3702 previously defined symbol signifies replacement of cos 8 by sill 6 in the 2p(r,,lSH interaction elements and change in sign in the SHHfactors. Note, in the E H method D and b are identical, which is not necessarily so in the K E H formulation. The energies of the bz orbitals in the C N D O and special EH cases are obtained in expansion as follows.
-
.-
E
Now
X
=
-x
=
-'* dz(l +
-
2x
'
*) (A 16)
+
= (2 g)/2 = ( b + 8)/2, since f i = 0 and EZ. The same replacements for bar quantities as
before are made; for the CNDO applications exchange of the bond orders P2pru,lsH for P2pr>1,1sH are also made.
Semiempirical Molecular Orbital Theory and Molecular Geometry. I1* Analytic Procedures for Charge Redistribution Methods' S. Ehrenson
Contribution f r o m the Chemistry Department, Brookhaven National Laboratory, Upton, New York 11973. Received December 26, 1968 Abstract: Analysis has been carried out to obtain closed-form energy and electron distribution expressions for the first-row atom dihydrides as represented by extended-Hiickel theory modified by self-consistent charge redistribution procedures, and by the similar but theoretically more soundly grounded SCF CND02 method. Previous examination of this group of molecules revealed deficiencies in the ability of two important variations of zerothorder Hiickel theory to represent energy upon geometry dependences. From the results obtained here, the specific way in which the charge imbalances established in the zeroth-order results are modified by the variations in electronegativities of the atoms so charged may be followed. Good approximations to the self-consistent energies and charge densities are obtained in both the Huckel and CNDO methods by familiar closed form summation procedures. Exactly how the original and the redistribution parameters interact to change the energy upon geometry dependence are revealed and the reasons why the Hiickel redistribution procedure is inherently inferior to the SCF redistribution of the CND02 method are examined. The approximate cancellation of hydrogen-hydrogen electron and nuclear repulsions are recognized in the CNDO angle deformation problem and the influence of this cancellation on making the CNDO and extended Huckel procedures appear similar is discussed.
I
n the preceding paper2 analytic expressions were developed for the orbital and total electronic energies of the first-row dihydrides within the framework of two important variations of the extended Huckel method. 3 , 4 The major purpose of the development was t o ascertain how the various specific interactions recognized by the theory contribute t o the computed energies, and how these contributions change with changes in molecular geometry. It seems clear from the results obtained, and those of other investigators,j that such semiempirical theories should be generally better able t o approximate the energetics of geometry changes from equilibrium configurations involving displacements of nonnearest neighbor atoms ( i e . , bond angle variations) than of neighboring more strongly bound atoms. In neither case, however, would confidence in the energy surfaces generated by these methods as originally formulated seem warranted. It appears from some recent studies, though, that considerable improvement of these independent electron methods is possible if charge redistribution techniques which alter the atom-type or standard molecule parameters are employed. Such fairly uncomplicated pro(1) Research performed under the auspices of the U. s. Atomic Energy Commission. (2) S . Ehrenson, J . A m . Chem. SOC.,91, 3693 (1969). (3) R. Hoffmann, J . Chem. Phys., 39, 1397 (1963), and later papers. (4) M. D. Newton, F. P. Boer, and W. N. Lipscomb, J . A m . Chem. SOC.,88, 2353 (1966), and succeeding papers in the same volume. ( 5 ) CY. L. C. Allen and J. D. Russell, J. Chem. Phys., 46, 1029 (1967).
Journal of the American Chemical Society / 91:14 / July 2, 1969
cedures specifically as applied to the EH method of Hoffmann6 have yielded interesting results and appear t o be gaining a measure of acceptance in the study of geometrical isomerization of large hydrocarbon molecules and ions (mainly the angular configuration problem).' Parameter adjustment based on charge distributions, which is entirely consistent with the philosophy of the original methods, represents a way, albeit an averaged way, of imposing self-consistent field conditions on the zeroth-order wave functions, and stands the chance of successfully approximating the results of more complete theories if integral variations with geometry change are approximately paralleled. Angular effects are certain on these grounds to be more favorably treated than bond stretches or contractions. It is not surprising that the CNDO approximations,8 which are true SCF methods as applied to wave functions obtained from complete but simplified Hamiltonians, are also capable, without specific modification for the task, of reproducing molecular energy variations with bond angle change quite successfully. Moreover, these methods, originally claimed to not as satisfactorily account for energy changes (6) D. G. Carroll, A. T. Armstrong, and S. P. McGlynn, ibid., 44, 1865 (1966). (7) Cf. J. E. Baldwin and W. D. Fogelsong, J . Am. Chem. SOC.,90, 4311 (1968). (8) J. A. Pople and G. A. Segal, J . Chem. P h ~ s . 44, , 3289 (1966).
