Nonlinear polyatomic molecules

These electronic energies dependence on the positions of the atomic centres cause them to be referred to as electronic energy surfaces such as that depicted below in figure B3.T1 for a diatomic molecule. For nonlinear polyatomic molecules having atoms, the energy surfaces depend on 3N - 6 internal coordinates and thus can be very difficult to visualize. In figure B3.T2, a slice tln-oiigh such a surface is shown as a fimction of two of the 3N - 6 internal coordinates. 

Nonlinear polyatomic molecules require further consideration, depending on their classification, as given in Section 9.2.2. In the classical, high-temperature limit, the rotational partition function for a nonlinear molecule is given by... 

Rotational motion of a general (nonlinear) polyatomic molecule accounts for three degrees of freedom. The partition function in this case was given by Eq. 8.67. It is easy to verify that... 

There are nv b = 3nat - 5 vibrational frequencies for a linear molecule, and nvib = 3nat - 6 for a nonlinear polyatomic molecule. 

However, for a general nonlinear polyatomic molecule of N atoms, only 3N — 6 of these modes correspond to physical vibrations with et 0 (cf. Sidebar 3.8). The remaining six null eigenvectors iq satisfying... 

From a molecular viewpoint, we know that heat capacity is closely connected to internal modes of molecular vibration. According to the classical equipartition theorem (Sidebar 3.8), a nonlinear polyatomic molecule of Aat atoms has ftmodes = 3Aat — 6 independent internal modes of vibration, each of which would contribute equally to heat capacity... 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

The Jahn-Teller theorem is important in considering the electronic states of polyatomic molecules. Jahn and Teller proved in 1937 that a nonlinear polyatomic molecule cannot have an equilibrium (minimum-energy) nuclear configuration that corresponds to an orbitally degenerate electronic term. Orbital degeneracy arises from molecular symmetry (Section 1.19), and the Jahn-Teller theorem can lead to a lower symmetry than... 

For a nonlinear polyatomic molecule the rotation partition function becomes... 

VQo, which would be part of the Q-branch of the spectrum. A Q-branch can occur for non-E diatomics and for nonlinear polyatomic molecules. The R-branch in Fig. 14.7 consists of transitions with AJ = +1 simultaneous with An = +1 vibrational transition. Specifically, the R(l) R(2)... lines represent the transitions /=1 J = 0, J = 2- /=l..., respectively. Analogously, the P-branch represents transitions with AJ = —T. P(l)is/ = 0 J = 1, etc. The spacing between successive lines in the P-branch are slightly greater than those in the R-branch, as determined by differences between BoJ(J + + 1)... 

IV. Nonlinear Polyatomic Molecule (Rigid Rotator, Harmonic Oscillator]... 

Polyatomic molecules have up to three different moments of inertia, corresponding to rotations about three axes (Fig. 20.6). The rotational spectra for nonlinear polyatomic molecules are more complex than the example just illustrated, but their interpretation is carried out in the same way and has enabled chemists to determine with high accuracy the molecular geometries for many small polyatomic molecules. 

The electronic spectrum of a nonlinear polyatomic molecule is very complicated. In addition to three modes of rotation with distinct moments of inertia, there are 3N — 6 modes of vibration. While some of these may be forbidden in the infrared or Raman spectrum on the basis of symmetry, there is no rule to forbid their appearance in the electronic spectrum, which is extraordinarily complex as a consequence. For our purposes here, we mention only a few fundamental points and present one example. 

If, instead of being structureless reactants, A and B are nonlinear polyatomic molecules containing and atoms, respectively, the partition functions for the transition-state complex and reactants may be written... 

For nonlinear polyatomic molecules, no orbital angular-momentum operator commutes with the electronic Hamiltonian, and the angular-momentum classification of electronic terms cannot be used. Operators that do commute with the electronic Hamiltonian are the symmetry operators Or of the molecule (Section 12.1), and the electronic states of polyatomic molecules are classified according to the behavior of the electronic wave function on application of these operators. Consider H2O as an example. 

Linear triatomic molecules also have two significant rotational degrees of freedom nonlinear molecules have three. For nonlinear polyatomic molecules, the number of vibrational degrees of freedom is 3N — 6, where N is the number of atoms in the molecule. 

The distribution of the iN — 6 vibrational symmetry coordinates of a nonlinear polyatomic molecule among the irreducible representations of its symmetry point group can be determined by standard methods. [7] Ordinarily, not all of the symmetry species will be represented and several of them will include more than one coordinate. If the molecule belongs to a commutative symmetry point group, all of them will be assigned to one-dimensional symmetry species. If its group is non-commutative, and therefore has representations that are two-or three-dimensional, some of its vibrations may be degenerate these are best discussed separately. 

In this chapter, we extend our treatment of rotation in diatomic molecules to nonlinear polyatomic molecules. A traditional motivation for treating polyatomic rotations quantum mechanically is that they form a basis for experimental determination for bond lengths and bond angles in gas-phase molecules. Microwave spectroscopy, a prolific area in chemical physics since 1946, has provided the most accurate available equilibrium geometries for many polar molecules. A background in polyatomic rotations is also a prerequisite for understanding rotational fine structure in polyatomic vibrational spectra (Chapter 6). The shapes of rotational contours (i.e., unresolved rotational fine structure) in polyatomic electronic band spectra are sensitive to the relative orientations of the principal rotational axes and the electronic transition moment (Chapter 7). Rotational contour analysis has thus provided an invaluable means of assigning symmetries to the electronic states involved in such spectra. 

In a polyatomic molecule with N nuclei, 3N independent coordinates are required to specify all of the nuclear positions in space. We have already seen in the preceding chapter that rotations of nonlinear polyatomics about their center of mass may be described in terms of the three Euler angles < >, 0, and x- Three additional coordinates are required to describe spatial translation of a molecule s center of mass. Hence, there will be3N — 6 independent vibrational coordinates in a nonlinear polyatomic molecule. In a linear polyatomic molecule, the orientation may be given in terms of two independent angles and (f). Linear polyatomics therefore exhibit 3N — 5 rather than 3N — 6 independent vibrational coordinates. 

---