THOMAS H. BROWN AND ROBERT L. TAYLOR
2316
Table V : Summary of Thermodynamic Data for the Reaction 2Al(l) AlFs(g) = 3AlF(g)
+
Second-law treatment
AHfoins(AIF(g))
93.35 1 . 7 8 kcal. 6 2 . 1 5 f 1 . 3 7 cal./deg. - 6 5 . 7 0 f 0 59 kcal./mole
AHfOm(AIF(g):
- 6 0 . 6 5 f 0 59 kcal./mole
AH,Oins ASroins
Third-law values 91.234 i 0 . 4 3 kcal.
-66 41 2= 0 . 1 4 kcal./
mole S0tz7a(A1F(g))
- 6 1 . 3 6 f 0 . 1 4 kcal./ mole
6 4 . 1 6 f 0 . 4 6 cal./(deg.
mole) S"z~dAlF(g))
5 1 . 9 3 i 0 . 4 6 cal./(deg. 6 1 . 4 0 cal./(deg.
mole)
mole)
on the validity of the measured equilibrium constants which have been shown to be highly consistent. The values obtained in this study of the reaction of liquid aluminum and gaseous aluminum fluoride by means of a transpiration technique are in excellent agreement with the values reported by Gross3 in 1954 based on an experimental determination of AH* of A1F3(g) and by Witt and Barrow5 in 1959 by means of an effusion technique coupled with optical spectroscopy. The excellent agreement of the second- and third-law values lends credence to the reliability of the presently reported thermodynamic values for AlF(g).
The Use of Nonlinear Estimation Techniques in Simple Molecular Orbital Calculations'"
by Thomas H. B r o w n l b I B M Watson Laboratory, Columbia Univereity, New York, New York
and Robert L. Taylor Union Carbide Corporation, Computer Center, New York, New York (Received January 19, 1966)
Nonlinear estimation techniques have been applied to simple a-electron molecular orbital calculations which seek to have self-consistent Coulomb integrals and charge densities. Having analytic expressions for the various derivatives involved allows one to use the Newton-Raphson-Gauss estimation procedure, rather than iteration by substitution, as is customarily done. The present procedure converges very rapidly to the self-consistent solution, in contrast with the slow convergence or even divergence of the previous method.
Nonlinear estimation techniques can be of considerable value when applied to problems in molecular quantum mechanics which seek a self-consistent solution.za In this report we demonstrate the usefulness of this method when applied to simple molecular orbital calculations. In recent years a number of refinements have been made to the simple Huckel molecular orbital procedure.2b Usually an attempt has been made to maintain The Journal of Physical Chemistry
a number of simplifying assumptions of the Huckel technique while introducing a certain amount of sophis~~
~~~~~~
~~
~
(1) (a) This research was initiated when both authors were employed by the Union Carbide Gorp.; (b) author t o whom correspon6ence should be addressed. ( 2 ) (a) A general discussion of these applications will be available elsewhere; T. H. Brown and R. L. Taylor, to be published; (b) see for example, A. Streitwieser, Jr., "Molecular Orbital Theory for Organic Chemists," John Wiley and Sons, Inc., New York, N. Y . , 1961.
