406
SHIH-TUNG KINGAND JOHN OVEREND I
I
I
I
I
I
Figure 5 . Walden products for picric acid in mixed solvents. The symbols have the same meaning as i n Figure 3.
basis; the line is drawn with a slope of 4. Except for the point for the lowest water concentration (0.9OOjC, where experimental errors are large), the assumption that water solvates picric acid in ethanol in the same way as in acetonitrile seems verified ( K s = 0.4). It would be interesting to extend the water-ethanol and water-acetonitrile curves t o the pure-water point, which lies offscale to the left and down on Figure 3; pKa for picric acid n water16is 0.2-0.3.
Finally we consider the mobilities. Figure 5 is a plot of the Walden product against the mole fraction of acetonitrile (or ethanol, in the case of the water-ethanol mixtures). The product is abnormally high in the two mixtures containing water, on account of the added conductivity due to proton jumps in the waterrich systems. The product is also high in ethanol (0.815) compared with the value of about 0.5 in solvent mixtures where proton jumps cannot occur. The abrupt drop which appears on the initial addition of water to picric acid in ethanol confirms the result that water forms a more stable solvate with the picrate ion than does ethanol and suggests that the ethanol complex is somewhat labile, exchanging solvate ethanol with free ethanol rather easily, so that the migrating species is effectively the bare anion. The dependence of mobility on composition is much smaller in the acetonitrile-ethanol system, where proton jumps contribute little to conductance beyond 0.5 mol fraction of acetonitrile.
(16) The value of 0.3 was obtained from R. G. Bates and G. Schwarzenbach, Ezperienta, 10, 482 (1954), and the value of 0.2 was obtained from G. Kortilm and H. Wilski, Z . Phys. Chem. (Frankfurt am Main), 2, 256 (1954).
The Importance of Lone-Pair Electrons in the Intramolecular Potential Function of Group V Hydrides and Trihalidesla by Shih-Tung Kinglb and John Overend Molecular Spectroscopy Laboratory, School of Chemistry, University of Minnesota, Minneapolis, Minnesota (Received August $3, 1968)
66466
.I potential function for XY, inolecules is described in wliicli the
nonbonded interaction between the Y atonis nnd the lone pair on the rentral X atoiii is included as a pseudo-Urey-Bradley force ronstailt. Numerical values of this interaction parameter are determined for a range of inolecules where X is %t group V atom and Y is H, I?, C1, Br, or 1. The lone-pair, Y force is attractive when Y = 13 and repulsive in the remaining cases.
Introduction Pariseau, Wu, and Overend have shown that the Urey-Bradley force field (UBFF) can be modified to take into account the forces between the hydrogen atoms and the lone pair of electrons in the ammonia molecule. 2& These forces, represented as Urey-Bradley interactions, enter the internuclear potential energy through the factoring of the kinetic and potential energy matrices, and contribute in this way to the vibrational frequencies of the molecule. It is therefore posThe Journal o j Physical Chemzstrv
sible, in principle, to determine the magnitude of the forces between the lone pair and the hydrogens from the vibrational frequencies by adjusting the force constants in the internuclear potential. I n their treatment of the problem, however, Pariseau, Wu, and Overend2&did (1) (a) Based on a dissertation submitted by S-T. King in partial fulfillment of the requirements for the degree of Ph.D. (b) Chemical Physics Research Laboratory, The Dow Chemical Go., Midland, Mich. (2) (a) M. Pariseau, E. \Vu, and J. Overend, J. Chem. Phys., 39, 217 (1963); (b) J. Overend and J. R. Scherer, ibid., 33, 446 (1960).
407
INTRAMOLECULAR POTENTIAL FUTU'C'SION OF GROUP V HYDRIDES not adjust the force constants associated with the lone pair of electrons since the relationships between these force constants and the ones in the working symmetry coordinates are nonlinear and cannot be handled in the usual formalism of perturbation theory.2b Rather, they used the model described above to select certain non-Urey-Bradley force constants with which to augment the basic UBFF and later interpreted these nonUrey-Bradley force constants in terms of the lone-pair interactions. Curtis and 19uirhead3 have applied a similar procedure to NF3 molecule. Because there are only four observed frequencies which can be used for NF3, they cannot uniquely determine more than four force constants from the observed frequencies. Only a range of values was estimated for each force constant. We have now programmed a "stepwise adjustment" algorithm which allows the refinement of force constants, even though they be curvilinearly related to the working force constar k t 8. We have reexamined the hydrides of the group V elements and have extended the treatment to the group V halides. We present here the results of these calculations.
