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Parameter method in the calculation of force constants - application to e' species force field of phosphorous pentafluoride
1 Journal of Molecular Structure, 26 (1975) l-15 @ Elsevier Scientific Publishing Company, Amsterdam
- Printed in
The Netherlands
PARAMETER METHOD IN THE CALCULATION OF FORCE CONSTANTS - APPLICATION TO E' SPECIES FORCE FIELD OF PHOSPHOROUS PENTAFLUORIDE T. R. ANANTHAKRISHNAN Deportment
(Received
of
Physics,
AND G. ARULDHAS
University of Kerala, Kariauattom,
695581 Trivandrum
(India)
16 May 1974)
ABSTRACT
A parameter method is developed for mapping all the mathematically possible force fields fitting the experimental data on the vibration frequencies and Coriolis coupling constants in the case of vibrational species of order three. The method is applied to the E' vibrations of phosphorous pentafluoride and all the possible force fields fitting the frequencies and Coriolis data, forboth assignments viz. v (equatorial in-plane bending) > v (axial bending) andviceversa, are mapped. The vibrational amplitudes /(P-F,-,) corresponding to these various force fields are also plotted. Comparison of the plot with the electron diffraction result conclusively settles the assignment in favour of v (axial bending) > v (equatorial in-plane bending). Comparison of the vibrational amplitudes [(P-F,,) and [(Fe‘, - - - F,,), for the remaining sets of force fields, with the electron diffraction results leads to a remarkable restriction of the ranges of parameters involved_ The final results yield a mapping for the force constants in the range F,,(E') = 4.18 to 5.79 mdyn A-‘, I;;,(E') = 0.23 to 0.57 mdyn A, F3s(E’) = 4.77 to 2.33 mdyn A, F,,(E') = -0.37 to 0.57 mdyn, F13(E') = -0.51 to 1.42 mdyn and F,,(E')=Oto -0.44 mdyn A. The result is discussed in the light of the various force fields suggested for the molecule by different authors.
INTRODUCTION
The complete determination of the general valence force field for a certain symmetry species vibration of a molecule with n vibrational frequencies requires +n(nt 1) force constants. Consequently the problem is indeterminate and Taylor El] has shown that all the solutions compatible with the vibration frequencies can be expressed as a function of +z(n - 1) parameters. Thus, any physically meaningful
2
approach to the probIem warrants the use of additional data [2] like Coriolis coupling constants [3], isotopic frequencies [4], etc. In one of our recent works [3], we have made a parametric approach to the problem of degenerate vibrations with n = 2, by associating Coriolis couphng constants as additional experimental data. In this paper, we present the extension of such parameter techniques to degenerate species with n = 3. The method is discussed with a special reference to the E’ vibrations of PF,, in which a great deal of indeterminacy prevails both in the assignment
THE PARAMETER
and in the force field [5-71.
APPROACH
The mathematical formulations are summarised as follows. The force constants I; and the Coriolis coupling constants 5 are ultimately governed by the normal coordinate transformation L through the following matrix relations. @)-AL-’ L-1
cE-’
= F
= 5
(2)
The symbols in these equations have their usual meanings [8,9]. In the parameter approach, the L matrix is split into two parts, one characteristic of the molecular structure and the other a periodic function of parameters vii such that L = TA
(3)
Here T is a trianguIar matrix defined by the relation T Y$= G, with G as the usual Wilson G matrix [IO]. A is an orthogonal matrix so as to satisfy the Wilson condition IlO] L.E = G and can be represented by means of Jpz(n - I) parameters [I ] in a vibrational problem of order n. In general, A can be written [ll, 121 as the product of &(n- 1) matrices A,, each dependent on a single angular parameter (~ii such that A =17A,
i,j = 1, 2, -
l
a, n, i cj
(4)
where A, is a unit matrix except for the iith and&h terms being equal to cos Cpi~, the ijth term being equal to -sin ‘pii and the jith term being equal to sin vii. Thus, the Aii’S are rotation matrices in the ij plane and the variousvalues of ‘pii along with eqns. (4), (3) and (1) serve to determine the complete set of mathematically possible force fields, compatible with the vibrational frequencies. For degenerate species, however, the Coriolis coupling constants cii are often available as experimental data. Substitution of eqn. (4) into eqns. (3) and (2) in turn, yields
3 where J = T-l C T-’ is a matrix dependent only on the molecular geometry and atomic masses. The zeta sum rule (Cx,i = ~iJiir since A is an orthogonal matrix) reduces the number of independent cii elements to (n- 1) and brings in an equal number of constraints on the $z(n- 1) parameters, thus leaving the number of independent parameters as ~(KZ - f )(n-2). Thus, for the n = 2 case, the parameter Cpiishould possess a unique value, the evaluation of which has already been discussed earlier [3]. For the n = 3 case, the force fieId fitting the vibrational frequencies as well as the Coriofis data should therefore be expressible as a function of a single parameter. There are three rotation matrices A, in the third-order problem and a restriction on their sequence as A = AU
A13
(6)
A23
limits the variation of each parameter y>iito within the region -n/4 < ‘pie d n/4 in order that a given assignment may hold good, as shown by Torok II 1 J. Substitution of eqn. (6) into eqn. (5) yields _ _ c A23A13A12JAZ2Af3
A23
5
=
If we fix qr2 constituting A,, matrix $ such that
(7)
as the independent parameter and define a
,j = A,, JAI2
(8)
the form of eqn. (7) becomes
,.S
-
A23A132
4343
=
(9)
4
Expansion of eqn. (9) yields j,,
cos’~~~i-.j~~
(p-q)
sin2 cp13+ti13
sin2p23+r
(p-q)COS2tp,3fr
coscp13Sinq13
=
COS2rp23+2SCOS
v)23
sin
5023 =
sin2Cp23--2SCOS
023
sin
Cp23 =Jt33
T~I
c22
(10) (11) (12)
where p = J,, 4 =
sin2 q213
zs,,
+
j33COS2(P13
U3)
sinV]13 cosfp13
r = .f,, S=
9 23 cosfP13
- J 12 Sin4013
The solution for pl 3 can be written from eqn. (10) as -f,34t313-(.2",,-5,l)(~33-rlr)l* aaSP
= J33-
5 11
(14).
