Page 290 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry

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Encyclopedia of Physical Science and Technology EN009N-447 July 19, 2001 23:3

Microwave Molecular Spectroscopy 807

follows, the factor 1/h will be included in the energy ex- where E is a diagonal matrix of eigenvalues. The transfor-
pression, so that E will be expressed in frequency units. mation matrix T, which diagonalizes the energy matrix,
According to quantum mechanics, gaseous molecules ro- yields the eigenfunctions. Each column of T corresponds
tate at only certain rates and have discrete values of rota- to the set of expansion coefficients a n (or eigenvector) for
tional energy and angular momentum. When a quantum of the eigenfunction ψ in the basis {φ n }. These define the
energy hν is absorbed, the molecule is raised to a higher set of asymmetric-top functions and are required to calcu-
energy level and rotates at the next allowed rate. The type late the average values of an operator in the asymmetric
of transition that can occur is governed by the electric rotor basis. It may be noted that diagonalization of H is
dipole moment matrix element or transition moment, equivalent to solving the secular determinate

|H − IE|= 0, (20)
∗
µ ij = ψ µψ j d τ = (i|µ| j), (16)
i
which is particularly useful when the order of H is small.
where ψ and ψ j are, respectively, the wavefunctions for Here I is a unit matrix.
∗
i
the lower and upper states of the transition. The i and
j denote the set of quantum numbers required to char-
A. Diatomic and Linear Molecules
acterize these states. Only those transitions are allowed
for which the above matrix element is nonvanishing. This The Hamiltonian operator for a rigid diatomic or linear
places certain restrictions on the changes possible in the molecule is
quantum numbers, and these restrictions are called selec- 2
= P /2I, (21)
tion rules. The basic problem of rotational spectroscopy
is the evaluation of the quantized rotational energy lev- where P is the total angular momentum and I the moment
els and selection rules by quantum mechanical techniques of inertia. The operator P Z corresponds to the projection
and the subsequent assignment of the observed transi- of P along a space-fixed Z axis. According to quantum
2
tions to particular levels characterized by certain quantum mechanics, since the operators , P , and P Z commute
numbers. with each other, they have a common set of eigenfunc-
The general procedure for evaluating the quantum me- tions, which we denote by ψ J,M ≡|J, M). Classically,
chanical energy levels starts with the formulation of the these quantities are constants of motion. The matrix el-
Hamiltonian , expressed in terms of the angular momen- ements of the angular momentum are
tum operators and coordinates, if required. For rigid linear 2 2
(J, M|P |J, M) = h J(J + 1),
and symmetric-top molecules, the Hamiltonian operator (22)
is simple enough that the energy eigenvalue equation may (J, M|P Z |J, M) = hM.
be solved directly,
Therefore, the rotational energy is given by
ψ = Eψ, (17)
E J = BJ(J + 1), (23)
to give the eigenfunctions ψ and the energy levels E.For 2
where E J is in frequency units and B = h/8π I.For
many problems of interest such as an asymmetric top, the
linear molecules, the rotational energy levels are charac-
above eigenvalue equation cannot be solved directly. In terized by two quantum numbers J and M, which are
such cases, the wavefunctions may be expressed as linear restricted to certain integral values:
combinations of a complete set of known functions {φ n },
such as a set of symmetric top functions, J = 0, 1, 2,..., M = 0, ±1, ±2,..., ±J. (24)
In the absence of external fields, the rotational energies do
ψ = a n φ n . (18)
not depend on M, as Eq. (23) implies, and all levels are
According to quantum mechanics, the energy levels are (2J +1)-fold degenerate. However, when an external field
now found by diagonalization of the Hamiltonian or en- isapplied,thisdegeneracyislifted,andtheenergydepends
ergy matrix H = [ n,n ], where the elements are the ma- on the space orientation quantum number M. A similar

trix elements of in the particular basis {φ n }. With the condition holds for symmetric and asymmetric tops. For
availability of high-speed computers, it is usually a rel- rotational absorption of radiation, the selection rule
atively simple matter to diagonalize a symmetric matrix
J → J + 1 (25)
by appropriate numerical methods. Diagonalization of the
matrix representation of the operator gives the eigenvalues applies, and the rotational frequencies of a rigid linear
of the operator molecule are given by
¯
THT = E, (19) ν = E J+1 − E J = 2B(J + 1). (26)