The Potential Energy Function The primitive force field is written, in terms of the coordinates defined in Figure 1, as 3
28
=
+ K r ( A r J 212 N u P A ~f i H a ( A a J z + ZFa'qu,OAqu, +
C[2Kr'ArI i= 1
Fu(Aqa2j21
+ 2Ko'At + K e ( A t ) 2 +
3
C[2Hp'Apt
3. H a ( A p J 2 4-
2=1
2Fp'qp,0Aqai
+ Fp(Aqd21
(13
The first six terms arise from the molecular skeleton and represent forces between the nuclei; the last six terms are the interactions of the nuclei with the lone pair of
electrons, taken as a point mass. Symmetry coordinates are defined as
= 3-'"(Aa1
&(")
#3(a1)
=
+
a ( A ~ i Aaz
+ Aa3) + b(AP1 f AD2 + Ap3)
= 2-'/'(-A&
- Aa3)
+ Ap3)
= 6-1/2(2Ar1- Ar2
- Ar3)
- Aa2 - Aa3)
S2(e'b)= 6-'"(2Aal
and
S3(etb) = 6-'/'(-2Apl
1- Ap2 + AB,)
(2)
-
where a = 2 sin a (coszfl - COS a) (COS a l),6 = 2 sin p cos p (1 - cos a ) 2are coefficients in the redundant coordinate X4. Although these symmetry coordinates are not orthogonal, the definition does separate exactly the coordinates of the lone pair and allows the coordinates and momenta of the lone pair to be split out of the Hamiltonian. Aforeover, when this is done, the reS2("), SI('I'), maining symmetry coordinates, SI("), S2(e1a), and S2('lb) are the symmetry coordinates usually written for a pyramidal XYs molecule. The treatment of the potential energy is described fully in ref 2a. Briefly, the UBFF of the primitive model shown in Figure 1 is set up and the Urey-Bradley redundancies are removed to give a 9 X 9 matrix F in the coordinates defined in Figure 1. After removing the coordinates and the momenta associated with the lone pair, we obtain an F matrix which is factored into two 2 X 2 blocks. The elements of the first block, used with the coordinates SIQ)and ,S2(")are
E
Fzz(&') = Faa
/
At
= 2-'/'(Aaz
FE(*')
/
=
Sl(e'a) = 2 - ' / ' ( A r ~ - Ar3)
F11"') e
+ + &$ + Aaz + Aa3)
Sl(a')= ~ - " ' ( A Y ~ Arz
= F,,
+ 2F,,'
- 3(Ftr)'/FEt
- (a/b)F,p f 3(a/b)F$7,p/F,, + 2Faa' - 4 ( a / b ) F a ~ +'
2FTa'
(3)
(4)
+
(a/b)2(F&3 2Fp,') - 3(a/b)2(&3)2/4E ( 6 ) qfl.
--/, i-
/ x.
I
/\\
and the elements of the second block, used with the are coordinates XI@)and
- F,,' - (FTpj2/Fpp- Fpp') (6) F I ~ '= ~ ) -Fret + FapfFrp/(FpB - Fpp') (7) F22(') = P uu - Faa' - ( F a ~ ' > ~ / ( F a oFpp') (8) FII(*)= F,r
The F' in eq 3-8 represents the interaction between the internal coordinates with different subscripts. For Figure 1. Internal coordinates used for the XYa molecule wjth the lone pLtir of electrons indicated by e.
(3) E. (1966).
C. Curtis and J. S. Muirhead, J . Phys. Chem., 70, 3330 Volume 73,Number I February 1060
408 example, FaP' is element in F matrix for AaCand Ab,, i # j . These matrix elements are used with an ordinary symmetrized G matrix for the calculation of normal coordinates and vibrational frequencies. The procedure is straightforward and is carried out with standard computer programs. Since the relationships in eq 3-8 are nonlinear, difficulties arise if we wish to adjust the parameters in the primitive force field (ie., the parameters in eq 1) to the vibrational frequencies using the customary first-order perturbation approach. Pariseau, Wu, and Overend avoided this problem by using the lone-pair model to suggest relationships between the elements of F, and did not attempt to adjust the primitive force constants. In the present work we have used, instead of the leastsquares adjustment, a stepwise procedure based on an algorithm described by Po-wellJ4and in this way have been able to determine numerical values for the force constants defined in eq 1.