4 The solution for tancpz3 =
following from eqn. (11) can be written as
(~23
-s+rsZ-(p-9--e22)(p-r,,)llf
(15)
P-9-522
Similarly the solution for qt3 following from eqn. (12) can be written as tan
923
=
~JII~=-(~-cr-r,,)(~-;,,)If
06)
r--c33
In the actual calcufation, cprs wiff have two values, say qlsA and cp13s, for each vaIue of (p12 according as the plus and minus sign preceding the square root in eqn. (14). The p, q, r, s elements in eqn, (13) can be cafculated for each value of yrl3 and used to calculate qz3 from eqn. (15) or (16). (Both these equations lead to the same solution as a consequence of the zeta sum rule.) Once again, there witi be two values of (~23 for each value of cprs. The two solutions of p23 arising from the value of p1 3Acan be called pZJAAand ~~~~~~The two solutions for ~23 arising from the value of cplsB can be called 923BA and c;p,,,. Thus, there are values, which in turn lead to four sets of matrices four sets Of 912, Y)13&23 A - A12A13A23 for a given value of cp12. The emergence of the four sets of A matrices for a given arbitrary value of 40f 2 in a general case is illustrated as follows. (;013Aeqn, Fixrp,2e9"*(14)
~
(15)Or(16)+
a
( qr3Beqn.
(15)Of(t6)+
,,_~923AA=.~
\
q23AB==)A
=
4243AA23AA
=
A12A1.3~~23.a
<~23BA3A2 Y)23BB=+A
= =
A,2A13BA238A A,Z
A,33A23BB
The suffixes following each A, matrix indicates the suffix in the parameter q involved in the matrix. Each of these A matrices leads to a certain force fields, which can be cafctdated through eqns. (3) and (1). The method thus provides a systematic approach to mapping of all the force constants compatibIe with the vibrational frequencies and Coriolis data as a function of a single parameter 9 r Z. Further, one gets four such mappings in generat, owing to the four A matrices emerging from the calcufations. Since the A matrix has been taken in the order A12A13A23, the value of the parameter rplz can be limited in the range --n/4 to + ~14, in anaIysing a given assignment of frequencies I1 11. Similar conditions are valid for cpl3 and 923 also and can often enabie further restrictions in the region of analysis for the pii parameters. In cases where additional experimental data like vibrational amplitudes, absolute intensities or centrifugal distorsion constants are available, further Iimiting of the ranges of parameters can be achieved such that the force field fits such data also.
5 PROBLEMS
WI-H
THE
E’
SPECIES
VIBRATIONS
OF
PF,
The trigonal bipyramidal molecule belongs to D,, symmetry and has vibrational representation r = 2A; +2A;‘+3E’+E”. The spectrum is well studied [13-151 and the analysis of the vibrational bands of the E’ species has yielded the values for the Coriolis coupling constants C,,(E’) .and ZZ2(E’) as 0.77+0.05 and 0.31+0.05 respectively [5]. The Raman band contour of the v3(E’) fundamental has also been recently analysed [16], which finally yields a Coriolis constant associated with the band lying in the range 02 Cs3(E’)> -0.30 thus supporting the above values. The vibrational amplitudes for the bonded and nonbonded distances between different atom pairs are also available from electron diffraction studies 1171. Raman intensities of the A’, bands have been recently analysed and used to evaluate uniquely the force field associated with the species
[181-
Despite the large amount of literature concerning such molecules, there remains an ambiguity in XY, types regarding the assignment of the frequencies v,(E’)and vj(E’) to the axial XY2 and equatorial in-plane XY, bending modes. Even if the assignment is completely known, a general valence force field for the E’ species would involve six Fii elements and the two zeta values along with the three vibrational frequencies provide insufficient data for unique fixing of the force field. However, there are contributions from the E’ species vibrations towards some of the vibrational amplitudes, which can provide a small share of useful information in unraveliing the E’ species force field. In an attempt to work out a force field in PF, fitting all the experimental data, Levin [5] has concluded that the axial bending frequency should be greater than the equatorial in-plane bending frequency. A force field fitting the vibrational frequencies as well as the Coriolis data for the E’ species, yet differing remarkably from that of Levin, has been reported by Lockett et al. [6] almost simultaneously. In calculating the force field Levin has assumed F,,(E’) = 0 while Lockett et al. have kept this element at -0.18 mdyn. An assumption on one of the Fii elements thus being arbitrary, Lockett has cautioned against the settling of the assignment probiem by Levin. Recently, we have also reported [7] a force field for the E’ species which fitted the frequencies as well as the Coriolis coupling constants (and even the vibrational amplitudes) extremely well, with very small values for the three interaction force constants.
APPLICATION
OF THE
PARAMETER
METHOD
The bond Iengths, the symmetry coordinates, the G matrix and the C matrix used for PF, are those employed by Levin [S]. The axial and equatorial bonds are designated as di and ri and the axial and in-plane angles are designated
as /Iii and czijrespectively- The symmetry coordinates S,(E), S,(E’) and S,(E’) involve changes in ri, ccii and Bii respectively. In applying the parameter method of mapping all the possible force fields fitting the vibrational frequencies and Coriolis constants, we first assume that. (case A) the equatorial in-plane bending frequency v(dc) to be greater than the axial bending frequency v(@. In a second approach, we shah reanalyse the problem under the assumption (case S) Y(P) > V(E)_ However, since there is no controversy over the assignment of v,(Z) to the equatorial stretching mode, eqns. (10) and (14) are not affected by the ambiguity of the assignment involved with the bending modes. The Jmatrix for the E’ species of PF, has a very convenient structure with Jir = -J& =0.4794, Jss = 1, Jrz ==J2r =0.8782 and Jz3 =.i3r = Jz3 =J& =0 Thus, one gets simplified expressions for the j elements (eqn. (8)) as jr,
= -L
= Jr, cos2 ~9r~tJr~ sin2 cp12
JIr2 = j zI = Jr, cos29p,,-J11
sin2 qt2
+ J33 I J,,
=
(17)
1 l
=
J3f
=
&3
=
33,
Fig. 1. Variationof ~~~elements values of p12_)
=
0
with cp12. C&, = I and $3
These results simpbfy eqn. (14) to the form tan (p13 = (‘;r:.:‘)
= j,,
= jz3 = j,,
= 0 for ail
7
The variation of the jij elements with qpl2 is platted in Fig. 1. The vaiue of cl I (E’) is taken to be 0.77 and eqn. (18) indicates ‘pi 3 will be imaginary for all cases of jtI > 0.77. The condition_?,, < 0.77 restrictsvlz to be less than lO”51’. Further, pr 3 has to be less than 45” in order that the given assignment (of vl(E’) to the > 0.54 from St (E’) mode) may hold good [l 1] and this leads to a constraint;,, eqn. (18), which is equivalent to ptl > 2”. Thus, one need consider only the range 912 = 2” to lO”51’ in order to map all the possible force fields fitting the experimental data in the E’ species vibrations of PF, .Thevari&ion of cpI3 with p12 is plotted in Fig. 2,
1
430’
Fig. 2..Variationof p13 with(pZ2.
