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Dipole moment matrix for vibration - rotation transitions in C3v molecules
G. Tarrago, O.N. Ulenikov, G. Poussigue
To cite this version:
G. Tarrago, O.N. Ulenikov, G. Poussigue. Dipole moment matrix for vibration - ro- tation transitions in C3v molecules. Journal de Physique, 1984, 45 (9), pp.1429-1447.
�10.1051/jphys:019840045090142900�. �jpa-00209883�
Dipole moment matrix for vibration - rotation transitions in C3v molecules
G. Tarrago, O. N. Ulenikov (*) and G. Poussigue
Laboratoire d’Infrarouge, L.A. 194, Université de Paris-Sud, Bâtiment 350, 91405 Orsay Cedex, France
(Rep le lerfivrier 1984, accepti le 26 avril 1984)
Résumé. 2014 Le moment dipolaire transformé des molécules C3v a été calculé jusqu’au second ordre d’approximation
sur la base de ses propriétés d’invariance. Sa dépendance par rapport aux composantes du moment angulaire total
a été explicitée. La matrice moment dipolaire a été écrite pour permettre le calcul des intensités de vibration- rotation dans le cas des polyades vibrationnelles fortement interagissantes. L’importance de la cohérence des
facteurs de phase pour un calcul rigoureux des intensités a été soulignée.
Abstract.
2014The transformed dipole moment or C3v molecules has been worked out up to second order of approxi-
mation on the basis of its invariance properties. Its dependence on the total angular momentum components has been made explicit. The dipole moment matrix has been written with a view to calculate vibration-rotation intensities in the case of strongly interacting vibrational polyads. The importance of a coherent phase system for
correct intensity calculations has been emphasized.
Classification
Physics Abstracts
33.10
1. Introduction
The line strengths in absorption vibration-rotation spectra of C3v molecules are, in the absence of external
field, given by
where ( JaTa I I y’ z I J’ a’ f a’ > represents the M-reduced (’) transition moment obeying the relation
JaT and J’ a’ T’ represent the lower and upper levels of the transition defined by their symmetry species T
and T’, the quantum numbers J and J’ of total angular momentum, a and a’ representing all other required
quantum numbers.
Using the symmetry properties defined in references [1, 2] for the C3v group, the dipole moment has the species A2, so that the relation T’ = F x A2 is imposed and all the transitions must strictly obey the selection rules
as well as
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019840045090142900
We observe that Q and a’(= 1,2) are only defined
for the degenerate representation, E : El and E2 in
equation (3a) and through all the paper represent its
two components. For transitions between E levels,
the matrix element in equation (1) is indifferently J (XE 1 I I pi I I J’ (x’ E2 > or J (XE2 I uil I I J’a’El >.
N is the number of absorbing molecules per unit
volume; v is the frequency of the transition;
EJar is the energy in the lower level; gr the statis- tical weight of this level, including all the nuclear spin degeneracies due to the sets of three identical nuclei.
If this total degeneracy is distributed over nAl, nA2 and nE spin functions of species A1, A2 and E respec-
tively, then (2) 9A = gA2 = nA, 1 + nA2; gE = 2 nE-
Q(T) is the partition function at the temperature T of the absorption, taking into account the nuclear spin degeneracies consistently with gr.
Usually the eigenfunctions I JoeFa > and I J’a’F’a’>
are calculated on the basis of a transformed vibration- rotation Hamiltonian HtvR, derived from the true one,
HVR, by a set of contact transformations, i.e.,
such transformations making it possible to work with
matrix representations which are diagonal as far as possible with respect to the main vibrational quantum numbers v,, [3, 4]. Then the required operator y’ z in equation (1) is obtained from the usual dipole moment component in the space fixed frame, i.e.,
by the same set of transformations as applied to HvR
in equation (4). Symbol À,za. in equation (5) represents direction cosines and /.t,, the dipole moment compo- nents in the molecule-fixed frame.
