924
PAUL R. SMITHAND JAMES W. RICHARDSON
Bonding and Hybridization in the Nitrogen Molecule1
by Paul R. Smith and James W.Richardson General Atomic Division of General Dynamics Corporation, Special Nuclear Efects Laboratory, Sun Dkgo, California, and the Department of Chemistry, Purdue University,Lafayette,Indiana (Received September 19, 1966)
Molecular orbital wave functions from successively improved LC-STO SCF calculations for the Nz molecule are compared. In addition to the usual physical properties and Mulliken population analyses, the total electronic charge distribution and the density difference function 6p(R) = ~ M ( R-) PA@) are considered. Although improvements in the wave function lead to difficulties with the population analysis, they do give a more satisfactory picture of the bond charge density. With such “flexibilization,” the electronic charge density is found to increase in the internuclear region over that of the superposed atoms; general agreement with Ruedenberg’s “clustering” hypothesis is found. By changing the “valence state” of the constituent atoms, PA changes also. A 2s-2pa hybridization of about 10% minimizes the average value over all space of ISp(R)I.
Introduction A knowledge of the charge distribution in a molecule is essential for understanding the chemical behavior of the molecule. The characteristics of the molecular charge distribution, particularly when compared with the charge densities of the noninteracting constituent atoms, reflect the nature of the chemical bonding mechanism and provide insight into traditional concepts, such as localized chemical bonds. The recent progress toward better approximations in molecular wave functions has increased their complexity and created a need for methods of comparison and evaluation. For some molecules, the techniques of improving self-consistent field molecular orbital (SCF-MO) wave functions have been carefully studied and a significant increase in accuracy has been achieved by expanding the basis set. The general improvement in computed observable properties is accompanied, however, by increasing difficulties in applying other techniques of analysis which attempt to correlate molecular orbitals with useful chemical concepts, such as localized chemical bonds and hybridization. There has been some utilization of density matrices2 and approximate natural spin orbitals13but most of the studies have used Mulliken’s population analysis4 as their basis. Recently, Mulliken has surveyed this problem and has drawn attention to the need for careful prior judgment in specifying an extended basis set in order that apparently misleading physical The Journal of Physical Chemistry
interpretations not be made from the population analysk6 For a great many systems, Roux and collaborators,6--11 Rosenfeld,12Bader and Henneker,l3-lS Smith and Richardson,ls and Politzerl’~’*have studied the -~
~~~
~~
~~
~~
~
(1) (a) This work was supported primarily by the Advanced Re-
search Projects Agency under a contract with Purdue University and partly by General Atomic; (b) presented in part at the Symposium on Molecular Structure and Spectroscopy, The Ohio State University, Columbus, Ohio, June 10-14, 1963. (2) (a) R. McWeeny, Proc. Roy. SOC.(London), A232, 114 (1954); (b) P.-0. Lowdin, Phys. Rev., 97,1474,1490 (1956); (c) R. McWeeny, Rev. Mod. Phys., 32, 335 (1960). (3) (a) P.-0. Lowdin and H. Shull, Phys. Rev., 101, 1730 (1956); (b) H. Shull, J . Am. Chem. Soc., 82, 1287 (1960). (4) R. S. Mulliken, J . Chem. Phys., 23, 1833 (1955). (5) R. S. Mulliken, ibid., 36, 3428 (1962). (6) M.Roux, 6 . Besnainou, and R. Daudel, J . Chim. Phys., 54, 218, 939 (1956). (7) M.Roux, ibid., 5 5 , 754 (1960). (8) M. Roux, ibid., 57, 53 (1960). (9) S. Bratoz, R. Daudel, M. Roux, and 3%.Allavena, Rev. Mod. Phys., 32, 412 (1960). (10) M. Roux, M. Cornille, and G. Bessis, J . Chim. Phys., 5 8 , 389 (1961). (11) M. Roux, M. Cornille, and L. Burnelle, J . Chem. Phys., 37, 933 (1962). (12) J. L. J. Rosenfeld, Acta Chem. Scand., 18, 1719 (1964). (13) R.F. W.Bader, J . Am. Chem. SOC.,86, 6070 (1964). (14) R. F. W. Bader and W. H. Henneker, ibid., 87, 3063 (1965). (15) R. F. W. Bader and W. H. Henneker, ibid., 88, 280 (1966). (16) P. R. Smith and J. W. Richardson, J . Phys. Chem., 69, 3346 (1965).
