Excited Vibronic States Research Articles

Pseudo Jahn-Teller origin of puckering in cyclohexahomoatomic molecules E6 (E = S, Se, Te) and restoring S6 planar ring configuration

The pseudo Jahn-Teller effect (PJTE) is employed to explore the origin of the puckering structure of cyclohexasulfur (S6), cyclohexaselenium (Se6) and cyclohexatellurium (Te6) and their nondegenerate and degenerate vibronic excited states and their planar structure instabilities have investigated. The ab initio geometry optimization and frequency calculations show that all these cyclohexahomoatomic molecules chose D6h symmetry in the planar configuration, and according the S6 and Se6 experimental structure, the chair form of the molecules is stable structure. The vibronic coupling between the ground state 1A1g and excited state 1B2g is the cause of chair puckering in all these series compounds and the numerical solutions of the PJTE (1A1g+1B2g)⊗b2g problems describe their instability. The adiabatic potential energy surfaces (APES) cross sections of low-lying electronic states along the b2g puckering normal coordinates have calculated by the state-average complete active space self-consistent field (SA-CASSCF) method. The calculation results show that, the chair puckering instability in the S6 from unstable planar configuration with D6h symmetry to a stable D3d distorted geometry, is stronger than others, whereas it is weaker in Te6. Additionally, coordination two canions (X = H+, He2+) to the S6 chair structure restore the planarity of S6 puckered ring in the S6X2 systems, although the D6h symmetry in S6 planar ring configuration changes to the Cs symmetry in the systems.

Molecular-Resolution Interrogation of a Porphyrin Monolayer by Ultrahigh Vacuum Tip-Enhanced Raman and Fluorescence Spectroscopy.

Tip-enhanced Raman scattering (TERS) and optically excited tip-enhanced fluorescence (TEF) of a self-assembled porphyrin monolayer on Ag(111) are studied using an ultrahigh vacuum scanning tunneling microscope (UHV-STM). Through selectively exciting different Q-bands of meso-tetrakis- (3,5-ditertiarybutylphenyl)-porphyrin (H2TBPP), chemical information regarding different vibronic excited states is revealed by a combination of theory and experiment; namely, TERS and time-dependent density functional theory (TDDFT) simulations. The observed TEF spectra suggest a weak coupling of H2TBPP to the substrate due to the bulky t-butyl groups and a possible alternative excited state decay path. This work demonstrates the potential of combining TERS and TEF for studying surface-mounted porphyins on substrates, thus providing insight into porphyrin-sensitized solar cells and catalysis.

Study of the vibronic coupling in the first excited state of MgCH3 radical

A methodology based on the crude adiabatic approximation is used to study the complete linear and quadratic vibronic coupling in the first excited state of MgCH 3 radical. Firstly we evaluate 30 integrals intervening in the formulation of the vibronic coupling and which are used to build the representation of the hamiltonian. Secondly, we diagonalize this representation to evaluate the vibronic levels. For the lowest excited vibronic states, the implied modes are Q 1 (symmetric C-Mg stretching) and Q 4 ( CH 3 rocking). Energy gaps A 1 -A 2 and A 2 - e resulting of the splitting due to the Jahn-Teller coupling E U e = A 1 + A 2 + e are evaluated to 126 and 208 cm -1 , respectively. Finally we surround essential coupling parameters to simplify the study of highly excited vibronic states. In these states, the involved modes are Q 5 (antisymmetric C-H stretching) and Q 6 (antisymmetric C-H bending). The energy gaps are evaluated to 22 cm -1 ( A 2 – A 1 ) and 34 cm -1 ( A 2 - e ) for Q 5 mode, 9 cm -1 ( A 1 -A 2 ) and 16 cm -1 ( A 2 - e ) for Q 6 mode.

