On the calculation of unique force constants

JOURNAL OF MOLECULARSPECTROSCOPY37, 494-516 (1971)

On the Calculation

of Unique

R. L. REDINGTON Depart.ment of Chemistry,

Force

AND A. L. KHIDIR

Texas Tech Universiiy,

Constants’

ALJIBURY~

Lubbock,

Texas 79.&X?

The vibrational F matrix can be written in the parameterized form F = f,:-’ Cf-’ AC’L,‘, where L, is determined from t,he G matrix alone; A is from the vibrat,ion frequencies; and C is an orthogonal mat’rix of !\‘(A’ - 1)/Z paramet ers +,j The restoring forces for internal coordinates or symmetry coordinates are calcttlated as a function of the 4;j parameters. Conditions on the restoring forces and energy are proposed that can be summarized in the equation F,t,,,, = points (e.g., CL TiDr:“. The critical points of F,t, or certain “int,ersection” acceptable T&Z 112= T.&i” for N = 2) det,ermine the several mathematically sets of +ij parameters. The molecular “true” F matrix has near-maximum parameterized restoring forces for coordinates associated with the highest vibrat.ion frequencies. In the F..tp eqttation, the T, are geomet.rical factors derived rtsing the virial theorem (e.g., 2’1 = r and TZ = 2 cos0 for CzF triatomic molecules) and the Dii are cofactors of the diagonal force constants. The method is illustrated with two-, three- and four-dimensional examples including F&l, NF,, ONF, H&e and other molecules. The force constants determined ttsing the new method with the vibration frequencies of a single molecttle compare very well with those determined by conventional methods using isotopic srtbst,it.ut ion, Coriolis coupling const,ants, and centrifugal distortion cotistaiits. For many molecules the F mat,rix is determined rmiquely by knowledge ot the vibrational assignment, stnce only one F matrix solution occttrs with the proper assignment; in other cases, a certain minimal amount of additional information beyond the vibration frequencies and geometry may be necessary in order to select the “true” F matrix. For example, t,he DJX_ celltrifugal distortion constants can isolate the trrte F matrices for the nitrosy hatides. ON THE

CALCULATION

OF UNIQUE

FORCE

Under the assumptions of infinitesimal vibrational force constant,s, it is well-known t’hat t’he vibrational can be calculated by solving t’he secular equation (1

CONSTANTS

amplitudes frequencies

and quadratic of a molecule

j

/ FG - BX j = 0. The

reverse

problem,

i.e., the

calculation

of thr

F matrix

using

only

the

G

1Presented, in part, at the Sout,hwest Kegional Meeting, American Chemical Societ,p, Tulsa, Oklahoma, December 4-7, 1969. 2 Permanent. address : Department of Chemistry, University of Baghdad, Baghdad Iraq. 494

CALCULATION

OF FORCE

CONSTANTS

495

matrix and a set. of observed X values has not yet been completely solved. Considering an F matrix of rank N, the difficulty is that there are N(N + 1)/Z force constants associated with only N values of A so t,hat it is possible t,o gener ate an infinite number of F matrices (2) which all yield the observed x values. Only one of t,hese can describe the t.rue molecular potential energy function; in the present article, a new condition to specify this “true” F mat’rix is proposed. Several criteria have been previously proposed for approximating F. Using the parametric representation to be described, with N = 2, Strey (2) selectsed that value of t.he paramet’er & which maximizes t*he bond stretching force constant fr for HTO, H,S, HBe, and their deuterated cogeners on the one hand and t,hat, +1Zwhich minimizes the bending constant fa for NOs, SOa and Cloy on t.he ot.her. Billes (3) chose $ij = 0, while Herranz and Castano (4) suggest,ed maximization of the trace of the F matrix-a procedure discussed in detail by Freeman (5). Several additional proposals for obtaining “unique” approximations to the t#rue F matrix have been given (6-8) and Pfeiffer (9) recently compared several of the met#hods. Our criterion arises from considerations of minimum energy and the slopes of the vibrational potential energy surface. The discussion focuses at)tention on t,hese restoring forces rather than the diagonal force constants as t,he significant, quantities and results in considerable improvement over the above met,hods in attaining agreement wit.11 eonvemional litera ture force constants. In addit#ion, much of the earlier work has been restricted by necessity or by choice t’o t,wo-dimensional problems. The present procedure is demonstrated with two-, three- and four-dimensional examples. In relat,ed work the properties of the F inverse, or compliance, matrix were studied by Decius (10). He showed that the compliance constants, closely related to t.he restoring forces discussed below, are uniquely defined. Compliance constant.s cornrast with force const’ant,s because the latter depend numerically on the nature of the force field chosen to express the problem. Very recently this fact was emphasized by Jones and Ryan (11)) who encourage the calculation of compliance constants as a routsine part of vibrational analysis. The above papers make no att,empt to derive conditions fixing either force constants or compliance constants on the basis of the vibrational frequencies of a single molecule or on a very limited set of experimental dat)a. COMPUTATIONAL

APPROACH

The approach to the problem can be quickly summarized (1, 2). The NXN transformat,ion matrix connecting N internal coordinates S and the N normal coordinates Q is S = LQ.

(2)

RED INGTON

496

The L matrix

mnst

satisfy

AND

AW IBURY

the condit’ion G = LL+

and allows the generation

of the F matrix

(3)

through

the equation

F = (L+)-‘AL-‘,

(4)

where A is the diagonal matrix construct.ed from the observed X values. A mathematically acceptable L matrix, called L, , is calculated from the diagonal eigenvalue matrix r and eigenvector matrix V of G from the equation L, = lv2. The infinity

of mathematically

(5)

L matrices

acceptable

are given by

L = L,C, where C is an orthogonal matrix dependent rameters. For N = 2, the matrix is

c= In general, C can be written pendent on a single parameter

Cl2 =

where Cij is a unit matrix

on N(N

-

1)/2

independent

1

Cos $12 - sin 413 sin +12 cos 412 [

as a product & . Thus,

C = IICii,i,j

(6)

= 1,2,

(7)

.

of N( N -

1)/2

...

i < j,

,N,

pa-

matrices-each

de-

(8)

except for the terms Cii = COS 4ij = Cjj , Cij =

-sin

4i, = -Cji.

