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The use of population localised orbitals to interpret molecular orbital calculations
Chemical Physics 44 (1979) 151-162 0 North-Holland Publishing Company
THEUSEOFPOPULATIONLOCALISEDORBITALSTOINTERPRET MOLECULARORBITALCALCULATIONS W.S. VERWOERD Department of Physics, University of South Africa, Pretoria 0001, Republic of South Africa
Received 23 May 1979; in timal form 10 August 1979
An efficient and general method is derived to calculate population localised mokcular orbit& (LMO’s) from agiven SCP eigenvector matrix, by reduction to an eigenvalue problem. Applications to both localised molecules (NH3 and CzH2) and deiocalised ones (BaHe, CeHe and butadiene) arc discussed in some detail. It is shown that unequal occupation of atomic cnergy levels leads to nonorthogonal LMO’s.The consequences cf nonorthogonal atomic hybrid orbitals are discussed, formulas for their overlap in terms of atomic occupation numbers are derived and it is shown that the occupation numbers are connected to LMO atomic orbital coefficients by various sum rules.
I. Introduction The use of localised molecular orbitals (LMO's) in molecular theory has received considerable attention in the past. The central idea is to use the freedom allowed by the determinant form of the many-electron wavefunction, to make linear combinations of the HartreeFock single electron molecular orbit& leading to a greater degree of spatial localisation. In other words, one has to find a linear transformation from the original set of occupied molecular orbitals (MO’s) which are expressed as a matrix of coefficients in an atomic orbital (AO) basis, to a new matrix defining the LMO’s. While a variety of criteria for the localisatiou and of methods for calculating LMO’s have been proposed, there seems to be general agreement that there are four main uses of localised orbitals: (1) To facilitate the interpretation of MO calculations in terms of bonds and other chemical concepts. (2) To find orbitals which may be transferred from one molecule to another containing a similar local atomic arrangement. (3) To simplify the treatment of correlation by findlug a description which minimises interorbital correlation effects. (4) To calculate self-consistent wavefunctions in large systems.
The most generally used localisation methods appear to be those of Edmiston and Reudenberg [I ] and Boys [2] and computer programs for their implementation are now generally available [3,4] . The particular method selected, however, depends to some extent on the purpose of calculating LMO’s. In the present context we wish to focus attention on the first of the above mentioned uses, and in contrast to calculations emphasizing the other points one would prefer a quick and simple method which is as independent of the particular MO procedure as possible, and which may be applied afterwards to various different completed MO Cal&ations. From this point of view, “energy” localisation methods like those mentioned above suffer from the drawback that they require the calculation of a large number of two-electron interaction integrals. This is cumbersome and may only become tractable for larger molecules if a simplifying assumption like the neglect of overlap, made in the MO calculation, is also applied to the localisation [.5], but which couples the two stages of the calculation together. For this reason, locallsation methods based on“population localisation” methods like those of Peters [6] * + See also ref. [S] _
152
MS. ~renr~oerdlPoplrlatiorllocalised orbitals
and Magnasco and Perico [7J which do not employ two-electron integrals are more suitable for our purpose. In these, subsets of atomic orbitais in the basis set which are centred on a specific atom or pair of atoms are defined as local sets, and to construct the LMO’s one chooses linear combinations of MO’s which maximize the contribution of the local sets. As a general purpose method appIicabIe to all moIecules the approach of Peters [6] is hampered by the rather ad hoc way in which the transformation matrix has to be determined from the symmetry properties of the particular system_ On the other hand, while t!re work of Magnasco and Perico [7] is based on a systematic generally applicable procedure, a comparison by Trindle and Sinanoghi [5] of the methods of refs. [I ,73 shows that despite its simp!er structure the latter method is not suitable for larger molecules due to the problems with the convergence of its iteration cycle. In this article we therefore present a new mathematical approach to the same physical idea of population localisation. The formalism that we use is described in section 2. The method is completely general, it needs only the MO coefficient matrix so that it may be directly applied to both ab initio and semi-empirical calculations, and we have applied it to molecules ranging from diatomic to a 35atom molecule with a 74 dimensional orbital space without experiencing any convergence problems. A FORTRAN program implementing the procedure has been deposited with the QCPE [9]. The calculation time required is only a fraction of the time requited by a self-consistent field MO calculation (for example, 5% per LMO of the time for a single SCF loop of the relatively fast MIND0/3 semi-empirical program [8] on the 35atom molecule) so that it should prove useful as a standard aid for the interpretation of MO calculations. Another objection against population localisation raised by Trindle and SinanoBu [5], is that it leads to larger deviations of atomic hybrid directions from the internuclear vector than energy locaiisation: an average of 9” against 3” for CNDO calculations on some small molecules. To the extent that MIND0/3 calculations on the same molecules are comparable, the value of 4O resulting from our version of the method thus seems quite acceptable. The reason for such deviations is discussed further in the last section. One feature of our procedure which should be pointed out at the outset, is that the LMO’s correspond-
ing to bonds at different locations are in genera1 not orthogonal. In this respect it corresponds to the localisation methods of Gilbert [IO) and of Adams [ll] , both of which utilise modifications to the molecular hamiltonian, and the possibility of non-orthogonality was also mentioned by Peters [6] _Most of the other studies (and notably those of refs. [ 1,7]) enforce an orthogonality requirement on their LMO’s. However, it was pointed out by Adams [ 111 that this may limit the degree of localisation attained and reduce the transferability of the resulting LMO’s; and a decrease in localisation will obviously also impair the usefulness of LMO’s for other purposes. One of the results to be derived in this article is to show that the overlap between LMO’s is ciosely connected to the difference in occupation number of AO’s forming hybrid orbitals, and that for such hybridised atoms the requirements of orthogonality and localisation are therefore incompatible. The use of non-orthogonal LMO’s is thus often unavoidable, but because the required overlap is well-defined there is no arbitrariness in the actual LMO’s obtained, supporting an empirical observation of Peters [6] . If one only wishes to study a given bond in a molecule, it is possible to sidestep the orthogonality issue by only projecting out the one or more LMO’s representing this bond, while the rest of the MO space may in principle be orthogonalised to these without full iocalisation. One is then dissecting the molecular electron cloud into a “local” part connected with the particular bond, and a “non-local” remainder. Further LMO’s?nay be extracted from the remainder, but will be less than maximally localised due to the shrinking dimension of the orbital space. Presumably this may be useful to generate fully orthogonal LMO’s for a small number of bonds in a large molecule for the purpose of transferability studies. A proposal for incorporating such a progressive orthogonalisation is outlined in section 2, but we have not investigated the matter any further. In section 3 we demonstrate that our method, using the single-bond-at-a-time approach, may be used to answer in a direct way qualitative questions about the existence and multiplicity or otherwise of localised bonds in molecules. Applications are shown to both traditionally locabsed systems like saturated hydrocarbons, and less localised ones like B,H, and benzene. More quantitative applications of our LMO’s to study
W.S. lfer~r~er~J?op~~latio~i
the relation between bond length and hybridisation, are reported in a subsequent paper [ 121. Section 4 is devoted to a more detailed study of the non-orthogonality question. In particular, we show that the overlaF between LMO’s is a well defined phenomenon, and that the connection between LMO orbital coefficients and A0 occupation numbers is given by an equation which can be well represented by simple renormalised sum rules. These supply the basis for the applicability of some generally used qualitative arguments on molecular geometry which are usually based on an assumption of ortllogonality, an example of which is given.
153
localised orhitals
has been found, it is used to construct the vector L=EX.
(2.2)
Since N is identical to the non-local components of L, it is clear that to make L as “local” as possible, the appropriate criterion for an optima! approximate solution to eq. (2.1) is that it slrould minimise Nthr, the square magnitude of N. It is easily seen that this condition is equivalent to the “normal” equation (2.3)
R’RX=O:
Unfortunately, the set of solutions of this equation is identical to that of eq. (1) (as may be slrown using an eigenvalue expansion
2. Transformation to local orbitals We assume as given an eigenvector matrix E, of whicll the column vectors are the LCAO coefficients of the occupied molecular orbitals. The problem to be solved is to find a linear combination of these column vectors which is as “local” as possibie. We define the atomic orbitals belonging to the atoms which participate in a given bond, as its local orbitals and a!1 the others as non-local. Our approach is then to describe the bond by a LMO which contains as little as possible non-local contributions to its probability density. Note that in accordance with the philosophy outlined in section 1, we treat only one (possibly multiple) bond at a time instead of finding a complete linear transformation of E. Translated into matrix algebra, to study a given bond we first convert E to a reduced matrix R by deleting all the rows corresponding to local AO’s. The columns of R are to be combined to form a sum equal to zero, and if we form a column vector X from the necessary linear combination coefficients this may be expressed as the equation N=RX=O.
(2-l)
If R is a square singular matrix, eq. (2.1) has a solution which may be found by conversion to a non-homogeneous equation and subsequent matrix inversion, or an equivalent procedure. In the present connection, however, neither the squareness of R nor the existence of exact solutions is guaranteed. To find the appropriate generalisation, we note that once some approximate solution of eq. (2.1)
of R).This
means that, apart
from exact solutions of eq. (2.1), NtN is minimised only by the trivia! solution X = 0, and we have to exclude this possibility explicitly to find other solutions. To do this, we note that because E has orthogonal columns, L’L = Xf.E*EX=
X’X
(2.4)
so that if L is to be normalised, the minimisation must be carried out subject to the constraint
XfX= 1.
(2.5)
Using X as a Lagrange multiplier it means that (NtN - hXtX) must be minimised, leading to the equation RtRX=
AX.
