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Anharmonic constants for degenerate modes of symmetric top molecules
JOURNAL
OF MOLECULAR
SPECTROSCOPY
(1990)
139,278-285
Anharmonic Constants for Degenerate Modes of Symmetric Top Molecules JOSEF PLfvA Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802
Anharmonic constants x,, and g,,, for the doubly degenerate modes of any symmetric top (or linear) molecule have been obtained in terms of the independent cubic and quartic constants, i.e., of the coefficients of the symmetry-allowed combinations of products of the components q: = qt, C_iq,, of the degenerate normal coordinates. Examples of application to planar XY, (D3b) and to allene-type X,Y, ( DZd) molecules are discussed and modifications of the formulas in the presence of Fermi resonances are indicated. 0 1990 Academic press. h. INTRODUCTION
The vibrational energy E, of a semirigid polyatomic molecule, in absence of anharmonic resonances, is usually written in terms of the harmonic frequencies w, and anharmonic constants x,,~ and g,,, as
In his seminal work on the rovibrational energies of polyatomic molecules, H. H. Nielsen (1-3) derived general expressions for the anharmonic constants in terms of the cubic and quartic potential constants in an expansion for the potential energy in a power series in dimensionless normal coordinates. His formulas have later been amended by other authors (4,5). The specific form of the cubic and quartic terms of the potential depends on the point group of the molecule and on the species of the normal coordinates involved. Henry and Amat (6) published tables of the symmetry-allowed (i.e., totally symmetric) cubic and quartic products of normal coordinates and of the relations existing among their coefficients (potential constants) for all symmetric top molecules. As a consequence of these relations, only certain combinations of the products of components qs_ of degenerate normal coordinates will have distinct coefficients, The different allowed combinations can, however, yield different contributions to the anharmonic constants. The purpose of this paper is to present expressions for the anharmonic constants in terms of those potential constants which are the distinct, and therefore the only independent and potentially determinable, coefficients of the symmetry-allowed combinations of normal coordinate products. The second part of the paper discusses two specific examples and details modifications of the formulas in the presence of Fermi resonances.
278
0022-2852190 $3.00 Copyright
0
1990 by Academic
All rights of reproduction
Press
Inc.
in any form rexwed.
ANHARMONIC EXPRESSIONS
CONSTANTS
FOR ANHARMONIC
279
CONSTANTS
Following the designations and conventions used by Henry and Amat (6), index n (or n’, n”) will be used for nondegenerate coordinates and index t (or t’, t”) for doubly degenerate coordinates; point groups C’,,, D,w,, DNd (N odd), and DcNIZjd (N/2 even) will be designated as “Type I” groups, while C,, C’,,, SzN (N odd), and S, (N/2 even) will be “Type II” groups; restricted sums are used in the potential energy expansion, e.g., for the cubic part, (2) so that each term only occurs once in the sum; for additional conventions see Ref. (6). Using Henry and Amat’s results, the allowed combinations of cubic and quartic products involving degenerate coordinate components qt,, qt2 have been rewritten in terms of qj = qr, + iq,, and are listed with their independent coefficients in Table I; only those quartic terms which have nonvanishing matrix elements diagonal in the quantum numbers v and I and thus contribute to the anharmonic constants x are shown. Numerical superscripts (in brackets) distinguish the different potential constants associated with different terms having the same coordinate indices n, t, t’, t”. Second order contributions from the cubic part of the potential, VI, involving the sum of all of the allowed cubic terms listed in Table I were calculated using standard perturbation theory. Combining these with the diagonal matrix elements of the quartic terms, V:“, resulted in the expressions for the various types of anharmonic constants given in Table II. The formulas for the constants x,,, and x,, I are, of course, identical to Nielsen’s results ( 1-3). The anharmonic constants involving degenerate indices t, t’ are written in terms of the superscripted cubic and quartic constants defined in Table I. Note that some of the terms contain contributions from sums of quartic constants k$j or k$rt and/or from sums of squares of the cubic constants k!$, k.$ etc., some of which have different coefficients ( LY,- Sj) . The summation over the index j indicated in the formulas extends, of course, only over those values for which the corresponding constants exist for the molecule: for example, in the terms involving.the cubic constants kj$, the sum will be overj = 1 and 2 orj = 3 and 4 (depending on the values of m and m’) for a molecule belonging to a type II group, but onlyj = 1 orj = 3 (no sum) for type I symmetry. Constants occurring only in sums over the index j, i.e., as combinations of more than one constant, are not separately determinable. In contrast, for the constants k,$ and k$jt the index j is associated with the species of the coordinate qn and thus these constants are, at least in principle, determinable individually. The existing formulas (l-5) for anharmonic constants do not reproduce all the terms shown in Table II correctly, in particular for the constants g,, and g,,T. EXAMPLES
As a first example, consider a planar XYJ molecule of D3h symmetry, which has 4 normal modes, v1 (A’1), ~2(A$), v3 (E’), and ~4(E’). Using Watson’s method (8) for calculating the number of potential constants required, we find there are 9 cubic and 19 quartic constants in the potential energy expansion. The allowed cubic terms and
280
JOSEF PLiVA TABLE
I
Allowed CubicandQuartic PotentialEnergy Terms for Doubly Degenerate Modesof Symmetric Top Molecules (qr - E,,,; qr, - A?,,,,;qro- Em.)a Cubic Terms
Speciesof s
m In'm"
anym
I II
N > 3, m
BI
4m = N
I
II
N = 4p
B2
4m = N
I
II
N = 4p
Ai
m' =m
I
II
NZ3,m
m’
I II
N>3
m+m'= N/2
I
II
N = 4, 6,..