(9)

The 4ij parameters, plus the X values, serve to determine the N(N + 1)/2 values of Fkl . To illustrate, Hz0 has four independent force constants which are determined by the three frequencies and by the value chosen for &-the parameter associated with the Al secular equation. The parameteriaed F matrix is calculated as a function of the +ij parameters using Eqs. (4) and (6) and the restoring forces are calculated from the F matrix as shown below. The following plausible assumptions concerning these restoring forces form the basis of our criterion for fixing unique values for the dij parameters and, hence, the F matrix. ASSUMPTIONA. It is assumed that a parameterized restoring force Mk must be directed parallel to the Sk internal coordinate axis. This is accomplished by holding firm a given displacement of Sk but relaxing all other internal coordinates Sj so as to attain the minimum molecular potential energy. This operation introduces constraints between the Qij parameters. ASSUMPTIONB. It is assumed that, the restoring force ilfk exert’ed by the mole-

CALCULATION

OF FORCE

CONSTANTS

497

cule for each Sk is as large as possible. Thus, we consider the largest available values that, exist for each Mk in t)he continuous range of choices that are calculated using the & parameters. Since it is impossible t(o assign the maximum experimentally compatible values t,o all of the 111ksimultaneously a compromise balance involving all the restoring forces is necessary. ASSUMPTION C. It is assumed that the ill, are validly compared with one another only under identical pot,ential energy. The balance is expressed through equations connecting the total molecular energy and the $;j parameters via t.he rest#oring forces by means of the virial theorem. The Generalized Restoring Forces The potential and kinetic energies are expressed classically using internal coordinates (1). Assuming a quadratic potential energy curve, the restoring force for Sk is expressed in terms of a restoring force constant Fk by the equations

(ioj The rest,oring force associated uniquely with each internal coordinate (or symmeky coordinate) is isolated by displacing the coordinate Sk and then relaxing every other coordinat,e to att,ain the minimum pot’ential energy so t’hat the entire nonzero restoring force acts along Sk . This relaxation process generates a set of linear equations connecting t’he internal coordinates through t,he equations

av -=

a&

o

. . . , N, 1 # k

I = 1,2,

>

which allows the potential energy to be easily expressed For example, considering a two-dimensional problem, 2V = F,S,2 + 2Fk4Sz becomes

under

the specified

conditions

+ FznSz

(11) in terms

of Sk alone.

(12)

either

21; = [Fu-

g]S:

2V = [pz? -

g]

= FlSl

(13a)

S,2 = FzS;.

(13b)

or

In the case of N dimensions,

it is not difficult

to show that

271 = F&’ for any Sk , where the force constant F

k-

(14)

Fk is

= D+FD @k

_ IFI -&k

(15) .

R.EDINGTON

49s

AND

ALJIBURY

D is a column mat’rix consisting of the elements Dkl , the cofactors of t’he Fkl (‘force constants” are the reciprocals of matrix elements. Thus, the restoring the diagonal elements of F-l. As a first, step in finding a good approximate P’ mat’rix, it is convenient to maximize the restoring force for the coordinate most strongly associated with the highest, frequency vibration. Other coordinates are considered in order of descending vibration frequency. This ordering is necessary to prevent t’he Cij matrices from permuting t’he vibrational assignment. Each FI: is a funct(ion of the ~$ij parameters and the extremal properties for the first to be examined, labeled Fk , are determined from the equations dFs -=

,,

i,

I

.j

=

1,

. . . , N, i < j.

2,

@ij

On substituting Eq. (15) into Eq. (16) and noting $ij , it is found that the simpler equations 0,

i,

j

=

I,

2,

that

. . . ,

( F / is independent

of the

N, i < j

define the critical points of Fk . There are N(N - 1)/2 = P of these equations, however, they are insufficient to independently determine any “unique” F matrix if N 2 3. This is because one or more arbitrary parameters arise in t’he solutions to Eq. (17), whereby the maximum value for Fk traces a curve or surface in the &, space. The presence of one or more relationships (depending on N) between the +ij parameters can, perhaps, be shown most simply by minimizing FL: rather t,han maximizing Fk . Standard matrix manipulations, paying particular recognition t’o commutation properties, suggest t,hat there is one free +ii parameter for N = 3 and two free parameters for N = 4, 5, when only one Fk is maximized absolutely. The single arbit’rary 4ij parameter that arises for the three-dimensional problem can be used as the abscissa on plots for the display of the other 9;,~ values, force constantSs or restoring forces. These plots exhibit the effects of t,he constraints that, are imposed on these functions by maximizing one of the rest,oring forces. For the four-dimensional problem, with two freely specified ~ij values, a second restoring force must be maximized (subject, to the constraints of the first restoring force maximization) in order to determine one of the tmo free cpij parameters. The remaining & is then used to display t,he force constants and/or otSher informat’ion as above. Restohy

Forxe Balance

It’ is not possible to determine the F matrix by successive maximization CJf the restoring forces alone (all but) the first one are constrained) because t,he last t’wo, at’ least, cannot be simultaneously maximized with respect to the final +ij pnrameter. In addit’ion, anot’her reason that the true F matrix will not usually occur

CALCULATION

OF FORCE

499

CONSTANTS

with an absolut,e maximum value for any restoring force is that the maximization conditions eliminat~emixing of the given internal (or symmetry) coordinate with the other coordinates. Thus, unless the normal coordinate and internal coordinate are identical in t)he real molecule, the t.rue F matrix will occur wit.1~restoring forces balanced at somewhat less than their absolute maximum values. Additional considerations are needed t’o specify the balance between all the restoring forces. The total molecular energy E, electronic plus nuclear, at an internuclear configuration slightly displaced from its equilibrium position is writ,ten in terms of the vibrational potential function V and equilibrium energy Eo as E = Eo + V. Alternatively,

(18)

it is written as

(19) using the virial theorem for polyatomic molecules (lSl4). The use of the virial theorem is discussed in the next section. We stipulate that the molecular energy be given a constant reference value V, + Eo = E, for comparing each restoring force. Then, given a value for Fn , the magnitude of each Sk is defined by the equations E, -

Eo = V, = .l/i F.A2,

k = 1, 2, ...

, N.

(20)

Henceforth, all energy and force derivatives are evaluated at these fixed values of reference energy and the subscript r is dropped. The dependence of Sk on the dij paramet.ers is determined from this equation. It is not possible to maximize the restoring forces with respect t.o $;f for all N separately displaced nuclear configurations simultaneously because of the constraints introduced between the 4ii for each maximization. Therefore, the following construct is used. An amount of energy equal to NE = Et is divided equally among N molecules, each of t#hem with a different internal (or symmetry) coordinate displaced by any amount given in Eq. (20). The total energy of the N real molecules is independent of t,he $ij parameters, so that &?Z,,/&+ii = 0 must be true for all #cj . These conditions serve t.o det.ermine all of the +ij parameters. The derivatives are expressed in terms of the restoring force Mb as -

z =g (g) rg) =

0,

i,j = 1,2, ...,N, i
(21)

There are P = N(N - 1)/2 of these equations. The (dE/dIUk) derivatives are obtained using t,he virial theorem, as discussed in t.he following section, and the (aMJa&~) are calculated directly from the parameterized F matrices.