(2.6)
From eqs. (2.4) and (2.5) it is found that N’N=XtRfRX=h,
(2.7)
showing that the eigenvalues h of R t R may be interpreted as the minima! values of NtN, i.e. the residual non-local content of L. Whatever the dimensions of R, the matrix RtR is square and has positive definite eigenvalues so that the problem reduces to finding its lowest eigenvalue h, and corresponding eigenvectors. If this is A = 0, eq. (2.6) reduces to eqs. (2.3) and (2.1) and Xis an exact solution. Otherwise, eq. (2.6) is the desired generahsation and X, immediately specifies the degree of localisation that can be attained. The case that 1, is degenerate corresponds physically to multiple bonds or to non-bonded orbitals localised on one of the participating atoms. It may also happen that other eigenvalues are so close to ho that they are not significantly worse solutions. To decide this issue,
154
WS. VenvoerdjPopulation localised orbitals
it is useful to note that the normalisation of the columns of E implies that the matrix elements of R t R are of order of magnitude unity. Its eigenvalues are therefore also typically of this order so that it is reasonable to treat eigenvalues within a range of +O.Ol or so as degenerate. In practice it is our experience that the non-local content of classical bonds is usually not more than 1% while the non-local content between neighbouring atoms where no classical bond exists, is typically about 40%. Tne criterion above seems adequate in the light of these numbers, but a comparison of the total number of electrons and the number of localised orbitals found can always be used to trace any LMO not directly found by our procedure. The diagonalisation of RtR yields multiple solutions as a series of orthonormal vectors Xi. When substituted in eq. (2), we find that the corresponding LMO’s are also orthogonal. because of the ortbogonality of E: LtL.=X!EtEX , . =XfX =S.. (2.8) I ij I]’ i I I Although they therefore certainly describe different bonds, it may be that a further unscrambling of these I.&IO’sinto u-bonds, r-r-bonds and non-bonded orbitals is desired. We treat this sub-problem in a similar way as the original localisation problem, but working now in the subspaceof the local bond set of L;s and concentrating on their local atomic orbital components. First any non-bonded orbitals are found by looking for orbitals localised on one of the participating atoms only. The rest of the L{s are orthogonalised to any of these found (which may be done easily by using the other eigenvectors of the sub-problem), and the subspace contracted once more to these truly “bonded” LMO’s. Finally we find the n-bonds by calculating orbitals in this space which have minimal s-orbital content, and a final orthogonahsation is carried out. If there are also d-orbitals present, a further similar step may be necessary at this stage to unscramble rrand S-bonds. The procedure is then repeated for the next bond required.‘Because it has a different set of local orbitals, the. reduced matrix R will be different for each bond and a different eigenvalue problem has to be solved. Consequently neither the linear combination vectors X, nor the LMO’s L, of different bonds will be orthogonal in general. If the process is carried to the determi-
nation of a full set of LMO’s, the X, vectors may be collected together to form a full linear (but not orthogonal) transformation X of E. This is further discussed in section 4. As regards computational aspects, let us assume that E is a tz X m matrix, i.e. there are n AO’s in the basis set and m occupie-d MO’s. Eq. (2.6) then requires the diagonalisation of the nr X m matrix Rf R, which is usually considerably smaller than the n X n energy matrix which had to be diagonalised in the original MO calculation. If there are also 112differently localised LMO’s, as in a saturated hydrocarbon, there are actually m different such eigenvalue problems, but often on!y a fraction of these need to be solved explicitly because of molecular symmetry properties. Even if all are solved, only the single lowest eigenvector need to be found in each case. It may therefore be expected that the required computational effort for calculating LMO’s will be much smaller than that for the MO calculation. As remarked earlier, this is borne out by practical experience on semi-empirical calculations, and the ratio would be even more favourable for ab initio calculations where the diagonalisation effort becomes relatively less important. In the introduction we have mentioned the possibility of a progressive orthogonalisation scheme. This is already inherent in our formalism and may be implemented by the following modification of our procedure. Instead of finding only the lowest solution of eq. (2.6), one calculates the complete eigenvector matrix X and substitutes this into eq. (2.2). The columns of the resulting matrix L which correspond to ho are the LMO’s of the given bond, while the remainder are orthogonalised to them by eq. (2.8). The same procedure repeated on the remainder matrix for the next bond will then generate LMO’s for it which is orthogonal to (but less local than) all previous LMO’s.
3. Examples of local and non-local orbitals In this section we wish to demonstrate the interpretative use of localised orbitals, by showing how the conventional description of some well-known molecules in terms of localised or partly localised bonds between specific atoms can be ‘obtained, using only the MO matrix and without relying on preconceived ideas about valency, hybridisation, etc.
155
ItiS. Verwoerd/Population localized orbitafs
Table 1 Localised bonds. Orbital coefficients of local orbitals projected from the molecular orbit& of a MIND013 calculation. Orbitals are allocated in the order s, px. p,, and pr on each atom, and atoms are taken in the order occurring in the chemical formula Molecule: Bond: Local orbitals:
NH3 NH l-5
atomic orbital no: 1 2 3 4 5 6 7 8 9 10
-0.731 -0.282 0.361 -0.506 0.0 0.0 0.0 -
residue
0.000
C2H2
NH 1-S ---
HH 5-6
cc l-8
cc l-8
-0.306 0.657 -0.036 0.051 -0.686 0.0 0.0
0.000 -0.564 -0.439 0.000 0.494 -0.495 0.000
0.610 0.358 0.000 0.000 0.610 -0.359 0.000
-
0.000
-
-
0.000 0.000
0.000 0.000 -0.162 -0.688 0.000 0.000 -0.162 -0.688 0.000 0.000
0.000 0.000 0.688 -0.162 0.000 0.000 0.688 -0.162 0.000 0.000
0.000
0.000
0.000
0.000
0.511
The first point to be established is that there is a one-to-one correspondence between conventional bonds and LMO’s, e.g. that no localised orbital can be constructed between atoms that are not classically bonded. It has already been shown in previously mentioned work that “intrinsic” locelisation methods such as those of refs. [I ,2] yield such a bond structure. However,.population localisation also requires an “extrinsic” input in the sense that a set of local AO’s has to be selected a priori. Since a localised peak anywhere in space may be built up from a complete set of delocalised functions, it is not immediately obvious that the present method can make a clear-cut distinction between bonded and non-bonded atoms. Secondly we consider the delocahsation of bonds. A standard text book approach to the idea of localised bonds is to start with the construction of atomic hybrid orbitals appropriate to the observed molecular geometry, and to combine them into two-center bonding and antibonding molecular orbitals. While such an approach is successful for the description of paraffins, it becomes ambiguous for conjugated molecules like benzene and fails to explain the observed tendency for bonds in a molecule like butadiene to be much more localised than in benzene [13-l 51. Such difficulties have led authors like Dewar [13,14] to adopt the attitude that localised orbitals constitute only an analog model of physicat reality, the applicability of which must be decided empirically.
--
cc l-8
In the present study we approach the matter from the opposite side, starting with a set of molecular orbit& and projecting out localised orbitals from which the atomic hybrids can finally be obtained. Our examples below show that the above mentioned problems do not occur here, and that the conclusions about the localisation or otherwise of bonds reached by direct calculation are in accordance with the empirical indications. Thus a more positive view of the physical significance of LMO’s than that expressed above seems justified. The sets of molecular orbitals used in the exampies are all taken from MIND0/3 [8] calculations of the appropriate molecules. 3.1. Ammonia (AWJ) dur first example is the ammonia molecule. To study the NH bond, the local set is defined as the nitrogen 2s, 2p,, 2pr and 2p, orbitals and the hydrogen Is. Our procedure projects out two occupied orbitals, given by the first two columns of table 1. From the given non-local residue density it is seen that both are completely localised within the tolerance shown, and inspection shows that the first is a non-bonded lone pair orbital while the second is a bonding orbital with some electron excess on the N atom. Consideration of the best attempt to locahse an or-
W.S. VenvoerdfPoplrlatbrl localised orbitals
156 Table 2 Non-localised
bonds. Orbital coefficients ---.
set out as in table 1; for benzene only the carbon orbit&
Molecule: Bond: Local orbit&
BA BH l-4,9
BHB 1-9
CbH6 cc l-8
atomic prbiti no. 1 2 3 4 5 6 7 !
0.409 0.386 -0.054 -0.000 0.198 0.061 -0.450 0.001 0.648
0.296 0.436 0.005 O.ODO 0.295 0.081 -0.429 0.667 0.001
0.514 0.480 -0.042 0.000 0.514 -0.480 -0.042 -0.012 0.000
-0.016 -0.047 -0.047 0.077 0.080 .-
0.000 0.000 0.000 0.000 0.000
-0.038 0.003 0.000 -0.005 0.001 0.002 0.000 0.005 0.009 0.002 0.000 -0.012 0.038 0.003 0.000
0.000 0.000 0.258 0.000 0.000 0.000 -0.129 0.000 0.000 0.000 -0.129 0.000 0.000 0.000 -0.258
0.000
0.006
0.167
10 11 12 I3 14 15 16 17 18 19 20 21 22 23 24 residue
-
-
0.262
bital on two of the H atoms (shown in table 1, third column) illustrates the first point mentioned above. The lowest residue value that can be achieved is 51%, which means an approximately equal division of the electron density between localised and non-localised orbitals, i.e. no localisation at all. The contrast between localisable and non-localisable orbitals in this case is something of an extreme because of the symmetry and small size of the molecule, and more typical values would be a residue of less than 1% for an ordinary localised bond and about 40% for a non-bond. However, there can clearly be no confusion between the two cases. In principle all the localised bonds could be found without any knowledge of the atomic valencies or arrangement by testing all the possible A0 subsets for localisation. The intuitive “extrinsic!’ input mentioned above therefore only serves to reduce labour and is not basic to the procedure.
cc 1-S
0.000 0.000 0.000 0.645 0.000 0.000 O.OOG 0.645 0.000
are shown
C4H6 cc 1-8
cc 1-8
0.552 0.440 0.004 -0.085 0.527 -0.456 0.011 -0.008 -0.085
0.067 0.053 0.004 0.699 0.060 -0.058 0.004 -0.003 0.694
-0.002 0.008 -0.006 0.018 -0.010 0.003 -0.011 0.000 0.001 0.002 -0.010 -0.002 ~-0.011 0.0015
,
-0.014 0.009 0.086 -0.008 0.033 -0.025 -0.073 0.000 0.001 0.004 0.020 0.000 0.005 0.015
3.2. Acetylene (C2H2) The case of multiple bonding is illustrated by the last three columns of table I_ Specifying the eight carbon orbitals of C2H2 as the local set, yields three localised bonding orbitals of which the first is immediately recognized as the u-bond and the others are mutually orthogonal n-bonds. Once again there is a one-to-one correspondence between classical bonding language and localised orbitals. While the information that there is a triple bond can of course also be extracted from the bond order matrix, LMO’s seem to be a more vivid and direct way to obtain it.
