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Resonance integrals in semi-empirical MO theories
__ __ Volume 54, number 3
RESONANCE
CHEMICALPHYSICSLElTERS
INTEGRALS
IN SEMI-EMPIRICAL
15 March 1978
MO THEORIES
S. DE BRUIJN Institute of Theoretical chemistr_v. University of Amsterdam Amsterdam. l%e Netherlands
.
Received 18 November 1977 Revised manuscript received 13 January 1978
Several shortcomings of semi-empirical methods based on the neglect of differential overlap (NDO) hpproxrmatron can be ascrrbcd to inconsistencies in the basic formulae of the methods. _4 rewed treatment results, without any addtttondl .ISsumptions, in e formalism which describes strong bonds as well es weak, stereospecrfic mterxtions m molecules and molecular complexes.
1. Introduction Since the publication of the CNDO/l method [ 1] in 1965 a considerable number of semi-empirical MO methods has been developed_ Attempts to improve the original version of the theory have exploited the possibilities inherent in modifications of the crude ND0 approximation (INDO [2] , NDDO [3] , MIND0 [4] )_ A further improvement of the accuracy of ND0 results is due to calibration procedures in which a wide variety of experimental data is used, as in the case of MlND0/3 [5]. Though the latter method can be regarded as virtually the best that can be obtained within the framework of conventional ND0 theories, the necessity of an
awareness of its limitations was stated by Dewar [5,6], confirmed by Pople [7] and Hehre 181, and documented by further calculations [9-l I] _ Some cases where MIND0/3 results are inaccurate were formulated by Dewar: the method underestimates the strain in small rings; on the other hand, it also underestimates stabilities, e.g. those of compact globular compounds [5]. In one of the problems studied by Bantle and Ahlrichs [ 1 l] the difficulties are ascribed to the presence of a three-centre four-electron bond; and though an analysis [ 121 of the inability of CNDO/2 and INDO to cope with non-bonded, i.e. weak, interactions does not directly refer to MIND0/3, it will prove to be equally significant as the work reported in ref. [ll] -
Now, if within the mathematical framework of MIND0/3 no further significant progress is possible [5], then the formulae themselves should be reconsidered (cf_ ref. [7] )_ In the present paper we focus the attention on one assumption whose validity is taken for granted in all ND0 methods proposed until now, where the one-electron two-centre integral fl is assumed to be a function of two atomic orbitals @A, VB) and the distance between the corresponding nuclei, i e. PpA,Vu =f@A*"ByRAB)-
(1)
We shall show that this expression is inappropnate in the important case of polyatomic molecules, because it ignores the dependence of fl on several highly relevant variables. Our proposed revision of the definition of /3 in terms of A0 integrals ~111 lead to a qualitative explanation of some systematic inaccuracres of MIND013 results. However desuable a revision of the ND0 formalism may be, it will be necessary to keep in mind two of the criteria for a decent semi-empirical method as formulated by Pople and Beveridge [ 131: is it possible to modrfy (1) and yet to keep a theory that is both unbiased and simple? It will become clear that our proposed modifications result in energy expressions which are as unbiased as those which occur in present-day ND0 theories. A part of their simplicity, however, has to be sacrificed; as a compensation for the additional labour 399
Volume 54, number 3
CHEMICAL
we get energy expressions which show clearly that interactions in polyatomic molecules are more complicated and more varied than one might conclude from the formulae obtained from earlier ND0 theories.