NONLINEAR ESTIMATION TECHNIQUES IN MOLECULAR ORBITALCALCULATIONS
2317
tication into the calculations. The self-consistency Specifically, we wish to determine a set of 6,-values procedures suggested by Wheland and Mann3 and by which minimize a particular expression in the leastMuller, Picket>t,and ;\lulliken* represent one such apsquares Here the 6,-values have the same proach. A simplified version of this procedure has been significance as in the previous method. We now conused extensively by Streitwieser and c o - ~ o r k e r s ~ * sider ~ ~ ~ two ~ expressions for determining the charge density among others It is this latter approach which we a t atom i. The first of these is eq. 3, while the second shall use as an example in this report. is obtained by rearranging eq. 2, i.e. Streitwieser, in what he has called the “w-technique,” 6 qt = &, - 2 considers the ‘proposition that atoms having unequal (4) W charge densities should have unequal Coulomb integrals, contrary to the simple Huckel theory. SpecifBy subtracting eq. 4 from eq. 3, we obtain the quantities ically, in the a-technique it is assumed that the various Y , given in eq. 5 . It is clear that the condition Y, = 0, Coulomb integrals have the form shown in eq. 1. at =
(YO
+
w(&t
-~JPo
(1)
Making the usual simplifications, we may write the difference from the normal value of the Coulomb integral for atom i , 6,, as in eq. 2 , where &, is the core charge on 61
= a(&, -
nt)
(2)
the atom in question, and g , is the computed electron charge density on the same atom. It is thus assumed that the difference in a Coulomb integral is directly proportional to the net charge density, w being the constant of proportionality. Several values for w have been suggested in the literature, some having been chosen to give a “best fit” to experiment.2a The coniputatiorial aspect of the w-technique, wherein its serious disadvantages lie, is basically iteration by simple substitution.2aj5 An appropriate value for w and a fixed set of &values are first chosen; then an initial set of charge densities q,“’ is guessed and used to deterniine the 6 f c 0 )via eq. 2 . The secular equation is solved, and a set of charge densities, qr(’),is computed from the eigenvectors by eq. 3. If the g , ( l ) do not agree Qr
=
Cn,Cil2
(3)
3
with the q t ( 0 ) , a new set of 6,“) are computed from eq. 2 , and the process is continued. The problem is said to converge when the charges computed from the eigenvectorsby eq. 3 are identical with those used to determine t’he secular equation via eq. 2 , that is, when q t ( k f l ) = Q , ‘ ~ ) ,for all i. The disadvantages of this technique are (l) that col’verge*’ce is not attained and (2) that, when convergence is attained, it may be reached very slowly. A discussion of the divergence of such procedures can be found el~ewhere.~ We consider the problem of self-consistency from the standpoint of estimation.8 The end will involve the sanle model for self-consistency as used in the w-technique (eq. and 2, but a superior computational technique.
(5) for all i, represents the same criterion of self-consistency as is used in the w-technique. We have thus reduced the problem of self-consistency to a nonlinear estimation problem, i . e . , finding a set of &-values which minimize the magnitude of the Y,-values. At this point a variety of estimation procedures is available. We choose to make use of analytic expressions, which have been discussed in another context, for the derivatives, bY,/b6,.lo These are based upon general expressions for the derivatives of eigenvalues and eigenvectors with respect to arbitrary molecular integrals appearing in the secular equations. As we shall see, the present problem is sufficiently linear that the use of these derivatives coupled with a straightforward application of the Newton-Raphson-Gauss estimation procedure provides a rapid method of obtaining a self-consistent solution; that is, for example, where b is the vector of 6,the first refinement of values, is given by (3) G. W. Wheland and D. E. Mann, J . Chem. Phys., 17, 264 (1949). (4) N. Muller, L. W. Pickett, and R. S. Xlulliken, J . Am. Chem. Soc., 76, 4770 (1954). (5) A. Streitwieser, Jr., and P. M. Nair, Tetrahedron,5 , 149 (1959). (6) A. Streitwieser, Jr., J . Am. Chem. Soc., 8 2 , 4123 (1960); A. Streitwieser. Jr., J. I. Brauman, and J. B. Bush, Tetrahedron, 19, SUPPl. 2, 379 (lg63). (7) See for example, (a) P. 0 . Lowdin, J . Mol. Spectry., 1 0 , (1963) ~ ; (b) L. R. Turner, Ann. N. Y . Acad. Sci.. 86, 817 (1960). (8) We use the term nonlinear estimation t o refer t o t h e general procedure whereby various linear least-square methods are used to approximate t h e solution of a nonlinear problem. In general, the procedures are iterative, continuing until convergence is reached in the least-squaresense. &,for example, ref. 7b; C . E. P. Box, Ann. S . Y . Acad. Sci., 86, 792 (1960); and H . 0. Hartley, Technometrics. 3, 269 (1961). . . (9) Wheland and Mann in their calculation of fulvene (ref. 3 ) give results obtained by a procedure which is somewhat similar t o one iteration of t h e technique proposed here. Their method is similar in t h a t it also considers t h e interrelations between charge densities and Coulomb integrals and utilizes atom polarizabilities which are similar t o t h e derivatives Z i j used here. (See ref. 10.) We would like t o thank a referee for clarifying their calculation. (10) T. H. Brown and R. L . Taylor, J . Chem Phys.. t o he published.
