The triatomic Eckart-frame kinetic energy operator in bond coordinates

Nuclear wave-packet-propagation-based study of the electron-coupled, proton-transfer process in the charge-transfer state of FHCl exhibiting three electronic states in full-dimensional space

The charge-transfer (CT) excited state of FHCl (F⁺H-Cl^-), generated by the photodetachment of an electron from its precursor anion (FHCl^-) by a photon energy of ∼9.5 eV, is a realistic prototype of two bidirectional-coupled reaction pathways, namely the proton-transfer (PT) and electron-transfer (ET) channels, that produce F + HCl and FH + Cl combinations, respectively. The early-time dynamics of the CT was studied via the time-dependent propagations of nuclear wave packets comprising three nonadiabatically coupled electronic states defined within a three-dimensional space. The detailed analyses of the early-time dynamics revealed an interesting phenomenon in which the onset of PT was ∼80 fs earlier than that of ET, indicating that PT dominated ET in this case. A more significant finding was that the proper adjustment of the electronic-charge distribution for the onset of ET was obtained ∼80 fs after the onset of PT; this adjustment was mediated by the initial movement of the H atom, i.e., the F-H vibration mode. To avail experimental observables, the branching ratio, χ = PT/(PT + ET), and absorption spectrum generating the neutral FHCl molecule from its precursor anion were also simulated. The results further demonstrated the dependences of the χs and spectrum on the change in the initial vibration level of the precursor anion, as well as the isotopic substitution of the connecting H atom with deuterium, tritium, and muonium.

Umbrella inversion of ammonia redux

Umbrella inversion of ammonia is a prototypical example of large-amplitude vibrational motion, described with a symmetric double-well potential. The transition state of the latter corresponds to a planar (D_3h) molecular geometry, whereas the two equilibrium configurations are equivalent (C_3v) pyramidal structures, with the nitrogen atom being either 'above' or 'below' the plane of the hydrogen atoms. As commonly understood, inversion motion of ammonia corresponds to the coherent, anharmonic, vibrational motion of the molecule, which shuttles back and forth between the two potential wells; that is, oscillation of the nitrogen atom from one side of the H₃ plane to the other, via coherent tunneling. However, this intuitively appealing view of umbrella inversion results from a reduced description of the dynamics, which includes only the inversion vibrational coordinate and fully neglects all the other molecular degrees of freedom. As such, this textbook picture of inversion motion ignores the fact that the two equilibrium structures of ammonia are superimposable, and can only be distinguished by labelling the identical hydrogen nuclei. A correct description of umbrella inversion, which incorporates nuclear permutations, requires the inclusion of other molecular modes. Indeed, it is well known that the quantum symmetrization postulate engenders entanglement between ammonia's nuclear-spin, inversion, and rotation. Using the explicit expressions of the corresponding zeroth-order eigenstates, we clearly show that the inversion density of any multilevel wavepacket of ammonia, including the case of perfectly aligned molecules, is symmetrically distributed between the two potential wells, at all times. This follows from a rigorous demonstration based on the evaluation of the expectation values of the inversion coordinate or equivalent projection operators. However, provided that these wavepackets involve inversion-rotation levels with opposite parity, the inversion density may exhibit dynamical spatial localization. In the latter case, the space-fixed inversion density or, equivalently, the expectation values of the projections of the inversion coordinate on the space-fixed axes, may oscillate between opposite directions in the space-fixed frame. Nevertheless, in all cases, localization of ammonia in a single potential well is impossible, even partially or transiently. This is equivalent to saying that the nitrogen atom has the same probability (one-half) to be on either side of the H₃ plane, for any wavepacket of the molecule and at all times-a conclusion which is in perfect accord with the principle of the indistinguishability of identical particles (nuclei).