Case A: v(t~) > v(B) The frequencies associated with the two bending modes have been reported [5] as 533 cm-’ and 179 cm-‘. Analysis of the 533 cm-l band has yielded [5] r;i value centred at 0.31. Thus, in this case, the assignment of the 533 cm- ’ band to the equatorial bending mode, prescribed by the symmetry coordinate S,(E’), shouId mean substitution of the zeta value associated with this band into eqn. (11). Consequently, substitution of this zeta value into eqn. (15), (solution of eqn. (1 f)) yields the possible values of (p2s for various valuesof v)l s .Thevariation of 9%3 and 923 values with q12 is shown graphicahy in Fig. 3. It is interesting to note that ah values of 923, and pp23sB be outside the +-n/4 to -n/4 region. Further, a smafi portion of the vahres of (p23ABand qZSBA in the range lO”51’ 3 rp,, 2 9”50’ also he outside the region. Thus, we need consider only the two sets of A matrices, viz. those for parameter sets q12:, ~~~~~ 923~sand 4112, p13B, cp2sBA.The force constants corresponding to these two sets
8
-80 L Fig. 3. Variation of q13 and qz3 with pllz in case A. The portions AB, BC, DE, EF, represent variation of q3tara, Q123gAtQ)23&, “p3jBs respectively, with qx2_ The curve with broken lines represents variation of 93% 3 with q, 2.
3‘
4’
5”
6‘
7‘
8’ -
9’ @12
Fig. 4a. Variation of Ffl eltements and I@-F,,) with pl,, for case A. (Parameters involved qs2. FX~A, 9~3_+~.)The units of the FtJ elements are as foIIowsr Fxz(E') in mdyn/A, F,,(E') in mdyn A, F&E') in mdyn A, F,,(E')in mdyn, F,.(E')in mdyn, and Fz3(l?) in mdyn A.
9
Experimental
value
= 0.041
r0.002
A
8-
Fig. 4b. Variation of Fri elements and I(P-F,,) with q ,2 for case A. (Parameters involved q12, P),~~, pzJsa_) Units for the FIi elements are given in the footnote to Fig. 4a.
have been calculated employing eqns. (6), (3) and (1) in turn and have been plotted as a function of parameter (p12 in Fig. 4(a) and Fig. 4(b). Case B: v(p) >
V(E)
In this case, the assignment of the 533 cm - ’ band goes to the axial bending mode prescribed by symmetry coordinate S,(E’), thus leading to the substitution of the corresponding Tii value (0.31) associated with this band into eqns. (12) and (16). The variation of ‘pz3 with cpl 3 is analysed using eqn. (16) and graphs relating (prs and (pz3 values with 4oi2 are drawn (Fig. 5). In this case all the values of q23AB and (P~~BA lie within the 7r/4 region. In addition, a small portion of the VahIeS of ~23~ and qzSBB in the range lO”51’ > ‘PI2 2 9”50’ also lie within the allowed region. The force fields corresponding to the four different sets of cpI 2, cpI 3, (p2 3 can be plotted as a function of 5o12, thus leading to four such mappings. However, choice of cpls as abscissa
10
UJ13
.
Ol-
f$3 46- --_
I3 ----
__
C
’
--__
3’
4’
5.
6’
% -4d-
____----
7’
__---
__--
8’
_\ 9’16
__--
MH
5.
5. .32 ‘. , ill’ , _*
E -6d-8d-F
Fig. 5. Variation of 4113and qz3 with p12 in case B. The portions AB, BC, DE, EF represent broken lines variation of pZ3AA, qZ3BA, qZ3ABy q23SS respectively, with p12. - the curve with represents variation of q13 with q112.
_:_____z__?i2__:~~e .03 Experimental due
=0041tQ002A
7-
,yF;2(E’) F&E’)
Fig. 6a. Variation of F,,elements and &P-F.,) with q,, for case B. (Parameters involved ~~2, qI 38, at 1x3 < 0 and v12, ~~3~~ q23AA at ~1,~ > 0.) Units for the Ffrelements are given in the footnote to Fig. 4a.
P)~~BB
11
I%. 6b. Variation of F~Jelements and &P-F& with
in case B. (Parameters invoIved tpt2, quota, and VIZ. Q~ISA~ 9323~~ at q13 > 0.) Units for the FfJelements are given in the footnote to Fig. 4a.
9238~
at
913
<
fp#J
0
enabies us to dispense with two separate additional diagrams, as can be seen from Fig. 6(a) and Fig. 6(b). The ranges of pi2 and 923 values are also indicated in the abscissae of these figures.
SOLUTION
OF THE
ASSiGNMENT
PROBLEM
The large muitiphcity of the force field solutions, as is evident from Figs. 4(a), 4(b), 6(a) and 6(b) fitting the vibrational frequencies as well as the Cor:olis data, owes its existence to the inadequacy of the experimental data, as already mentioned. The data on the mean amplitudes of vibration [17], along with the recent Raman intensity results [18], should provide some fruitful insight towards the eiimination of the spurious solutions. The mean amplitudes of vibration are related to theelements of a matrix C defined [19] in genera1 as z1= ==tAZ
(19)
with A,
=
_.w.!_ coth 8x2Vi
!-!!k
2kT
12 The terms in eqn. (20) have their usual meaning [19]_ The value of the vibrational amplitude Z(P-F,J arises from the contributions of the Z matrix elements from the A; and E’ species as
Investigations on the Raman intensities of the vr(A;) and v,(A;) bands have enabled unique fixing of the A; species force field [ 181 at Fr ,(A;) = 7.15 mdyn and F,,(A’,) = 0.9 mdyn a-‘. This leads to A-‘, F&A;) = 5.09 mdyn A-’ the unique evaluation of the L matrix and of the I: matrix elements. We have thus obtained Z,,(A;) = 0.001171 AZ, Z12(A;) = -0_0001175 A2 and Zz2(A;) = 0.001440 A’. The Z matrix elements corresponding to ali the E’ force fields mapped (Figs. 4 and 6) have been calculated through eqns. (6), (3) and (19). The vibrational amplitudes Z(P-F,J obtained from eqn. (21) are also plotted for the different force fields in the same Figs. 4 and 6, along with the experimental value 1171. The curve for the case A (v(cc) > v(p)) h as no intersection with the experimental value, while the portion of the curve corresponding to case B (v(p) > v(a)) lies within the experimental value in the range 8O36’ d (p12 d lO”51’. This result conclusively proves the correct assignment of the bending modes as v(axia1 bending) > v(equatoria1 in-plane bending).