SHIH-TUNG KIKGAND JOHN OVEREND
0.5
I
-'-/
Ha
Group V Trihydrides In our simple model, we think of the primary interaction between the lone pair and the remainder of the molecule as a coulombic interaction with the hydrogen nuclei; this interaction is expressed in the force constant Fp. The other force constants Hp (lone-pair-bending) and K (tension of the tetrahedral angle6) are considered to be of secondary importance; setting them equal to zero corresponds to neglecting the forces, over and above the coulombic forces which oppose the bending of thc lone-pair orbital from its symmetrical configuration. K , may be thought of as the effective stretching force constant of the lone pair, ie., as a2V/dC;Z when the lone pair of electrons is pulled off the central nucleus. It seems reasonable that the value of K Obe quite large and we have, as a first approximation, set it equal to infinity. We did explore the effect on the other force constants of varying K,. Sumerical results are given in Table I, and in Figure 2 we show plots of the refined force constants of NHa RS a function of the assumed value of K,. Our conclusion is that the assuniption K , = m does not significantly affect the values of the other force constants determined in the present study. The total number of force constants was reduced further6 by setting F,' = -1/loF, and Fp' = -l/loFp. The force constants K , (X-H bonding), H , (HXH bending), F , (H, H interaction), and F p were adjusted to fit the eight harmonic frequencies of XH3 and XU, given in ref 2a. The results are displayed in Tables I and 11. The calculated frequencies differ from the observed ones by less than lyG. It is not clear what should be used for the effective distance of the lone pair from the parent atom. We have carried out c$culations using the Bohr radius of the H atom (0.54 A) and the covalent bond distance of the center atom. We show both sets of results in The Journal of Physical Chemistry
5
10
'' K,
20
00
Figure 2. "8, values of force constants K,, Ha,F a , and F p as functional of the assumed value of K,. Hp = K = 0; 6, the distance of the lone pair from the nitrogen nucleus, is taken to be 0 3 4 A. Units: K,, K,, Fa, and F p in mdyn A/, H a in mdyn A/(radian)Z.
Table I. However, we prefer the second assumption and have used this in most of the prescnt work. If, as seems reasonable, the H atom in the XI38 molecule has a partial positive charge,' the interaction between the lone pair of electrons and the hydrogen nucleus is attractive and the force constant Fa has negative values. The results of our calculation are consistent with this; cf. Table I. The value of Fp changes from about -1.0 to -0.1 mdyn/A as we change X from N to Sb. It is interesting to examine the numerical relationship between the force constant Fp and the spectroscopic data from which the force constants are determined. The change in eachovibrational frequency, corresponding to !0.1 mdyn/A change in K,, Fa,and Fp or aO.l mdyn A/(radian)z change in Ha,is given in Table 111. In each molecule, the antisymmetric deformation v4 does not depend on Fp; the reason is obvious when one considers that the normal coordinate corresponds mainly to the symmetry coordinate Sz(e)in which the Powell, Computer J., 7, 155, 303 (1965). (5) B. Crawford, Jr., and J. Overend, J. Mol. Spectrosc., 12, 307
(4) AM.J. D.
(1964). (6) Y. LMorino, K. 20, 726 (1952).
Kuchitsu, and T. Shirnanouchi, J . Chem. Phys.,
(7) V. Schornaker and C. 5. Lu, J. Amer. Chem. Soc., 72, 1182 (1950).
409
INTRAMOLECULAR POTENTIAL FUNCTION OF GROUP V HYDRIDES Table I: Force Constants of Group V Hydrides, Adjusted to Fit the Harmonic Frequencies of XHa and XDa in Ref 2" Ha,
mdyn/A
mdyn A/ (radian)2
6.946 3.277 2.936 2 334
0.487 0.455 0.595 0.427
7.048 3.314 2.946 2.348 7.040 3.282 2.920 2.315
0.524 0.477 0.606 0.441 0 540 0.551 0.671 0.529
KF,
Molecules
For
K, NDa PHs, PDa AsHal AsDa SbH,, SbDs "8,
I
Fa?