3.3. Diborane (BZHJ Next we turn to a classical example of a non-local-
W.S. Ven,,oerdlPopulatioll
ised bond. If we assume ignorance of the 3centre bond in diborane and ask for a localised bond between a boron and one of the bridge hydrogens, the resulting orbital is shown in the first column of table 2. The non-local residue of 26% is by far too large to qualify as a local bond, and yet it is sufficiently small to suggest that a partial localisation has been attained. Inspection of the orbital coefficients shows that most of the remainder is located on the second B atom, strongly suggesting a 3-centre bond. This is confirmed by including the orbitals of all three atoms in the localised set, which yields the well-defined and symmetrical three-centre LMO in the second column of table 2. There is of course also a similar bond via the other bridging hydrogen, while the other four BH bonds are represented by ordinary well-localised LMO’s. 3.4.
Benzene (C6H6)
151
localised orbitah
If we use our program to find the LMO’s on the two end carbons, only a u-bond is projected out (table 2, column 5). However, inspection of the next best LMO (last column in the table) shows that there is also a n-bond which only just misses classification as a localised orbital according to our somewhat arbitrary 1% criterion because of a 1.3% density contribution from the other n-bond. While the total non-local residue of 1.5% is an order of magnitude larger than that of the ubond, it is also an order of magnitude smaller than the minimum that could be obtained for the s-bonds in benzene. The empirical conclusion that a localised bond description is much more valid for butadiene than for benzene, is thus well supported although a noticeable delocalisation is still present. As in all the examples, it should be remembered that this conclusion is still dependent on the validity of the MIND013 approximation. However, we believe
that they amply demonstrate the usefulness of LMO’s A calculation of all the localised bonds in benzene yields only six CH bonds and six CC u-bonds, one of which is shown in table 2 (third column). Since there are 15 occupied MO’s in the original SCF calculation, three are still missing, and the previous example suggests that they may be traced by inspecting the unsuccessful localisation attempts. In column 4 of the table we show the next best attempt at CC bond localisation, and clearly this has a a-bond character and is independent of the CC o-bonds. Furthermore, it has contributions from all six carbon atoms, so that in order to find the total number of such LMO’s we follow the same procedure as before and extend the local set to include all the carbon AO’s. This duly delivers nine LMO’s, namely three s-bond orbitals and six with obond character. Since the results given in the table show that id contrast to the u-bonds the n-bonds cannot be localised on two atoms at a time, we are thus led (without external information) to the standard view of three delocalised n-bonds supplementing the localised u-bond framework_ 3.5. 1.3 Butadiene (C4H6) In the butadiene molecule, the standard representation has two double bonds on either side of a CC single bond, and the intuitive approach to localisation would lead one to expect a similar interaction and thus delocalisation as in benzene. On the other hand, empirical evidence argues against deiocalisatidn [ 151.
to interpret molecular orbital calculations.
4. Nonorthogonality
and atomic occupations
From the examples in the previous section it is seen that an LMO Vi>representing a bond between atoms a and b takes the typical form llij = Cailha j) + Cbilhbi) + lnj),
(4-I)
where Iha,) and lhbi) are normalised hybrid orbitals on a and b, and [ni> is the total non-local residue. Let . us consider the ideal case of perfect localisation, i.e. Ini> = 0 for all i, which is exact for some symmetrical molecules like CH4 and approximated within a fraction of 1% by many others. It is then clear that overlap between different LMO’s can only occur when they share a common atom, in which case Clil’jY= CarCai (h,ilh,i)-
(4.2)
It is seen that nonorthogonality of the localised orbitals implies that the hybrid orbitals of a given at0.m are not orthogonal and vice versa. The standard treatment of hybridisation usually assumes that hybrid orbitals are orthogonal_ If we consider a tetragonal bonding situation, for example, the bond directions determine the mutual relationship of the three parbital amplitudes and orthogonality of the hybrids can only be brought about by a specific
WS. l’cr,voerdfPopulatio)~ localised orbitals
158
choice of the relation between s- and p-amplitudes. In this way’; ihe “classical” hybridisation of 25% s-content for tetragonal sp3 hybrids or similarly for trigonal and digonal hybrids is obtained. However, as pointed out for example by Dewar [ 141, this classical choice also implies an equal occupation of the atomic s and p states, which is unreasonable in view of the difference in their energies. It would therefore seem that in order to accommodate such differences in occupation numbers, we have to allow the hybrid orbitals and thus also the LMO’s, to be nonarthogonal. This is supported by the observation that the LMO’s obtained by our method from MO calculations which do have different s- and p-level occupation numbers, are in fact not orthogonal. However, once the possibility of overlap between hybrids is allowed, it is not clear any more how the bond order matrix and occupation numbers are to be calculated. For that matter, one may doubt that these quan!ities or even the normalisation of such hybrids are uniquely defined. Our purpose in this section is to clear these matters up and show that there is a well-defined relation between the orbital overlap and atomic occupation numbers, and that the latter may be quite practically calculated from non-orthogonai LMO’s. Let us consider specifically the occupation nk of atomic orbital k; a similar argument applies to offdiagonal bond order elements. For the original MO calculation, it is given by “k = 2 C’E~~E~~ = 2(EEtjkk, P
(4.3)
where E is then X m matrix of occupied MO’s, and n is the number of atomic orbitals, m the number of occupied MO’s. By collecting together al1 the localised orbitals L, as its column vectors, we now construct an X m local orbital matrix L which is related to E by the following generalisiition of eq. (2.2) L=EX,
(4.4)
and X is the m X m matrix constructed by collecting the corresponding coefficient vectors X, together. If the set of vectors Lhcorrespond to the individual classical bonds, they may in genera1 be assumed to be linearly independent so that the inverse of X exists and we may solve eq. (4.4) to obtain
EEf
Lt =
= Lx-lx-l’
L(XtX)-1
Lt.
(4.5)
It is easily seen that eq. (4.4) implies that eq. (2.4) holds not only for.the vectors but also for the matrices L and X, so that EEt =L(L’L)-lLf.
(4.6)
Note that while L is non-square in general and has no inverse, the existence of (L’L)-1 is assured by that of x-1. The matrix Lt L defines the overlap between the various LMO’s (not to be confused with the standard A0 overlap matrix S) and has the elements (Lf L)ii = ~ii = Li’Lj = ~!,il,,
(4.7)
so that it is a symmetrical matrix with diagonal elements ail equal to one, and in the case of orthogonal orbitals it reduces to the unit matrix. If we denote the elements of 0-l by wV and of L by $, we find from eqs. (4.3) and (4.6) that “k = 2 Cl~~W, t 2 C C IkjlkiWii _ Ci I i
(4.8)
At this point it is instructive to consider the specific example of theCH, molecule. The atomic orbital basis set consists in this case of the s, px, p,, and pr orbitals on the carbon atom, and the four s-orbit& of the hydrogen atoms. There are four LMO’s which, from symmetry requirements, yield the L matrix defined by
is Lt=
s S
s
-p
-p
-p
u
0 0
-P P P
P -P p
p0 po -p 0
a00 000’ 0 0
0) (4-9) a
This gives an overlap matrix with all the off-diagonal elements equal to a common value .Q=sz -p2
(4.10)
(which, because of Iocalisation, depends only on the AO’s on the common atom) and havinga determinant D equal to D=lnl=1-6Qk3@-3Q4=(1
-Q)3(1+3G). (4.11)
The inversion of fi is straightforward and we find that
159
N.S. Ver~c~oer~/PopllZation Iocaliredorbitak
wj/ =D-1
(1 - 3sP + 2s23)
=
(1 -#(l
+ 2fi), for i=j;
= D-1
(-a
(4.12)
- 2~22 -I-~3)
= -52(1
-
~92,
for i + j. From eq. (4.8) we calculate that, in an obvious notation, 89 “s=~+~
8p2 %=r-_¶
nH=(l
2(1 + 2S2)fJ2 -9)(1+3Q)’ (4.13)
Usually the nett occupation of the various basis orbitals is known from the original LCAO calculation. Using eq. (4.10), we may then solve for SI to find that !Tl= (n, - Q/(8
- 3n, - np).
(4.14)
Note that the overlap of the localised orbitals, and hence via eq. (4.13) also the LMO’s themselves, are uniquely determined by the occupations. Using the’se values, the entire procedure for projecting out the LMO’s that was described in the previous section may be side-stepped in this simple symmetric case by the use of eqs. (4.13) and (4.14), and then quantities like the atomic hybridisation and the relative distribution of electrons between the atoms participating in the bond, may be calculated.
Eq. (4.14) also shows explicitly that a difference in s- and p-orbital dccupations necessarily gives rise to non-orthogonal localised orbitals.
Finally, we remark that eq. (4.13) has the form n = n’/P -f(Q)1 ,
(4.15)
where n’ is the occupation obtained when overlap is ignored. The “renormalisation” factor differs from unity by some functionf(Q) which has a linear behaviour for G < 1. It is easily seen from eq. (4.14) that a is in fact quite small, n = 0.1 being a typical value for carbon and silicon, and values in excess of about 0.25 are only attainable at extreme occupation differences. Let us now return to the case of a general molecule. We focus attention on a specific atom, A, and wish to derive the connection between the occupation nk of a given atomic orbital Jk) centered on A, and the amplitude Zkiwith which it contributes to the local orbitals [similar to eq. (4.13)] _ Jf A contributes na orbitals to the basis set, we may
arrange them in such a way that the first n, rows of L correspond to the atomic orbitals of A, and the first nb columns to the local orbitals in which A participates. In the usual case that all LMO’s represent twocentre bonds, we have n, = nb. Henceforth, we neglect the non-local contributions @‘of section l), so that in its first n, rows L will contain non-zero elements only in the first nb columns. The sums in eq. (4.8) may then be restricted to the range i, j Gn,,, and we now show that a similar restriction may also be enforced approximately when c&ulating the Q.. Firstly, all local orbitals which do not overlap with those of A may be omitted without approximation, because such orbitals only give rise to separate blocks on the diagonal of n and which may be factored out. Apart from A’s “own” LMO’s, we then remain only with those associated with the atoms to which A is bonded. In order to assess the approximation involved when the latter group of LMO’s is disregarded as well, we consider the following expressions for the Oij (accurate to second order in the aji) found by direct expansion: m (4.16)
Oii
z D-l
1) h,lfi, 1
(4.17)
, l+i,j,i+j.(4.18)
It is seen that if the summations are restricted to h, I< nb, we are only disregarding terms of second and higher order in the overlap, which is expected to be small in view of the results for CH,. In fact, most of the neglected terms are zero in any case because LMO’s can only overlap when they share a common atom. Lastly, when eqs. (4.16)-(4.18) are substituted in (4.8), we see that it is a similar set of terms which is neglected in the numerator and the denominator of no., reducing the effect of the approximation even further. The degree of simplification brought about depends on the bonding situation of atom A. If it is a carbon atom with four bonds in tetragonal directions, for example, we are left with a 4 X 4 matrix with off-diago-
ICS. I’er~oerdlPopuIa[ioll localised orbirak
160
the ttk using these Slii values in eq. (4.8), and also tak-
nal elements ~~ = SiSi
-
pipi.