ELIMINATION
OF ANOMALOUS
FORCE
FIELDS
The Z(P-F,,) data not only solve the assignment problem, but also reduces the range of analysis as 8”36’ < 40r2 < lO”51’. Among the other vibrational amphtudes Z(P-F,,) depends only on the Z eiements of A; and A’; species as
C@‘-J312 =
&2(4)+&&G’)
(22)
2
The parallel mean square amphtudes for the nonbonded distances between atom pairs F,, - * - F_, FCq * - - F,, and F,, - * - F,, have been evaluated by the method of Morino and Hirota [20]. The final expressions emerging for these constants are
IW,
.--F,,,)12
NK,
**-F,,)]2=0.1619 +0.2569 to.2511 +0.1678
U(F,,
= &,(A;)+
---L)12 =
=2264;)
1.6273 ZI,(E’)+0.1670 C,,(A;)+0.2570 C11(A;I)+0.08802 ~,,(E’)+O.l573 C,,(E”)
C,,(E’)-00.5781
Z,,(A;)+O.4O8l &,(A;‘)+O.3007 &@‘)+0.4515
C,,(A’,) &(A;I) &,(I?)
Z,,(E’) (23)
(24)
(25)
13
from the E’ species force field, the Thus, while Z(F,, - - - F’,) has no contribution W=,q - * - Fes) and the Z(Fes - - - Fax) data have contributions from the E’ species
force field. Since Z,,(A;) is uniquely fixed at 0.001172 A’, the variation of I(F,, - - - Fe,) has been evaluated in the range q12 = 8”30’ to lo”51 using eqn. (23), for the case v(p) > v(a). But it is noted that the contribution from the E’ species towards this value is aIways a constant, thus leading to a value of Z(F_, - * * Fe.,) = 0.094 A for the above range of (;pr2, as against the experimental value of 0.081 +O.OOS A, in agreement with the result reported by Levin [SJ. Use of eqn. (24) to analyse the variation of Z(F,, - - - F,,) with fpX2 causes difficulty, since the A’,’ species force field is not uniquely settled as there are only the two vibration frequencies as experimental data. However, along with the known value of Z,,(A;) = 0.001440 A2, eqn. (22) enables us to calculate that the range of .X1,(A;‘) is from 0.001922 to 0.002610 A2 in order to suit the electron diffraction result of Z(P-F,,) = 0.~3~0.002 A. The value of C,,(A;‘) and Z22(AT) corresponding to this range of Z,,(A’~) can be calculated from the secular equation [ 191 IZG--” -EA[
= 0
(26)
Substitution of these X values into eqn. (24) yields the contribution from the A;’ species towards [Z(F,, - * - F,,)]’ to be 0.002324$-0.000 133 A2. As the contribution from the A; species (I;,,(A;), Crz(A;), C22(A;) values are given earlier) and the E” species (Et r (E”) = 0.007 131 A2) are accurately known, eqn. (24) yields a mapping for the variation of Z(Fecl- - - F,,) as a function of ‘plz. Figure 7 gives a summary of such results catculated for the range 8” < (p12 d 10”51’ for the two sets of force fields mapped in Fig. 6(a) and Fig. 6(b). The resufts favour the case corresponding to Fig, 6(b), since the values in the other case lie well beyond the experimental limits.
-061
_---.._---__--_..-_ __ ___- --__ --...-L
‘0 i
I
8’
9’
IO” -GJ
If’ 12
Fig. 7. Variation of&F,, * - - F,,) with qx2. (-X-X-X-) corresponds to force field in Fig. 6a. (-0-O-O-j corresponds to force field in Fig. 6b. Value from electron studies for Z(FCg* * - Fes)= 00.059f0.003 A.
14 DISCUSSION
OF RESULTS
The setthng of the assignment in favourof v(axial bending) > v(equatoria1 in-plane bending) is in agreement with the result obtained by BarteII 1211recently. The result aIso supports the general conclusion that the axiai bending mode occurs in general at a higher frequency than the in-plane bending mode in a11 trigonal bipyramidal molecules, as observed by Wafker [22], through an ab initio approach to a hypothetical molecule of type XYs. Analysis of the assignment problem using mean Amplitudes of vibration [S] produced practically insensitive results for the Z(P-F& value, for both assiguments. However, the force field set used to represent the assignment v(equatoria1 in-pIane) > v(axia1) is pureiy diagonai and does not fit the Coriolis data. (For example, the
The force field set given by Lockett et al. [6’j Iies in ~hevicinityof (pz2 = lO”51’ = 0) in Fig. 6(a), However, comparison with vibrationaI amplitudes enabIes
us to discard all the sets in the figure as already shown. The force field sets by Levin IS] lie in the vicinity of (pls = -21” as far as the diagonal force constants are concerned and in the vicinity of pPf3 = - 15” as far as the off-diagonaI force constants are concerned (see Fig. 6(b)). The non-occurrence of Levi& resuh in an exact vertical may be traced to the fact that, while all the force fields in Fig. 6(b) fit a Coriolis data cl l(E’) = 0.77, Levin’ssetyieId a value of cl ,(E’) =0.83. (it is quite well known that the force constants are very sensitive to Coriolis data.) However, a set fitting the Coriofis data extremely well, reported by us earlier f7], Iies quite along a vertical at p13 = - 16”.
ACKNGWLEDGEMENT
It is a pleasure to acknowledge the many interesting and helpful discussions with Dr. C, P, G~~~ava~la~ha~in connection with this work.