In this way, whatever the vibration-rotation bands under investigation (including interacting polyads),
a set of suitable Si functions can be found, with which
is associated an effective operator y’ z written as a
serial expansion in terms of dimensionless coordi- nates (3) qs, conjugate momenta ps and components Pa
(1) For molecules XH3, nA1
=4, nA2 = 0, nE = 2; then 9A, = gA2 = 9E
=4.
For molecules XD3, nAl
=10, nA2 = l, nE
=8; then 9A, = gA2 = 11; gE
=16, etc....
(3) Through all this paper, the indices n, r and t specify A, non-degenerate, A2 non-degenerate, and E degenerate
vibrations respectively; the index s is used for any vibration, without consideration of symmetry. A2 vibrations are not common in C3v molecules, but an interesting example is
the torsion in species like CH3 - CX3.
of the total angular momentum in the molecule-fixed frame. For convenience, Ps and P« will be used as
reduced, dimensionless momenta (instead of PslFi and P’.1h).
Analytical expressions for such an effective opera- tor y’ z and matrix elements can be found in the litera- ture for some specific vibration-rotation bands of C3v
molecules [5-8]. Our purpose in the present paper is rather to reach p% through the properties of the sym- metry, hermiticity and time reversal invariance requir-
ed for this operator (or pz) in the C3v group. A general expression is derived for this operator in terms of all qs, ps, P« and Az.. It is valid whatever the set of Si
functions chosen to carry out the energy calculations;
only the coefficients involved in the expression are
concerned with these Si functions. An expansion in
orders of magnitude of the terms in y’ z is then proposed
and all contributions retained to second order of
approximation.
Matrix elements for y’ z are then derived using the
usual zero-order vibration-rotation functions. The
phase factors involved in these matrix elements are
chosen to be consistent with those previously intro-
duced for energy matrix elements [2].
As a next step, basis functions carrying the full
symmetry of the C3v group are built, using the same simple rules of tensorial products as for the building
of p%. Easily introduced as a computational step, they
make it possible to work with matrices factorized as
far as possible. This point is of crucial interest in the treatment of large vibrational polyads where the
dimension of the matrices to be dealt with increases
rapidly with J, due to the vibrational degeneracies.
It is clear that the final matrix elements of p%
-
in C3v symmetry adapted functions
-might be
derived using the Wigner-Eckart theorem and the
3c_, 6 c_, 9 c- symbols defined for the C3v group [9].
Such a way has been successfully used by M. Loete
to derive dipole moment matrix for Td molecules [10].
But presently a reduction relatively to C3v does not actually simplify the calculations; more, our approach step by step makes easier the comparison with results
derived elsewhere in the treatment of C3v molecules.
On the other hand, the reduction in the subgroup (F)O(3) of the full molecular symmetry group (F)O(3) x C3v is used to drop very conveniently the dependence
in the magnetic quantum number M of the investigated dipole moment matrix elements.
The present approach to transition moments and intensities has the objective to introduce as exhausti- vely as possible the rotational dependence in intensity
calculations. No attempt is made in this paper to relate the Herman-Wallis type coefficients appearing
in y’ z to the molecular parameters. Some of them
have been calculated in the case of isolated bands [5-8].
For interacting bands, these coefficients have of course to take into account specifically the form of the
adopted Si functions.
2. Expansion of ulz for C3v molecules.
According to the symmetry species A2 of the dipole moment [1, 2], iez can be written in tensorial form as
assuming that a VI, RNn and A" are Hermitian operators as well as y’ [11].
The symbol (9 means a tensorial product between
two irreducible tensors in the C3v group.
RN" are functions of the operators Pa, defined by
their symmetry species F and total power N in Pa ;
the index n completes the labelling of R when more
than one operator R exists for given values of T
and N.