(4.19)
where we have made use of the fact that since the bond directions are specified by the relative contributions of the p_,, p,, and pz orbital!, the same interrelationship fhat holds in the row vectors of eq. (4.9), is valid here. When A is a carbon atom participating in a multiple bond, the orthogonal n-bonding orbitals may be factored out and a similar 3 X 3 or 2 X 2 matrix remains. Even for these small matrices eq. (4.8) is still too complicated to supply a direct link between the local amplitudes and the occupation numbers. We therefore proceed by considering an iterative solution of eq. (4.8) in which the wii in a given step is obtained from the liri of the previous iteration. For the first iteration, we assume all the Rii to be equal to the common value !TZthey assume for the ful-
ly synmetric case, so that wii reduces to eq. (4.12) for the 4 X 4 case, and may be factored out of the summations in eq. (4.8). A useful relation between the ~ii may be obtained by summing the resulting equation over the four values of k belonging to atom A. Since by construction the only overlap of the first four columns occurs in the fust four rows, we have
(4.20) The assumption of a common Sz value in the zeroth iteration amounts to taking the s2 and p* values obtained from eqs. (4.13) and (4.14) as estimates of the average values of sf and pf, i.e.
ing into account (4.16) to (4.18), it is seen that changes from the first iteration values only occur in correction terms of order R or smaller, all of which involve &is over the !$ or products of them so that fluctuations will be expected to average out partially, and finally changes in the numerator and denominator will again tend to compensate_ We may therefore expect the first iteration to be already a good approximation. The same philosophy may also be applied to the 3 X 3 and 2 X 2 cases. Assuming bonds which are equally spaced in a plane, matrices analogous to eq. (49) may be written down for CH, and CH, which gives rhe following common overlap values in terms of sand parbital amplitudes: Q;2,= sz -Gp”,
respectively. Inversion of the reSulting overlap matrix yields (for N = 2,3): w!!)= II
[I t(N= +/U
l)a2N]/(1 --cN)(l
Using eq. (4.13) to eliminate 12,and nr, from the left hand side, the indicated summation of eq. (4.8) then yields (4.22) This implies that the average value of Rii stays unchanged although the individual values may fluctuate about it. If we now calculate approximate values for
+NQ),
- fiN)(l +NQ)>
i=j; i+j_
(4.24) Substituting this back into eq. (4.8), expressions for ns and n,, may be obtained in each case which is combined with eq. (4.23) to find that a/,, = (12s- np)/[2N - (N - l)rz, - np] .
(4.25)
Finally, the arguments for generalising this to an arbitrary molecule are similar to those for the 4 X 4 case. In applying eq. (4.8) to the orbitals on a given atom, it is assumed that the Wii may still be calculated from eqs. (4.24) and (4.25) so that it takes the final form N
N
(4.21)
(4.23)
!A2 =s2 -p*,
2[1+(N-2)aN] =(I -i-+q)[l
5?-4n~r_.l 1=1 h-7 +(N-
_ i< i hZQ
(4.26)
l)ii?~]Q.
Eqs. (4.24), (4.25) and (4.26) hold for all three cases, namely for the number of equally spaced bonds N equal to 2,3 and 4. In order to evaluate the validity of these equations (and of the approximations leading up to them) we have compared the value of nk calculated from eq. (4.26) with that holding for the original MO calculation, for different atoms in a number of molecules.
IV.S. 1~~erwoerdjPopulatiorl localised orbitals
Firstly, a set of hydrocarbon molecules (saturated and unsaturated) of increasingsize (ranging from CH4 to C,,H,z) was used in order to test the effect of neglecting overlap with neighbouring bonds. Secondly, carbon atoms in these, as well as in some small molecules containing C1and F substituents, were chosen to maximize the deviations from equal lzj values which holds in the symmetric case. The agreement found is usually within l%, or 3% at worst. Since the deviations from non-locality are also of this order of magnitude, it seems pointless to carry the above mentioned iteration any further. In fact, we find that with very little additional loss of accuracy we may make the further approximation that, for the 4 X 4 case, q
sisj=;Es,?, cc pipj~-+&l~. i
’
(4.27)
which becomes exact in the symmetric case, as seen from eq. (4.9). Making similar approximations for the 2 X 2 and 3 X 3 cases, it is found that eq. (4.26) reduces to the simple form 2 cs;
= [I +(N-
l)~]“s’
(4.28)
(4.29) These equations represent renormalised sum rules like eq. (4.1 S), but now generalise_’ to arbitrary molecules. A detailed application of the sum rules to interpret hybridisation trends in hydrocarbon molecules, will be discussed elsewhere [12] _In particular, we note the result found there that while eqs. (4.26), (4.28) and (4.29) are numerically quite similar, the presence of the cross-terms between different LMO’s in eq. (4.26) is important to explain some of the more subtle differences between related MO calculations. It has been noted by Peters [6] that non-orthogonality of hybrid orbitals may seriously disturb some of the generally accepted notions about molecular geometry. However, the presence of sum rules like equations (4.28) and (4.29) restores the basis of at least those qualitative arguments which are founded only on the requirement that the sum of squared amplitudes must add up to a constant total. As an example, we consider
an often used argument that connects
16:
the s-orbital con-
tent of hybrid orbitals to the bond directions 1161. The argument is usually based on the observation that the s-orbital content of classical trigonal sp’ orbitals, for example, is higher than for tetragonal sp3 orbit-
als. As previously pointed out, however, the classical hybridisation is a special case where orbitals have beer1 constructed to be orthogonal without taking actual occupation numbers into account. The direct applicability of this argument to the general case is therefore questionable, but eqs. (4.28) and (4.29) supply the necessary generalisation. Provided that the right hand side stays unchanged, the transition from tetragonal to trigonal bonds may be understood by noting that as one orbital becomes purely p-like to participate in a rrbond, the s-content of the other orbitals must increase to maintain the sum rule of eq. (4.28). Actual calculations show that II, is in fact virtually the same (e.g. in C,H, and C2H4) while the decrease in N from 4 to 3 is largely compensated for by an increase in ,r2, implicit in eq. (4.25). In this case it is thus t:ue that changes in the bond directions are accompanied by a change in the hybridisation. However, since the s-orbital is spherically symmetric
it can clearly not influence the bond directions on its own. The above mentioned connection therefore does not exist in cases where the sf values change because of a change in ~zs,such as may be brought about by changes in the electronegativity of neighbouring atoms. Finally, we note that the sum rules may explain the observed tendency in both our own and in other calculations [5] of atomic hybrids to deviate slightly from internuclear directions. The necessity to maintain balanced occupations of the different atomic p-states clearly places limitations on the hybrid directions. Such an interpretation is supported by our finding that when the MO calculation is repeated with surrounding atoms displaced slightly from their optimised positions, the hybrids on the central atom stay largely unchanged with only a very slight following tendency discernible. Thus the energy gained by increased overlap with
neighbouring atoms when the hybrid direction is adjusted, is outweighed by the loss due to internal readjustments. It seems plausible that it may similarly be advantageous to allow slight deviations of hybrid directions in the equilibrium state of the molecule if this dan lead to a better balance in the electron distribution of a given atom.
162
KY_ Vemoerd/Popzrlation localised orbitals
References [I] C. Edmiston and K. Reudenberg, Rev. Mod. Phys. 35 (1963) 457; J. Chem. Phys. 43 (1965) 597. [Z] S.F. Boys, Rev. Mod. Phys. 32 (1960) 296. [3] J.A. Hunter, QCPE 10 (1978) 355. [4] D. Boerth, J.A. Hashmall and A. Streitweiser, QCPE 10 (1978) 354. [S] C. Trindle and 0. Sinanosu, J. Chem. Phys. 49 (1968) 65. [6] D. Peters, 3. Chem. Sot. (1963) 2003, and subsequent papers. [7] V. Magnasco and A. Perico, J. Chem. Phys. 47 (1967) 971. [Sj M.J.S. Dewar. H. Metiu, P. Student, A. Brown, R.C. Bimgham, D.H. Lo, C.A. Ramsden, H. Kollmar, P. Weiner and P.K. BiscoF, QCPE 10 (1976) 309.
[P] Quantum Chemistry Program Exchange, Indiana University, Bloomington, Indiana 47401, USA. [lo] T.L. Gilbert, in: Molecular orbitals in chemistry, physics and biology, eds. P.-Q. Lijwdin and B. PuUman (Academic Press, New York, 1964) p_ 405. [ 111 W.H. Adams, J. Chem. Phys. 42 (1965) 4030. [ 121 W.S. Verwoerd, to be published. [ 131 M.J.S. Dewar and R.C. Dougherty, The PM0 theory of organic chemistry (Plenum Press, New York, 1975). [ 141 M.J.S. Dewar, The molecular orbital theory of organic chemistry (McGraw-Hill, New York, 1969). [15] M.J.S. Dewar, Hyperconjugation (Ronald Press, New York, 1962). [16] W. Month, in: Advances in solid state physics, ed. H J. Queisser (Pergamon Press, Oxford, 1973) p. 247; and many other references.
