Internal displacement coordinate potential energy

The actual calculation consists of minimizing the intramolecular potential energy, or steric energy, as a function of the nuclear coordinates. The potential-energy expressions derive from the force-field concept that features in vibrational spectroscopic analysis according to the G-F-matrix formalism [111]. The G-matrix contains as elements atomic masses suitably reduced to match the internal displacement coordinates (matrix D) in defining the vibrational kinetic energy T of a molecule ... 

The vibrational potential energy of a molecule can be expanded as a function of internal displacement coordinates, qjt in the following way ... 

For rigid molecules it is convenient to express the potential energy by an expansion in powers of the internal displacement coordinates. A similar expansion can also be applied in the case of nonrigid molecules. However, the potential function is expanded only in the small amplitude coordinates, Rt t= 1, 2,... 3 N—717 36),... 

Consider an ideally infinite helical chain whose crystallographic repeat contains N chemical repeat units, each with P atoms. A screw symmetry operation transforms one chemical unit into the next, with a being the rotation about the helix axis and d the translation along the axis. Let r" denote the ith internal displacement coordinate associated with the nth chemical repeat unit. The potential energy, by analogy with Eq. (3), is given by... 

The potential energy is expanded as a power series in internal displacement coordinates R the coefficients are the force constants in the chosen coordinate representation. 

The determination of these normal frequencies, and the forms of the normal vibrations, thus becomes the primary problem in correlating the structure and internal forces of the molecule with the observed vibrational spectrum. It is the complexity of this problem for large molecules which has hindered the kind of detailed solution that can be achieved with small molecules. In the general case, a solution of the equations of motion in normal coordinates is required. Let the Cartesian displacement coordinates of the N nuclei of a molecule be designated by qlt q2,... qsN. The potential energy of the oscillating system is not accurately known in the absence of a solution to the quantum mechanical problem of the electronic energies, but for small displacements it can be quite well approximated by a power series expansion in the displacements ... 

The amide VI mode is mainly CO out-of-plane bend in terms of the potential energy distribution from various internal coordinates. However, the N and H atoms are also displaced and influence the relative intensity of this mode. 

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 potential energy is written in terms of internal coordinates or here explicitly atom-atom distances, and it takes the form of a sum of terms as in (2.2). All these distances (including those within the molecule and those between different molecules) form the components of a column vector R. The harmonic force constant matrix in terms of the corresponding displacements r is As already noted, however, it is advantageous to use cartesian displacement coordinates since the kinetic energy matrix G is then diagonal. It is therefore necessary to express also the potential 4> in terms of the cartesian displacements (vector x) and to write the force constant matrix accordingly ... 

Changes in interatomic distances or in the angles between chemical bonds, or both, can be used to provide a set of 3A — 6 (or 37V — 5 for linear molecules) internal coordinates (Sec. 2-6), i.e., coordinates which are unaffected by translations or rotations of the molecule as a whole. These are particularly important because they provide the most physically significant set for use in describing the potential emwgy of the molecule. The kinetic energy, on the other hand, is more easily set up in terms of cartesian displacement coordinates of the atoms (Sec. 2-6). A relation between the two types is therefore needed. 

The proof starts with a demonstration that the potential energy cannot depend upon the coordinate Oij. For suppose the molecmle is subjected to a small internal rotation, This means that the /3th atom of the (7th top is given a displacement... 

These coordinates represent changes of (n — 1) internal coordinates Rp the changes which minimize the potential energy of the system for a given displacement of Ri (compare with the method of reaction coordinate). The method uses the force constants of molecular vibrations (interaction compliance) calculated somewhat differently than in Eq. (1.4), namely as the forces that need to be applied to achieve measurable distortions in with the potential energy minimized with respect to other coordinates ... 

In order to express the potential energy in terms of Cartesian displacement coordinates, we first introduce vectors X(fcj) and Rik ), respectively describing the phonons in terms of Cartesian and internal coordinates, where... 

Bartell (66-69) has considered in detail interactions between non-bonded atoms as a factor in secondary isotope effects. His treatment is similar in several respects to that outlined in Section IIIB. Specifically, he calculates the average over the lowest vibrational level of the potential energy due to H H, C H, and C C nonbonding interactions in the reactant and in the product or a model of the transition state. The isotope effect on the difference is simply ASF of our eq. (III-28), in which the displacements from equilibrium of the distances between nonbonded atoms are chosen as internal coordinates. 

The classical kinetic energy of the system has now been separated into the effect of displacement of the center of mass of the system, with momentum P and that of the relative movement of the two particles, with momentum p. In the absence of external forces, the interaction of the two (spherical) particles is only a function of (heir separation, r. That is, the potential function appearing in Eq. (37) depends only on the internal coordinates x, y, z. 

---