2. Justifications of the ND0 approximation Our analysis will be based throughout on some consequences of the orthogonalisation procedure introduced by Ldwdm [ 141 and appiied by Pople [I], i.e. A.= *s-l/’
,
(2)
where K represents the non-orthogonal minimal basis set of Slater orbitals. It is well known that in the h basis all overlap integrals vanish, and that the neglect of all two-electron integrals except “(wlvv) is justitied, at least to first order in S, by Mulliken’s approximation [ 151 (PI “$S,“{(IyII
+bvl3 -
(3)
The most elaborate discussion of (2) and (3) is to be found in Fischer-Hjalmars’ analysis of the ZDO approximation [ 16]_ Before we proceed to apply (2) we consider a few alternative arguments for the elimination of all terms in (pvl from the energy expression. In the first place we mention the procedure preferred by Dewar [5,17], which is quite similar to the one applied in the very first version of the ZDO approxlmation [ 181. Without any refererlce to an orthogonalized basis set Dewar argues that in the LCAO MO version of the expression for the electronic energy E= (*IHl*)l(*liP),
(4)
the numerator is built up from atomic orbital multicentre integrals, whereas the overlap integrals appear in the denominator. In view of (3) we may assume that (4) remains approximately the same if we neglect both the overlap and the multicentre integrals. An argued exception is made for the one-electron integrals (ylh 1~); the most efficient explanation of this exception is that Mulliken’s approximation (3) does not apply to (JL1V2 Iv) because of the absence of a charge distribution (~.lvI. All surviving A0 integrals are now to be evaluated by means of purely empirical methods. In our opinion Dewar’s justification of the ND0 approximation is as good as Pople’s, except for one feature: it offers no guidance with regard to the next steps to be 400
15 March J978
PHYSICS LETTERS
taken; immediately
after the simplification of (4) we seas of sheer param-
are set adrift on the unmapped
etrisation. Now it may be argued that this separate discussion of Dewar’s treatment of (4) is superfluous, because it could be replaced by the simplification based on (2) without any change to his MIND0 method. Sure, but that is only part of the truth. From a semi-empirical point of view Dewar’s approach is so flexible that he might have effectively obtained the same conclusions as in the present paper, but for one incorrect statement in ref. [S] : contrary to Dewar’s expressed opinion it will appear in the following section that p can be made to depend on bond angles without a violation of the requirement of rotational invariance. In the second place we give some a_ttention to a possible refinemenr of (2). It is well known that the Mulliken approximation when applied to o-orbitals is less accurate than in the case of sr-orbitals. We may try to improve our justification of the neglect of differential overlap by replacing Sell2 in (2) by S-1/3U, the unitary matrix U being chosen in such a way that the equivalence of atomic orbit& coupled by a molecular symmetry operation is maintained. From the allowed matrices we can choose U such as to minimalise the inaccuracy of (3), a conceivable criterion being the minimalisation of the sum of the absolute values of the transformed (y lpo) with p # v and/or p # u. We have two objections against such an improved orthogonalisation. First, the unknown matrix U precludes the general analysis we wish to offer; second, the refinement is superfluous: many of the multicentre integrals may emerge from a standard orthogonalisation treatment with values of several tenths of an eV without seriously affecting the accuracy of an ND0 method. As an example we discuss the integrals Xblw) which occur in the energy formula E=CP,,
AhPa
with a coefficient P,,P,,_
If for any A0 ZJ,Pu, can be
assumed to be approximately constant for a large set of molecules in different states where the same JL is found on the neighbouring atom, then the energy term &~~~~(“I~JII) is effectively dependent on Pw done, vy hOly(vv) can be incorporated in the empiri-
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CHEMICAL
15
PHYSICS LETTERS
cal value of flflv. For further examples we refer to a similar analysis of energy terms in s-electron theory [ 193 _We mention in passing that the assumption of a constant Pvv in the discussion of such minor terms seems to be quite reasonabie for organic molecules. It should not be used when one compares different states
March
1978
one s-orbital only. In the given configuration S13 is considerably smaller than S17 and Sz3.
of metallo-organic systems, where extensive opposite migrations of n- and u-electrons occur frequently 1201.
3.
Long distance interactions in polyatomic molecules There is a perfectly sound principle underlying all
well-considered
parametrisation
methods
in semi-em-
pirical theories: the empirical parameter is expressed in terms of A0 integrals, the expression obtained is used in order to describe the parameter as a plausible function of the relevant variables, and the rest is a problem of calibration, i.e. fitting to a given, appropriate type of curve. This principle was correctly applied to a minority of chemical problems, viz. to diatomic molecules exclusively. If we can take it as established that &, is the empirical counterpart of “h 12, then it is easily found that, up to first order in S, Ddir = hhdir = W h,U lur
- t S,“($,
+ h”,),
In all semi-empirical methods the mutual orthogcnality of the orbitals X, and 5 is duly taken into account when PI3 1s computed on the basis of (6) and (7); however, the fact that both of them should be regarded as crthcgcnalised with respect to X7 IS not considered. In order to calculate %t3 we expand S-1/3 = (1 + A)-*/* up to second order in A ; maintaining all those terms in which no more than two off-diagonal elements of S and h occur we find that
+ it,,
-
$,,h,3 _
