2586
J . Am. Chem. SOC.1987, 109, 2586-2588
Vibrational Analysis and Mean Bond Displacements in M(XY& Complexes Cynthia J. Jameson Contribution from the Department of Chemistry, University of Illinois at Chicago, Chicago, Illinois 60680, Received July 3, 1986
Abstract: An empirical quadratic GVFF potential for M(XY)6 molecules, coupled with explicit anharmonicities in the form of a Morse potential and a Urey-Bradley interaction between each nearest-neighbor nonbonded pair of X atoms, is used to model the stretching anharmonicities of the MX and XY bonds. Mean V-C, C-0, Co-C, and C-N bond displacements and mean square amplitudes are calculated for V(co)6- and CO(CN),~-.
N M R chemical shifts of transition metal nuclei are very sensitive to the metal-ligand distance. Measures of the sensitivity of transition metal shielding to the metal-ligand distance are the observed large chemical shifts with temperature and upon isotopic substitution. For example, 59C0shifts of 1.4 to 3 ppm/deg and 5'V shifts of 0.3 to 1.5 ppm/deg have been rep0rted.l Isotope shifts are also large: -4.7, -6, and -10 ppm on D substitution in (CpM(CO),H] for M = 'lV, 93Nb, and lS3W,respectively.2 Furthermore, the temperature coefficients of the shifts of 51Vin various vanadium carbonyl complexes show an interesting correlation with the chemical shifts at 300 K. A theory to interpret these very large shifts requires the knowledge of the mean bond length changes in these molecules in the form of various rovibrational averages, ( A r ) and ( ( A r ) 2 ) .V(CO),- is a reasonable prototype of these octahedral complexes. In this paper we use the vibrational frequencies of V(CO)6- to determine an empirical quadratic force field which is consistent with the ones which have been established for the analogous M(CO)6 neutral molecules (M = Cr, Mo, W). We augment this with cubic force constants calculated using an anharmonic model for stretching and nonbonded interactions, a model which has been successful in reproducing the 10 stretching-mode anharmonicities that are presently known for SF6. We calculate the thermal averages ( Ar) and ( ( A r ) 2 )for V-C and C-0 bonds in V ( C 0 ) c (13/12C,l8/l60) and also for Co-C and C-N bonds in Co(CN),'- (13/12C,15/14N) for comparison. The complete quadratic force field for the metal carbonyls Cr(C0)6, M o ( C O ) ~ and , W(CO)6 has been established by a comprehensive study of the vibrational spectra of the molecules M(12C160)6, M(13C160)6, and M(12ClsO)6(M = Cr, Mo, W),3 These studies lead to the important result that the interaction force. constants are reasonably transferable from one M(CO)6 molecule to another. The force field for V(CO)6- has not been reported. Only the frequencies for the 12C160 species are known from the work of Abel et aL4 However, it has been shown that in the M(CO)6 series (M = Cr, Mo, W) most of the interaction force constants have equal or nearly equal magnitudes irrespective of M.3 Therefore, we will determine the force. field for V(CO)6- with the assumption that these interaction force constants which are invariant in the Cr, Mo, and W hexacarbonyls can be used for V(CO)6- to establish the off-diagonal symmetry force constants. That is, we will use the same values for fMc,co, ffMc,co., fCO,g$f C O . d , fCO,d,,fMC,@', fMC,?', fMC.oN?f a g , fop"? and f o p , as were reported for Cr(C0)6. The displacement coordinates are desig(1) Benedek, G. E.; Englman, R.; Armstrong, J. A. J . Chem. Phys. 1963, 39, 3349-3363. Jameson, C. J.; Rehder, D.; Hoch, M. J . Am. Chem. SOC., following paper in this issue. (2) Hoch, M.; Rehder, D. Inorg. Chim. Acta 1986,111, L13. Naumann, F.; Rehder, D.; Pank, V. J . Organomer. Chem. 1982, 240, 363. McFarlane, H. C. E.; McFarlane, W.; Rycroft, D. S. J . Chem. SOC.,Dalton Trans. 1976, 1616. (3) Jones, L. H.; McDowell, R. S.; Goldblatt, M . Inorg. Chem. 1969, 8, 2349-2363. (4) Abel, E. W.; McLean, R. A. N.; Tyfield, S. P.; Braterman, P. S.; Walker, A. P.; Hendra, P. J. J . Mol. Spectrosc. 1969, 30, 29-50.
