Anharmonic effects

In order to demonstrate the accuracy of the potential energy surface described by GAP outside the harmonic regime, we calculated the temperature dependence of the optical phonon mode of the T point in diamond. In fact, the low temperature variation of this quantity has been calculated using Density Functional Perturbation Theory by Lang et al. [9]. The ab initio calculations show excellent agreement 

We note that even at 0 K there are anharmonic effects present due to the zero-point motion of the nuclei. We accounted for the quantum nature of the nuclei by rescaling the temperature of the molecular dynamics runs, by determining the temperature of the quantum system described by the same phonon density of states whose energy is equal to the mean kinetic energy of the classical molecular dynamics runs. The scaling function for the GAP model is shown in Fig. 6.13. 

We are aware that at low temperatures this approximation is rather crude, and the correct way of taking the quantum effects into account would be solving the Schrodinger equation for the nuclear motion. However, we note that the anharmonic correction calculated by Lang et al. [9] by Density Functional Perturbation Theory and our value show good agreement (Table 6.3). 

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Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

Vibrational energy, which is associated with the alternate extension and compression of die chemical bonds. For small displacements from the low-temperature equilibrium distance, the vibrational properties are those of simple harmonic motion, but at higher levels of vibrational energy, an anharmonic effect appears which plays an important role in the way in which atoms separate from tire molecule. The vibrational energy of a molecule is described in tire quantum theory by the equation... 

Modification of inertia of hydrogen-only rigid bodies is a simple and safe way to balance different frequencies in the system, and it usually allows one to raise to 10 fs. Unfortunately, the further increase appears problematic because of various anharmonic effects produced by collisions between non-hydrogen atoms [48]. 

The approach to the evaluation of vibrational spectra described above is based on classical simulations for which quantum corrections are possible. The incorporation of quantum effects directly in simulations of large molecular systems is one of the most challenging areas in theoretical chemistry today. The development of quantum simulation methods is particularly important in the area of molecular spectroscopy for which quantum effects can be important and where the goal is to use simulations to help understand the structural and dynamical origins of changes in spectral lineshapes with environmental variables such as the temperature. The direct evaluation of quantum time- correlation functions for anharmonic systems is extremely difficult. Our initial approach to the evaluation of finite temperature anharmonic effects on vibrational lineshapes is derived from the fact that the moments of the vibrational lineshape spectrum can be expressed as functions of expectation values of positional and momentum operators. These expectation values can be evaluated using extremely efficient quantum Monte-Carlo techniques. The main points are summarized below. 

As discussed for the two complexes [V(CO)6] and Cr(CO)6 these constant shifts obtained for BP86 are just about the order of magnitude of anharmonic effects present in such compounds see Spears, 1997. 

In order to obtain better agreement between theory and experiment, computed frequencies are usually scaled. Scale factors can be obtained through multiparameter fitting towards experimental frequencies. In addition to limitations on the level of calculation, the discrepancy between computed and experimental frequencies is also due to the fact that experimental frequencies include anharmonicity effects, while theoretical frequencies are computed within the harmonic approximation. These anharmonicity effects are implicitly considered through the scaling procedure. 

Starting from the normal mode approximation, one can introduce anharmonicity in different ways. Anharmonic perturbation theory [206] and local mode models [204] may be useful in some cases, where anharmonic effects are small or mostly diagonal. Vibrational self-consistent-field and configuration-interaction treatments [207, 208] can also be powerful and offer a hierarchy of approximation levels. Even more rigorous multidimensional treatments include variational calculations [209], diffusion quantum Monte Carlo, and time-dependent Hartree approaches [210]. 

In order to assign more IR signals of 4a, ab initio calculations on Hbdmpza (3b) and 4a were performed. It is well known for the chosen HF/6-31G basis set that calculated harmonical vibrational frequencies are typically overestimated compared to experimental data. These errors arise from the neglecting anharmonicity effects, incomplete incorporation of electron correlation and the use of finite basis sets in the theoretical treatment (89). In order to achieve a correlation with observed spectra a scaling factor (approximately 0.84-0.90) has to be applied (90). The calculations were calibrated on the asymmetric carboxylate Vasym at 1653 cm. We were especially interested in... 

Keywords Anharmonic effects Displacive phase transition Isotope effects KDP-type ferroelectrics Order-disorder phase transition... 

From Reference 16. For CH3CI normal modes are frequencies corrected by the anharmonicity effects for measured frequencies. 

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