Ar’ represents the direction cosines defined by their species T’ (= A2 or E).
aVc* represents functions of the vibrational opera- tors qs and ps ; the symmetry species C* is connected to the species C of the rotational part by the relation C* = C x A2. The index a labels all the possible operators aV’ within a given C*; this index a is connected to N in the sense that the overall power of each term in equation (6) must be even in all momenta p., and Pa (time reversal invariance). adN."’
are effective real coefficients all depending on mole-
cular parameters : electric charge distribution cons-
tants, as well as geometrical and intramolecular
potential parameters via the contact transformations.
Orders of magnitude can now be assigned to the
various terms in equation (6), in relation to the usual series expansion of y’ z as :
with
The convergence of the terms S1 > S2 ... is the
same as in the expansion of H VRI i.e.,
as extensively discussed in reference [3].
Concerning the convergence in the expansion (8),
we observe that (p(l)/p(O))2 represents roughly the
ratio between the line intensities in fundamental vibrational bands and in pure rotation transitions,
(Jl2)/JlI»)2 the ratio between harmonic or combina- tion bands and fundamental bands, etc.
Current evidence suggests that, the dipole moment expansion (8) appears to be generally less convergent
than the energy expansion, so that it seems reasonable
to put
Analytical expansion of equation (7) in terms of equations (9) and (10) allows us to relate the orders of magnitude of the terms in p% to their dependence
in q, p, P and k (subscripts dropped) as reported in
table I.
As shown in this table, quadratic as well linear terms in Pa are involved in the expansion of /.i’ z to
second order. The quadratic terms, in APP and APPp,
contribute to transitions with zero quantum of vibration (pure rotation) and one quantum (Av., = 1) respectively. The linear terms, in APp and APqp, are
involved in transitions with one quantum (Av., = 1)
and two quanta (L åvs = 2) of vibration respectively.
But, within second order, no rotational correction needs to be considered for transitions with three quanta (y Av,, = 3) or four quanta (L Av,, = 4). We
also note that the scale from zero to two in the orders of magnitude retained in table I for it’ z corresponds
to intensities as a scale from zero to four.
That such a scale is not an overestimate is apparent from recent vibration-rotation studies more and more
concerned with vibrational polyads covering a large
range in frequencies as well as intensities. An example
is given by the recent investigation of the absorption
spectrum of methane in the range 2 250-3 260 cm-’
[12], involving no less than five bands (fundamentals
and overtones) with relative intensity ratios as large
as 100 000/1. Moreover, the high sensitivity of detec-
tion now available in microwave experhnents allows
one to observe very weak rotational transitions. An
example is given by the transitions with AK = ± 3 observed in the pure rotational spectrum of PH3 [13-16], generated by terms quadratic in Pa in the expansion of y’
All the elementary operators required for the build- ing of y%’according to equation (6) are presented in
table II, with their symmetry species and orientations in the C3v group [2]. The tensorial product of any
two of them, named A"T and Br’, will be given by
Table I.
-Magnitude of the terms (I in the expansion of y’
(8)ln the table, A and P represent any component of direction cosines and angular’momentum,
q and p any vibrational coordinate and momentum.
According to Ref. [1], each commutator can be written as [ , ] = [ , IV + [
,]R,
,the first term on
the right being in any case one order of magnitude larger than the second one; for i[Si,W z (0) 1, only
this second term contributes, the first one being zero.
Table II.
-Symmetry properties of operators and functions in the C3, group (a).
(a )According to the definitions given in Reference [2] for symmetry operations, J represents the inversion of the space fixed frame ; Cz (.L) the rotation of + 23 of the molecule fixed frame around z axis, and oxz
the reflection of the molecule fixed frame with respect to xz plan. Even and odd permutations within each set of three identical nuclei are associated to Cz(3 ) and ozx operations respectively.
where [C] represents the dimension of the representation C and the 3c-symbols are defined in table III. Among
these symbols the relations hold
with (- l)r = 1 ifr=Al or E and - 1 if r = A2.
