Molecular Orbital Set Determined by a Localization Procedure B Y HAREL WEINSTEIN * AND RUBEN PAUNCZ Department of Chemistry Technion Israel Institute of Technology Haifa Israel Received 12th September 1968 A starting set of molecular maximum overlap orbitals is localized by means of an external procedure using local density maximization as transformation criterion. For a group of hydride molecules (LiH BH BH3) improved localization of the starting orbitals leads to a set having increased overlap with LCAO SCF calculated molecular wave functions and improved molecular energies. The possible use of the easily obtained localized orbitals as a starting set for more elaborate calcula- tions is considered. The interest in localized molecular orbitals (L.M.O.) originally arose from the early observation that wave functions obtained from molecular orbital calculations do not exhibit direct connection to well-established chemical concepts.LMO describ- ing electronic densities localized in well-defined regions of a molecular system are however found to be closer to the chemical picture and therefore represent an important factor in the conceptual analysis of the results obtained from the molecular orbital method. Moreover since they also seem to be useful for different refinements of existing calculation methods various attempts have been made to find effective procedures for calculating LMO. Using the definition of Edmiston and Ruedenberg localization procedures can be classified as being intrinsic or external in character according to the localization criteria proposed.Orbitals obtained by both procedures exhibit particular properties which might favourably be used when dealing with correlation effects in many- electron systems. Thus localized orbitals are expected to have maximal intra- orbital correlation and divide the total electronic distribution into groups which can be later dealt with by means of special methods.3* 4 9 The localization method proposed by Edmiston and Ruedenberg being intrinsic in character uses the maxi- mization of the sum of orbital self-repulsion energies as localization criterion. The idea is based on the suggestion of Lennard-Jones and Pople6 and has been used successfully in localizing SCF starting orbital^.^ The method however necessitates the calculation of a large number of two-electron integrals. Another method proposed by Boys * is simpler computationwise but more limited in its range of application.Recently Magnasco and Perico suggested an even simpler procedure using localiza- tion criteria obtained by imposing an extremum principle on the sum of local electron populations within the molecule. Nevertheless their results are in fairly close agreement to the ones obtained by Edmiston and Ruedenberg although the localiza- tion criterion is an external one. The closeness of these results indicates that the * This work is part of a thesis submitted to the Senate of the Technion Israel Institute of Tech- nology in partial fulfillment of the requirements for the degree of M.Sc. 23 24 MOLECULAR ORBITAL SET BY A LOCALIZATION PROCEDURE connection between properties of localized molecular orbitals and the chemical concepts might be independent of the intrinsic or external features of the localization criteria emphasizing the importance of the easier external procedures.The aim of the present work is to investigate the possibility of using only an external localization criterion similar to the " localization functions " of Magnasco and Perico in order to obtain a set of molecular orbitals which can serve as good starting wave functions for more elaborate calculations. Since by the use of such a procedure an orbital set is obtained without cumbersome calculations of two-electron integrals results being close enough to those obtained from an SCF procedure would motivate the use of the resulting localized functions for these purposes. LOCALIZATION CRITERIA AND STARTING MOLECULAR SET The molecular model which has been successfully used in chemistry considers the electron pair as being the fundamental structural unit in the construction of the local electron densities known as inner-shells chemical bonds and lone pairs.Since the partition of the molecular density used here will be based on chemical evidence it will use doubly occupied molecular orbitals to describe the local densities known in chemistry. The set of orbitals will thus consist of b molecular functions describing bonds between pairs of atoms i molecular orbitals describing inner shells and I lone pairs the total number of molecular orbitals being b+i+l= n where n = +N = half the number of electrons in the molecule. (2.2) The choice of localization criteria is based on the partition of the total electronic population into subtotal electronic densities and local densities described by defined orbitals.According to Mulliken,lo an orbital @(j) defined by an LCAO-method where the summations include all atoniic orbitals Y and s on atoms k and I respectively. The total orbital density can be further decomposed into