Volume.69, Number 7 J u l y 1966
THOMAS H. BROWNAND ROBERTL. TAYLOR
2318
~
~~
Table 11: a-Protonated Pyrrole, Yz-Technique, w = 1.2
where Z,
=
bYJb6,.
Thus, having chosen a value for
w and a fixed set of @-values,we proceed as follows.
(I) An initial set of Coulomb integrals, is chosen; these for the purposes of comparison can be chosen to reproduce the Huckel charge densities via eq. 4. ( 2 ) The secular equation is solved for the coefficients of the atomic orbitals, C,,. (3) The Yt(Oj are computed, and, if they are not sufficiently small, the derivatives dY,/b6, are computed. (4) Values for 6 ( ( l j are obtained by using the Newton-Raphson-Gauss procedure. (Note that a t this point, if we wish to introduce further sophistication, it is possible by modification of the Newton-Raphson-Gauss procedure to guarantee that the Y2C1)will not be larger than the Yzco) in the least-squares sense, where Y2 = x Y f 2 . In acI
tual practice this precaution seems unnecessary.) (5) The procedure then continues from step 2 until a set of values for Y,@)is obtained that are sufficiently small. (In general two iterations are generally sufficient to reduce all of the Y,-values to the order of to 10-7.) The technique as described above has been used successfully on a wide variety of examples, including all of the divergent cases given in ref. 5, i.e., the benzyl cation, etc. The superiority of the present method is best shown by some comparisons. We consider first aprotonated pyrrole. This is a four-atom problem with four electrons and a net charge of 1 For simplicity, we set all resonance integrals equal; i.e., @C-N = @C-C = Po. Using w = 1.2, the w-technique diverges, as is shown for four iterations in Table I, whereas the technique described herein converges rapidly and without difficulty to a self-consistent answer. See Table 11. (Since the yi-values are the quantities being iterated, the charge densities a t each iteration are ambiguous. Values of the charge densities as computed by both eq. 3 and eq. 4 are therefore included in Table 11.) For the case of w = 1.0, the w-technique again diverges; in addition, though the correct charge densities are 1.3120, 0.8153, 0.9829, and 0.8897, starting the w-
+.
Iteration
Eq.
0
4 3
1
2
Iteration
0 1 2 3 4
41
1.000
1.598 1.036 1.603 0.993
42
1.000 0.626 I . 003 0.630 1.013 All diverge
The Journal of P h y k a l Chemistry
w =
qa
1.0000 1.5980
1.0000 0.6256
1.0000 1.0369
1.0000 0.7395
4 3
1.3815 1,3029
0.7798 0.8267
0,9737 0.9670
0.8661 0.9033
4 3
1.3415 1.3415
0.8015 0.8015
0.9760 0.9760
0.8810 0.8810
Q4
technique, with such sets of charge densities as 1.310, 0.817, 0.985, 0.888, and 1.314, 0.813, 0.981, 0.892, still produces divergence. I n other words, divergence in the w-technique cannot be blamed on an extraordinarily bad initial guess. An initial bad guess does not affect the method described here for the problem discussed. The same solution is reached in two additional steps when the initial charges are chosen to be the ridiculous values of 5001, 5001, -4999, and -4999. This is equivalent to saying that the problem converges in approximately the same number of iterations, no matter whether the initial charges are chosen 1.0, 1.0, 1.0, and 1.0, or the unrealistic values of 0.0, 0.0, 2.0, and 2.0.
Table 111: @-Technique,Butadiene Cation, w = 1.4" Iteration
QI
qa
1 2 3 4
0.652 0.729 0,669 0.716 0.679
0.848 0.771 0,831 0.784 0.821
..
...
...
0
..
...
...
