Spin—orbit and vibronic perturbations on the chiroptical spectra of dissymmetric pseudo-tetragonal metal complexes

The influence of spin—orbit and vibronic interactions upon the chiroptical properties of nearly degenerate dd transitions in metal complexes of pseudo-tetragonal symmetry is investigated. A model system is considered in which three nearly degenerate dd excited states are coupled via both spinorbit and vibronic interactions. Vibronic interactions among the three nearly degenerate dd electronic states are assumed to arise from a pseudo-Jahn—Teller (PJT) mechanism involving three different vibrational modes (each nontotally symmetric in the point group of the undistorted model system).A vibronic hamiltonian is constructed (for the excited states of the model system) which includes linear coupling terms in each of the three PJT-active vibrational modes as well as a linear coupling term in one totally symmetric mode of the system and a spin—orbit interaction term. Wavefunctions and eigenvalues for the spin—orbit/vibronic perturbed excited states. of the model system are obtained by diagonalizing this hamiltonian in a basis constructed of uncoupled vibrational and electronic (orbital and spin) wavefunctions.Rotatory strengths associated with transitions to vibronic levels of the perturbed system are calculated and “rotatory strength spectra” are computed assuming gaussian shaped vibronic spectral components. Calculations are carried out for a number of vibronic and spin—orbit coupling parameters and for various splitting energies between the interacting electronic states. The calculated results suggest that chiroptical spectra associated with transitions to a set of nearly degenerate dd excited states of a chiral transition metal complex cannot be interpreted directly without some consideration of the effects introduced by spin—orbit and vibronic perturbations. These perturbations can lead to substantial alterations in the sign patterns and intensity distributions of rotatory strength among vibronic levels derived from the interacting electronic states and it is generally not valid to assign specific features in the observed circular dichroism spectra to transitions between states with well-defined electronic (orbital and spin) identities.Our theoretical model is conservative with respect to the total (or net) rotatory strength associated with transitions to levels derived from the three interacting electronic states; the vibronic and spin—orbit coupling operators are operative only within this set of states. That is, the total (or net) rotatory strength associated with these transitions remains invariant to the vibronic and spin—orbit coupling parameters of the model.     

Vibronic effects in elastic scattering of electrons

A method for obtaining general equations for the scattered intensities from vibronic systems is given. The approximate formulas obtained are used to calculate the effects on the electron diffraction pattern for molecules with doubly degenerate electronic E terms interacting with e-type vibrations (Ee-type of problem). The results of the approximate calculations are compared to more precise results, based on numerical solution of the vibronic problem.     

Multimode Jahn-Teller and pseudo-Jahn-Teller interactions in the cyclopropane radical cation: complex vibronic spectra and nonradiative decay dynamics

The complex vibronic spectra and the nonradiative decay dynamics of the cyclopropane radical cation (CP+) are simulated theoretically with the aid of a time-dependent wave packet propagation approach using the multireference time-dependent Hartree scheme. The theoretical results are compared with the experimental photoelectron spectrum of cyclopropane. The ground and first excited electronic states of CP+ are of X2E' and A2E' type, respectively. Each of these degenerate electronic states undergoes Jahn-Teller (JT) splitting when the radical cation is distorted along the degenerate vibrational modes of e' symmetry. The JT split components of these two electronic states can also undergo pseudo-Jahn-Teller (PJT)-type crossings via the vibrational modes of e', a1' and a2' symmetries. These lead to the possibility of multiple multidimensional conical intersections and highly nonadiabatic nuclear motions in these coupled manifolds of electronic states. In a previous publication [J. Phys. Chem. A 2004, 108, 2256], we investigated the JT interactions alone in the X2E' ground electronic manifold of CP+. In the present work, the JT interactions in the A2E' electronic manifold are treated, and our previous work is extended by considering the coupling between the X2E' and A2E' electronic states of CP+. The nuclear dynamics in this coupled manifold of two JT split doubly degenerate electronic states is simulated by considering fourteen active and most relevant vibrational degrees of freedom. The vibronic level spectra and the ultrafast nonradiative decay of the excited cationic states are examined and are related to the highly complex entanglement of electronic and nuclear degrees of freedom in this prototypical molecular system.     

