CommunicationsA Self-consistent co-Technique

PRATIP R. RAY & NIRAD G. MUKHERJEE

Department of Chemistry. University College of ScienceCalcutta 700009

Received 1 December 1976; accepted 3 January 1977

A self-consistent m-technlque, which includes thecharge dependence of the coulomb integrals while mini-mtstng the energy of a system, has been developed.As the method uses a steepest descent procedure, theconvergence is guaranteed. Test calculations show~reatly improved convergences for the allyl and benzylcations.

THE "It-charge density is not uniform in the non-alternant hydrocarbons and alternant hydro-

carbon ions. The Huckel calculations for suchsystems should thus be less valid than those foralternant hydrocarbons in which the uniform chargedistribution guarantees a self-consistent fieldl,!One simple way for achieving a self-consistentfield for the non-alternants and the a lt ernantions is through the application of the so-called<>l-technique3. Though the method is not new andhas been widely applied in the past to varioussystems with considerable success, especially inthe case of the non-alternants. some interestinganalyses of the technique have been made in recentyears4,6. The <>l-technique is notorious for its poorconvergence and oscillations especially near theoptimum charge density values". Many techniquest-"have been devised from time to time to step up theconvergence, but almost all of them have ignoredthe important fact that while minimizing the energyof a system one must consider the charge dependenceof the coulomb integrals+ This serious omissionleads to charge distributions which are not strictlyself-consistent. In what follows we consider thisand present a method for the optimization of thecharge density. As the method uses a steepestdescent technique, the convergence is guaranteedhere. A detailed account of the principles of thesteepest descent technique has been given byMeWeeny". We shall, therefore, avoid any elabora-tion on the principles and always refer to McWeerl} 9

for the details. The method has been developedfor even-electron svstems. An extension to odd-electron systems should, however, not pose anygreat difficulty.

Self-consistent charge distrib'ution - The "It-electronenergy (E) of an even-electron system is given byEq. (1)E=2trHR •.. (1)where H,,=oco+<>l~o(I-2Rrr)

H,s=~o, when rand s are neighbours=0, otherwise

In the above expression OCoand ~o are standardcoulomb and resonance integrals respectively and

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w is a dimensionless constant the value of which isusually taken as 1·4 to provide a best fit with theobserved and calculated ionization potentials. TheR matrix is defined as equal to TTt where the LCAOcoefficients of respective MO's are collected in thecolumnS of the T matrix. The R matrix, whichhas the property R2 =R, is a measure of electrondensity and bond order as P"=2R,, gives thetotal number of electrons on atom rand P,.=2R,sgives the bond order between atoms rand s.

As H is really a function of R. a first order variationin energy for changing R to R'=R+8R is given byexpression (2)8E=2tr(8R)H +2trR(8H) ... (2)

A little mathematical manipulation will showthat expression (2) can be written as8E=2t,.8RW ... (3}where W =H +D. The elements of Dare givenby the relationDjj= (-2CJ)~o)Rij8ii ... (4)It is obvious that Wt=W. As W depends on theR matrix the problem must, therefore, be solvediterativelv.

Any variation in R, compatible with the auxiliarycondition R2=R. may be expressedt as8R=(V +Vt)+(VVt -vtVJ+... ... (S}whereV=(1-R;ARA being an arbitrary non-singular matrix.

Substituting (5) in (3) and simplifyirg one gets8E=4tr[(1-R;WKtA ... (7)For a steepest descent down the energy surfaceA must have a value''A=-A(l-R)WR ... (8)where A is a positive number yet to be determined.

The change in R corresponding to steepest descentof the energy Surface is th1;:58R= -A(S+st) + (sst-~ts)where s= (1- R)W R.

As (l-R)R=O, we can'SR=-AL-A2LMwhere L=s+st ar.d M=s-st.

In order to find optimum A we consider the energychange up to the second orderSE =2tr( _AI_A2 LM)W +2.A2trI 11-where Nij= (-2CJ)~o)LijSij.

Setting d('SE;OA)=0, one obtainstrLW

A= 2(trLN -trLMW) ... (12)The correction (10) thus leads to the best l-descent

approximation to Rand E. The iterative processis continued until self-consistency is achieved.Any departure from idempotency of R can be

... (6)

... (9)

write expression (9) as... (10)

... (ll}

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eliminated by a simple iterative process prescribedby jlcWeelly9.

Applicability of the method - Although a steepestdescent procedure guarantee" convergence, veryoften the convergence is rather poor. It is thusnecessary to test whether a particular methodinvolving a steepest descent procedure leads toreasonable convergence to be of any prLlctical value.For this we have done two test calculations theresults of which seem quite encouraging.

