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Vibration-rotation theory and isotopic substitution in nonlinear triatomic molecules
JOURNAL
OF MOLECULAR
SPECTROSCOPY
58,
344-366 (1975)
Vibration-Rotation Theory and Isotopic Substitution in Nonlinear Triatomic Molecules PAUL M. PARKER Department of Physics,
Michigan
St&e University, East Lansing,
Michigan
48824
In the interpretation of high-resolution data it is helpful to be able to relate the experimentally determined molecular constants, such as rotational constants, centrifugal distortion constants, and vibration-rotation interaction constants of a given molecule to those of its isotopically substituted variants. Results concerning the symmetric nonlinear trlatomic molecule (XYX) have been given through the second order of approximation by previous workers. In this paper, extensions of the work to the fourth order of approximation and to the nonsymmetric nonlinear triatomic molecule (XYZ) are considered. Theseextensions are found to increase the algebraic complexity of the problem considerably, and results can generally be given in implicit form only. The approach used is one that emphasizes the isotopic invariants of the problem, i.e., those expressions which remain unchanged when applied to the various isotopic modifications of a given molecule under the assumption that the molecular force field and the molecular geometry remain unchanged under isotopic substitution. I. INTRODUCTION
The problem considered in this article is the following one. Given the molecular constants of a certain nonlinear triatomic molecule, how does one calculate corresponding molecular constants of an isotopically substituted variant of this molecule without the approximation that relative changes of mass produced by the substitution are small. Except in a few favorable instances, the answers that emerge are unfortunately not very straightforward. Nevertheless, the problem is of considerable interest in the analysis of high-resolution data since its consideration can form the basis for more accurate, more consistent, and more complete determinations of molecular constants. The basic assumption made in isotopic substitution studies is that the substitution does not produce a nuclear size effect on the molecular force field, at least to a certain order of approximation. As a consequence, the molecular equilibrium geometry remains unchanged and the potential energy function is the same for all isotopic variants of a given molecule (I). When the potential energy is expressed in an appropriate set of mass-independent internal coordinates, the coefficients of the resulting function are invariant under isotopic substitution. For the purposes of this presentation I find it useful to distinguish, somewhat arbitrarily, between molecular constants that appear in the vibration-rotation Hamiltonian (“II-constants”) and the more fundamental constants (“F-constants”) that appear in vibration-rotation energy expansions. Among the R-constants, one 344 Copyright
@
1975 by Academic
All rights of reproduction
Press. Inc.
in any form reserved.
TRIATOMIC
ISOTOPIC SUBSTITUTION
345
has: 1=, I#, and I,, the equilibrium principal moments of inertia, of order of approximation zero. R-constants of order two are: the centrifugal distortion constants T, the vibration-rotation interaction constants (r, and the anharmonic vibrational energy constants x. In order four, one has the centrifugal distortion constants @, the vibrational corrections to the second-order centrifugal distortion constants p, and vibration-rotation interaction constants P. The F-constants include the atomic masses, the equilibrium geometry parameters, and the harmonic potential constants K,,,. Beyond order zero, one includes appropriate sets of cubic, quartic, etc., potential constants. In lieu of the complete set of harmonic potential constants (KBB,},one may use the set of normal frequencies {X,1}. However, as is well known, the set (x,4] has fewer members than the set (L). The former set must therefore be augmented by further, independent F-constants chosen from the set of coefficients (tisat) of a transformation to normal coordinates and/or from the set of Coriolis coupling coefficients ({./). The augmentation of (x,t) to obtain a set equivalent to (K,,p) is not unique. A basic feature of vibration-rotation energy theories is that, on the basis of a particular set of assumptions, they give the H-constants in terms of the F-constants; symbolically, LI = H(F). Experimental analysis yields numerical values for a-constants. Subsequently, one attempts to obtain numerical values for the F-constants; i.e., one seeks F = F(H). It may have been recognized that with the particular type of H-constants listed above I have adopted the vibration-rotation theory of Amat, Nielsen, and Goldsmith (Z), which is based on the Darling-Dennison Hamiltonian. II. GENERAL
REMARKS
Isotopic substitution relationships of the most immediate and direct applicability in data analysis are of the type which gives an B-constant of the substituted molecule Rs in terms of the corresponding constant of the reference molecule a,
where f(m) is a definite function of the masses and possibly the equilibrium geometry of the reference molecule and_f(&) is the same function for the substituted molecule. As an example, for the XYX molecule it has been found (3, 4) that the centrifugal distortion constant 7Z2Z2 (molecule in xy-plane) obeys the relation s = ~ZZZZ
G2/w>*1Tzzzz,
(2)
where M is the reduced mass of the molecule, ~1 = 2mlm2/(ml+
2m2),
with ml the mass of the apex atom. The content of Eq. (2) can also be stated concisely as fi2rZZ2= invariant.
(3) more (4)
The invariance refers, of course, to isotopic substitution. Unfortunately, statements of this type can be given only in a few instances, specifically, for the equilibrium principal moments (or for the equilibrium rotational constants) of XYX and for four of the seven nonzero 7’s of XYX. For the remaining 7’s one has to settle for
346 more general statements
PAUL M. PARKER
of the type C jj(m)Hi
invariant.
For example, it has been shown by Steenbeckeliers
(5)
(5) that
(p/m2>(g2rzEOZ - p2~zzzz)invariant,
(6)
g = I.c(rnl+ 2m2 sin201,)/ml cos2~.,
(7)
where
and where 2a, is the equilibrium apex angle. In addition to this general approach, some authors (6, 7) have pursued the development of sum and product rules for the 7%. Some of these rules are applicable to the XYZ as well as to the XYX type of molecule. If direct relationships between H-constants cannot be established in suflicient number to effect a complete comparison of reference molecule and substitute, then it becomes necessary to work directly with the F-constants, i.e., to develop statements of the type C ji(m)Fi invariant. (8) , Such statements can be developed for the potential constants, but only in isolated instances is it possible to give such a statement for an individual potential constant, as can be done for each of the four harmonic potential constants of XYX. After establishing how the F-constants change under isotopic substitution, vibrationrotation theory can be used to determine how the corresponding H-constants are affected. Of course, this can be done only to the extent that theoretical expressions for the H-constants in terms of the F-constants are available. Schematically, the computational procedure in favorable instances is thus H--, while in the less favorable
HZ,
but much more common instances H--,F+FS--,HS.
(9) one must proceed (10)
Since in the second of the above procedures the step H ---)F implies that an appropriately complete theoretical analysis of the molecule is possible, it is reasonable to expect that in many instances a similar analysis HS -+ Fs is equally possible for the variant. In this case the indicated procedure would be H-+Ft,FStHS;
(11)
i.e., the comparison is made on the level of the F-constants. Two general strategies may be pursued. One may accept the basic assumptions as valid and the predicted isotopic correspondences are programmed into the analysis for improved consistency and accuracy. Or, one may analyze isotopically related molecules independently and then ascertain how closely the isotopic correspondences are obeyed to a given order of approximation. This would yield a measure for the validity of the assumptions and/or the quality of the data fit.
TRIATOMIC
347
ISOTOPIC SUBSTITUTION
The following three molecular configurations will be distinguished: (a) the symmetric (C,,) molecule XYX, (b) the unsymmetrically substituted symmetric molecule XYX’, where X and X’ are distinct isotopes of a given element, and (c) the general nonsymmetric (C,) molecule XYZ. Meaningful comparisons result when the reference molecule and the isotopic variant: are both Case (a); are both Case (b); are both Case (c); or one is Case (a) and the other is Case (b). I will first discuss the potential energy function for the above three cases in the harmonic approximation. I then proceed to a consideration of the second-order centrifugal distortion constants 7. Finally, a discussion of the molecular force field beyond the harmonic approximation is given. III. THE XYX
Let the equilibrium geometry it is seen that a = f’s sina6,
MOLECULE
of the molecule c = r’s COW.,
For internal coordinates I choose to adopt the projections of the instantaneous length directions, as described by Herzberg (a), viz, sr = (x2 - xl) S2 =
be as shown in Fig. 1, from which
(Xl -
a2 + c2 = r,=.
(12)
central force coordinates representing of the bonds along their equilibrium
sina
+
(yl
-
y2) cosa,,
x3) sinh
+
(yl
-
yd
cow,,
03)
ss = (x2 - xa). The Cartesian coordinates are the instantaneous position coordinates of the three nuclei in the principal axes system of the equilibrium inertia tensor. In his study of the centrifugal distortion constants 7, Steenbeckeliers (5) adopted valence bond coordinates, but his results are easily duplicated with the above central force coordinates, whose use proves more convenient later on for the formulation of the anharmonic potential energy function. The Eckart conditions are expressed by
FIG. 1. Equilibrium
PM1
+
m2O2 +
x2) =
0,
my1
+
mz(y2 +
ya> =
0,
(x2 +
x2) =
(ys -
configuration
(14) y2) tm%?,
and principal axes (CC,y) of the nonlinear XYX
molecule.
348 and mass-adjusted
PAUL M. PARKER
symmetry
coordinates
are introduced
as follows (9).
24 = (+mz)t(xz - XP), v = ,u+Cyr - t
(rz + Y311,
(15)
7J.J = I.dcx1 - 3(x2 + x3)1, where cc was given in Eq. (3) and ~8 = r (ml + 2~8~sin2ffB)/(m1 + 2mz) sink,. For the quadratic
part of the potential
(16)
energy V, one has
2V = SF.9 = i!?KU = C&Q,
(17)
where the tilde denotes the transpose, s = (S1, sz, S,), i.7= (24,o,w), 0 =
and the Q’s are the normal coordinates. eral, the symmetric matrices
(18)
(QI, Q2,Qd, For the coefficient matrices
one has, in gen-
and A is the diagonal matrix with nonzero elements X1, X2, and X3 which are the squares of the normal frequencies (~1, ~2, and ~3, respectively. The Czr symmetry of the molecular force field is expressed by requiring that Fz2 = FII
and
F23 = F13,
(20)
or that k13 = &?a = 0,
(21)
with Eqs. (20) implying Eqs. (21) and vice versa. The elements of the matrix F are assumed to be invariant under an isotopic substitution that preserves the Czo (XYX) symmetry. It has been shown by Pliva (10) that setting up a potential energy function whose coefficients are invariant under isotopic substitution requires, in general, the use of a set of mass-independent internal coordinates detined along the directions of the straight lines joining the instantaneous positions of the nuclei, not their equilibrium positions. However, in the harmonic approximation (only), the latter type of internal coordinate is adequate. Hence, for the study of the quadratic part of the potential energy function, the coordinates S, defined by Eqs. (13) are satisfactory. The transformations between S, U, and Q are given by S=TU, where I? is the orthogonal
matrix
U = I’Q,
S =
(TOQ,
(22)
(9) (23)
TRIATOMIC
Table I. Matrix Elements of the Coordinate Transfomtions Eqs. (22) for the XIX Molecule Elements of Tr --
51
349
ISOTOPIC SUBSTITUTION
= t21
Defined by
= (sinae)/J2m2
t12 = t22 = CCOSU‘~,/J; t13 = - t23 = - CvG;sinae)/u t31 = h/mz t t 32 = 33 = ' Elemmts Of T
-1
:
t;;
= tit = t;;
= 0
t;: = $77 t-l = t;; =
Ji
J;/2cosae
= _ t;; = _ u/2+inae
-1 t23 = - (4Sanae)/2
Elements of (Tr)-': (Tr);i = (Tr);: = (&inY)/Zcosa
e
(Tr);; = (Tr);; = ("&0SY)/2COSCr e (Tr);: = - (Tr);: = - 1.1/26$na
e
(TI?;: = [+($7?JcosY
- (v$7sinYtanae)/21
(Tr);: = [-(v$ZZsi*Y
- (&osYtanae)/21
(TT);; = 0
Using Eqs. (13)-(15), one can work out are listed in Table I. For the three I’-’ = T, T-l, and (TI’)-1 = i?‘, with also listed in Table I. Writing all coordinates of Eq. (17) in
the elements of the matrix T whose elements respective inverse transformations, one has the elements of the last two transformations terms of S, one obtains
2V = .$FS = s(F-‘KT-‘)S
= ~[(f~)-lA(Tr)-l-JS,
(24)
which allows the making of the three comparisons F = F-~KT-~,
(25)
F = (f~)-lA(Tr)-l,
(26)
K =
f-lAr-l,
or
K-l =
r-T.
Introducing or
the Cz. symmetry of the harmonic force field by requiring (21) hold, one obtains from Eq. (25) the set of invariants
(27) that Eqs. (20)
FII = F22 = (IA/~ cos‘4tc)k2z+ (I.L’/~P~sin2ae)kaa, FS9 = (m2/2)kll + b/4
cot2+22
- C(~ZP/~)* tancuJkl2,
Fa = (p/4 cos2a$k22 - (JA~/~PSsin2aJkss, Fla = F28 = - (~14 cos(yb cota,)krz + [(w~zc(/2)~/2 cosa.Ikn.