REFERENCES 1 W_ J. Tayfor, J. CIIenz.P&s., 18 (1950) f301. 2 D. E. Freeman, f. Mof. Specfr5sc., 27 (f968) 27, 3 T. R. Ananthaktishnan and G. Aru’ldhas, f. Mol. Srrrrct., 13 (1972) 163. 4 T. R. Ananthakrishnan, C. P. Girijavallabhan and G. Arutdhas, 3. MO!_ Sirrrcf., I6 (1973) $49. 5 I. W. Levin, J. h-fol. Spectrosc., 33 jlQ70) 6 1. 6 P. Lockett, W- Fowler and P. M. Wilt, J. CJ~ew PJtys., 53 (1970) 02. 7 T. R. Ananthakrishnan and G. Aruldhas, Crirr. SC& 42 (1973) 311. 8 E,. B. Wilson, Jr., J. C. De&s and P. C. Cross, Moleei&r Vibmrions, McGraw-Hill, New York, 1955. 9 J, H. Meal and S. R. Polo, 3. C&m. Pltys., 24 (tQ56) 1126. IO E. B. Wilson, Jr., J. Clrern. P&vs., 9 f I941 1 76. 11 F. Torok, Acra Citinr. Acad, Sci. Hwg., 52 (1967) 205. I2 F. Tarok and P, Puiay, J. .&foL Srrrtct,, 3 (1969) 1. 13 J. E. Griffiths, R. P. Carter and R. R. Wotmes, J. C&m. PJtys-, 41 (1964) 863. f4 L C. Noskins and R. C. Lord, .i. C&n. P&zt 46 (1967) 2402. 15 F- A, Mirier and R. C. Capwelt, SpecrrocJdm. A&a, Part A, 27 (1971) X27. I6 J. D. Witt, L. A. Carreira and J. R. Durig, f. Mof. Strflct., 18 (1973) 1.57. I7 K. W. Hansen and L. S. Barteli, Ilrorg. Chem., 4 (1965) 1775. 18 Rager S. Gay, Bernardo Fontal and Thomas, G. Spiro, Inorg. CJzew, 12 (1973) 1881. 19 S. J. Cyvin, Molecrrlar Vibrarions and Mean Sq are A~~~Jir~~des~EIsevier, Amsterdam, 1968. 20 Y. Morino and H. Wirota. J_ Clretlr.p&s., 23 <1955) 737. 21 L. S. Bartefl, Inorg, Cffer?z..,9 ($970) $594. 22 W. Waiker, J. Mo1. Specrrosc., 43 (1972) 41 I.
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The Netherlands
PARAMETER METHOD IN THE CALCULATION OF FORCE CONSTANTS - APPLICATION TO E' SPECIES FORCE FIELD OF PHOSPHOROUS PENTAFLUORIDE T. R. ANANTHAKRISHNAN Deportment
(Received
of
Physics,
AND G. ARULDHAS
University of Kerala, Kariauattom,
695581 Trivandrum
(India)
16 May 1974)
ABSTRACT
A parameter method is developed for mapping all the mathematically possible force fields fitting the experimental data on the vibration frequencies and Coriolis coupling constants in the case of vibrational species of order three. The method is applied to the E' vibrations of phosphorous pentafluoride and all the possible force fields fitting the frequencies and Coriolis data, forboth assignments viz. v (equatorial in-plane bending) > v (axial bending) andviceversa, are mapped. The vibrational amplitudes /(P-F,-,) corresponding to these various force fields are also plotted. Comparison of the plot with the electron diffraction result conclusively settles the assignment in favour of v (axial bending) > v (equatorial in-plane bending). Comparison of the vibrational amplitudes [(P-F,,) and [(Fe‘, - - - F,,), for the remaining sets of force fields, with the electron diffraction results leads to a remarkable restriction of the ranges of parameters involved_ The final results yield a mapping for the force constants in the range F,,(E') = 4.18 to 5.79 mdyn A-‘, I;;,(E') = 0.23 to 0.57 mdyn A, F3s(E’) = 4.77 to 2.33 mdyn A, F,,(E') = -0.37 to 0.57 mdyn, F13(E') = -0.51 to 1.42 mdyn and F,,(E')=Oto -0.44 mdyn A. The result is discussed in the light of the various force fields suggested for the molecule by different authors.
INTRODUCTION
The complete determination of the general valence force field for a certain symmetry species vibration of a molecule with n vibrational frequencies requires +n(nt 1) force constants. Consequently the problem is indeterminate and Taylor El] has shown that all the solutions compatible with the vibration frequencies can be expressed as a function of +z(n - 1) parameters. Thus, any physically meaningful
2
approach to the probIem warrants the use of additional data [2] like Coriolis coupling constants [3], isotopic frequencies [4], etc. In one of our recent works [3], we have made a parametric approach to the problem of degenerate vibrations with n = 2, by associating Coriolis couphng constants as additional experimental data. In this paper, we present the extension of such parameter techniques to degenerate species with n = 3. The method is discussed with a special reference to the E’ vibrations of PF,, in which a great deal of indeterminacy prevails both in the assignment
THE PARAMETER
and in the force field [5-71.
APPROACH
The mathematical formulations are summarised as follows. The force constants I; and the Coriolis coupling constants 5 are ultimately governed by the normal coordinate transformation L through the following matrix relations. @)-AL-’ L-1
cE-’
= F
= 5
(2)
The symbols in these equations have their usual meanings [8,9]. In the parameter approach, the L matrix is split into two parts, one characteristic of the molecular structure and the other a periodic function of parameters vii such that L = TA
(3)
Here T is a trianguIar matrix defined by the relation T Y$= G, with G as the usual Wilson G matrix [IO]. A is an orthogonal matrix so as to satisfy the Wilson condition IlO] L.E = G and can be represented by means of Jpz(n - I) parameters [I ] in a vibrational problem of order n. In general, A can be written [ll, 121 as the product of &(n- 1) matrices A,, each dependent on a single angular parameter (~ii such that A =17A,
i,j = 1, 2, -
l
a, n, i cj
(4)
where A, is a unit matrix except for the iith and&h terms being equal to cos Cpi~, the ijth term being equal to -sin ‘pii and the jith term being equal to sin vii. Thus, the Aii’S are rotation matrices in the ij plane and the variousvalues of ‘pii along with eqns. (4), (3) and (1) serve to determine the complete set of mathematically possible force fields, compatible with the vibrational frequencies. For degenerate species, however, the Coriolis coupling constants cii are often available as experimental data. Substitution of eqn. (4) into eqns. (3) and (2) in turn, yields
3 where J = T-l C T-’ is a matrix dependent only on the molecular geometry and atomic masses. The zeta sum rule (Cx,i = ~iJiir since A is an orthogonal matrix) reduces the number of independent cii elements to (n- 1) and brings in an equal number of constraints on the $z(n- 1) parameters, thus leaving the number of independent parameters as ~(KZ - f )(n-2). Thus, for the n = 2 case, the parameter Cpiishould possess a unique value, the evaluation of which has already been discussed earlier [3]. For the n = 3 case, the force fieId fitting the vibrational frequencies as well as the Coriofis data should therefore be expressible as a function of a single parameter. There are three rotation matrices A, in the third-order problem and a restriction on their sequence as A = AU
A13
(6)
A23
limits the variation of each parameter y>iito within the region -n/4 < ‘pie d n/4 in order that a given assignment may hold good, as shown by Torok II 1 J. Substitution of eqn. (6) into eqn. (5) yields _ _ c A23A13A12JAZ2Af3
A23
5
=
If we fix qr2 constituting A,, matrix $ such that
(7)
as the independent parameter and define a
,j = A,, JAI2
(8)
the form of eqn. (7) becomes
,.S
-
A23A132
4343
=
(9)
4
Expansion of eqn. (9) yields j,,
cos’~~~i-.j~~
(p-q)
sin2 cp13+ti13
sin2p23+r
(p-q)COS2tp,3fr
coscp13Sinq13
=
COS2rp23+2SCOS
v)23
sin
5023 =
sin2Cp23--2SCOS
023
sin
Cp23 =Jt33
T~I
c22
(10) (11) (12)
where p = J,, 4 =
sin2 q213
zs,,
+
j33COS2(P13
U3)
sinV]13 cosfp13
r = .f,, S=
9 23 cosfP13
- J 12 Sin4013
The solution for pl 3 can be written from eqn. (10) as -f,34t313-(.2",,-5,l)(~33-rlr)l* aaSP
= J33-
5 11
(14).