m+m'= N/2
I II
N = 4, 6,..
3m = N
I II
N = 3p
3m = N
II
N = 3p
Bi
4
=m
2mtm'=N 2m+m'= N m' = 201 m'
Use
N
Al
A2
a)
Group
q
2m
I
II II
N
q
3, ?5
N = 3, h5
I II
N>5
II
N?5
mtm'tm":N
I II
N = 3, 25
m+m'+m"=N
II
N = 3, >5
m"= m+m'
I II
NL5
m"= m+m'
II
NL5
m=l for speciesE or E, , m=2 for E2 etc.; add subscriptsg or u to the
speciessymbols for groups with a center of synm&xy , or superscripts ’ or I’ for groups with a plane uh andodd principalaxis $
; all
products must.have
an even nuober of s, or q" coordinates.For type II groups,subscripts1 arxi2 should be omitted in col. 2. Terms existingfor linearmoleculesare indicated by - in the last column giving the foldnessof the principalsynxnetry axis G. Only the parts (qiqc)(qt,qt,) in the last four quarticterms listedyield diagonal matrix elements.
those quartic terms contributing to the anharmonic constants are listed in Table III. Only type ( 1) cubic constants are present for this molecule (so their superscripts could be omitted), but the quartic part, Vr’, contains two different terms,
ANHARMONIC
281
CONSTANTS
TABLE I-Continued
m
DiagonalguarticTerms
m'
Group
(%-A
anym
I
Is--B
anym
I II
N
q
4, 6,..
k&tq;q;F
m f N/4
I II
N
q
3, 25, m
~~tt{[(q~f2+(qE)21z+4(q~q~)2j/8
m = N/4
I II
N = 4p
~~~,r-[(q~)a-(q~)al=/8)
m = N/4
I
N =
Qp
N
3, 25, _
kNdaq:qG
k&v
Cq;q;)Cq&)
k&'rq;q;l+q;q;a k&*
c
[ Cq;s )‘+($I
+
I II
q
jm'# m, m'+m f N/2 I II
NL5 N = 6, 8,..
m'f m, )‘I
II
N t3, w
[m'#m,m'+m=N/21 m'= m # N/4
1'14
[q;q;~+q~q;,l'/4
kit;,t'([(q~)a+~s,)‘l
an’= m f N/4
II
N
m’+m
= N/2
II I II
N
I
N = 6, 8,..