BONDING AND HYBRIDIZATION IN THE NITROGEN MOLECULE
character of the function 6P(R) = PM(R)
- PA(R)
(1)
where PY(R) is the total electronic charge density at the point R of the molecule M and PA@) is the electronic charge density a t the same point R which would result if the constituent atoms of the molecule M were superposed at the molecular equilibrium distance. I n a careful analysis and interpretation of the nature and cause of chemical bond formation, Ruedenberg has ascribed some considerable importance to such a function.Ig Ruedenberg suggests, among other things, that the real source of energy lowering during bond formation occurs when electrons pull in closer, or cluster nearer, to the nuclei. The objectives of this study are (1) to compare the actual form of wave functions and charge densities with results of corresponding population analyses, (2) to utilize the 6p(R) function to detect the clustering phenomenon, and (3) to utilize the 6p(R) function in determining an alternative, more general definition of hybridization in a molecule. The Nz molecule in its ground state was selected for the present study because it has a variety of bonding interactions and because there are several sets of 1 4 0 wave functions available which represent a wide range in flexibility and polarization as indicated by the choice of basis orbital^.^ Wave Functions for the
NzMolecule
The wave functions considered here represent various stages of approximation to a closed-shell HartreeFock (HF) solution of the Schrodinger equation. Four successively improved sets of single-determinant MO wave functions were studied in detail: (1) the Scherr SCF-MO wave function,20which uses a minimal set of Slater-type orbitals (STO’s) with the free-atom orbital exponents ({’s) fixed by Slater’s rules; (2) the Best Limited (BL) SCF-MO’s (or Best Minimal in Mulliken’s most recent terminologyz1) of in which the orbital exponents of the Scherr function are varied in order to minimize the total molecular energy and thus achieve the best over-all t’s; (3) the “double-t” SCF-R/IO’s of R i c h a r d ~ o n in , ~ ~which the basis set is expanded by doubling the number of second quantum STO’s included; and (4)a more nearly exact ~ set of Hartree-Fock SCF-340’s of Cade, et U Z . , ~ which for most, practical purposes give the most accurate wave function in the orbital approximation. The wave function used here is for R = R,(exptl) not R = Re(HF). Some comparisons of theoretical and predicted ex-
925
perimental properties are given in Tables I, 11,and 111. A steady decrease is seen in the total-energy error toward the residual correlation energy of 16.1 ev. Also seen is a steady increase in a predicted “bond energy,” defined here as the difference between the energies of the Nz molecule and two N atoms, the atomic and the molecular wave functions each having the same degree of improvement. Although there is no special justification for this definition in general, it is relevant to some calculations of 6p(R) reported below. Table I: Comparisons of Some Physical Properties Wave
function
AEa
D*b
Q”
eQad
Scherr R a n d BL6 2r Hartree-Fock’ Experimental
27.5 25.9 21.8 16.1
1.20 2.61 3.48 5.20 9.90
-1.90 -1.02 -1.38
1.49 3.49 4.41
...
...
...
-1.11
4.65
a Error in computed total energy, in ev. * Dissociation energy (in ev) computed as indicated in text. Molecular Quadrupole moment, in au. Computed as indicated by C. Greenhow and W. V. Smith, J . Chem. Phys., 19, 1298 (1951). The necessary integrals were obtained from a molecular p r o p erties integral program written by Dr. A. C. Wahl for the IBM 704 computer at Argonne National Laborat,ories. Quadrupole coupling constants, in Mc; calculated values assume the nuclear cm. The necessary quadrupole moment to be 0.02 X atomic data were calculated from Roothaan’s Best Atomic wave Based on R = Re (exptl) = 2.068 au. functions (see ref 27).