(28)
350
PAUL M. PARKER
It will be recognized that an invariant expression remains invariant when multiplied by an arbitrary factor that is not explicitly or implicitly mass-dependent (or, more precisely, by a factor that is independent of the mass(es) isotopically replaced), and that a linear combination of invariants is also an invariant as long as the linear combination constants are mass-independent. Using these observations in connection with Eqs. (28), it is possible to deduce the equivalent set of four invariants: m&rl,
r2&kw
&m&tk12,
&zz,
(29)
Equation (26) yields invariances equivalent to Eqs. (28) expressed in terms of the three normal frequencies and the normal-coordinates transformation angle y in place of the four independent nonzero k,,t. Equations (27) relate the normal frequencies and the angle y to the harmonic potential constants. The second form, viz, K-l = I’-lf is found to be very useful as it yields expressions that appear directly in the r’s, and it was this observation together with the simple form of the k-invariantes, Eqs. (29), that allowed Steenbeckeliers (5) to obtain the T-invariances listed in his Table III. The indicated isotopic variations of the r’s have been observed to be well satisfied in a number of specific instances (6, 11). Knowing the invariants, Eq. (29), means that given the potential constants k,,t of the reference molecule, one can immediately obtain the potential constants k,,ts of the substituted molecule, and knowing these, one can obtain further molecular parameters of the substituted molecule insofar as they depend only on the harmonic potential constants. Thus for the squares of the normal frequencies one has (9) X1.2
=
4&l
kzz) f
+
+[(kn
-
kzzY + 4ku2]+,
XB = ka,
and hence, with the aid of Eq. (29), and with fi2 = m.JrnP, Xr,z! = W&r
+ L&z) f
+[(&ku
-
j&d2
(30)
etc., + 4,ci&kn2]+,
(31)
Xss = j22j?,s-1kxs. The Teller-Redlich product rules (I) are implicit plicit dependence of s = sin?, c = cosy on the known (P), one can similarly write
ss= &(l/V2)(1 f.78= +p
-
-
in these relations. harmonic potential
Since the exconstants is
(ritzku - fikzz)/[ (G’zzkn - j.ik2J2 + 4~zPzkZJ~)
*,
(32)
(ss)“]t
It may be verified trigonometrically tan%
that = 2klz/(kll
- Rzz),
and hence the angle ys can also be specified through tan2-P
= 2(%$)*k12/(&kl~
- fikz2).
It appears advantageous to take the range of y as -3~ < y < +&r which then necessarily encompasses the entire range of tan2y but could require negative values of siny. According to Steenbeckeliers (5), the isotopic invariants of the equilibrium principal moments of inertia are 1=/P,
Idm2,
Id&
(33)
TRIATOMIC
ISOTOPIC
(9) of the substituted
These can be used to express the Coriolis constants r13x =
@Iz/j.IL)V
s 123 =
-
-
(gI,/pI,)t~
351
SUBSTITUTION
molecule as
(&/?%L)b",
(34)
- (&//?&I*)wJ.
The centrifugal distortion constants r which depend on the quadratic, but no higher order, potential constants will be discussed in Section VI. It should be appreciated that all relations between constants of the original and the substituted molecule are, in a sense, rigorous since one and the same theoretical formulation must be expected to be equally applicable to both isotopic modifications of a given molecule. IV. THE XYX’ MOLECULE
The equilibrium configuration of the XYX’ molecule, with X and X’ distinct isotopes of the same element, is shown in Fig. 2. Equations (12) hold as given previously, since the geometry of XYX’ is assumed to be the same as that of XYX. Discussion of this configuration is based on the formulation presented by Chan and Parker (IL’), henceforth referred to as CP. This publication uses results by a number of previous workers, most notably Shaffer and Schuman (13) and Posener (14). Taking b = a throughout specializes CP to the XYX’ case. The principal axes (x, r) of the equilibrium inertia tensor are rotated with respect to the axes (5, g) through an angle 0 given by (13) tan28 = 2ml(m3 - m~)ac/[4m3m3a2 + ml(m3 + m3) (a” - c”)] and the principal
equilibrium
moments
(35)
are
I, = [4m3m3a2 + ml(m3 + m3b,21/M =
r:[4rnsrn3
sir&, + rnl(rnz -I- m3>3/M,
(36)
with M = ml-j- mz-l- ma.
(37)
Furthermore, 1, = $(I, - I’), (38)
I, = +(I* + I’), with I’ = f { [2ml(m3
-
m3)ac)z +
[4m2m3a2 +
ml(m3 +
m3) (a2
-
c2)121 &lM,
(39)
J
Y
ml
8 ae ;
x
\I r.
Y jC
II!!& ma
(1.
a
m2
FIG. 2. Equilibrium configuration and principal axes (x, y) of the nonlinear XYX’ molecule; X and X’ are isotopes.
PAUL M. PARKER
352
where the choice of sign is taken depending on whether I, > I# or I, < I,, for the choice of axes made in Fig. 2 with -&r < 13< &r. For the internal coordinates again to represent the instantaneous lengths of the three sides of the configuration triangle for small amplitudes of vibration, they must now be defined originally in terms of the (2, g) coordinate system as S1 = (zz - Z-1)sina. + (?jl - Qz) cos+ SZ = (2, - 53) sinol, + (jj1 - 53) cos+ &=
(40)
(L&-523).
In terms of the (x, y) coordinates
these become
s1 = (x2 - xl) sin@. - 0) +
(y1 -
y2) COdaa
&
(yl -
y8) cos(~~+
=
(xl -
X8) sin(&+
S8 =
(x2 -
x8)cod
6) -k
-
(y2 -
y8)
-
e>,
(41)
e),
sine.
The associated F-matrix preserves the symmetry of the XYX force field, FZZ = F11 and Fz8 = Fra, and hence there result four invariants for XYX’, the same number as for XYX. The potential constants K,, are defined through CP Eq. (60), i.e., 2v = DKU = knu2 + kzzv2+ k88ti + 2knuv + 2k18Uw+ 2kz8vW, where u = (2c, v, w) represents
the intermediate
(42)
coordinates
u = x2 - x8, 0=
y1 -
(m2yZ-k
m8y8)/&2+
m8),
w = Xl -
(m%%f
m8%8)/(m2+
ma),
(43)
defined in the principal axes system, and K is the symmetric matrix of the potential constants, Eq. (19). It appears convenient to define an auxiliary set of potential constants &,I in terms of intermediate coordinate system as
8 = fh tB=&-
-
coordinates
6 = (ZZ,ii, 2~) defined in the (2, g)
(m292-k
m888)/(m2+
ma),
(m252-t
m8%8)/(m2+
ma),
(W
and 2v = i?gu
=
ih2+
ii22ij2f
&88@+
2&2%+
2kl8a?.&+-
2%2&a
(45)
Writing O=MJ,
(46)
one may determine the transformation matrix 8 and its inverse as follows. The transformation equations between (x, y) and (3, 9) are xi = Zi cos8 + gi sine, yi = --Zi sine + V&co&,
(47)
TRIATOMIC
for the ith atom. Introducing
353
ISOTOPIC SUBSTITUTION
these into Eqs. (43) gives, with the aid of Eqs.
(44),
that u = ti cos49+ (g2 - 83) sine, v = B cos8 - Zz,sin@,
(4s)
w = ti co& + ii sine. Since the Eckart conditions, CP Eqs. (29)-(31), have the same form in the barred and the unbarred Cartesian coordinates, CP Eqs. (35)-(40) hold in the barred as well as the unbarred Cartesian and intermediate coordinates. Therefore, CP Eqs. (39) and (40) give the expressions for 92 and 53 which can be used in Eqs. (48) above and thus, the matrix elements of 8-l are determined, and hence also those of 0. They are listed in Table II. Of course, for m2 = ma, one has that 0 = 0, and 8 and 8-’ are equal to the unit matrix of order three. In terms of 8, the relationship between the potential constants K,,, and &8~ is then given through K = 6Re, (49) R = &‘K0-‘. For the various coordinate
transformations,
S=Du,
one has U = I’Q.
lT=OU,
(50)
The transformation matrix D with elements &.I and its inverse L)-l with elements &-l = casl are given in Table III. The matrix elements of D are determined by introducing CP Eqs. (35)-(40) in barred coordinates and with 0 = 0 into Eqs. (40). The matrix elements of D-* follow by the standard procedure for constructing the inverse. The elements of the matrix I’, denoted by TU, are those of the transformation from intermediate to normal coordinates, s = 1, 2, 3,
u = C m,Q., v = C m2eQq
s = 1, 2, 3,
w = C nasQs,
s = 1, 2, 3,
(51)
and cannot be given in closed form for the XYX‘ and XYZ configurations be determined by the diagonalization procedure. For the potential energy function, one now has 2V = SFS = 6~0
but must
= OKLJ = @IQ
(52)
and 2V = .i?FS = ?@i?YES. Therefore,
the invariants,
expressed in terms of the &,I, are obtainable F = ij%D-‘,
(53) through (54)
with the i,,, related to the k,., by Eqs. (49). The relationship between the k,,l and the normal frequencies also follows from Eqs. (52) with the aid of Eqs. (SO) and it is R = f%W-1,
K-1 = l?A-lf.
(55)
354
PAUL M. PARKER T!ableII.
Matrix Elements of the Transformation defined by Eq. (46) for the XYX' and XYZ Molecules
+aents
of 0:
= 1/case
%l
e e 0 21 = 31 = 0 = case 22 = e33 e23 = - 832 = sine %2 = (wane/m2m3)912 e13
= ()itane/m2m3)613
P = m,(m2+m3)/M
+12
For XYX':
$13 For XYZ:
= [(m3-m2)c0se+ (m2+m3)cotaesine]/2 = [ (m,-m,)
sine - (UI~+III~)CCA~~C~S~)/Z
$12 = t(m3a-m2b)cose+ c(m2+m31sine]/(a+b) $13
-1 Elements of 0 :
= I(m3a-m2b)sin3- c(m2+m3)cosel/(a+b)
-1 -1 -1 ell = ez2 = e33 = c06e e-l = e-I 21 31 =.O e-l= _ e-l= - sine 23 32 e-l = - (wine/m2m3)$;: 12 e-l = + (winB/m2m3)$;: 13 u = ml(m2+m3)/M -1 412 -1 413 -1 $12 -1 $13
For mx':
For Xx&:
Writing out eqs. (54) explicitly and ~13 = 1 gives
= (m3-m2)/2 = r(m2cm3)cotaJ/2 = $a-m2b)/(a+b) = c(m2+m3)/(a+b)
and recognizing
from Table
III
that
cl1 = cl2 = 0
F11 =
C212L22+
2CzlC3lE23 +
C312E33,
(56)
2722 =
C222i22+
k22G32k23 +
c322k33,
(57)
F33 =
kl1-k
F12 =
62lc22&22f
F13 =
c21&2
+
c23f22 +
c33223) +
c31@13 +
c23a23 +
c33J533),
(60)
F23 =
c22(&12 +
c23&22 +
c33&23) +
c32(fl3 +
c23&23 +
c33&33).
(61)
c232h22+
C332G33+
(c2lc32+
2C23&2 +
c22631)k23+
2c33&3 f
2c23C33~23,
calcazkrr,
(58)
(59)
The coefficients c,8~ are given explicitly in Table III. The C2# symmetry of the molecular force field requires that F22 = F11 and F23 = F13, which leads to two con-
TRIATOMIC
355
ISOTOPIC SUBSTITUTION
Table III. Matrix Elements of the Coordinate Transformation D Defined by Eq. (50) for the XIX' Molecule
Elements Of D:
dll = m3a/(m2+m3Pe d21 = m2.V(m2+m3)re d
d3l = 1
32 = d33 - O
d12 = !Jc@lll+2m2)/2mlm*r, d22
= ~c(mlt2m3)/2mlm3',
d13 - - (a/r,)[l+(uc2/2m2a2~l d23
= + (a/r,)[l+(!Jc2/2m3A
IJ= ml(m2+m3)/'M -1 Elements Of D :
Cl1 = Cl2 = 0 =21 c22
=31 =
"33
1
=
= v'r (2m3a2+w21/cm31z r 2 2 = p're(2m2a +vc )/crn21z
'23 =
'32
53
- 21Pa(Ma2+m1c2)/cMIz - m2(ml+2m3kxe/MIZ
= + m3(ml+2m2)ar,/MIZ = 2va2(m3-m2)/~TZ
p' = m2m3/(m2+m3) Iz is given by Eq. (36)
straints
among five of the six harmonic (C212 (~21 -
C222)522+
c22)(&2 +
629522 +
potential
2(C2101 csafzr) +
constants,
C22caz)kza+ (~31 -
(cn2 -
c32)(&3 +
viz, Ca22)kaa = c2&2a +
0,
c&a)
(62) =
0. (63)
As a result of these constraints, there are four linearly independent invariants. The above equations for FII, &a, Fu, and FM, subject to Eqs. (62) and (63), constitute such a set of invariants. To effect a comparison in the manner of Eq. (10) between two isotopically related molecules, both of type XIX’, one assumes as known the set of six potential constants of one of the two molecules. This set presumably satisfies the constraints, Eqs. (62) and (63). The set of six potential constants of the substituted molecule is determined through the set of six equations: Eqs. (62) and (63), and the invariance equations resulting from Eqs. (56), (58), (59), and (60). The new set of potential constants (KS) is used in the secular determinant equation, CP Eq. (64), IU/.@) -
(=?I
to give the Xas and n,.a S of the substituted are those cited in CP Eqs. (53)-(59). [There read a.] With the Ls and n..~ s determined,
= 0,
(64)
molecule. The matrix elements of bs) is an erratum in CP Eq. (59): a2 should one can calculate the Coriolis constants
356
PAUL. M. PARKER
of the substituted
cc = /&" =
molecule,
m(m2+
CP Eq. (SS), as
m)/M,
~mlm2cos(&,
@)+
-
II, M = [mrm2 sin&
mlm8cos(&+
6)]/21ucod
- 0) - mrm8 &(a.