Conformational Gating of Charge Separation in Porphyrin Oligomer-Fullerene Systems

The rate of the photoinduced charge-separation in C60-terminated butadiyne-linked porphyrin oligomers Pn (n = 4, 6) is strongly influenced by their molecular conformation. In these systems, the presence of the butadiyne linkers gives rise to a broad distribution of conformations in the ground state, due to an almost barrierless rotation of individual porphyrin units in the oligomer chain. The conformational states of these oligomers, either twisted or planar, could be selected by varying the excitation wavelength, thereby providing different initial excited states for charge separation. Charge separation in the different conformers was followed using both steady-state and 2D time-resolved emission using a streak camera system. Singular value decomposition (SVD) analysis applied on streak camera data provides here a powerful tool to study the conformational dependence of the charge separation in long PnC60 systems. Both the kinetics and spectral changes accompanying charge separation could be analyzed for different populations of conformation. From this analysis we show that, for both systems studied, twisted conformations undergo faster charge separation than planar conformations. This disparity in charge separation rates was ascribed mainly to the difference in driving force for charge separation between twisted and planar conformations. Charge separation was also studied in oligomers PnC60 coordinated to an octadentate ligand T8 that hinders the rotation of porphyrin subunits. The semicircular complexes PnC60-T8 show dramatic changes in their spectral properties, as well as slow excitation wavelength independent rate of charge separation and corresponding low efficiency compared to their linear counterparts. This slow charge separation rate was attributed to fast relaxation to the lowest excited vibronic state and lack of driving force for charge separation in these close to planar semicircular systems; i.e., the template systems behave like “normal” donor–acceptor systems without slow conformational relaxation. This work illustrates how control of conformation can be used to tune the rate of charge separation.

Photo excitation and laser detachment of C60− anions in a storage ring

We have studied the photo physics of C60(-) anions in the electrostatic storage ring ELISA with ions produced in a plasma source and cooled and bunched in a He filled ion trap. A previous study using delayed electron detachment as a signal of resonance-enhanced multiphoton electron detachment (REMPED) has been repeated both at room temperature and with the trap cooled to liquid nitrogen temperature. However, wavelength dependence of the overlap of the strongly focused laser beam with the ion beam introduces distortions of the absorption spectrum. We have therefore applied a new method, combining the IR light with a slightly delayed, powerful UV pulse (266 nm). After absorption of three UV photons, the ions decay by delayed (thermal) electron emission, and time spectra are recorded for varying wavelength. The fraction of ions heated by absorption of a single IR photon is then extracted from a principal component analysis of these spectra. In good agreement with the earlier REMPED experiment, an origin band for transitions between the two lowest electronic levels of the anion, with t1u and t1g symmetry, is observed at 9380 cm(-1), with strong sidebands from excitation of the two A(g) and eight H(g) vibrations. As before, a hot band is observed at about 9150 cm(-1) and assigned to a transition from an excited vibronic Jahn-Teller state. However, an earlier observed band at higher energy, interpreted as a transition from this excited state to an excited vibronic state in the t1g electronic level, is much weaker in the new measurements and could be an H(g) vibrational sideband. Also earlier studies of direct laser detachment from C60(-) in the storage ring ASTRID have been revisited, with ions cooled by liquid nitrogen in the ion trap. We confirm the previous measurement with a determination of the threshold for s-wave detachment at 2.664 ± 0.005 eV, slightly lower than a recent value of the electron binding, 2.683 ± 0.008 eV, obtained from the energy spectrum of photo electrons. The detachment yield is observed to increase sharply for photon energies above a threshold at 2.78 eV, maybe caused by either Feshbach resonances or a shape resonance due to the presence of a bound electron state in the continuum.

Electronic Spectroscopy of Diatomic VC

Resonant two-photon ionization spectroscopy has been applied to diatomic VC, providing the first optical spectrum of this molecule. The ground state is determined to be a (2)Δ3/2 state that arises from the 1σ(2)1π(4)2σ(2)1δ(1) configuration. The r0" ground-state bond length is 1.6167(3) Å. The manifold of excited vibronic states in the visible portion of the spectrum is quite dense, but two possible vibrational progressions have been identified. It is noted that VC joins CrC, NbC, and MoC as species in which the metal ns-based 3σ orbital is unoccupied, resulting in large dipole moments in the ground states of these molecules. In the corresponding 5d metal carbides, however, the 3σ orbital is occupied, leading to different ground electronic states of the 5d congeners, TaC and WC.