500

REDINGL’ON

APiD ALJIRURY

These equations have the appearance of a set of P linear, homogeneous equations in the N “unknowns” (&ZS/&V,), where the “trivial” solution (in this case all aJJk/‘a+ij = 0) is not, possible. However, various other direct solutions arc’ possible. Application of the Virial Theorem The virial theorem is vitally concerned with the forces that act on a set, of particles. As is frequently illustrated with the example of a diatomic molecule (la), t,he virial theorem provides a specific relationship between an external force that is applied to hold the molecule in a displaced internuclear configuration and the internal properties of the molecule. These properties are T for the average electronic kinetic energy, U for the average total potent,ial energy, and E = T + U, the total internal energy of the molecule. The internal restoring forces provided by tJhe molecule exactly balance the applied external forces. The virial theorem, as generalized for a polyat,omie molecule (IS, ld), can be written using Cartesian coordinates or it can be written as

where the (aE/dri,) terms represent external forces directed along the interatomic distances ~ij . The rij may be summed over all possible values for the molecule. Alternatively, the sum may be limited to just those few (arbitrarily selected) interatomic distances which, when combined with appropriak angle coordinates, completely describe the molecular geometry. Under this description the angle coordinates do not enter into the virial theorem (14). For the present application, internal coordinate Sk is displaced and held fixed by an external force while all other internal coordinates are relaxed to yield minimum potential energy (Eq. (11) ) . Equation (22) is applied, but first it is rewritten as

(23) where (aE/drij) = (aV/&,i) and notation is added to C’ t)o emphasize that U$’ is calculated from the true ground-state electronic wavefunction for the kth displaced internuclear configuration. U$’ is independent of the cpij parameters and, of course, Eq. (23) is valid for the true +ij parameters only. The derivative term(s) in Eq. (23) represent t,he external restraining forces that are directed along the inbernuclear distances and which are required to maintain the molecule at the displaced internuclear configuration. It is clear from t’heir definitions that the virial theorem expressions for the Mk restoring forces used in the present work become very simple. Considering a bent kiatomic molecule, t,he virial t,heorem expressions for the three displaced

CALCULATION

OF FORCE

CONSTANTS

501

configurations are (rl , rz bond stretches, r3 distance opposite 0) : 2E = U:;’ -

rl 2

2E = u,$’ -

r2 2

1

= U:;’ -

r1Ml,

(24d

= lJA,“’-

r2Mz,

(24b)

2

All terms on the right side are evaluated at the kth displaced molecular configuration. To sufficient accuracy (and dropping the s) T1 = rt, T2 = r:, and T3 = [(ry2 + r”z”)sin ~“/rlorzo],the values for the equilibrium geometry. The derivation of Eq. (24~) considers the restoring force components shown in Fig. 1 for the angle bending internal coordinate. Because it is stipulated that there are no forces directed along rl or r2 , the forces acting on atom 1 and atom 2 are uncoupled (i.e., as reactive forces) and are considered separately. Idealistic external forces are applied to atoms 1 and 2 to balance these restoring forces but no external force is applied to atom 3, the apex atom. The desired force component (aV/&,) then has the magnitude 1F1 1cos a1 + ) F2 J cos 012. Using this force with the equations F1 = (aV/at?)/ rl and Fz = (dV/ao)/r2 leads directly to Eq. (24~). The force ( aV/ar3) matches the corresponding internal restoring force of the molecule under the stipulated conditions which direct, for example,

FIG. 1. Restoring forces for the bending internal coordinate. The angle 0 = LYI+ at is considered to be displaced from its equilibrium value and the bond distances ~1 and rt are relaxed to achieve minimum potential energy. The arrows indicate the restoring forces localized on atoms 1 and 2, while a restoring force on atom 3 is not, drawn. Unit infinitestimal displacements 05~ and dr? show the restoring force directions, perpendicular to TI and r2, and emphasize the relationship bet.ween these restoring forces and a displacement of 0.

REDINGTON

502

AND

AWIBURY

that it must approach zero rather than infinity when the angle 0 approaches 180”. For CI,, triatomic molecules the corresponding equations utilize symmetry coordinates (& : symmetric stretch, Sa : angle bend, S’s : antisymmetric stretch) and are

(25b)

The bracketed terms are Ml, Af, , and fif, , evaluated at the displaced positions labeled SI , X2 , and S3 , respectively, so that T1 = r”, Tz = 2 sin @“,and TX = r”. The Tk values for other molecules are evaluated analogously. On considering calculations for molecules such as the met’hyl halides note must be made of redundancy conditions.

F-steep: Solution to Equation (21) Considering

N = 2, with T1 = (aE/&V,)

and MI = FlSl,

Eq.

(21) reduces

to

on considering Eq. (20) into Eq. (26) yields

and it,s derivatives.

Further,

subst,ituting

Eqs.

(20)

or

T,~~?"(aD,,/a4,,) + ~~~~~~~~~~~~~~~~~~ = 0. Solving Eq. (28) is equivalent we call F-steep;

These critical points, form part- of t.he set sohrtion for t,he true More generally, in

t’o finding

the critical

(28)

pointIs for t,lle funct,iw that

solutions to Eqs. (21) and usually t,wo or four in number, of solutions which must include among t,heir members the F matrix. its N-dimensional form the F,,, function is written as