’
A general improvement is clearly seen in total energy and in a few available experimental quantities, which depend on the molecular charge distribution. Shapes of the Various MO’s and Corresponding Population Analysis The Nz molecule has the ground-state electronic configuration iugz iuuz 2 4 2q,2
inu4
3rg2
Appropriate contour maps, as described below, were drawn for each of the occupied M O ’ S . ~ The ~ drawings (17) P. Politzer, J . Phys. Chem., 70, 1174 (1966). P. Politzer and R. E. Brown, J . Chem. Phys., 45, 451 (1966). K. Ruedenberg, Rev. M o d . Phys., 34, 326 (1962). C. W. Scherr, J . Chem. Phys., 23, 569 (1955). R. S. Mulliken, Rev. Mod. Phys., 32, 232 (1960). B. J. Ransil, ibid., 32, 245 (1960). 3. W. Richardson, J . Chem. Phys., 35, 1829 (1961). (24) P. E. Cade, K. D. Sales, and A. C. U‘ahl, ibid., 44, 1973 (1966). (25) For a visual presentation of these MO contour maps, see ref 16 and A. C. Wahl, Science, 151, 961 (1966).
(18) (19) (20) (21) (22) (23)
Volume 71, Number 4
March 1967
PAULR. SMITHAND JAMES W. RICHARDSON
926
Table 11: Comparison of Orbital Energies and Experimental Ionization Potential (in au) Orbits1
- 15.72176 - 15.71965 - 1.45241 -0.73066
1Ug
1 vu 2% 2 U“ 1Tu 34
zt
BL
Soherr
- 15.80452 - 15.80219 - 1.47922 -0.75409 -0.60486 -0.56759
-0.57951 -0.54451
-15.70512
- 15.70192 - 1.49301 -0.76287 -0.61378 -0.62225
HF
Exptl
- 15.68195 -15.67833 - 1.47360 -0.77796 -0.61544 -0.63495
... ... ... -0.69 -0.62 -0.57
-~ I I
Table III : Overlap Populations and s-pa Hybridization
t
f MO
2% 20, 3% 1Tu
Total Hybridization
Soherr
BL
2Y
0.85 -0.37 -0.08 0.88 1.28
0.83 -0.68 0.07 0.91 1.12
0.78 -2.05 0.10 0.95 -0.22
26%
21%
2%
I I
I
+ I
1 II I
I
II
!
represent a cross section of the three-dimensional wave function on a plane passing through the internuclear axis. The comparisons detailed below are made chiefly between the Scherr and 2f functions. The BL functions were intermediate in all cases and, except for 20,, the 2f and HF-MO’s are essentially the same. Upon improvement of the basis set, the most marked change is observed in the 20, MO, which becomes considerably more concentrated in the internuclear region. Note that this is in contradiction to the moderate decrease in the overlap population for this MO, which becomes even more concentrated in the internuclear region in the HF case. Although the use of population analyses in correlating the bonding and antibonding character of various molecular orbitals and total overlap populations has been atl;empted,4*20*26 the general validity of the population analyses remains questionable. That these analyses must be used with reservation is indeed borne out by the very negative overlap population of the 2a, MO of the 21 wave function, which leads to an over-all negative value for the total overlap population even though the over-all shape of the 2a, is virtually unchanged ‘tiy improvements in the wave functions. To emphasize this point, drawings from the Scherr and the 2{ approximations are superimposed in Figure 1. Despite the uniformity in Figure 1, there is a drastic change in the overlap population from the reasonable -0.37 to the disastrous -2.05. This extreme negative value causes even the total overlap population to become negative. The Journal of Physical Chemistry
Figure 1. Superposition of the 2u,, MO as obtained from the Scherr wave function and the 26 wave function for Nz. For clarity, the solid lines represent the 26 wave function and the dashed lines represent the Scherr wave function.
The 3ug MO remains essentially the same when the wave function is improved, even though the orbital energy of this most loosely bound MO is seen to fluctuate most widely. There is only a slight change in the overlap population from -0.08 to 0.10. The lr, MO has its familiar shape and retains a strong resemblance to the free atom 2pa. As the total wave function is improved, the M O becomes slightly more diffuse, even though a moderate increase in the overlap population is observed. In order to see how the overlap population of the 2uu MO changed so much, first note that the ~ ( 2 0 . ~ ) may be written as (p(2a,) = hy
- hy+
(2)
where hy is the sum of all STO’s centered on nucleus a and hy+ is the mirror-image sum on nucleus b. Graphs of hy for the Scherr and 2f MO’s are given in Figure 2. In the Scherr function, the magnitude of the 2uu MO is decreased in the internuclear region by considerable 2 s 2 p a hybridization; Le., the 2s A 0 is strongly polarized outward. Subtraction of hy+ from hy decreases it a small amount mor2 and leads to the moderate negative overlap population. On the other hand, in the 2{ function, the 2s is polarized only in a region nearer the nucleus (by hybridization of the (26)
S. Frsga and B. J. Ransil, J . Chem. Phya.,
34,727 (1961).