n(j,rksl) = partial overlap populatioii = 2N(j)cjrkcjSlSrkSI (2.5) (2.6) and n(j,rk) = net atomic population = ~ ( j ) I cjrk I ' Unlike populations representing sums over all MO the n(j,rk) and iz(j,rm) popula- tions are non-invariant quantities with respect to orthogonal transformations. The localization procedure we present here being based on a series of such orthogonal transformations will be expected to increase the numerical value of certain local electron densities defined through the n( j r k q ) and n(j,r,t) partial densities.The confinement of the electronic densities to certain previously defined regions of the molecule is achieved by restricting the summations in the total density eqn. (2.4) to certain groups of atomic orbitals only. H . WEINSTEIN AND R. PAUNCZ 25 The definitions of the resulting localization functions are similar to those given by Magnasco and Perico where to yield the electronic density described by orbital @(j). shells the localization function will take the form and Aj2) represent indices of certain groups of atomic orbitals considered If the molecular orbital @(j) belongs to the first i wave functions describing inner where Aj = I(A,) represents the inner shell atomic orbital of atom A. For @(j) representing a bond orbital which describes the interatomic charge density in the region between the bonded atoms B and C the localization function becomes where Ajl) = V(Bj) and Aj2) = V(Cj) are indices of the valence atomic orbitals of atoms B and C respectively.An orbital @(j) considered to belong to the group of I lone-pair wave functions yields a localization function given as (2.10) where Aj G V(Dj) is the group of valence atomic orbitals on atom D. So far there is a strong correspondence between the localization procedure derived for LCAO-SCF-MO given by Magnasco and Perico and the one presented here. The crucial difference lies in the fact that we do not start with an SCF solution but with a conveniently chosen basis set obtained from a procedure being much simpler than SCF.Given a group of such molecular wave functions it may be considered as belonging to a subspace of the general Hilbert space X . If the molecular orbitals are obtained by an LCAO-procedure then they are said to be elements of the subspace dcYf spanned by the atomic wave functions. Different molecular orbitals formed by linear combinations of the same set of atomic basis functions form different subspaces of the same space d. Let A and 9 represent two such subspaces of d. According to the basic projection theorem,l' each element fc%' can be uniquely decomposed into two components with where is the orthogonal complement of the subspace 4 and f o is called the projection off on A. Consider now a case in which the subspace .Acd is spanned by molecular wave functions obtained by an LCAO-SCF procedure and 9 is also a subspace of a? defined by some arbitrary wave-functions.The decomposition of every vector in 9 according to eqn. (2.1 1) being unique and the norm off being split according to the same decomposition procedure I f 1 = i f 0 I 2 + If1 I 2 (2.13) f = f o +fl (2.11) f 0 c 4 and flc.Al (2.12) 26 MOLECULAR ORBITAL SET BY A LOCALIZATION PROCEDURE the overlap between the two spaces 9' and d e a n be defined as 09"4 = c I .fk I (2.14) wherefk are the molecular basis vectors of the 9 space and ft is their orthogonal projection on the A space. If a certain subspace Y f is now defined in 9 orthogonal transformations between elements of 9' and 9 can define a new subspace 9. The orthogonal transformations can be chosen so as always to cause an increase in the sum (2.15) 1 The resulting subspace 9 will thus have an increased overlap with the 4 space.Consequently the aim of this work will be achieved if the transformations needed in the localization of a certain starting set of molecular orbitals (Y) will also lead to an increase of the defined overlap between the new space of localized functions @ and the LCAO-SCF functional space. THE LOCALIZATION PROCEDURE The localization method described has been applied to a group of hydride mole- cules LiH BH and BH3. As a starting point in each calculation use was made of a basis set of molecular orbitals obtained as eigenvectors in the diagonalization procedure of the atomic overlap matrix. These orbitals defined by Lykos and Schmeising,12 represent a set of maximum overlap orbitals.These authors showed that for homonuclear systems the Hiickel MO are identical to maximum overlap molecular orbitals. Moreover MOO have also been shown l2 to be in better agree- ment with the SCF-MO without overlap than the Huckel MO in a general case merely because nonclosest neighbour interactions are included in their construction. Taking also into consideration the readiness with which they can be obtained MOO have been considered as offering a good starting set. The number of molecular maximum overlap orbitals coincides with the number of basis atomic orbitals which can usually be taken to be larger than half the number of electrons in the molecule. The first step in the