CALCULATION

OF FORCE

CONSTANTS

503

Locating the critical points of this function with respect to all of the & parameters corresponds to solving Eqs. (21) with no restrict,ions on coordinate mixing. However, maximizat’ions are performed on one or more Fk because of the convenient display they provide for the various quant’ities of interest) and because they quickly isolat,e the proper locale of the true force constant solut#ion in & space. In addition to t’he Fst, “critical point” solutions for Eqs. (21) there are additional possibilities. These occur when t’he Sk coordinates are given ‘?iatural” values-namely, values given directly by the Tk coefficients. The reference potential energy has been stipulated to be some constant V, for all coordinates (Eqs. (20) ). If the Sk , which are infinitesimal displacement coordinates, are nevertheless assigned the mathemat,ically acceptable values Tk , the force constants Fk , and t’he 4ij parameters are constrained by Eq. (20). The Fk are invariant and Eqs. (21) are satisfied if t#hereference energy is stipulated. Rather t’han being equal, the A%and Tk are actually taken as simply proportional with the same proportjionalitty constant for all coordinates. Considering the two-dimensional problem, the condition &/T1 = X9/Ts is fulfilled if curves for the two subfunctions of Fstp (Eq. (29) ) inbersect. For example, TIDr,“’ = TzDil”. Under t#hiscondition T1/X1 = Tp/& since the int’ersection condition is equivalent’ly written as FIT: = FzTz, a result to be compared with Eq. (20). Examples of these “intersection” solutions occur in our calculat’ions. Thus, for ONF the rabher startling condition (3Ia) i.e.,

Fl(t;U1)’

= F&:U$

= F&t&U$

= 2V,

(31b)

yields the t’rue F matrix, where the lower case notation tk is used instead of Tk to indicate dimensionless quantities. Thus, at this intersection point X1 = ttU1, Sz = tlU2 , and S, = tltztaUa , where the Uk represent unit infinit’esimal displacements carrying t’he dimensions of AS’,. Conditions analogous to Eqs. (31) do not determine unique values for the +ij parameters unless V, is known. This is because the P - 1 independent equations for the P & parameters define a line, rather than a point, in t’he &i space. In t’hree dimensions the value for V, is defined under the condition of maximum restoring force for the highest vibration frequency (previous section). This restoring force (say Mhi = 44,) is maximized and the appropriate intersection point of the other two coordinates along the axis of the free &j parameter ( i.e., t:DTl” = t&t3Dyii2 in the above example) is found. This yields SV, , Fz , and FB ; relocating F1 t,o satisfy Eq. (31b) then gives a solution to Eqs. (21) and gives an F matrix virt>ually identical t’o the literature comparison. While the method is not presently capable of independently selecting the “criti-

REDINGTON

504

AND

ALJIBURY

cal point” or “intersection” that is actually taken by a molecule from the limited set of mathematically acceptable possibilities, it does provide a direct route to these few possibilities. As will be seen, they often include a single, prominent member that is easily identified as the “true” solution with no further effort. CALCULATIONS

BND

DISCUSSION

In this se&ion the conditions given above are used to determine F mat,rices for several well-characterized molecules. The results are compared with literature values derived by conventional methods using isotopic frequencies, centrifugal distortion constants, and Coriolis coupling constants. They are also compared with approximation met’hods (6-8) in several instances. The computations are st*raightforward : (a) For two-dimensional secular equations the critical points of FStp (Eq. (29) ) and any intersections are determined as a function of t,he single parameter +I2 . The F mat,rix is calculated using Eq. (4). (b) For three-dimensional secular equat’ions the restoring force for t,he highest frequency (As,,) is maximized (Eq. (17)) using 412 as t,he arbitrary parameter. The Fstp critical points are det’ermined and any “intersection” points along the curve for n/hi are found. With two restoring forces fixed by any two-way intersect,ions along t’his curve, they are then held constant and Fhi examined t’o see which three-way intersect’ion solutions analogous t,o Eqs. (31) exist. (c) Concerning the four-dimensional secular equation (for H&L), Alhi is maximized using & and 43_, as the arbitrary paramet,ers. Then, for each $PJ value, that, r&a which maximizes t.he rrst’oring force for the second highest frequency is found. The remaining t’wo rest’oring forces are det,ermined using the F *tl, maximum solution according to Eq. (29) for F3 and F4 . While this will not be the true solut’ion, it must occur near to it for hhe given example and the results compare well with the lit,erature values. The ordering used for the +ij matrices is C&&23 in t,hree dimensions, where the C,j are defined in Eq. (9). The i and j values do not necessarily correspond to the vibrational coordinat,e numbering. In our calculations, the coordinates were generally numbered in t.he order of descending eigenvalues for t’he G matrices. The coordinat’e numbering is changed to standard notation in the figures and tables, however, the original notation for the +ij parameters is retained. H,O, H,S

and Isotopic

Species,

Table I

The diagonal force constants agree well with those obtained using isotopic frequencies (15)) while the off-diagonal constants t,end to be smaller. The fVol interaction t’erm is very sensitive to 4p2 , as seen in Fig. 2; for comparison, Nibler and Pimentel (15) have constructed plots illustrating t’he delicate sensitivity of

CALCULATION

OF FORCE TABLE

505

CONSTANTS

I

FORCE CONSTANTS FOR H20 AND H&3

8.437 8.423

-0.113 -0.219

0.750 0.749

0.020 -0.006

Fstpa Fat,

DzO

8.414 8.453

-0.136 -0.100

0.756 0.760

0.039 0.235

F at!J

HzS

4.281

-0.013

0.424

0.006

DzS

4.269 4.281

-0.024 -0.018

0.425 0.434

0.011 0.144

F atpa F StP

Hz0 HDO

8 !/‘i = R, Tz = 2 sin 8; frequencies b Reference

and geometry

listed in Ref.

Isotopesb

Isotopesb

15.

15.

/I

I

40

80 L FIG. 2. F matrix elements mum value for F-H fall very tions are indicated by dots. with the proper vibrational

IL

-%

N

18

I

for the A1 block of Hz0 plotted versus 412 The F,t, and miniclose together for this molecule. Locations of other F,t, soluThe “true” solution has the near maximum restoring force FI assignment.

the off-diagonal constants for these molecules to the standard determination using isotopic frequencies. The F matrix for HDO was obtained independently using the three-term Fstp equation. The calculations are more complicated for HDO than for the Czo species because two conditions (beyond F,,,) have to be satisfied in order to preserve the desired symmetry of the F matrix. The frm interaction constant for HDO is negative though very small. This constant is extremely sensitive to the +112 parameter and a decrease of 0.5” brings the constant into agreement with the Hz0 and DzO values. It is interesting that a better F matrix occurs for HDO at the maximum restoring force for the highest frequency vibration (one degree from the given solution), but that this point does not satisfy Eqs. (21). There is more than one curve in $12that satisfies the symmetry of the HDO F matrix.