Wavepacket propagations for the early time dynamics of proton-coupled electron transfer in the charge-transfer state of NH3Cl complex

A charge-transfer (CT) excited state of NH₃Cl, generated by photo-detachment of an electron from the anionic NH₃Cl^- precursor, can be represented as H₂N⁺-H-Cl^- and proceeds to two chemical reactions: one reaction generating NH₂ and HCl resulting from a proton transfer (PT) and the other reaction producing NH₃ and a Cl atom resulting from an electron transfer (ET); both are coupled to form a typical proton-coupled electron transfer (PCET) process. The early time dynamics of this CT were studied using time-dependent wavepacket propagation on three nonadiabatically coupled electronic states in a reduced three-dimensional space. The electronic states were treated using the XMS-CASPT2/aug-cc-pVTZ ab initio methodology. The population dynamics of the three coupled electronic states were analyzed in detail to reveal the initial stage of the PCET process up to ∼100 fs, while the branching ratio, χ = PT/(ET+PT), was determined after wavepacket propagations of up to 2000 fs. Another main result is the dependence of χ on the vibration levels of the initial precursor anion and the isotope substitution of the connecting H atom with deuterium and tritium. Our study reveals the detailed microscopic features of the PCET process embedded in the CT state of the NH₃Cl complex and certain systematic dependences of the branching ratio χ on the above factors.

On neglecting Coriolis and related couplings in first-principles rovibrational spectroscopy: Considerations of symmetry, accuracy, and simplicity. II. Case studies for H2O isotopologues, H3+, O3, and NH3

For centuries, it has been known that vibrational and rotational degrees of freedom are in general not separable. Nevertheless, surprisingly little is known about the best strategies for approximately separating these degrees of freedom in practice-even in the case of semirigid molecules, where the separation is most meaningful. There is also some confusion in the literature about the proper way to quantify the magnitude of the Coriolis (i.e., rotation-vibration) coupling in rovibrational Hamiltonians or its effect on the rovibrational eigenenergies. In this study, a vibrational-coordinate-independent metric is proposed to quantify the magnitude of the Coriolis contribution to the rovibrational Hamiltonian. The impact of Coriolis coupling on the rovibrational eigenenergies is computed numerically exactly, using both full and various truncated Hamiltonians. The role played by the choice of the vibrational coordinate system-and especially by the choice of "embedding" or body-fixed frame-is examined extensively, both numerically and analytically. This investigation targets several molecular prototypes, all of which serve as important benchmarks for the high-resolution spectroscopic community. Most of these are triatomic molecules, including water (H₂¹⁶O), its deuterated isotopologues (D₂¹⁶O and HD¹⁶O), H₃⁺, and ozone (¹⁶O₃), but the tetratomic ammonia molecule (¹⁴NH₃) is also investigated. These studies provide important insight into the nature of Coriolis coupling under various circumstances. The findings of this study also have significant practical ramifications, vis-à-vis the use of simplifying numerical approximation techniques in nuclear-motion computations.

On neglecting Coriolis and related couplings in first-principles rovibrational spectroscopy: considerations of symmetry, accuracy, and simplicity

The rotation-vibration (Coriolis) coupling contribution to variationally computed rovibrational energy levels is investigated, employing triatomic AB[Formula: see text] molecules as models. In particular, calculations are performed for H[Formula: see text][Formula: see text]O, across a range of vibrational and rotational excitations, both with and without the Coriolis contribution. A variety of different embedding choices are considered, together with a hierarchy of increasingly severe approximations culminating in a generalized version of the so-called "centrifugal sudden" method. Several surprising and remarkable conclusions are found, including that the Eckart embedding is not the best embedding choice.

Development of a potential energy surface for the O3–Ar system: rovibrational states of the complex

Collisional stabilization is an important step in the process of atmospheric formation of ozone.

Eckart-Sayvetz conditions revisited

It is shown that vibrational displacements satisfying the Eckart-Sayvetz conditions can be constructed by projection of unconstrained displacements. This result has a number of interesting direct and indirect ramifications: (i) The normal coordinates corresponding to an electronic state or an isotopologue of a molecule are transformed to those of another state or isotopologue by a linear and, in general, non-orthogonal transformation. (ii) Novel interpretation of axis switching. (iii) One may enhance the separation of rotational-large-amplitude internal motions and the vibrational motions beyond that offered by the standard use of the Eckart-Sayvetz conditions. (iv) The rotational-vibrational Hamiltonian given in terms of curvilinear internal coordinates may be derived with elementary mathematical tools while taking into account the Eckart conditions with or without enhancement.