+ 0)]/2M
sin%,
(66)
cos8 sink..
Knowledge of the harmonic potential constants of the substituted molecule also allows determination of the centrifugal distortion constants G, as discussed in Section VI. When the comparison is made between a molecule of type XYX and one of type XYX’, it is important to note that the conventional definition of the intermediate coordinates of XYX, Eqs. (15), is inconsistent with the corresponding definition for XYX’, Eqs. (43). Hence, since the harmonic potential constants are defined through 2V = DKU, there results an inconsistency in the definition of the harmonic potential constants. This discrepancy must be taken into account whenever expressions applying to XYX’ are specialized to XYX according to the replacement scheme kll(XYX’) + 3m2hl(XYW, k22(Xyx')
-+Pk22(XJm,
k,,(XYX')
-+Fcrkaa(XYX),
k12(XYX’)
+
k,,(XYX’)
--) 0,
kza(XYX’)
3
(4pm2)%2(XYX),
0.
Also, specializing to XYX, 0 = 0, 8 = 0-i = 1, &,.f = K,,p, and the elements matrix D-l listed in Table III become Cl1
=
Cl2 =
Gas =
c21 = c22 = l/(2 c2a = -3
(67)
0,
Cl8= 1,
COSCYJ,
tancu,,
c31 = -632 = -p/2pa
of the
(68)
s&,
where p and ~3 are given by Eqs. (3) and (16), respectively. With these observations in mind, it will be seen that for XYX the constraints, Eqs. (62) and (63), reduce to the requirement that kla = k23 = 0, which was anticipated and used in Section III, and Eqs. (56)-(61) reduce to Eqs. (28). Equating the four invariances, Eqs. (56), (58), (59), and (60), for XYX’ relates, then, the four potential constants of XYX to the six potential constants of XYX’. Since the six potential constants of XYX’ are constrained by the two conditions, Eqs. (62) and (63), the full set of harmonic potential constants of XYX’ is determinable from that of XYX, and vice versa.
357
TRTATOMIC ISOTOPIC SUBSTITUTION
It is possible to solve the four invariance XYX with the following result. KII = (2/m)[Fs3
equations
for the potential
constants
of
+ 2F 18sinad + +(FII + Fr2) sirk],
k22
=
(2/P)(Fn+
F12)
ha
=
(~&&(FII
- FH) sin%,,
cos2cY.,,
KIZ = C(coscr,)/(accm2)l][2F13
(6%
+ (Fu + Fu) sinru,].
The expressions FII, F12, Fu, and Faa are those for the isotopically related XYX’. This set of equations will be found useful for determining the potential constants of XYX from those of XYX’. V. THE XYZ MOLECULE
The equilibrium configuration of the XYZ molecule is shown in Fig. 3 and represents the most general triatomic configuration. The discussion given in CP applies without simplifying features. Geometrically, a = rs2 sine+
b = ?,I sin&
G = 182co&Y*= r’s1co@,, r,r2 = b2 + c29 The principal
axes system
(70)
r,z2 = a2 + c2.
(x, r) is reached from (5, 8) by a rotation
B given through
tan28 = T/Q,
(71)
with r = 2mrc(msa - mzb), Q = ma(mnl and the equilibrium
+
m&r2 + m2(mt + ma)b2 - mr(m2 + ms)8 -I- 2mma4
principal
moments
of inertia
(72)
are given by
I, = 3(1* - I’), IV = HIS + I’),
(73)
I, = I, + Iti = [Q + 2mr(m2 + m8)c’l/M, where I’ = f(r*;+
@y//M.
m3
FIG. 3. Equilibrium configuration and principal axes (x, y) of the nonlinear XYZ molecule.
(74)
358
PAUL M. PARKER
For the internal
coordinates
one now has
Sl = (2, -
21)sinBe+ (81- g2>co&
s2 = (31 - 5%)sinar, + (gr - jj3) s3 =
(22 -
Sl =
(x2 -
XI) sin@,
SZ =
(a
x3) sinbe+
(75)
cosa3,
23),
or, in terms of (x, r),
-
-
Sa = (x2 - x8) coSe For the potential
energy function,
0) +
(YI -
~2) coscBO -
O),
0) +
(~1 -
~3) cc&+
e),
(76)
(ye - ya) sine.
once again,
2V = SFS =
i?RO =
DKU = @iQ,
(77)
with the intermediate coordinates U and 0 as defined previously in Eqs. (43) and (44), respectively. The matrices F, K, and R now have no restrictions on their elements other than that they are real and symmetric. The harmonic potential function therefore is specified by six independent constants. For the several coordinate transformations involved, again, S=De,
U = l?Q.
O=WJ,
(78)
The procedure for determining D and D-l is similar to that described for XYX’ and leads to the elements listed in Table IV. The procedure for determining 0 and 8-l is also similar to that described for XYX’, and the results are included in Table II. The general form of I’ is specified by Eqs. (51). Its elements cannot be given analytically in closed form. The invariants of the harmonic potential energy function are given by the six Eqs. (56)-(61) in terms of the &,.p. The &,, are related to the Rsrt by Eqs. (49). Of course, the coefficients clsg in Eqs. (56)-(61) are now those for XYZ found in Table IV. The constraints, Eqs. (62) and (63), do not apply and the six invariants represent a system of six equations for the six potential constants {krr,s) or ( k,,tS} of the substituted molecule in terms of those of the reference molecule. As for XYX’, the XSs and naetS are determined from the secular equation, Eq. (64), and the {,,rs are given by Eq. (6.5), where now Jr =
sin0 + c co@ +
[wwn2(b
cc“’ = [mlm,(b
mm3(
--a
sin0 + c cos8)]/M(a
cosd - c sine) - mlma(a cos0 + c sirM)]/ilf(a VI. CENTRIFUGAL
According to CP Eqs. (116)-(123), stants can be written in the form -21,47,W
= (L)(u) (t.),
DISTORTION
+ b) co.s&
(79)
CONSTANTS
the second-order
centrifugal
distortion
con-
a = x, Y>s>
- 2~,2182TL%zk¶8 = (L) (u) @a) = (la) (Q) (L), -2I,2I Y2fzwu = &)(d(~w),
+ b) cost?,
*
LY,@ = 2, y cyclic,
(go)
TRIATOMIC
ISOTOPIC SUBSTITUTION
Table IV. Matrix Elements of the Coordinate Defined by Eq. (78) for the XYZ Molecule
Elements
of D:
dll d21 djl
d12 d
22
d13 d23
= m3b/(m = m2a/(m = 1
2 2
+m +m
3 3
d32
= UC 'ml+
)r jr
359
Transformation
D
el e2
= d33
= 0
(a+bh21/mlm2+el
(a+b)
= ~c~bml+~atb~m31/mlm3re2~a+b~ -
~b(atb)m2+~c21/m2rel(a+b)
= +
[a(a+b)m3+~c21/m3re2('+b)
=
P = ml(m2+m3)/M
Elements
of
D
-1: Cl1 = Cl2 = 0
El3 = 1 2
=21
l/cm31z
= u'rel[a(a+b)m3+w 2
c22 '23
= u're21bla+b)m2+!-c
=
=31 = '32 =33
l/cm21z
- !_~'(a+b)[Mab+inlc21/cMIZ - m21bml+(a+b)m31rel/MI
= + m3[aml+(a+b)m21fe2/MI
z z
= ~'(a+b) [(b-a)ml+(bm3-am2)l/MIZ
v' = m2m3/(m2+m3) Iz is given
where u is the 3 X 3 symmetric
matrix
by Eqs.
(72) and
(73)
with elements
and where each of S, s’, s” may take the values 1, 2, 3. The matrices (0 are 1 X 3 matrices with elements given by CP Table I for XYZ, and with elements given in Table V of this paper for XYX’ and XYX. For the XYX case, due allowance has been made for scaling the harmonic potential constants in accordance with Eqs. (67). The planarity relations (1.5, 16) are implicit in the expressions for the centrifugal distortion constants given here. Since the isotopic invariances have been formulated in terms of the harmonic potential constants, it appears to be advantageous to express the centrifugal distortion constants directly in terms of the potential constants. This can be done by the use of Eq. (55) in the form K-1 = IWf, which holds for all three molecular configurations elements involved, Eq. (82) becomes
(82) considered.
In terms of the matrix
PAUL M. PARKER
360
Table V. Elements of the Raw Matrices (t) Occurring in Eqs. (80)
4am2m3sin2e For XYX':
G,) =
.2 2aml(m3-m2)su 8 _ 2cml(m2+~3)sinB
2cml(m2+m3)
( (m2+m3)cose '
.M case
t
,
0,
I
,
0,
'
4any3coE.e
6,) =
(m,+m,)
6,)
=
=
M.
(ix)
Y.
2m1
(m*+rn,)
+
1 1
(m3-m2) sine
2cml(m2+m3)COse
M
M
1
(Q
Chan and P. M. Parker, J. Mol. Spectrosc. 42, 53 (1972).
FOX XIZ:
See
For XIX:
W,) = (0, 21Jreco.we,0)
&,I
(ty) = (2m2r,sina,,0, 0)
and comparison
M
2aml(m3-m2)cose 2cml(m2+m3)sine + M M
4m2m3dne
6,)
M cos8
= (0, 0, -21Jrecosae)
&,I = (TX) + G,,
with Eq. (81) shows that therefore
K-' = u. For the elements
of u occurring
(84)
in Eqs. @SO),one has, therefore,
UII = (k&3
- Rd)/Det
explicitly:
K,
ua = (K&a - ha2)/Det K, (~33= (K&Z - W)/Det
K,
K,
ua2 =
(h&23 - WdlDet (h&la - hh)/Det
aal
@I&H - h&)/Det
K.
412 =
UZI =
US
=
ala
=
=
(85)
K,
Since the set of harmonic potential constants of a substituted molecule can be obtained from the set of the reference molecule in the manner described in this article, the set of centrifugal distortion constants of the substituted molecule can be calculated from Eqs. (80). An alternate form for Det K can be given with the aid of CP Eqs. (71), (72), and (79, which show that Det K = h&As Det p. (86) After lengthy
calculation
one finds for XYZ that
Det K = X&A&(m~
+ m#(l + tanV)I,/i315(a
+ b)2,
(87)
where I, is given by Eqs. (72) and (73). The corresponding expression for XYX’ is found by replacing (a + b)* by 4u2 in Eq. (87), and by using I, as given by Eq. (36). Finally, for XYX, Eq. (87) reduces to Det K = X$da, when once more allowance is made for scaling as specified by Eqs. (67). Equation is an immediate consequence of Eqs. (30) as well.
(88) (88)
TRIATOMIC
ISOTOPIC SUBSTITUTION
VII. GENERALIZED
INTERNAL
361
COORDINATES
For the study of the isotopic variation of H-constants other than those discussed in previous sections, one needs to know the isotopic variation of the cubic potential constants, and, also, in some instances, the quartic potential constants, as listed in Table VI. The present availability of theoretical expressions in the Nielsen-AmatGoldsmith formulation is also indicated in Table VI. Determinations of cubic and quartic potential constants have been made principally through the theoretical expressions for the LY’Sand x’s (17, 18). As mentioned in Section III, it has been shown by Plfva (10) that in order to obtain an anharmonic potential function with invariant coefficients, one must specify the function in generalized internal coordinates which have reference to the instantaneous positions of the nuclei. Such a set is Sf"
= [(x2 - x1)" +
sze" =
[(Xl - xaj*+
s3gW = [(LO, -
x3)2
+
02 - Ql",
(y2
-
ys)*y.
by ICY; with LY= x, y and
Denoting instantaneous displacements from equilibrium i = 1,2,3, it is readily found for XYZ that x2
-
Xl
=
(6x2
-
6~~) +
(89)
(Yl - r*>*1*,
ye1 sin@,
- 01,
y2 - y1 = (6yz - 6y1) - rs1 cos(Bs - 01, x1 - xa =
(6~~ - 6x3) +
yl - y3 =
(6~~ -
Qa)
x2 -
x3 =
(6x2 -
6x3)+
rdcosd
y2 -
y3 =
(6y2 -
Qa) -
re3.d4
+
yea sin(a, +
01,
ye2 co&k+
69,
Table VI. coefficientsof the TriatomicVibration-Rotation Hamiltonian
. Dependenceon potentialconstants: Order
CUbiC
a
2
Yes
no
YeSa
d,e Yes
X
2
Yes
Yes
yesa
YeSd
no
b.c Yes
Yes
Yes
yesb
no
Yes
Yes
no
no
caeffs A-
Q
4
P
4
6
4
Yes
puartic
Theoreticalexpressions available: xyx
XYX', mz
partiallyC
B. T. Darlingand D. M. Dennison,Phys. Rev. z, 128 (1940). b. M. Y. Cban, L. Wilardjo,and P. M. Parker,J. E(ol.SpeCtToSc.40, 473 (1971).The resultsgiven in this referenceare very complicated.AP discussedin Reference (c) below, it is possible that these resultscan eventuallybe given in much simplifiedform. C. D. A. Sumberg and P. M. Parker,J. Mol. Spectrosc.s, 459 (1973). d. W. H. Shafferand R. P. Schuman,J. Chem. Phys. 12, 504 (1944); H. H. Nielsen, Rev. md. Phys. 2, 90 (1951). e. M. Y. Chan and P. M. Parker,J. Mol. Spectrosc.42, 449 (1972).
a.