3703
accompanying nearest neighbor distance variations, may in fact, with certain well-defined exceptions, yield fair energy variation approximations and thereby provide useful insights into this important type of problem. Extensive application of the CND02 method in this direction is u n d e r ~ a y . ~ As an extension t o the work described in ref 2, the first-row closed-shell dihydrides, MH2, are examined in the framework of the two methods discussed, Le., the charge redistributed extended Huckel, CREH, and the true SCF C N D 0 2 methods. The approach is the same as previously adopted in that approximate analytic expressions for the orbital, electronic, and total energies, and where necessary charge distributions, are derived and their changes with angular geometric variation assessed. Where possible, comparisons with Walsh's rules will again be drawn, under the recognition, however, that the electronic and thereby the total energies are not the simple proportional functions of the orbital energies they were accepted to be in the previous treatments discussed.
are in the cases of interest, the charge densities may be expanded to good approximation about the values obtained from the zeroth-order solutions with the original VSIP's. Recognizing that xaa,,B, E, and E will be quite insensitive to balanced changes in the diagonal elements (Le., Aazs e A a z p Z - A a H , accompanying charge redistribution, cide infra), a remarkably compact expansion may be obtained.
Theory and Mechanics of Charge Redistribution in the EH Method The symmetry orbital approach detailed in ref 2 is again used. Employing the same notation, the following exact relationships for the individual symmetry orbital contributions t o the total charges for the six and eight-valence electron cases may be obtained either from the orthogonalized matrices'O or more directly but somewhat more laboriously from the orginal matrices.
(6)
AqM 2
F{
i = 2(1 \ q2s
+
q2p.
+
- qMo =
(& + A e x ) ( T 'C/ 2
Here
Y
P A - x
q2p.
E qhl
=
+
q2p,
2P
+ + q2p.
AA
q2pY)
=
+
(5)
+ [Po/(Ao - x ) ] ~+ [Qo/(Do - x)]']l"
[I
P
=
[l
+ (2Qo/d0)~]"~
The quantities A A , A D , and A b are, of course, A a t S , and AaZp.. The efficacy of this approximation is demonstrated in Table I where sizable and unmatched Table I. Tests of the Charge Density Difference Expansion, Eq 3a3b 1.2 2.4 4.8
-1.35 -2.7 -5.4
0.127(0.132) 0.166(0.148) 0.293(0.280) 0.298 (0.263) 0.330(0.296) 0.628 (0.559) 0.606 (0.526) 0.590 (0.591) 1.196 (1.017)
a A l l energies in eV: A0 = -21.4, D o = Do = -11.4, and = - 13.6. b The first value in each column employing eq 3 and 4, the values in parentheses from eq 5 with the expansion values of 6 (ref 2, eq 9) also employed in the equating of A - x and-D - x to a 6 and d 6, respectively. ACYU= AA = A D = AD, (CY&
-
D - x
+
+
(5J2 +(
=
A - x A - x
D - x
(3)
D - x
These equations are generally appropriate for the EH calculations ; the density in the out-of-plane orbital, qaP,,is zero and two respectively in the six- and eightelectron cases. The orbital energies, x, are for the 3a1 and 2b2 levels, the most antibonding of the particular symmetries in eq 1-4. Where the quantities P / ( A - x), Q / ( A - x ) , and o/(D - x ) are small compared to unity as they typically (9) J. A. Pople, private communication. (10) C. A . Coulson and H. C. Longuet-Higgins, Proc. Roy. Soc. (London), A191, 39 (1947).
+
shifts in diagonal elements about the zeroth-order parametrization for a singlet CHa-like molecule are tested. It is apparent that the expansion overestimates the charge difference component of the al symmetry somewhat and underestimates that of the ba symmetry, for the smaller A d s . At larger A d s both are underestimated but not seriously even though rather large changes are forced.'' The third line of Table I, it should be recognized, corresponds t o -0.4 electron excess on M if a linear version of a-upon-q dependence, which is somewhat rougher than often used,' is invoked and assumes a rather drastic and unrealistic dependence for the hydrogen matrix elements as well. It may therefore be concluded that the charge densities, at least in the systems of present interest, will exhibit roughly linear dependence upon the VSIP values employed. If, conversely, the a values are themselves realistically adjustable through equations (1 1) Despite results shown in Table I which appear to be to the contrary, the truncated expansion of eq 3 should generally be of better mathematical quality than that of eq 4. Approximations of the offdiagonal, diagonal quotient terms compensate here to make both estimations appear equally good.