Vibration induced memory effects and switching in ac-driven molecular nanojunctions

We investigate bistability and memory effects in a molecular junction weakly coupled to metallic leads with the latter being subject to an adiabatic periodic change of the bias voltage. The system is described by a simple Anderson-Holstein model and its dynamics is calculated via a master equation approach. The controlled electrical switching between the many-body states of the system is achieved due to polaron shift and Franck-Condon blockade in the presence of strong electron-vibron interaction. Particular emphasis is given to the role played by the excited vibronic states in the bistability and hysteretic switching dynamics as a function of the voltage sweeping rates. In general, both the occupation probabilities of the vibronic states and the associated vibron energy show hysteretic behaviour for driving frequencies in a range set by the minimum and maximum lifetimes of the system. The consequences on the transport properties for various driving frequencies and in the limit of DC-bias are also investigated.

Control of vibronic excitation using quantum-correlated photons

We theoretically investigate the two-step excitation of a molecular vibronic state using quantum-correlated photons with time delay in order to control the population of the vibronic excited state. A Morse oscillator having three sets of vibronic states, namely, the ground state, intermediate states, and excited states, is used to evaluate the efficiency of the two-step excitation process. We show that we can efficiently and selectively excite only a target state by using correlated photons and can control the excitation population of the target state by adjusting the delay time of the correlated photons. The potential of controlling a chemical reaction using correlated photons is also discussed.

Dynamics and Mechanisms of Interfacial Photoinduced Electron Transfer Processes of Third Generation Photovoltaics and Photocatalysis

Photoinduced electron transfer (PET) across molecular/bulk interfaces has gained attention only recently and is still poorly understood. These interfaces offer an excellent case study, pertinent to a variety of photovoltaic systems, photo- and electrochemistry, molecular electronics, analytical detection, photography, and quantum confinement devices. They play in particular a key role in the emerging fields of third-generation photovoltaic energy converters and artificial photosynthetic systems aimed at the production of solar fuels, creating a need for a better understanding and theoretical treatment of the dynamics and mechanisms of interfacial PET processes. We aim to achieve a fundamental understanding of these phenomena by designing experiments that can be used to test and alter modern theory and computational modeling. One example illustrating recent investigations into the details of the ultrafast processes that form the basis for photoinduced charge separation at a molecular/bulk interface relevant to dye-sensitized solar cells is briefly presented here: Kinetics of interfacial PET and charge recombination processes were measured by fs and ns transient spectroscopy in a heterogeneous donor-bridge-acceptor (D-B-A) system, where D is a Ru(II)(terpyridyl-PO3)(NCS)3 complex, B an oligo-p-phenylene bridge, and A nanocrystalline TiO2. The forward ET reaction was found to be faster than vibrational relaxation of the vibronic excited state of the donor. Instead, the back ET occurred on the micros time scale and involved fully thermalized species. The D-A distance dependence of the electron transfer rate was studied by varying the number of p-phenylene units contained in the bridge moiety. The remarkably low damping factor beta = 0.16 angstroms(-1) observed for the ultrafast charge injection from the dye excited state into the conduction band of TiO2 is attributed to the coupling of electron tunneling with nonequilibrium vibrations redistributed on the bridge, giving rise to polaronic transport of charges from the donor ligand to the acceptor solid oxide surface.

Ab initio rovibronic states of the thioketenyl radical in its X2Π electronic ground state

We present an ab initio study of the thioketenyl (HCCS) radical in its degenerate X(2)Π electronic ground state. All rotational and vibrational degrees of freedom are taken into account including the electronic orbital and spin angular momenta. The structure of the rovibronic levels and the nature of the corresponding wave functions show resonances even at very low energies due to strong couplings between the bending, rotation, and spin terms in the Hamiltonian. Assignments from the dominant contributions of the eigenvectors are discussed in parallel with previously published data. The rotational structures for the first excited vibronic states are computed as well as transition intensities for the fundamental one.

Cavity Ringdown Spectroscopy of PH2 Radical in 465–555 nm

Absorption spectra of jet-cooled PH2 radicals were recorded in the wavelength range of 465–555 nm using cavity ringdown spectroscopy. The PH2 radicals were produced in a supersonic jet by pulsed direct current discharge of a mixture of PH3 and SF6 in argon. Seven vibronic bands with fine rotational structures have been observed and assigned as 000, 2n0, and 2n1 (n = 1–3) bands of the Ã2A1−X̃2B1 electronic transition. Rotational assignments and rotational term values for each band were re-identified, and the molecular parameters including rotational constants, centrifugal distortion constants, and spin-rotation interaction constants were also improved with reasonably high precision. In addition, large perturbations observed in each quantum number of total angular momentum of the a axis level of the excited vibronic states were briefly discussed.

