Indian Journal of ChemistryVol. 31A, January 1992, pp. 1-5 .~

A bond orbital approach to theoretical studies of molecular properties froma fully localised SCF method

BKSen"Department of Chemistry, Cotton College, Guwahati 781 001

andBMajee

Department of Chemistry, North Bengal University, Darjeeling 734430

Received 18 April 1991; revised 3 September 1991; accepted 3 October 1991

The charge distributions, dipole moments and J-J coupling constants of some simple o-bonded mole-cules have been calculated using a newly developed method within the framework of bond-orbitalSCFMO approach that includes only the theoretically justified parameters and optimised hybrid orbitals.The results agree well with ab initio and some semi-empirical results. The computing times are extremelyshort.

The problem of finding a reliable approach basedon localised description of molecules has receivedconsiderable attention because the bond-conceptallows one to rationalise many molecular propertiesin terms of intrinsic bond contributions!", In recentyears, although rigorous all-electron ab initioschemes and several delocalised semi-empircaltechniques have been developed for describing themoiecular systems with varying degree of success,there is still scope for further development on thebasis of localised bond-description of the molecules.In this field, several approaches=!', varying widelyin complexities and nature of approximations in-volved, are available.

We have developed a reliable method based onbond-orbital SCF MO model that with the use ofminimum of empirical parameters and computa-tional efforts allows one to rapidly determine the na-ture of hybridisation in each bond and the relatedproperties even for very large molecules.

In this paper, our study is restricted mainly to cal-culation of atomic charge distributions, J-J couplingconstants and dipole moments of some simple 0-bonded systems. The reliability and internal con-sistency of the results have been checked by com-parison with results obtained from other localised,delocalised and ab initiocalculations.

Apparently, the application of this model is res-tricted to only those molecules which can be pre-dominantly described by localised structures.

MethodologyThe present model is based on the concept of loc-

alised two-centre MO's referred to as bond orbitals.The bond orbitals are independent one-electronmodels and the matrices representing the Fock ma-trix, overlap matrix and density matrix in a hybridbasis can be factorised into 2 x 2 blocks corre-sponding to various bonds constituting the mole-cule!".

Let us consider the molecular orbitals of themolecule under examination as strictly two-centrebond orbitals of the form,1/Jj= C"j,p"A+ C.j,pvB ... (1)where subscript j represents j-th bond orbital and,p"A and ,pvB are hybridised atomic orbitals attachedto the atoms A and B respectively. The hybrid orbi-tals are assumed to be orthogonal to each other.

In the block form, the equations for Fock matrixelements take the form,

F"v= H"v +I P,\,![CuvIU) - l!2('uvl VA)]).

).•oetc.

+I PJ.u[2CuvIAa)-(J.lAlva)]l

2 INDIAN J CHEM, SEC. A, JANUARY 1992

only the first order terms in overlap. All other inte-grals, second order in overlap, are neglected.

(b) The diagonal core Hamiltonian matrix ele-ments Hpp are evaluated by using CND012 (ref. 14)procedure. Adopting the S-expansion technique"for density matrix elements, we can rearrange thediagonal Fock matrix element, in the following form,r::= - 112 (I!' + A!') + 112(1 - ~!')(.u.ul.u.u)

B

+ L: L: (Pv. - n;)(.u.ul A.A.)B J.

... (3)

where P's are Mulliken's gross populations definedby P!,p = (np - qp) and n!, is the number of electronsin the orbital rj>w

(c) For off-diagonal core matrix elements Hpv, weapply the Goeppert-Meyer procedure" for the esti-mation of nuclear interaction integrals, retaining on-ly one-centre penetration integrals. Taking thespherically averaged values for all one- and two-centre repulsion integrals, one ultimately obtainsoff-diagonal Fock matrix element in the form,F:'B= - 114[3 (I!' + Iv) + (ZA - 2)YAA+ (ZB- 2)YBB

- (ZA+ ZB- 2)YABlS!,v-112 P!'vYAB ... (4)= fJ°!,vS!'v - 112P!'vYAB; u, von the same bond.

where ZA and ZB are the core charges on atoms Aand B respectively. Assuming ff'!'v as atomic par-ameters invariant to hybridisation, we replace theionisation energies by their average values. For anatom A with a sand p basis,

I~v = (Is + 3 Ip)/4The bonding parameter then becomes,fJ°!,v= - 114 [3u:+ I~vl + (ZA - 2) YAA+ (ZB- 2) YBB

- (ZA+ ZB- 2)YABl = 1!2[fJoA + fJoBlK ... (5)where K is a scaling factor having the range of values0.6-0.9. The mono atomic parameters are then de-termined by choosing certain standard bonds.

