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Molecular Structure 31.4 Nuclear Motion 483

right-hand side of (31.91) are accomplished by expand-

ing the angular dependence of H



in d-matrices and

using angular momentum coupling algebra [31.158].



gives the interaction potential of the ground state

noble gas atom and the I

2S+1

Λ state of AB.Itrep-

resents the potential coupling in the diabatic basis

2S+1

Λ. These matrix elements are derived from

the corresponding adiabatic potential energy surfaces,

determined using the techniques of Sect. 31.2. Since the

van der Waals interaction is weak, the counterpoised

method discussed in Sect. 31.2.4 should be used. When

only one electronic state of a given symmetry is involved

(point group C

in the case of a triatom), the adiabatic

and diabatic potentials are taken as equal. This is the

case if, for example, only the

Π state of AB is con-

sidered. When more than one electronic state of a given

sysmmetry is involved, for example, when both

Π and

electronic states of AB are included, the adiabatic

potentials are used to determine the diabatic state po-

tentials using an approximate adiabatic → diabatic state

transformation [31.159,160].

The solution to (31.90) can be obtained by either

variational methods or by direct integration [31.153].

Results for Ar-OH [31.161], Ar-BH [31.155], and

Ar-CH [31.162], based on potential energy surfaces

determined from multireference contracted CI wave

functions, are encouraging. In these studies, the

r-dependence of the potential energy surfaces was

neglected.

31.4.4 Reaction Path

The rovibrational spectrum reﬂects the molecular struc-

ture in the vicinity of an equilibrium structure of the

molecule. However a chemical reaction samples a much

broader range of molecular structures. The evolution of

molecular structure can be characterized by the reaction

path, which represents a minimum energy path along

the potential energy surface connecting the reactants and

products. The reaction path Hamiltonian [31.163, 164]

enables dynamical aspects of a chemical reaction to

be inferred from a characterization of the reaction

path.

The reaction path R(s) on potential energy sur-

face E

(R) is the curve in mass weighted Cartesian

coordinates given parametrically in terms of the arc

length s and is deﬁned as that solution to the differential

equation

= g

I†

≡ R



(s) (31.92)

which approaches the saddle point from below [31.17,

165], where the arc length is given by



i=1

. (31.93)

At each point s, R(s) represents the molecular structure

in terms of 3N mass weighted Cartesian coordinates.

(In this subsection, the notation suppresses the differ-

ence between Cartesian and mass weighted Cartesian

coordinates.)

The reaction path can be obtained by integrat-

ing (31.92). Integration begins at the saddle point along

the direction of the single negative eigenvalue of F

which is related to the reaction path curvature dR

/ds

≡







−

(

†



)





I†

(31.94)

Thus integration of (31.92) gives the steepest descent

path from the saddle point. This integration requires,

in principle, only knowledge of the energy gradient,

(R), which is readily available using analytic gradient

techniques (see Sect. 31.2). This approach works quite

well when the reaction path is used only to investigate

the topology of a potential energy surface. However if

quantities such as the curvature are desired, procedures

that follow the reaction path more closely are required.

Such procedures require higher derivatives of the energy,

as fully discussed in [31.165].

In a nonadiabatic process, the reaction path neces-

sarily involves more than one potential energy surface.

In the case of nonadiabatic reactions involving avoided

intersections, the reaction path is difﬁcult to deﬁne since

the propensity for an intersurface transition spreads

out over a range of nuclear coordinates. However,

for nonadiabatic reactions that proceed though a sur-

face of intersection of two potential energy surfaces,

the reaction path passes through the minimum en-

ergy point on the surface of intersection of the two

potential energy surfaces [31.166]. An example of

such a process is provided in Sect. 31.5. The min-

imum energy point on the surface of intersection

satisﬁes (31.72) with no geometric constraints. For

a spin-forbidden reaction, the solution of (31.72) yields

R+ ξ

(R) = 0, so that g

(R) = ξ

/(1+ ξ

(R),

i. e., the gradients on the two surfaces are either par-

allel or anti-parallel depending on the value of ξ

The relationship between the gradients in the vicinity

Part C 31.4

484 Part C Molecules

of an intersection of two states of the same sym-

metry is not as simple. The wave functions are not

differentiable at a point of intersection. In the vicin-

ity of the point of intersection, the condition g

(R) +

(R) + ξ

(R) ≈ 0 holds, so that g

(R) andg

(R)

will not be simply related. Finally, note that the adia-

batic correction, deﬁned in Sect. 31.1, is large (and

positive) in the vicinity of the conical intersection, so

that the nuclear wave function exhibits a node at this

point.