Temperature dependence of resonance Raman spectra of carotenoids

To understand the mechanism of the photoprotective and antioxidative functions of carotenoids, it is essential to have a profound knowledge of their excited electronic and vibronic states. In the present study we investigate the most powerful antioxidants: β-carotene and lutein by means of resonance Raman spectroscopy. The aim was to study in detail their Raman spectra in solution at room temperature and their changes as a function of temperature. To measure the spectra in their natural environment pyridine has been used as a solvent. It has been chosen because of its polarizability ( n = 1.5092) which is close to that of membrane lipids and proteins. The temperature dependence of the most intensive ν 1 band in the range from 77 K to 295 K at 514.5 nm excitation has been obtained. It was found that in pyridine the C C stretching frequency, its intensity, line shape, and line width are very sensitive to the temperature (the sensitivity being different for the two studied carotenoids). The observed linear temperature dependence of the C C stretching frequency is explained by a mechanism involving changes of the vibronic coupling and the extent of π-electron delocalization. The different behavior of the temperature-induced broadening of the ν 1 band and its intensity for the two studied carotenoids can be associated with the different nature of their solid matrices: glassy for β-carotene and crystalline-like for lutein, owing to their different chemical structures.

Isotope Effects on the Nonequilibrium Dynamics of Ultrafast Photoinduced Proton-Coupled Electron Transfer Reactions in Solution.

The hydrogen/deuterium isotope effects on the ultrafast dynamics of photoinduced proton-coupled electron transfer (PCET) are investigated with a recently developed nonadiabatic dynamics approach. An ensemble of surface hopping trajectories is propagated according to a Langevin equation on electron-proton vibronic free energy surfaces that depend on a collective solvent coordinate. The calculations illustrate that ultrafast PCET reactions could exhibit a significant normal isotope effect, where PCET is faster for hydrogen than deuterium, but could also exhibit a negligible isotope effect or even a slight inverse isotope effect. The isotope effect is very small or absent when highly excited electron-proton vibronic states dictate the nonadiabatic dynamics and increases with greater participation of lower vibronic states. Thus, although the presence of a significant isotope effect strongly suggests that proton motion is coupled to electron transfer, the absence of an isotope effect does not exclude the possibility that proton transfer accompanies electron transfer in ultrafast photoinduced charge transfer processes.

Survival of molecular information under surfaced‐enhanced resonance Raman (SERRS) conditions

AbstractIn the present paper, we discuss the molecular information that can be derived from surface‐enhanced resonance Raman Scattering (SERRS) experiments performed with different excitation wavenumbers, which are close to resonance with an excited electronic state of the molecule [surface‐enhanced Raman dispersion spectroscopy (SERADIS)]. We specifically consider the situation, where a molecule is physisorbed to a site characterized by a local electric field with a direction independent of the direction of the external, exciting field. The molecular information available in this experimental situation is compared with the information available in a corresponding Raman dispersion spectroscopy (RADIS) experiment performed on a free molecule or a molecule physisorbed to a site, where the local field is isotropic. The consequences for resonance Raman scattering (RRS) and RADIS, when the molecule is adsorbed in the highly anisotropic hot spot (HS), are discussed; here it is shown that only the molecular information originating from the symmetric part of the scattering tensor can survive in SERRS and in SERADIS. Besides, it is shown that the depolarization ratio can no longer be used to discriminate between totally and non‐totally symmetric modes in the polarized surface‐enhanced Raman scattering (SERS) spectra. These results have implications for the resonance Raman spectra, but even more important for the application of the resonance Raman effect in the investigation of excited vibronic molecular states, in general, and in the investigation of electronic states in larger bio‐molecules, such as the various metallo‐porphyrins. Copyright © 2009 John Wiley & Sons, Ltd.

The effect of solvent on the excited vibronic states and first hyperpolarizability of “push–pull” merocyanines

The absorption and fluorescence spectral properties of several merocyanines with very high first hyperpolarizabilities in fluid solutions are reported. These molecules contain a cyanodicyanomethylidenedihydrofuran electron acceptor group and with either a pyridinylidene or quinolinylidene electron donor. The molecules also contain a bulky substituent group which has been incorporated in order to hinder molecular aggregation. No effect of merocyanine concentration on the absorption spectral distributions of these compounds was observed in any of the solvents studied indicating there was no molecular aggregation in solution. The absorption spectra of these compounds display changes in their vibronic structure in solvents with different polarities/polarizabilities while the fluorescence spectra exhibit the standard solvent polarity stabilizing behaviour. The first hyperpolarizabilities decreased with decreasing solvent polarity for merocyanines with a quinolinylidene electron donor and this trend was reversed for compounds with a pyridinylidene electron donor. This is consistent with changes in electron delocalization within the merocyanines induced by changes in the dielectric of the solvent.