I
rndyn/A
mdyn/A
€9
k.
98,
A
= 3.0
mdyn/b 0.451 0.295 0.076 0.119
-0.757 -0.381 -0.219 -0.245
0.54 0.54 0.54 0.54
1.3138 1.7703 1.8780 2.0643
-1.061 -0.468 -0.240 -0.275 -0.966 -0.283 -0.107 -0.124
0.54 0.54 0.54 0.54 0.6 1.1 1.21 1.41
1.3138 1.7703 1.8780 2.0643 1.3566 2.2163 2.4157 2.7626
K, = 0.361 0 273 0.067 0.110 0.324 0.198 0.014 0.053 I
I
The force constants and the quantities of 5 and qp are defined in Figure 1and eq 1.
Table 11: Group V Hydrides. Harmonic Frequencies (in cm-l) Compared with (i) Those Calculated Using the Force Constants in Table I and (ii) Those Calculated Using a Simple UBFF (A& = (Uobsd Y&d) x 1ooo/V,b,d)
-
7 -
AsHa
SbH,
XHs-
7
Av%
Av%
X -D --
I _
Av%
Av%
(ii)
Vobad
(i)
(ii)
0.12 -0.94 -0.35 0.07
-3.33 -7.64 3.56 10.65
2495 792.6 2651.6 1225
-0.12 0.99 0.36 -0.10
-3.79 -5.37 4.41 10.34
2448 1045 2390 1153
-0.39 0.39 -0.02 0.11
-1.64 -2.27 2.70 3.19
1760 759 822
0.41 0.41 -0.38 -0.08
-0.98 -2.92 2.35 3.00
2 3 4
2209.2 973 * 3 2225.8 1012
-0.01 0.13 0.08 -0.04
0.23 -1.01 -0.07 1.09
1571.2 696.3 1582 4 718.6
0.01 -0.13 -0.08 0.04
0.10 -1.14 -0.26 1.19
1 2 3 4
1989 796 1975 845
0.07 -0.07 -0.05 -0.07
-0.11 -2.22 0.19 2.29
1409 569 1404 600
-0.07 0.07 0.05 0.07
-0.33 -2.00 0.27 2.44
k
vobsd
(9
1 2 3 4
3506 1021.5 3577 1691
1 2 3 4 1
motion of the hydrogen atoms is on a circle passing through their equilibrium positions and in this normal coordinate the hydrogen nuclei remain a t almost constant distance from the lone pair. The strongest dependence on Fa is shown by V I and VZ, the totally symmetric stretching and deformation modes, and it is tliese frequencies which principally determine Fa. I n Table 11, we also show the percentage errors in the frequencies when the basic UBFF is used without the extra force constant Fa. The large percentage errors, compared with those obtained when Fp is admitted, clearly show that the introduction of Fa makes a substantial improvement in the potential function and suggest that the forces associated with the lone
...
I
pair do have a significant effect on the vibrational spectmm. The present values of Fa may be compared with those expected from the electron hydrogen atom potential derived by Gorin, Walter, and Eyringste
where e is the electronic charge, a. is the Bohr radius, and r is the distance of the electron from the hydrogen (8) E.