31.5 Reaction Mechanisms: A Spin-Forbidden Chemical Reaction

This section presents an example to illustrate how

the ideas outlined in the preceding sections can

be used to obtain a mechanistic description of

a chemical reaction which could not be deduced on

intuitive grounds. Considered here is the ground state

reaction

(

)

+ N

(

)

→ HCN

(



)

→ HCN

(

14A



)

→ HCN

(

)

+ N

(

)

(31.95)

which is of considerable importance in the chemistry of

planetary atmospheres [31.168, 169] and hydrocarbon

ﬂames, having been suggested as the initial step in the

production of prompt or Fenimore NO [31.169–172].

However, because the reaction is spin-forbidden, its

importance has been questioned.

The reaction will be interpreted in terms of the inter-

mediate complex model [31.173,174]. In this model, the

reaction can occur despite a small probability for inter-

system crossing by repeatedly traversing the 2A



–4A



surface of intersection. The repeated traversals result

from a local minimum on the 2A



potential energy sur-

face. This mechanism is operative for the spin-forbidden

oxygen quenching reactions [31.173,174]

(

)

+ N

(

)

→ N

∗

→ O

(

)

+ N

(

)

(31.96)

(

)

+ CO

(

)

→ CO

∗

→ O

(

)

+ CO

(

)

(31.97)

which are important in the chemistry of the atmosphere.

To demonstrate the feasibility of reaction (31.95)as

an intermediate complex assisted chemical reaction it

is necessary to (i) determine the energetically relevant

portion of the doublet-quartet surface of intersection,

(ii) determine the spin-orbit interaction coupling the

doublet and quartet surfaces, (iii) characterize a local

minimum on the doublet surface from which the surface

of intersection is accessible, (iv) characterize a path from

the reactant channel to this region, and (v) determine the

exit channel path on the quartet surface. The results of

electronic structure calculations at the MR-CI level, ad-

dressing these points [31.167,175–178] are summarized

below.

Local Extrema on the 2A



–4A



Surface of Intersec-

tion.

Points in the 2A



–4A



surface of intersection were

determined from the solution to (31.72) [31.126, 175].

The minimum energy point on the surface of intersec-

tion, labelled C

mex

, has approximate C

symmetry,

and is shown in Fig. 31.3. It is the region of this

crossing point that must be accessed from the doublet

surface.

a) b)

Fig. 31.3a–c Key bond distances in Å for (a) R

dative

min

(CH) =

1.082, R

dative

min

(CN

) = 1.340, R

dative

min

) = 1.143;

(b) R

min

(CH)= 1.071, R

min

(CN

)= 1.311; R

min

1.735; R

mex

(CH) = 1.076, R

mrex

(CN

) = 1.3, R

mex

) = 2.221; (c) R

(CH) = 1.093, R

(CN

) =

1.515, R

) = 1.214. The notation is as deﬁned

in Sect. 31.5.Thedashed line in structure

H is out of the plane of the paper.

(a), (b) after [31.124],

Part C 31.5

Molecular Structure 31.5 Reaction Mechanisms: A Spin-Forbidden Chemical Reaction 485

The Spin-Orbit Interaction. In the double group

corresponding to C

symmetry, the 4A



and



wave functions each carry degenerate irre-

ducible representations, Kramer’s doublets [31.179].

The following pairs of nonrelativistic zeroth-

order wave functions can be used to span

these degenerate representations (i) Ψ





(

)







(

−

)

, (ii) iΨ





(

)

,iΨ





(

−

)

and (iii) iΨ





(

)

,iΨ





(

−

)

,wherethe

value has been given parenthetically. In this

case, all nonvanishing matrix elements connecting the

components of the 4A



and 2A



states can be ex-

pressed in terms of the single real-valued matrix

element [31.139]

(



, 2A



)

≡

iΨ























(31.98)

which was found to be ≈ 12.5cm

−1

[31.126,175]inthe

vicinity of the minimum energy crossing point deter-

mined above.

Local Extrema on the 2A



Potential Energy Surface.