THEUSEOFPOPULATIONLOCALISEDORBITALSTOINTERPRET MOLECULARORBITALCALCULATIONS W.S. VERWOERD Department of Physics, University of South Africa, Pretoria 0001, Republic of South Africa
Received 23 May 1979; in timal form 10 August 1979
An efficient and general method is derived to calculate population localised mokcular orbit& (LMO’s) from agiven SCP eigenvector matrix, by reduction to an eigenvalue problem. Applications to both localised molecules (NH3 and CzH2) and deiocalised ones (BaHe, CeHe and butadiene) arc discussed in some detail. It is shown that unequal occupation of atomic cnergy levels leads to nonorthogonal LMO’s.The consequences cf nonorthogonal atomic hybrid orbitals are discussed, formulas for their overlap in terms of atomic occupation numbers are derived and it is shown that the occupation numbers are connected to LMO atomic orbital coefficients by various sum rules.
I. Introduction The use of localised molecular orbitals (LMO's) in molecular theory has received considerable attention in the past. The central idea is to use the freedom allowed by the determinant form of the many-electron wavefunction, to make linear combinations of the HartreeFock single electron molecular orbit& leading to a greater degree of spatial localisation. In other words, one has to find a linear transformation from the original set of occupied molecular orbitals (MO’s) which are expressed as a matrix of coefficients in an atomic orbital (AO) basis, to a new matrix defining the LMO’s. While a variety of criteria for the localisatiou and of methods for calculating LMO’s have been proposed, there seems to be general agreement that there are four main uses of localised orbitals: (1) To facilitate the interpretation of MO calculations in terms of bonds and other chemical concepts. (2) To find orbitals which may be transferred from one molecule to another containing a similar local atomic arrangement. (3) To simplify the treatment of correlation by findlug a description which minimises interorbital correlation effects. (4) To calculate self-consistent wavefunctions in large systems.
The most generally used localisation methods appear to be those of Edmiston and Reudenberg [I ] and Boys [2] and computer programs for their implementation are now generally available [3,4] . The particular method selected, however, depends to some extent on the purpose of calculating LMO’s. In the present context we wish to focus attention on the first of the above mentioned uses, and in contrast to calculations emphasizing the other points one would prefer a quick and simple method which is as independent of the particular MO procedure as possible, and which may be applied afterwards to various different completed MO Cal&ations. From this point of view, “energy” localisation methods like those mentioned above suffer from the drawback that they require the calculation of a large number of two-electron interaction integrals. This is cumbersome and may only become tractable for larger molecules if a simplifying assumption like the neglect of overlap, made in the MO calculation, is also applied to the localisation [.5], but which couples the two stages of the calculation together. For this reason, locallsation methods based on“population localisation” methods like those of Peters [6] * + See also ref. [S] _
152
MS. ~renr~oerdlPoplrlatiorllocalised orbitals
and Magnasco and Perico [7J which do not employ two-electron integrals are more suitable for our purpose. In these, subsets of atomic orbitais in the basis set which are centred on a specific atom or pair of atoms are defined as local sets, and to construct the LMO’s one chooses linear combinations of MO’s which maximize the contribution of the local sets. As a general purpose method appIicabIe to all moIecules the approach of Peters [6] is hampered by the rather ad hoc way in which the transformation matrix has to be determined from the symmetry properties of the particular system_ On the other hand, while t!re work of Magnasco and Perico [7] is based on a systematic generally applicable procedure, a comparison by Trindle and Sinanoghi [5] of the methods of refs. [I ,73 shows that despite its simp!er structure the latter method is not suitable for larger molecules due to the problems with the convergence of its iteration cycle. In this article we therefore present a new mathematical approach to the same physical idea of population localisation. The formalism that we use is described in section 2. The method is completely general, it needs only the MO coefficient matrix so that it may be directly applied to both ab initio and semi-empirical calculations, and we have applied it to molecules ranging from diatomic to a 35atom molecule with a 74 dimensional orbital space without experiencing any convergence problems. A FORTRAN program implementing the procedure has been deposited with the QCPE [9]. The calculation time required is only a fraction of the time requited by a self-consistent field MO calculation (for example, 5% per LMO of the time for a single SCF loop of the relatively fast MIND0/3 semi-empirical program [8] on the 35atom molecule) so that it should prove useful as a standard aid for the interpretation of MO calculations. Another objection against population localisation raised by Trindle and SinanoBu [5], is that it leads to larger deviations of atomic hybrid directions from the internuclear vector than energy locaiisation: an average of 9” against 3” for CNDO calculations on some small molecules. To the extent that MIND0/3 calculations on the same molecules are comparable, the value of 4O resulting from our version of the method thus seems quite acceptable. The reason for such deviations is discussed further in the last section. One feature of our procedure which should be pointed out at the outset, is that the LMO’s correspond-
ing to bonds at different locations are in genera1 not orthogonal. In this respect it corresponds to the localisation methods of Gilbert [IO) and of Adams [ll] , both of which utilise modifications to the molecular hamiltonian, and the possibility of non-orthogonality was also mentioned by Peters [6] _Most of the other studies (and notably those of refs. [ 1,7]) enforce an orthogonality requirement on their LMO’s. However, it was pointed out by Adams [ 111 that this may limit the degree of localisation attained and reduce the transferability of the resulting LMO’s; and a decrease in localisation will obviously also impair the usefulness of LMO’s for other purposes. One of the results to be derived in this article is to show that the overlap between LMO’s is ciosely connected to the difference in occupation number of AO’s forming hybrid orbitals, and that for such hybridised atoms the requirements of orthogonality and localisation are therefore incompatible. The use of non-orthogonal LMO’s is thus often unavoidable, but because the required overlap is well-defined there is no arbitrariness in the actual LMO’s obtained, supporting an empirical observation of Peters [6] . If one only wishes to study a given bond in a molecule, it is possible to sidestep the orthogonality issue by only projecting out the one or more LMO’s representing this bond, while the rest of the MO space may in principle be orthogonalised to these without full iocalisation. One is then dissecting the molecular electron cloud into a “local” part connected with the particular bond, and a “non-local” remainder. Further LMO’s?nay be extracted from the remainder, but will be less than maximally localised due to the shrinking dimension of the orbital space. Presumably this may be useful to generate fully orthogonal LMO’s for a small number of bonds in a large molecule for the purpose of transferability studies. A proposal for incorporating such a progressive orthogonalisation is outlined in section 2, but we have not investigated the matter any further. In section 3 we demonstrate that our method, using the single-bond-at-a-time approach, may be used to answer in a direct way qualitative questions about the existence and multiplicity or otherwise of localised bonds in molecules. Applications are shown to both traditionally locabsed systems like saturated hydrocarbons, and less localised ones like B,H, and benzene. More quantitative applications of our LMO’s to study
W.S. lfer~r~er~J?op~~latio~i
the relation between bond length and hybridisation, are reported in a subsequent paper [ 121. Section 4 is devoted to a more detailed study of the non-orthogonality question. In particular, we show that the overlaF between LMO’s is a well defined phenomenon, and that the connection between LMO orbital coefficients and A0 occupation numbers is given by an equation which can be well represented by simple renormalised sum rules. These supply the basis for the applicability of some generally used qualitative arguments on molecular geometry which are usually based on an assumption of ortllogonality, an example of which is given.
153
localised orhitals
has been found, it is used to construct the vector L=EX.
(2.2)
Since N is identical to the non-local components of L, it is clear that to make L as “local” as possible, the appropriate criterion for an optima! approximate solution to eq. (2.1) is that it slrould minimise Nthr, the square magnitude of N. It is easily seen that this condition is equivalent to the “normal” equation (2.3)
R’RX=O:
Unfortunately, the set of solutions of this equation is identical to that of eq. (1) (as may be slrown using an eigenvalue expansion
2. Transformation to local orbitals We assume as given an eigenvector matrix E, of whicll the column vectors are the LCAO coefficients of the occupied molecular orbitals. The problem to be solved is to find a linear combination of these column vectors which is as “local” as possibie. We define the atomic orbitals belonging to the atoms which participate in a given bond, as its local orbitals and a!1 the others as non-local. Our approach is then to describe the bond by a LMO which contains as little as possible non-local contributions to its probability density. Note that in accordance with the philosophy outlined in section 1, we treat only one (possibly multiple) bond at a time instead of finding a complete linear transformation of E. Translated into matrix algebra, to study a given bond we first convert E to a reduced matrix R by deleting all the rows corresponding to local AO’s. The columns of R are to be combined to form a sum equal to zero, and if we form a column vector X from the necessary linear combination coefficients this may be expressed as the equation N=RX=O.
(2-l)
If R is a square singular matrix, eq. (2.1) has a solution which may be found by conversion to a non-homogeneous equation and subsequent matrix inversion, or an equivalent procedure. In the present connection, however, neither the squareness of R nor the existence of exact solutions is guaranteed. To find the appropriate generalisation, we note that once some approximate solution of eq. (2.1)
of R).This
means that, apart
from exact solutions of eq. (2.1), NtN is minimised only by the trivia! solution X = 0, and we have to exclude this possibility explicitly to find other solutions. To do this, we note that because E has orthogonal columns, L’L = Xf.E*EX=
X’X
(2.4)
so that if L is to be normalised, the minimisation must be carried out subject to the constraint
XfX= 1.
(2.5)
Using X as a Lagrange multiplier it means that (NtN - hXtX) must be minimised, leading to the equation RtRX=
AX.