Gorin, J. Walter, and H. Eyring, J. Chem. Phys., 6, 824
(1938). (9) W. J. Kausmann, J. E. Walter, and H. Eyring, Chem. Rev., 26, 370 (1940). Volume 79, Number B Februaru 1060
410
SHIH-TUNG KIIUGAND JOHN OVEREND
Table I11 : The Sensitivity of the Vibrational Frequencies to the Force Constants in Table I (The Change in Each Frequency, in em-1, Corresponding to an Increase in Each Force Constant of 0.1, in Appropriate Thits, Is Given. j i
&
Ha
Fa
FP
1 2 3 4
24.46 0.85 25.55 0.02
0.04 107.21 0.20 120.48
64.46 32.36 12.08 51.30
19.42 26.61 5.19 0.00
1 2 3 4
17.65 0.58 18.84 0.00
0.01 81.57 0.83 87.03
44.51 25.09 7.16 38.04
14.80 19.94 3.82 0.00
1 2 3 4
34.60 1.03 36.11 0.13
0.18 77.01 0.01 73.99
80.32 58.05 15.58 72.71
26.49 26.82 10.08 0.04
1 2 3 4
24.94 0.65 26.00 0.07
0.02 56.36 0.00 52.83
55.12 43.44 10.14 52.44
19.88 19.18 7.26 0.02
1 2 3 4
38.27 0.03 38.09 0.00
0.00 71.39 0.00 71.06
80.89 78.92 14.52 85.89
31.22 21.39 11.00 0.00
I 2 3 4
27.25 0.01 27.13 0.00
0.01 51.17 0.01 50.44
56.15 57.22 9.91 61.18
22.66 15.13 7.83 0.00
I 2 3 4
42.15 0.30 42.52 0.03
0.05 68.33 0.00 66.55
92.85 86.53 17.26 98.92
33.01 29.04 12.43 0.01
1 2 3 4
29.95 0.20 30.21 0.02
0.02 48.76 0.00 47.18
65.02 62.12 11.96 70.27
23.72 20.60 8.83 0.00
Q
The Journal o/ Physical Chemistry
P O
Figure 3. Plot of hydrogen-electron force constant, Fp us. the group V trihydrides.
Group V Trihalides The model taken for the potential energy of the halides XY3 is the same as previously taken for the hydrides. In this case there are no isotopic frequency data and only four force constants ( K y , H a , F a , and Fp) can be determined. We make the assumptions, as previously checked for NH,, K , = a, and Hb = K = 0. Since reliable estimates of the harmonic frequencies are not available, we have adjusted the force constants to the observed frequencies (Table V). Several starting sets of force constants were tried; all converged either to the values given in Table IV or t o apparently unreasonable values.
Table IV: Force Constants of Group V Halides, Adjusted to Fit the Observed Frequencies Given in Table V
mdyn/w
Ha, mdyn b/ (radian)a
Fa mdyn/A
rndyn/i
2 746 3.913 1.795 1.'725 1.277 3.010 1.591 1.393
0.364 0.288 0.092 -0.227 -0.251 0.548 0.406 0.084
1.589 0.809 0.499 0,524 0.384 0.306 0.228 0.286
1.496 1.520 0.600 0.378 0.295 1.465 0.544 0.412
K,,
I
The factor of 6 is to take account of the effective number of electrons in the lone-pair orbital. Ideally, it should be 2 but may be less due to distortion of the electron distribution in the hydrogen orbital or the partial delocalization of the lone pair. In Figure 3,(d21"/ br2) is plotted against r = 49 for different values of E . The values of F p for SH3,PH3,AsH3, and SbHa are also shown in this plot. Even though the uncertainty is probably quite high, particularly for the heavier molecules where Fp makes a smaller contribution to the frequencies, the points do fall pleasingly close t o the theoretical line.
i
pp for
hboleoule
nucleus. The force constant Fs is just the second derivative of VH,i.e.
I 2 5
I
I 5
I O
Fp
!
f,
a
reY,
0.6 1.1 1.1
1.717 2.263 2.729 2.848 3.078 2.495 2.945 3.080
1.1.
1.1 1.21 1.21 1.21
w
The positive values of Fp in Table IV are interpreted as evidence of a repulsive force between the halogen atoms and the lone pair, in contrast with the situation in the hydrides where the interadion is attractive. The attractive force in the hydrides must originate in the incomplete screening of the proton implying an effective residual positive charge resident on the hydrogen nucleus. The apparent repulsive force between the lone pair and the halogens in the group V
411
INTRAMOLECULAR POTENTIAL FUNCTION OF GROUPV HYDRIDES halides suggests that the halogen atom carries an effective negative charge, a result which is consistent with the electronegative character o,f the halogens. The value of Fa for KFa (1.496 mdyn/A) is much lower than the calculated value from Slateor-type orbitals by Curtis and RIuirhead3 (3.65 mdyn/A; fl = 0.2)) but the F. - Frepulsive force constant for NFa in Table I V ( F a = 1.559 mdyn/8) falls into the range estimated by Curtis and Muirhead3 (1.0 1.9 mdyn/b). The comparisons between the observed frequencies and the calculated frequencies with or without the lonepair interaction are given in Table V. The tests of the sensitivity of the vibrational frequencies to the dif-
ferent force constant are given in Table VI. The results are similar to that of the hydrides.