Two local minima on the 2A



potential energy surface

have been found [31.175–177]. The local minimum con-

ﬁguration labeled C

dative

min

pictured in Fig. 31.3 represents

a datively bonded structure in which N

donates a pair

of electrons to the empty CH(1π) orbital. This can be

thought of as a reactant channel structure, since the N–

N and C–H bond lengths are similar to those in the

isolated molecules. This point on the 2A



potential en-

ergy surface is stable by 20.2 kcal/mol relative to the

doublet asymptote in the reactant channel. The second

minimum on the 2A



potential energy surface is also

pictured in Fig. 31.3 and is seen to have C

symme-

try. This conﬁguration point, denoted C

min

,isstableby

22.3 kcal/mol relative to the doublet asymptote in the

reactant channel. Note that C

min

and C

mex

differ only in

the length of the N–N bond.

Reaction Path. From the above data it might ap-

pear that both the C

min

and C

dative

min

structures are

involved in the mechanism of reaction (31.95). How-

ever, the energy of the transition state connecting

these two minima is quite large [31.177], so that the

reaction path avoids C

dative

min

[31.167]. The true reac-

tion path involves C

min

, C

mex

, and a reactant channel

transition state C

that connects C

min

and the re-

actant channel species CH + N

. C

is also pictured

CH(a

–1

)+N

CH(X

Π)+N

–21

HCN+N(

A''

Fig. 31.4 Reaction path for CH(X

Π ) + N

) →

HCN(X

) + N(

S ) after [31.167]. Energy scale, which

is approximate, is in kcal/mol

in Fig. 31.3. It involves the nonplanar cis- approach of

HC to N

[31.167]. The total reaction path is presented

in Fig. 31.4 [31.178]. Note that following intersys-

tem crossing, the system encounters a ﬁnal transition

state before proceeding to the spin-forbidden prod-

ucts.

From this qualitative analysis, a bimodal picture

of the reaction of ground state CH with N

emerges.

CH(

Π ) can be removed by either (i) forming a complex

dative

min

which is stabilized by a third body colli-

sion or (ii) forming the spin-forbidden products with

the reaction mechanism indicated in Fig. 31.4.This

mechanism has been deduced entirely from the com-

putational results cited above. The key computational

ﬁndings responsible for this mechanism are (i) the

existence and geometrical relationship between C

min

and C

mex

, (ii) the large barrier separating C

dative

min

and

min

, and (iii) the entrance channel path across C

to C

min

Since there is apparently no or little barrier to form-

ing C

dative

min

[31.177], at low temperatures CH will be

removed exclusively by this process at a rate with

a small or negative temperature dependence, indicat-

ing a barrierless reaction. At higher temperatures, the

rate constant is expected to exhibit a more usual

Arrhenius type behavior as the spin-forbidden chan-

nel can be accessed [31.180]. These expectations are

consistent with previous low temperature measure-

ments [31.169, 181] and high temperature shock tube

results [31.172, 182, 183]. The above reaction path for

the spin-forbidden reaction has been used in a two in-

ternal coordinate model for reaction (31.95) [31.180]

that also supports the proposed description of this

reaction.

Part C 31.5

486 Part C Molecules

31.6 Recent Developments

Added by Mark M. Cassar. If the motion of nuclei within

a molecule causes more than one Born-Oppenheimer

potential energy surface to intersect, then nonadiabatic

transitions become an essential process in molecu-

lar dynamics. These transitions are at the heart of

many biological processes, such as light harvesting

by plants [31.184, 185], which depends on electronic

excitation transfer, and a host of processes in the

upper atmosphere. In recent years, it has become

apparent that conical intersections (see Sect. 31.3.3

for a deﬁnition) of two states of the same sym-

metry are a fundamental element of electronically

nonadiabatic phenomena [31.186]. As an example,

conical intersections can facilitate transitions from

upper to lower electronic states. Transitions of this

nature provide efﬁcient pathways to lower electronic

states, and are thus amenable to experimental study

(e.g., through dissociation products of photoexcita-

tion [31.187]).

Although points of conical intersection, which are

not individual but form seams, can be located in

a straightfoward manner, ﬁnding points of minimum

energy has proved more difﬁcult. This difﬁculty has

been attributed to the erratic behavior of the parameters

used in the search algorithm. This seemingly intrinsic

problem which would preclude extrapolation – a conse-

quence of the singular nature of the intersection itself –

has been avoided through the use of extrapolatable func-

tions [31.188]. This recent work introduces functions

that vary smoothly (i. e., are well-behaved) as one moves

along the intersection seam, and hence allow the use of

extrapolation techniques.