0002-786318711509-2586$01.50/0
Table I. Valence Force Constants for V(CO)6-
valence stretching
valence angle bending force constants, mdvn A rad-I
force constants, mdyn
A-I
fco
15.030 2.160 f'do.co, 0.285 0.180 -0.025 f fhlC,MC' 0.360 JMC,CO 0.683 f'MC,CO' -0.052 f fhlc.cw -0.097
J6
= 0.4595
fpp = 0.1005 fpg" = 0.0005 f p p = -0.005
fMC
(fa - f a n , , ) = 0.50 - J&) = 0.075 (f,,.. = 0.01
&+
Table 11. Svmmetrv Force Constants and Freauencies for V(C0L-
symmetry force
constants' Ai.. D
frequencies, cm" v(12c160)6-~ ( 1 3 ~ 1 6 0 ) ~ -~ ( 1 2 ~ 1 8 0 ) ~ obsdb calcd calcd calcd
31,
16.35 2036 2.42 374 0.38 E, 333 14.64 w3 1908 393 344 2.57 334 0.69 356 Fig 355 0.358 Flu 366 14.85 w6 1895 650 YT7 1.80 460 Ygg 0.55 0.65 92 399 Y6, 0.78
2034.5 378.1
1987.0 371.9
1989.1 364.6
1907.7 391.4
1863.3 385.0
1864.8 377.5
356.1 1897.9 655.1 454.4 91.8
345.4 1855.1 643.1 445.5 91.3
351.6 1853.2 65 1.9 445.9 87.9
517 84
518.2 85.0
499.6 84.7
5 15.7
506
505.7 67.0
489.2 66.6
501 .O 63.8
1.
322 312
368 369 378
-0.18 -0.3 gg9 -0.21 3 1 0 ~ 0 0.36 ~ 1 1 , I l 0.52 910.11 -0.52 312.12 0.57 3 1..,.. 3 I3 O.35 370
F2,
F2"
312.13
80.5
-O.ll
"Units are mydn A-' except mdyn rad-' for y(68,F6,, ?78, and 3 7 9 , and mdyn A rad-2 for 3~ 3881 5 % 5%. Slo.lo~ ~ l o , l l , 911,11, 3 1 2 ~ 2 , 312,13, and 313,13. The definitions of the symmetry coordinates are the same as in ref 3. *From ref 4. Only the frequencies for vibrations I , 3, and 6 have been corrected for anharmonicity; that is, harmonic frequencies rather than observed frequencies are given for these vibrations only.
nated in the same way as in ref 3. The CMC angles are labeled a and the MCO angle is labeled /3. The quadratic force constants (shown in Table I) are the second derivatives of the potential energy with respect to AT,,, ARC,, AcucMc, and Ap,co. The fundamental frequencies for V(CO)6- are then used to determine the following force constants: fco, f"co,co., and fco,co., 0 1987 American Chemical Society
J . A m . Chem. Soc., Vol. 109, No. 9, 1987 2587
Mean Bond Displacements in M(XY)6Complexes
Table 111. Evaluation of Approximate Cubic Force Constants for M(CO),-Type Molecules" Fijk
term
f,, f 5::.
Ar13 Ar12Ar2 ( r k ~ ) ~ Ariz(riAaiJ) Arl ArArAalJ
faon
f,,, fd* f,,,
Arl (:I
ARI (d3V/Ia%idRjdR,), all in mydn 1 of ref 7.