(2.6)
From eqs. (2.4) and (2.5) it is found that N’N=XtRfRX=h,
(2.7)
showing that the eigenvalues h of R t R may be interpreted as the minima! values of NtN, i.e. the residual non-local content of L. Whatever the dimensions of R, the matrix RtR is square and has positive definite eigenvalues so that the problem reduces to finding its lowest eigenvalue h, and corresponding eigenvectors. If this is A = 0, eq. (2.6) reduces to eqs. (2.3) and (2.1) and Xis an exact solution. Otherwise, eq. (2.6) is the desired generahsation and X, immediately specifies the degree of localisation that can be attained. The case that 1, is degenerate corresponds physically to multiple bonds or to non-bonded orbitals localised on one of the participating atoms. It may also happen that other eigenvalues are so close to ho that they are not significantly worse solutions. To decide this issue,
154
WS. VenvoerdjPopulation localised orbitals
it is useful to note that the normalisation of the columns of E implies that the matrix elements of R t R are of order of magnitude unity. Its eigenvalues are therefore also typically of this order so that it is reasonable to treat eigenvalues within a range of +O.Ol or so as degenerate. In practice it is our experience that the non-local content of classical bonds is usually not more than 1% while the non-local content between neighbouring atoms where no classical bond exists, is typically about 40%. Tne criterion above seems adequate in the light of these numbers, but a comparison of the total number of electrons and the number of localised orbitals found can always be used to trace any LMO not directly found by our procedure. The diagonalisation of RtR yields multiple solutions as a series of orthonormal vectors Xi. When substituted in eq. (2), we find that the corresponding LMO’s are also orthogonal. because of the ortbogonality of E: LtL.=X!EtEX , . =XfX =S.. (2.8) I ij I]’ i I I Although they therefore certainly describe different bonds, it may be that a further unscrambling of these I.&IO’sinto u-bonds, r-r-bonds and non-bonded orbitals is desired. We treat this sub-problem in a similar way as the original localisation problem, but working now in the subspaceof the local bond set of L;s and concentrating on their local atomic orbital components. First any non-bonded orbitals are found by looking for orbitals localised on one of the participating atoms only. The rest of the L{s are orthogonalised to any of these found (which may be done easily by using the other eigenvectors of the sub-problem), and the subspace contracted once more to these truly “bonded” LMO’s. Finally we find the n-bonds by calculating orbitals in this space which have minimal s-orbital content, and a final orthogonahsation is carried out. If there are also d-orbitals present, a further similar step may be necessary at this stage to unscramble rrand S-bonds. The procedure is then repeated for the next bond required.‘Because it has a different set of local orbitals, the. reduced matrix R will be different for each bond and a different eigenvalue problem has to be solved. Consequently neither the linear combination vectors X, nor the LMO’s L, of different bonds will be orthogonal in general. If the process is carried to the determi-
nation of a full set of LMO’s, the X, vectors may be collected together to form a full linear (but not orthogonal) transformation X of E. This is further discussed in section 4. As regards computational aspects, let us assume that E is a tz X m matrix, i.e. there are n AO’s in the basis set and m occupie-d MO’s. Eq. (2.6) then requires the diagonalisation of the nr X m matrix Rf R, which is usually considerably smaller than the n X n energy matrix which had to be diagonalised in the original MO calculation. If there are also 112differently localised LMO’s, as in a saturated hydrocarbon, there are actually m different such eigenvalue problems, but often on!y a fraction of these need to be solved explicitly because of molecular symmetry properties. Even if all are solved, only the single lowest eigenvector need to be found in each case. It may therefore be expected that the required computational effort for calculating LMO’s will be much smaller than that for the MO calculation. As remarked earlier, this is borne out by practical experience on semi-empirical calculations, and the ratio would be even more favourable for ab initio calculations where the diagonalisation effort becomes relatively less important. In the introduction we have mentioned the possibility of a progressive orthogonalisation scheme. This is already inherent in our formalism and may be implemented by the following modification of our procedure. Instead of finding only the lowest solution of eq. (2.6), one calculates the complete eigenvector matrix X and substitutes this into eq. (2.2). The columns of the resulting matrix L which correspond to ho are the LMO’s of the given bond, while the remainder are orthogonalised to them by eq. (2.8). The same procedure repeated on the remainder matrix for the next bond will then generate LMO’s for it which is orthogonal to (but less local than) all previous LMO’s.
3. Examples of local and non-local orbitals In this section we wish to demonstrate the interpretative use of localised orbitals, by showing how the conventional description of some well-known molecules in terms of localised or partly localised bonds between specific atoms can be ‘obtained, using only the MO matrix and without relying on preconceived ideas about valency, hybridisation, etc.
155
ItiS. Verwoerd/Population localized orbitafs
Table 1 Localised bonds. Orbital coefficients of local orbitals projected from the molecular orbit& of a MIND013 calculation. Orbitals are allocated in the order s, px. p,, and pr on each atom, and atoms are taken in the order occurring in the chemical formula Molecule: Bond: Local orbitals:
NH3 NH l-5
atomic orbital no: 1 2 3 4 5 6 7 8 9 10
-0.731 -0.282 0.361 -0.506 0.0 0.0 0.0 -
residue
0.000
C2H2
NH 1-S ---
HH 5-6
cc l-8
cc l-8
-0.306 0.657 -0.036 0.051 -0.686 0.0 0.0
0.000 -0.564 -0.439 0.000 0.494 -0.495 0.000
0.610 0.358 0.000 0.000 0.610 -0.359 0.000
-
0.000
-
-
0.000 0.000
0.000 0.000 -0.162 -0.688 0.000 0.000 -0.162 -0.688 0.000 0.000
0.000 0.000 0.688 -0.162 0.000 0.000 0.688 -0.162 0.000 0.000
0.000
0.000
0.000
0.000
0.511
The first point to be established is that there is a one-to-one correspondence between conventional bonds and LMO’s, e.g. that no localised orbital can be constructed between atoms that are not classically bonded. It has already been shown in previously mentioned work that “intrinsic” locelisation methods such as those of refs. [I ,2] yield such a bond structure. However,.population localisation also requires an “extrinsic” input in the sense that a set of local AO’s has to be selected a priori. Since a localised peak anywhere in space may be built up from a complete set of delocalised functions, it is not immediately obvious that the present method can make a clear-cut distinction between bonded and non-bonded atoms. Secondly we consider the delocahsation of bonds. A standard text book approach to the idea of localised bonds is to start with the construction of atomic hybrid orbitals appropriate to the observed molecular geometry, and to combine them into two-center bonding and antibonding molecular orbitals. While such an approach is successful for the description of paraffins, it becomes ambiguous for conjugated molecules like benzene and fails to explain the observed tendency for bonds in a molecule like butadiene to be much more localised than in benzene [13-l 51. Such difficulties have led authors like Dewar [13,14] to adopt the attitude that localised orbitals constitute only an analog model of physicat reality, the applicability of which must be decided empirically.
--
cc l-8
In the present study we approach the matter from the opposite side, starting with a set of molecular orbit& and projecting out localised orbitals from which the atomic hybrids can finally be obtained. Our examples below show that the above mentioned problems do not occur here, and that the conclusions about the localisation or otherwise of bonds reached by direct calculation are in accordance with the empirical indications. Thus a more positive view of the physical significance of LMO’s than that expressed above seems justified. The sets of molecular orbitals used in the exampies are all taken from MIND0/3 [8] calculations of the appropriate molecules. 3.1. Ammonia (AWJ) dur first example is the ammonia molecule. To study the NH bond, the local set is defined as the nitrogen 2s, 2p,, 2pr and 2p, orbitals and the hydrogen Is. Our procedure projects out two occupied orbitals, given by the first two columns of table 1. From the given non-local residue density it is seen that both are completely localised within the tolerance shown, and inspection shows that the first is a non-bonded lone pair orbital while the second is a bonding orbital with some electron excess on the N atom. Consideration of the best attempt to locahse an or-
W.S. VenvoerdfPoplrlatbrl localised orbitals
156 Table 2 Non-localised
bonds. Orbital coefficients ---.
set out as in table 1; for benzene only the carbon orbit&
Molecule: Bond: Local orbit&
BA BH l-4,9
BHB 1-9
CbH6 cc l-8
atomic prbiti no. 1 2 3 4 5 6 7 !
0.409 0.386 -0.054 -0.000 0.198 0.061 -0.450 0.001 0.648
0.296 0.436 0.005 O.ODO 0.295 0.081 -0.429 0.667 0.001
0.514 0.480 -0.042 0.000 0.514 -0.480 -0.042 -0.012 0.000
-0.016 -0.047 -0.047 0.077 0.080 .-
0.000 0.000 0.000 0.000 0.000
-0.038 0.003 0.000 -0.005 0.001 0.002 0.000 0.005 0.009 0.002 0.000 -0.012 0.038 0.003 0.000
0.000 0.000 0.258 0.000 0.000 0.000 -0.129 0.000 0.000 0.000 -0.129 0.000 0.000 0.000 -0.258
0.000
0.006
0.167
10 11 12 I3 14 15 16 17 18 19 20 21 22 23 24 residue
-
-
0.262
bital on two of the H atoms (shown in table 1, third column) illustrates the first point mentioned above. The lowest residue value that can be achieved is 51%, which means an approximately equal division of the electron density between localised and non-localised orbitals, i.e. no localisation at all. The contrast between localisable and non-localisable orbitals in this case is something of an extreme because of the symmetry and small size of the molecule, and more typical values would be a residue of less than 1% for an ordinary localised bond and about 40% for a non-bond. However, there can clearly be no confusion between the two cases. In principle all the localised bonds could be found without any knowledge of the atomic valencies or arrangement by testing all the possible A0 subsets for localisation. The intuitive “extrinsic!’ input mentioned above therefore only serves to reduce labour and is not basic to the procedure.
cc 1-S
0.000 0.000 0.000 0.645 0.000 0.000 O.OOG 0.645 0.000
are shown
C4H6 cc 1-8
cc 1-8
0.552 0.440 0.004 -0.085 0.527 -0.456 0.011 -0.008 -0.085
0.067 0.053 0.004 0.699 0.060 -0.058 0.004 -0.003 0.694
-0.002 0.008 -0.006 0.018 -0.010 0.003 -0.011 0.000 0.001 0.002 -0.010 -0.002 ~-0.011 0.0015
,
-0.014 0.009 0.086 -0.008 0.033 -0.025 -0.073 0.000 0.001 0.004 0.020 0.000 0.005 0.015
3.2. Acetylene (C2H2) The case of multiple bonding is illustrated by the last three columns of table I_ Specifying the eight carbon orbitals of C2H2 as the local set, yields three localised bonding orbitals of which the first is immediately recognized as the u-bond and the others are mutually orthogonal n-bonds. Once again there is a one-to-one correspondence between classical bonding language and localised orbitals. While the information that there is a triple bond can of course also be extracted from the bond order matrix, LMO’s seem to be a more vivid and direct way to obtain it.