Numerically constructed internal-coordinate Hamiltonian with Eckart embedding and its application for the inversion tunneling of ammonia

It is shown that the use of an Eckart-frame embedding with a kinetic energy operator expressed in curvilinear internal coordinates becomes feasible and straightforward to implement for arbitrary molecular compositions and internal coordinates if the operator is defined numerically over a (discrete variable representation) grid. The algorithm proposed utilizes the transformation method of Dymarsky and Kudin to maintain the rotational Eckart condition. In order to demonstrate the applicability and flexibility of our approach the non-rigid ammonia molecule is considered and the corresponding rotational-vibrational energy levels and wave functions are computed using kinetic energy operators with three different embeddings. Two of them fulfill the Eckart conditions corresponding to a trigonal pyramidal (C3v) and a trigonal planar (D3h) reference structure and the third one is a non-Eckart frame. The computed energy levels are, of course, identical, and the structure of the three different wave-function representations are analyzed in terms of the rigid rotor functions for a symmetric top. The possible advantages of one frame representation over another are discussed concerning the interpretation of the rovibrational states in terms of the traditional rigid rotor labels.

Dipole moments of HDO in highly excited vibrational states measured by Stark induced photofragment quantum beat spectroscopy

We report here a measurement of electric dipole moments in highly vibrationally excited HDO molecules. We use photofragment yield detected quantum beat spectroscopy to determine electric field induced splittings of the J=1 rotational levels of HDO excited with 4, 5, and 8 quanta of vibration in the OH stretching mode. The splittings allow us to deduce mua and mub, the projections of dipole moment onto the molecular rotation inertial axes. We compare the measured HDO dipole moment components with the results of quantitative calculations based on Morse oscillator wave functions and an ab initio dipole moment surface. The vibrational dependence of the dipole moment components reflect both structural and electronic changes in HDO upon vibrational excitation; principally the vibrational dependence of the O-H bond length and bond angle, and the resulting change in orientation of the principal inertial coordinate system. The dipole moment data also provide a sensitive test of theoretical dipole moment and potential energy surfaces, particularly for molecular configurations far from equilibrium.

Body frames in the separation of collective angles in quantum N-body problems

The application of the concept of body-fixed reference frames, proposed by C. Eckart [Phys. Rev. 47, 552 (1935)], to the problem of the separation of three collective angles in quantum N-body problems is analyzed based on the technique recently developed by Meremianin and Briggs [Phys. Rep. 384, 121 (2003)]. Special attention is paid to the body frame defined by the "second Eckart condition" which minimizes vibro-rotational couplings near the equilibrium position. The important case of the Eckart frame for three-body systems is considered in detail. The connection of the basis vectors of the Eckart frame with Jacobi vectors is derived. All results of this work are valid for an arbitrary choice of internal (body-frame) coordinates.

The molecular Hamiltonian

The molecular Hamiltonian represents one of the most basic concepts in spectroscopy and molecular reaction dynamics. Its derivation is notoriously difficult owing to the use of a rotating reference frame which, in turn, is necessary to define the concept of vibration and rotation. In this article, we review the construction of the molecular Hamiltonian in normal mode and in internal coordinates. For normal mode coordinates, the Watson Hamiltonian including its modification for linear molecules is derived using an approach based on classical mechanics and the Podolsky transformation. The method is subsequently used to derive the molecular Hamiltonian in terms of Jacobi and valence coordinates. Results are presented for the triatomic system and for the extension toward N-atom systems with N > or = 3.

The rovibrational Hamiltonian for ammonia-like molecules

A new exact quantum mechanical rovibrational Hamiltonian operator for ammonia-like molecules is derived. The Hamiltonian is constructed in a molecular system of axes, such that its z' axis makes a trisection of the pyramidal angle formed by three bond vectors with the vertex on the central atom. The introduced set of the internal rovibrational coordinates is adapted to facilitate a convenient description of the inversion motion. These internal coordinates and the molecular axis system have a remarkable property, namely, the internal vibrational angular momentum of the molecule equals zero. This property significantly reduces the Coriolis coupling and simplifies the form of the Hamiltonian. The correctness of this Hamiltonian is proved by a numerical procedure. The orthogonal Radau vectors allowing us to define a similar molecular axis system and the internal coordinates are considered. The Hamiltonian for the Radau parameterization takes a form simple enough to carry out effectively variational calculations of the molecular rovibrational states. Under the appropriate choice of the variational basis functions, the Hamiltonian matrix elements are fully factorizable and do not have any singularities. A convenient method of symmetrization of the basis functions is proposed.