506

REDINGTON

AND

TABLE

ALJIBURY II

FORCE CONSTANTS FOR F20, CLO, AND O3

FyO

4.160

0.804

0.749

0.156

F st,la

F10

3.964 3.950 3.784

0.822 0.806 0.850

0.719 0.724 0.718

0.150 0.194 0.172

FbtPh Isotopes, C.i>.c C.D.c (centrifugal distori.ion)

Cl?0

2.792 2.75 2.882

0.449 0.40 0.308

0.450 0.46 0.423

0.169 0.15 0.166

F &”

5.674 5.701 5.188

1.498 1.523 1.354

1.292 1.285 1.334

0.319 0.332 0.246

F*@

0,

Isotopes’ C.D.’ Isotopes, C.D.”

C.D.‘)

i? TI = R, Ts = 2 sin 0; geometry listed in Ref. 17 and frequencies in Ref. 18 as corrected for Fermi resonance. They differ by 17.4, 9.4, and 27.8 cm-l from the valrles rlsed in lines 2-4. b Frequencies listed in Ref. 19, the same as used in the literature comparison force constants. r llefereuce 17. d Frequencies from Ref. 20; geometry from Ilef. 21. c Reference 20. * Reference 21. g Frequencies from Ref. 23; geometry ‘1 Reference 22.

from Llef. 22.

The values of the +12 parameters are: H,O, 173.71”; D,O, 168.4”; H,S, 164.98”; Des, 135.23’; HDO, 6.44’, with 413 = 6.64” and 42;s = 88.10”. The symmetric strekh was labeled coordinate 1. F,O, Cl20 and O3 , Table II The vibrational properties of these molecules differ from t,hose of the tristomic hydrides discussed above and, as pointed out by Sawodny (lC), force constants determined using the minimum value of F?, as an arbitrary criterion for determining +12 are particularly inaccurate (1.525 % in FI1). These molecules provide a critical test of the F stp equation because of their large kinetic and potential energy interaction terms; the force constant values are very sensitive fun&ions of 412 and the literature force constants for these geometrically simple molecules are reliable. The agreement wit’h the literature values is excellento about 1 or 2 % for the diagonal elements. In each of these cases there is only one F stl, solution with the correct assignment point) though several other solutions t)o Eqs. (21) exist (the Fat,, maximum for each. The #Al?values are; FzO, (Nebgen et al frequencies), 116.11”; F,O, 116.18”; C&O, 106.98”; 03 , 124.81’. The symmetric stretchwas labeled coordinate 1 in the calculations.

CALCULATION

OF FORCE TABLE

507

CONSTANTS

III

FORCE CONSTANTS FOR NFI AND PF, Flz(md)

NFa

PFs

5.478 5.765 6.131 6.14 6.53 6.90 5.873 6.23

0.544 0.570 0.632 0.618 0.72 0.81 0.214 0.41

F??_

I

(mdA)

(m?TA)

Fdmd)

1.435 1.368 1.288 1.293 1.23 1.18 0.827 0.802

3.405 3.364 3.404 3.39 3.36 3.55 4.879 4.98

-0.334 -0.316 -0.329 -0.330 -0.31 -0.40 -0.095 -0.19

0.893 0.901 0.899 0.896 0.91 0.88 0.503 0.50

F ,tpa Isotopesb Isotopes, C.D.c Isotopes, C.D.d “Eigenvector”e ((PE”I FEQ Microwaveh

& T1 = R, Tz = 2 sin B for each secular equation. Geometry from Ref. 26, frequencies from Ref. 28. b Reference 28. c Reference 29. d Reference 26, 27. e Result,s of “eigenvect.or” approximation method, Ref. 8c,d as quoted in Ref. 16. method, Ref. 8e as quoted in Ref. 16. ‘ Results of “potential energy” approximation g Geomery from Ref. 30, frequencies from M. K. Wilson and S. R. Polo, J. Chem. Phys. 20, 1716 (1952). h Reference 30.

NFB , PF, , Table III The E block force constants for NF, have been reliably determined (d&29), however, t,here may still be some possible uncertainty concerning t’he Al force constants because of their sensitivity to the input data. They have been determined from isotopic frequency values, either alone (28) or combined with centrifugal distortion constants (26, 27, 29). The literature diagonal force constant values change by about 6 % on including an accurate isotopic frequency shift Avs with the vibrational frequency data (28) (cf., lines 2, 3, and 4 of Table III), while the diagonal F,,, force constants also differ from this best set by about 5 %. Allan et al did not weight’ their F matrix with the centrifugal distortion constants and none of the calculations used harmonic frequencies. The E block force constants uniformly agree very well. The force constants for the A, secular equation of NF, are plotted as a function of & in Fig. 3. The unique F matrix is determined by an “intersection” point and represents the only solution with the correct vibrational assignment. This is also the case for the E block. These curves are representative of twodimensional secular equations with large vibrational interactions. The principal difference from Fig. 2 for Hz0 is that the “phase angle” between the F1, and F2, curves is much larger. The interactions cause other inverse F matrix solutions to go awry, as shown in Table III for two of the better methods. The new results

REDINGTON

508

AND

AWIBURY

FIG. 3. F matrix elements for the AI block of NFI plotted versus $12. Locations of other F,t, solutions that occur for NF, are indicated. Only the “true” solution has the propel vibrational

assignment.

of Hirota and Morino (SO) for PFs are compared with the Fstp calculation in Table III, where agreement to about 5 % is seen for the diagonal force constants. Their F matrix, determined exclusively from microwave data (SO, Sl), cannot reproduce the experimental vibration frequencies. The difference in these two F matrices is consistent with the lack of perfect agreement between the Fstp force constants which are derived exclusively from vibrational data and those calculated from centrifugal distortion constants (cf., Table II). Some of the difficulties that can arise in force constant calculations using centrifugal distortion constant data alone are discussed by Pierce and his colleagues (17, 22). The 412 values are: NFz(A1), 104.17’; NR(E), 162.30’; PF,(A,), 118.54”; PFI(E), 158.96”. The stretching coordinates are coordinate 1 for each secular equation. Whereas intersection solutions uniquely determine F matrices for both NR symmetry blocks, Fstp maxima define the PFs solutions. Again, there is no ambiguity in the choice of solution as only one possibility exists with the conventionally anticipated assignment. Nitrosyl Halides, Table IV A severe test of the Fstp criterion is provided by the high reliabilit,y of the literature force constants (SS-S4) for these complex molecules and by the sensitivity of the force constants to the &j parameters. The “intjersection” solutions described below for NOX are not necessarily typical of N = 3 problems; they are certainly not typical for the other systems that we have investigated-all of which seem to possess simple Fst, “maximum” solutions. The Fstp critical point solution yields the incorrect vibrational assignment for ONBr and ONCI and force constants that are far from the established literat,ure values for ONF (e.g., 15.81, 1.87, and 1.96 versus 15.92, 2.26, and 1.82 for foN,