Ehrenson
Analytic Procedures for Charge Redistribution Methods
3104 Table 11. Charge Redistribution Effects in CH2 upon Variation of Internal Angle"
4-
k??S
0
qzPz)dl)
q2P.dl)
45 50 54.3 60 70 80
3.143 3.179 3.212 3.252 3.318
1.106 1.046 1.010 0.976 0.944
qtot
qtot
4.249 (4.239) 4.225 (4.216) 4.221 4.228 (4.217) 4.262 (4.250) (4.285)
-
4.063 (4.068) 4.058 (4.062) 4.058 4.060 (4.064) 4.070 (4.076) (4.088)
ACd
-f-
-f-
5
0.746 0.694 0.686 0.723 0,836
0.8104 0.7744 0.7488 0.7220 0.6870
1.0755 1.1896 1.2413 1.2635 1.2421
2.228 2.072 2.020 2.028 2.042
0 45 50 54.3 60 70 80
id')
-1 2 s
-1 2 P z
0.0707 0.0733 0.0759 0.0798 0.0871
0.0497 0.0430 0.0374 0.0298 0.0163
Aa2,(l) = Aa2pz(1)= A a ,?Pi ( 1 )
=
AaZp,(1)= A(y(1)
Aa(1) = D o ( q M ( l )- qM(0))= DoAqMvr")
+
(7)
+
A q M ( " = 12sA~2s(1) 12pzA~2p,(1)
l*pzA~Zp,(= LACY( '1 (8)
+ oqkr(') ( A + o q M ( ' ) + DwAqM(l))]= Dw[AqM") + (1 - D)AqM(')] (9)
Aa(2) = D [ A
n-1
j-0
It is quickly recognized that the cycling procedure under the conditions outlined is resolvable in terms of geometric series and that consequently, since q(M)
qC-1 -
q(0)
- 4 (0)
n
=
(1 1)
j=l
0.1291 0.1245 0.1228 0.1228 0.1262
0.2495 0.248 0.2361 0.2424 0.2296
( a e ) - ~ ( 6 o ) ) 0 ~ ~ ~ (E(0) - E(60))mt,t 1.154(0.511) 0.548 (0.216) 0.228 0.0 (0.0) 0.088 (0.095) (0.279)
a All energies in eV; I and L values in electronslev; q and f values are unitless; 0 in degrees. columns obtained analytically. Values in parentheses are exact computer results for comparison.
which are linear in the orbital densities, then analytic evaluations of the convergence properties accompanying the self-consistent field cycling procedure may be conveniently made. The equation systems are similar to those encountered previously in analysis of wtechnique effects in T-electron theory; l 2 similar notation is therefore used. The cycling equations are condensed as follows with inclusion of the damping factor, D , on the adjustment of the a's as it would be imposed in the computational iterative procedure to guarantee convergence.
-L
-l2&
1.349 (0.710) 0.519 (0.149) 0.147
0.0 (0.0) 0 . 4 1 1 (0.575) (1.296)
First entries are sums of the first two
In CH2 ('A1 state, rCH = 1.094 A, 28 = 120"), 12s, and are from eq 5, -0.0798, -0.0298, and -0.1228 electron/eV, respectively, and L is therefore -0.2324. With w = 11.9 eV/electron,13 the above analysis indicates the largest value of D under which convergence may be obtained is 0.531 and that qMm will be 4.060. The value of qMm obtained by complete computer analysis using D = 0.4 is 4.064. The limiting value of D for convergence obtained analytically was roughly verified; a computer run with D of 0.6 steadily diverged. Similarly for H 2 0 (rOH = 0.960 A, 28 = 120"), I*,, lZp2,and are -0.0474, -0.0200, and -0.1511 electron/eV. Since and olP are not equal (16.9 and 17.4 evjelectron, obtained as the slope for a linear relationship connecting the VSIP's of neutral and uninegative oxygen atoms6), a weighted L value of -0.2171 is obtained to be used with w = 17.4. The largest D value for convergence is therefore predicted t o be 0.419 and the analytic qMm value from eq 12 is 6.190. By computer analysis with D = 0.4, qMm is 6.241. With D at 0.6, the computer cycling diverged, as in the CH2 runs but much more rapidly, indicating the analytic assessments of maximum D for convergence to be trustworthy. The initial computed densities, qhx"), are 6.906 and 6.905 by analysis (eq 3 and 4) and full computation; the comparable numbers for CH? are 4.228 and 4.217. Tables I1 and I11 present the analytically derived parameters for CH2 and H 2 0 which determine the selfconsistent densities and, hence, the adjusted diagonal elements and molecular energies. It is easily shown that
= Aq(11 +
L DwAq( I ) 1 - [LDw
Aq(
+ (1 - D ) ] =-1 - LW
(12)
+
for l L D o 1 - D1 < 1. Note, the converged value of the density is properly independent of the damping factor employed to guarantee convergence in the cycling. We are now in a position to analyze the factors contributing to production of the self-consistent densities are how these factors will be affected in detail upon geometric change in the molecular structures.
