Equilibrium position, molecular vibrations

Energy calculations and geometry optimizations ignore the vibrations in molecular systems. In this way, these computations use an idealized view of nuclear position. In reality, the nuclei in molecules are constantly in motion. In equilibrium states, these vibrations are regular and predictable, and molecules can be identified by their characteristic spectra. 

Let us now turn our attention to liquid water. Just as in ice I, molecular motions may be divided into rapid vibrations and slower diffusional motions. In the liquid, however, vibrations are not centred on essentially fixed lattice sites, but around temporary equilibrium positions that are themselves subject to movement. Water at any instant may thus be considered to have an I-structure. An instant later, this I-structure will be modified as a result of vibrations, but not by any additional displacements of the molecules. This, together with the first I-structure, is one of the structures that may be averaged to allow for vibration, thereby contributing to the V-structure. Lastly, if we consider the structure around an individual water molecule over a long time-period, and realize that there is always some order in the arrangement of adjacent molecules in a liquid even over a reasonable duration, then we have the diffusionally averaged D-structure. 

A molecule is composed of a certain number N of nuclei and usually a much larger number of electrons. As the masses of the electrons and the nuclei are significantly different, the much lighter elections move rapidly to create the so-called electron cloud which sticks die nuclei into relatively fixed equilibrium positions. The resulting geometry of die nuclear configuration is usually referred to as the molecular structure. The vibrational and rotational spectra of a molecule, as observed in its infrared absorption or emission and the Raman effect, are determined by this molecular geometry. 

For small displacements molecular vibrations obey Hooke s law for simple harmonic motion of a system that vibrates about an equilibrium configuration. In this case the restoring force on a particle of mass m is proportional to the displacement x of the particle from its equilibrium position, and acts in the opposite direction. In terms of Newton s second law ... 

For a given normal mode of frequency v we may write the polarizability as the sum of the polarizability in the equilibrium position Mq and the induced polarizability due to molecular vibrations. 

The dynamical problem to be solved in describing molecular vibrations is analogous to the calculation of the motion of a set of masses connected by springs. The equations of motion can be stated, according to classical mechanics, by applying Newton s second law to a set of atoms acted on by forces acting counter to displacements from a set of equilibrium positions. 

Here t0 is an average vibrational period in a surface potential well having a value of the order of subpicoseconds for a simple molecule, and U() is a depth of the potential, which strongly depends on the type of interaction. For strong interaction, t exceeds 1 s or more, while in the case of no interaction, such as elastic collision, it is less than T(>. If we assume the Lennard-Jones potential for the interaction potential and expand it at the equilibrium position r0, t() is given by relevant molecular parameters as... 

Our model for molecular vibration is a set of N point masses (the nuclei), each of which vibrates about an equilibrium position (which is... 

A few words should be said about the difference between resonance and molecular vibrations. Although vibrations take place, they are oscillations about an equilibrium position determined by the structure of the resonance hybrid, and they should not be confused with the resonance among the contributing forms. The molecule does not resonate or vibrate" from one canonical structure to another. In this sense the term resonance is unfortunate because it has caused unnecessary confusion by invoking a picture of vibration. The term arises from a mathematical analogy between the molecule and the classical phenomenon of resonance between coupled pendulums, or other mechanical systems. 

Fig. 8 A local molecular vibration. The empty circles show the equilibrium positions of the atoms. The energies ea, ep and the overlap integral tap are perturbed.

Actually, the nuclei are not stationary, but execute vibrations of small amplitude about equilibrium positions it is these equilibrium positions that we mean by the fixed nuclear positions. It is only because it is meaningful to speak of (almost) fixed nuclear coordinates that the concepts of molecular geometry or shape and of the PES are valid [12]. The nuclei are much more sluggish than the electrons because they are much more massive (a hydrogen nucleus is about 2,000 more massive than an electron). 

But if we examine the localized near the donor or the acceptor crystal vibrations or intra-molecular vibrations, the electron transition may induce much larger changes in such modes. It may be the substantial shifts of the equilibrium positions, the frequencies, or at last, the change of the set of normal modes due to violation of the space structure of the centers. The local vibrations at electron transitions between the atomic centers in the polar medium are the oscillations of the rigid solvation spheres near the centers. Such vibrations are denoted by the inner-sphere vibrations in contrast to the outer-sphere vibrations of the medium. The expressions for the rate constant cited above are based on the smallness of the shift of the equilibrium position or the frequency in each mode (see Eqs. (11) and (13)). They may be useless for the case of local vibrations that are, as a rule, high-frequency ones. The general formal approach to the description of the electron transitions in such systems based on the method of density function was developed by Kubo and Toyozawa [7] within the bounds of the conception of the harmonic vibrations in the initial and final states. 

The rotational energy concerns molecular rotation around its gravity center, vibrational energy is the result of periodic displacement of atoms of the molecule away from the equilibrium position, and electronic energy is generated by electron movement within the molecular bonds. Rotational levels have lower energy than vibrational ones and thus are lower in energy. Upon photon absorption, electronic, rotational, and vibrational levels... 

Most spectroscopic studies involve the lowest energy vibration-rotation levels, and the determination of the values of the molecular parameters at or near the equilibrium position. This is equally true of most theoretical studies indeed there are many published accurate ab initio calculations of equilibrium properties which do not even extrapolate with the correct analytical form to the dissociation asymptote. Calculations which... 

The V-B coupling Hamiltonian to first order in the three HOD dimensionless normal coordinates is Hv b = —2, c], l , where F, is the inter-molecular force due to the solvent exerted on the harmonic normal coordinate, evaluated at the equilibrium position of the latter. This force obviously depends on the relative separations of all molecules, and on their relative orientations. In the most rigorous quantum description of rotations, this term would depend on the excited molecule rotational eigenstates and of the solvent molecules. Instead rotation was treated classically, a reasonable approximation for water at room temperature. With this form for the coupling, the formal conversion of the Golden Rule formula into a rate expression follows along the lines developed by Oxtoby (2,53), with a slight variation to maintain the explicit time dependence of the vibrational coordinates (57),... 

Molecules are never motionless. They are performing vibrations all the time. In addition, the gaseous molecules, and also the molecules in liquids, are performing rotational and translational motion as well. Molecular vibrations constitute relative displacements of the atomic nuclei with respect to their equilibrium positions and occur in all phases, including the crystalline state, and even at the lowest possible temperatures. The magnitude of molecular vibrations is relatively large, amounting to several percent of the intemuclear distances. Typically, there are about 1012-1014 vibrations per second. 

In an actual crystal the atoms are in permanent motion. However, this motion is much more restricted than that in liquids, let alone gases. As the nuclei of the atoms are much smaller and heavier than the electron clouds, their motion can be well described by small vibrations about the equilibrium positions. In our discussion of crystal symmetry, as an approximation, the structures will be regarded as rigid. However, in modem crystal molecular structure determination atomic motion must be considered [19], Both the techniques of structure determination and the interpretation of the results must include the consequences of the motion of atoms in the crystal. 

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