proposed procedure therefore consists in the partition of the starting set 9 into 9' and (9-9'). For diatomic molecules the subset 9' was chosen to contain functions yielding maximal numerical values of the respective localization functions Lj.Each element in the defined subset 9' of size n represented a molecular orbital describing the electronic charge distribution in a well-defined molecular region. The molecular charge concentrations were defined according to a certain chemical picture of the molecule. However the total charge partition is by no means unique and several empirical " charge maps " can be investigated as to which of them yields the most accurate results. Once the set 9' was defined orthogonal transformations between pairs of orbitals were performed each pair being composed of an element fc9" and an element g c (9 - 9'). Every new f obtained by the transformation changed 9'' into a new subset which was further treated in the same way.The transformations were thus completed for all possible pairs subject to the restriction that the new f obtained yield a higher numerical value for the corresponding localization function Lj. Since the transformation matrix T cos 0 -sin 6 T = ( sin 6 cos 6 H . WEINSTEIN AND R . PAUNCZ 27 was made dependent 011 the parameter 0 it was possiblc to construct it function of 0 from which an optimal value for the rotational angle in each 2 x 2 transformation was found l 3 by solving the equation dLj(0)/dO = 0. For each diatomic molecule considered an atomic basis set of the same magnitude was defined consisting of STO with energetically optimized exponents. The basis functions were chosen to be of Is 2s and 2pa type on the heavy atom and included only one 1s orbital on the hydrogen.The LiH molecule was considered to be described by two doubly occupied mole- cular orbitals one of them describing an inner shell around the Li atom and the other generating the bonding charge distribution in the region between the two atoms. The exponents of the atomic STO as well as the value for the interatomic distance were taken from Ransil’s calculations of heteronuclear diatomic molecules,14 in order to make possible an ulterior comparison of the results. The molecular energies calculated for the set of orbitals obtained in several stages of the localization pro- cedure numerical values for the overlap function OgA defined in (2.15) where d represents the LCAO-SCF functions reported by Ransil and numerical values of the localization functions,* are listed in table 1.The BH molecular orbital set was considered to consist of three doubly-occupied molecular orbitals forming an inner shell centered on the B nucleus a bond molecular orbital and an orbital describing a lone pair charge distribution. Table 2 contains results obtained at various stages of the localization procedure. For the polyatomic boron hydride the choice of the 9’-set was first made accord- ing to a maximum-population criterion made possible by the cumbersome calculation of the charge density matrix for a multiconfigurational wave function using maximum overlap orbitals as basis functions. The 9’-set turned out to be identical with the one obtained from a choice using maximal numerical values of corresponding localiza- tion functions as a criterion.This was encouraging and the possibility was con- sidered to rely completely on maximum values of localization functions as the criterion to be used for the definition of the 9’-set. The atomic basis STO were chosen to be of Is 2s 2p, 2p and 2p,-type on the B atom and to contain one 1s type function for every H atom in the molecule. Of the eight functions contained in the 9-space four were considered to form the 9 ’ - subspace. They included one molecular orbital describing each 2-electronic bond in the molecule and one doubly occupied orbital forming an inner shell around the B nucleus. The exponents of the basis atomic orbitals and the inter-nuclear distance were taken to be identical to the constants used by Pipano l5 in an LCAO-SCF and a full C.I.calculation of BH3 to facilitate comparison of the obtained results. Table 3 contains the results of the localization procedure applied to the BH3 molecule. DISCUSS I ON An observation common to all the investigated example is that the starting molecular set yields energies which are in poor agreement with the LCAO-SCF calculated energies and have the smallest overlap with the SCF orbital set. In each case by performing the localization of the orbitals one obtains an improvement of the calculated energies and overlap with the LCAO-SCF space which accompanies the requested increase in the values of the localization functions. One therefore observes the gratifying relation between localization and improvement of the molecular orbital set. However the required improvement in molecular energy as compared * Ptotd represents values of localization functions Pi defined by Magnasco and Perico ; Pspeciec are the numerical values of the localization functions Lj defined in this work.28 MOLECULAR ORBITAL SET BY A LOCALIZATION PROCEDURE 00 2 W 00 00 0 9 0 4 - 0 0 0 3 8 3 0 E x I4 .r( 1. H f r; I b I I * m e H c\1 00 m W 6\ m ? I w h d 8 Q\ 00 9 I H 2 c .