Optimisation of hybridisationFor closed shell system, the total electronic ener-

gy (E) in the Hartree- Fock SCF procedure in singledeterminant approximation is given by,

E = 1/2 L: P!'v (Hpv + Fpv)p,v

... (6)

Let us consider a basis rj>which we shall assume tobe hybrid formed by mixing the AO's which consti-tute the basis rj>' of the Hartree-Fock procedure. Thetwo basis rj> and rj>' will be related to each other

through an orthogonal transformation matrix (T)byEq.7,rj>= T'r/>' ..• (7)

The total energy and electron density at any pointmust remain invariant to such a change of basis pro-vided both sets of matrices P, H and F are obtainedthrough full LCAO MO SCF calculations 17. How-ever, in a localised MO approximation, the F-matrixis not really of block diagonal form. Because of this,we assume that the P-matrix in a localised approxi-mation differs negligibly from one that is obtained ina full LCAO MO calculation using the same hybrid.

Using Eqs 6 and 7, we can now writeE=e+t1e ... (8)where e is the electronic energy in the localised ap-proximation and t1e is the very small difference inenergy beween E and e.

Apparently, the most possible localised descrip-tion will result only when the hybrid basis rj> is sochosen that lt1el is as small as possible.

The hybrid orbital rj>p on an atom A containing sand porbitals (STO) is given by,

112 ( )112 (9)tP",=p!,s SA+ I-p!,s PA ...where Pps denotes the s-character of the hybrid or-bital.

Energy minimisation with respect to variations inP ,subject to the constraints imposed due to the or-thogonality of the hybrid on a given centre, can beachieved by minimising the function

G=e+ L:atp!'iA,i.!,

... (10)

where at are yet undetermined Lagrangian multi-pliers.

The minimisation and then expansion of thisfunction lead to,

+ higher order terms ... (11 )where (de! dp;.Jo refers to the value of the derivativeat some initially chosen approximation to p's, andt1p's are the differences between the optimum valueand initial value of p's.

Neglecting higher order terms, we can write Eq.(11) in the matrix form,le'llt1p, a] = le'l ... (12)

SEN et al.: STUDIES OF MOLECULAR PROPERTIES FROM FULLY LOCALISED SCF METHOD 3

In a localised approximation, these matrix ele-ments are then obtained assuming that the repulsionintegrals are invariant to hybridisation and E" = 0,for a lone pair and non-bonded orbitals.

Results and DiscussionWe have used the numerical integration tech-

niques to estimate the overlap and repulsion inte-grals. The scaling factor K is determined by the rela-tion

where A ~v is the average electron affinity. The set ofbonding parameters {J°A calculated by our method islisted in Table 1.

For obtaining optimised two centre MO's for amolecule containing only sand p orbitals, all hybridsare initially assumed to be sp'. The net charges asso-ciated with the bonding orbitals are then calculatedby using independent 2 x 2 blocks corresponding todifferent bonds. In the next step, hybrids are opti-mised at different centres and the process is iterated.

(i) Atomic charge distributionsThe net atomic charges of twelve molecules ob-

tained by our method are presented in Table 2, to-gether with those obtained by the methods of Ba-rone et a/.1O, CND018 and STO-3G19. Forhydrocar-bon molecules, our method, like other methods,yields a C - -H + bond polarity. As we go from me-thane to ethane, there is a slight increase in charge asin Del Re method 5-6. In H20, substitution of H byCH3 makes. the net atomic charge on oxygen morepositive. The charge variation is well reproduced asin the CNDO and STO- 3G calcualtions. The N - Hbond in NH3 is polarised as N- -H+ and the effect ofmethyl substitution on nitrogen is well reproduced.However, the result shows a small net positivecharge on carbon atom.

Examination of atomic charges of fluoro-com-pounds shows that charge transfer takes place fromcarbon to more electronegative fluorine atoms.More the number of substituent fluorine atoms, the-more will be net positive charge on carbon atom andless the negative charge on fluorine atoms. Thesetrends agree well with those obtained from othermethods. While a small net positive charge on hy-drogen is at variance with results from STO-3G cal-culations, other methods give a very small positiveor zero value. The common interpretation of fluo-rine, nitrogen and hydroxyl group as negative induc-tive type substituents, is reflected well in our calcu-

Table I-Values of {lOA (in joules x 1019)

