Jump to ContentJump to Main Navigation
Algebraic Theory of Molecules$
Users without a subscription are not able to see the full content.

F. Iachello and R. D. Levine

Print publication date: 1995

Print ISBN-13: 9780195080919

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780195080919.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 07 December 2021

Many-Body Algebraic Theory

Many-Body Algebraic Theory

Chapter:
(p.133) Chapter 6 Many-Body Algebraic Theory
Source:
Algebraic Theory of Molecules
Author(s):

F. Iachello

R. D. Levine

Publisher:
Oxford University Press
DOI:10.1093/oso/9780195080919.003.0009

For molecules with many atoms, the simultaneous treatment of rotations and vibrations in terms of vector coordinates r1, r2, r3, . . . , quantized through the algebra . . .G ≡ U1(4) ⊗ U2(4) ⊗ U3(4). . . . . . .(6.1). . . becomes very cumbersome. Each time a U(4) algebra is added one must go through the recoupling procedure using Racah algebra, which, although feasible, is in practice very time consuming. An alternative treatment, which can be carried out for molecules with any number of atoms, is that of separating vibrations and rotations as already discussed in Sections 4.2-4.5 for triatomic molecules. For nonlinear molecules, there are three rotational degrees of freedom, described by the Euler angles α, β, γ of Figure 3.1, and thus there remain 3n − 6 independent vibrational degrees of freedom, where n is the number of atoms in the molecule. For linear molecules, there are two rotational degrees of freedom, described by the angles α, β, and thus there remain 3n − 5 independent vibrational degrees of freedom, some of which (the bending vibrations) are doubly degenerate. In this alternative treatment, the algebraic theory of polyatomic molecules consists in the separate quantization of rotations and vibrations. Each bond coordinate is then a scalar, and the corresponding algebra is that of U(2). In polyatomic molecules, the geometric symmetry of the molecule also plays a very important role. For example, the benzene molecule, which is the example we discuss in this book has the point group symmetry D6h. A consequence of the symmetry of the molecule is that states must transform according to representations of the appropriate symmetry group. In terms of coordinates, this implies that one must form internal symmetry coordinates. These are linear combinations of the internal coordinates.

Keywords:   C-H overtone spectrum, Poschl-Teller potential, bending vibrations, dipole function, infrared transitions, octahedral molecules, symmetry-adapted operators, transition operator

Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us .