PAUL M. PARKER
362
where rs3 = a + b. These expressions specialize to XYX’ by taking f.,l=
rs2 =
Ye,
ra3 =
2a,
Pb = a,,
(91)
and, additionally, B = 0 for XYX. Substituting Eqs. (90) into Eqs. (89) and expanding in Taylor series to terms cubic in Gcrigives that SB” = s. + (R?/rer) - (S,R,2/2r,,4) + ’ * -,
s = 1, 2, 3,
(92)
where R1 = [(6x2 - 6x1) CO@@- e) - (6y1 - 6yz) sin@ - e)], Rz = C&r - 6x8) cos(cr, + 8) - (6yr - @a) sin(cy. + e)],
(93)
Ra = [(6x3 - 6x3) sine + (6~s - &ya)co&J,
and the S, are the internal coordinates introduced previously. In terms of the 6ai they are S1 = rdl + (6x3 - 6x1) sin@, - e> - (6~3 - 6yl) COS(& - e), Sa = r.2 +
(6x1 - 6x3) sin@, + e> + @yl - 6ya) COS(LY. + e),
(94)
SS = rb8+ (6x2 - 6x3) coti - (6yz - 6y3) sine. In terms of the k,
the three Eckart conditions are equivalent to 6x1 = 6Yl = -4,26x2 +
(m2bx2 + m36x3)/ml,
-(mdy2+
maSyd/m,
(95)
Adxa + A,dya + A,dya = 0,
where A=2 = r,,lrna cos& - e), A,3 = re2m3COS(CY, + e), A,2 = rs1m2sin@, - e),
.
(96)
A,s = --r,zrns sin& + e). Application of Eqs. (91) readily specializes all of the above equations to the XYX’ and XYX cases. To obtain the R, explicitly in terms of the S,, it is necessary to solve the six equations, Eqs. (94) and (95), for the six &xi as functions of the S., and substitute these expressions into Eqs. (93). For XYX, this procedure gives RI = #[(G - tam,)& - (G + t.anaJS~ + (ma,>&!, R2 = 3[- (G f tanaJS1 + (G - tam&
i- (mxJ&],
(97)
R3 = 3 (GmJma sina (SZ - Si), where G = (2mz sincr, cos(yJ/(mr + 2m2 sin*ar,).
(98)
It does not seem practical to carry out the above procedure algebraically in order to obtain the corresponding expressions for the R, of XYX’ and XYZ, which are rather complicated. If data for the equilibrium geometry of the molecule are available,
TRIATOMIC
ISOTOPIC SUBSTITUTION
363
the coeihcients of the S, in the expressions for the R, can be obtained numerically without dithculty. The mass dependence of these coefficients needs to be retained explicitly in order to arrive at the proper invariance equations. VIII. ANHARMONIC
In terms of the generalized tion is
POTENTIAL
internal
ENERGY FUNCTION
coordinates
SP,
the potential
energy
V = VzO” + Vae” + IlqBen+ . . . ,
func-
(99)
where 2V#*n = C C F,,,S,g”S,+‘” B 8‘
= C C 2F,,S,@5We”/(l B 58’
+ S,,t),
2Va=” = C C C Fss,s,,SsgenSB,genSB,,gen) * <-8’58”
(W
For XYZ, there are six coefficients F.,?, ten coefkients Fs.ll)~~,and fifteen coefficients and XYX’, the following restrictions hold because of the symmetry equivalence of SP and S!P. F l,,a~,ll,l.For XYX
Fn = F22,
F la = Fza;
and F III = F222,
I7112 = F122,
F 118= F223,
Flaa
=
Fzaa;
W)
and F 1111 = F2222,
I71112 = F1222,
F 113a =
271113 = F2223,
Fnza = F122a,
Flaaa= Fzaaa.
Fzzaa, W)
There are thus four, six, and nine independent potential constants in TIP’, VP, and Vdga, respectively, for XYX and XYX’. To obtain the potential energy function in the linearized internal coordinates S,, Eqs. (92) are introduced into Eqs. (100) and all terms quadratic, cubic, and quartic in the S, are retained. The result is v = vt+ va+ v4+ ***, (103) where 2V2 = C C 2F,,S&/(l II 5s’
which is the harmonic part of the potential above. Furthermore, it is found that
+ S,,t) = 3%S, energy function
as used in Sections
ow
III-V
364
PAUL M. PARKER
+ C C 2F,,f a 58’
(RstRs~2/resrca~)-
(S&RaY2rer2) -
In be
manner of CP Eq.
(SS,,R,~2/2r,,~2)1/(1
+ b).
(1%)
(109), the anharmonic potential constants are intro-
duced through (107) 2Va = 4M C C C C k~~~.~~glll*Q~Q~IQ~~~Q,~~~, 8
(108)
where k age*
= (X,A,A~~~)tk,~~,w,
* = (XIXII~X*)~XIrl~)*k,l~S)I.,,I, k al)r*,ror’, N = m/i@,
(109)
M = mgi,
and where ksatllt and kLIJ.JfIttt are in wavenumber units. The invariants for the sets of cubic and quartic potential constants are obtained by first transforming Eqs. (107) and (108) to the S. coordinates by means of Eq. (22), Q = @I’)-‘S
for XYX,
(110)
or by means of Eqs. (50) or (78), Q = (DW)-‘S
for XYX’
and XYZ.
(111)
The resulting expressions are then arranged in powers of S, and the coefficients of lie powers of S, are equated from Eqs. (107) and (105), and from Eqs. (108) and (106), respectively. For XYX, the conditions represented by Eqs. (101) and (102) are satisfied automatically, but for XYX’ they must be imposed as constraints. For XYZ the constraints do not hold. The procedure described is thus linally yielding a set of invariants from which the cubic potential constants of the isotopically substituted molecule can be determined. It is seen that this generally requires a knowledge of the complete set of cubic and harmonic potential constants of the reference molecule. Similarly, the determination of the quartic potential constants of the substituted molecule requires a knowledge of the quartic, cubic, and harmonic potential constants of the reference molecule. This parallels the general conclusions reached for XYX by previous investigators (17, 19). IX. CUBIC POTENTIAL
CONSTANTS OF XYX
The configuration most feasible to pursue algebraically is that of XYX. As a specific illustration of the general procedure and as the case of most immediate interest, the set of six cubic potential constants of XYX will be discussed further in this section.
36.5
TRIATOMIC ISOTOPIC SUBSTITUTION
Eq. (105) is
For XYX, 2vs = FdS?
+ S29 + FlllslS2(Sl+
S2) + FnaSa(S12 + S23
-kFl23SlS2Sa -k Flaa&+(S~ -b S2) d- F3arSa3 + (2F11hJ
(SlR? + S2R2*) +
+ (F33/@%&*
(2F12/yJ (S&2* + S2R19
+ 2FlaI: (Ra2/2a) (SI + S2) +
(Sa/y,) (RI* + R2*)],
where RI, Rz, and R3 are given by Eqs. (97). Introducing these explicitly lecting like powers of S,, the resulting expression assumes the general form 2Va = (F111+ flll)(S?
+ s2a) +
(F123 + /123)SS2S3
+ (F133 + f1a3)Sa2(S1 + S2) +
FM + F112,
and col-
(Fll2 + j112)&S*(.% + S2)
+ (F118 + _fl13)S3(S12 + S2*) +
where the f..$.l~ six invariants is responding ones Fin, Fllz, F113, advantageous to
(112)
(Faaa + f333)Sa*,
(113)
can be worked out in a straightforward manner. A complete set of obtained by equating the coefficients (F1ll + fin), etc., to the corof Eq. (107) and solving the resulting equations for the invariants F123, FWJ, and Fw. Carrying out the lengthy details, one finds it form the following equivalent set of six invariants. F 193,
F123 + 2Fll3,
F 333, 3F111 -
F112,
F 123 - 2Fll3.
(114)
It is found that in the expressions for the first four of the above equivalent invariants, terms involving FII, F12, F13, and Fa3 appear only with mass-independent coefficients and can therefore be dropped. After some further development, the following four invariants involving K111*,Kz~z*,&z*, and &ZZ* are found. $[saknl*
+ c%zz* + sac&z* + sc*klzz*],
pmz*[3s*cknl* - 3sc*k222*+ 428
- s2)kn2* - ~(2s~ - c2)klzz*],
(115)
/.dmmz[3sc*kn~* + 3s2ck22z*- c(2s2 - 8)knz* - s(2c2 - s*)kwz*], m$[c3klll*
- s3k222*- sc2k112*+ s2ck122*],
where, as before, s = siny and c = cosy. The remaining associated with KISS*and k233*. These are 4NccZ(m2/2)~~3-1(-ck13a* + skzaa*)+ (3f111 - f112) sinsoh 2Np%B-W133*
+ ck233*) -
two invariants
(fl23 -
are those
2fila) sin*%,
(Ila)
(3fi11 - fl12) sin2cybcos+,
where (3flll -fll2)
= (4/y3(3G2(F~1
+ Fl2) -
G(Fll
-
F12) tana, + C(Gml/2m2)2 csc3~,y713),
(jl23 -
2f113) = (4/r@) (.- (G seccu3 (Fll -
F12) -
G3F13 -
(117)
C(Gml/2m2)2 csch,Y;‘88},
and where FII, F12, FM, and Faa are those given by Eqs. (28). It is thus demonstrated that the cubic potential constants of the substituted molecule can be calculated from the cubic harmonic potential constants of the reference molecule.
366
PAUL M. PARKER
A particularly simple case is represented by the comparison of the ozone molecules I803 and 18O3 for which my formulation applies with ml = m2. With this condition, G in Eqs. (97) becomes mass-independent, and as a consequence, all f,8~8~~ become linear combinations of the harmonic invariants F,,l with mass-independent coefficients. They can therefore be deleted completely from the expressions involving the F,,I,I~. Furthermore, p and ~3 become proportional to m2. The angle y becomes invariant. As a consequence, the set of cubic invariants can be rearranged to the equivalent set of six invariants { mlfksstsll}. Similarly, one finds for the quartic potential constants the set of nine invariants {m&sa~s~~r~~~}. For the harmonic part of the problem one has the invariant sets (m2L3) and {m&). These findings have already been published by Barbe, Secroun, and Jouve (20). These authors also found very good agreement between the observed and calculated values of the quartic potential constants of ozone which would seem to indicate that the general approach described in the present study may prove to be useful. ACKNOWLEDGMENTS I thank Dr. G. Strey for the useful comments he made to me at the Third InternationalSeminaron High Resolution Infrared Spectroscopy at Liblice, Czechoslovakia,where part of this work was first reported. I thank Miss Iris Gomez for her able assistancein preparingthe manuscript. RECEIVED: May 1.5,1975 REFERENCES 1. G. HEPZBERG, “Infrared and Raman Spectraof Polyatomic Molecules,” pp. 227-238, Van Nostrand, New York, 1945. 2. G. AMAT,H. H. NIELSEN,ANDG. TARRAGO,“Rotation-Vibration of Polyatomic Molecules,” Dekker, New York, 1971. 3. W. E. SMITH,Azcstral.J. Phys. 12, 109 (1959). 4. V. T. ALEKSANYAN, A. P. ALEKSANDROV, ANDM. R. ALIEV,Opt. Spectrosc.26,292 (1969). 5. G. STEENBECKELIERS, Ann. Sot. Sci. (Bruxelles)85, 163 (1971). 6. A. P. ALEKSANDROV, M. R. ALIEV,ANDV. T. ALEKSANYAN, Opt. Spectrosc.29, 568 (1970). 7. A. P. ALEKSANDROV ANDM. R. ALIEV,J. Mol. Spectrosc. 47, 1 (1973). 8. G. HEIUBERG, “Infrared and Raman Spectra of Polyatomic Molecules,” p. 143, Van Nostrand, New York, 1945. 9. K. T. CHUNG ANDP. M. PARKER, J. Chem.Phys. 43,3869 (1965). 10. J. PLfv.4,Coil. Czech. Chew. Comm. 23, 1839 (1958). 11. J. BELLET,W. J. LAFFERTY, ANDG. STEENBECKELIERS, J. Mol. Spectrosc. 4’1,388 (1973). 12. M. Y. CHANANDP. M. PARKER, J. Mol. Spectrosc. 42, 53 (1972). 13. W. H. SHAFFER ANDR. P. SCEUMAN, J. Chem. Phys. 12, 504 (1944). 14. D. W. POSENER, Ph.D. thesis, MassachusettsInstitute of Technology, 1953. 1.5. T. OKAANDY. MORXNO, J. Mol. Spectrosc. 6, 472 (1961). 16. J. M. DOWLING, J. Mol. Spectrosc. 6, 550 (1961). 17. K. KUCBITSU ANDY. MORINO, Bull. Chm. Sot. Jap. 38,814 (1965). 18. A. BARBE,C. SECROUN, ANDP. JOWE, J. Plays. 33,209 (1972). 19. K. KUIZHITSU ANDL. S. BARTELL, J. Chew Phys. 36,246O (1962). 20. A. BARBE,C. SECROUN, AM) P. JOUVE,J. Mol. Spectrosc. 49, 171 (1974).
