Non-linear molecule

To describe the orientations of a diatomic or linear polyatomic molecule requires only two angles (usually termed 0 and ([)). For any non-linear molecule, three angles (usually a, P, and y) are needed. Hence the rotational Schrodinger equation for a nonlinear molecule is a differential equation in three-dimensions. 

There are 3M-6 vibrations of a non-linear molecule containing M atoms a linear molecule has 3M-5 vibrations. The linear molecule requires two angular coordinates to describe its orientation with respect to a laboratory-fixed axis system a non-linear molecule requires three angles. 

So, for any atom, the orbitals can be labeled by both 1 and m quantum numbers, which play the role that point group labels did for non-linear molecules and X did for linear molecules. Because (i) the kinetic energy operator in the electronic Hamiltonian explicitly contains L2/2mer2, (ii) the Hamiltonian does not contain additional Lz, Lx, or Ly factors. 

Atoms, linear molecules, and non-linear molecules have orbitals which can be labeled either according to the symmetry appropriate for that isolated species or for the species in an environment which produces lower symmetry. These orbitals should be viewed as regions of space in which electrons can move, with, of course, at most two electrons (of opposite spin) in each orbital. Specification of a particular occupancy of the set of orbitals available to the system gives an electronic configuration. For example,... 

For non-linear molecules of the spherical or symmetric top variety, pf j(Rg) (or dpf j/dRa) may be aligned along or perdendicular to a symmetry axis of the molecule. The selection rules that result are... 

For non-linear molecules, when treated as rigid (i.e., having fixed bond lengths, usually taken to be the equilibrium values or some vibrationally averaged values), the rotational Hamiltonian can be written in terms of rotation about three axes. If these axes (X,Y,Z) are located at the center of mass of the molecule but fixed in space such that they do not move with the molecule, then the rotational Hamiltonian can be expressed as ... 

In Section 4.3.f it was shown that there are 3N — 5 normal vibrations in a linear molecule and 3N — 6 in a non-linear molecule, where N is the number of atoms in the molecule. There is a set of fairly simple rules for determining the number of vibrations belonging to each of the symmetry species of the point group to which the molecule belongs. These rules involve the concept of sets of equivalent nuclei. Nuclei form a set if they can be transformed into one another by any of the symmetry operations of the point group. For example, in the C2 point group there can be, as illustrated in Figure 6.18, four kinds of set ... 

In linear molecules only the component of orbital momentum normal to the figure axis is destroyed, that along the figure axis being retained. In non-linear molecules with strong interatomic interactions the concept of orbital angular momentum loses its significance. 

It is then shown that (excepting the rare-earth ions) the magnetic moment of a non-linear molecule or complex ion is determined by the number of unpaired electrons, being equal to ms = 2 /S(S + 1), in which 5 is half that number. This makes it possible to determine from magnetic data which eigenfunctions are involved in bond formation, and so to decide between electron-pair bonds and ionic or ion-dipole bonds for various complexes. It is found that the transition-group elements almost without exception form electron-pair bonds with CN, ionic bonds with F, and ion-dipole bonds with H2O with other groups the bond type varies. 

It is instructive to examine further the approximate semi-classical form for R7 shown above because, when viewed as a rate of transition between two intersecting energy sur ces, one anticipates that connection can be made with the well known Landau-Zener theory (10). For a non-linear molecule with N atoms, the potentials (Q) depend on 3N-6 internal degrees of fi eedom (for a linear molecule, Vj f depend on 3N-5 internal coordinates). The subspace S... 

See also the theoretical description of a micro reactor for optical photocatalytic dissociation of non-linear molecules in [140]. Here, a mathematical model for a novel type of micro reactor is given. Rotating non-linear molecules at excitation of valent vibrations are considered, having a magnetic moment. Resonance decay of molecules can be utilized with comparatively weak external energy sources only. 

Andreev, V. V., Microreactor for optical dissociation of non-linear molecules, in Proceedings of the VDE World Microtechnologies Congress, MICRO.tec 2000, 25-27 September 2000, pp. 779-781, VDE Verlag, Berlin, EXPO Hannover (2000). 

Non-polar, linear molecules (e.g. O2, CI2) Non-polar, non-linear molecules (e.g. CH4, C6H6) Strongly polar molecules (e.g. CH3OH, S02, HC1)... 

The internal degrees of freedom are associated with the rotation and vibration of the molecule. A linear molecule has 2 degrees of rotational motion and a non-linear molecule has three. The remaining (3N — 5) or (3N — 6) degrees of freedom describe the motion of the nuclei with respect to each other. For example, the linear CO2 has (3N — 5) = 4 vibrational degrees of freedom and the non-linear SO2 has three. The mode... 

To illustrate the above concept, we consider the possible and observed peaks for H2S and CS2. H2S is a non-linear molecule and is expected to have 3N—6 = 3 spectroscopic peaks. The diagram below shows the three possible vibrations as a symmetrical stretch, and asymmetric stretch and a motion called scissoring (Fig. 5.3). 

This is a non linear molecule and vibrational degree of freedom can be calculated as follows Number of atoms (N) = 12 Total degree of freedom = 3 x 12 = 36 Translational = 3 Rotational = 3... 

There are two important rules involving harmonic vibrational frequencies that are well known to spectroscopists. They are important in the present context because they permit the simplification of some of the statistical mechanics results for iso-topomers in Chapter 4. The first rule, the Teller-Redlich (TR) product rule, follows straightforwardly from Equation 3.A1.13 (Appendix 3.A1) if one remembers that A = 4n2vf and that there are six frequencies for the non-linear molecule which... 

For a non-linear molecule, there are only 3N — 5 frequencies and two moments of inertia which are equal. 

Equation 4.76 is written for a non-linear molecule with 3N - 6 vibrations. For linear molecules (including diatomics), there are 3N - 5 vibrations, two of the moments of inertia, say Ia and Ib, are equal, and there is no third moment of inertia. Using u = hv/kT, we obtain a more compact formula. 

Figure 1. Translation, rotation, and vibration of a diatomic molecule. Every molecule has three translational degrees of freedom corresponding to motion of the center of mass of the molecule in the three Cartesian directions (left side). Diatomic and linear molecules also have two rotational degrees of freedom, about rotational axes perpendicular to the bond (center). Non-linear molecules have three rotational degrees of freedom. Vibrations involve no net momentum or angular momentum, instead corresponding to distortions of the internal structure of the molecule (right side). Diatomic molecules have one vibration, polyatomic linear molecules have 3V-5 vibrations, and nonlinear molecules have 3V-6 vibrations. Equilibrium stable isotope fractionations are driven mainly by the effects of isotopic substitution on vibrational frequencies.

---