4 The solution for tancpz3 =
following from eqn. (11) can be written as
(~23
-s+rsZ-(p-9--e22)(p-r,,)llf
(15)
P-9-522
Similarly the solution for qt3 following from eqn. (12) can be written as tan
923
=
~JII~=-(~-cr-r,,)(~-;,,)If
06)
r--c33
In the actual calcufation, cprs wiff have two values, say qlsA and cp13s, for each vaIue of (p12 according as the plus and minus sign preceding the square root in eqn. (14). The p, q, r, s elements in eqn, (13) can be cafculated for each value of yrl3 and used to calculate qz3 from eqn. (15) or (16). (Both these equations lead to the same solution as a consequence of the zeta sum rule.) Once again, there witi be two values of (~23 for each value of cprs. The two solutions of p23 arising from the value of p1 3Acan be called pZJAAand ~~~~~~The two solutions for ~23 arising from the value of cplsB can be called 923BA and c;p,,,. Thus, there are values, which in turn lead to four sets of matrices four sets Of 912, Y)13&23 A - A12A13A23 for a given value of cp12. The emergence of the four sets of A matrices for a given arbitrary value of 40f 2 in a general case is illustrated as follows. (;013Aeqn, Fixrp,2e9"*(14)
~
(15)Or(16)+
a
( qr3Beqn.
(15)Of(t6)+
,,_~923AA=.~
\
q23AB==)A
=
4243AA23AA
=
A12A1.3~~23.a
<~23BA3A2 Y)23BB=+A
= =
A,2A13BA238A A,Z
A,33A23BB
The suffixes following each A, matrix indicates the suffix in the parameter q involved in the matrix. Each of these A matrices leads to a certain force fields, which can be cafctdated through eqns. (3) and (1). The method thus provides a systematic approach to mapping of all the force constants compatibIe with the vibrational frequencies and Coriolis data as a function of a single parameter 9 r Z. Further, one gets four such mappings in generat, owing to the four A matrices emerging from the calcufations. Since the A matrix has been taken in the order A12A13A23, the value of the parameter rplz can be limited in the range --n/4 to + ~14, in anaIysing a given assignment of frequencies I1 11. Similar conditions are valid for cpl3 and 923 also and can often enabie further restrictions in the region of analysis for the pii parameters. In cases where additional experimental data like vibrational amplitudes, absolute intensities or centrifugal distorsion constants are available, further Iimiting of the ranges of parameters can be achieved such that the force field fits such data also.
5 PROBLEMS
WI-H
THE
E’
SPECIES
VIBRATIONS
OF
PF,
The trigonal bipyramidal molecule belongs to D,, symmetry and has vibrational representation r = 2A; +2A;‘+3E’+E”. The spectrum is well studied [13-151 and the analysis of the vibrational bands of the E’ species has yielded the values for the Coriolis coupling constants C,,(E’) .and ZZ2(E’) as 0.77+0.05 and 0.31+0.05 respectively [5]. The Raman band contour of the v3(E’) fundamental has also been recently analysed [16], which finally yields a Coriolis constant associated with the band lying in the range 02 Cs3(E’)> -0.30 thus supporting the above values. The vibrational amplitudes for the bonded and nonbonded distances between different atom pairs are also available from electron diffraction studies 1171. Raman intensities of the A’, bands have been recently analysed and used to evaluate uniquely the force field associated with the species
[181-
Despite the large amount of literature concerning such molecules, there remains an ambiguity in XY, types regarding the assignment of the frequencies v,(E’)and vj(E’) to the axial XY2 and equatorial in-plane XY, bending modes. Even if the assignment is completely known, a general valence force field for the E’ species would involve six Fii elements and the two zeta values along with the three vibrational frequencies provide insufficient data for unique fixing of the force field. However, there are contributions from the E’ species vibrations towards some of the vibrational amplitudes, which can provide a small share of useful information in unraveliing the E’ species force field. In an attempt to work out a force field in PF, fitting all the experimental data, Levin [5] has concluded that the axial bending frequency should be greater than the equatorial in-plane bending frequency. A force field fitting the vibrational frequencies as well as the Coriolis data for the E’ species, yet differing remarkably from that of Levin, has been reported by Lockett et al. [6] almost simultaneously. In calculating the force field Levin has assumed F,,(E’) = 0 while Lockett et al. have kept this element at -0.18 mdyn. An assumption on one of the Fii elements thus being arbitrary, Lockett has cautioned against the settling of the assignment probiem by Levin. Recently, we have also reported [7] a force field for the E’ species which fitted the frequencies as well as the Coriolis coupling constants (and even the vibrational amplitudes) extremely well, with very small values for the three interaction force constants.