927
BONDING AND HYBRIDIZATION IN THE NITROGEN MOLECULE
a L 4 5 .:I ~~~
.f.,+
j t . 2 .-I:,
:05,
:Ol,
Figure 2. Comparison between hy(Scherr) and hy(21). See text for discussion. Upper plane, Scherr approximation; lower plane, 26 approximation.
Figure 3. Total electronic charge density for the No molecule in the Scherr approximation. The numbers correspond to the density in units of electron per cubic atomic unit. The value 200 represents the density a t the nitrogen nucleus.
larger-p 2pu STO of the double-p function) and instead the 2s becomes much more diffuse. Now the 2a, MO becomes small in the internuclear region only by a large cancellation of hy by hy+. There is thus found the large negative overlap population and also the greatly decreased s-pa hybridization noted in Table 111. In other words, expansion of the 2s A 0 at one center has partially replaced hybridization at the other center. In many respects this situation is like that found, for example, in some studies of H F and discussed by M ~ l l i k e n . ~There, when additional hydrogen STO’s but not fluorine AO’s were included in the basis set, the hydrogen STO’s assumed a disproportionate share of the bonding and an unreasonable charge distribution was obtained from the population analysis. In this case, a reasonable situation returns only when expansion of the basis set is judiciously balanced between H and F.
Molecular Charge Distributions A contour drawing of the total electronic charge density in a plane passing through the nuclei is given in Figure 3. The curves from the Scherr approximation are given; those from the other functionsare quite similar.
Although all the charge densities obtained from the various approximations are nearly the same, their resemblance is only superficial. The distinction between the charge densities becomes appreciable and significant when the difference function 6p(R) is examined. Figures 4 and 5 represent 6p(R) for the Scherr, BL, 2p, and H F wave functions. The same convention was followed here as in Table I ; e.g., 6p(R) for the BL function is the difference between the densities from Roothaan’s Best Atomic2? wave function in the 4S state and Ransil’s Best Limited molecular function. Although such a convention is arbitrary, it is consistent and does eliminate some artificial disparities, such as those arising near the nuclei when different orbital exponents are used for the 1s AO’s or at large distances where even small differences in 2s and 2 p p’s cause a large change in charge densities. If, for example, the 2p atomic density is subtracted from the Scherr molecular density, 6p(R)is greatly changed in character. The density difference function, 6p(R), for the Scherr wave function was first sketched out by ROUX.~ The most conspicuous feature is the decrease in electronic charge in the region between the nuclei and the increase in electronic charge at the ends of the molecule. There is, however, a slightly positive annular ring which was taken to support the notion of bent
Figure 4. 6 p ( R ) for the Scherr and Best Limited approximations for Nz. The dotted lines are the zero contours, solid lines are positive contours, and dashed lines are negative contours. Contours are drawn a t intervals of 0.02 electron per cubic atomic unit. Upper, 6 p ( R ) Scherr approximation; lower, 6p(R) Best Limited approximation. ~~
(27) C. C. J. Roothaan, Technical Report, Laboratory of Molecular Structure and Spectra, Department of Physics, The University of Chicago, Chicago, Ill., 1955, p 24.
Volume 71, Number 4
March 1867
928
PAULR. SMITHAND JAMESW. RICHARDSON
.
,_-_
I
,--1.
Figure 5. 6 p ( R ) for the 2-t and Hartree-Fock approximations for Nz. Upper, 6p(R)2-t approximation; lower, 8 p ( R ) Cade-Sales-Wahl approximation.