3.3. Diborane (BZHJ Next we turn to a classical example of a non-local-
W.S. Ven,,oerdlPopulatioll
ised bond. If we assume ignorance of the 3centre bond in diborane and ask for a localised bond between a boron and one of the bridge hydrogens, the resulting orbital is shown in the first column of table 2. The non-local residue of 26% is by far too large to qualify as a local bond, and yet it is sufficiently small to suggest that a partial localisation has been attained. Inspection of the orbital coefficients shows that most of the remainder is located on the second B atom, strongly suggesting a 3-centre bond. This is confirmed by including the orbitals of all three atoms in the localised set, which yields the well-defined and symmetrical three-centre LMO in the second column of table 2. There is of course also a similar bond via the other bridging hydrogen, while the other four BH bonds are represented by ordinary well-localised LMO’s. 3.4.
Benzene (C6H6)
151
localised orbitah
If we use our program to find the LMO’s on the two end carbons, only a u-bond is projected out (table 2, column 5). However, inspection of the next best LMO (last column in the table) shows that there is also a n-bond which only just misses classification as a localised orbital according to our somewhat arbitrary 1% criterion because of a 1.3% density contribution from the other n-bond. While the total non-local residue of 1.5% is an order of magnitude larger than that of the ubond, it is also an order of magnitude smaller than the minimum that could be obtained for the s-bonds in benzene. The empirical conclusion that a localised bond description is much more valid for butadiene than for benzene, is thus well supported although a noticeable delocalisation is still present. As in all the examples, it should be remembered that this conclusion is still dependent on the validity of the MIND013 approximation. However, we believe
that they amply demonstrate the usefulness of LMO’s A calculation of all the localised bonds in benzene yields only six CH bonds and six CC u-bonds, one of which is shown in table 2 (third column). Since there are 15 occupied MO’s in the original SCF calculation, three are still missing, and the previous example suggests that they may be traced by inspecting the unsuccessful localisation attempts. In column 4 of the table we show the next best attempt at CC bond localisation, and clearly this has a a-bond character and is independent of the CC o-bonds. Furthermore, it has contributions from all six carbon atoms, so that in order to find the total number of such LMO’s we follow the same procedure as before and extend the local set to include all the carbon AO’s. This duly delivers nine LMO’s, namely three s-bond orbitals and six with obond character. Since the results given in the table show that id contrast to the u-bonds the n-bonds cannot be localised on two atoms at a time, we are thus led (without external information) to the standard view of three delocalised n-bonds supplementing the localised u-bond framework_ 3.5. 1.3 Butadiene (C4H6) In the butadiene molecule, the standard representation has two double bonds on either side of a CC single bond, and the intuitive approach to localisation would lead one to expect a similar interaction and thus delocalisation as in benzene. On the other hand, empirical evidence argues against deiocalisatidn [ 151.
to interpret molecular orbital calculations.
4. Nonorthogonality
and atomic occupations
From the examples in the previous section it is seen that an LMO Vi>representing a bond between atoms a and b takes the typical form llij = Cailha j) + Cbilhbi) + lnj),
(4-I)
where Iha,) and lhbi) are normalised hybrid orbitals on a and b, and [ni> is the total non-local residue. Let . us consider the ideal case of perfect localisation, i.e. Ini> = 0 for all i, which is exact for some symmetrical molecules like CH4 and approximated within a fraction of 1% by many others. It is then clear that overlap between different LMO’s can only occur when they share a common atom, in which case Clil’jY= CarCai (h,ilh,i)-
(4.2)
It is seen that nonorthogonality of the localised orbitals implies that the hybrid orbitals of a given at0.m are not orthogonal and vice versa. The standard treatment of hybridisation usually assumes that hybrid orbitals are orthogonal_ If we consider a tetragonal bonding situation, for example, the bond directions determine the mutual relationship of the three parbital amplitudes and orthogonality of the hybrids can only be brought about by a specific
WS. l’cr,voerdfPopulatio)~ localised orbitals
158
choice of the relation between s- and p-amplitudes. In this way’; ihe “classical” hybridisation of 25% s-content for tetragonal sp3 hybrids or similarly for trigonal and digonal hybrids is obtained. However, as pointed out for example by Dewar [ 141, this classical choice also implies an equal occupation of the atomic s and p states, which is unreasonable in view of the difference in their energies. It would therefore seem that in order to accommodate such differences in occupation numbers, we have to allow the hybrid orbitals and thus also the LMO’s, to be nonarthogonal. This is supported by the observation that the LMO’s obtained by our method from MO calculations which do have different s- and p-level occupation numbers, are in fact not orthogonal. However, once the possibility of overlap between hybrids is allowed, it is not clear any more how the bond order matrix and occupation numbers are to be calculated. For that matter, one may doubt that these quan!ities or even the normalisation of such hybrids are uniquely defined. Our purpose in this section is to clear these matters up and show that there is a well-defined relation between the orbital overlap and atomic occupation numbers, and that the latter may be quite practically calculated from non-orthogonai LMO’s. Let us consider specifically the occupation nk of atomic orbital k; a similar argument applies to offdiagonal bond order elements. For the original MO calculation, it is given by “k = 2 C’E~~E~~ = 2(EEtjkk, P
(4.3)
where E is then X m matrix of occupied MO’s, and n is the number of atomic orbitals, m the number of occupied MO’s. By collecting together al1 the localised orbitals L, as its column vectors, we now construct an X m local orbital matrix L which is related to E by the following generalisiition of eq. (2.2) L=EX,
(4.4)
and X is the m X m matrix constructed by collecting the corresponding coefficient vectors X, together. If the set of vectors Lhcorrespond to the individual classical bonds, they may in genera1 be assumed to be linearly independent so that the inverse of X exists and we may solve eq. (4.4) to obtain
EEf
Lt =
= Lx-lx-l’
L(XtX)-1
Lt.
(4.5)
It is easily seen that eq. (4.4) implies that eq. (2.4) holds not only for.the vectors but also for the matrices L and X, so that EEt =L(L’L)-lLf.
(4.6)
Note that while L is non-square in general and has no inverse, the existence of (L’L)-1 is assured by that of x-1. The matrix Lt L defines the overlap between the various LMO’s (not to be confused with the standard A0 overlap matrix S) and has the elements (Lf L)ii = ~ii = Li’Lj = ~!,il,,
(4.7)
so that it is a symmetrical matrix with diagonal elements ail equal to one, and in the case of orthogonal orbitals it reduces to the unit matrix. If we denote the elements of 0-l by wV and of L by $, we find from eqs. (4.3) and (4.6) that “k = 2 Cl~~W, t 2 C C IkjlkiWii _ Ci I i
(4.8)
At this point it is instructive to consider the specific example of theCH, molecule. The atomic orbital basis set consists in this case of the s, px, p,, and pr orbitals on the carbon atom, and the four s-orbit& of the hydrogen atoms. There are four LMO’s which, from symmetry requirements, yield the L matrix defined by
is Lt=
s S
s
-p
-p
-p
u
0 0
-P P P
P -P p
p0 po -p 0
a00 000’ 0 0
0) (4-9) a
This gives an overlap matrix with all the off-diagonal elements equal to a common value .Q=sz -p2
(4.10)
(which, because of Iocalisation, depends only on the AO’s on the common atom) and havinga determinant D equal to D=lnl=1-6Qk3@-3Q4=(1
-Q)3(1+3G). (4.11)
The inversion of fi is straightforward and we find that
159
N.S. Ver~c~oer~/PopllZation Iocaliredorbitak
wj/ =D-1
(1 - 3sP + 2s23)
=
(1 -#(l
+ 2fi), for i=j;
= D-1
(-a
(4.12)
- 2~22 -I-~3)
= -52(1
-
~92,
for i + j. From eq. (4.8) we calculate that, in an obvious notation, 89 “s=~+~
8p2 %=r-_¶
nH=(l
2(1 + 2S2)fJ2 -9)(1+3Q)’ (4.13)
Usually the nett occupation of the various basis orbitals is known from the original LCAO calculation. Using eq. (4.10), we may then solve for SI to find that !Tl= (n, - Q/(8
- 3n, - np).
(4.14)
Note that the overlap of the localised orbitals, and hence via eq. (4.13) also the LMO’s themselves, are uniquely determined by the occupations. Using the’se values, the entire procedure for projecting out the LMO’s that was described in the previous section may be side-stepped in this simple symmetric case by the use of eqs. (4.13) and (4.14), and then quantities like the atomic hybridisation and the relative distribution of electrons between the atoms participating in the bond, may be calculated.
Eq. (4.14) also shows explicitly that a difference in s- and p-orbital dccupations necessarily gives rise to non-orthogonal localised orbitals.
Finally, we remark that eq. (4.13) has the form n = n’/P -f(Q)1 ,
(4.15)
where n’ is the occupation obtained when overlap is ignored. The “renormalisation” factor differs from unity by some functionf(Q) which has a linear behaviour for G < 1. It is easily seen from eq. (4.14) that a is in fact quite small, n = 0.1 being a typical value for carbon and silicon, and values in excess of about 0.25 are only attainable at extreme occupation differences. Let us now return to the case of a general molecule. We focus attention on a specific atom, A, and wish to derive the connection between the occupation nk of a given atomic orbital Jk) centered on A, and the amplitude Zkiwith which it contributes to the local orbitals [similar to eq. (4.13)] _ Jf A contributes na orbitals to the basis set, we may
arrange them in such a way that the first n, rows of L correspond to the atomic orbitals of A, and the first nb columns to the local orbitals in which A participates. In the usual case that all LMO’s represent twocentre bonds, we have n, = nb. Henceforth, we neglect the non-local contributions @‘of section l), so that in its first n, rows L will contain non-zero elements only in the first nb columns. The sums in eq. (4.8) may then be restricted to the range i, j Gn,,, and we now show that a similar restriction may also be enforced approximately when c&ulating the Q.. Firstly, all local orbitals which do not overlap with those of A may be omitted without approximation, because such orbitals only give rise to separate blocks on the diagonal of n and which may be factored out. Apart from A’s “own” LMO’s, we then remain only with those associated with the atoms to which A is bonded. In order to assess the approximation involved when the latter group of LMO’s is disregarded as well, we consider the following expressions for the Oij (accurate to second order in the aji) found by direct expansion: m (4.16)
Oii
z D-l
1) h,lfi, 1
(4.17)
, l+i,j,i+j.(4.18)
It is seen that if the summations are restricted to h, I< nb, we are only disregarding terms of second and higher order in the overlap, which is expected to be small in view of the results for CH,. In fact, most of the neglected terms are zero in any case because LMO’s can only overlap when they share a common atom. Lastly, when eqs. (4.16)-(4.18) are substituted in (4.8), we see that it is a similar set of terms which is neglected in the numerator and the denominator of no., reducing the effect of the approximation even further. The degree of simplification brought about depends on the bonding situation of atom A. If it is a carbon atom with four bonds in tetragonal directions, for example, we are left with a 4 X 4 matrix with off-diago-
ICS. I’er~oerdlPopuIa[ioll localised orbirak
160
the ttk using these Slii values in eq. (4.8), and also tak-
nal elements ~~ = SiSi
-
pipi.