The XAS model of dissolved Cu(II) and its significance to biological electron transfer

The standard model for dissolved Cu(II) portrays the complex ion as an axially elongated, equatorially planar octahedron. Using EXAFS and MXAN analyses of copper K-edge XAS spectra, new structural models for dissolved [Cu(aq)]2+ and [Cu(amm)]2+ have been determined. These structures uniformly depart from the octahedral model in favour of an axially elongated square pyramidal core. MXAN results also indicate that the equatorial ligands need not be coplanar with copper. Further structural elements include a -z axially localized scatterer at ∼3 Å. Even more distant scatterers imply second shell solvent organization, which can vary with the medium. Preliminary results from new extended, k = 18 Å-1, higher resolution copper K-edge XAS data sets are reported. The low symmetry of dissolved Cu(II) ion contradicts the central thesis of the rack-induced bonding hypothesis of copper electron transfer proteins. The asymmetry of biological copper is not a frozen vibronic excited state enforced by a rigid protein scaffold, but is entirely in harmony with the structural ground state of the dissolved aqueous Cu(II) complex ion.

Theory of Proton-Coupled Electron Transfer in Energy Conversion Processes

Proton-coupled electron transfer (PCET) reactions play an essential role in a broad range of energy conversion processes, including photosynthesis and respiration. These reactions also form the basis of many types of solar fuel cells and electrochemical devices. Recent advances in the theory of PCET enable the prediction of the impact of system properties on the reaction rates. These predictions may guide the design of more efficient catalysts for energy production, including those based on artificial photosynthesis and solar energy conversion. This Account summarizes the theoretically predicted dependence of PCET rates on system properties and illustrates potential approaches for tuning the reaction rates in chemical systems. A general theoretical formulation for PCET reactions has been developed over the past decade. In this theory, PCET reactions are described in terms of nonadiabatic transitions between the reactant and product electron-proton vibronic states. A series of nonadiabatic rate constant expressions for both homogeneous and electrochemical PCET reactions have been derived in various well-defined limits. Recently this theory has been extended to include the effects of solvent dynamics and to describe ultrafast interfacial PCET. Analysis of the rate constant expressions provides insight into the underlying physical principles of PCET and enables the prediction of the dependence of the rates on the physical properties of the system. Moreover, the kinetic isotope effect, which is the ratio of the rates for hydrogen and deuterium, provides a useful mechanistic probe. Typically the PCET rate will increase as the electronic coupling and temperature increase and as the total reorganization energy and equilibrium proton donor-acceptor distance decrease. The rate constant is predicted to increase as the driving force becomes more negative, rather than exhibit turnover behavior in the inverted region, because excited vibronic product states associated with low free energy barriers and relatively large vibronic couplings become accessible. The physical basis for the experimentally observed pH dependence of PCET reactions has been debated in the literature. When the proton acceptor is a buffer species, the pH dependence may arise from the protonation equilibrium of the buffer. It could also arise from kinetic complexity of competing concerted and sequential PCET reaction pathways. In electrochemical PCET, the heterogeneous rate constants and current densities depend strongly on the overpotential. The change in equilibrium proton donor-acceptor distance upon electron transfer may lead to asymmetries in the Tafel plots and deviations of the transfer coefficient from the standard value of one-half at zero overpotential. Applications of this theory to experimentally studied systems illustrate approaches that can be utilized to tune the PCET rate. For example, the rate can be tuned by changing the pH or using different buffer species as proton acceptors. The rate can also be tuned with site-specific mutagenesis in biological systems or chemical modifications that vary the substituents on the redox species in chemical systems. Understanding the impact of these changes on the PCET rate may assist experimental efforts to enhance energy conversion processes.