- x kind of transform Ekin Epot M.O.O. 24.853762 - 59.515638 inner shell and lone p. optim 24.907302 - 57.140541 lone p. optim. 24.082540 - 57.603794 SCF 25.1246 - 58.01482 Function exponent TABLE 2.-BH MOLECULE ‘total i ; 1.24875204 I ; 0.41 5760702 overlap with SCF Epo t 2 el. Etotal 8.530937 - 23.984939 0.97533 16 b ; 1,88720029 i ; 1*99701091 5.910709 - 24.17653 0-978223 b ; 1.88720029 I; 1-00865878 i ; 1 -24875204 7.1 25723 - 24.249531 0.985320 b ; 1 ~88720029 1 ; 1.88691773 5.6686 -25.0756 * 1 SH 1 S B 2SB 2PB ; 1.1860 4.6805 1 -2955 1.3168 ; ‘specific i ; 1.24875204 b ; 0.792066 163 1 ; 0.41 5760702 i ; 1.99701091 b ; 0.7920661 63 I ; 140865878 i ; 1 -24875204 b ; 0.792066163 I ; 1.88691773 internuclear distance 2-329 a.u.; internuclear repulsion 2.146 am. * Rand l4 TABLE 3 .-BH3 MOLECULE Epot 2e1. Etotal 16.641091 -23.0930 overlap with:SCF 0.7464549 0.789704 0.920575 0.920899 ‘total i ; 1 -9554295 1 b ; 0.869304179 b ; 2-8 1892274 b ; 1.45779479 ‘specific i ; 1.95542951 b ; 0.267928363 b ; 048800799 b ; -0’858374978 Ekin Epot 27’222153 - 74.104605 kind of transform M.O.O. i ; 1 -9554295 1 b ; 0.956055994 b ; 2.77342052 b ; 1.37954702 i ; 1 m9554295 1 b ; 0.271768867 b ; 0.27563364 b ; 0.508858310 14.594344 - 24.736825 27’080747 - 73.560276 bond opt.I i ; 1 -9554295 1 b ; 0.272450727 b ; 0.535184892 b ; 0.513980190 i ; 1 -9554295 1 b ; 0,985927778 b ; 1.76734181 b ; 1.37230965 14.143330 - 25.639230 25.851089 - 72.782009 bond. opt. I1 i ; 1.96536577 b ; 0.985927778 b ; 1.767341 84 b ; 1.37230965 i ; 1 *96536577 b ; 0.543528790 b ; 0.535184892 b ; 0.508858314 inner shell and bond opt. 14.163857 - 25.654446 25.977 1 18 - 72.94378 1 i ; 1.98581803 b ; 1.16591427 b ; 1.33254770 b ; 0.841003964 14.197128 - 26.3 17308 25.887759 - 73.586555 SCF* 2PzB 1-30 ; 2SB 2PXB 2PyB 1.30 1.30 1-30 Function exponent internuclear distance 2.383 a.u. ; internuclear repulsion 7.14836 a.u. * A. Pipano l 5 H. WEINSTEIN AND R. PAUNCZ 31 to the initial values strongly depends upon the possibility of performing a satisfactory number of orthogonal transformations-which implies a requirement of a large 9’- set.For the BH molecule only a small improvement of the starting results is achieved mainly because the (9-9’) set contained only one molecular orbital because of the very small atomic basis used. For the LiH and BH3 molecules the improvement is sensibly larger as compared to BH since the two transformed sets 9’ and (9-9‘) were equal in size. The result of the localization procedure is also found to be encouraging as to the connection existing between chemical concepts and the results of molecular orbital calculations in view of the possible construction of a good starting molecular orbital set by the simple use of chemical concepts. Since the handling of the starting functions and of the results in the presented procedure is simple and the procedure showed up to be dependent on the magnitude of the basis set used fairly large sets of functions can be used probably causing a further improvement in the results and determining a final minimal set which can then be used in more accurate calculations.In view of the fact that determination of the best minimal set of functions when starting with an arbitrarily chosen basis is not always easily achieved with more accurate calculations the readiness with which the determination is achieved with the presented procedure only emphasizes its usefulness. Also since accurate calculation methods based on localized electron pair functions can be sensibly simplified if the molecular system is localizable and such information being quite difficult to obtain for some systems,16 the presented localization procedure seems to be adequate to use in localizability investigations and subsequent starting functional-basis construction.C. Edmiston and K. Ruedenberg Rev. Mod. Physics 1963 35 457. J. M. Parks and R. G. Parr J. Chem. Physics 1958 28 335. E. Kapuy Acta Physica Acad. Sci. Hung. 1958 9 237 ; Acta Physica Acad. Sci. Hung. 1961 13,461. R. McWeeny Proc. Roy. SOC. A 1959,253 242. J. E. Lennard-Jones and J. A. Pople Proc. Roy. SOC. A 1950 202 166. C. Edmiston and K. Ruedenberg in Quantum Theory of Atoms Molecules arid the Solid State ed. P. 0. Lowdin (Academic Press Inc. New York 1966) p. 263. S. F. Boys in Quantum Theory of Atoms Molecules and the Solid State ed. P. 0. Lowdin (Academic Press Inc. New York 1966) p. 253. V. Magnasco and A. Perico J. Chem. Physics 1967 47,971. lo R. S. Mulliken J. Chem. Physics 1955 23 1833. B. A. Lengyel in Adv. in Quantum Chemistry ed. P. 0. Lowdin (Academic Press Inc. New York 1968) p. 14. P. G. Lykos and H. N. Schmeising J. Chem. Physics 1961 35 288. * J. E. Lennard-Jones and J. A. Pople Proc. Roy. SOC. A 1951 210 190. l 3 M. Rossi J. Chem. Physics 1965 43 2918. 14B. J. Ransil Rev. Mod. Physics 1960 32 239. * l 6 E. Kapuy J. Chem. Physics 1966 44 956. A. Pipano DSc. Thesis (Technion Israel Inst. of Technology).