II
q
3, ?5, m
m’ = m = N/4
I II
N
m'= m = N/4
I
N = 4p
q
4p
+ 4(q;q;)(q;Bq;.))/8 k;&v
I-[(q;)'+(q;)'lI(q& )'+(q;~ j21 +
II
+ 4(q;q&q;vq;W8
k&q:q;q4fqi
and k$[qZqi + q5q:]2/4, involving the indices 3344.Expressions for the anharmonic constants are shown in the same table: all the constants are the same as given by the amended Nielsen formulas (4, 5) for this molecule, one of the simplest symmetric tops. The expression for xj4 contains the combination of the quartic constants k& + k’*’ 3344/2;thus these two constants are not separately determinable. Several Fermi resonances, i.e., interactions due to cubic terms, can occur: ( 1) Between the A ; fundamental v1 and the A ‘, component of the overtone 2~8 through the matrix element (u,~& I kI::q,q:q: Iv1 _+ 1, v4 r 2, ld) which has to be removed from and g4, with the result that the contributions to these constants from x14, x44, (kiz)2 are altered as indicated in Table III. (2) Between the E’ fundamental u3 and the E’ component of the overtone 2& which could result from the off-diagonal matrix element (v313v4141k:~~[q:(qq)2 + qT(qi)2]/21v3 _t 1, 1, T 1, v4 + 2, Z4k 2), the removal of which modifies the constants x 44, x34,gd4,and g34 as shown. ( 3 ) Between v3 and the E’ combination vl + v4, could occur due to (v~v31~v414~k~~~ql(q~q~ +q;q4+)/2jv, _t I, 213T 1, 13 + 1, v4 -t 1, Z4 * l), resulting in changes in the (k I$)” contributions to the constants x i3, x14, x34, and g34 indicated in Table III. No cases of Fermi resonances seem to have been recognized in the spectra of molecules of this type (boron trihalides, SO3, etc.), although for some of the molecules the levels which might be involved in such resonances are not far apart. A second example to be discussed is an allene-type molecule, X3Y4 (or Y2.X = X = XY2) belonging to the point group Dz~. There are 11 normal modes, ~~-3(A,), ~4 (B,), ~5-7(Bz), and &_I1 (E), and a total of 101 cubic and 437 quartic
JOSEF PLiVA
282
TABLE II Expressions for Anharmonic Constants in Terms of Independent Cubic and Quartic Potential Constants for a Symmetric Top Moleculea
8 The potential
constants
COIM~AW.S
(xi= 1, (x2= 3/4, da= 6,~ Coriolis
are defined
of the molecule
1 if
a3= l/4;
; Dab’
~9,= 1, Bj= l/2
m and m’< m”
oonstants
in Table
I
; B and C are the rotational
(“st”b+“c)(~s+~-~c)(~a-~+~c)~~s-~-~c) (j>l);
yi=y2=
1, y3=y,=-1;
; 61=62=
1,
, d3r hb=-l if m or m’ > m” . For the nonvanishing
< see Ref.
(7)
.
potential constants, those contributing to Table IV. Here we have three different stants kji::) (j = 1, 3, 4) and two types for (j = 4, 5 ) . Expressions for the anharmonic
the anharmonic constants being listed in types of cubic terms involving the conthe quartic terms k{dl (j = 2, 3) and k& constants involving degenerate modes are
ANHARMONIC
TABLE Potential Cubic
Energy Terms and Anharmonic
and Qwrtic
VI= kiiiq:
Potential
+ ki,,ql<
&ergy
III
Constants
for Planar X Y,
(&,)
Molecule
Terms:
+ k:j;q,qjq;
+ k;j;[(qj)3+(q,)31/2
283
CONSTANTS
+ k;:;q,q:q;
+ k;;:q,(q;q;+q;q:)/2
+ k::;[(q:)3+(q;)31/2
+ k:::[(qj)‘q:+(q;)‘q;1/2
+ k~~~Iqj(q~)‘+q~(q;)“1/2 “(21, 2
k ii,,4
+
+
+
k:;
+
k;;:,(
:,$q;q;
Anhanwnic k::k
Xit= St
Btt=
(k;;;,
t
(k;::~+k;~:,/2,
u1/2v;
(2)
:
(x,,)
(3) (%’ Dl*
; ti’=
3,4
(k;;;,
)
4,3
)‘wt,
(0;,-0;-0:)/2Ditt,
a/w t tw t /w P]
(k~~~,‘(8w~-3~~)/4~,(4wz_o:)
-2(k:::)20,/(40:-ot)
- k;;;k:::/w,
+ C(c~+)2(~3/o,t~~/~ol/2 -
)‘(80;-$,,
(k:;;)‘oJ4(
)/4wt,
4o;-w:)
(40;~o’t,
) t c((.&)’
+ 4(k:::)‘o~/03(40:-u~)
t
(k~j~)20103~,/2D~3,
+ 2c;,c:,1
-(k;::)s[
:,+
& i
.