The study of conical intersections has also been ex-

tended to include three-state intersections and ﬁrst-order

relativisticeffects. Accidental conical intersections (i. e.,

those not required by symmetry arguments) of three

states of the same symmetry, which may provide

an efﬁcient mechanism for radiationless decay, have

been shown to exist [31.189]. Spin-orbit effects are

essential for the proper description of the nuclear dy-

namics in molecules containing an odd number of

electrons [31.190]. The topography of conical intersec-

tions changes in this case, requiring the development of

new algorithms [31.191].

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Part C 31

491

Molecular Sym

32. Molecular Symmetry and Dynamics

Molecules are aggregates of two or more nuclei

bound by at least one electron. The nuclei

of most stable molecules can be imagined

to be points in a more or less rigid body

whose relative positions are constrained by

an electronic bonding potential. This potential

depends strongly upon the electronic state as

described in Chapt. 31. Most of this discussion

is about stable molecules in their electronic

ground state. In Sect. 32.6 some comments are

made about molecules with excited, or “loose”,

parts.

32.1 Dynamics and Spectra

of Molecular Rotors.............................. 491

32.1.1 Rigid Rotors.............................. 492

32.1.2 Molecular States Inside and Out .. 492

32.1.3 Rigid Asymmetric Rotor

Eigensolutions and Dynamics ..... 493

32.2 Rotational Energy Surfaces

and Semiclassical Rotational Dynamics .. 494

32.3 Symmetry of Molecular Rotors............... 498

32.3.1 Asymmetric Rotor Symmetry

Analysis ................................... 498

32.4 Tetrahedral-Octahedral

Rotational Dynamics and Spectra .......... 499

32.4.1 Semirigid Octahedral Rotors

and Centrifugal Tensor

Hamiltonians............................ 499

32.4.2 Octahedral and Tetrahedral

Rotational Energy Surfaces ......... 500

32.4.3 Octahedral and Tetrahedral

Rotational Fine Structure............ 500

32.4.4 Octahedral Superﬁne Structure ... 502

32.5 High Resolution Rovibrational Structure 503

32.5.1 Tetrahedral Nuclear Hyperﬁne

Structure .................................. 505

32.5.2 Superhyperﬁne Structure

and Spontaneous Symmetry

Breaking .................................. 505

32.5.3 Extreme Molecular Symmetry

Effects ..................................... 506

32.6 Composite Rotors and Multiple RES ....... 507

32.6.1 3D-Rotor

and 2D-Oscillator Analogy .......... 509

32.6.2 Gyro-Rotors

and 2D-Local Mode Analogy ....... 510

32.6.3 Multiple Gyro-Rotor RES

and Eigensurfaces ..................... 511

References .................................................. 512

32.1 Dynamics and Spectra of Molecular Rotors

Motions that stretch or compress the bonds are called

vibrational motions, and give rise to spectral resonances

in the infrared region of the spectrum. Typical fun-

damental vibrational quanta (ν

) lie between 80 cm

−1

(the lowest GeBr

mode) and 3020 cm

−1

(the high-

est CH

mode). (A 1000 cm

−1

wave has a wavelength

of 10 µm and a frequency deﬁned by the speed of

light: 29.979 2458 THz.) Vibrational amplitudes are

usually tiny since zero-point motions or vibrations

involving one or two quanta (ν = 0, 1, 2,...)are

constrained by the steep bonding potential to less

than a few percent of the bond lengths, but high

overtones may lead to dissociation, i. e., molecular

breakup.

Overall rotation of molecules in free space is un-

constrained, and gives rise to far-infrared or microwave

pure rotational transitions or sidebands on top of vi-

brational spectra. Typical rotational quanta (2B) lie

between 0.18 cm

−1

(5.4 GHz) for SF

and 10.6cm

−1

for CH

. Individual molecules are free to rotate or

translate as a whole while undergoing tiny but usually

rapid vibrations. Vibrating molecules may be thought

of as tumbling collections of masses held together by

‘springs’ (the electronic vibrational potential or force

ﬁeld), and are called semirigid rotors. The coupling of

rotational and vibrational motion is called rovibrational

coupling and includes centrifugal and Coriolis coupling,

which will be introduced in Sect. 32.6.

Part C 32

492 Part C Molecules

This discussion of molecular dynamics and spec-

tra mainly involves molecular rotation and prop-

erties of rotationally excited molecules, particu-

larly those with high rotational quantum number

J = 10–200. However, the discussion also applies

to molecules in excited vibrational states, and

even certain cases of molecules in excited elec-

tronic states. The analysis of vibronic (vibrational-

electronic), rovibrational (rotation–vibration), or rovi-

bronic (all three) types of excitation can be very

complicated [32.1–5] and is beyond the scope

of this article, but these problems can all ben-

eﬁt from the elementary considerations described

here.