CALCULATION

OF FORCE TABLE

509

CONSTANTS

IV

FORCE CONSTANTSFOR NOX

NOF

ONCl ONBr

16014NF ‘8CPNF 180’4NF lsO’5NF ‘601*NC1 l80’5N Cl ‘60’5NCl 16014NBr lSO15NBr 16014NBr ‘8016NBr

16.006 15.92 15.094 15.26 14.962 14.961 15.25

2.223 2.26 1.292 1.27 1.087 1.133 1.13

1.832 1.82 1.314 1.32 1.151 1.126 1.13

2.278 2.36 1.528 1.53 1.158 1.302 1.47

0.383 0.27 -0.155 0.10 -0.207 -0.290 0.11

0.232 0.21 0.093 0.12 0.082 0.053 0.10

F stps

16.000 15.986 16.011 16.025 15.095 15.086 16.101 14.960 14.964 14.946 14.975

2.222 2.221 2.223 2.224 1.292 1.292 1.292 1.086 1.087 1.130 1.135

1.833 1.833 1.832 1.830 1.314 1.313 1.315 1.150 1.151 1.126 1.125

2.271 2.262 2.283 2.295 1.529 1.528 1.528 1.159 1.157 1.289 1.314

0.375 0.361 0.393 0.404 -0.156 -0.147 -0.162 -0.211 -0.203 -0.308 -0.272

0.231 0.230 0.234 0.233 0.092 0.095 0.092 0.081 0.082 0.051 0.055

F St”

a Tl = RNO , Ta = RNX , Tz = (R? + R2')sin 6/R,R2.Geometry same as literature comparisons. b Refs. 32-34. c Close to “true” solution. This matches solutions for ONCl or ONF. d “True” F stpsolution.

Fi of literature

comparisons

Isotopes,

C.D.b

F StP Isotopes,

C.D.

Fstpc (close) Fstgd (true) Isotopes,

C.D.

and frequencies

the

best, as do above

solutions as well and js , respectively). However, there are “intersection” a”,“,ie single credible Fstp critical point solution for each molecule. The magnitudes of the Ti constants allow two or more reasonable “intersection” solutions for each molecule (Tz and T3 are close together for ONCI and ONBr while t’he product TIT2 is close to Ts2 for ONF). The spurious solutions are not discussed further than to say that the closest yield diagonal force constants differing from the true ones by several percentage points and that the “true” solutions are easily selected for the nitrosyl halides because of their very close agreement with the known literature force constants; if the true solutions had not been previously established they could have been unambiguously determined at t,he present time given the vibration frequencies for a single molecule and, most sensitively, the experimental DJK value for each molecule. solutions are illustrated in Figs. 4-6. Figures 4 and 5 The F,,, “intersection” show the diagonal force constants and the restoring forces with maximum F,o , the case of ONBr over the entire range of dIz-the arbitrarily chosen free parame-

f

510

REDINGTON

=0

-80

LL

AND

-40

ALJIBURY

0

I

@,12

40

80

FIG. 4. Diagonal F matrix elements for ONBr plotted versus ale with maximum restoring force for the ON stretching coordinate. The “true” F matrix will not occur on this plot but the approximate Fzz and F33 values will be in the range 412 - -17--22” (cf., Fig. 5).

I

I

I

-80

-40

1

@I2

I 0

I

40

I

FIG. 5. The three restoring forces for ONBr. The restoring force for the coordinate with highest vibration frequency is maximized. The “true” F matrix will evolve from the range $12 - -17--22”, limited by the vibrational assignment and maximum Fe range. The vallle of FJF3 is determined from the data on this plot for intersection points such as given in Eq. (31).

ter. Considering the bending internal coordinate to be associated most strongly with the second highest vibration frequency (the actual situatSion), t’he acceptable Fstp intersection solutions must derive from the narrow (-3”) range of +12extending from roughly the maximum point of Fs t,o just, beyond t,he intersection point of Fe and FNX seen for ONBr in Fig. 5. If the vibrational assignment’ were reversed the solution would occur near themaximum of FxS (cf., Fig. 5 for ONBr) and would likely have been the Fstp maximum solution. Ratios of the t; values ( T1, Tz, and T, = 1.13, 1.9626, 1.52; 1.139, 2.1217, 1.975; 1.15, 2.1910, 2.14; for ONF, ONCl and ONBr, respectively), used with

CALCULATION

OF FORCE

311

CONSTANTS

-1 16

28 2o

-@I2

24

FIG. 6. Force constants for 16014NF. The restoring force constants F, and F3 are constant at 1.8071 and 1.8826, respectively. The three-way intersection (Eq. (31)) occurs at $I2 = -28.69”.

Eq. (20), determine the possible intersection solutions. For example, with 8’1 maximized, F2 and F3 are functions of only one +ij parameter (cf., Fig. 5 for ONBr), where it is found that the ratio FJFS = t,/t, exists for a +u that leads to the proper vibrational assignment. This point fixes FZ and Fs (i.e., VJ, which are then held constant while Fl is allowed to decrease from its maximum value until Eqs. (20) are satisfied. This is shown in Fig. 6 for ONF, where the high sensitivity of the force constants to t’he +ii parameters is again very apparent. Nevertheless, the part of t,he L-l matrix that refers to coordinates 2 and 3 is rather insensitive to t.his part of t.he calculat’ion and the vibrat.ional assignment is formed along t’he curve for maximum F1 . It was determined that the intersections F1(t12U1)2 = Fz(t33U2)2 = Fs(&t~U3)~; Fl(tltzU1)2 = Fz(ts3C.T2)2 = FB(t22t3U3)2; Fl(t?t3U1)2 = F2(t;3U2)2 = F3(ts3U3)2 for OiYF, ONCl, and OKBr, respectively, agree very well with t.he restoring forces calculated from t,he lit,ernture F mat,rices. However, the “true” solution for ONBr is F1 (t~‘U~)” = Fz(t32U2)2 = F3(t42U3)2 as found on comparing with t’he microwave data, in particular. While it might appear that a large number of intersectSion points analogous to the above points can occur, t,his is note act.ually t,he case. The agreement of these points with t’he literature force const’ants is very remarkable considering the sensit,ivit8y of the FEZto the +ij parameters. The upper port’ion of Table IV lists t’he average values of the F matrix solutions usmg Fstp along with the literature comparisons. Vibration frequencies