-1 1.343 -0.203 -0.236 -0.122 -0.170 -0.162 -125.8
-1 1.313 -0.093 -0.109 -0.056 -0.078 -0.075 -108.9
mb
fRRR
OFrJk
CO(CN),'-
v(c0)6-
(l/rMC)(F3 + 3 F - 3 F 1 - 3aMCKMC (1/4rMCW3 - F + (1/4rMc)(F3 - 3 F - F1 ( 1 / 4 r ~ c ) ( F 3+ 3 F - 3F9 ( I / ~ ~ M C )+( F ~+ 3 F 1 ( 1 / 4 r d F 3 + F - F? -3acoKco or -3acNKcN
A-2, F'= (dV/'/aq)o/qo, F 1 (d2V/dq2)o,F31 qO(Ia3V/Iaq3)),. bNotethe typographical error in this term in eq
from wl, w3, and Wg;fMC,fCMC,MC'9 andfMC,MC,from w2, ~ 4 and % fp, f o p ,fop,and for, from w51c08r wI0, and wI2; and finally the linear combinations cf, -fa,,.), cfmd -fe,...), and cf,. -fJe) from w g , wI,, and ~ 1 3 .Since ~ 1 was 3 not observed, we used the same symmetry force constant 313,13 as in Cr(C0)6. This fixes the sum cf, -f,,,.)- 2cf,,, -fa,...). The complete set of force constants is given in Tables I and 11. These are consistent with the set for C r - , Mo-, and W(CO), in that the values for V ( C O ) 6 -are not drastically different and the relative magnitudes of cis vs. trans force constants and other such trends are preserved. The elements of the Gs matrix have been given by Jones et aL5 Solution of the GF matrix problem reproduces the fundamental frequencies of V ( C O ) 6 -to within *2 cm-'. Frequencies for the I3C and " 0 isotopomers calculated with this set of force constants are also given in Table 11. In order to calculate mean V-C and C-O bond displacements, we need some reasonable estimate of the anharmonicity of the bonds. Here we extend the method of Krohn and Overend for SF6,6 which we have successfully applied to other molecules of this type (SeF,, TeF,, PtCI;-, and ptBr62-).7 We assume, as they did, that the anharmonicity can be described by a stretching Morse anharmonicity combined with nonbonded interactions. In this way we can derive the expressions for the cubic force constants as shown in Table 111. The Morse parameters avc = 1.7 11 A-' and aco = 2.416 8,-' are calculated by the method of Herschbach and Laurie' using the bond lengths r,(CO) = 1.146 8, and r,(VC) = 1.931 A in V(CO),- from the X-ray cr stal structure.' Kvc = 2.16 mdyn/A and Kco = 15.03 mdyn/ were estimated from thefMc andfco values found in this vibrational analysis (see Table I). We neglect all contributions of nonbonded interactions to the cubic force constants involving the C - 0 stretch, so the entire anharmonicity of the CO bond is due to Morse anharmonicity. Terms other than those types shown in Table I11 are neglected. The nonbonded interaction constant F(C.-C) in Table 111 is obtained from the symmetry force constant 3 2 2 = KMC 4F. For V(CO),- F = 0.065 mdyn/8, and F' and F3 are taken to be -0.0065 and -0.65 mdyn/A according to the usual recipe in which F' = -0.1 F, F3 == -1 OF. The vibrational contributions to the mean bond displacements are calculated using the method of Bartelllo as implemented in our previous work," from which the following is easily derived: 0,;
8'
+
Table IV. Mean Bond Displacements and Mean Square Amplitudes for the V-C Bond and the C-0 Bond in V(CO),'"
T
V('2C'60)6- 300 V(13C160)6- 300 V('2C180)6- 300
V('3C160)6- 300 V('2C'80).5- 300 V('2C160)c 200 400 a
A.
