On the bonding in B2H6 and the lone pairs in H2O: The use of localized molecular orbitals in spin-coupled calculations

Journal of Molecular Structure (Theochem), 229 (1991) 155-162 Elsevier Science Publishers B.V., Amsterdam 155 ON THE BONDING IN BzH6 AND THE LONE PAIRS IN H,O: THE USE OF LOCALIZED MOLECULAR ORBITALS IN SPINCOUPLED CALCULATIONS DAVID L. COOPER Department of Chemistry, University of Liverpool, P.O. Box 147, Liverpool L69 3BX (U.K.) JOSEPH GERRATT Department of Theoretical Chemistry, University of Bristol, Cantocks Close, Bristol BS8 ITS (U.K.) MARIO RAIMONDI Dipartimento di Chimica Fisica ed Elettrochimica, Universit& di Milano, Via Go&i 19, 20133 Milan0 (Italy) (Received 15 March 1990) ABSTRACT The implementation of a recently proposed procedure for generating localized SCF molecular orbitals makes it possible to carry out spin-coupled calculations on the bridging region of B,Hs and on the non-bonding electrons in H,O. The description that emerges for B,Hs is fairly similar to the conventional view of three-centre two-electron “banana bonds”, but the orbital picture for the non-bonding electrons in H,O turns out to be significantly different from the classical valence bond idea of doubly-occupied lone pairs. INTRODUCTION Many different criteria have been developed for localizing SCF or CASSCF orbitals, of which the best known is probably Boys’ scheme [l] for self-consistent-field (SCF) wavefunctions. In practice, most of these procedures turn out to be expensive or unreliable, or both, and some of them require additional “external” criteria. The obvious disadvantages of localized molecular orbitals (LMOs ) include the non-uniqueness of any localization criterion, as well as the orthogonalization tails exhibited by orthogonal LMOs. Nonetheless, it is clear that a reliable, inexpensive scheme could be very useful when constructing wavefunctions that take into account electron correlation only for the more “interesting” parts of a (large ) system. In the present work we describe our implementation of a very simple localization procedure discussed by Pipek and Mezey [ 21, and we pre- 0166-1280/91/$03.50 0 1991- Elsevier Science Publishers B.V. 156 sent the results of modern valence bond (VB) calculations on the bridging region of BzH6 and on the non-bonding electrons in H20. POPULATION LOCALIZATION SCHEME Consider molecular orbitals I,Y~ expanded in a basis of atom-centred functions xP according to Vi = C cipXp P where, for convenience, we assume that all the tip are real. We use the symbol S,, for the overlap integrals between the basis functions, and it is useful to define Rjq = f CcjpSpp P The basic idea of the population localization scheme [ 2 ] is to maximize the quantity in which (i 1PA 1i) is the contribution made by each electron in molecular orbital I,Yi to the Mulliken population on centre A. The matrix elements of PA can be written where qeA signifies that the summation is restricted to basis functions on centre A. Obvious advantages of this method are that no additional localization criteria are invoked and that the only integrals required are S,,. We define Ast, Bst and CAstas follows: %,=C(SIPAI~)'-CA~~'/~ A A and cA,,=(slPAIS)-_(tlPAIt) In our implementation, the maximization of 2 is carried out using consecutive 2 x 2 rotations on pairs of orbitals (and on the matrix R): Is’) =cos(x Is) +sincr It) It’) = -sinew Is) +cosa It) 157 with In general, we have found that it is preferable to localize core orbitals and doubly-occupied valence orbitals separately, rather than to introduce mixing between these two groups. Within each group, we first rotate all pairs of orbitals through a small angle, typically a! = n/1000, in order to destroy any symmetry that is present. Our criterion for convergence, which is normally satisfied after a very small number of inexpensive iterations, is that o! should not exceed 10 -’ for any pair of orbitals. One of the least subjective localization procedures is the method of Edmiston and Ruedenberg [ 31, in which the Coulomb self-energy is maximized. In general, Pipek and Mezey [2] have found that the LMOs from the population localization and Edmiston-Ruedenberg methods are almost identical, but that the new scheme is very much less expensive. APPLICATIONS Single-configuration restricted Hartree-Fock (RHF) calculations were carried out at the equilibrium geometries of BzH6 and of Hz0 using basis sets of TZVP quality. These basis sets for (B,O/H) consisted of (10s 6p/5s) primitive spherical gaussians contracted [ 41 to [ 5s 3p/3s], and augmented with polarization functions with dn = 0.5, do = 1.28 and p,= 1.0. The six valence molecular orbitals of BzH6 were then localized using the strategy described in the previous section. Four of the LMOs were found to be equivalent, with each corresponding to a different terminal B-H bond; one of these is shown in Fig. 1 (a) as contours of 1t,u1’ in the plane containing the four terminal H atoms. We refer to these orbitals as &-+6, where ly, and y2 are the core orbitals (essentially B (1s 2) ) . The two remaining valence LMOs (I,+ and vs) are equivalent to one another, and involve the bonding between the two boron atoms and the two bridging hydrogens. In view of the fact that orbitals (y,-& correspond to rather ordinary B-H bonds, it seems very reasonable to concentrate our attention on the four electrons involved in the B-H-B bridges. For this purpose, we use the spin-coupled approach to electronic structure [ 51, which represents the modern development of VB theory. In the present case, the four “active” electrons were described by four distinct, singly-occupied, non-orthogonal orbitals V/~-V/~,with the remaining electrons accommodated in the doubly occupied molecular orbitals (MOs ) v/1-v/6.Both modes of spin coupling for four electrons with a total spin of zero were included. The spin-coupled wavefunction YS,Mfor this system can be written in the form [5a] 158 (4 (b) I Fig. 1. Orbitals in B,H,: (a) ty3in the plane containing the four terminal H atoms; (b) @I in the plane containing the two B-H-B bridges; (c) & in the plane