118
; (xl,)
-(k;;:)?2(u1+2W,)
-(k;::)=[ +(k;::,‘[
; $,-
+ &
]/8
; (x3,)
-(k:::)=/2(oit20J
g
1,:
; (ix,,)
l
“3/(“i+y,)
k;:;,q;qjq;
+(kj::)2/8(~1+2~,J
(x,,)
(g,,)
+
:
Resonances :
k:;:,qfq;q;
k&Cs;s;,Cq;s;)
-
B(Cyat )‘[o
+ 21(k;;;)‘/40t
(ix,,) ~~12~:
( t =
, gtt,
-
+4(k:j:)‘o~/wl(40~-w~)
(1)
+
-2(k:::)‘oJ(4u;-o:)
+ CI(&)2 Fermi
k:::+(q)Q2
(k:::)‘oi(~:-~:-~:)/4D~,,
-k&/2
+
)“(8wZ-30Z,)/40t,(40~-oZ,)
+ (k;$ 8_=
kj;&fq;q,
-2(k;;;)‘o,/(4$-o:)
-15($;;)2/4wt
-
+
, xtt,
xnt
- k,22W"*
-
x3,=
k,,,,q,
+
-3k_k:&
3k&/2
+
k;j:,(q;q;)’
Constants
= k&
xtt=
q2q= I 2
k 1122
: hia)
1/(01+03+w,)
; (xlr)
-(k;::)‘(DI-Ds-D3W3
-(k~j~)=(DI+DstD3)/16
;
Ds’
+(k:::,‘[
;
(B,,)
1/(~1t03-o,)
;
; - & 3 a
]/2 4
-(k:::)s(DitD,-D,)/8
+(k:j:)‘(D,-D,+D,)/16 Die
l/(w,-0,-u,)
also shown in Table IV. The constants xtr, g II, and g,,’ are not correctly given by previously published formulas (Z-5). The cubic constants k$), are all individually determinable in principle, but only combinations of the quartic constants (3k$ + kji:))/4 and (k NI (4)‘r’ + kjl:jt,) are determinable.
284
JOSEF
PLiVA
TABLE IV Potential Cubic
Energy Terms and Anharmonic
and Quartic
Potential
mergy
Constants
for Allene-Type
(Du)
Molecule
Texms:
Many different Fermi resonances can occur for molecules some have been detected in the spectra of allene, C3H4 (9, resonance between a B2 vibration v, and the Bl component to (%Gl-&?‘,‘4n[(q:)* - (q;)*l Iv, T 1, ut 5 2, It + 2),
of this type and, indeed, 10) and C,D, (11). The of an overtone 2v, is due and removal of the con-
ANHARMONIC
CONSTANTS
285
tributions of this matrix element will modify the anharmonic constants xnt, &, and g, as indicated in Table IV. CONCLUSIONS
Writing the expressions for the anharmonic constants in terms of the independent cubic and quartic potential constants allows one to readily obtain correct formulas for x,, , x,, , x,, I, g,, , and g, I for the degenerate modes of any symmetric top molecule. This also makes it possible to recognize which of the potential constants may be separately determinable. The formulas published by previous authors (4, 5), which do not take explicitly into account the relations existing among the coefficients of the various symmetry-allowed cubic and quartic products of degenerate normal coordinate components, yield correct results only for molecules with a three-fold symmetry axis and for linear molecules, but fail to correctly reproduce some of the terms for molecules with four-fold and higher symmetry axes. ACKNOWLEDGMENT I am much indebted to Dr. J. K. G. Watson for valuable comments on the manuscript of this paper. RECEIVED:
September 29, 1989 REFERENCES
1. H. H. NIELSEN,Phys. Rev. 60,794-g 10 ( 194 1). 2. H. H. NIELSEN,Rev. Mod. Phys. 23,90-136 ( 1951). 3. H. H. NIELSEN,in “Handbuch der Physik” (S. Fluegge, Ed.), Vol. XXXVII/ 1,pp. 173-3 13, SpringerVerlag, Berlin/New York, 1959. 4. Y. MORINO, K. KUCHITSU,AND S. YAMAMOTO,Spectrochim. Acta Part A 24, 335-352 ( 1968). 5. I. M. MILLS, in “Molecular Spectroscopy: Modern Research” (K. Narahari Rao and C. W. Mathews. Eds.), pp. 115-140, Academic Press, New York, 1972. 6. L. HENRY AND G. AMAT, J. Mol. Spectrosc. 5,319-325 (1960): 15, 168-179 ( 1965). 7. L. HENRY AND G. AMAT, Cah. Phys. 14,230-256 (1960). 8. J. K. G. WATSON, J. Mol. Spectrosc. 41, 228-230 (1972). 9. J. PL~VAAND C. A. MARTIN, J. Mol. Spectrosc. 91,218-237 (1982). IO. Y. OHSHIMA,S. YAMAMOTO,K. KUCHITSU,T. NAKANAGA, H. TAKEO,AND C. MATSUMURA,J. Mol. Spectrosc. 117, 138-151 (1986). II. J. PLfvA AND K. ROUSAN, J. Mol. Spectrosc. 129, 254-267 ( 1988).
