Nuclear motions vibrational methods

Molecular modeling, also known as molecular mechanics, is a method to calculate the structure and energy of molecules based on nuclear motions. Electrons are not considered explicitly, but rather it is assumed that they will find their optimum distribution once the positions of the nuclei are known. This assumption is based on the Bom-Oppenheimer approximation of the Schrodinger equation. The Bom-Oppenheimer approximation states that nuclei are much heavier and move much more slowly than electrons. Thus, nuclear motions, vibrations and rotations can be studied separately from electrons the electrons are assumed to move fast enough to adjust to any movement of the nuclei. 

A meaningful comparison of kinetic data obtained in different biological systems should be based on the determination of the respective contributions of the nuclear and electronic factors. The most direct method of separating these contributions consists in the measurement of the temperature dependence of the rate over the widest available range. In the following, we distinguish between experiments performed at room temperature, which are usually interpreted by assuming that all the nuclear motions coupled to the transfer may be described classically, and experiments performed at lower temperature, in which the quantified character of particular vibrational modes may appear. 

There have been a few recent studies of the corrections due to nuclear motion to the electronic diagonal polarizability (a ) of LiH. Bishop et al. [92] calculated vibrational and rotational contributions to the polarizability. They found for the ground state (v = 0, the state studied here) that the vibrational contribution is 0.923 a.u. Papadopoulos et al. [88] use the perturbation method to find a corrected value of 28.93 a.u. including a vibrational component of 1.7 a.u. Jonsson et al. [91] used cubic response functions to find a corrected value for of 28.26 a.u., including a vibrational contribution of 1.37 a.u. In all cases, the vibrational contribution is approximately 3% of the total polarizability. 

Application of 2D IR spectroscopy to PCET models of Section 17.3.2 is a logical starting point for this type of investigation. 2D methods can unravel the correlated nuclear motion in a PCET reaction and in principle decipher how vibrational coupling in the Dp/Ap interface couples to the ET event between the Ae/De sites. These data can identify the structural dynamics within the interface that promote PCET reactions in much the same way that local hydrogen bonding structure and dynamics mediate excited state PT reactions [239, 240]. In these experiments, the PCET reaction can be triggered by an ultrafast resonant visible laser pulse (as in a standard TA experiment) and a sequence of IR pulses may be employed to build a transient 2D IR spectrum. These experiments demand that systems be chosen so that the ET and PT events occur on an ultrafast timescale. 

The molecular mechanics method may be considered to derive from an extension of the Born-Oppenheimer approximation, in that first, the nuclear and electronic motions may be separated, and second, the nuclear motion may be broken down into contributions due to different vibrational modes. In attempting to describe the motions within a molecule with respect to classical concepts it is necessary to regard the molecule as consisting of a set of hard spheres linked by springs the vibration of the springs describes the motion within the molecule. 

The Bom-Oppenheimer separation of the electronic and nuclear motions is a cornerstone in computational chemistry. Once the electronic Schrodinger equation has been solved for a large number of nuclear geometries (and possibly also for several electronic states), the potential energy surface (PES) is known. The motion of the nuclei on the PES can then be solved either classically (Newton) or by quantum (Schrodinger) methods. If there are N nuclei, the dimensionality of the PES is 3N, i.e. there are 3N nuclear coordinates that define the geometry. Of these coordinates, three describe the overall translation of the molecule, and three describe the overall rotation of the molecule with respect to three axes. For a linear molecule, only two coordinates are necessary for describing the rotation. This leaves 3N - 6(5) coordinates to describe the internal movement of the nuclei, which for small displacements may be chosen as vibrational normal coordinates . 

The CP method may be considered as a semi-classical dynamics approach where the electrons are treated quantum mechanically while the nuclear motion is treated classically. The latter implies that for example zero point vibrational effects are not included, nor can nuclear tunnelling effects be described this requires fully quantum methods, as described in the next section. 

The vibration-rotation spectra and/or the rotational spectra in excited vibrational states provide the af constants and, when all the a/ constants are determined, the equilibrium rotational constants can be obtained by extrapolation. This method has often been hampered by anharmonic or harmonic resonance interactions in excited vibrational states, such as Fermi resonances arising from cubic and higher anharmonic force constants in the vibrational potential, or by Coriolis resonances. Equihbrium rotational constants have so far been determined only for a limited number of simple molecules. To be even more precise, one has further to consider the contributions of electrons to the moments of inertia, and to correct for the small effects of centrifugal distortion which arise from transformation of the original Hamiltonian to eliminate indeterminacy terms [11]. Higher-order time-independent effects such as the breakdown of the Bom-Oppenheimer separation between the electronic and nuclear motions have been discussed so far only for diatomic molecules [12]. 

An alternative route is based on time-dependent approaches, where the standard statistical mechanics formalism relies on Fourier transform of the time correlation of vibrational operators [54—57]. These approaches can provide a complete description of the experimental spectrum, that is, the characterization of the real molecular motion consisting of many degrees of freedom activated at finite temperature, often strongly coupled and anharmonic in namre. However, computation of the exact quantum dynamics evolution of the nuclei on the ab initio potential surface is as prohibitive as the quantum/stationary-state approaches. In fact, even a semiclassical description of the time evolution of quanmm systems is usually computationally expensive. Therefore, time correlation methods for realistic systems are usually carried out by sampling of the nuclear motion in the classical phase space. In this context, summation over i in Eq. 11.1 is a classical ensemble average furthermore, the field unit vector e can be averaged over all directions of an isotropic fluid, leading to the well-known expression... 

It is shown that the well-known contributions to the molecular terms corresponding to the energies of the motion of the electrons, the nuclear vibrations and the rotations of a molecule can be obtained systematically as the terms of a power expansion of the fourth root of the ratio of the electron mass to the (average) nuclear mass. The method yields, among other things, an equation for the rotations, which is a generalisation of the ansatz of Kramers and Pauli (top with built-in flywheel). Furthermore it provides a justification of the considerations by Franck and Condon on the intensities of the band lines. The situation is illustrated by the example of two-atomic molecules. 

In addition, the shallow potential energy surfaces are intrinsically more sensitive than the surfaces associated with local modes in the mid-IR region. Experimentally, THz vibrational frequencies are indeed found to be sensitive to small perturbations of the molecular environment. In computations, they are likewise sensitive to the accuracy of detail of the model, as well as the numerical accuracy of the method. These factors increase the difficulty of making a descriptive vibrational assignment of the nuclear motions, and underline the advantages of studying well resolved spectra of small crystalline systems, for which accurate models may be devised. 

Secondly, we have not discussed in any detail the effects of nuclear motion. Methods used to calculate these vibrational corrections, for both zero-point vibrational effects and temperature effects, have been described elsewhere in this book. There are, however, other effects that should also be considered. We have not discussed the role of the purely vibrational contributions to molecular (electric) properties (Bishop 1990), which in certain cases can be as large as the electronic contributions (Kirtman et al. 2000). Moreover, in conformationally flexible molecules one has to consider the effects of large nuclear motions. For instance, for a proper comparison with experiment, it may not be sufficient to perform an ab initio calculation for a single molecular structure. In experiment one will always observe the average value of the different thermally accessible isomers (rotamers, conformers), and in order to allow for a direct comparison with these experimental observations, a Boltzmann average of the properties of these isomers must be computed. This is particularly important when the properties of the isomers are very different, possibly even differing in sign (Pecul et al. 2004). 

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