OF MOLECULAR
SPECTROSCOPY
58,
344-366 (1975)
Vibration-Rotation Theory and Isotopic Substitution in Nonlinear Triatomic Molecules PAUL M. PARKER Department of Physics,
Michigan
St&e University, East Lansing,
Michigan
48824
In the interpretation of high-resolution data it is helpful to be able to relate the experimentally determined molecular constants, such as rotational constants, centrifugal distortion constants, and vibration-rotation interaction constants of a given molecule to those of its isotopically substituted variants. Results concerning the symmetric nonlinear trlatomic molecule (XYX) have been given through the second order of approximation by previous workers. In this paper, extensions of the work to the fourth order of approximation and to the nonsymmetric nonlinear triatomic molecule (XYZ) are considered. Theseextensions are found to increase the algebraic complexity of the problem considerably, and results can generally be given in implicit form only. The approach used is one that emphasizes the isotopic invariants of the problem, i.e., those expressions which remain unchanged when applied to the various isotopic modifications of a given molecule under the assumption that the molecular force field and the molecular geometry remain unchanged under isotopic substitution. I. INTRODUCTION
The problem considered in this article is the following one. Given the molecular constants of a certain nonlinear triatomic molecule, how does one calculate corresponding molecular constants of an isotopically substituted variant of this molecule without the approximation that relative changes of mass produced by the substitution are small. Except in a few favorable instances, the answers that emerge are unfortunately not very straightforward. Nevertheless, the problem is of considerable interest in the analysis of high-resolution data since its consideration can form the basis for more accurate, more consistent, and more complete determinations of molecular constants. The basic assumption made in isotopic substitution studies is that the substitution does not produce a nuclear size effect on the molecular force field, at least to a certain order of approximation. As a consequence, the molecular equilibrium geometry remains unchanged and the potential energy function is the same for all isotopic variants of a given molecule (I). When the potential energy is expressed in an appropriate set of mass-independent internal coordinates, the coefficients of the resulting function are invariant under isotopic substitution. For the purposes of this presentation I find it useful to distinguish, somewhat arbitrarily, between molecular constants that appear in the vibration-rotation Hamiltonian (“II-constants”) and the more fundamental constants (“F-constants”) that appear in vibration-rotation energy expansions. Among the R-constants, one 344 Copyright
@
1975 by Academic
All rights of reproduction
Press. Inc.
in any form reserved.
TRIATOMIC
ISOTOPIC SUBSTITUTION
345
has: 1=, I#, and I,, the equilibrium principal moments of inertia, of order of approximation zero. R-constants of order two are: the centrifugal distortion constants T, the vibration-rotation interaction constants (r, and the anharmonic vibrational energy constants x. In order four, one has the centrifugal distortion constants @, the vibrational corrections to the second-order centrifugal distortion constants p, and vibration-rotation interaction constants P. The F-constants include the atomic masses, the equilibrium geometry parameters, and the harmonic potential constants K,,,. Beyond order zero, one includes appropriate sets of cubic, quartic, etc., potential constants. In lieu of the complete set of harmonic potential constants (KBB,},one may use the set of normal frequencies {X,1}. However, as is well known, the set (x,4] has fewer members than the set (L). The former set must therefore be augmented by further, independent F-constants chosen from the set of coefficients (tisat) of a transformation to normal coordinates and/or from the set of Coriolis coupling coefficients ({./). The augmentation of (x,t) to obtain a set equivalent to (K,,p) is not unique. A basic feature of vibration-rotation energy theories is that, on the basis of a particular set of assumptions, they give the H-constants in terms of the F-constants; symbolically, LI = H(F). Experimental analysis yields numerical values for a-constants. Subsequently, one attempts to obtain numerical values for the F-constants; i.e., one seeks F = F(H). It may have been recognized that with the particular type of H-constants listed above I have adopted the vibration-rotation theory of Amat, Nielsen, and Goldsmith (Z), which is based on the Darling-Dennison Hamiltonian. II. GENERAL
REMARKS
Isotopic substitution relationships of the most immediate and direct applicability in data analysis are of the type which gives an B-constant of the substituted molecule Rs in terms of the corresponding constant of the reference molecule a,
where f(m) is a definite function of the masses and possibly the equilibrium geometry of the reference molecule and_f(&) is the same function for the substituted molecule. As an example, for the XYX molecule it has been found (3, 4) that the centrifugal distortion constant 7Z2Z2 (molecule in xy-plane) obeys the relation s = ~ZZZZ
G2/w>*1Tzzzz,
(2)
where M is the reduced mass of the molecule, ~1 = 2mlm2/(ml+
2m2),
with ml the mass of the apex atom. The content of Eq. (2) can also be stated concisely as fi2rZZ2= invariant.
(3) more (4)
The invariance refers, of course, to isotopic substitution. Unfortunately, statements of this type can be given only in a few instances, specifically, for the equilibrium principal moments (or for the equilibrium rotational constants) of XYX and for four of the seven nonzero 7’s of XYX. For the remaining 7’s one has to settle for
346 more general statements
PAUL M. PARKER
of the type C jj(m)Hi
invariant.
For example, it has been shown by Steenbeckeliers
(5)
(5) that
(p/m2>(g2rzEOZ - p2~zzzz)invariant,
(6)
g = I.c(rnl+ 2m2 sin201,)/ml cos2~.,
(7)
where
and where 2a, is the equilibrium apex angle. In addition to this general approach, some authors (6, 7) have pursued the development of sum and product rules for the 7%. Some of these rules are applicable to the XYZ as well as to the XYX type of molecule. If direct relationships between H-constants cannot be established in suflicient number to effect a complete comparison of reference molecule and substitute, then it becomes necessary to work directly with the F-constants, i.e., to develop statements of the type C ji(m)Fi invariant. (8) , Such statements can be developed for the potential constants, but only in isolated instances is it possible to give such a statement for an individual potential constant, as can be done for each of the four harmonic potential constants of XYX. After establishing how the F-constants change under isotopic substitution, vibrationrotation theory can be used to determine how the corresponding H-constants are affected. Of course, this can be done only to the extent that theoretical expressions for the H-constants in terms of the F-constants are available. Schematically, the computational procedure in favorable instances is thus H--, while in the less favorable
HZ,
but much more common instances H--,F+FS--,HS.
(9) one must proceed (10)
Since in the second of the above procedures the step H ---)F implies that an appropriately complete theoretical analysis of the molecule is possible, it is reasonable to expect that in many instances a similar analysis HS -+ Fs is equally possible for the variant. In this case the indicated procedure would be H-+Ft,FStHS;
(11)
i.e., the comparison is made on the level of the F-constants. Two general strategies may be pursued. One may accept the basic assumptions as valid and the predicted isotopic correspondences are programmed into the analysis for improved consistency and accuracy. Or, one may analyze isotopically related molecules independently and then ascertain how closely the isotopic correspondences are obeyed to a given order of approximation. This would yield a measure for the validity of the assumptions and/or the quality of the data fit.
TRIATOMIC
347
ISOTOPIC SUBSTITUTION
The following three molecular configurations will be distinguished: (a) the symmetric (C,,) molecule XYX, (b) the unsymmetrically substituted symmetric molecule XYX’, where X and X’ are distinct isotopes of a given element, and (c) the general nonsymmetric (C,) molecule XYZ. Meaningful comparisons result when the reference molecule and the isotopic variant: are both Case (a); are both Case (b); are both Case (c); or one is Case (a) and the other is Case (b). I will first discuss the potential energy function for the above three cases in the harmonic approximation. I then proceed to a consideration of the second-order centrifugal distortion constants 7. Finally, a discussion of the molecular force field beyond the harmonic approximation is given. III. THE XYX
Let the equilibrium geometry it is seen that a = f’s sina6,
MOLECULE
of the molecule c = r’s COW.,
For internal coordinates I choose to adopt the projections of the instantaneous length directions, as described by Herzberg (a), viz, sr = (x2 - xl) S2 =
be as shown in Fig. 1, from which
(Xl -
a2 + c2 = r,=.
(12)
central force coordinates representing of the bonds along their equilibrium
sina
+
(yl
-
y2) cosa,,
x3) sinh
+
(yl
-
yd
cow,,
03)
ss = (x2 - xa). The Cartesian coordinates are the instantaneous position coordinates of the three nuclei in the principal axes system of the equilibrium inertia tensor. In his study of the centrifugal distortion constants 7, Steenbeckeliers (5) adopted valence bond coordinates, but his results are easily duplicated with the above central force coordinates, whose use proves more convenient later on for the formulation of the anharmonic potential energy function. The Eckart conditions are expressed by
FIG. 1. Equilibrium
PM1
+
m2O2 +
x2) =
0,
my1
+
mz(y2 +
ya> =
0,
(x2 +
x2) =
(ys -
configuration
(14) y2) tm%?,
and principal axes (CC,y) of the nonlinear XYX
molecule.
348 and mass-adjusted
PAUL M. PARKER
symmetry
coordinates
are introduced
as follows (9).
24 = (+mz)t(xz - XP), v = ,u+Cyr - t
(rz + Y311,
(15)
7J.J = I.dcx1 - 3(x2 + x3)1, where cc was given in Eq. (3) and ~8 = r (ml + 2~8~sin2ffB)/(m1 + 2mz) sink,. For the quadratic
part of the potential
(16)
energy V, one has
2V = SF.9 = i!?KU = C&Q,
(17)
where the tilde denotes the transpose, s = (S1, sz, S,), i.7= (24,o,w), 0 =
and the Q’s are the normal coordinates. eral, the symmetric matrices
(18)
(QI, Q2,Qd, For the coefficient matrices
one has, in gen-
and A is the diagonal matrix with nonzero elements X1, X2, and X3 which are the squares of the normal frequencies (~1, ~2, and ~3, respectively. The Czr symmetry of the molecular force field is expressed by requiring that Fz2 = FII
and
F23 = F13,
(20)
or that k13 = &?a = 0,
(21)
with Eqs. (20) implying Eqs. (21) and vice versa. The elements of the matrix F are assumed to be invariant under an isotopic substitution that preserves the Czo (XYX) symmetry. It has been shown by Pliva (10) that setting up a potential energy function whose coefficients are invariant under isotopic substitution requires, in general, the use of a set of mass-independent internal coordinates detined along the directions of the straight lines joining the instantaneous positions of the nuclei, not their equilibrium positions. However, in the harmonic approximation (only), the latter type of internal coordinate is adequate. Hence, for the study of the quadratic part of the potential energy function, the coordinates S, defined by Eqs. (13) are satisfactory. The transformations between S, U, and Q are given by S=TU, where I? is the orthogonal
matrix
U = I’Q,
S =
(TOQ,
(22)
(9) (23)
TRIATOMIC
Table I. Matrix Elements of the Coordinate Transfomtions Eqs. (22) for the XIX Molecule Elements of Tr --
51
349
ISOTOPIC SUBSTITUTION
= t21
Defined by
= (sinae)/J2m2
t12 = t22 = CCOSU‘~,/J; t13 = - t23 = - CvG;sinae)/u t31 = h/mz t t 32 = 33 = ' Elemmts Of T
-1
:
t;;
= tit = t;;
= 0
t;: = $77 t-l = t;; =
Ji
J;/2cosae
= _ t;; = _ u/2+inae
-1 t23 = - (4Sanae)/2
Elements of (Tr)-': (Tr);i = (Tr);: = (&inY)/Zcosa
e
(Tr);; = (Tr);; = ("&0SY)/2COSCr e (Tr);: = - (Tr);: = - 1.1/26$na
e
(TI?;: = [+($7?JcosY
- (v$7sinYtanae)/21
(Tr);: = [-(v$ZZsi*Y
- (&osYtanae)/21
(TT);; = 0
Using Eqs. (13)-(15), one can work out are listed in Table I. For the three I’-’ = T, T-l, and (TI’)-1 = i?‘, with also listed in Table I. Writing all coordinates of Eq. (17) in
the elements of the matrix T whose elements respective inverse transformations, one has the elements of the last two transformations terms of S, one obtains
2V = .$FS = s(F-‘KT-‘)S
= ~[(f~)-lA(Tr)-l-JS,
(24)
which allows the making of the three comparisons F = F-~KT-~,
(25)
F = (f~)-lA(Tr)-l,
(26)
K =
f-lAr-l,
or
K-l =
r-T.
Introducing or
the Cz. symmetry of the harmonic force field by requiring (21) hold, one obtains from Eq. (25) the set of invariants
(27) that Eqs. (20)
FII = F22 = (IA/~ cos‘4tc)k2z+ (I.L’/~P~sin2ae)kaa, FS9 = (m2/2)kll + b/4
cot2+22
- C(~ZP/~)* tancuJkl2,
Fa = (p/4 cos2a$k22 - (JA~/~PSsin2aJkss, Fla = F28 = - (~14 cos(yb cota,)krz + [(w~zc(/2)~/2 cosa.Ikn.
