Force constant - adiabatic

However, Badger s rule is fully confirmed by the adiabatic force constants of polyatomic molecules, which explains its usefulness even for today s research. 

The analysis of adiabatic force constants and adiabatic frequencies associated with the internal coordinates that describe the reaction complex provides direct insight into how the reaction complex changes along s. 

The adiabatic force constant associated with coordinate R3 does not change during the reaction, indicating that the CH bonds of the CH3 radical are spectator bonds. The changes in the R1 and bending angle force constants are related to the changes in the RP vector and the bifurcation of the RP. 

The force constants kj,kc and the dimensionless Renner parameters r, c ate defined by the adiabatic potentials for the components of the II state at pure trans (Vj, Vj) and pure cis (V, V ) bending vibrations,... 

Born-Oppenheimer method Ipi-tyschemI A method for calculating the force constants between atoms by assuming that the electron motion is so fast compared with the nuclear motions that the electrons follow the motions of the nuclei adiabatically. born ap-an.hTm ar. meth ad ... 

The term H e is the electron correlation operator, the term H p corresponds to phonon-phonon interaction and H l corresponds to electron-phonon interaction. If we analyze the last term H l we see that when using crude approximation this corresponds to such phonons that force constant in eq. (17) is given as a second derivative of electron-nuclei interaction with respect to normal coordinates. Because we used crude adiabatic approximation in which minimum of the energy is at the point Rg, this is also reflected by basis set used. Therefore this approximation does not properly describes the physical vibrations i.e. if we move the nuclei, electrons are distributed according to the minimum of energy at point Rg and they do not feel correspondingly the R dependence. The perturbation term H) which corresponds to electron-phonon interaction is too large... 

As we have shown in [21,14] this quasiparticle transformation leads from crude adiabatic to adiabatic Hamiltonian. The Hamiltonian (39) is adiabatic Hamiltonian. Note that the force constant for harmonic oscillators is given as second derivative of Escf at point R . We shall call the corresponding phonons the adiabatic phonons. 

Further we can proceed similarly as in the case of adiabatic approximation. We shall not present here the details, these are presented in [21,22]. We just mention the most important features of our transformation (46-50). Firstly, when passing from crude adiabatic to adiabatic approximation the force constant changed from second derivative of electron-nuclei interaction ufcF to second derivative of Hartree-Fock energy Therefore when performing transformation (46-50) we expect change offeree constant and therefore change of the vibrational part of Hamiltonian... 

If T0 is the adiabatic flame temp, the energy released by foe decompn of unit wt of proplnt, called Force Constant , is defined by... 

The remaining terms in equations (10.38) are the adiabatic corrections to the bond length, and are given by Watson [103] in terms of the Bom Oppenheimer force constant /as... 

In most of the more recent classical approaches [18], no allusion to Ehrenfest s (adiabatic) principle is employed, but rather the differential equations of motion from classical mechanics are solved, either exactly or approximately, subject to a set of initial conditions (masses, force constants, interaction potential, phase, and initial energies). The amount of energy, AE, transferred to the oscillator is obtained for these conditions. This quantity may then be averaged over all phases of the oscillating molecule. In approximate classical and semiclassical treatments, the interaction potential is expanded in a Taylor s series and only the first two terms are retained. 

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