Classification of energy levels for polyatomic molecules

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Classification of energy levels for polyatomic molecules

H. Berger

To cite this version:

H. Berger. Classification of energy levels for polyatomic molecules. Journal de Physique, 1977, 38

(11), pp.1371-1375. �10.1051/jphys:0197700380110137100�. �jpa-00208707�

CLASSIFICATION OF ENERGY LEVELS FOR POLYATOMIC MOLECULES

H. BERGER

Laboratoire de

Spectronomie

Moléculaire

(*),

^{Faculté des}

Sciences,

²¹⁰⁰⁰

Dijon,

^France

(Reçu

^{le 9}

juin

1977,

accepté

^{le 22}

juillet 1977)

Résumé. ²⁰¹⁴ Nous décrivons une nouvelle façon d’aborder la classification des niveaux d’énergie

d’une molécule

polyatomique :

^la

symétrie

^des fonctions rovibrationnelles est étudiée dans le groupe d’invariance G x O(3) de l’hamiltonien. La classification des niveaux d’énergie, ^les règles ^{de sélec-}

tion et le calcul des

poids

statistiques nucléaires sont envisagés pour

plusieurs

types de molécules.

Abstract. ²⁰¹⁴ The classification of energy levels for ^a

polyatomic

molecule is considered in a new

way : the symmetry of rovibrational functions is studied in the invariance group G ^x O(3) ^{of the}

full Hamiltonian. The classification of energy levels, the selection rules and the computation ^of

nuclear statistical

weights

^are given for several types of molecules.

Classification Physics ^Abstracts 31.1S ^- 33.20E - 33.20F

1. Introduction. ^- As

yet

^{there is no universal}

agreement

among theoreticians on the classification of _energy levels for

spherical top

molecules Jahn

[1], Hougen [2-5]

^and

Moret-Bailly [6-8]

^{have set} up different notations. If notation is not the crucial

point

^{in the}

study

^of

molecules,

^{it is} nevertheless

highly

inconvenient that a

given

^{level of a rotation-}

vibration band be labelled

differently according

^to

the author or the _process

(dipole absorption

^or

Raman

scattering). Recently,

the notation of the 3.39

fin

methane line has raised _many discussions

[9- 10].

Two recent papers led us to consider this ^new classification : in the

first,

Louck and Galbraith

[11]

have

given

^an

important

contribution to the

theory

of the vibration-rotation Hamiltonian demonstrat-

ing

the fundamental role of the Eckart

frame ;

^{in the} second Hilico,

Berger

^{and Loëte}

[12], using

^tensor

techniques,

^have

given general

formulae for the calculation of transition moments in the case of

spherical top

molecules. In both _papers it _appears that the invariance _group of the full Hamiltonian is for a

polyatomic

molecule the direct

product

group,

designated

^here

by

^{G x}

(1)0(3),

where G is the invariance _group of the Eckart frame and

0(3)

the invariance _group of the

laboratory

^{fixed frame.}

It is then

justified

^{to set} up ^a classification for the energy

levels,

derived from this new

viewpoint.

. 2. Inconveniences of

présent

classifications for

sphe-

rical

top

molécules. ^- For detailed discussions of conventions

adopted

^{the reader is referred to the}

recent paper of Husson

[ 13] :

^{here we} shall recall

only

(*) Equipe de Recherche associée au C.N.R.S.

the main

points

^{of the}

Hougen’s

^and

Moret-Bailly’s

classifications.

In the

study

^{of a molecule two} frames of reference

are

used,

the first is bound to the molecule

(Rm)

and the second to the

laboratory (R1).

^{Let us examine}

some differences between

Hougen’s

^{and Moret-}

Bailly’s viewpoints.

2.1 HOUGEN’S TREATMENT. ^- In

Hougen’s

^{treat- ,}

ment appears the _group of

feasible

operators : ^let G’ be this _group. G’ is a

subgroup

^{of the}

permutation-

inversion _group

(Longuet-Higgins [14])

and it is

isomorphic

^to the molecular

point

group G.

