Hamiltonian vibrational

Note that only the polynomial factors have been given, since the exponential parts are identical for all wave functions. Of course, any linear combination of the wave functions in Eqs. (D.5)-(D.7) will still be an eigenfunction of the vibrational Hamiltonian, and hence a possible state. There are three such linearly independent combinations which assume special importance, namely,... 

The vibrational Hamiltonian for the ground eleetronie state of the N2 moleeule within this approximation is given by ... 

The energy levels are described by quantum theory and may be found by solving the time-independent Schroedinger equation by using the vibrational Hamiltonian for a diatomic molecule [13,14]. 

Treating Singularities Present in the Sutcliffe-Tennyson Vibrational Hamiltonian in Orthogonal Internal Coordinates. 

The Hamiltonian (7.149) contains also a rotational term, BC(0(3U)). This term can be treated in the same way as the vibrational terms. The complete Hamiltonian is obtained by adding to the vibrational Hamiltonian of Eq. (7.157) three... 

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. 

The equilibrium electronic energy is a constant, and we shall drop it from the Hamiltonian this does not affect the eigenfunctions of (6.44) and simply decreases the eigenvalues by Ue (Problem 1.6). We write the vibrational Hamiltonian as... 

Since the vibrational Hamiltonian is a sum of terms, each of which involves only one coordinate, the separation-of-variables theorem of Sec- ... 

The electronic Hamiltonian (assuming we are dealing with the electronic Schrodinger equation for the equilibrium nuclear configuration) and the vibrational Hamiltonian are each specified relative to the equilibrium nuclear configuration, which defines the molecular point group. The point... 

Since a symmetry operation sends the molecular framework into a configuration physically indistinguishable from the original one, it does not affect either the electronic or the vibrational Hamiltonian, which are... 

Adding the kinetic and potential energies (6.24) and (6.23) and replacing classical quantities by operators, we have as the (approximate) quantum-mechanical vibrational Hamiltonian for a polyatomic molecule... 

Since the wave functions form a complete set, we conclude that 0R commutes with the electronic Hamiltonian and with the vibrational Hamiltonian ... 

In the energy range 0-16,000 cm-1, the vibrational Hamiltonian of this molecule can be modeled by a Dunham expansion without anharmonic resonances of the classical form [112]... 

Figure 16. Scattering resonances of the full rotational-vibrational Hamiltonian describing the dissociation of CO2 on a LEPS surface obtained by equilibrium point quantization with (2.8). The resonances with 7 = 0,..., 10 are given by dots. Their close vicinity explains the formation of hyphens , i.e., unresolved sequences of dots. Note that rotation is very slightly destabilizing in the present model. The successive hyphens are the bending progressions with V2 = 0,. .. 5. The solid line is given by the Lyapunov exponent of the symmetric-stretch periodic orbit 0 expressed as an imaginary energy.

From the fact that r Ncf 3% is the isometric group of the NC (4.1) it follows by the same reasoning as for SRMs that e is the symmetry group of the rotation-vibration hamiltonian. Though the representation (4.11) is commonly used in vibrational spectroscopy55-S7 it only seldom has been characterized as a group of isometric transformations57. ... 

When the function V(r) is used in the vibrational Hamiltonian instead of the simple harmonic V(x), the quantised vibrational energy levels are... 

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