In Table 1 we present the electron densities onthe terminal atoms of allyl cation after successiveiterations using the present method. The resultsobtained by the conventional o-technique" andthose improved by Hakala's method" are alsoincluded in Table 1. A comparison among thethree sets shows a great improvement in the con-vergence by the present method.

Conventional w-teclll1ique is known to lead toa divergence for benzyl cat ioi.s. Convergence canonly be achieved by the application of ratherelaborate methods involving averaging after succes-sive iterations? or including w'- and {o"-techniqueslo.In Table 2 we present the electron densities onthe first atom (outside the ring) of benzyl cationafter successive iterations using the present method.A comparison with the values given in lit erntnretagain shows a great improvement in the convergenceby the present method.

In order to see whether the charge densitiescalculated by the present method give a betterpicture of the actual -e-electron charge densitiesthe authors have started calculations of dipolemoments of a series of non-alternants using thecharge densities obtained by the present method.The results of those calculations, which will bepresented in a later paper, are expected to be veryhopeful indeed. The values of dipole momentsof fulvene and azu lcn« so far calculated appeareven better than those10 obtained by (0'- and w"-

TABLE 1 - ELECTRON DENSITIES ON THE TERMINALCENTRES OF ALLYL CATION AFTER SUCCESSIVE ITERATIONS

No. of Present As in As initerations method ref. 3 ref. 8

0 0·500 0·500 0·5001 0·601 0·621 0·6212 0·598 0·534 0·5793 0·598 0·597 0·569

10 0·571

TABLE 2 - CHARGE DENSITIES ON THE FIRST ATOM (i.e.THAT OUTSIDE THE RING) OF BENZYL CATION AFTER

SUCCESSIVE ITERATIONS

No. of Present cu- Technique' Arithmeticiterations method mean'

0 0·5714 0·5714 0·57141 0·3680 0·1697 0·37012 0·3510 0·5697 0·39383 0·3182 0·1532 0·37264 0·3158 0·5879 0·39485 0·3140 0·1212 0·3728

techniques. In conclusion we may say that as theconvergence here is quite good, the charge distri-butions and hence the dipole moments are betterthan those obtained by the conventional w-tecitniqueand as the method avoids repeated diagonalizationof the Hamiltonian matrix, generally a timc-cor.si.m-ing step in a self-consisted field calculation, thepresent method seems to have a significant promiseand hence should be extensively appliEd.

One ofthc authors (N.G.lVI.)is indebted to Prof. R.McWceny of the University of Sheffield, UK, formaking him acquainted with the principles of thesteepest descent technique ar.d to the UGC, NewDelhi, for financial assistance. The authors arcalso grateful tc the Centre of Computer Science,University of Calcutta, for computer facility.

References

1. COULSON, C. A. & RUSHBROOKE, G. S., Proc. CambridgePhil Soc., 36 (1940), 193.

2. MCVVEENY, R., Molecular orbitals in chemistry, physicsand biology, a tribute to R. S. Mulliken, edited byP. O. Lowdin & B. Pullman (Academic Press, NewYork), 1964, 304.

3. STREITWIESER, (Jr) A., Molecular orbital theory fororganic chemists (John Wiley, New York), 1961.

4. HARRIS, F. E., J. chem, Phys., 48 (1968), 4027.5. GOODISMAN, J., Theoret. Chim. Acta (Bern, 36 (1974),

117.6. STREITWIESER (Jr), A. & NAIR, P. M., Tetrahedron, 5

(1959), 149.7. COULSON, C. A. & VVILLE, F., Tetrahedron, 22 (1966),

3549.8. HAKALA, R. W., Intern. J. Quantum Chem. (Symposium),

1 (1967), 227.9. MCWEENY, R., Proc. ray. Soc., A235 (1957), 496.

10. STREITWIESER (Jr), A., HELLER, A. & FELDMAN, M., J.phys. Chem., 68 (1964), 1224.

Floating Spherical Gaussian Orbital (FSGO)Studies with a Model Potential: Methane,

Silane & Germane

S. P. MEHANDRU & N. K. RAY*

Department of Chemistry, University of Delhi, Delhi 110007

Received 30 December 1976

A Gaussian-based model potential is used withinFSGO formalism to study the equilibrium geometriesof CH.. SiH. and GeH.. The predicted bond lengthsare in excellent agreement with the experimental values.

VALENCE electron studies with model potentialshave drawn considerable attention in recent

years1-IS• Most of these calculations have beenmade within the LCAO-SCF-MO formalism andonly a few have been carried out in the frameworkof floating orbital basis4,5,10,16-IS.

In the present communication we have used themodel potential suggested by Schwartz andSwitalski-" within FSGO formalisms? to study theequilibrium geometries of CH4, SiH4 and GeH4•

This method has been used quite successfully earlierby Ray and Switalski to study the first row atom

. *Author to whom all correspondence should be made.

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