However,

the action of an element of G’ on the Eckart frame is not

always

^{the same as that of one} of G. The diffe-

rence between both _groups is the

following :

^{to an}

improper

rotation of G is associated the element of G’

which is the

product

^{of a} pure rotation of the Eckart frame and of the inversion of the

laboratory-fixed

frame.

Thus,

^{it is not}

surprising

^{that the}

symmetry

proper- ties of functions or operators of the transition moment be different

depending

^{on which} group is

used ;

for

example,

^the

R,-components

of the electric

dipole

moment are of

symmetry Ai

^{in G and}

A2

^{in G’.}

We think that it is

important

^{not to mix the} sym- metry

operations

^{related to} the molecule-fixed frame and those related to the

laboratory

^{fixed frame.}

They

are

distinguished clearly

^if one uses a

larger

group such as G ^x

(1)0(3).

It is clear that a measure made in the

laboratory

^does

not

depend

^on the way the Eckart frame is bound to the molecule and observables of

quantum

^mecha-

nics,

^{as the} energy, the

Rl-components

of transition moments are to be invariant in G.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197700380110137100

1372

2.2 MORET-BAILLY’S TREATMENT. - In this scheme the symmetry

properties

^are considered in the mole- cule _group G

(Td

^for

CH4).

^With

respect

^{to the Hou-}

gen’s

^treatment where the rotational functions have the _g

parity

in the rotation-inversion group, Moret-

Bailly

^considers

^that,

^{for each J}

^value,

there exist two

independent

rotational functions of

parity

g and u.

Thus,

^a rotational level is

degenerate

^with

respect ^to

parity

^{and must be}

doubly

labelled in the group Td.

By

^means of this double

labelling

it would be

possi-

ble to

explain

^{the different}

^I.R.,

^{M.W. or Raman}

transitions,

^{but it is not} convenient. So that a

unique symmetry species

^be

assigned

^{to each} energy

level, Moret-Bailly attributes, by convention,

^the

parity

g to the

ground

vibrational level and the

parity

^{of the}

excited levels is determined so that the selection rules be verified.

Unfortunately

this convention is

inconvénient ;

the excited vibrational levels are labelled

differently

in

absorption

and in Raman

scattering.

^Moreover

for the

interpretation

of microwave transitions in ^a

given

vibrational

level,

^{it is} necessary ^{to use} the double

labelling.

3.

Symmetry properties

of rovibrational functions in Td x

(’) 0(3).

^- The total wavefunction of a state is

usually

^{written as a}

product

of four functions :

The electronic function

tp e

is invariant in the

ground

electronic state of

CH4.

^’

The translational function

tpTr

^{does not involve}

internal coordinates of the molecule and is therefore invariant under the

symmetry operations concerning

the molecule.

tpRV

is the rovibrational function and

tpNS

^{the nuclear}

spin

^function.

3.1 SYMMETRY IN Td. - 3 .1.1

Symmetry of ’PT.

^-

The

hydrogen

^{nuclei are} fermions and the total wavefunction of a state

is,

^from Pauli exclusion

prin- ciple, expressed

ⁱⁿ group

theory language,

^{as the basis}

of the

representation T(1)

^{of the}

permutation

^group

S4 (in Young’s

^notation

[15]),

^{that is}

A2

^{in Td}

(the

group

S4

^is

isomorphic

^to

Td).

3.1.2

Symmetry of tp NS’

^- The nuclear

spin

functions are obtained

by coupling

ⁱⁿ

SU(2)

^the

nuclear

spin

functions of each nucleus. Wilson

[16]

. made the

study

^{of their}

symmetry properties

^{in the}

proper

subgroup

^{of the}

point

group ; but it is

possible

to make this

study

^{in the}

permutation

group

S4

as Mizushima

[15]

^{or in the} group Td

[17].

Let us recall the result of Landau and Lifshitz

[17]

which

gives

the character of the

representation generat-

ed

by ’FNS

^{for a finite} group g

where i is the nuclear

spin

^{of a set E of nuclei}

which : permute

^{in an}

operation

g of _g.

In methane

"CH4e ^’PNs

^{has the symmetry}

5Ai+E+3F2.