32.1.1 Rigid Rotors

As a ﬁrst approximation, and for the purposes of dis-

cussing basic molecular dynamics and spectra, one may

ignore vibrations and model stable molecules as ‘stick-

and-ball’ structures or rigid rotors. Then the Hamiltonian

has just three terms:

H = AJ

+ BJ

+CJ

. (32.1)

Here {J

, J

} are rotational angular momentum op-

erators, and the rotational constants are half the inverses

of the principal moments of inertia I

of the body:

A =

, B =

, C =

. (32.2)

This implies that the J-coordinate system being used

is a special one ﬁxed to the rotor’s body and aligned

to its principal axes, an elementary body, or Eckart,

frame.

Many molecules, particularly all diatomic mol-

ecules, have two of these rotational constants equal, say,

A = B. Such rotors are called symmetric tops, and their

Hamiltonian can be written in terms of the square of

the total angular momentum J· J and one other body

component J

H = BJ

+ BJ

+CJ

= BJ

+ BJ

+(C − B)J

= B J· J+(C − B)J

(32.3)

This gives a simple formula for the symmetric top rota-

tional energy levels in terms of the quantum numbers J

for the total angular momentum and K for the body

z-component:

E(J, K) = BJ( J +1) +(C − B)K

(32.4)

However, this eigenvalue formula may be a little too

simple, since it hides the structure of the eigenstates or

eigenfunctions. Indeed, the full Schrödinger angular dif-

ferential equation based upon the Hamiltonian (32.1)is

more lengthy. One should remember that H is written in

a rotating body coordinate frame that must be connected

to a star-ﬁxed, or laboratory, frame in order to get the

full theory.

32.1.2 Molecular States Inside and Out

Rotor angular momentum eigenfunctions can be ex-

pressed as continuous linear combinations of rotor

angular position states |αβγ  deﬁned by Euler angles

of the lab azimuth α, the polar angle β of body z-axis,

and the body azimuth, or ‘gauge twist’, γ . The eigen-

functions are,





√

2J +1

8π

2π



dα



sin β dβ

2π



dγ D

∗

(αβγ)|αβγ  , (32.5)

where the rotor wave functions D

∗

are just the

conjugates of the Wigner rotation matrices described

in Sect. 32.3.1, and row and column indices M and K ,

respectively, are the lab and body components of the

angular momentum [32.5–7].

An important feature of polyatomic molecules is that

their angular momentum states have two kinds of az-

imuthal quantum numbers. In addition to the usual lab

component of momentum M associated with the lab co-

ordinate α (α and β are usually labeled ϕ and ϑ), there

is a body component K associated with the Euler co-

ordinate γ , the body azimuthal angle of the laboratory

Z-axis relative to the body z-axis.

The physics of atomic or diatomic angular momen-

tum states has no internal or “body” structure, so the

quantum number K is always zero. Unless one sets

K = 0, the energy formula (32.4) blows up for a point

particle because z-inertia for a point is zero and C is

inﬁnite. Also, the dimension of the angular momen-

tum state multiplet of a given J is larger than the usual

(2J +1) found in atomic or diatomic molecular physics.

In polyatomic rotors, the number of states for each J is

(2J +1)

, since both quantum numbers M and K range

between -J and +J.

A further important feature is that the molecular rotor

wave functions contain, as a special (K = 0) case, all

Part C 32.1

Molecular Symmetry and Dynamics 32.1 Dynamics and Spectra of Molecular Rotors 493

the usual atomic spherical harmonics Y

complete with

correct normalization and phase, since

√

4πY

(ϕϑ) = D

(ϕϑ·)

∗

√

2l +1). (32.6)

This is part of a powerful symmetry principle: group

representations are quantum wave functions, and sym-

metry analysis is an extension of Fourier analysis – not

just for translations as in Fourier’s original work, but for

any group of symmetry operations. The usual Fourier

coefﬁcients e

ikx

are replaced by the D functions in the

rotational Fourier transform embodied by (32.5).