(28)
350
PAUL M. PARKER
It will be recognized that an invariant expression remains invariant when multiplied by an arbitrary factor that is not explicitly or implicitly mass-dependent (or, more precisely, by a factor that is independent of the mass(es) isotopically replaced), and that a linear combination of invariants is also an invariant as long as the linear combination constants are mass-independent. Using these observations in connection with Eqs. (28), it is possible to deduce the equivalent set of four invariants: m&rl,
r2&kw
&m&tk12,
&zz,
(29)
Equation (26) yields invariances equivalent to Eqs. (28) expressed in terms of the three normal frequencies and the normal-coordinates transformation angle y in place of the four independent nonzero k,,t. Equations (27) relate the normal frequencies and the angle y to the harmonic potential constants. The second form, viz, K-l = I’-lf is found to be very useful as it yields expressions that appear directly in the r’s, and it was this observation together with the simple form of the k-invariantes, Eqs. (29), that allowed Steenbeckeliers (5) to obtain the T-invariances listed in his Table III. The indicated isotopic variations of the r’s have been observed to be well satisfied in a number of specific instances (6, 11). Knowing the invariants, Eq. (29), means that given the potential constants k,,t of the reference molecule, one can immediately obtain the potential constants k,,ts of the substituted molecule, and knowing these, one can obtain further molecular parameters of the substituted molecule insofar as they depend only on the harmonic potential constants. Thus for the squares of the normal frequencies one has (9) X1.2
=
4&l
kzz) f
+
+[(kn
-
kzzY + 4ku2]+,
XB = ka,
and hence, with the aid of Eq. (29), and with fi2 = m.JrnP, Xr,z! = W&r
+ L&z) f
+[(&ku
-
j&d2
(30)
etc., + 4,ci&kn2]+,
(31)
Xss = j22j?,s-1kxs. The Teller-Redlich product rules (I) are implicit plicit dependence of s = sin?, c = cosy on the known (P), one can similarly write
ss= &(l/V2)(1 f.78= +p
-
-
in these relations. harmonic potential
Since the exconstants is
(ritzku - fikzz)/[ (G’zzkn - j.ik2J2 + 4~zPzkZJ~)
*,
(32)
(ss)“]t
It may be verified trigonometrically tan%
that = 2klz/(kll
- Rzz),
and hence the angle ys can also be specified through tan2-P
= 2(%$)*k12/(&kl~
- fikz2).
It appears advantageous to take the range of y as -3~ < y < +&r which then necessarily encompasses the entire range of tan2y but could require negative values of siny. According to Steenbeckeliers (5), the isotopic invariants of the equilibrium principal moments of inertia are 1=/P,
Idm2,
Id&
(33)
TRIATOMIC
ISOTOPIC
(9) of the substituted
These can be used to express the Coriolis constants r13x =
@Iz/j.IL)V
s 123 =
-
-
(gI,/pI,)t~
351
SUBSTITUTION
molecule as
(&/?%L)b",
(34)
- (&//?&I*)wJ.
The centrifugal distortion constants r which depend on the quadratic, but no higher order, potential constants will be discussed in Section VI. It should be appreciated that all relations between constants of the original and the substituted molecule are, in a sense, rigorous since one and the same theoretical formulation must be expected to be equally applicable to both isotopic modifications of a given molecule. IV. THE XYX’ MOLECULE
The equilibrium configuration of the XYX’ molecule, with X and X’ distinct isotopes of the same element, is shown in Fig. 2. Equations (12) hold as given previously, since the geometry of XYX’ is assumed to be the same as that of XYX. Discussion of this configuration is based on the formulation presented by Chan and Parker (IL’), henceforth referred to as CP. This publication uses results by a number of previous workers, most notably Shaffer and Schuman (13) and Posener (14). Taking b = a throughout specializes CP to the XYX’ case. The principal axes (x, r) of the equilibrium inertia tensor are rotated with respect to the axes (5, g) through an angle 0 given by (13) tan28 = 2ml(m3 - m~)ac/[4m3m3a2 + ml(m3 + m3) (a” - c”)] and the principal
equilibrium
moments
(35)
are
I, = [4m3m3a2 + ml(m3 + m3b,21/M =
r:[4rnsrn3
sir&, + rnl(rnz -I- m3>3/M,
(36)
with M = ml-j- mz-l- ma.
(37)
Furthermore, 1, = $(I, - I’), (38)
I, = +(I* + I’), with I’ = f { [2ml(m3
-
m3)ac)z +
[4m2m3a2 +
ml(m3 +
m3) (a2
-
c2)121 &lM,
(39)
J
Y
ml
8 ae ;
x
\I r.
Y jC
II!!& ma
(1.
a
m2
FIG. 2. Equilibrium configuration and principal axes (x, y) of the nonlinear XYX’ molecule; X and X’ are isotopes.
PAUL M. PARKER
352
where the choice of sign is taken depending on whether I, > I# or I, < I,, for the choice of axes made in Fig. 2 with -&r < 13< &r. For the internal coordinates again to represent the instantaneous lengths of the three sides of the configuration triangle for small amplitudes of vibration, they must now be defined originally in terms of the (2, g) coordinate system as S1 = (zz - Z-1)sina. + (?jl - Qz) cos+ SZ = (2, - 53) sinol, + (jj1 - 53) cos+ &=
(40)
(L&-523).
In terms of the (x, y) coordinates
these become
s1 = (x2 - xl) sin@. - 0) +
(y1 -
y2) COdaa
&
(yl -
y8) cos(~~+
=
(xl -
X8) sin(&+
S8 =
(x2 -
x8)cod
6) -k
-
(y2 -
y8)
-
e>,
(41)
e),
sine.
The associated F-matrix preserves the symmetry of the XYX force field, FZZ = F11 and Fz8 = Fra, and hence there result four invariants for XYX’, the same number as for XYX. The potential constants K,, are defined through CP Eq. (60), i.e., 2v = DKU = knu2 + kzzv2+ k88ti + 2knuv + 2k18Uw+ 2kz8vW, where u = (2c, v, w) represents
the intermediate
(42)
coordinates
u = x2 - x8, 0=
y1 -
(m2yZ-k
m8y8)/&2+
m8),
w = Xl -
(m%%f
m8%8)/(m2+
ma),
(43)
defined in the principal axes system, and K is the symmetric matrix of the potential constants, Eq. (19). It appears convenient to define an auxiliary set of potential constants &,I in terms of intermediate coordinate system as
8 = fh tB=&-
-
coordinates
6 = (ZZ,ii, 2~) defined in the (2, g)
(m292-k
m888)/(m2+
ma),
(m252-t
m8%8)/(m2+
ma),
(W
and 2v = i?gu
=
ih2+
ii22ij2f
&88@+
2&2%+
2kl8a?.&+-
2%2&a
(45)
Writing O=MJ,
(46)
one may determine the transformation matrix 8 and its inverse as follows. The transformation equations between (x, y) and (3, 9) are xi = Zi cos8 + gi sine, yi = --Zi sine + V&co&,
(47)
TRIATOMIC
for the ith atom. Introducing
353
ISOTOPIC SUBSTITUTION
these into Eqs. (43) gives, with the aid of Eqs.
(44),
that u = ti cos49+ (g2 - 83) sine, v = B cos8 - Zz,sin@,
(4s)
w = ti co& + ii sine. Since the Eckart conditions, CP Eqs. (29)-(31), have the same form in the barred and the unbarred Cartesian coordinates, CP Eqs. (35)-(40) hold in the barred as well as the unbarred Cartesian and intermediate coordinates. Therefore, CP Eqs. (39) and (40) give the expressions for 92 and 53 which can be used in Eqs. (48) above and thus, the matrix elements of 8-l are determined, and hence also those of 0. They are listed in Table II. Of course, for m2 = ma, one has that 0 = 0, and 8 and 8-’ are equal to the unit matrix of order three. In terms of 8, the relationship between the potential constants K,,, and &8~ is then given through K = 6Re, (49) R = &‘K0-‘. For the various coordinate
transformations,
S=Du,
one has U = I’Q.
lT=OU,
(50)
The transformation matrix D with elements &.I and its inverse L)-l with elements &-l = casl are given in Table III. The matrix elements of D are determined by introducing CP Eqs. (35)-(40) in barred coordinates and with 0 = 0 into Eqs. (40). The matrix elements of D-* follow by the standard procedure for constructing the inverse. The elements of the matrix I’, denoted by TU, are those of the transformation from intermediate to normal coordinates, s = 1, 2, 3,
u = C m,Q., v = C m2eQq
s = 1, 2, 3,
w = C nasQs,
s = 1, 2, 3,
(51)
and cannot be given in closed form for the XYX‘ and XYZ configurations be determined by the diagonalization procedure. For the potential energy function, one now has 2V = SFS = 6~0
but must
= OKLJ = @IQ
(52)
and 2V = .i?FS = ?@i?YES. Therefore,
the invariants,
expressed in terms of the &,I, are obtainable F = ij%D-‘,
(53) through (54)
with the i,,, related to the k,., by Eqs. (49). The relationship between the k,,l and the normal frequencies also follows from Eqs. (52) with the aid of Eqs. (SO) and it is R = f%W-1,
K-1 = l?A-lf.
(55)
354
PAUL M. PARKER T!ableII.
Matrix Elements of the Transformation defined by Eq. (46) for the XYX' and XYZ Molecules
+aents
of 0:
= 1/case
%l
e e 0 21 = 31 = 0 = case 22 = e33 e23 = - 832 = sine %2 = (wane/m2m3)912 e13
= ()itane/m2m3)613
P = m,(m2+m3)/M
+12
For XYX':
$13 For XYZ:
= [(m3-m2)c0se+ (m2+m3)cotaesine]/2 = [ (m,-m,)
sine - (UI~+III~)CCA~~C~S~)/Z
$12 = t(m3a-m2b)cose+ c(m2+m31sine]/(a+b) $13
-1 Elements of 0 :
= I(m3a-m2b)sin3- c(m2+m3)cosel/(a+b)
-1 -1 -1 ell = ez2 = e33 = c06e e-l = e-I 21 31 =.O e-l= _ e-l= - sine 23 32 e-l = - (wine/m2m3)$;: 12 e-l = + (winB/m2m3)$;: 13 u = ml(m2+m3)/M -1 412 -1 413 -1 $12 -1 $13
For mx':
For Xx&:
Writing out eqs. (54) explicitly and ~13 = 1 gives
= (m3-m2)/2 = r(m2cm3)cotaJ/2 = $a-m2b)/(a+b) = c(m2+m3)/(a+b)
and recognizing
from Table
III
that
cl1 = cl2 = 0
F11 =
C212L22+
2CzlC3lE23 +
C312E33,
(56)
2722 =
C222i22+
k22G32k23 +
c322k33,
(57)
F33 =
kl1-k
F12 =
62lc22&22f
F13 =
c21&2
+
c23f22 +
c33223) +
c31@13 +
c23a23 +
c33J533),
(60)
F23 =
c22(&12 +
c23&22 +
c33&23) +
c32(fl3 +
c23&23 +
c33&33).
(61)
c232h22+
C332G33+
(c2lc32+
2C23&2 +
c22631)k23+
2c33&3 f
2c23C33~23,
calcazkrr,
(58)
(59)
The coefficients c,8~ are given explicitly in Table III. The C2# symmetry of the molecular force field requires that F22 = F11 and F23 = F13, which leads to two con-
TRIATOMIC
355
ISOTOPIC SUBSTITUTION
Table III. Matrix Elements of the Coordinate Transformation D Defined by Eq. (50) for the XIX' Molecule
Elements Of D:
dll = m3a/(m2+m3Pe d21 = m2.V(m2+m3)re d
d3l = 1
32 = d33 - O
d12 = !Jc@lll+2m2)/2mlm*r, d22
= ~c(mlt2m3)/2mlm3',
d13 - - (a/r,)[l+(uc2/2m2a2~l d23
= + (a/r,)[l+(!Jc2/2m3A
IJ= ml(m2+m3)/'M -1 Elements Of D :
Cl1 = Cl2 = 0 =21 c22
=31 =
"33
1
=
= v'r (2m3a2+w21/cm31z r 2 2 = p're(2m2a +vc )/crn21z
'23 =
'32
53
- 21Pa(Ma2+m1c2)/cMIz - m2(ml+2m3kxe/MIZ
= + m3(ml+2m2)ar,/MIZ = 2va2(m3-m2)/~TZ
p' = m2m3/(m2+m3) Iz is given by Eq. (36)
straints
among five of the six harmonic (C212 (~21 -
C222)522+
c22)(&2 +
629522 +
potential
2(C2101 csafzr) +
constants,
C22caz)kza+ (~31 -
(cn2 -
c32)(&3 +
viz, Ca22)kaa = c2&2a +
0,
c&a)
(62) =
0. (63)
As a result of these constraints, there are four linearly independent invariants. The above equations for FII, &a, Fu, and FM, subject to Eqs. (62) and (63), constitute such a set of invariants. To effect a comparison in the manner of Eq. (10) between two isotopically related molecules, both of type XIX’, one assumes as known the set of six potential constants of one of the two molecules. This set presumably satisfies the constraints, Eqs. (62) and (63). The set of six potential constants of the substituted molecule is determined through the set of six equations: Eqs. (62) and (63), and the invariance equations resulting from Eqs. (56), (58), (59), and (60). The new set of potential constants (KS) is used in the secular determinant equation, CP Eq. (64), IU/.@) -
(=?I
to give the Xas and n,.a S of the substituted are those cited in CP Eqs. (53)-(59). [There read a.] With the Ls and n..~ s determined,
= 0,
(64)
molecule. The matrix elements of bs) is an erratum in CP Eq. (59): a2 should one can calculate the Coriolis constants
356
PAUL. M. PARKER
of the substituted
cc = /&" =
molecule,
m(m2+
CP Eq. (SS), as
m)/M,
~mlm2cos(&,
@)+
-
II, M = [mrm2 sin&
mlm8cos(&+
6)]/21ucod
- 0) - mrm8 &(a.