OF MOLECULAR
SPECTROSCOPY
(1990)
139,278-285
Anharmonic Constants for Degenerate Modes of Symmetric Top Molecules JOSEF PLfvA Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802
Anharmonic constants x,, and g,,, for the doubly degenerate modes of any symmetric top (or linear) molecule have been obtained in terms of the independent cubic and quartic constants, i.e., of the coefficients of the symmetry-allowed combinations of products of the components q: = qt, C_iq,, of the degenerate normal coordinates. Examples of application to planar XY, (D3b) and to allene-type X,Y, ( DZd) molecules are discussed and modifications of the formulas in the presence of Fermi resonances are indicated. 0 1990 Academic press. h. INTRODUCTION
The vibrational energy E, of a semirigid polyatomic molecule, in absence of anharmonic resonances, is usually written in terms of the harmonic frequencies w, and anharmonic constants x,,~ and g,,, as
In his seminal work on the rovibrational energies of polyatomic molecules, H. H. Nielsen (1-3) derived general expressions for the anharmonic constants in terms of the cubic and quartic potential constants in an expansion for the potential energy in a power series in dimensionless normal coordinates. His formulas have later been amended by other authors (4,5). The specific form of the cubic and quartic terms of the potential depends on the point group of the molecule and on the species of the normal coordinates involved. Henry and Amat (6) published tables of the symmetry-allowed (i.e., totally symmetric) cubic and quartic products of normal coordinates and of the relations existing among their coefficients (potential constants) for all symmetric top molecules. As a consequence of these relations, only certain combinations of the products of components qs_ of degenerate normal coordinates will have distinct coefficients, The different allowed combinations can, however, yield different contributions to the anharmonic constants. The purpose of this paper is to present expressions for the anharmonic constants in terms of those potential constants which are the distinct, and therefore the only independent and potentially determinable, coefficients of the symmetry-allowed combinations of normal coordinate products. The second part of the paper discusses two specific examples and details modifications of the formulas in the presence of Fermi resonances.
278
0022-2852190 $3.00 Copyright
0
1990 by Academic
All rights of reproduction
Press
Inc.
in any form rexwed.
ANHARMONIC EXPRESSIONS
CONSTANTS
FOR ANHARMONIC
279
CONSTANTS
Following the designations and conventions used by Henry and Amat (6), index n (or n’, n”) will be used for nondegenerate coordinates and index t (or t’, t”) for doubly degenerate coordinates; point groups C’,,, D,w,, DNd (N odd), and DcNIZjd (N/2 even) will be designated as “Type I” groups, while C,, C’,,, SzN (N odd), and S, (N/2 even) will be “Type II” groups; restricted sums are used in the potential energy expansion, e.g., for the cubic part, (2) so that each term only occurs once in the sum; for additional conventions see Ref. (6). Using Henry and Amat’s results, the allowed combinations of cubic and quartic products involving degenerate coordinate components qt,, qt2 have been rewritten in terms of qj = qr, + iq,, and are listed with their independent coefficients in Table I; only those quartic terms which have nonvanishing matrix elements diagonal in the quantum numbers v and I and thus contribute to the anharmonic constants x are shown. Numerical superscripts (in brackets) distinguish the different potential constants associated with different terms having the same coordinate indices n, t, t’, t”. Second order contributions from the cubic part of the potential, VI, involving the sum of all of the allowed cubic terms listed in Table I were calculated using standard perturbation theory. Combining these with the diagonal matrix elements of the quartic terms, V:“, resulted in the expressions for the various types of anharmonic constants given in Table II. The formulas for the constants x,,, and x,, I are, of course, identical to Nielsen’s results ( 1-3). The anharmonic constants involving degenerate indices t, t’ are written in terms of the superscripted cubic and quartic constants defined in Table I. Note that some of the terms contain contributions from sums of quartic constants k$j or k$rt and/or from sums of squares of the cubic constants k!$, k.$ etc., some of which have different coefficients ( LY,- Sj) . The summation over the index j indicated in the formulas extends, of course, only over those values for which the corresponding constants exist for the molecule: for example, in the terms involving.the cubic constants kj$, the sum will be overj = 1 and 2 orj = 3 and 4 (depending on the values of m and m’) for a molecule belonging to a type II group, but onlyj = 1 orj = 3 (no sum) for type I symmetry. Constants occurring only in sums over the index j, i.e., as combinations of more than one constant, are not separately determinable. In contrast, for the constants k,$ and k$jt the index j is associated with the species of the coordinate qn and thus these constants are, at least in principle, determinable individually. The existing formulas (l-5) for anharmonic constants do not reproduce all the terms shown in Table II correctly, in particular for the constants g,, and g,,T. EXAMPLES
As a first example, consider a planar XYJ molecule of D3h symmetry, which has 4 normal modes, v1 (A’1), ~2(A$), v3 (E’), and ~4(E’). Using Watson’s method (8) for calculating the number of potential constants required, we find there are 9 cubic and 19 quartic constants in the potential energy expansion. The allowed cubic terms and
280
JOSEF PLiVA TABLE
I
Allowed CubicandQuartic PotentialEnergy Terms for Doubly Degenerate Modesof Symmetric Top Molecules (qr - E,,,; qr, - A?,,,,;qro- Em.)a Cubic Terms
Speciesof s
m In'm"
anym
I II
N > 3, m
BI
4m = N
I
II
N = 4p
B2
4m = N
I
II
N = 4p
Ai
m' =m
I
II
NZ3,m
m’
I II
N>3
m+m'= N/2
I
II
N = 4, 6,..
