 (2008)]. The present article is devoted to another tetra-atomic molecule, namely, the H(2)CN molecule, which just like the C(2)H(2)(+) ion, is characterized by Renner-Teller intersections along its collinear axis. Indeed, we revealed the existence of Renner-Teller intersections along the collinear axis, but in contrast to the C(2)H(2)(+) case a shift of one atom from the collinear arrangement did not form Jahn-Teller intersections. What we found instead is that the noncollinear molecule was not affected by the shift and kept its Renner-Teller character. Another issue treated in this article is the extension of (the two-state) Berry (topological) phase to situations with numerous strongly interacting states. So far the relevance of the Berry phase was tested for systems characterized by two isolated interacting states, although it is defined for any number of interacting states [M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984)]. We intend to show how to overcome this limitation and get a valid, fully justified definition of a Berry phase for an isolated system of any number of interacting states (as is expected).

A Born–Oppenheimer Expansion in a Neighborhood of a Renner–Teller Intersection

We perform a rigorous mathematical analysis of the bending modes of a linear triatomic molecule that exhibits the Renner–Teller effect. Assuming the potentials are smooth, we prove that the wave functions and energy levels have asymptotic expansions in powers of \({\epsilon}\) , where \({\epsilon^4}\) is the ratio of an electron mass to the mass of a nucleus. To prove the validity of the expansion, we must prove various properties of the leading order equations and their solutions. The leading order eigenvalue problem is analyzed in terms of a parameter \({\tilde{b}}\) , which is equivalent to the parameter originally used by Renner. For \({0< \tilde{b}< 1}\) , we prove self-adjointness of the leading order Hamiltonian, that it has purely discrete spectrum, and that its eigenfunctions and their derivatives decay exponentially. Perturbation theory and finite difference calculations suggest that the ground bending vibrational state is involved in a level crossing near \({\tilde{b}=0.925}\) . We also discuss the degeneracy of the eigenvalues. Because of the crossing, the ground state is degenerate for \({0< \tilde{b}< 0.925}\) and non-degenerate for \({0.925< \tilde{b}< }\) .

Photodissociation of CO2− in water clusters via Renner-Teller and conical interactions

The photochemistry of mass selected CO(2) (-)(H2O)(m), m=2-40 cluster anions is investigated using 266 nm photofragment spectroscopy and theoretical calculations. Similar to the previous 355 nm experiment [Habteyes et al., Chem. Phys. Lett. 424, 268 (2006)], the fragmentation at 266 nm yields two types of anionic products: O(-)(H2O)(m-k) (core-dissociation products) and CO(2) (-)(H2O)(m-k) (solvent-evaporation products). Despite the same product types, different electronic transitions and dissociation mechanisms are implicated at 355 and 266 nm. The 355 nm dissociation is initiated by excitation to the first excited electronic state of the CO(2) (-) cluster core, the 1 (2)B(1)(2A") state, and proceeds via a glancing Renner-Teller intersection with the ground electronic state at a linear geometry. The 266 nm dissociation involves the second excited electronic state of CO(2) (-), the 2 (2)A(1)(2A') state, which exhibits a conical intersection with the 3 (2)B(2)(A') state at a bent geometry. The asymptotic O(-) based products are believed to be formed via this 3 (2)B(2)(A') state. By analyzing the fragmentation results, the bond dissociation energy of CO(2) (-) to O(-)+CO in hydrated clusters (m> or =20) is estimated as 2.49 eV, compared to 3.46 eV for bare CO(2) (-). The enthalpy of evaporation of one water molecule from asymptotically large CO(2) (-)(H(2)O)(m) clusters is determined to be 0.466+/-0.001 eV (45.0+/-0.1 kJ/mol). This result compares very favorably with the heat of evaporation of bulk water, 0.456 eV (43.98 kJ/mol).

Scattering at the intersections of potentials

The scattering process influenced by the intersection of potential energy surfaces is studied. Exact quantum-mechanical solutions to the two-dimensional two- and three-state models have been found for the collision energy equal to the potential energy at the intersection point. The models considered are the Jahn-Teller and Renner-Teller intersections and the triple intersection of the double cone with a plane. Closed analytical expressions for the scattering amplitudes have been obtained. The results reveal the geometric phase effect and the nonadiabatic phase impact on the differential cross section.

