Internal modes - adiabatic

(28) determines the form of internal vibrations Vn because it defines onedimensional subspaces within the full configuration space. The motion in an one-dimensional subspace can be described by vector Vn, which can be found by linearization (e.g. via a Taylor expansion at point A=0) of Eq. (28). If needed, the time dependence of X can be found using generalized momenta 

A set of equations similar to (27) can be obtained by applying a completely different approach [18]. One can displace parameter q from its equilibrium value (qn = 0), keep it frozen and equal to a constant qn - At the same time, all other parameters qm can relax until the molecular energy attains its minimum. Hence, parameter qp leads the corresponding motion as described by Eq. (30) 

(31) can be easily solved using the method of Lagrange multipliers d 

In quantum chemical calculations, the vibrational problem is normally described in the harmonic approximation. Assuming that the vibrational problem has been solved, potential energy and each internal parameter qn can be expressed as function of Nvib normal mode coordinates Q, [1-6] 

The superscript n denotes the solution for internal parameter q where 

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Clearly, the assets of a useful, in itself noncontradictory, and physically based CNM analysis are the internal vibrational motions and their properties as well as the amplitudes that relate internal modes to normal modes. As shown in the previous section, the adiabatic internal modes an are the appropriate candidates for internal modes. Adiabatic modes are based on a dynamic principle, they are calculated by solving the Euler-Lagrange equations, they are independent of the composition of the set of internal coordinates to describe a molecule, and they are unique in so far as they provide a strict separation of electronic and mass effects [18,19]. Therefore, they fulfil the first requirement for a physically based CNM analysis. 

The internal mode vector Vp can be defined with the help of the c-vectors (Eq. 22) as is implicitly assumed within the PED analysis [25-27]. Alternatively, one can use the adiabatic internal modes ap which are led by the associated internal parameters qn as internal vibrational modes. The latter are preferred since they have a better physical justification than vectors Cp, which should pay off when defining the amplitude/ nn [18-20]. 

Clearly, the best correlation pattern complying exactly with the expected Lorentzian form is obtained in the case of the AvAF amplitudes in connection with a comparison of frequencies with adiabatic internal frequencies (Og. Adiabatic internal modes, the amplitude definition of Eq. (64) and the force constant matrix f as a suitable metric for comparison provide the right ingredients for a physically well-founded CNM analysis. 

ANALYSIS OF VIBRATIONAL SPECTRA IN TERMS OF ADIABATIC INTERNAL MODES... 

A characterization of vibrational normal modes in terms of adiabatic internal modes is straightforward with the definitions given in the previous sections. As an example, the vibrational modes of cyclopropane [28] will be discussed. They have been calculated at the FIF/6-31G(d,p) level of theory and they are compared with experimental frequencies in Table 1. 

Analysis of the normal modes of cyclopropane using adiabatic internal modes. ... 

Characterization of normal modes in terms of adiabatic internal modes for difluorodioxirane and dioxirane. ... 

As indicated for 1 and 2, the CNM analysis in terms of adiabatic internal modes makes it rather simple to correlate the vibrational spectra of related molecules and to discuss the influence of substituents, heteroatoms, and structural changes in terms of the internal mode frequencies. In the following section, we will provide further examples how vibrational spectra of different molecules can be correlated with the help of the CNM analysis. 

The CNM analysis in terms of adiabatic internal modes has been carried out to correlate the calculated vibrational spectra of the three dehydrobenzenes, namely ortho- (3), meta- (4) and para-henzyne (5), with the vibrational spectrum of benzene (6). Investigation of dehydrobenzenes with the help of infrared spectroscopy is of considerable interest at the moment since these molecules have been found to represent important intermediates in the reaction of enediyne anticancer drugs with DNA molecules [34-37]. Both 4 and 5 are singlet biradicals and, therefore, they are so labile that they can only be trapped at low temperatures in an argon matrix upon photolytic decomposition of a suitable precursor [38-40]. 

Kraka and co-workers [41] have calculated the vibrational spectra of 3, 4, and 5 at the GVB(l)/6-31G(d,p) level of theory where in each case the biradical nature of the dehydrobenzenes was described by the two-configuration approach of GVB. In Tables 4, 5, and 6, a CNM analysis of the calculated spectra based on calculated adiabatic internal modes is presented. 

ADIABATIC INTERNAL MODES FROM EXPERIMENTAL FREQUENCIES... 

In a similar way as dipole derivatives with respect to normal coordinates are obtained from dipole derivatives with regard to Cartesian coordinates, one can also obtain dipole derivatives with respect to the internal coordinates associated with the adiabatic internal modes ... 

Eq. (95) gives the conditions for obtaining generalized adiabatic internal modes ang(s)[22] ... 

Dr>n(s) denotes an element of a Wilson B-type matrix D that connects normal coordinates with internal coordinates. Solving Eq. (95), generalized adiabatic internal modes and related force constants kn (s), mass mn fs), and frequency (On (s) are obtained by Eqs. (97) [22],... 

The generalized adiabatic internal modes are essential for the unified reaction valley analysis (URVA) developed by Konkoli, Kraka, and Cremer to investigate reaction mechanisms and reaction dynamics [22,52]. As an example for the application of the generalized adiabatic internal modes, the hydrogenation reaction of the methyl radical is shortly discussed here ... 

There are immediately a number of applications of adiabatic internal modes that lead to a new dimension in the analysis of vibrational spectra. For example, the adiabatic vectors ap are perfectly suited to present a set of localized internal modes that can be used to analyze delocalized normal modes. This has led to the CNM analysis (Sections 7 and 8) of calculated vibrational spectra of molecules as was discussed in Section 9. With the CNM analysis it is rather easy to correlate the vibrational spectra of different molecules (Section 10). With the help of perturbation theory and calculated normal modes, the determination of adiabatic modes and the CNM analysis can be extended to experimental spectra (Section 12). 

Adiabatic internal modes can be defined for equilibrium points on the PES as well as for all points along the RP where in the latter case it is required that the harmonic part of the energy in equation (28b) is minimized with regard to displacements in the 3K — L — 1 )-dimensional vibrational space while relaxing all internal parameters but one. In this way, generalized adiabatic modes aifs) together with the corresponding adiabatic frequencies a>k s) and force constants kkis) are calculated where k denotes a particular internal coordinate Rk. 

Konkoli, Kraka, and Cremer have shown that the basis vectors Uk correspond to the internal modes that characterize the movement along the RP and, therefore, represent the equivalent to the adiabatic internal modes which are used for the analysis of the transverse normal vibrational modes. Accordingly, an amplitude A., based on the matrix M can be defined as ... 

Modes with a strong dependence on s can be decomposed in terms of adiabatic internal modes to unravel their dependence on certain geometrical features of the reaction complex. 

Analysis of the RP curvature k(s) helps to identify those path regions with strong curvature and a coupling between translational and transverse vibrational modes. For this purpose, the curvature is investigated in terms of normal mode-curvature coupling coefficients and adiabatic internal mode-curvature coupling amplitudes At.,. 

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