Table V : Observed Frequencies Compared with (i) Those
Table VI : The Sensitivity of the Vibrational Frequencies to the Different Force Constant (The Frequency changes When Each Force Constant Is Increased by 0.1.)
-
Calculated with the Lone-Pair Interaction (Table IV), Av% = (Yobsd - Ycalod) X I O O / Y ~ M and (ii) Those Calculated with a Simple UBFF Avi%
Aui%
i
vi(obsd), om-'
(0
(ii)
1 2 3 4
1031 642 907 497
-1.14 3.97 1.75 -2.81
8.74 8.38 -4.37 -7.39
1 2 3 4
892 487 860 344
-4.97 6.56 4.12 -3.06
4.43 17.72 -3.02 -9.03
1
511 258 484 190
-2.23 3.66 2.04 -1.69
6.66 9.64 -4.18 -6.97
4
380 162 400 116
-1.32 1.72 0.92 -1.17
7.30 7.64 -3.64 -4.31
1 2 3 4
303 111 325 79
0.05 -0.01 0.00 0.03
8.00 5.04 -4.91 -4.82
1 23
4
707 34 1 644 273
-1.55 4.70 1.89 -1.20
1 2 3 4
405 194 370 158
-1.05 1.80 1.52 -0.66
1 1 3 4
[284] 128 275 98
-1.58 1.52 1.27 -0.48
2 3 4 1 2 3
a E. L. Pace and L. Pierce, J. Chem. Phys., 23, 1248 (1955). L. Dayennette, J . Chim. Phys., 58, 487 (1961). See F. A. Miller and W. K. Baer, Xpectrochim. Acta; 17, 112 (1961).
Estimated from
(VI
+
~ 4 = )
382 cm-1.
NHF, and NDF, Estimates of the vibrational frequencies of KHF, and KDFa are available,1° and the geometry has been determined.ll It therefore appeared feasible and useful t o teiit the lone-pair potential function on this molecule. The procedure is similar to that used for the more symmetrical molecules and will not be described in any detail. Table VI1 shows the force constants of SHF2, which has a different set of internal coordinates t)han that of more symmetrical molecules, ie., a! is the HNF
i
K,
1 2 3 4
7.50 0.02 13.21 0.00
1 2 3 4
Ha
Fa
FP
5.30 14.74 4.52 14 58
7.77 13.82 1.09 12.71
11.68 2.16 3.15 0.00
6.63 0.05 9.25 0.03
1.74 12.41 0.82 15.17
8.41 15.86 1.50 15.82
8.89 3.66 2.22 0.01
1 2 3 4
8.02 0.01 11.74 0.03
2.16 8.86 1.19 9.17
8.31 17.83 1.10 16.72
12.09 3.50 2.88 0.01
1 2 3 4
6.97 0.00 10.82 0.03
3.25 7.11 1.88 6.56
5.62 14.92 0.64 13.29
12.82 3.48 2.60 0.01
1 2 3 4
7.40 0.00 12.02 0.02
3.93 5.47 2.13 5.10
5.44 14.91 0.75 12.67
14.81 2.67 2.29 0.00
1 2 3 4
7.02 0.19 9.17 0.01
0.56 9.78 0.22 14.19
11.97 17.10 2.71 18.20
7.83 3.34 2.10 0.00
1 2 3 4
7.79 0.05 10.08 0.01
0.75 7.59 0.32 8.49
18.63 18.63 1.90 18.00
9.61 2.89 2.58 0.00
1 2 3 4
6.15 0.01 8.75 0.01
1.19 5.68 0.68 6,03
6.64 14 80 0.85
9.10 2.73 2.15
14,20
0.00
I
(10) J. J. Comeford, D. E. Mann, L. J. Schoen, and D. It. Lide, Jr., J . Chern. Phys., 38, 461 (1963). (11) D. R. Lide, ibid., 38, 466 (1963),report8 r N F = 1.400 -&, ?NE = 1.026 A, QFNF = 102.9',