+ 0)]/2M
sin%,
(66)
cos8 sink..
Knowledge of the harmonic potential constants of the substituted molecule also allows determination of the centrifugal distortion constants G, as discussed in Section VI. When the comparison is made between a molecule of type XYX and one of type XYX’, it is important to note that the conventional definition of the intermediate coordinates of XYX, Eqs. (15), is inconsistent with the corresponding definition for XYX’, Eqs. (43). Hence, since the harmonic potential constants are defined through 2V = DKU, there results an inconsistency in the definition of the harmonic potential constants. This discrepancy must be taken into account whenever expressions applying to XYX’ are specialized to XYX according to the replacement scheme kll(XYX’) + 3m2hl(XYW, k22(Xyx')
-+Pk22(XJm,
k,,(XYX')
-+Fcrkaa(XYX),
k12(XYX’)
+
k,,(XYX’)
--) 0,
kza(XYX’)
3
(4pm2)%2(XYX),
0.
Also, specializing to XYX, 0 = 0, 8 = 0-i = 1, &,.f = K,,p, and the elements matrix D-l listed in Table III become Cl1
=
Cl2 =
Gas =
c21 = c22 = l/(2 c2a = -3
(67)
0,
Cl8= 1,
COSCYJ,
tancu,,
c31 = -632 = -p/2pa
of the
(68)
s&,
where p and ~3 are given by Eqs. (3) and (16), respectively. With these observations in mind, it will be seen that for XYX the constraints, Eqs. (62) and (63), reduce to the requirement that kla = k23 = 0, which was anticipated and used in Section III, and Eqs. (56)-(61) reduce to Eqs. (28). Equating the four invariances, Eqs. (56), (58), (59), and (60), for XYX’ relates, then, the four potential constants of XYX to the six potential constants of XYX’. Since the six potential constants of XYX’ are constrained by the two conditions, Eqs. (62) and (63), the full set of harmonic potential constants of XYX’ is determinable from that of XYX, and vice versa.
357
TRTATOMIC ISOTOPIC SUBSTITUTION
It is possible to solve the four invariance XYX with the following result. KII = (2/m)[Fs3
equations
for the potential
constants
of
+ 2F 18sinad + +(FII + Fr2) sirk],
k22
=
(2/P)(Fn+
F12)
ha
=
(~&&(FII
- FH) sin%,,
cos2cY.,,
KIZ = C(coscr,)/(accm2)l][2F13
(6%
+ (Fu + Fu) sinru,].
The expressions FII, F12, Fu, and Faa are those for the isotopically related XYX’. This set of equations will be found useful for determining the potential constants of XYX from those of XYX’. V. THE XYZ MOLECULE
The equilibrium configuration of the XYZ molecule is shown in Fig. 3 and represents the most general triatomic configuration. The discussion given in CP applies without simplifying features. Geometrically, a = rs2 sine+
b = ?,I sin&
G = 182co&Y*= r’s1co@,, r,r2 = b2 + c29 The principal
axes system
(70)
r,z2 = a2 + c2.
(x, r) is reached from (5, 8) by a rotation
B given through
tan28 = T/Q,
(71)
with r = 2mrc(msa - mzb), Q = ma(mnl and the equilibrium
+
m&r2 + m2(mt + ma)b2 - mr(m2 + ms)8 -I- 2mma4
principal
moments
of inertia
(72)
are given by
I, = 3(1* - I’), IV = HIS + I’),
(73)
I, = I, + Iti = [Q + 2mr(m2 + m8)c’l/M, where I’ = f(r*;+
@y//M.
m3
FIG. 3. Equilibrium configuration and principal axes (x, y) of the nonlinear XYZ molecule.
(74)
358
PAUL M. PARKER
For the internal
coordinates
one now has
Sl = (2, -
21)sinBe+ (81- g2>co&
s2 = (31 - 5%)sinar, + (gr - jj3) s3 =
(22 -
Sl =
(x2 -
XI) sin@,
SZ =
(a
x3) sinbe+
(75)
cosa3,
23),
or, in terms of (x, r),
-
-
Sa = (x2 - x8) coSe For the potential
energy function,
0) +
(YI -
~2) coscBO -
O),
0) +
(~1 -
~3) cc&+
e),
(76)
(ye - ya) sine.
once again,
2V = SFS =
i?RO =
DKU = @iQ,
(77)
with the intermediate coordinates U and 0 as defined previously in Eqs. (43) and (44), respectively. The matrices F, K, and R now have no restrictions on their elements other than that they are real and symmetric. The harmonic potential function therefore is specified by six independent constants. For the several coordinate transformations involved, again, S=De,
U = l?Q.
O=WJ,
(78)
The procedure for determining D and D-l is similar to that described for XYX’ and leads to the elements listed in Table IV. The procedure for determining 0 and 8-l is also similar to that described for XYX’, and the results are included in Table II. The general form of I’ is specified by Eqs. (51). Its elements cannot be given analytically in closed form. The invariants of the harmonic potential energy function are given by the six Eqs. (56)-(61) in terms of the &,.p. The &,, are related to the Rsrt by Eqs. (49). Of course, the coefficients clsg in Eqs. (56)-(61) are now those for XYZ found in Table IV. The constraints, Eqs. (62) and (63), do not apply and the six invariants represent a system of six equations for the six potential constants {krr,s) or ( k,,tS} of the substituted molecule in terms of those of the reference molecule. As for XYX’, the XSs and naetS are determined from the secular equation, Eq. (64), and the {,,rs are given by Eq. (6.5), where now Jr =
sin0 + c co@ +
[wwn2(b
cc“’ = [mlm,(b
mm3(
--a
sin0 + c cos8)]/M(a
cosd - c sine) - mlma(a cos0 + c sirM)]/ilf(a VI. CENTRIFUGAL
According to CP Eqs. (116)-(123), stants can be written in the form -21,47,W
= (L)(u) (t.),
DISTORTION
+ b) co.s&
(79)
CONSTANTS
the second-order
centrifugal
distortion
con-
a = x, Y>s>
- 2~,2182TL%zk¶8 = (L) (u) @a) = (la) (Q) (L), -2I,2I Y2fzwu = &)(d(~w),
+ b) cost?,
*
LY,@ = 2, y cyclic,
(go)
TRIATOMIC
ISOTOPIC SUBSTITUTION
Table IV. Matrix Elements of the Coordinate Defined by Eq. (78) for the XYZ Molecule
Elements
of D:
dll d21 djl
d12 d
22
d13 d23
= m3b/(m = m2a/(m = 1
2 2
+m +m
3 3
d32
= UC 'ml+
)r jr
359
Transformation
D
el e2
= d33
= 0
(a+bh21/mlm2+el
(a+b)
= ~c~bml+~atb~m31/mlm3re2~a+b~ -
~b(atb)m2+~c21/m2rel(a+b)
= +
[a(a+b)m3+~c21/m3re2('+b)
=
P = ml(m2+m3)/M
Elements
of
D
-1: Cl1 = Cl2 = 0
El3 = 1 2
=21
l/cm31z
= u'rel[a(a+b)m3+w 2
c22 '23
= u're21bla+b)m2+!-c
=
=31 = '32 =33
l/cm21z
- !_~'(a+b)[Mab+inlc21/cMIZ - m21bml+(a+b)m31rel/MI
= + m3[aml+(a+b)m21fe2/MI
z z
= ~'(a+b) [(b-a)ml+(bm3-am2)l/MIZ
v' = m2m3/(m2+m3) Iz is given
where u is the 3 X 3 symmetric
matrix
by Eqs.
(72) and
(73)
with elements
and where each of S, s’, s” may take the values 1, 2, 3. The matrices (0 are 1 X 3 matrices with elements given by CP Table I for XYZ, and with elements given in Table V of this paper for XYX’ and XYX. For the XYX case, due allowance has been made for scaling the harmonic potential constants in accordance with Eqs. (67). The planarity relations (1.5, 16) are implicit in the expressions for the centrifugal distortion constants given here. Since the isotopic invariances have been formulated in terms of the harmonic potential constants, it appears to be advantageous to express the centrifugal distortion constants directly in terms of the potential constants. This can be done by the use of Eq. (55) in the form K-1 = IWf, which holds for all three molecular configurations elements involved, Eq. (82) becomes
(82) considered.
In terms of the matrix
PAUL M. PARKER
360
Table V. Elements of the Raw Matrices (t) Occurring in Eqs. (80)
4am2m3sin2e For XYX':
G,) =
.2 2aml(m3-m2)su 8 _ 2cml(m2+~3)sinB
2cml(m2+m3)
( (m2+m3)cose '
.M case
t
,
0,
I
,
0,
'
4any3coE.e
6,) =
(m,+m,)
6,)
=
=
M.
(ix)
Y.
2m1
(m*+rn,)
+
1 1
(m3-m2) sine
2cml(m2+m3)COse
M
M
1
(Q
Chan and P. M. Parker, J. Mol. Spectrosc. 42, 53 (1972).
FOX XIZ:
See
For XIX:
W,) = (0, 21Jreco.we,0)
&,I
(ty) = (2m2r,sina,,0, 0)
and comparison
M
2aml(m3-m2)cose 2cml(m2+m3)sine + M M
4m2m3dne
6,)
M cos8
= (0, 0, -21Jrecosae)
&,I = (TX) + G,,
with Eq. (81) shows that therefore
K-' = u. For the elements
of u occurring
(84)
in Eqs. @SO),one has, therefore,
UII = (k&3
- Rd)/Det
explicitly:
K,
ua = (K&a - ha2)/Det K, (~33= (K&Z - W)/Det
K,
K,
ua2 =
(h&23 - WdlDet (h&la - hh)/Det
aal
@I&H - h&)/Det
K.
412 =
UZI =
US
=
ala
=
=
(85)
K,
Since the set of harmonic potential constants of a substituted molecule can be obtained from the set of the reference molecule in the manner described in this article, the set of centrifugal distortion constants of the substituted molecule can be calculated from Eqs. (80). An alternate form for Det K can be given with the aid of CP Eqs. (71), (72), and (79, which show that Det K = h&As Det p. (86) After lengthy
calculation
one finds for XYZ that
Det K = X&A&(m~
+ m#(l + tanV)I,/i315(a
+ b)2,
(87)
where I, is given by Eqs. (72) and (73). The corresponding expression for XYX’ is found by replacing (a + b)* by 4u2 in Eq. (87), and by using I, as given by Eq. (36). Finally, for XYX, Eq. (87) reduces to Det K = X$da, when once more allowance is made for scaling as specified by Eqs. (67). Equation is an immediate consequence of Eqs. (30) as well.
(88) (88)
TRIATOMIC
ISOTOPIC SUBSTITUTION
VII. GENERALIZED
INTERNAL
361
COORDINATES
For the study of the isotopic variation of H-constants other than those discussed in previous sections, one needs to know the isotopic variation of the cubic potential constants, and, also, in some instances, the quartic potential constants, as listed in Table VI. The present availability of theoretical expressions in the Nielsen-AmatGoldsmith formulation is also indicated in Table VI. Determinations of cubic and quartic potential constants have been made principally through the theoretical expressions for the LY’Sand x’s (17, 18). As mentioned in Section III, it has been shown by Plfva (10) that in order to obtain an anharmonic potential function with invariant coefficients, one must specify the function in generalized internal coordinates which have reference to the instantaneous positions of the nuclei. Such a set is Sf"
= [(x2 - x1)" +
sze" =
[(Xl - xaj*+
s3gW = [(LO, -
x3)2
+
02 - Ql",
(y2
-
ys)*y.
by ICY; with LY= x, y and
Denoting instantaneous displacements from equilibrium i = 1,2,3, it is readily found for XYZ that x2
-
Xl
=
(6x2
-
6~~) +
(89)
(Yl - r*>*1*,
ye1 sin@,
- 01,
y2 - y1 = (6yz - 6y1) - rs1 cos(Bs - 01, x1 - xa =
(6~~ - 6x3) +
yl - y3 =
(6~~ -
Qa)
x2 -
x3 =
(6x2 -
6x3)+
rdcosd
y2 -
y3 =
(6y2 -
Qa) -
re3.d4
+
yea sin(a, +
01,
ye2 co&k+
69,
Table VI. coefficientsof the TriatomicVibration-Rotation Hamiltonian
. Dependenceon potentialconstants: Order
CUbiC
a
2
Yes
no
YeSa
d,e Yes
X
2
Yes
Yes
yesa
YeSd
no
b.c Yes
Yes
Yes
yesb
no
Yes
Yes
no
no
caeffs A-
Q
4
P
4
6
4
Yes
puartic
Theoreticalexpressions available: xyx
XYX', mz
partiallyC
B. T. Darlingand D. M. Dennison,Phys. Rev. z, 128 (1940). b. M. Y. Cban, L. Wilardjo,and P. M. Parker,J. E(ol.SpeCtToSc.40, 473 (1971).The resultsgiven in this referenceare very complicated.AP discussedin Reference (c) below, it is possible that these resultscan eventuallybe given in much simplifiedform. C. D. A. Sumberg and P. M. Parker,J. Mol. Spectrosc.s, 459 (1973). d. W. H. Shafferand R. P. Schuman,J. Chem. Phys. 12, 504 (1944); H. H. Nielsen, Rev. md. Phys. 2, 90 (1951). e. M. Y. Chan and P. M. Parker,J. Mol. Spectrosc.42, 449 (1972).
a.