Laser-induced fluorescence excitation and dispersed fluorescence spectroscopy of the Ã(1B1)–X̃(1A1) transition of dichlorocarbene

The A(1B1)–(1A1) transition of jet-cooled CCl2 has been investigated using both laser-induced fluorescence (LIF) excitation and dispersed fluorescence (DF) spectroscopy. DF spectra were taken from over 90 emitting vibronic states over the breadth of the LIF spectrum. 68 ground state vibrational levels were assigned in the C35Cl2 isotopomer and 47 in C35Cl37Cl. The ground state frequencies were fit to an anharmonic potential yielding: ω″1 = 736 cm−1, ω″2 = 338 cm−1, ν″3 = 760 cm−1, x″11 = −3.19 cm−1, x″22 = −0.29 cm−1, x″12 = −1.80 cm−1, x″13 = −5.70 cm−1 and x″23 = −3.80 cm−1. This is the first gas phase measurement of ν″3. Despite the close frequency match between ν″1 and 2ν″2 there was no evidence for Fermi resonance. The intensities of transitions in the DF spectra were analysed to identify the nature of the emitting state. Levels below T00 + 2500 cm−1 in the A state could be assigned simply as combinations and overtones of ν′1 and ν′2. Between T00 + 2500 and T00 + 5300 cm−1 the emission revealed an increasing mixing between levels within the same polyad, i.e., (m,n,0), (m + 1,n − 2,0), (m + 2,n − 4,0) etc, presumably due to Fermi resonance. This mixing becomes so extensive that simple assignment of the emitting states is not possible. Above T00 + 5300 cm−1 only K′ = 0 states could be identified, indicating that the Renner–Teller intersection between the and A states had been exceeded. This intersection is therefore 22 600 cm−1 above the ground state, in excellent agreement with a previous theoretical calculation of 23 000 cm−1.

Electronic spectroscopy of jet-cooled CFCl: Laser-induced fluorescence, dispersed fluorescence, lifetimes, and C–Cl dissociation barrier

The Ã(1A″)–X̃(1A′) transition of jet-cooled chlorofluorocarbene (CFCl) has been measured by laser-induced fluorescence (LIF) excitation and dispersed fluorescence spectroscopy. Over 170 vibronic transitions were measured in the LIF spectrum, consisting of cold bands and hot bands of carbenes containing both Cl35 and Cl37 isotopes. Dispersed fluorescence spectroscopy was used both to map the ground-state vibrational levels and to provide confirmation of the vibronic identity of the emitting level. A predictor–corrector method was used to progressively assign almost all of the vibronic transitions, resulting in the positive assignment and measurement of almost every bound vibrational state within the Ã-state manifold. The vibrational structure is modeled well by a Morse potential with frequencies ν1′=1229 cm−1, ω2′=399.2 cm−1, and ω3′=748.0 cm−1 for CF 35Cl and 1235 cm−1, 397.0 cm−1, and 744.5 cm−1 for the same three vibrations in CF 37Cl. The standard diagonal and cross-anharmonicity constants for a three-coordinate Morse potential were also measured for each isotopic species. Dispersed fluorescence spectroscopy provided a map of ground-state vibrational levels up to about 4000 cm−1. Franck–Condon factors were modeled well by a simple, one-dimensional harmonic potential, and these were also used to confirm assignment of many transitions. The fluorescence lifetime of the excited vibronic states decreased markedly from a consistent 650 ns for most states, to &amp;lt;20 ns for the highest lying observed state. In addition, the Franck–Condon analysis indicates that higher lying members of progressions were missing in the LIF spectrum. This strongly indicated the presence of a nonradiative pathway that opens for energies above T00+4073 cm−1. Analysis of the rotational structure of many transitions indicated that the molecule was not reaching the Renner–Teller intersection, where the à and X̃ states are degenerate. We attribute the nonradiative channel to cleavage of the C–Cl bond directly on the à state, in exact analogy with the observed process in CFBr. The height of the barrier, and the vibrational frequencies are all in reasonable agreement with recent ab initio values.

Rotational State Dependence of the Predissociation Dynamics in H2O, D2O (C̃1B1 and β1A1)

The two photon photodissociation of H2O, D2O (C̃1B1) from individually selected J'Ka'Kc' rotational levels in the (000) vibrational level has been investigated experimentally using a line-narrowed, tunable KrF laser. The results establish a strong dependence on the magnitude of 〈Ja'2〉1/2 ≅ Ka', both for the heterogeneous predissociation path and for the subsequent branching on the β1'A1 potential at the Renner-Teller intersection with Ã1B1 near linearity.