APPLICATION
OF THE
PARAMETER
METHOD
The bond Iengths, the symmetry coordinates, the G matrix and the C matrix used for PF, are those employed by Levin [S]. The axial and equatorial bonds are designated as di and ri and the axial and in-plane angles are designated
as /Iii and czijrespectively- The symmetry coordinates S,(E), S,(E’) and S,(E’) involve changes in ri, ccii and Bii respectively. In applying the parameter method of mapping all the possible force fields fitting the vibrational frequencies and Coriolis constants, we first assume that. (case A) the equatorial in-plane bending frequency v(dc) to be greater than the axial bending frequency v(@. In a second approach, we shah reanalyse the problem under the assumption (case S) Y(P) > V(E)_ However, since there is no controversy over the assignment of v,(Z) to the equatorial stretching mode, eqns. (10) and (14) are not affected by the ambiguity of the assignment involved with the bending modes. The Jmatrix for the E’ species of PF, has a very convenient structure with Jir = -J& =0.4794, Jss = 1, Jrz ==J2r =0.8782 and Jz3 =.i3r = Jz3 =J& =0 Thus, one gets simplified expressions for the j elements (eqn. (8)) as jr,
= -L
= Jr, cos2 ~9r~tJr~ sin2 cp12
JIr2 = j zI = Jr, cos29p,,-J11
sin2 qt2
+ J33 I J,,
=
(17)
1 l
=
J3f
=
&3
=
33,
Fig. 1. Variationof ~~~elements values of p12_)
=
0
with cp12. C&, = I and $3
These results simpbfy eqn. (14) to the form tan (p13 = (‘;r:.:‘)
= j,,
= jz3 = j,,
= 0 for ail
7
The variation of the jij elements with qpl2 is platted in Fig. 1. The vaiue of cl I (E’) is taken to be 0.77 and eqn. (18) indicates ‘pi 3 will be imaginary for all cases of jtI > 0.77. The condition_?,, < 0.77 restrictsvlz to be less than lO”51’. Further, pr 3 has to be less than 45” in order that the given assignment (of vl(E’) to the > 0.54 from St (E’) mode) may hold good [l 1] and this leads to a constraint;,, eqn. (18), which is equivalent to ptl > 2”. Thus, one need consider only the range 912 = 2” to lO”51’ in order to map all the possible force fields fitting the experimental data in the E’ species vibrations of PF, .Thevari&ion of cpI3 with p12 is plotted in Fig. 2,
1
430’
Fig. 2..Variationof p13 with(pZ2.
Case A: v(t~) > v(B) The frequencies associated with the two bending modes have been reported [5] as 533 cm-’ and 179 cm-‘. Analysis of the 533 cm-l band has yielded [5] r;i value centred at 0.31. Thus, in this case, the assignment of the 533 cm- ’ band to the equatorial bending mode, prescribed by the symmetry coordinate S,(E’), shouId mean substitution of the zeta value associated with this band into eqn. (11). Consequently, substitution of this zeta value into eqn. (15), (solution of eqn. (1 f)) yields the possible values of (p2s for various valuesof v)l s .Thevariation of 9%3 and 923 values with q12 is shown graphicahy in Fig. 3. It is interesting to note that ah values of 923, and pp23sB be outside the +-n/4 to -n/4 region. Further, a smafi portion of the vahres of (p23ABand qZSBA in the range lO”51’ 3 rp,, 2 9”50’ also he outside the region. Thus, we need consider only the two sets of A matrices, viz. those for parameter sets q12:, ~~~~~ 923~sand 4112, p13B, cp2sBA.The force constants corresponding to these two sets
8
-80 L Fig. 3. Variation of q13 and qz3 with pllz in case A. The portions AB, BC, DE, EF, represent variation of q3tara, Q123gAtQ)23&, “p3jBs respectively, with qx2_ The curve with broken lines represents variation of 93% 3 with q, 2.
3‘
4’
5”
6‘
7‘
8’ -
9’ @12
Fig. 4a. Variation of Ffl eltements and I@-F,,) with pl,, for case A. (Parameters involved qs2. FX~A, 9~3_+~.)The units of the FtJ elements are as foIIowsr Fxz(E') in mdyn/A, F,,(E') in mdyn A, F&E') in mdyn A, F,,(E')in mdyn, F,.(E')in mdyn, and Fz3(l?) in mdyn A.
9
Experimental
value
= 0.041
r0.002
A
8-
Fig. 4b. Variation of Fri elements and I(P-F,,) with q ,2 for case A. (Parameters involved q12, P),~~, pzJsa_) Units for the FIi elements are given in the footnote to Fig. 4a.
have been calculated employing eqns. (6), (3) and (1) in turn and have been plotted as a function of parameter (p12 in Fig. 4(a) and Fig. 4(b). Case B: v(p) >
V(E)
In this case, the assignment of the 533 cm - ’ band goes to the axial bending mode prescribed by symmetry coordinate S,(E’), thus leading to the substitution of the corresponding Tii value (0.31) associated with this band into eqns. (12) and (16). The variation of ‘pz3 with cpl 3 is analysed using eqn. (16) and graphs relating (prs and (pz3 values with 4oi2 are drawn (Fig. 5). In this case all the values of q23AB and (P~~BA lie within the 7r/4 region. In addition, a small portion of the VahIeS of ~23~ and qzSBB in the range lO”51’ > ‘PI2 2 9”50’ also lie within the allowed region. The force fields corresponding to the four different sets of cpI 2, cpI 3, (p2 3 can be plotted as a function of 5o12, thus leading to four such mappings. However, choice of cpls as abscissa
10
UJ13
.
Ol-
f$3 46- --_
I3 ----
__
C
’
--__
3’
4’
5.
6’
% -4d-
____----
7’
__---
__--
8’
_\ 9’16
__--
MH
5.
5. .32 ‘. , ill’ , _*
E -6d-8d-F
Fig. 5. Variation of 4113and qz3 with p12 in case B. The portions AB, BC, DE, EF represent broken lines variation of pZ3AA, qZ3BA, qZ3ABy q23SS respectively, with p12. - the curve with represents variation of q13 with q112.
_:_____z__?i2__:~~e .03 Experimental due
=0041tQ002A
7-
,yF;2(E’) F&E’)
Fig. 6a. Variation of F,,elements and &P-F.,) with q,, for case B. (Parameters involved ~~2, qI 38, at 1x3 < 0 and v12, ~~3~~ q23AA at ~1,~ > 0.) Units for the Ffrelements are given in the footnote to Fig. 4a.
P)~~BB
11
I%. 6b. Variation of F~Jelements and &P-F& with
in case B. (Parameters invoIved tpt2, quota, and VIZ. Q~ISA~ 9323~~ at q13 > 0.) Units for the FfJelements are given in the footnote to Fig. 4a.
9238~
at
913
<
fp#J
0
enabies us to dispense with two separate additional diagrams, as can be seen from Fig. 6(a) and Fig. 6(b). The ranges of pi2 and 923 values are also indicated in the abscissae of these figures.