containing the two B-H-B bridges. The contours of the square modulus are shown for each orbital. where 0;: is the perfect-pairing spin function for 12 electrons with a total spin of zero. The two spin-coupling coefficients, cO,k,were optimized simultaneously with the four spin-coupled orbitals, @P,which were expanded in the basis comprising LMOs ~,Y-I,Y~ and all the virtual MOs (68 orbitals in all). There were no constraints on the overlaps between these orbitals or preconceptions as to their form. The electron correlation taken into account by this simple spin-coupled wavefunction results in an improvement in the total energy of ca. 61 kJ mol-l 159 over the RHF solution. We find that two of the spin-coupled orbitals are involved in one of the B-H-B bridges, and have an overlap with one another of 0.85 (see Table 1). One of them, &, is shown in Fig. 1 (b) and closely resembles the superposition of sp3-like hybrids stemming from the two boron centres, but it is distorted towards the line joining the two boron atoms. The other orbital, q&,is a distorted H (1s) function and is shown in Fig. 1 (c). The dominant mode of spin coupling (99.3% of the wavefunction in the basis of Kotani et al. [ 61) corresponds to singlet coupling of the spins of these two electrons. Orbitals G3and @4are the counterparts in the other B-H-B bridge, and can be transformed into the first pair by symmetry operations of the molecular point group. From the overlap integrals listed in Table 1, it is clear that it would have been unreasonable to impose any constraints of orthogonality between these orbitals. The picture that emerges from our spin-coupled calculations for the bridging region of B2H6 is very appealing, and has much in common with the conventional view of three-centre two-electron B-H-B “banana bonds”. One difference from the schematic representations in many text books is the distortion of the orbitals towards the B-B axis. We turn now to the calculations on H20. The four valence LMOs consist of two O-H bonding orbitals ( I,U~ and y3) and two non-bonding orbitals ( v4 and v5), where v/1is essentially 0 (1s 2) . Orbital I,U~ is shown in Fig. 2 (a). Our main interest is in the four non-bonding electrons, usually assigned in classical VB descriptions to doubly-occupied lone-pair orbitals, pointing above and below the molecular plane. With this in mind, spin-coupled calculations were carried out for these four electrons, with the remaining six electrons accommodated in the doubly-occupied MOs t,q-+~~.The four spin-coupled orbitals qjPwere optimized as completely general linear combinations of v4-v5 and all the virtual MOs (a total of 28 orbitals). In this case, the spin-coupled wavefunction provides an energy improvement over SCF theory of ca. 73 kJ mol-‘. All the spin-coupled orbitals are found to point away from the hydrogen atoms. Two of them, q.jland ti2,point up from the molecular plane - they are shown in Fig. 1. Orbitals @3and & are the counterparts pointing down, and are related to the first two by symmetry operations of the C,, point group. Orbitals TABLE 1 Overlap integrals between spin-coupled orbitals for B,H, (the orbitals are identified in the text) 91 $62 1 0.85 1 $3 $4 0.38 0.21 1 0.21 0.16 0.85 1 160 Fig. 2. Orbitals in H,O: (a) yz in the molecular plane (~4 mirror); (b) & in the CT,mirror plane; (c) & in the CT,mirror plane. For each orbital, we show contours of the square modulus. q&and & are rather more compact that $I and q&,and point along a direction much closer to the C, rotation axis. As can be seen from Table 2, the orbitals in each “pair”, i.e. (q&&) and (q&,&), have an overlap with one another of 0.59, which is in fact similar to the overlap between $I and q&,and is smaller than the overlap between & and #4. Nonetheless, the most important mode of spin coupling (93.9% in the Kotani basis) corresponds to pairing up the spins within each pair, although the other contribution is clearly not negligible. The picture revealed by these spin-coupled calculations is very different from the ideas of classical VB theory. From our experience with many other systems, 161 TABLE 2 Overlap integrals between spin-coupled orbitals for Hz0 (the orbitals are identified in the text) $3 91 h? $3 $4 1 0.59 1 0.61 -0.17 1 -0.17 -0.81 0.59 1 44 we can predict that reagents approaching the oxygen atom will interact initially with one of the larger non-bonding orbitals (& or &) and will thus appear to attack “at the position expected for a lone-pair”. Nonetheless, we can see that the form of the four spin-coupled orbitals, when taken together, is totally consistent with the well-known finding that the total electron density around the oxygen atom does not show particularly prominent maxima in the positions expected from the classical VB lone-pair description [ 71. CONCLUSIONS The population localization procedure proposed by Pipek and Mezey [Z] appears to be a very useful tool, especially when combined with the spin-coupled method, in that it allows us to concentrate on the bonding in part of a molecule. In the present work, this combination of techniques leads to a description of the bridging region of BzH6 which is fairly similar to the conventional “banana bond” view, but to a description of the non-bonding electrons in HZ0 which is quite different from that employed in classical VB theory. More extensive calculations are underway to confirm that descriptions such as these are reproduced by spin-coupled wavefunctions in which electron correlation is included for all the valence electrons. Methods similar to those described here are currently being used to model cycloaddition reactions, particularly those involving 1,3-dipoles, and to elucidate the bonding in complexes involving transition metal atoms in low oxidation states, such as Cr (C0)5H2. It should also now be possible to investigate the electronic structure of larger boron hydride systems. #Results will be reported in due course. REFERENCES 1 2 3 4 S.F. Boys, Rev. Mod. Phys., 32 (1960) 296. J. Pipek andP.G. Mezey, J. Chem. 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