Quasi-relativistic treatment of the low-lying KCs states

The potential energy curves, permanent and transition dipole moments as well as spin-orbit and angular coupling matrix elements between the KCs electronic states converging to the lowest three dissociation limits were evaluated in the basis of the spin-averaged wavefunctions corresponding to pure Hund’s coupling case ( a). The quasi-relativistic matrix elements have been obtained for a wide range of internuclear distance by using of small (9-electrons) effective core pseudopotentials of both atoms. The core-valence correlation has been accounted for a large scale multi-reference configuration interaction method combined with semi-empirical core polarization potentials. The static dipole polarizabilities of the ground X 1 Σ + and a 3 Σ + states were extracted from the closed-shell coupled-cluster energies by the finite-field method. Among the singlet and triplet Σ + states manifold the pronounced avoided crossing effect between repulsive walls of the (2,3) 3 Σ + states has been discovered and analyzed by finite-difference calculation of radial coupling matrix elements. The resulting transition dipole moments and potentials were used to predict radiative lifetimes and emission branching ratios of excited vibronic states while the calculated angular coupling matrix elements were transformed to Λ -doubling constants of the (1,2) 1 Π states and magnetic g -factor of the ground state. The accuracies of the present results are discussed by comparing with experimental data and preceding calculations.

Analysis of kinetic isotope effects for proton-coupled electron transfer reactions.

A series of rate constant expressions for nonadiabatic proton-coupled electron transfer (PCET) reactions are analyzed and compared. The approximations underlying each expression are enumerated, and the regimes of validity for each expression are illustrated by calculations on model systems. In addition, the kinetic isotope effects (KIEs) for a series of model PCET reactions are analyzed to elucidate the fundamental physical principles dictating the magnitude of the KIE and the dependence of the KIE on the physical properties of the system, including temperature, reorganization energy, driving force, equilibrium proton donor-acceptor distance, and effective frequency of the proton donor-acceptor mode. These calculations lead to three physical insights that are directly relevant to experimental data. First, these calculations provide an explanation for a decrease in the KIE as the proton donor-acceptor distance increases, even though typically the KIE will increase with increasing equilibrium proton donor-acceptor distance if all other parameters remain fixed. Often the proton donor-acceptor frequency decreases as the proton donor-acceptor distance increases, and these two effects impact the KIE in opposite directions, so either trend could be observed. Second, these calculations provide an explanation for an increase in the KIE as the temperature increases, even though typically the KIE will decrease with increasing temperature if all other parameters remain fixed. The combination of a rigid hydrogen bond, which corresponds to a high proton donor-acceptor frequency, and low solvent polarity, which corresponds to small solvent reorganization energy, allows the KIE to either increase or decrease with temperature, depending on the other properties of the system. Third, these calculations provide insight into the dependence of the rate constant and KIE on the driving force, which has been studied experimentally for a wide range of PCET systems. The rate constant increases as the driving force becomes more negative because excited vibronic product states associated with low free energy barriers and relatively large vibronic couplings become accessible. The ln[KIE] has a maximum near zero driving force and decreases significantly as the driving force becomes more positive or negative because the contributions from excited vibronic states increase as the reaction becomes more asymmetric, and contributions from excited vibronic states decrease the KIE. These calculations and analyses lead to experimentally testable predictions of trends in the KIEs for PCET systems.

Solving non-Born–Oppenheimer Schrödinger equation for hydrogen molecular ion and its isotopomers using the free complement method

The Schrödinger equations for the hydrogen molecular ion (H(2)(+)) and its isotopomers (D(2)(+), T(2)(+), HD(+), HT(+), and DT(+)) were solved very accurately using the free iterative complement interaction method, which is referred to in short as the free complement (FC) method, in the non-Born-Oppenheimer (non-BO) level, i.e., in the nonrelativistic limit. Appropriate complement functions for both electron and nuclei were generated automatically by the FC procedure with the use of the non-BO Hamiltonian, which contains both electron and nuclear operators on an equal footing. Quite accurate results were obtained not only for the ground state but also for the vibronic excited states. For example, we obtained the ground-state energy of H(2)(+) as -0.597 139 063 123 405 074 834 134 096 025 974 142 a.u., which is variationally the best in literature. The difference in the nuclear spin states of (1)S (para) and (3)P (ortho) of H(2)(+) and some physical expectation values for several of the isotopomers shown above were also examined. The present study is the first application of the FC method to molecular systems with the non-BO Hamiltonian.