512

REDINGTON AND AWIBURY

calculated using these average force constants return most frequencies to 0.2 or 0.3 cm-’ and the largest percentage error in any case is considerably less than 0.1%. While this frequency fit is very good, it is inferior to the fit provided by the literature F matrices, which are determined by best-fitting the isotopic frequencies and modified by fitting the available cenkifugal distortion data in addition. The F,, force constants fit the centrifugal distortion constant data a shade bett,er than the literature force constants, but, neither set provides perfect experimental agreement. The calculated D.,, value for ONBr clearly favors the so-called “true” Fatp solution over the listed alternative, as it yields near experimental agreement here (so does Jones), as opposed to a more t,han 12 % error. Literature and Fbt, force constant’s yield a common D, value t,hat, misses the experimental mark by about 7.5 %. The liberat,ure force constants are not so well-det.ermined for ONCl and ONBr as t.hey are for ONF because less experimental data is available. In part(icular, in cont’rast to ONF, calculat’ions reveal t’hat, neither ONCl nor ONBr literature F mat(rices can be very precisely reproduced using the parametric represent’ation (they do not exactly reproduce the observed vibration frequencies). This point is made to indicat,e that there may be some possible significance to the negative values for foN,o determined using F,t, for ONCl and ONBr. In conclusion, t,he FstB force constants closely reproduce the literature values and the observed experimental data. The lat,ter fit is sufficiently sensitive that, thr force con&ants could have been quit,e reliably determined using very much less data than 1~~s used for the literature comparison. The +ij parameters are (412 , 413, 423) : ONF, -28.73”, -7.409”, 24.474”; ONCl, -23.32O, -6.592’, 15.660”; OxBr (true), -24.46’, -4.958’, 11.744”; ONBr (near), -25.38”, -3.950”, 11.841’. In the calculations the coordinates were numbered: 8 = 1. ON = 2 and NX = 3. H:l&oyen PersulJide, Table V The Fs3 and F41 diagonal force constants using the given procedure are displayed in Fig. 7 as a function of +12 . The F11 and Fy2 force constants are nearly constant over the given region; F1 and F, are exactly constanO while F, and F4 may be judged directly from the given plot because of the small diagonal force constants. The &j angles are (& , 41~, 4~~4 , 423, +ar , &) : - 180.080”, - 1.776”, - 174.18S”, -1.556”, -0.028”, -0.720”. In the calculation, t’he coordinates were numbered: 1 (torsion), 2 (SH), 3 (SSH), 4 (SS). The force constant)s listed in Table V utilize the straightforward Fst, solution for the 3 x2 secular equation. The Fstp constants for DzS, were not calculated because some of the vibration frequencies are not yet available. Fairly reliable calculated vibrational frequencies for D,S, and HDSS could be expect,ed from t,he F matrix determined using the H,S, dat,a alone.

CALCULATION

OF FORCE TABLE

Foac~

fr f l, fa

f 00

4.05 md/d 4.08-4.09 -0.001 md/ii 0 0.82 rnda 0.83-0.85 0.013 mdd 0

CONSTANTS

V

CONSTANTS

FOR

Hi!&

fR

2.50 md/d

frR

f,

2.52-2.62 0.091 md.%

fry

f TO f’ Tc.

0.0926 0.008 md fO.10 0.004 md

s Ra f my

0 f RY

9 Geometry bined. ‘OReference

from Ref. 35, frequencies

513

0.003 md,iA f0.30 0.0004 md 0 0.088 md fO.10 0.026 md,k 0 0.019 md 0

F,t, n Isot.opes”

from Ref. 36, 4 X 4 and 2 X 2 Fst, results com-

36.

@,2 -178

-180

-180.25

FIG. 7. Diagonal force constants for HQSZ with the restoring forces for SH stretching and SSH bending simultaneously maximized. The RI and Fzz force constants are nearly constant over the interval plotted in the figure and the off-diagonal elements are all small. assignment The circle indicates the remarkably small range of 412 , where the vibrational is correct. The torsion and SS stretching assignments are reversed for all other angles.

Other Mobcules

The present cules-notably isotopic species. ture values in

method is being further deveIoped and tested using other moleformaldehydes, the methyl halides, binary fluorides and their The force constants for these systems agree well with the literathose cases where comparison is possible. The nitrosyl halides

514

:;I

REDINGTON

AND

F22

H

(CO)

ALJIBURY

Hz0

;

cu8

LP

-2 LL 8

6:

D2C0

4-

-1

D2C0

2-

1

H $0

01

I

I

-80

/

I

-40

I

1

@l2

0

I

I

40

FIG. S. Diagonal F matrix elements of A, block for formaldehyde ing force for CH stretching is maximized.

I

10

molecules.

The rest

or.

I

I

:=

n

4

g 816 Liz- ,

H2C0 $

(CH)

t

F,

(HCH)

t

I

I

-40

-20

t 2

3

I I

0

20

@12 FIG. 0. The three restoring forces for H&O. vibration is constant at, its maximum value.

The restoring forre for the highest frequency

seem to provide the most complex situation to date and would have yielded, perhaps, the most difficult to judge solutions if vibration frequencies for a single isotopic molecule had been the only available data. The diagonal force constants and restoring forces for formaldehyde are shown in Figs. 8 and 9, since they appear to be more typical of 3X3 solutions than the corresponding curves for the nitrosyl halides. The figures are calculated using the data listed by Shimanouchi and Suzuki (37) and the derived force constants

CALCULATION

OF FORCE

CONSTANTS

515

agree to within a few percent of values calculated by these authors using isotopic frequencies. Force constants for the FstD maximum do not occur in Figs. 8 and 9, but close by, at a point in &j space with near-maximum F, and F, restoring forces (cf., Fig. 9). CONCLUSIONS

