Polyatomic Molecules

Abstract

Section 4.1 presents the arbitrarily shaped, rigid rotor, and Sect. 4.2. introduces normal coordinates used to treat vibrations. Section 4.3 addresses the symmetries of point groups as a key principle to classify polyatomic molecules. The specialties of the electronic structure of polyatomic molecules are introduced in Sect. 4.4 by way of example, starting with H₂O. We then introduce the important concept of orbital hybridization by discussing sp ³ MOs for CH₄ and NH₃ (Sect. 4.4.2). Finally, in Sect. 4.5 conjugated hydrocarbon systems are introduced. A simple, quantitative description is provided by the Hückel approximation (HMO).

Diatomic molecules, treated in the previous chapter, provide only a first step into the world of real molecules. Anyone who wishes to obtain an idea about that world should at least browse through the present chapter – even though it is, admittedly, somewhat demanding. The spectroscopy of triatomic and polyatomic molecules is largely determined by their symmetry, and all concepts introduced in Chap.  3 have to be generalized. The motion of, say, $\mathcal{N}_{\mathrm{nu}}$ atomic nuclei is characterized by three rotational and $3\mathcal{N}_{\mathrm{nu}}-6$ vibrational degrees of freedom (aside from the trivial three translational motions). Correspondingly, the description of electronic states becomes significantly more complex.

Acronyms and Terminology

AO:

‘Atomic orbital’, single electron wave function in an atom; typically the basis for a rigorous structure calculation.

APES:

‘Adiabatic potential energy surfaces’, potential energy hyper-surface determined in Born-Oppenheimer approximation.

BO:

‘Born Oppenheimer’, approximation, the basis when solving the Schrödinger equation for molecules (see Sect. 3.2).

E1:

‘Electric dipole’, transitions induced by the interaction of an electric dipole with the electric field component of electromagnetic radiation.

FT:

‘Fourier transform’, see Appendix I in Vol. 1.

good quantum number:

‘Quantum number for eigenvalues of such observables that may be measured simultaneously with the Hamilton operator (see Sect. 2.6.5 in Vol. 1)’.

HF:

‘Hartree-Fock’, method (approximation) for solving a multi-electron Schrödinger equation, including exchange interaction.

HITRAN:

‘High-resolution transmission molecular absorption database’, http://www.cfa.harvard.edu/hitran (Rothman et al. 2009).

HMO:

‘Hückel molecular orbital method’, approximation for describing molecular orbitals in conjugated hydrocarbon molecules (Sect. 4.5).

HOMO:

‘Highest occupied molecular orbital’.

IP:

‘Ionization potential’, of free atoms or molecules (in solid state physics the equivalent is called “workfunction”).

IR:

‘Infrared’, spectral range of electromagnetic radiation. Wavelengths between $760\operatorname{nm}$ and $1\operatorname{mm}$ according to ISO 21348 (2007).

isotopologue:

‘Molecules that differ only in their isotopic composition’, http://en.wikipedia.org/wiki/Isotopologue.

JT:

‘Jahn and Teller’, have first treated in 1937 the symmetry breaking effect, now referred to by their names.

JTE:

‘Jahn-Teller effect’, symmetry breaking effect first treated by Jahn and Teller in 1937.

LCAO:

‘Linear combination of atomic orbitals’, linear superposition of atomic, single electron wave functions to form a molecular orbital (MO).

LUMO:

‘Lowest unoccupied molecular orbital’.

MO:

‘Molecular orbital’, single electron wave function in a molecule; typically the basis for a rigorous molecular structure calculation.

PJTE:

‘Pseudo-Jahn-Teller effect’, vibronic coupling for nearly degenerate molecular states, leading to symmetry breaking.

RHF:

‘Restricted Hartree-Fock’, assuming all spatial wave functions in a given closed shell to be equal when computing atomic wave functions.

UV:

‘Ultraviolet’, spectral range of electromagnetic radiation. Wavelengths between $100\operatorname{nm}$ and $400\operatorname{nm}$ according to ISO 21348 (2007).

VIS:

‘Visible’, spectral range of electromagnetic radiation. Wavelengths between $380\operatorname{nm}$ and $760\operatorname{nm}$ according to ISO 21348 (2007).

VUV:

‘Vacuum ultraviolet’, spectral range of electromagnetic radiation. part of the UV spectral range. Wavelengths between $10\operatorname{nm}$ and $200\operatorname{nm}$ according to ISO 21348 (2007).