SOLUTION
OF THE
ASSiGNMENT
PROBLEM
The large muitiphcity of the force field solutions, as is evident from Figs. 4(a), 4(b), 6(a) and 6(b) fitting the vibrational frequencies as well as the Cor:olis data, owes its existence to the inadequacy of the experimental data, as already mentioned. The data on the mean amplitudes of vibration [17], along with the recent Raman intensity results [18], should provide some fruitful insight towards the eiimination of the spurious solutions. The mean amplitudes of vibration are related to theelements of a matrix C defined [19] in genera1 as z1=
(19)
with A,
=
_.w.!_ coth 8x2Vi
!-!!k
2kT
12 The terms in eqn. (20) have their usual meaning [19]_ The value of the vibrational amplitude Z(P-F,J arises from the contributions of the Z matrix elements from the A; and E’ species as
Investigations on the Raman intensities of the vr(A;) and v,(A;) bands have enabled unique fixing of the A; species force field [ 181 at Fr ,(A;) = 7.15 mdyn and F,,(A’,) = 0.9 mdyn a-‘. This leads to A-‘, F&A;) = 5.09 mdyn A-’ the unique evaluation of the L matrix and of the I: matrix elements. We have thus obtained Z,,(A;) = 0.001171 AZ, Z12(A;) = -0_0001175 A2 and Zz2(A;) = 0.001440 A’. The Z matrix elements corresponding to ali the E’ force fields mapped (Figs. 4 and 6) have been calculated through eqns. (6), (3) and (19). The vibrational amplitudes Z(P-F,J obtained from eqn. (21) are also plotted for the different force fields in the same Figs. 4 and 6, along with the experimental value 1171. The curve for the case A (v(cc) > v(p)) h as no intersection with the experimental value, while the portion of the curve corresponding to case B (v(p) > v(a)) lies within the experimental value in the range 8O36’ d (p12 d lO”51’. This result conclusively proves the correct assignment of the bending modes as v(axia1 bending) > v(equatoria1 in-plane bending).
ELIMINATION
OF ANOMALOUS
FORCE
FIELDS
The Z(P-F,,) data not only solve the assignment problem, but also reduces the range of analysis as 8”36’ < 40r2 < lO”51’. Among the other vibrational amphtudes Z(P-F,,) depends only on the Z eiements of A; and A’; species as
C@‘-J312 =
&2(4)+&&G’)
(22)
2
The parallel mean square amphtudes for the nonbonded distances between atom pairs F,, - * - F_, FCq * - - F,, and F,, - * - F,, have been evaluated by the method of Morino and Hirota [20]. The final expressions emerging for these constants are
IW,
.--F,,,)12
NK,
**-F,,)]2=0.1619 +0.2569 to.2511 +0.1678
U(F,,
= &,(A;)+
---L)12 =
=2264;)
1.6273 ZI,(E’)+0.1670 C,,(A;)+0.2570 C11(A;I)+0.08802 ~,,(E’)+O.l573 C,,(E”)
C,,(E’)-00.5781
Z,,(A;)+O.4O8l &,(A;‘)+O.3007 &@‘)+0.4515
C,,(A’,) &(A;I) &,(I?)
Z,,(E’) (23)
(24)
(25)
13
from the E’ species force field, the Thus, while Z(F,, - - - F’,) has no contribution W=,q - * - Fes) and the Z(Fes - - - Fax) data have contributions from the E’ species
force field. Since Z,,(A;) is uniquely fixed at 0.001172 A’, the variation of I(F,, - - - Fe,) has been evaluated in the range q12 = 8”30’ to lo”51 using eqn. (23), for the case v(p) > v(a). But it is noted that the contribution from the E’ species towards this value is aIways a constant, thus leading to a value of Z(F_, - * * Fe.,) = 0.094 A for the above range of (;pr2, as against the experimental value of 0.081 +O.OOS A, in agreement with the result reported by Levin [SJ. Use of eqn. (24) to analyse the variation of Z(F,, - - - F,,) with fpX2 causes difficulty, since the A’,’ species force field is not uniquely settled as there are only the two vibration frequencies as experimental data. However, along with the known value of Z,,(A;) = 0.001440 A2, eqn. (22) enables us to calculate that the range of .X1,(A;‘) is from 0.001922 to 0.002610 A2 in order to suit the electron diffraction result of Z(P-F,,) = 0.~3~0.002 A. The value of C,,(A;‘) and Z22(AT) corresponding to this range of Z,,(A’~) can be calculated from the secular equation [ 191 IZG--” -EA[
= 0
(26)
Substitution of these X values into eqn. (24) yields the contribution from the A;’ species towards [Z(F,, - * - F,,)]’ to be 0.002324$-0.000 133 A2. As the contribution from the A; species (I;,,(A;), Crz(A;), C22(A;) values are given earlier) and the E” species (Et r (E”) = 0.007 131 A2) are accurately known, eqn. (24) yields a mapping for the variation of Z(Fecl- - - F,,) as a function of ‘plz. Figure 7 gives a summary of such results catculated for the range 8” < (p12 d 10”51’ for the two sets of force fields mapped in Fig. 6(a) and Fig. 6(b). The resufts favour the case corresponding to Fig, 6(b), since the values in the other case lie well beyond the experimental limits.
-061
_---.._---__--_..-_ __ ___- --__ --...-L
‘0 i
I
8’
9’
IO” -GJ
If’ 12
Fig. 7. Variation of&F,, * - - F,,) with qx2. (-X-X-X-) corresponds to force field in Fig. 6a. (-0-O-O-j corresponds to force field in Fig. 6b. Value from electron studies for Z(FCg* * - Fes)= 00.059f0.003 A.
14 DISCUSSION
OF RESULTS
The setthng of the assignment in favourof v(axial bending) > v(equatoria1 in-plane bending) is in agreement with the result obtained by BarteII 1211recently. The result aIso supports the general conclusion that the axiai bending mode occurs in general at a higher frequency than the in-plane bending mode in a11 trigonal bipyramidal molecules, as observed by Wafker [22], through an ab initio approach to a hypothetical molecule of type XYs. Analysis of the assignment problem using mean Amplitudes of vibration [S] produced practically insensitive results for the Z(P-F& value, for both assiguments. However, the force field set used to represent the assignment v(equatoria1 in-pIane) > v(axia1) is pureiy diagonai and does not fit the Coriolis data. (For example, the
The force field set given by Lockett et al. [6’j Iies in ~hevicinityof (pz2 = lO”51’ = 0) in Fig. 6(a), However, comparison with vibrationaI amplitudes enabIes
us to discard all the sets in the figure as already shown. The force field sets by Levin IS] lie in the vicinity of (pls = -21” as far as the diagonal force constants are concerned and in the vicinity of pPf3 = - 15” as far as the off-diagonaI force constants are concerned (see Fig. 6(b)). The non-occurrence of Levi& resuh in an exact vertical may be traced to the fact that, while all the force fields in Fig. 6(b) fit a Coriolis data cl l(E’) = 0.77, Levin’ssetyieId a value of cl ,(E’) =0.83. (it is quite well known that the force constants are very sensitive to Coriolis data.) However, a set fitting the Coriofis data extremely well, reported by us earlier f7], Iies quite along a vertical at p13 = - 16”.
ACKNGWLEDGEMENT
It is a pleasure to acknowledge the many interesting and helpful discussions with Dr. C, P, G~~~ava~la~ha~in connection with this work.
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