10 11
0.691 0.698 0 . 695b
0.809 0.802 0 ,805b
m
From ref. 5.
Table IV: Table I : w-Technique, a-Protonated Pyrrole,
qa
Q1
' An estimate given in ref. 5.
Y'J-Technique,Butadiene Cation,
w =
1.4
1.2
qs
44
1,000 1.037 0.859 1.140 0.761
1,000 0.739 1.102 0,627 1.233
Iteration
Eq*
Q1
0
4 3
0.6520 0.7287
0.8480 0.7713
1
4 3
0.6947 0.6950
0.8053 0.8050
2
4 3
0.6949 0.6948
0.8051 0,8051
Ql
KINETICSOF URANYL IONHYDROLYSIS AND POLYMERIZATION
We lastly wish to compare the two techniques on a problem for which the w-technique converges. We choose an example given by Streitwieser and XairI6the butadiene cation. The two methods are compared in Tables I11 and IV, where the initial charge densities are the same. The superiority of the present method is clear ; the w-technique still produces sizable variations in the third significant figure, even after 11 iterations, whereas the Y2-technique gives a reasonable answer in only one iteration. I t should be pointed out that it is
2319
not fair to make a direct comparison of the number of iterations the two methods take. The iterations of the Y2-technique are more complex than those of the wtechnique ; nonetheless, there is an appreciable saving in time with the present technique, in addition to its obvious advantage of succeeding where the w-technique fails. Acknowledgment. The authors wish to thank Dr. AI, Karplus, Dr. V. Schomaker, and Dr. E. B. Whipple for several helpful discussions.
Kinetics of Uranyl Ion Hydrolysis and Polymerization'
by M. P. Whittaker, E. M. Eyring, and E. Dibble2 Department of d h a i s t r y , University of Utah, Salt Lake City, Utah
(Received January 20, 1966)
Temperature jump relaxation times have been observed in acidic, aqueous solutions of uranyl ion that have been identified with the equilibrium 2 U 0 2 +
+
+ 2H20 + (U02)2kz
k -1
2H+. The rate constant k2 at 25" is found to have the value 116 M-I set.-'. This result is compared with the dimerization of chromate and vanadate ions.
Recently a number of workers have reported equilibrium constants for the hydrolysis and polymerization of uranyl ion in aqueous s0lution.3-~ In principle, such data permit the determination of rate constants for several over-all reactions using relaxation methods. A similar kinetic study of aqueous boric acid polymerization has been rep0rted.I The hydrolysis mechanism and equilibrium constants or uranyl ion in aqueous nitrate media proposed by Baes and Aleyer4 and also those proposed by Dunsmore, Hietanen, and Sill6nj were used as a basis for explaining relaxation times observed with a Joule heating-type temperature jump apparatus. The hydrolytic species and their corresponding equilibrium constants reported by Rush and Johnson6 were the basis for our calculations of rate constants in aqueous perchlorate medium. The calculated rate constants were essentially the same regardless of which set of equilibrium constants was used in their calculation provided that the
reaction responsible for the observed relaxation time was assumed to be the formation of dimer, (U02)2(OH)3+ or (U02)2(OH)22+ depending upon the hydrolysis scheme, from monomer.
Experimental For our experiments with uranyl nitrate, samples of (1) This research was supported in part by an equipment grant from the University of Utah research fund and by Grant AM-06231 from the National Institute of Arthritis and Metabolic Diseases of the C . S.Public Health Service. (2) National Science Foundation Undergraduate research participant.
(3) R. M . Rush, J. S. Johnson, and K . A. Kraus, Inorg. Chem., 1, 378 (1962). (4) C. F. Baes, Jr.. and N . J. Meyer, ibid., 1, 780 (1962). ( 5 ) H . S.Dunsmore, S. Hietanen, and L. G . SillBn, Acta Chem. Scand.. 17, 2644 (1963). (6) R. hl. Rush and J. S. Johnson, J . P h y e . Chem.. 67, 821 (1963). (7) J. L. Anderson, E. M . Eyring, and M. P. Whittaker, ibid., 6 8 , 1128 (1964).
Volume 69, Number 7
Julu 1966