PAUL M. PARKER
362
where rs3 = a + b. These expressions specialize to XYX’ by taking f.,l=
rs2 =
Ye,
ra3 =
2a,
Pb = a,,
(91)
and, additionally, B = 0 for XYX. Substituting Eqs. (90) into Eqs. (89) and expanding in Taylor series to terms cubic in Gcrigives that SB” = s. + (R?/rer) - (S,R,2/2r,,4) + ’ * -,
s = 1, 2, 3,
(92)
where R1 = [(6x2 - 6x1) CO@@- e) - (6y1 - 6yz) sin@ - e)], Rz = C&r - 6x8) cos(cr, + 8) - (6yr - @a) sin(cy. + e)],
(93)
Ra = [(6x3 - 6x3) sine + (6~s - &ya)co&J,
and the S, are the internal coordinates introduced previously. In terms of the 6ai they are S1 = rdl + (6x3 - 6x1) sin@, - e> - (6~3 - 6yl) COS(& - e), Sa = r.2 +
(6x1 - 6x3) sin@, + e> + @yl - 6ya) COS(LY. + e),
(94)
SS = rb8+ (6x2 - 6x3) coti - (6yz - 6y3) sine. In terms of the k,
the three Eckart conditions are equivalent to 6x1 = 6Yl = -4,26x2 +
(m2bx2 + m36x3)/ml,
-(mdy2+
maSyd/m,
(95)
Adxa + A,dya + A,dya = 0,
where A=2 = r,,lrna cos& - e), A,3 = re2m3COS(CY, + e), A,2 = rs1m2sin@, - e),
.
(96)
A,s = --r,zrns sin& + e). Application of Eqs. (91) readily specializes all of the above equations to the XYX’ and XYX cases. To obtain the R, explicitly in terms of the S,, it is necessary to solve the six equations, Eqs. (94) and (95), for the six &xi as functions of the S., and substitute these expressions into Eqs. (93). For XYX, this procedure gives RI = #[(G - tam,)& - (G + t.anaJS~ + (ma,>&!, R2 = 3[- (G f tanaJS1 + (G - tam&
i- (mxJ&],
(97)
R3 = 3 (GmJma sina (SZ - Si), where G = (2mz sincr, cos(yJ/(mr + 2m2 sin*ar,).
(98)
It does not seem practical to carry out the above procedure algebraically in order to obtain the corresponding expressions for the R, of XYX’ and XYZ, which are rather complicated. If data for the equilibrium geometry of the molecule are available,
TRIATOMIC
ISOTOPIC SUBSTITUTION
363
the coeihcients of the S, in the expressions for the R, can be obtained numerically without dithculty. The mass dependence of these coefficients needs to be retained explicitly in order to arrive at the proper invariance equations. VIII. ANHARMONIC
In terms of the generalized tion is
POTENTIAL
internal
ENERGY FUNCTION
coordinates
SP,
the potential
energy
V = VzO” + Vae” + IlqBen+ . . . ,
func-
(99)
where 2V#*n = C C F,,,S,g”S,+‘” B 8‘
= C C 2F,,S,@5We”/(l B 58’
+ S,,t),
2Va=” = C C C Fss,s,,SsgenSB,genSB,,gen) * <-8’58”
(W
For XYZ, there are six coefficients F.,?, ten coefkients Fs.ll)~~,and fifteen coefficients and XYX’, the following restrictions hold because of the symmetry equivalence of SP and S!P. F l,,a~,ll,l.For XYX
Fn = F22,
F la = Fza;
and F III = F222,
I7112 = F122,
F 118= F223,
Flaa
=
Fzaa;
W)
and F 1111 = F2222,
I71112 = F1222,
F 113a =
271113 = F2223,
Fnza = F122a,
Flaaa= Fzaaa.
Fzzaa, W)
There are thus four, six, and nine independent potential constants in TIP’, VP, and Vdga, respectively, for XYX and XYX’. To obtain the potential energy function in the linearized internal coordinates S,, Eqs. (92) are introduced into Eqs. (100) and all terms quadratic, cubic, and quartic in the S, are retained. The result is v = vt+ va+ v4+ ***, (103) where 2V2 = C C 2F,,S&/(l II 5s’
which is the harmonic part of the potential above. Furthermore, it is found that
+ S,,t) = 3%S, energy function
as used in Sections
ow
III-V
364
PAUL M. PARKER
+ C C 2F,,f a 58’
(RstRs~2/resrca~)-
(S&RaY2rer2) -
In be
manner of CP Eq.
(SS,,R,~2/2r,,~2)1/(1
+ b).
(1%)
(109), the anharmonic potential constants are intro-
duced through (107) 2Va = 4M C C C C k~~~.~~glll*Q~Q~IQ~~~Q,~~~, 8
(108)
where k age*
= (X,A,A~~~)tk,~~,w,
* = (XIXII~X*)~XIrl~)*k,l~S)I.,,I, k al)r*,ror’, N = m/i@,
(109)
M = mgi,
and where ksatllt and kLIJ.JfIttt are in wavenumber units. The invariants for the sets of cubic and quartic potential constants are obtained by first transforming Eqs. (107) and (108) to the S. coordinates by means of Eq. (22), Q = @I’)-‘S
for XYX,
(110)
or by means of Eqs. (50) or (78), Q = (DW)-‘S
for XYX’
and XYZ.
(111)
The resulting expressions are then arranged in powers of S, and the coefficients of lie powers of S, are equated from Eqs. (107) and (105), and from Eqs. (108) and (106), respectively. For XYX, the conditions represented by Eqs. (101) and (102) are satisfied automatically, but for XYX’ they must be imposed as constraints. For XYZ the constraints do not hold. The procedure described is thus linally yielding a set of invariants from which the cubic potential constants of the isotopically substituted molecule can be determined. It is seen that this generally requires a knowledge of the complete set of cubic and harmonic potential constants of the reference molecule. Similarly, the determination of the quartic potential constants of the substituted molecule requires a knowledge of the quartic, cubic, and harmonic potential constants of the reference molecule. This parallels the general conclusions reached for XYX by previous investigators (17, 19). IX. CUBIC POTENTIAL
CONSTANTS OF XYX
The configuration most feasible to pursue algebraically is that of XYX. As a specific illustration of the general procedure and as the case of most immediate interest, the set of six cubic potential constants of XYX will be discussed further in this section.
36.5
TRIATOMIC ISOTOPIC SUBSTITUTION
Eq. (105) is
For XYX, 2vs = FdS?
+ S29 + FlllslS2(Sl+
S2) + FnaSa(S12 + S23
-kFl23SlS2Sa -k Flaa&+(S~ -b S2) d- F3arSa3 + (2F11hJ
(SlR? + S2R2*) +
+ (F33/@%&*
(2F12/yJ (S&2* + S2R19
+ 2FlaI: (Ra2/2a) (SI + S2) +
(Sa/y,) (RI* + R2*)],
where RI, Rz, and R3 are given by Eqs. (97). Introducing these explicitly lecting like powers of S,, the resulting expression assumes the general form 2Va = (F111+ flll)(S?
+ s2a) +
(F123 + /123)SS2S3
+ (F133 + f1a3)Sa2(S1 + S2) +
FM + F112,
and col-
(Fll2 + j112)&S*(.% + S2)
+ (F118 + _fl13)S3(S12 + S2*) +
where the f..$.l~ six invariants is responding ones Fin, Fllz, F113, advantageous to
(112)
(Faaa + f333)Sa*,
(113)
can be worked out in a straightforward manner. A complete set of obtained by equating the coefficients (F1ll + fin), etc., to the corof Eq. (107) and solving the resulting equations for the invariants F123, FWJ, and Fw. Carrying out the lengthy details, one finds it form the following equivalent set of six invariants. F 193,
F123 + 2Fll3,
F 333, 3F111 -
F112,
F 123 - 2Fll3.
(114)
It is found that in the expressions for the first four of the above equivalent invariants, terms involving FII, F12, F13, and Fa3 appear only with mass-independent coefficients and can therefore be dropped. After some further development, the following four invariants involving K111*,Kz~z*,&z*, and &ZZ* are found. $[saknl*
+ c%zz* + sac&z* + sc*klzz*],
pmz*[3s*cknl* - 3sc*k222*+ 428
- s2)kn2* - ~(2s~ - c2)klzz*],
(115)
/.dmmz[3sc*kn~* + 3s2ck22z*- c(2s2 - 8)knz* - s(2c2 - s*)kwz*], m$[c3klll*
- s3k222*- sc2k112*+ s2ck122*],
where, as before, s = siny and c = cosy. The remaining associated with KISS*and k233*. These are 4NccZ(m2/2)~~3-1(-ck13a* + skzaa*)+ (3f111 - f112) sinsoh 2Np%B-W133*
+ ck233*) -
two invariants
(fl23 -
are those
2fila) sin*%,
(Ila)
(3fi11 - fl12) sin2cybcos+,
where (3flll -fll2)
= (4/y3(3G2(F~1
+ Fl2) -
G(Fll
-
F12) tana, + C(Gml/2m2)2 csc3~,y713),
(jl23 -
2f113) = (4/r@) (.- (G seccu3 (Fll -
F12) -
G3F13 -
(117)
C(Gml/2m2)2 csch,Y;‘88},
and where FII, F12, FM, and Faa are those given by Eqs. (28). It is thus demonstrated that the cubic potential constants of the substituted molecule can be calculated from the cubic harmonic potential constants of the reference molecule.
366
PAUL M. PARKER
A particularly simple case is represented by the comparison of the ozone molecules I803 and 18O3 for which my formulation applies with ml = m2. With this condition, G in Eqs. (97) becomes mass-independent, and as a consequence, all f,8~8~~ become linear combinations of the harmonic invariants F,,l with mass-independent coefficients. They can therefore be deleted completely from the expressions involving the F,,I,I~. Furthermore, p and ~3 become proportional to m2. The angle y becomes invariant. As a consequence, the set of cubic invariants can be rearranged to the equivalent set of six invariants { mlfksstsll}. Similarly, one finds for the quartic potential constants the set of nine invariants {m&sa~s~~r~~~}. For the harmonic part of the problem one has the invariant sets (m2L3) and {m&). These findings have already been published by Barbe, Secroun, and Jouve (20). These authors also found very good agreement between the observed and calculated values of the quartic potential constants of ozone which would seem to indicate that the general approach described in the present study may prove to be useful. ACKNOWLEDGMENTS I thank Dr. G. Strey for the useful comments he made to me at the Third InternationalSeminaron High Resolution Infrared Spectroscopy at Liblice, Czechoslovakia,where part of this work was first reported. I thank Miss Iris Gomez for her able assistancein preparingthe manuscript. RECEIVED: May 1.5,1975 REFERENCES 1. G. HEPZBERG, “Infrared and Raman Spectraof Polyatomic Molecules,” pp. 227-238, Van Nostrand, New York, 1945. 2. G. AMAT,H. H. NIELSEN,ANDG. TARRAGO,“Rotation-Vibration of Polyatomic Molecules,” Dekker, New York, 1971. 3. W. E. SMITH,Azcstral.J. Phys. 12, 109 (1959). 4. V. T. ALEKSANYAN, A. P. ALEKSANDROV, ANDM. R. ALIEV,Opt. Spectrosc.26,292 (1969). 5. G. STEENBECKELIERS, Ann. Sot. Sci. (Bruxelles)85, 163 (1971). 6. A. P. ALEKSANDROV, M. R. ALIEV,ANDV. T. ALEKSANYAN, Opt. Spectrosc.29, 568 (1970). 7. A. P. ALEKSANDROV ANDM. R. ALIEV,J. Mol. Spectrosc. 47, 1 (1973). 8. G. HEIUBERG, “Infrared and Raman Spectra of Polyatomic Molecules,” p. 143, Van Nostrand, New York, 1945. 9. K. T. CHUNG ANDP. M. PARKER, J. Chem.Phys. 43,3869 (1965). 10. J. PLfv.4,Coil. Czech. Chew. Comm. 23, 1839 (1958). 11. J. BELLET,W. J. LAFFERTY, ANDG. STEENBECKELIERS, J. Mol. Spectrosc. 4’1,388 (1973). 12. M. Y. CHANANDP. M. PARKER, J. Mol. Spectrosc. 42, 53 (1972). 13. W. H. SHAFFER ANDR. P. SCEUMAN, J. Chem. Phys. 12, 504 (1944). 14. D. W. POSENER, Ph.D. thesis, MassachusettsInstitute of Technology, 1953. 1.5. T. OKAANDY. MORXNO, J. Mol. Spectrosc. 6, 472 (1961). 16. J. M. DOWLING, J. Mol. Spectrosc. 6, 550 (1961). 17. K. KUCBITSU ANDY. MORINO, Bull. Chm. Sot. Jap. 38,814 (1965). 18. A. BARBE,C. SECROUN, ANDP. JOWE, J. Plays. 33,209 (1972). 19. K. KUIZHITSU ANDL. S. BARTELL, J. Chew Phys. 36,246O (1962). 20. A. BARBE,C. SECROUN, AM) P. JOUVE,J. Mol. Spectrosc. 49, 171 (1974).