Vibrational kinetic energy operator

J. H. Frederick and C. Woywood, General formulation of the vibrational kinetic energy operator in internal bond-angle coordinates. J. Chem. Phys. Ill, 7255-7271 (1999). 

The definitions of the terms can be found in [30], but it is sufficient here to note that Ka represents the vibrational kinetic energy operator, Ep the electronic energy, Vn the nuclear repulsion operator, and the terms b and bo are elements of a matrix closely related to the inverse of the instantaneous inertia operator matrix. It should also be noted that the y terms arise from the interaction of the rotational with the electronic motion and tend to couple electronic states, even those diagonal in k. 

For the studies on benzene described in the following sections, the Hamiltonian was formulated in rectilinear coordinates. The pure vibrational kinetic energy operator was treated exactly (but nonquadratic vibrational angular momentum terms tt,tt, Cori-olos, and rotational terms were neglected), but the price to be paid is that the anharmonic potential contains a large number of terms. Development of the vibrational anharmonic Hamiltonian is described in the next three sections. 

Csaszar, A.G., Handy, N.C. Exact quantum-mechanical vibrational kinetic-energy operator of sequentially bonded molecules in valence internal coordinates, J. Chem. Phys. 1995,102, 3962-7. 

The vibrational kinetic energy operator is shown here for completeness but is neglected in the derivation of the Holstein polaron (stationary solution). 

Explicit forms of the coefficients Tt and A depend on the coordinate system employed, the level of approximation applied, and so on. They can be chosen, for example, such that a part of the coupling with other degrees of freedom (typically stretching vibrations) is accounted for. In the space-fixed coordinate system at the infinitesimal bending vibrations, Tt + 7 reduces to the kinetic energy operator of a two-dimensional (2D) isotropic haiinonic oscillator. 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

Pj and p2 represent the displacement vectors of the nuclei A and D (the corresponding polar coordinates are p1 cji, and p2, < )2, respectively) p, and pc are the displacement vectors and pT, r and pc, <[)f the corresponding polar coordinates of the terminal nuclei at the (collective) trans-bending and cis-bending vibrations, respectively. As a consequence of the use of these symmetry coordinates the nuclear kinetic energy operator for small-amplitude bending vibrations represents the kinetic energy of two uncoupled 2D harmonic oscillators ... 

After the separation of the kinetic energy operator due to the center-of-mass motion from the Hamiltonian, the Hamiltonian describes the internal motions of electrons and nuclei in the system. These in the BO approximation can be separated into the vibrational and rotational motions of the nuclear frame of the molecule and the electronic motion that only parametrically depends on the instantenous positions of the nuclei. When the BO approximation is removed, the electronic and nuclear motions become coupled and the only good quantum numbers, which can be used to quantize the stationary states of the system, are the principle quantum number, the quantum number quantizing the square of the total (nuclear and electronic) squared angular momentum, and the quantum number quantizing the projection of the total angular momentum vector on a selected direction (usually the z axis). The separation of different rotational states is an important feamre that can considerably simplify the calculations. 

As is well known, the vibrational Hamiltonian defined in internal coordinates may be written as the sum of three different terms the kinetic energy operator, the Potential Energy Surface and the V pseudopotential [1-3]. V is a kinetic energy term that arises when the classic vibrational Hamiltonian in non-Cartesian coordinates is transformed into the quantum-mechanical operator using the Podolsky trick [4]. The determination of V is a long process which requires the calculation of the molecular geometry and the derivatives of various structural parameters. 

For a diatomic species, the vibration-rotation (V/R) kinetic energy operator can be expressed as follows in terms of the bond length R and the angles 0 and < ) that describe the orientation of the bond axis relative to a laboratory-fixed coordinate system ... 

A complete treatment of this derivation can be found in Ref. [19]. The first three terms in the kinetic energy operator indicates the presence of a 3D harmonic oscillator, and the final two terms indicate the presence of a 2D rotator (as for the hydrogen atom). A similar conclusion was made by Auberbach et al. [22] where they use a semi-classical quantisation method and the molecule is said to undergo a unimodal distortion and then the semi-classical Hamiltonian is found to be separated into two parts - a harmonic oscillator part with three vibrational coordinates and a rotational part with two rotational coordinates. However, more progress in terms of specifying the wavefunctions of the system can be made by following a different approach. 

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