A mixed adiabatic—diabatic representation for the vibronic coupling problem of polyatomic molecules

A model consisting of two electronic states with different symmetries and an arbitrary number of vibrational modes which couple these states is considered. After a series of canonical transformations the hamiltonian is brought into a form suited to study effects of anharmonic and vibronic couplings. Divergences typical for non-adiabatic coupling terms are avoided.     

The influence of vibronic interactions on the chiroptical spectra of dissymmetric pseudo tetragonal metal complexes

The influence of vibronic interactions on the chiroptical spectra associated with a threesome of nearly degenerate electronic excited states in a dissymmetric molecular system is examined on a formal theoretical model. The model considers two vibrational modes to be effective in promoting pseudo Jahn-Teller (PJT) type interactions between the three closely spaced electronic excited states. Formal expressions are developed for the rotatory strengths of individual vibronic levels derived from the coupled electronic states. Two mode (vibrational)-three state (electronic) vibronic Hamiltonians are constructed (basis set size, 63–108, depending upon interaction parameters used) and diagonalized for a large number of different parameter sets representative of various vibronic coupling strengths, electronic energy level spacings, oscillator (vibrational mode) frequencies, and electronic rotatory strengths. Diagonalization of these vibronic Hamiltonians yields vibronic wave functions and energies which are then used to calculate rotatory strength spectra for the model system. The calculated results demonstrate the profound influence which vibronic interactions of the PJT type may have on the sign patterns and intensity distributions within the rotatory strength spectrum associated with a set of nearly degenerate electronic states. The implication of these results for the interpretation of circular dichroism spectra of chiral transition metal complexes with pseudo tetragonal symmetry are discussed.     

Ab initio study of dynamical E × e Jahn-Teller and spin-orbit coupling effects in the transition-metal trifluorides TiF3, CrF3, and NiF3

Multiconfiguration ab initio methods have been employed to study the effects of Jahn-Teller (JT) and spin-orbit (SO) coupling in the transition-metal trifluorides TiF(3), CrF(3), and NiF(3), which possess spatially doubly degenerate excited states ((M)E) of even spin multiplicities (M = 2 or 4). The ground states of TiF(3), CrF(3), and NiF(3) are nondegenerate and exhibit minima of D(3h) symmetry. Potential-energy surfaces of spatially degenerate excited states have been calculated using the state-averaged complete-active-space self-consistent-field method. SO coupling is described by the matrix elements of the Breit-Pauli operator. Linear and higher order JT coupling constants for the JT-active bending and stretching modes as well as SO-coupling constants have been determined. Vibronic spectra of JT-active excited electronic states have been calculated, using JT Hamiltonians for trigonal systems with inclusion of SO coupling. The effect of higher order (up to sixth order) JT couplings on the vibronic spectra has been investigated for selected electronic states and vibrational modes with particularly strong JT couplings. While the weak SO couplings in TiF(3) and CrF(3) are almost completely quenched by the strong JT couplings, the stronger SO coupling in NiF(3) is only partially quenched by JT coupling.     

Thermodynamic functions for vibronic systems

A method is introduced which simplifies the derivation of equations for the calculation of statistical sums for vibronic systems. Within the “independent ordering approximation” analytical expressions are found for thermodynamic functions for molecules with degenerate electronic ground state (E?β and E ? e types of degeneracies). Vibronic interactions are shown to yield an additional thermodynamic stabilization. The computations were carried out for VCl₄ and VBr₄ with doubly degenerate electronic ground states. For these molecules the effect is shown to be small.     

“Proximity effects” in radiative transitions: a model study of the role of vibronic coupling and static crystal field interactions

We report the results of a model study of the influence of vibronic coupling involving non-totally symmetric vibrations and static crystal field interactions on the spectral properties of molecules with close-lying excited electronic states. The presented results suggests that “proximity effects” brought about by solvent perturbation arise from two sources: (i) alterations in the energy separation between vibronically coupled electronic states and (ii) crystal field mixing of the isolated molecular electronic states. It is shown that crystal field mixing leads to the breakdown of the vibronic coupling scheme for non-totally symmetric vibrations in isolated molecules. This breakdown is shown to have a very pronounced effect on the spectral properties of molecules with close-lying excited electronic states. The effect of environmental perturbations on excited state frequencies, the breakdown of symmetry and polarization selection rules, and vibrational intensity distributions is discussed.     