1. A condition is proposed for generating relationships between the N(N 1)/2 parameters that serve to determine the F matrix for a molecule. Sufficient relationships arise t’hat all parameters can be determined from t)he frequency data for a single molecule. Unfortunately, spurious F matrices are also generated and no ab initio condition is proposed to isolate the true F matrix from this set so that an independent’ judgement based on some prior knowledge of t’he molecule must be used to dist,inguish between these. In most’ molecules that we have examined the vibrat,ional assignment is obvious t’o spectroscopists, and it is often found that only one P’ matrix occurs wit8h the proper assignment. However, for the nitrosyl halides more than one feasible F matrix solution exists. In this case experiment)al knowledge of D,, is sufficient to isolate the true F matrix given the three vibration frequencies. 2. The criterion (Eqs. (21) as expressed via Fstp , Eqs. (30) ) is based on the following plausible assumptions, which consider the separat)e parameterized restoring forces that arise for each internal coordinate when all the other internal coordinates are allowed t’o relax t*o attain minimum potential energy. These parameterized restoring forces are assumed to t’ake their near-maximum possible values and must be compared w&h one another at the same value of the potential energy. Their average energy, or the energy of N separately displaced molecules, is independent of the +il paramet’ers. The restoring forces for coordinates must be considered in order of descending vibrat’ion frequency in order to avoid permuting the vibrational assignment. 3. The force constant,s derived for different isotopic molecules using the new approach are nearly identical for heavy atom substitutdon but may differ for H-D substitution. The largest differences are in t#he off-diagonal element,s in all cases. An F matrix arises for each isotopic molecule in contrast to standard methods which best fit a single F matrix to all of the data. 4. The results of comparing force constant’s calculated via the given criteria with several well-established values from the literature are very encouraging. Calculation of force constants using this method seems feasible for small molecules, particularly so in those cases where some isotopic or other data exists but which is insufficient, for a complet,e standard force const.ant determination. ACKNOWLEDGMENTS The authors are grateful to the U.S. Army Research Office, Durham (Contract DAHC04 67 C 0069) and t,o the Robert A. Welch Foundation of Texas (Grant D-335) for financial support of this research. They are also indebted to the Computer Center of Texas Tech

516

REDINGTON

University for free computer Dr. W. N. Shepard. RECEIVED:

ANI)

time and facilities

ALJIBURY and are appreciative

of discussions

with

August 24, 1970 REFERENCES

1.

“Moleclrlar Vibrations,” McGraw-Ilill New York, 1955. d (i.STR~;Y, J. Mol. Sped-ox. 24, 87 (1967). 5. F. BII,I,IS, Acta Chink. (Budapest) 47, 53 (1966). 4. J. IIISRK.\NZand F. CASTANO, Spectrochim. Acta 22, 1965 (1966). 5. I>. 14:.FI{~:xM.IN, J. AVol. Spectrosc. 27, 27 (1968). 6. P. Pur..\~ and F. Tii~ii~;, Acla Chirrl. (Budapest) 44, 287 (1965); 46, 273 (1966). 7. P. PVL\Y, Acta China. (Budapest) 62, 49 (1967). 8. (a) A. F.\DINI, Z. ~\Tutz~rforsch.219, 426 (1966); (b) C. J. PISXOCS, U. HEIDBORN, and 8. ML~LLIX. b. Mol. Spectrosc. 30, 338 (1969); (c) W. SSWODNY, A. FADISI, and K. Bar,I,XIN, Spectrochim. Actu. 21, 995 (1965); (d) IT. J. RECHER and 11. MATTES, Spectrochim. Acta 23A, 2449 (1967); (e) H. J. BIXHIIXIUS, J. Chern. Phys. 38, 241 (1963). 11. I,. H. JON~:S and It. R. RY.ZN, J. Chem. Phys. 62, 2003 (1970). f2. (a) W. K.~uzM~N, “Quantum Chemistry”, Academic Press, New York, 1957; (b) P. 0. Lij\vDIN, J. Mot. Spectrosc. 3, 46 (1959). 13. 11. G. P‘\RR and J. E. BROWN, J. Chern. Phys. 49, 4849 (1968). 14. B. EL.\NDER, J. Chem. Phys. 61,469 (1969). 15. J. W. NIBLER and G. C. PIMENTEL, J. Mol. Spectrosc. 26,294 (1968). 16. W. SA~ODNY, J. Mol. Spectrosc. 30, 56 (1969). 17. L. PIGRCF:, N. DICI.~NNI, and R. H. JXKSON, J. Chem. Phys. 36,730 (1963). 18. J. W. NE:BGEN, F. I. METZ, and W. B. ROSE:, J. Mol Spectrosc. 20,99 (1966). 19. E. A. JONISS, J. Chem. Phys. 19, 337 (1951). 20. M. M. ROCH~IND and G. C. PIMI;NTE:L, J. Chem. Phys. 42, 1361 (1965). 21. Cr. E. HERBERICH, 1~. H. JACKSON, and 11. J. MILLEN, J. Chem. Sot. (A), 336 (1966). 12. L. PIERCE’,, J. Chem. Phys. 24, 139 (1956). 23. R. M. B~DG~:R and M. K. WILSON, J. Chem. Phys. 16,998 (1950). 24. It. J. L. PoPPL~:~+;LL, F. N. MASRI, and H. W. THOMPSON, Spectrochim. Actu 23A, 2747 (1967). 25. A.M. MIRRI and G. Cazzo~r, J. Chem. Phys. 47, 1197 (1967). 26. M. OTAKE, C. M~TSUMUR~~,and Y. MORINO, J. Mol. Spectrosc. 28, 316 (1968). 27. M. OTXI’:, E. HIK~T~ and Y. MORINO, J. Mol. Spectrosc. 28, 325 (1968). 28. A. ALL.\N, J. L. I)uNc:\N, J. H. Ho~r,ow.\y, and D. C. McKB;.\N, J. Mol. Spectrosc. 31, 368 (1969). 29. W. $.I\\ODNY, A. RUOFF, C. J. P~;.xocK, and A. MULLER, LVol. Phys. 14,433 (1968). SO. E. HIKOTI and Y. MORINO, J. Mol. Spectrosc, 33, 460 (1970). 31. A. &I. MIRRI, J. Chem. Phyys. 47,2823 (1967). 32. I,. H. -JONIS, It. R. RY~\N, and L. B. ASPREY, J. Chem. Phys. 49, 581 (1968). 53. R. I<. lly.1,~ and L. H. JONICS,J. Chem. Phys. 60, 1492 (1969). $4. J. LI.INE, I,. H. JONES, R. It. RYAN, and L. B. Asparcr, J. Mol. Spectrosc. 30, 489 (1969). 35. G. WIXNEWISSEK, M. WINNEWISSEH, and W. GORDY, J. Chem. Phys. 49,365 (1968). 36. B. P. WIN~~F.WISS~:Rand M. WINNHWISSER, Z. Nutwforsch. 23a, 832 (1968). 37. T. SHIM.\NOUCHIand I. S~JZUIS, J. Chem. Ph.+?. 42, 196 (1965).

PI. H. WILSON

JR., J. C. I>~:cIus, and P. C. Caoss,

Book Company,