91:14
2(6
+ 6)-
=
2(6
+ 8)' + AEo"
(14)
where AEo", which represents the correction to the total electronic energy forthcoming from the establishment of self-consistent charge densities, is obtained (13) H.Basch, A. Viste, and H. B. Gray, Theor. Chim. Acta, 3, 458 (1965).
(12) S. Ehrenson, J . Phys. Chem., 66,706, 712 (1962).
Journal of the American Chemical Society
and
1 July 2 , 1969
3705 Table 111. Charge Redistribution Effects in H 2 0 upon Variation of Internal Angle"
e
(q2a
+
q2P*)dl)
(qzP,)11(1)
3.513 3.543 3.568 3.600 3.651 3.687
1.464 1.398 1.352 1.306 1.253 1.226
45 50 54.3 60 70 80 90
qtot &I(=)
-1%
- l2PZ
6.190(6.256) 6.187 (6.246) 6.187 6.190(6.242) 6.197 (6.249) 6.205 (6.260) (6.265)
0.0441 0.0453 0.0462 0.0475 0.0495 0.0510
0.0378 0.0315 0.0264 0.0200 0.0100 0,0027
q t o t If(')
6.977(6.976) 6.941 (6.940) 6.920 6.906(6.905) 6.904(6.903) 6.913 (6.913) (6.918)
e
Acr
-f+
-f-
T
45 50 54.3 60 70 80 90
3.307 3.257 3.256 3.296 3.432 3.563
0.2116 0.1994 0.1900 0.1792 0.1640 0.1544
0.2906 0.3520 0.3960 0.4407 0.4864 0.5049
6.938 6.840 6.770 6.703 6.642 6.624
(QO)
- E(60))'tot
-L
-I2Pz
0.1574 0.1560 0.1538 0.1511 0.1479 0.1465 c,d
0.754 ( 0 . 5 8 3 ) 0.436 (0.322) 0.218 0.0 (0.0) -0.230(-0.118) -0.330(-0.132) -0.356(-0.126)
0.2380 0.2315 0.2251 0.2172 0.2060 0.1987
me) -~(60))"~~~ 1 ,604 (2.426) 0.619 (0.935) 0.169 0 . 0 (0.0) 0,476 (0.698) 1.186 ( I . 880) (2.454)
* Includes two electrons from the b1(2p,,,) orbital. From column 6, Table 111, ref 2. d The exact and a Footnotes of Table I1 apply. analytical EtOtvalues are themselves in good agreement. For example, E(60)0t,t's are respectively - 161.16 and - 161.62; E(60)"t, a useful comparison of electronic transition circular dichroism to ordinary absorption. Kuhn, Mathieu, and other^^-^ used g us. X plots to characterize various electronic transitions. The anisotropy factor was often found t o be constant throughout a single transition. Its variation was taken to lenote a change of transition in that interval. Later theoretical work has largely abandoned the concept of an anisotropy factor as a detailed function of wavelength. Condon6 first defined the factor as proportional to the ratio of integrated intensities. (1) NASA Predoctoral Fellow. (2) (a) W . Kuhn, Trans. Faraday Soc., 26,299 (1930); (b) Z.Physik. Chem., B8, 286 (1930). (3) W. Kuhn and H. L. Lehmann, ibid., 18, 32 (1932). ( 4 ) J. P . Mathieu, Ann. Phys., 3, 371 (1935). ( 5 ) T. M . Lowry, “Optical Rotatory Power,” Dover Publications, New York, N. Y., 1935, p 393. (6) E. U. Condon, Rec. Mod. Phys., 9, 432 (1937).
goi
=
4RoiIDoi
ROi is the rotatory strength of the i defined by
+
(1)
0 transition and is
Rot= Im{pot.mto} = 0.24 X 10where poi and mio are the electric and magnetic dipole transition moments, respectively, is the wavelength, Ae(X) = €](A) - €,(A), q(X) and €,(A) are the decadic molecular extinction coefficients for the left and right circularly polarized light, and the universal constants have been evaluated to give cgs units. Dei, the dipole strength Of the is defined by D~~ = voi.pio= 0.96 where E(X)
= [q(X)
Robinson, Weigang
x
1 0 - 3 3 S ‘ P d ~ (cgs) (3)
+ e,(X)]/Z.
1 Anisotropy Factor of Opticully Actice Ketones