Vibronic contributions to DICD in achiral molecules

The theory of dispersion-induced circular dichroism (the CD induced in a transition of an achiral species through long-range dispersive coupling with a chiral species) is extended to include vibronic terms. Symmetry rules are deduced for DICD-active vibronic states. It is shown that the intensity distribution over DICD-active vibrations within a given electronic band of the achiral species gives both insight into the mechanism through which the DICD appears, and vibronic spectral data not accessible through direct absorption studies. Applications to the carbonyl chromophore and comparison with recent experimental studies suggest that vibronic terms may predominate in certain cases over those expected from the purely electronic case.     

The Jahn-Teller and Barnett effects

In the usual formulation of the Jahn-Teller effect a simplification is made in going from the adiabatic to the crude adiabatic approximation in which the electronic parts of the vibronic wavefunction are assumed independent of the nuclear coordinates. This then neglects momentum coupling in the vibronic coupling matrix. The momentum coupling has been termed the molecular Barnett effect when the active vibration transforms as the irreducible representation of a rotation in the molecular point group. Experimental evidence for the molecular Barnett effect has recently been found. In this paper the various point groups in which momentum and Barnett coupling can occur are investigated. A vibration capable of momentum coupling is contained in the asymmetric direct product of the degenerate electronic state and, as with the Jahn-Teller effect, is possible in the orbitally degenerate electronic states of molecules of all non-linear point groups. A static distortion along such a coordinate will lift the electronic degeneracy. Unlike the Jahn-Teller effect, however, in some point groups a minimum complexity of the molecule is required before such coupling can occur. In particular it will be absent in the degenerate electronic states of such simple molecules of the form X₃(D_3h); XY₃(C_3v, D_3h); XY₄(T_d); and XY₆(O_h).     

Vibronic coupling in cyclopentadienyl radical: a method for calculation of vibronic coupling constant and vibronic coupling density analysis

A method of calculation of vibronic or electron-phonon coupling constant is presented for a Jahn-Teller molecule, cyclopentadienyl radical. It is pointed out that symmetry breaking at degenerate point and violation of Hellmann-Feynman theorem occur in the calculations based on a single Slater determinant. In order to overcome these difficulties, the electronic wave functions are calculated using generalized restricted Hartree-Fock and complete active space self-consistent-field method and the couplings are computed as matrix elements of the electronic operator of the vibronic coupling. Our result agrees well with the experimental and theoretical values. A concept of vibronic coupling density is proposed in order to explain the order of magnitude of the coupling constant from view of the electronic and vibrational structures. It illustrates the local properties of the coupling and enables us to control the interaction. It could open a way to the engineering of vibronic interactions.     

Theoretical study of the polarized infrared spectra of the hydrogen bond in 2‐furoic acid crystal dimer

This work presents a theoretical simulation of ν_{O? H} and ν_{O? D} band shapes in the polarized infrared spectra of 2‐furoic acid dimer crystals measured at liquid‐nitrogen temperature. The line shapes are studied theoretically within the framework of the anharmonic couplings between low‐frequency hydrogen‐bond vibrations and degenerate excited states of high‐frequency hydrogen vibrations in hydrogen‐bonded dimers and the anharmonic coupling between the first excited state of the fast mode and the harmonics or band combinations of some low‐frequency bending modes, which lead to Fermi resonances.This approach takes into account the adiabatic approximation, the intrinsic anharmonicity of the low‐frequency mode through a Morse potential, Davydov coupling triggered by resonance exchange between the excited states of the fast modes of the two hydrogen bonds involved in the cyclic dimer, and the direct and indirect damping of the fast‐stretching modes of the hydrogen bonds and of the bending modes. The infrared spectral density was calculated within the linear response theory by Fourier transform of the autocorrelation function of the transition dipole moment operator of the fast mode. Numerical results show that mixing of all these effects allows satisfactory reproduction of the main features of the experimental IR line shapes of crystalline H‐ and D‐bonded 2‐furoic acid at liquid‐nitrogen temperature and for different polarizations. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Nonadiabatic quantum dynamics of Br2 in solid Ar: A four-dimensional study of the B to C state predissociation

Quantum dynamics simulations are performed for a diatomics-in-molecules based model of Br₂ in solid Ar which incorporates four nuclear degrees of freedom and four electronic states. The nuclear motions comprise two large amplitude coordinates describing the Br₂ bond distance and an effective symmetry-preserving matrix mode. Two symmetry-lowering harmonic modes are added in the spirit of linear vibronic coupling theory. Initiating the dynamics on the B state by means of an ultrafast laser pulse, nonadiabatic transitions to the two degenerate C states are monitored and the effect of vibrational preexcitation in the electronic ground state is investigated.     