Atom -fjDA Atom - {lOA

H 6.4 C 8.2Li 1.4 N 12.0Be 2.5 0 18.4B 5.7 F 22.5

Table 2- Net atomic charges in different molecules

Molecule Atom Net atomic charge

Present V.Barone'""CND012(IK} HehrellO)work

CH4 C - 0.059 -0.006 -0.050 - 0.073H 0.015 0.002 0.013 0.018

C2H6 C -0.061 -0.008 -0.026H 0.020 0.003 0.009

H2O 0 -0.289 -0.330 -0.288 -0.406H 0.144 0.165 0.144 0.203

CHPH C 0.018 0.122 0.128 0.136H 0.054 0.009 0.003 0.0140 -0.267 -0.305 -0.247 -0.350H 0.124 0.174 0.143 0.211

NH3 N -0.199 -0.255 -0.228 -0.486H 0.066 0.085 0.076 0.162

CH3NH2 C -0.044 0.068 0.079 0.113H 0.038 0 - 0.021 -0.019N -0.189 -0.240 -0.204 -0.402H 0.059 0.086 0.078 0.151

HF H 0.287 0.230 0.228F -0.287 -0.230 -0.228

CH3F C 0.038 0.187 0.169H 0.079 0.001 -0.004F -0.272 -0.189 -0.157

CHF3 C 0.333 0.593 0.532H 0.236 -0.007 -0.018F -0.1897 -0.195 -0.171

CF4 C 0.653 0.674F -0.163 -0.169

OF2 0 0.101 0.120F -0.051 -0.060

NF3 N 0.279 0.336 0.147F -0.094 -0.112 -0.049

lated results. However, for methyl amine and me-thyl fluoride this feature is not reproduced well asin the CNDO and STO-3G calculations, thoughother localised calculations agree with our results.As regards the mixed compounds, OF2 and NF3, theresults are in excellent agreement with the CNDOcalcualtions.

The deviations may be attributed to (a) neglect ofthe effect of hybridisation on the electron repulsionintegrals which leads to high values of s-character of

4 INDIAN J CHEM, SEe. A, JANUARY 1992

lone pairs and consequently high values of lone pairmoment (as shown in Table 3), (b) use of experimen-tal value of ionisation energy for hydrogen ratherthan theoretical value used in CNDO and mostother semiempirical methods and lastly (c) the na-ture of approximations involved.

(ii) 13 C- H Spin- spin coupling constantThe coupling constant between the directly

bonded C-H atoms is obtained by considering thepossible variations of the s-electron density at thenucleus+"",Juc-H=(KIKo)3apc+b ... (13)where K and K 0 are the respective s-orbital expo-nents of the carbon orbitals in the molecular envi-ronment and in the neutral carbon in the sp' confi-guration; a and b are constants to be evaluated. Us-ing the Slater's rules for the orbital exponent and re-taining only the first order term in net charge Qc oncarbon atoms, we haveJuc- H = a( 1 + 0.24 Qc)Pc + bFor a number of compounds listed in Table 4,

JI3C-H= 636(1 + 0.24 Qc)Pc -10.9 (14)Juc - H was obtained by least-squares analysis witha correlation coefficient of 0.993, which is quitegood. This excellent correlation illustrates the relia-bility of our model.

(iii) Dipole momentIn a two-centre MO approximation, the electric

dipole moment u for a bond between fJ/J and fJv canbe written as the sum of two contributions due tonuclear charges and asymmetric distribution ofatomic populations.

The first contribution due to nuclear charge is,

uchg = 4.802 q" RAB ••• (15)where RAB is the bond distance.

The second contribution, due to asymmetric dis-tribution of atomic populations is,Uhyb = 7.337[p~/~1 - pul1l2

P",,/ ~ /J - P ~12(1- Pv)ll2 Pv./ ~vl ... (16)The calculated dipole moment values for twelvemolecules are listed and compared with values ob-tained by other methods in Table 3. These resultsshow generally high values in the case of moleculescontaining N, 0 and F atoms, although the increas-ing trend is well reproduced. This is, apparently, dueto high values of lone pair moment. Since the lonepair electrons occupy a hybridised orbital alignedalong the principal symmetry axis and directed awayfrom the bonded atoms, the dipole moment asso-

Table 3-·Dipole moments for o-bonded molecules (in Debye)

Molecule Present Exp. CNDOI2IH MNDOz4 SINDO,5calc. value value value valuevalue

CH3CH,CH) 0.12 0.08 0.00 0.00 0.03HF 2.21 1.83 1.85 1.99 1.44OF, 0.12 0.30 0.21 0.25NF3 0.24 0.24 0.43 0.20 0.20CH3F 2.36 1.86 1.66 1.76 1.55CHzF, 2.66 1.96 1.90 2.21 1.73CHF) 2.35 1.65 1.66 2.23 1.45H,O 2.35 1.85 2.10 1.78 1.89

CHPH 2.15 1.69 1.94 1.48 1.61

NH3 2.38 1.47 1.97 1.16 2.16CH3NHz 1.61 1.33 1.86 1.48 1.99CH3CF3 2.68 2.32 2.18

Table 4_13C - H coupling constants (J) in o-bonded molecules

Molecule J (I3C - H) in Hz

Calc. Exp."