3.1.3

Symmetry of ’PRv’

^{- From the}

symmetry

of functions

Y’T, ’Pe, ’PTr’ ’PNs

^{we can} deduce that of

tp RV using

the Pauli

principle

and it is _easy to see

that

only

^{states of}

symmetry A2, E, F,

^are

^{allowed ;}

this result has

already

^been

given by

^{Landau and}

Lifshitz

[17].

3.2 SYMMETRY IN Td x

(1)0(3).

^{- The} introduction of two frames in the

study

^{of a molecule comes from}

the

study

of the rotational motion and as well as the

use of a

larger

group than Td. Hilico et al.

[12]

^have

established that the rotational functions of the

sphe-

rical rotator can be

formally

considered as the _compo- nents of tensors of

(m)0(3)

^x

(1)0(3) ((m)0(3)

^corres-

ponds

^to the different _ways a frame can be bound to the

spherical rotator).

As

Moret-Bailly,

^we consider that rotational func- tions are

degenerate

^with respect ^to

parity

^and

belong

to the

representations

^{D(J-,J-) of}

(m)O(3)

^x

(’)0(3) (L

⁼ g ^or

u) ;

^the

possibility

^{D(Ju,Jg) is not considered}

because both _groups

0(3)

^{have the same}

generators.

The rotation-vibration Hamiltonian is

written,

to a first

approximation,

^{as the sum} of Hamiltonians of a

rigid

^{rotator and} of harmonic oscillators : JC =

Jeo

⁺

XRV.

^{From reference}

[12],

^the

eigenfunc-

tions of the Hamiltonian

Jeo,

^obtained

by coupling

the vibrational and rotational

functions,

^are

expressed

as tensors of

(’)0(3)

^x

(1)0(3) :

R comes from the

coupling

^{of the total momentum J}

with the

angular

vibrational momentum 1.

Taking

^{into account} the molecular

field,

^{the states}

are characterized

by

^their symmetry

C(n) arising

^from

the reduction in Td of the

representation

^D(Ry)

( D(Ry) t

^Td

= >’ Cni)

^{- The index n} is introduced to

distinguish

the various

representations

^of symme- try ^{C. A state} will then be labelled

by ^c(n)

^{in Td and}

7(,)

ⁱⁿ

(’)0(3) ;

it is convenient to denote it

JC (,r)

^*

Table 1 illustrates the

possible

labels for the _energy levels

by application

of the Pauli

principle (the

spe- cies E alone remains

degenerate

^with

respect

^to

parity)

and as a consequence the statistical

weights

^{for the}

states

A2, E, F 1 are 5, 2,

^3.

Table II

gives

the nomenclature in our notation and in that of

Hougen

^{for some} levels in the

ground

vibronic state. Several remarks are- to the made :

- Our

assignment

^{in the}

ground

^{vibronic state}

is consistent with that

given by

Landau and Lif- shitz

[17] (the equivalence

between the

parity

^notations

is _{g -}

( +)

^{and u}

=(-)).

^Oka

^[18]

discussed the

parity

of rotational levels

by using

^the

Longuet- Higgins

group, and found that the

Ai

^and

F2

^rota-

tional levels

correspond

^to

( - ) parity

^and

A2

^and

Fi

1

TABLE 1

Symmetries

and nuclear statistical

weights of

^rovi-

brational states

of

^the

XY4

^molecules

(Y

^with

spin 1/2).

In this table and in the

following

^{the star}

refers

^to

levels which are

forbidden by

the Pauli

principle.

TABLE II

Classification of

^{some levels in the}

ground

^state

for

the methane molecule

using

^{our notations and those}

of Hougen.

^{So as to convert the}

first

^{into the second}

it is

sufficient

^to

exchange

^the

subscripts

^{1 and}

2, if

^the

parity

^{is u}

(the

^{same rule}

applies for

^{an excited}

state).

rotational levels to

( + ) parity ;

except ^{for the E levels} his results are different from ours, but it is not sur-

prising

because the _groups we used are different.

Our agreement with Landau and Lifshitz. confirms the fundamental role of the

(’)0(3)

group for the

study

of space inversion.