(Ar) 12.0875 1 1.9602 12.08 10 9.8122 10.6661 11.5980 12.5900 13.6278 14.7013
((Ar)') 3.2451 3.2016 3.2380 2.7734 2.9453 3.1406 3.3535 3.5798 3.8167
4.2026 4.1904 4.2958 4.3077
0.1105 0.1105 0.0737 0.1474
4.3131 4.3009 4.3694 4.4551
1.2549 1.2513 1.2817 1.2862
Ar refers to the V-C bond and 4 R to the CO bond; all are in
(a3v/ankanianJ),
Ehrenfest's theorem, we obtain the set of coupled equations: .^
Thus,
(R)= F ' Z
(3)
The column vector ( R ) contains the 12 desired mean displacements (72,)of the CO and the MC bonds. The vector X contains the elements 12
Here W i stands for the curvilinear internal coordinates ARco or ArMc for i = 1 to 6 and i = 7 to 12, respectively, Aa for i = 13
0.4432 0.4432 0.4432 0.2954 0.3545 0.4136 0.4727 0.53 18 0.5909
to 24, and Ap for i = 25 to 36. r = r,(CO) for k = 1 to 6 and r = r,(MC) for k = 7 to 12. c k j = 1 if the bond to atom k is in the jth bond angle deformation; otherwise ckJ = 0. F,J stands for (a2v/aRianj) and Fkjj for given in Tables I and 111, respectively. As in previous work, in eq 7 we have neglected the averages (%?k%j%,)and also set the sums over ( A m ) and ( A b ) to zero. Upon setting ( a V / a z k ) = 0, i.e., applying
z k
k = 1 to 12 (1)
200 240 280 320 360 400
v('2c160)6-
(ArL 1 1.6443 11.5170 11.6379 9.5167 10.3115 11.1844 12.1172 13.0960 14.1104
=
c
36
i=1 j = 1 3
F.. 2(-%iBj)ckj 2r
36
+
36
FiJ
c z(n,nJ)(ckJ +
cki)
i=13 1=13
+
All the mean square amplitudes ( W , R J ) such as ( S r 1 2 ) or ( ArlAa12),etc., including all cross-terms are evaluated as =
C-L(Q,Z)LsJ
(5)
S
( 5 ) Jones, L. H. J . Mol. Specirosc. 1962, 8, 105-120. (6) Krohn, B. J.; Overend, J. J . Phys. Chem. 1984, 88, 564-574. (7) Jameson, C. J.; Jameson, A. K . J . Chem. Phys. 1986,85, 5484-5492. (8) Herschbach, D. R.; Laurie, V. W. J . Chem. Phys. 1961,35,458-463. (9) Wilson, R. D.; Bau, R. J . Am. Chem. SOC.1974, 96, 7601-7602. (10) Bartell, L. S. J . Chem. Phys. 1963, 38, 1827-1833; 1979, 70, 4581-4584. (11) Jameson, C. J.; Osten, H. J. J . Chem. Phys. 1984, 81, 4915-4921, 4288-4292, 4293-4299,4300-4305,
where (Q:)
= (h/8a2cws) coth (hcws/2kT)
(6)
The above equations therefore allow us to calculate ( ATMC) and (ARco) as a function of temperature and masses. We need the inverse F'of the force constant matrix for stretches only, for
2588 J . Am. Chem. Soc., Vol. 109, No. 9, 1987
cc?
I
I
I
I
I
Jameson I
AI
C O ( ~ ~ C ' ~ N ) , 300 ~' CO('*C'~N),~- 300 C0(12C14N)63- 200 240 280 320 360 400
- {Ar)300K A 0
71 : ' 190. A
I
A CO-C 0 C-N
220.
1 250.
I
I 310.
I
I 340.
I
280. 370. 400. TEMPERATURE, K Figure 1. Mean bond displacements in V(CO),- and CO(CN),~-.
which the redundancy condition is not a problem. For k = 1 to 6 we obtain the same value of (ARC,), and for k = 7 to 12 we obtain the same value of ( ArMc), as dictated by symmetry. The results of these calculations are given in Table IV. The rotational contributions to the mean bond displacements are calculated by the usual method." We find the temperature dependence of ( ARco) and ( Arvc) to be: (A
R
~- (ARco)2WK ~ ) ~ = 1.20 ~ ~ X ~ 8, due to vibration A due to rotation (86% rotation) plus 7.36 X
(Arvc)400- ( A r v ~ =) 4.59 ~ ~ ~ X 8, due to vibration plus 2.95 X IOw4 A due torotation (only 6 % rotation) We have shown in Figure 1 the temperature dependence of the VC and CO bond displacements. The mass dependence of the V-C and C-0 bond displacements at 300 K are given by: ( A ~ ) s I ~- -( IA~ ~~ ) s I ~= -1.27 I ~ ~X
8, for
l60
( Ar)siV-i2,-(~6o) - ( A~)sIV-I~C(I~O) =6X ( A R )1
- (A R )
= 9.4
isotopomers
A
X
8,
- ( A R ) I ~ c - I I=o 1.06 X
8,
2 ~ ~ 1 6 0
(AR)i2~-160
1 ~ ~ ~ 1 6 0
We have applied the same theoretical calculation to the Co(CN)63- ion using the quadratic force field of Jones et al.I2 and using the molecular geometry from X-ray data: r(Co-C) = 1.89 A and r(C-N) = 1.5 Morse parameters used are acoc = (12) Jones, L. H.; Memering, M. N.; Swanson, B. I. J . Chem. Phys. 1971, 54, 4666-4671.