bonding, as suggested by Pauling, 28 among others. Over-all, the description of the bonding in Nzin this instance seems to be in fair agreement with the usual Lewis formula for N2. As the wave function is improved by allowing the orbital effechive nuclear charges to vary, a more conventional picture emerges. Electronic charge moves into the central bonding region, an effect which is more pronounced in the 21 calculation and still more in the H F limit. There is no vestige of annular density, as was observed in the Scherr calculation. The characteristics of 6p(R) are seen to depend very strongly on the quality of the approximate wave function. Indeed, serious doubt is shed upon results published for other molecules using wave functions lacking orbital exponent adjustment and/or adequate provision for atomic promotion. Again, ii, is interesting to compare the extent of internuclear charge buildup with total overlap populations; as one goes up, the other goes down. I n total energy, there is only a 5.7-ev difference (in some 3000 ev) .between the Scherr and the 21 function. The De corresponding to Figure 4 is 1.20 ev, compared to 3.48 for Figure 5, as contrasted with the value 5.20 ev in the HF limit. Even though variations occur in the internuclear region when the wave function is improved, a consistent feature displayed by these various 6p(R) results is the significant electronic charge accumulation at the ends of the molecule. Clearly this must be related to hybridization of lone-pair electrons. This effect seems The Journal o j Physical Chemistry
to be primary; it apparently occurs before the internuclear charge buildup that resuIts from adjusting the effective nuclear charges. Approximate numerical integration of the 2t 6p(R) function gives the following increases: (1) an increase of 15% in charge density at the bond midpoint with a total increase of 0.09 electron within the internuclear region (Figure 5 ) bounded by the nodal contour, and (2) an increase up to 25% in charge density outside the bond along the internuclear axis, yielding a net increase of 0.16 electron at each end. When one considers the vigorous covalent bonding that N2 undergoes and the considerable electronic reorganization which is deemed to have taken place, these net charge dislocations seem rather small. Contrast, for example, an increase of less than 0.01 electron in the positive annular ring of the Scherr calculation and an increase of 0.03, 0.09, and 0.26 electron in the nodally bounded internuclear regions of the BL, the 2t, and the HF functions. Obviously, a poor choice in a basis set can lead to serious and misleading variations in the disposition of 6p(R). A clear demonstration of this same type of effect has been observed for various wave functions of the H F molecule.29 It is certainly clear that electronic correlation effects are of considerable importance in determining bond energies. One might suggest, therefore, that the character of 6p(R) might be quite different if calculated from wave functions in which correlation was taken into account. Brillouin30 and Moller and Plesset31 in their early investigations demonstrated that electron correlation induces only second-order corrections to the H F density. In a complicated system such as Nz, little can be said about the magnitude and sign of the correlation effect in different regions of space except that the error in the charge density for a Hartree-Fock wave function is of the same order as the error in the total energy. This would seem to indicate that introduction of electron correlation into the wave function would not change the character of the 6p(R) function and should eliminate any possibility of the annular-ring, bent-bond configuration being restored. One further point merits note. For all the degrees of orbital adjustment which were permitted in the wave functions, there still is found no increase of molecular charge density a t the nuclei over that of the superposed atoms. This apparent contradiction of Rueden(28) L. Pauling, “Nature of the Chemical Bond,” 3rd ed, Cornell University Press, Ithaca, N. T.,1960, p 136. (29) C. W. Kern and M. Karplus, J. Chem. Phys., 40, 1374 (1964). (30) (a) L. Brillouin, ActuaZiti&s Sei. et Ind., No. 71 (1933); (b) L. Brillouin, ibid., No. 159 (1934). (31) C. Moller and M. S. Plesset, Phys. Rev., 46, 618 (1934).
BONDING AND HYBRIDIZATION IN THE NITROGEN MOLECULE
berg’s clustering hypothesis is tempered in two ways. First, Ruedenberg’s Postulate Of the contraction of AO’s in passing to the valence state appropriate to the molecule, but in Nz, this valence state also involves some promotion out of the 2s A 0 into the 2P (and higher and a ‘Onsequent decrease in charge density at the nuclei (see below). Second, and much more important, the valence AO’s, which are second quantum AO’s, are being dealt with here. The largest magnitude effects of clustering or orbital contraction will not be at the nucleus as in Hz, but farther outward at the maxima of the second quantum AO’s. Thus the clustering phenomenon is indeed observed and it turns out to be synonymous with charge buildup in the bond region.