-

Considering

^{now an} excited vibrational state, how can we label the levels for a

given

^{J value ?}

In a

general

way the

symmetry

of levels is obtained

by coupling

^{in Td the}

representation ^(v)C

^which

gives

^the

symmetry

of the vibrational state with

D(J1;) !

^Td

(i

⁼ g ^or

u).

Here we assume that the tensorial extension from Td to

(’)0(3)

^can be made and that to

(v)C

we can

associate the

representation ^D(lp)

^of

(m)0(3) (Eq. (3)) ;

moreover we suppose that R

(Eq. (3))

^{is a}

^good

^quan-

tum number. Then for a

given

^R value the levels are

labelled as in Table II if _p is _g, but if _{p is} u the

parity

of levels has to be

changed (an example

^is

given

^on

figure

¹ where transitions between a

ground

^level

and a vibrational level of a

triply degenerate

^band

are

considered).

3 . 3 SELECTION RULES. ^- In the _{group G} x

(’)o(3)

a transition operator ^{is of} symmetry

Ai

^{x D(’fl)}

(Ai

^{is the trivial}

representation

^of

G) ;

^it expresses the fact that the

R,-components

^{of the transition}

FIG. 1. ^- Illustration of selection rules between ground ^and

excited states for a triply degenerate ^band (- - - ^IR transitions,

MW transitions, ^Raman transitions).

operator ^are invariant in G and transform in

(1)0(3) according

^{to the}

representation D(Kp);

^for

example

D(lu) for the electric

dipole

moment, D(og) ⁺

D (2g)

for the

polarizability, ^D(lu)

⁺

^{D (3u)}

^{for the}

hyper- polarizability...

A transition between states

JC(,)

^{and J’}

C’,,)

^will

be allowed if :

this relation

being

satisfied when :

In

particular

^we have the rule for

parity :

^{u H} g in

dipole absorption (IR

^or

MW)

^{and u H u or} g ^H g in Raman

scattering.

Figure

¹

depicts

these selection rules and as men-

tioned in Table II shows that it is _easy to go from our

notation to the

Hougen’s.

In our notation the

3.39 m

^methane

line, proposed

to define the standard

length,

would be denoted -:

P, Fip(u).

1374

4. Extension to the other molecules. ^- A similar classification can be used for the other

polyatomic molécules ;

^{let us summarize the}

grounds

^{of our}

classification :

1)

The rotational functions are, in a first

approxi- mation,

^the

components

^{of tensors of}

symmetry D(J1:,J1:)

in

(m)O(3)

^x

1>O(3)

^{and are}

degenerate

^with

"

respect

^to

parity (z

⁼ g ^or

u).

2)

^The energy levels are classified

according

^{to the}

symmetry

of rovibrational functions in the invariance group G ^x

(1)0(3).

3)

^{The Pauli}

principle

^{must be}

applied.

Now let us examine several cases :

4.1 SPHERICAL TOP

XY4 (Y

NUCLEI WITH SPIN

1).

- From eq.

(1), ’l’SN

^{is of}

symmetry

and

tp T

^of

symmetry Al’

Table III

gives

the allowed states and their statisti- cal

weights.

^{It is the case of}

CD4.

TABLE III

States allowed

by

^{the Pauli}

principle for

^the

XY4

molecules

(Y

^with

spin 1)

4.2 SPHERICAL TOP

XY 6 (Y

NUCLEI WITH SPIN

1/2).

- The invariance group is Oh ^x

(’)0(3)

^{and from}

(1)

it follows that

’FSN

^{is of}

symmetry

10 A,, + A2g + A2,. + 8 Eg + 6 Fl. + 3 F2,,

and

’PT

^of

symmetry A2,,.

The results of table IV with

regard

^to the statistical

weights

^{are in}

agreement

with those of Cantrell and Galbraith

[19].

TABLE IV

States allowed

by

^{the Pauli}

principles for

^the

XY6

molecules

(Y

^with

spin 1/2)

4.3 SYMMETRIC TOP MOLECULE

XY3 (Y

^NUCLEI

WITH SPIN

1/2).

^{- Let us} suppose that the

symmetry

group G is the _group

C3v (neglecting inversion).