From results of tables II and III, the rotational operators RNnF. different from zero for C3v molecules are
easily constructed. These are seven within N = 0, 1 and 2, the components of which are reported in table IV.
Table III.
-3c-symbols for the C3v group.
Table IV.
-Rotational operators RN different from zero for C3v molecules.
All possible products of these operators RNn with the direction cosines ;..r’ (T’ =- A2 or E) generate twenty Hermitian operators
as indicated in the first six columns of table V. The index r appears only when more than one set rr’ n is sui- table within given values of C and N. The twenty operators SNt are therefore labelled as follows
In columns 7 and 8 of table V appear the symmetry species and appropriate form in q and p of the vibra-
tional operators aVC* which can be associated with S operators in the expression (6) of p’z (within the second
order of approximation).
Table V.
-Operators Sc and ayc* involved in it’ (within second order).
Then, the exact expressions for the components of the operators Sc are given in table VI, where for
convenience (4) they appear as functions of the ele- mentary operators
and
On the other hand, explicit expressions for the components of operators ’V’ are given in table VII,
columns 2 and 3; the expected jumps on v", vr, vt, It
are indicated in column 4 and the types of transitions
generated in column 5. The index a is now developed, exhibiting the dependence on all q", qr, qt1, qt2 in subscript andp",Pr,Ptl,Pt2 in superscript. On the right part of table VII is indicated for each operator aVc*,
the set of associated operators Sc in the expansion
of p% (within the second order). Lastly, a cross in the
first column means that the operator aVc* contributes to uz as well as to uz.
In both tables VI and VII, the components of the tensorial operators of symmetry E are written for convenience (1) in complex form
and
The transitions generated by the operators aVC.
presented in table VII include pure rotational transi- tions within any vibrational state as well as vibrational bands, both cold and hot transitions, up to four quanta of vibration. Nevertheless, the aVC., operators are only developed to quadratic terms in q and p.
(4) The operators Pt, A:, are presently introduced only
for sake of simplicity, but in no case as tensorial components
corresponding to a complex representation of the C3v group.
Table VI.
-Components (a) of the rotational operators SNc,,.
Table VII.
-Components of vibrational operators ayc* and associated Sit: operators.
Table VII.
-(Continued).
(b)
Ro involves pure rotation transitions in any vibrational state
...
means that hot bands involving lower vibrational states higher than the ground.state are also included.
In terms of tensorial operators avC* and S’, the expression (6) of It’ can be written explicitly as follows
or
or, using the components defined in equations (16),
The expression for the non-transformed component pz can be straightforwardly derived by dropping
all the terms including p., or/and Pa. The terms to be retained within the second order (put with a cross in
table VII) are
where adN instead of ad§ means that the coefficient in pz can be slightly different from the corresponding one
in pi due to the dropping of the contributions coming from contact transformations.
The connection between d coefficients and the usual notation in terms of molecular dipole moment yz
and derivatives is as follows
3. Dipole moment matrix in zero order functions
In table VIII are reported the M-reduced matrix elements, in zero order functions, for rotational operators Ao, A:i, Po, Pt as well as all operator components SN involved in the expansion (18) of y’z. The way to derive these last elements from those of ;,0, &, * , Po, Pt, is explained in appendix A. The elements JK I I J’K’>
of table VIII involve F-functions which are specific for given values of AJ = J’ - J and AK = K’ - K. These functions are listed in table XIV.
In table IX appear the matrix elements for all the vibrational operator components a VC*l listed in table VII.
The phase factors for all these elements are consistent with those introduced in reference [2] for matrix
elements of the energy. They are also consistent with the conventions of Di Lauro and Mills [17] as far as ope-
rators Ao, A ±, Po, P ±, qn or qr, pn or Pr are concerned Nevertheless, for degenerate coordinates qti, qt2 and
Table VIII.
-M-reduced matrix elemlnts of rotational operators (a).
Table IX.