0.4158 0.4158 0.2772 0.3327 0.3881 0.4436 0.4990 0.5545
(AR)vib 4.2865 4.1986 4.2168 4.2949 4.2827
12.4949 12.6127 10.1171 11.0554 12.0796 13.1681 14.3051 15.4791
3.0896 3.1264 2.683 1 2.8447 3.03 1 1 3.2361 3.4551 3.6849
(AR) 4.3886 4.3006 4.3189 4.3630 4.4419
((AR)2, 1.2129 1.1882 1.1936 1.2117 1.2152 C-N bond; all are in
1.736 and UCN = 2.406 k', respectively. Kc0c and KCN are assigned the values 2.084 and 17.425 mdyn kl,estimated from f c K andfCN given by Jones.12 The nonbonded interaction constant F(C.-C) = 0.138 mdyn/8, is obtained from 922 = KMc+ 4F using the experimental value of the symmetry force constant YZ2. The results are presented in Table V. We find the temperature dependence of (ARcN) to be very similar to that of (ARco):
0 d
12.0791 12.I968 9.8399 10.7228 11.6915 12.7246 13.8061 14.9247
(AR)rot 0.1021 0.1021 0.1021 0.068 1 0.1361 ' A r refers to the Co-C bond and AR to the 10-3 A. C O ( ' ~ C ' ~ N ) ~300 ~CO("C'~N):300 C O ( ' ~ C ' ~ N ) ~300 ~C O ( ' ~ C ' ~ N ) , ~200 400
8
N
Table V. Mean Bond Displacements and Mean Square Amplitudes for the Co-C Bond and the C-N Bond in C O ( C N ) ~ ~ - "
( A R c N ) ~ "-~ (ARCN)200K = -1.2 X 8, due to vibration plus 6.8 X 8, due to rotation (120% rotation) (Arc0c)400 - ( Arcoc)200= 5.08 X 8, due to vibration plus 2.77 X 8, due to rotation (5% rotation)
The mass dependence of the Co-C and C-N bonds at 300 K are given by: ( A ~ ) S ~ C -~ (I&)59C0_13C ~ C
=
1.22
X
A for
14N isotopomers
( AT)S~C~-I~C(I~N) - ( AP)S~C~IZC(I= S N4.2 ) X
(AR)12@4~-
( A R ) I ~ C - I ~=N 8.8 X
(AR)12@4~- (AR)IIC-ISN= 6.9
X
8,
A A
It is worthwhile noting in Figure 1 that the temperature dependence of the M-C bond displacements is nearly two orders of magnitude larger than that of the C O or C N bond displacements, although the 13C-induced changes are only 1.4 times as large for the M-C bonds as for the C O or C N bonds. Since the electron distribution in transition metal complexes is known to be sensitive to the metal-ligand distance, these results indicate that the optical absorption bands of these complexes should exhibit a measurable temperature dependence. The change in the position of the lAl, IT,, absorption maximum in the Co(CN):- complex has been noted in aqueous solution, a change of about 4 nm in the range 0-90 OC.I4 The observed temperature and mass dependence of 5'V and 59C0 N M R chemical shifts in V(CO),- and Co(CN):- are interpreted in the following paper in terms of the calculated temperature and mass dependence of ( Arvc), ( ARco), ( Arcoc), and ( ARC, ).
-
Acknowledgment. This research has been supported in part by the National Science Foundation (CHE85-05725). (13) Curry, N.A.; Runciman, W. A . Acta Crystallogr. 1959, 12, 674. (14) Juranic, N. J . Chem. Phys. 1981, 7 4 , 3690-3693.