m+m'= N/2
I II
N = 4, 6,..
3m = N
I II
N = 3p
3m = N
II
N = 3p
Bi
4
=m
2mtm'=N 2m+m'= N m' = 201 m'
Use
N
Al
A2
a)
Group
q
2m
I
II II
N
q
3, ?5
N = 3, h5
I II
N>5
II
N?5
mtm'tm":N
I II
N = 3, 25
m+m'+m"=N
II
N = 3, >5
m"= m+m'
I II
NL5
m"= m+m'
II
NL5
m=l for speciesE or E, , m=2 for E2 etc.; add subscriptsg or u to the
speciessymbols for groups with a center of synm&xy , or superscripts ’ or I’ for groups with a plane uh andodd principalaxis $
; all
products must.have
an even nuober of s, or q" coordinates.For type II groups,subscripts1 arxi2 should be omitted in col. 2. Terms existingfor linearmoleculesare indicated by - in the last column giving the foldnessof the principalsynxnetry axis G. Only the parts (qiqc)(qt,qt,) in the last four quarticterms listedyield diagonal matrix elements.
those quartic terms contributing to the anharmonic constants are listed in Table III. Only type ( 1) cubic constants are present for this molecule (so their superscripts could be omitted), but the quartic part, Vr’, contains two different terms,
ANHARMONIC
281
CONSTANTS
TABLE I-Continued
m
DiagonalguarticTerms
m'
Group
(%-A
anym
I
Is--B
anym
I II
N
q
4, 6,..
k&tq;q;F
m f N/4
I II
N
q
3, 25, m
~~tt{[(q~f2+(qE)21z+4(q~q~)2j/8
m = N/4
I II
N = 4p
~~~,r-[(q~)a-(q~)al=/8)
m = N/4
I
N =
Qp
N
3, 25, _
kNdaq:qG
k&v
Cq;q;)Cq&)
k&'rq;q;l+q;q;a k&*
c
[ Cq;s )‘+($I
+
I II
q
jm'# m, m'+m f N/2 I II
NL5 N = 6, 8,..
m'f m, )‘I
II
N t3, w
[m'#m,m'+m=N/21 m'= m # N/4
1'14
[q;q;~+q~q;,l'/4
kit;,t'([(q~)a+~s,)‘l
an’= m f N/4
II
N
m’+m
= N/2
II I II
N
I
N = 6, 8,..