^The

symmetry

^of

’PT

^is

A2

^{and that of}

’PSN,

⁴

Ai

^{+ E.}

The states

A2(g), A2(u)

^and

E(u, g)

^are allowed with the statistical

weights 4, 4,

^2.

In this case the classification can be established with the chain

of groups (Fig. 2)

FIG. 2. ^- Symmetry classification for levels of a symmetric top molecule (group C3v).

4.4 HOMONUCLEAR LINEAR MOLECULES

X2°

^{- The}

point

group is

Dooh

⁼

Coov

^x

C;.

^{From the tensorial}

formalism in

(m)0(3)

^x

(1)0(3)

the rotational func- tions of this molecule are denoted

P«t,ùt>.

^{As the}

vibrational functions are

totally symmetric,

^{the rovi-}

brational states have in

Dooh

^the

symmetry

^of

Y ôt.

We summarize in table V the results for the

H2, N2,

and

O2

^molecules.

The

molecule 1602

^{is a}

particular

^case because the electronic function

is,

ⁱⁿ the electronic

ground

state, of

symmetry A2g’

TABLE V

States allowed

by

^the

Pauli principle for

^some linear molecules

5. Conclusion. ^- A

unique

classification of _energy levels for

polyatomic

^{molecules is} convenient

when

one works in different fields of

spectroscopy.

^More-

over, the introduction of the _group

(1)0(3)

may be

particularly interesting

^{in Stark or Zeeman}

spectros-

copy.

Namely, (’)0(3)

appears ^{as an} external

degene-

racy group and the

degeneracy

^{can be lifted}

by

^an

external field. Our classification shows

clearly

^the

levels which can

split

^{in a} first order Stark

effect ;

the selection rules for the Stark effect are the same as

those for electric

dipole transitions,

^from

(5)

^it

follows that the levels must have the same

symmetry

in G and a different

parity

ⁱⁿ

(’)0(3).

^In agreement with Watson

[20],

^we find that these are of

type

^E for

XY4 spherical tops

^and may have any

symmetry

for

XY6 tops.

The _group G ^x

(!)O(3)

^seems

appropriate

^{to the}

clear

description

of molecular motions and we

showed in this _paper that the _group of

feasible

operators ^{was not the}

only

group for

accomodating

the Pauli

principle

^{and for}

calculating

the statistical

weights.

References

[1] JAHN, ^H. A., ^{Proc. R.} Soc. London, Ser. A 168 (1938) ^495.

[2] ^{HOUGEN, J. T., J. Chem. Phys. 37 (1962) 1433.}

[3] HOUGEN, ^J. T., ^{J. Chem.} Phys. ³⁹ (1963) ^358.

[4] HOUGEN, ^J. T., ^J. Chem. Phys. ⁵⁵ (1971) ^1122.

[5] HOUGEN, ^J. T., Catalog of explicit symmetry operations for

the vibrational, rotational, and nuclear spin functions of methane, and the use of ^these operations ⁱⁿ determinating symmetry labels and selection rules to be published ⁱⁿ

MTP International Review of Science ; Physical Chemistry Series, edited by ^{D. A.} Ramsay, ^1975.

[6] MORET-BAILLY, J., Cah. Phys. ¹³ (1959) ^476.

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[12] HILICO, ^J. C., BERGER, ^{H. and} LOETE, M., Can. J. Phys. ⁵⁴ (1976) 1702.

[13] HussoN, N., ^Ann. Phys. ⁹ (1975) ^271.

[14] LONGUET-HIGGINS, ^H. C., Mol. Phys. ⁶ (1963) ^445.

[15] MIZUSHIMA, M., Theoretical Physics (John Wiley ^et Sons, N.Y.) ^1972.

[16] WILSON, ^E. B., ^J. Chem. Phys. ³ (1935) ^276.

[17] LANDAU, ^{L. and} LIFSHITZ, E., Quantum ^Mechanics (Editions Mir, Moscou) ^1967.

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[19] CANTRELL, C. D. and GALBRAITH, ^H. W., ^{J. Mol.} Spectrosc.

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[20] WATSON, ^{J. K.} G., ^{J. Mol.} Spectrosc. ⁵⁰ (1974) ^281.