CH4 121.25 125C2H6 127.46 126CH3'CH2'CH3 128.01 128CH]F 144.36 148CH2F2 179.11 185CHF] 238.63 238CHPH 138.25 141CH3OCH3 136.75 138CH2(OCH3lz 161.47 162CH]NH2 133.36 133(CH]lzNH 132.72 132(CH3hN 132.71 131

ciated with the non-centrality of such a lone pair islarge. However, this effect is mostly cancelled out inOF2 and NF3 molecules due to vectorial addition.

ConclusionThe results of our calculation show that the atom-

ic charges, dipole moments and J-J coupling con-stants for simple a-bonded molecules can be ac-counted for by a fully localised model based on theSCF MO scheme, and thus reflect the soundness ofthe approximations involved in the method. Eventhough the results are somewhat inferior to thoseobtained from some other sophisticated methods,they may be considered encouraging due to (a) sim-plicity of the method, (b) inclusion of only theoreti-cally justified parameters and (c) involvement of ex-tremely short time in computation. In future, we in-tend to extend the method to the study of small n-systems, and very large systems, such as polymersand molecules of biological interest.

SEN et 01.: STUDIES OF MOLECULAR PROPERTIES FROM FULLY LOCALISED SCF METHOD 5

ReferencesI Pauling L, The nature of the chemical bond, 3rd Edn (Cor-

nell University Press, N.Y.), 1960, ch. 7.2 Cottrell T L, The strengths of chemical bonds (Butterworths

Scientific Publications. London), 1958,268.3 Exner 0, Dipole moments in organic chemistry (George

Thieme, Stuttgart), 1975,27.4 Pople J A, Schneider W G & Bernstein H J, High resolution

nuclear magnetic resonance (McGraw-Hili Book Company.N.Y.), 1959,ch. 7&11.

5 Del Re G, J chem Sac, (1958) 4031.6 Nagy J & Reffy J, J organomet Chern, 22 (1970) 565.7 Sondorf C, CanJChem,33(1955) 1397.H Miertus S, Boca R, Felekan & Yalke L, J chem Phys, II

(1976)237.9 Gayoso J, Compt Rend, 280 (1975) Cl05.

IO Barone Y, Donady J, Ellinger Y, Subra R & Del Re G, Jchem Soc. Faraday Trans 11,75 (1979) 12, 1597.

II Brownell R, Diss Abstrlnt, 42 (1982) 10,4083.

12 DeI Re G, lnt Jquant Chern, 8 (1974) 95.13 Mulliken R S, J chem Phys, 44 (1949) 497; Mulliken R S. J

chem Phys, 23 (1955) 1833.14 Pople J A & Segal G A, J chem Phys, 44 (1966) 3289; Pople

J A, Santry 0 P & Segal G A, J chem Phys, 43 (1965) S129.15 Fischer-Hjalmers I. J chem Phys, 42 (1965) 1962.16 Goeppert Meyer & Sklar A L, J chem Phys, 6 (1938) 648.17 Lowdin P C, J chem Phys, 18 (1950) 365.18 PopJe J A, Beveridge 0 L, Approximate molecular orbital

theory(McGraw-HiII, N.Y.), 1970. 194.19 Hehre W J & Pople J A, JAm chem Soc, 92 (1970) 2191.20 Maksic Z B, Eckert-Maksic M & Randic M, Theoret Chilli

Acta, 70 (1971) 22.21 Aisenoy C Y, Gigeya H P & Geerlings p, Theoret chim Acta.

55 (1980) 87.22 Maciel G E, Mciver J W, Ostlund N S & Poplc J A, J Am

chem Soc, 1 (1970) 92.23 Muller N & Partichard 0 E, J chem Phys, 31 (1959) 768.24 Dewar M J S & Henry S R, J Am chem Soc. 5H(Il)HO) 100.25 Nanda 0 N & Julg K, Theoret Chim Acta, 57 (1980) 95.