6
Alternative Definition and Determination
of the Valence State Consideration of the Gp(R) function suggests that it might lend itself for use in determining valence states for atoms in a molecule. Instead of the usual criterion, one might define the valence state as that state (or combination of states) for each atom which, when all atoms are superposed, most nearly reproduces the molecular charge density. This definition has been discussed briefly by Ruedenberg. l 9 We first use this criterion here under the restriction that only one excited atomic state be mixed with the ground state, i e . , that only atomic 2s-2pa hybridization be permitted. The complications and possible indeterminacies which arise with greater flexibility will be treated subsequently. The difference function now to be considered is
929
I ( 4 = SlW;4ldv
(6) Although this criterion is arbitrary, preliminary studies indicate that minimizing
S [ W wI2dv for example, yields virtually the same results. Still to be examined is the possibility of incorporating some weighting function, such as ~ M ( R ) . The evaluation of eq 6 was accomplished by a simple numerical integration. Specifically, the function Gp(R;a) was evaluated at 1254 points in a quarterplane through the bond axis and multiplied at each point by the appropriate volume element and then the products were summed. To assess the accuracy of this procedure, the $Gp(R;a)dv, which is identically zero, was numerically integrated in the same way. In all cases, a very small positive residual of no more than 0.02 electron was found. Using the Scherr molecular function and hybridized Slater atoms, I(a) os: a is given in Figure 6. The minimum occurs at approximately 11% hybridization, contrasted with a value of 26% from population analyI
r
0
20
40
60
80
100
% HYBRIDIZATION ( N p SLATER)
Gp(R;a) = Pi@)
- p*(R;a)
(3)
which is exactly the same as eq 1, except that here the atoms are hybridized 100 a per cent. Specifically, ate is described by this promoted SI
Figure 6. Variation off16p(R;or)ldv with (Y, the degree of 2s-2pa hybridization, for the Scherr wave function.
1.30 I
1
ls)2 h)2 k)2 2pn)l 2 p 3 ’ in which the “lone pair” hybrid h
==
dl - 4 2 s )
1.10
-442pa)
(4)
=:
d42s)
+ dl - a ( 2 p a )
: IIF, v
1.00
? 0.90
and the orthogonal ‘(bonding” hybrid
k
I
0.80
(5)
0.70
Wow Gp(R;a) may be evaluated as a function of the
hybridization parameter a. The criterion selected here for optimal reproduction of the molecular charge density is to minimize the expression (eq 6).
0
20
40
I
,
60
80
1 100
% HYBRIDIZATION (2-6 N2)
Figure 7. Variation ofS/Gp(R;a)ldv with LY, the degree of 2s-2pa hybridization, for the 21 wave function.
Volume 71,Number 4
March 1067
930
PAULR. SMITH AND JAMES W. RICHARDSON
A
Figure 8. Variation of Gp(R;a)with CY, the degree of 2 s - 2 ~ ~ hybridization, for the Scherr wave function.
sis. The "error" indicated in the figure is plus or minus the value of JGp(R;a)dv. Results for a similar calculation using the 2c molecular wave function and hybridized HF atoms are presented in Figure 7; the minimum occurs at about 8% compared to 2% from the population analysis. It is quite satisfying that these two different approximations to the wave function now give much more nearly equal predictions to the valence state of the atoms. On the other hand, it is disturbing that the values obtained are so much smaller than those suggested by established notions. Perhaps what are revealed from this new definition of hybridization are the contributions from attached atoms to the effective charge polarization. In Figures 8 and 9, Gp(R;a) for the Scherr and the 2t functions are drawn, each at CY = 0, a = amin, and a = 0.2. The Gp(R;a) curves for a = amincorrespond to those defined by Ruedenberg, Le., for the noninteracting atoms in a promoted state. Now, for the Scherr approximation, we see a very striking example of clustering. At 0% hybridization, an apparent lack of contraction is observed because no provision
The Journal of Physical Chemistry
Figure 9. Variation of Gp(R;a)with a,the degree of 2s-2po hybridization, for the 2r wave function.
was made in the wave function for adequate atomic promotion, Le. , the orbital exponents were fixed. However, bonding of the promoted atoms produces an accumulation of charge in the internuclear region. It is tempting to extend .this procedure to refine the specification of the N atom promotion state by inclusion of perhaps atomic 3s, 3p, and 3d character. The immediate difficulty, however, is that of minimizing a function of several nonlinear parameters plus a set of constraints. A more fundamental problem is whether a unique valence state actually exists for the atoms. Just as in the case of the 2uu MO discussed earlier, it is quite possible that including 3s character, for example, in the promotion state of one atom may displace 2pu character from the promotion state of the other atom and vice versa. Acknowledgments. The authors wish to express their thanks to Dr. A. C. Wahl for computing the necessary quadrupole moment integrals and to Drs. P. Cade, K. Sales, and A. C. Wahl for releasing their Nz wave functions prior to publication.