-Matrix elements v I aVC*t1* I v’ > in zero order functions.
momenta Pt I pr2 our conventions are different, Le.,
instead of equations (12) and (13) of reference [17]. But the phase in equation (21) is consistent with the matrix elements used by Kwan [6].
With the help of tables VII, VIII and IX, any matrix element of plz can be straightforwardly derived Table VII summarizes all the contributions to be retained according to the values of Ovn, ev,, Av, and Al,; then tables
VIII and IX give matrix elements for rotational and vibrational parts of the contributions respectively.
For example, if we are interested in the matrix elements involved in pure rotation transitions, Le. with all Av, = 0, we have to retain in table VII all the contributions mentioned with RO in column 5. They are
The matrix elements for (p%ko are reported in table X (within the second order). They include terms in JK J’ K > responsible for the « allowed » pure rotation transitions, but also terms in JK J’ K ± 3 >
and JKI, J’ K ± 1, It + 2 > that generate « forbidden » transitions. Moreover, in the elements ( JK J’ K )
the rotational dependence up to terms quadratic in J and K are required within the second order. We note that the coefficient of the element ( JK J’ K ± 3 > in terms of molecular parameters was calculated by Wat-
son [18] for the case of an isolated vibrational state.
Table X.
-M-reduced matrix elements of p’z with all Av, = 0.
Then, in tables XI to XIII are reported the matrix elements of I.Ilz with Avn = ± 1, wr = ± 1 and Av, = + 1 respectively. They include contributions linear and quadratic in J and K required within the second order.
We note that the results of table XIII has been successfully used to analyse the intensities in the vs band
of 12CD3H from diode laser measurements [19].
Elements of uz with Av,, > 2 are not given explicitly because they would not exhibit anything more than
is already derived in references [6-8] for rotational contributions, i.e. linear terms in J and K. Only the phases
should be changed to be consistent with our conventions on Pt and ).± rather than those by Cross et al. [20]
used in references [6-8].
The functions F and G appearing in tables X to XIII are given in full in table XIV. In all of them m = J + 1 and - J is associated with J’ = J + 1 and J - 1 respectively.
4. Dipole moment matrix in basis functions adapted to C3v symmetry.
As well known, the zero order functions, i.e.,
can only be used as basis for matrices factorized into A, E, E blocks. In the investigation of molecules with strong vibration-rotation interactions, functions adapted to the full symmetry of the C3, group may be more
Table XI.
-M-reduced matrix elements of uz with wn = + 1.
Table XII.
-M-reduced matrix elements of pi with Av,, = + 1.
Table XIII.
-M-reduced matrix elements of pt z with wt = + 1.
Table XIV. - F and G functions.
convenient (with large dimensions of matrices but also quasi-degeneracy of the eigenvalues A,, A2)’ Such
functions were introduced especially for studies of CH3D [21], PH3 [22, 23], in the form of symmetrized func-
tions :
with k
I= K 1.
Such functions are irreducible tensor components in the C3, group, but are of additional interest in achiev-
ing a partial diagonalization of the second order hamiltonian through the introduction of quantum number
G = K - Lit.
t
Now, the tensorial method developed in the previous section foral z can be very efficient for building basis
functions such as
where both the total function and the vibrational part I vABb > as well as the rotational part I JkMDd > are
irreducible tensor components in the C3v group. In fact for all axially symmetric molecules, such an approach:
uses basis, functions in which vibrational and rotational parts have been separately symmetrized I vÀBb >
with respect to all and JkMDd ) with respect to K.
Functions I vÀBb > and I JkMDd> are constructed in appendix B and C respectively, with the symmetry properties and orientations defined in table II. Their main advantage over functions (24) is the ease with which they may be formed through simple rules of tensorial products, as a computational step in a program of analysis.
Using equations (17a) and (25), the M-reduced matrix elements of jilz in the C3, symmetry adapted func-
tions are calculated as
where T’ = r x A2 and C* = C x A2; the 3c-coefficients are those of table III.