On the construction of quasidiabatic state representations of bound adiabatic state potential energy surfaces coupled by accidental conical intersections: incorporation of higher order terms

The quadratic vibronic coupling model is an important computational tool for simulating photoelectron spectra involving strongly coupled electronic states in polyatomic molecules. However, recent work has indicated the need for higher order terms, with most of the initial studies focusing on molecules with symmetry-required degeneracies. In this study we report an extension of our approach for constructing fully quadratic representations of bound electronic states coupled by conical intersections, which allows for the inclusion of higher order terms, demonstrated here employing a quartic expansion. Procedures are developed that eliminate unphysical behavior for large displacements, a problem likely to be an endemic to anharmonic expansions. Following work on representing dissociative electronic states, Lagrange multipliers are used to constrain the constructed representation to reproduce exactly the energy, energy gradients, and∕or derivative couplings at specific points, or nodes, in nuclear coordinate space. The approach is illustrated and systematically studied using the four lowest electronic states of triazolyl, (CH)(2)N(3).     

Vibronic coupling in the ground and excited states of oligoacene cations

The vibrational coupling in the ground and excited states of positively charged naphthalene, anthracene, tetracene, and pentacene molecules is studied on the basis of a joint experimental and theoretical study of ionization spectra using high-resolution gas-phase photoelectron spectroscopy and first-principles correlated quantum-mechanical calculations. Our theoretical and experimental results reveal that, while the main contribution to relaxation energy in the ground state of oligoacene systems comes from high-energy vibrations, the excited-state relaxation energies show a significant redistribution toward lower-frequency vibrations. A direct correlation is found between the nature of the vibronic interaction and the pattern of the electronic state structure.     

Response functions for dimers and square-symmetric molecules in four-wave-mixing experiments with polarized light

Four-wave-mixing nonlinear-response functions are given for intermolecular and intramolecular vibrations of a perpendicular dimer and intramolecular vibrations of a square-symmetric molecule containing a doubly degenerate state. A two-dimensional particle-in-a-box model is used to approximate the electronic wave functions and obtain harmonic potentials for nuclear motion. Vibronic interactions due to symmetry-lowering distortions along Jahn-Teller active normal modes are discussed. Electronic dephasing due to nuclear motion along both symmetric and asymmetric normal modes is included in these response functions, but population transfer between states is not. As an illustration, these response functions are used to predict the pump-probe polarization anisotropy in the limit of impulsive excitation.     

Allene and pentatetraene cations as models for intramolecular charge transfer: vibronic coupling Hamiltonian and conical intersections

We consider the vibronic coupling effects involving cationic states with degenerate components that can be represented as charge localized at either end of the short cumulene molecules allene and pentatetraene. Our aim is to simulate dynamically the charge transfer process when one component is artificially depopulated. We model the Jahn-Teller vibronic interaction within these states as well as their pseudo-Jahn-Teller coupling with some neighboring states. For the manifold of these states, we have calculated cross sections of the ab initio adiabatic potential energy surfaces along all nuclear degrees of freedom, including points at large distances from the equilibrium to increase the physical significance of our model. Ab initio calculations for the cationic states of allene and pentatetraene were based on the fourth-order M?ller-Plesset method and the outer valence Green's function method. In some cases we had to go beyond this method and use the more involved third-order algebraic diagrammatic construction method to include intersections with satellite states. The parameters for a five-state, all-mode diabatic vibronic coupling model Hamiltonian were least-square fitted to these potentials. The coupling parameters for the diabatic model Hamiltonian are such that, in comparison to allene, an enhanced preference for indirect charge transfer is predicted for pentatetraene.