II
q
3, ?5, m
m’ = m = N/4
I II
N
m'= m = N/4
I
N = 4p
q
4p
+ 4(q;q;)(q;Bq;.))/8 k;&v
I-[(q;)'+(q;)'lI(q& )'+(q;~ j21 +
II
+ 4(q;q&q;vq;W8
k&q:q;q4fqi
and k$[qZqi + q5q:]2/4, involving the indices 3344.Expressions for the anharmonic constants are shown in the same table: all the constants are the same as given by the amended Nielsen formulas (4, 5) for this molecule, one of the simplest symmetric tops. The expression for xj4 contains the combination of the quartic constants k& + k’*’ 3344/2;thus these two constants are not separately determinable. Several Fermi resonances, i.e., interactions due to cubic terms, can occur: ( 1) Between the A ; fundamental v1 and the A ‘, component of the overtone 2~8 through the matrix element (u,~& I kI::q,q:q: Iv1 _+ 1, v4 r 2, ld) which has to be removed from and g4, with the result that the contributions to these constants from x14, x44, (kiz)2 are altered as indicated in Table III. (2) Between the E’ fundamental u3 and the E’ component of the overtone 2& which could result from the off-diagonal matrix element (v313v4141k:~~[q:(qq)2 + qT(qi)2]/21v3 _t 1, 1, T 1, v4 + 2, Z4k 2), the removal of which modifies the constants x 44, x34,gd4,and g34 as shown. ( 3 ) Between v3 and the E’ combination vl + v4, could occur due to (v~v31~v414~k~~~ql(q~q~ +q;q4+)/2jv, _t I, 213T 1, 13 + 1, v4 -t 1, Z4 * l), resulting in changes in the (k I$)” contributions to the constants x i3, x14, x34, and g34 indicated in Table III. No cases of Fermi resonances seem to have been recognized in the spectra of molecules of this type (boron trihalides, SO3, etc.), although for some of the molecules the levels which might be involved in such resonances are not far apart. A second example to be discussed is an allene-type molecule, X3Y4 (or Y2.X = X = XY2) belonging to the point group Dz~. There are 11 normal modes, ~~-3(A,), ~4 (B,), ~5-7(Bz), and &_I1 (E), and a total of 101 cubic and 437 quartic
JOSEF PLiVA
282
TABLE II Expressions for Anharmonic Constants in Terms of Independent Cubic and Quartic Potential Constants for a Symmetric Top Moleculea
8 The potential
constants
COIM~AW.S
(xi= 1, (x2= 3/4, da= 6,~ Coriolis
are defined
of the molecule
1 if
a3= l/4;
; Dab’
~9,= 1, Bj= l/2
m and m’< m”
oonstants
in Table
I
; B and C are the rotational
(“st”b+“c)(~s+~-~c)(~a-~+~c)~~s-~-~c) (j>l);
yi=y2=
1, y3=y,=-1;
; 61=62=
1,
, d3r hb=-l if m or m’ > m” . For the nonvanishing
< see Ref.
(7)
.
potential constants, those contributing to Table IV. Here we have three different stants kji::) (j = 1, 3, 4) and two types for (j = 4, 5 ) . Expressions for the anharmonic
the anharmonic constants being listed in types of cubic terms involving the conthe quartic terms k{dl (j = 2, 3) and k& constants involving degenerate modes are
ANHARMONIC
TABLE Potential Cubic
Energy Terms and Anharmonic
and Qwrtic
VI= kiiiq:
Potential
+ ki,,ql<
&ergy
III
Constants
for Planar X Y,
(&,)
Molecule
Terms:
+ k:j;q,qjq;
+ k;j;[(qj)3+(q,)31/2
283
CONSTANTS
+ k;:;q,q:q;
+ k;;:q,(q;q;+q;q:)/2
+ k::;[(q:)3+(q;)31/2
+ k:::[(qj)‘q:+(q;)‘q;1/2
+ k~~~Iqj(q~)‘+q~(q;)“1/2 “(21, 2
k ii,,4
+
+
+
k:;
+
k;;:,(
:,$q;q;
Anhanwnic k::k
Xit= St
Btt=
(k;;;,
t
(k;::~+k;~:,/2,
u1/2v;
(2)
:
(x,,)
(3) (%’ Dl*
; ti’=
3,4
(k;;;,
)
4,3
)‘wt,
(0;,-0;-0:)/2Ditt,
a/w t tw t /w P]
(k~~~,‘(8w~-3~~)/4~,(4wz_o:)
-2(k:::)20,/(40:-ot)
- k;;;k:::/w,
+ C(c~+)2(~3/o,t~~/~ol/2 -
)‘(80;-$,,
(k:;;)‘oJ4(
)/4wt,
4o;-w:)
(40;~o’t,
) t c((.&)’
+ 4(k:::)‘o~/03(40:-u~)
t
(k~j~)20103~,/2D~3,
+ 2c;,c:,1
-(k;::)s[
:,+
& i
.