The vibrational matrix elements involved in equation (26) are obtained from the elements of table IX through the transformation
with coupling coefficients defined in appendix B.
The M reduced rotational elements of equation (26) are derived from the matrix elements of table VIII through the transformation
with coupling coefficients defined in appendix C.
We note that the element (26) obeys the same selection rules (3a) and (3b) as the final transition moment.
Coming now to transition moments involved in equation (1), eigenfunctions JaTQ ) of Hv R are derived
after diagonalization of the energy matrix as
where JTtlU:AB;kD are the elements of the unitary eigenvector matrix (real with our choice of phase factors).
Transitions moments are given by
obeying the strict selection rules given by the equations (3a) and (3b).
5. Conclusion.
The formulation developed in the present paper introduces for the dipole moment matrix calculation a set of phase factors consistent with those introduced in reference [2 ] for the energy matrix.
A suitable method is now presented for analysis without ambiguity both of intensities and of frequencies
in electric dipole vibration-rotation spectra whatever the nature of the vibration-rotation interactions, ie., in
the general case of vibrational polyads.
The present approach to dipole transitions could be easily extended to the calculation of polarisability
tensor and matrix elements, as in an investigation of the scattering intensities in Raman spectra.
Appendix A.
According to table VI, all the operators S£u can be written as a sum of terms like
where Ri is only function of components Po, P±, and Ai represents one component Ao, Å+ or A-. In the full
rotation group (F)O(3), Ri is a scalar, whereas Ai (like ylz) has the behaviour of the zero component of a term of rank 1. In such conditions, the M-reduced matrix of Si can be easily derived as follows
Appendix B.
VIBRATIONAL FUNCTIONS ! I vABb >.
-The non degenerate oscillator functions I vn ) and I vr ) are irreducible tensors in the C3v group and can be written
or
As for degenerate modes, irreducible tensor components obeying the orientation defined in table II are
built from the harmonic oscillator functions I Vt’ It > as follows
with A, = III = 3 n + p (n = 0,1, 2, ...).
If p = 0, the two functions (3B1) and (3B2) have the species Al and A2 respectively; for Å.t = 0, only the
first component exists, i.e.,
If p = 1 or 2, the two functions (3B1) and (3B2) represent the two components El and E2 of a degenerate
state, i.e.,
For one oscillator, the step from zero order functions to the symmetrized ones is
From elementary tensor components defined by equation (4B), any vibrational function can be built by
tensorial products according to equation (11) of the text. The most general function is written I vÀBb > where
v represents the set of main quantum numbers vn, vr, vt and A involves the set of quantum numbers Àt, but also
the species in the intermediate couplings. So,
In the case of coupling between only two vibrations, equation (5B) is easily made explicit as
Appendix C.
ROTATIONAL FUNCTIONS JkMDd >. - From symmetric top functions JKM >, irreducible tensor components obeying the orientations defined in table II, can be built as
with k = K = 3 n + p (n = 0,1, 2, ...).
If p = 0, the two functions (1C1) and (1C2) have the symmetries A, and A2 respectively; for k = 0, only
one function exists which is Ai or A2 according as J is even or odd, ie.,
If p = 1 or 2, the two functions (1C1) and (1C2) represent the two components El and E2 of a degenerate
state, i.e.,
The step from the zero order functions to the symmetrized ones is
References
[1] HOUGEN, J. T., J. Chem. Phys. 37 (1962) 1433-1441.
[2] TARRAGO, G., Cahiers de Physique 19 (1965) 149-217.
[3] AMAT, G., NIELSEN, H. H., TARRAGO, G., Vibration-
Rotation of Polytatomic Molecules (Marcel Dek- ker, Inc. New York) 1971.
[4] MAES, S., J. Physique 27 (1966) 37-42.
[5] HANSON, H. M., NIELSEN, H. H., J. Mol. Spectros. 4
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