Molecular rotational analysis displays another im-

portant but little known aspect of symmetry analysis in

general. For every group of symmetry operations, such

as the external lab-based rotations familiar to atomic

physics, there is an independent dual group of internal

or body-based operations. The external symmetry of the

environment or laboratory is independent of the internal

symmetry of the molecular body, and all the operations

of one commute with all those of the other. The mo-

lecular rotation group is thus written as an outer product

R(3)

LAB

⊗ R(3)

BODY

of the external and internal parts,

and the degeneracy associated with this group’s repre-

sentations for a single J is (2J +1)

as mentioned above.

It is a special ⊗-product, however, since the J-number

is shared.

The inversion or parity operator I(r →−r) can be

deﬁned to be the same for both lab and body frames.

Including I with the rotational group R(3) gives the

orthogonal group O(3) = R(3) ⊗{1, I}. If parity is con-

served (e.g., no weak neutral currents), the fundamental

molecular orthogonal group is O(3)

LAB

⊗O(3)

BODY

How this symmetry breaks down and which lev-

els split depends upon both the perturbative laboratory

environment and the internal molecular structure.

A spherical top Hamiltonian is (32.1) with A = B = C.

This has a full O(3)

BODY

(spherical) symmetry since it

is just B J· J. Given that the rotor is in an O(3)

LAB

labo-

ratory (empty space), the original symmetry O(3)

LAB

⊗

O(3)

BODY

remains intact and the (2 J +1)

degener-

acy is to be expected. However, a symmetric rotor in

a lab vacuum has its internal symmetry broken down

to O(2)

BODY

if A = B = C, and the energies given

by (32.4) consist of internal quantum singlets for K = 0

and ±K doublets for K = 0. But each of these levels still

has a lab degeneracy of (2J +1) if O(3)

LAB

is still in

effect. So the (2J +1)

level degeneracies are each split

into multiplets of degeneracy (2J +1) and 2(2J +1) for

K = 0andK = 0, respectively. The resulting levels are

often labeled Σ, Π, ∆,Φ,Γ, ... in a somewhat inap-

propriate analogy with the atomic s, p, d, f, g, ... labels

of Bohr model electronic orbitals.

Only by perturbing the lab environment can one

reduce the O(3)

LAB

symmetry and split the M degen-

eracies. For example, a uniform electric ﬁeld would

reduce the O(3)

LAB

to an O(2)

LAB

, giving Stark split-

tings which consist of external quantum singlets for

M =0and±M doublets for M = 0. A uniform magnetic

ﬁeld would reduce the O(3)

LAB

to an R(2)

LAB

,giv-

ing Zeeman splittings into external quantum singlets for

each M. The analogy between atomic external ﬁeld split-

ting and internal molecular rotational structure splitting

is sometimes a useful one and will be used later.

32.1.3 Rigid Asymmetric Rotor

Eigensolutions and Dynamics

The general case for the rigid rotor Hamiltonian (32.1)

has three unequal principal moments of inertia

(A = B = C). This is called the rigid asymmetric top

Hamiltonian, and provides a ﬁrst approximation for

modeling rotation of low symmetry molecules, such

as H

0. Also, a number of properties of its eigenso-

lutions are shared by more complicated systems. The

dynamics of an asymmetric top is quite remarkable, as

demonstrated by tossing a tennis racquet in the air, ﬂat

side up. The corresponding quantum behavior of such

a molecule is also nontrivial.

Given the total angular momentum J, one may con-

struct a (2J +1)-dimensional matrix representation of

H using standard matrix elements of the angular mo-

mentum operators J

, J

,andJ

, as given in Chapt. 2.

The H matrix connects states with (2 J +1)-different

body quantum numbers K(−J ≤ K ≤ J ),butthema-

trix is independent of the lab quantum numbers M,so

there are (2J +1) identical H matrices; one for each

value of the lab quantum number M(−J ≤ K ≤ J ).

A plot of the 21 eigenvalues of (32.1)forJ = 10

is shown in Fig. 32.1. Here, the constants are set to

A = 0.2cm

−1

and C = 0.6cm

−1

, while B is varied be-

tween B = A, which corresponds to a prolate symmetric

top (an elongated cylindrical object) and B = C,which

corresponds to an oblate symmetric top (a ﬂattened

cylindrical object or discus). For all B values between

those of A and C, the object is asymmetric.

The left hand end



A = B = 0.2cm

−1

, C =

0.6cm

−1



of the plot in Fig. 32.1 corresponds to a pro-

late symmetric top. The symmetric top level spectrum

isgivenby(32.4). It consists of a lowest singlet state

corresponding to K = 0 and an ascending quadratic lad-

der of doublets corresponding to K =±1, ±2,... ,±J.

Part C 32.1