(4.19)
where we have made use of the fact that since the bond directions are specified by the relative contributions of the p_,, p,, and pz orbital!, the same interrelationship fhat holds in the row vectors of eq. (4.9), is valid here. When A is a carbon atom participating in a multiple bond, the orthogonal n-bonding orbitals may be factored out and a similar 3 X 3 or 2 X 2 matrix remains. Even for these small matrices eq. (4.8) is still too complicated to supply a direct link between the local amplitudes and the occupation numbers. We therefore proceed by considering an iterative solution of eq. (4.8) in which the wii in a given step is obtained from the liri of the previous iteration. For the first iteration, we assume all the Rii to be equal to the common value !TZthey assume for the ful-
ly synmetric case, so that wii reduces to eq. (4.12) for the 4 X 4 case, and may be factored out of the summations in eq. (4.8). A useful relation between the ~ii may be obtained by summing the resulting equation over the four values of k belonging to atom A. Since by construction the only overlap of the first four columns occurs in the fust four rows, we have
(4.20) The assumption of a common Sz value in the zeroth iteration amounts to taking the s2 and p* values obtained from eqs. (4.13) and (4.14) as estimates of the average values of sf and pf, i.e.
ing into account (4.16) to (4.18), it is seen that changes from the first iteration values only occur in correction terms of order R or smaller, all of which involve &is over the !$ or products of them so that fluctuations will be expected to average out partially, and finally changes in the numerator and denominator will again tend to compensate_ We may therefore expect the first iteration to be already a good approximation. The same philosophy may also be applied to the 3 X 3 and 2 X 2 cases. Assuming bonds which are equally spaced in a plane, matrices analogous to eq. (49) may be written down for CH, and CH, which gives rhe following common overlap values in terms of sand parbital amplitudes: Q;2,= sz -Gp”,
respectively. Inversion of the reSulting overlap matrix yields (for N = 2,3): w!!)= II
[I t(N= +/U
l)a2N]/(1 --cN)(l
Using eq. (4.13) to eliminate 12,and nr, from the left hand side, the indicated summation of eq. (4.8) then yields (4.22) This implies that the average value of Rii stays unchanged although the individual values may fluctuate about it. If we now calculate approximate values for
+NQ),
- fiN)(l +NQ)>
i=j; i+j_
(4.24) Substituting this back into eq. (4.8), expressions for ns and n,, may be obtained in each case which is combined with eq. (4.23) to find that a/,, = (12s- np)/[2N - (N - l)rz, - np] .
(4.25)
Finally, the arguments for generalising this to an arbitrary molecule are similar to those for the 4 X 4 case. In applying eq. (4.8) to the orbitals on a given atom, it is assumed that the Wii may still be calculated from eqs. (4.24) and (4.25) so that it takes the final form N
N
(4.21)
(4.23)
!A2 =s2 -p*,
2[1+(N-2)aN] =(I -i-+q)[l
5?-4n~r_.l 1=1 h-7 +(N-
_ i< i hZQ
(4.26)
l)ii?~]Q.
Eqs. (4.24), (4.25) and (4.26) hold for all three cases, namely for the number of equally spaced bonds N equal to 2,3 and 4. In order to evaluate the validity of these equations (and of the approximations leading up to them) we have compared the value of nk calculated from eq. (4.26) with that holding for the original MO calculation, for different atoms in a number of molecules.
IV.S. 1~~erwoerdjPopulatiorl localised orbitals
Firstly, a set of hydrocarbon molecules (saturated and unsaturated) of increasingsize (ranging from CH4 to C,,H,z) was used in order to test the effect of neglecting overlap with neighbouring bonds. Secondly, carbon atoms in these, as well as in some small molecules containing C1and F substituents, were chosen to maximize the deviations from equal lzj values which holds in the symmetric case. The agreement found is usually within l%, or 3% at worst. Since the deviations from non-locality are also of this order of magnitude, it seems pointless to carry the above mentioned iteration any further. In fact, we find that with very little additional loss of accuracy we may make the further approximation that, for the 4 X 4 case, q
sisj=;Es,?, cc pipj~-+&l~. i
’
(4.27)
which becomes exact in the symmetric case, as seen from eq. (4.9). Making similar approximations for the 2 X 2 and 3 X 3 cases, it is found that eq. (4.26) reduces to the simple form 2 cs;
= [I +(N-
l)~]“s’
(4.28)
(4.29) These equations represent renormalised sum rules like eq. (4.1 S), but now generalise_’ to arbitrary molecules. A detailed application of the sum rules to interpret hybridisation trends in hydrocarbon molecules, will be discussed elsewhere [12] _In particular, we note the result found there that while eqs. (4.26), (4.28) and (4.29) are numerically quite similar, the presence of the cross-terms between different LMO’s in eq. (4.26) is important to explain some of the more subtle differences between related MO calculations. It has been noted by Peters [6] that non-orthogonality of hybrid orbitals may seriously disturb some of the generally accepted notions about molecular geometry. However, the presence of sum rules like equations (4.28) and (4.29) restores the basis of at least those qualitative arguments which are founded only on the requirement that the sum of squared amplitudes must add up to a constant total. As an example, we consider
an often used argument that connects
16:
the s-orbital con-
tent of hybrid orbitals to the bond directions 1161. The argument is usually based on the observation that the s-orbital content of classical trigonal sp’ orbitals, for example, is higher than for tetragonal sp3 orbit-
als. As previously pointed out, however, the classical hybridisation is a special case where orbitals have beer1 constructed to be orthogonal without taking actual occupation numbers into account. The direct applicability of this argument to the general case is therefore questionable, but eqs. (4.28) and (4.29) supply the necessary generalisation. Provided that the right hand side stays unchanged, the transition from tetragonal to trigonal bonds may be understood by noting that as one orbital becomes purely p-like to participate in a rrbond, the s-content of the other orbitals must increase to maintain the sum rule of eq. (4.28). Actual calculations show that II, is in fact virtually the same (e.g. in C,H, and C2H4) while the decrease in N from 4 to 3 is largely compensated for by an increase in ,r2, implicit in eq. (4.25). In this case it is thus t:ue that changes in the bond directions are accompanied by a change in the hybridisation. However, since the s-orbital is spherically symmetric
it can clearly not influence the bond directions on its own. The above mentioned connection therefore does not exist in cases where the sf values change because of a change in ~zs,such as may be brought about by changes in the electronegativity of neighbouring atoms. Finally, we note that the sum rules may explain the observed tendency in both our own and in other calculations [5] of atomic hybrids to deviate slightly from internuclear directions. The necessity to maintain balanced occupations of the different atomic p-states clearly places limitations on the hybrid directions. Such an interpretation is supported by our finding that when the MO calculation is repeated with surrounding atoms displaced slightly from their optimised positions, the hybrids on the central atom stay largely unchanged with only a very slight following tendency discernible. Thus the energy gained by increased overlap with
neighbouring atoms when the hybrid direction is adjusted, is outweighed by the loss due to internal readjustments. It seems plausible that it may similarly be advantageous to allow slight deviations of hybrid directions in the equilibrium state of the molecule if this dan lead to a better balance in the electron distribution of a given atom.
162
KY_ Vemoerd/Popzrlation localised orbitals
References [I] C. Edmiston and K. Reudenberg, Rev. Mod. Phys. 35 (1963) 457; J. Chem. Phys. 43 (1965) 597. [Z] S.F. Boys, Rev. Mod. Phys. 32 (1960) 296. [3] J.A. Hunter, QCPE 10 (1978) 355. [4] D. Boerth, J.A. Hashmall and A. Streitweiser, QCPE 10 (1978) 354. [S] C. Trindle and 0. Sinanosu, J. Chem. Phys. 49 (1968) 65. [6] D. Peters, 3. Chem. Sot. (1963) 2003, and subsequent papers. [7] V. Magnasco and A. Perico, J. Chem. Phys. 47 (1967) 971. [Sj M.J.S. Dewar. H. Metiu, P. Student, A. Brown, R.C. Bimgham, D.H. Lo, C.A. Ramsden, H. Kollmar, P. Weiner and P.K. BiscoF, QCPE 10 (1976) 309.
[P] Quantum Chemistry Program Exchange, Indiana University, Bloomington, Indiana 47401, USA. [lo] T.L. Gilbert, in: Molecular orbitals in chemistry, physics and biology, eds. P.-Q. Lijwdin and B. PuUman (Academic Press, New York, 1964) p_ 405. [ 111 W.H. Adams, J. Chem. Phys. 42 (1965) 4030. [ 121 W.S. Verwoerd, to be published. [ 131 M.J.S. Dewar and R.C. Dougherty, The PM0 theory of organic chemistry (Plenum Press, New York, 1975). [ 141 M.J.S. Dewar, The molecular orbital theory of organic chemistry (McGraw-Hill, New York, 1969). [15] M.J.S. Dewar, Hyperconjugation (Ronald Press, New York, 1962). [16] W. Month, in: Advances in solid state physics, ed. H J. Queisser (Pergamon Press, Oxford, 1973) p. 247; and many other references.