118
; (xl,)
-(k;;:)?2(u1+2W,)
-(k;::)=[ +(k;::,‘[
; $,-
+ &
]/8
; (x3,)
-(k:::)=/2(oit20J
g
1,:
; (ix,,)
l
“3/(“i+y,)
k;:;,q;qjq;
+(kj::)2/8(~1+2~,J
(x,,)
(g,,)
+
:
Resonances :
k:;:,qfq;q;
k&Cs;s;,Cq;s;)
-
B(Cyat )‘[o
+ 21(k;;;)‘/40t
(ix,,) ~~12~:
( t =
, gtt,
-
+4(k:j:)‘o~/wl(40~-w~)
(1)
+
-2(k:::)‘oJ(4u;-o:)
+ CI(&)2 Fermi
k:::+(q)Q2
(k:::)‘oi(~:-~:-~:)/4D~,,
-k&/2
+
)“(8wZ-30Z,)/40t,(40~-oZ,)
+ (k;$ 8_=
kj;&fq;q,
-2(k;;;)‘o,/(4$-o:)
-15($;;)2/4wt
-
+
, xtt,
xnt
- k,22W"*
-
x3,=
k,,,,q,
+
-3k_k:&
3k&/2
+
k;j:,(q;q;)’
Constants
= k&
xtt=
q2q= I 2
k 1122
: hia)
1/(01+03+w,)
; (xlr)
-(k;::)‘(DI-Ds-D3W3
-(k~j~)=(DI+DstD3)/16
;
Ds’
+(k:::,‘[
;
(B,,)
1/(~1t03-o,)
;
; - & 3 a
]/2 4
-(k:::)s(DitD,-D,)/8
+(k:j:)‘(D,-D,+D,)/16 Die
l/(w,-0,-u,)
also shown in Table IV. The constants xtr, g II, and g,,’ are not correctly given by previously published formulas (Z-5). The cubic constants k$), are all individually determinable in principle, but only combinations of the quartic constants (3k$ + kji:))/4 and (k NI (4)‘r’ + kjl:jt,) are determinable.
284
JOSEF
PLiVA
TABLE IV Potential Cubic
Energy Terms and Anharmonic
and Quartic
Potential
mergy
Constants
for Allene-Type
(Du)
Molecule
Texms:
Many different Fermi resonances can occur for molecules some have been detected in the spectra of allene, C3H4 (9, resonance between a B2 vibration v, and the Bl component to (%Gl-&?‘,‘4n[(q:)* - (q;)*l Iv, T 1, ut 5 2, It + 2),
of this type and, indeed, 10) and C,D, (11). The of an overtone 2v, is due and removal of the con-
ANHARMONIC
CONSTANTS
285
tributions of this matrix element will modify the anharmonic constants xnt, &, and g, as indicated in Table IV. CONCLUSIONS
Writing the expressions for the anharmonic constants in terms of the independent cubic and quartic potential constants allows one to readily obtain correct formulas for x,, , x,, , x,, I, g,, , and g, I for the degenerate modes of any symmetric top molecule. This also makes it possible to recognize which of the potential constants may be separately determinable. The formulas published by previous authors (4, 5), which do not take explicitly into account the relations existing among the coefficients of the various symmetry-allowed cubic and quartic products of degenerate normal coordinate components, yield correct results only for molecules with a three-fold symmetry axis and for linear molecules, but fail to correctly reproduce some of the terms for molecules with four-fold and higher symmetry axes. ACKNOWLEDGMENT I am much indebted to Dr. J. K. G. Watson for valuable comments on the manuscript of this paper. RECEIVED:
September 29, 1989 REFERENCES
1. H. H. NIELSEN,Phys. Rev. 60,794-g 10 ( 194 1). 2. H. H. NIELSEN,Rev. Mod. Phys. 23,90-136 ( 1951). 3. H. H. NIELSEN,in “Handbuch der Physik” (S. Fluegge, Ed.), Vol. XXXVII/ 1,pp. 173-3 13, SpringerVerlag, Berlin/New York, 1959. 4. Y. MORINO, K. KUCHITSU,AND S. YAMAMOTO,Spectrochim. Acta Part A 24, 335-352 ( 1968). 5. I. M. MILLS, in “Molecular Spectroscopy: Modern Research” (K. Narahari Rao and C. W. Mathews. Eds.), pp. 115-140, Academic Press, New York, 1972. 6. L. HENRY AND G. AMAT, J. Mol. Spectrosc. 5,319-325 (1960): 15, 168-179 ( 1965). 7. L. HENRY AND G. AMAT, Cah. Phys. 14,230-256 (1960). 8. J. K. G. WATSON, J. Mol. Spectrosc. 41, 228-230 (1972). 9. J. PL~VAAND C. A. MARTIN, J. Mol. Spectrosc. 91,218-237 (1982). IO. Y. OHSHIMA,S. YAMAMOTO,K. KUCHITSU,T. NAKANAGA, H. TAKEO,AND C. MATSUMURA,J. Mol. Spectrosc. 117, 138-151 (1986). II. J. PLfvA AND K. ROUSAN, J. Mol. Spectrosc. 129, 254-267 ( 1988).