(7) We can also accept the working hypothesis that for cases other than those considered in ref. 1211 fldir is roughly proportional to SP,. What we have to show next is that in apolyatomic molecule the identification of fl and pdlr IS _ impossible; that it is, indeed, the oversimplification which is responsible for most of the failures of ND0 theories. Let us consider a triatomic molecule with a bondangle somewhere between 100” and 180” (smaller bondangles are more conveniently approached from the extreme case to be discussed in section 4)_ In order to keep both the derivation and the interpretation of the formulae as simple as possible we restrict the discussion to an elementaxy case: on each centre we consider
03)
using (6) we can rearrange this expression to yield x/l 13 =&
-$s,,Pg
+ ~Si3S~3(~l~~
where our superscript “dir” emphasizes that the equation applies to the direct interaction between two atoms. Eq. (6) was introduced and, for a few simple cases, evaluated by Mulliken [21] ; it is obvious that
_ ;
-$s&~ -
u12z +/l33)-
(9)
We see that Ah 13 is the sum of three positive and three negative terms, all of the same order of magnitude. It is impossible to predict off-hand even the sign of their sum; it is equally impossible to replace the expression by its first term, which is the only one that does not depend on atom 2. Any /&formula based on eq. (1) is invalid for the Interaction between non-neighbcur atoms. We conclude from (9) that the p which is to describe the interaction between second neighbcurs depends critically on aI1 elements that make up the geometry of the relevant triangle. Consequently, Dewar [S] would have been correct if he had introduced the bondangle 0 into his p-function. The idea that an angular dependence must violate the principle of rotational invariance is wrong: the interactions are not made to depend on the choice of the coordinate axes, but on the direction of the bonds. It is a consequence of these directed physical perturbations that the directions in an atom in a molecule are no longer equivalent;even s-crbitals lose their spherical symmetry on crthcgcnalisaticn. 401
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CHEMICALPHYSICSLETTERS
3
It is another question whether the explicit introduction of 0 is expedient. We do not think it is, for two reasons. First, the angIes are redundant parameters: the geometry of a molecule is determ?red by the set of interatomic distances, which we need anyway for the calculation of the overIap integrals. Second, 8 will be a highly inconvenient parameter in polyatomic molecules where the Iine connecting atoms 1 and 3 is an eIement of more than one triangle_ There is some computational evidence in favour of the validity of (9).
In n-electron
theory we have a situ-
ation which is comparable with the one described above: we consider one orbital per atcm, and all these orbitals are equivalent. Chong [22J found that in benzene %h,3 = +0.07 eV, a calculation [23] of PI3 based on finderberg’s relation [24] /3 =R-‘ds/$R gave the same result for benzene, and values between +0.03 and to.07 eV for the various second-neighbour interactions in naphthalene. Needless to say, positive p’s are at variance with the prescriptions of ali established ND0 methods: they can give onIy negative vaIues, the order
of magnitude being -1 eV for second neighbours. Now if eq. (1) is fimdamentahy wrong, how can it possibly have served as the starting point for an impressive series of successful calculations? In order to answer this question we need not immediately invoke the freedom inherent to a parametrisation. A consideration of the formulae
erful compensation
will reveal the presence
or a pow-
mechanism.
On investigating a number of available CNDO, INDO and MIND0 results we found that the sum of the sec-
ond-neighbour terms PMy&,, was non-negligible and positive, as a consequence of the predominance cf small negative bond orders. For a medium-sized molecule the sum of the computed second-neighbour interactions can be as Iarge as 5-10 eV. If the few p-values calculated from duly orthogonalised orbitah are iepresentative, then the whole computed second-neighbour interaction energy is spurious, and in a theory which aims at an accuracy of a few kcaI/moIe this constitutes a considerabie error. How can it be corrected? To begin with, we can identify two errors with the opposite sign. The first-neighbour interactions and the one-centre integrals are given by (10) and (11) respectively, i.e. %l 13 =@
- 3sz,@
f&3523(hll 402
+h,,
- $,,@I? -a,,);
(10)
15 March 1978
(11) In Ah12 there is a large leading term which is defined in agreement with (1); there are five small correction dir , and three of them terms, each with a factor S,, or f313 are positive. Therefore, PI2 values which are on the whole slightly too negative can be conveniently formulated in agreement with (1). Similarly, if %, r is identified with a free-atom parameter we neglect at least the two following terms, which are positive. The overall situation can be summarized as follows: in the usual evaluation of the second-neighbour terms we introduce a spurious positive energy, due to a large error in &3 and to small negative bond orders. In the terms in “h12 and hhll we neglect small positive terms in integrals which are to be multiplied by large positive bond orders. Even if these errors do not cancel it is quite conceivable that their resultant can be roughly represented
as a function
of the number of bonds in the
molecule. If that is the case, then a simple adaptation of the parameters is sufficient to obtain highly accurate total energies. The two Iast terms in (11) deserve some attention. If in the triatomic molecule the charge distribution is fairly uniform (P,, = l), it is readily shown that these terms do not contribute
to the energy. Therefore,
in
many organic molecules their neglect is of no consequence. If, however, there are appreciable differences between the charge densities on the atoms, then the fina! terms in (11) become important_ The neglect of these terms may be one of the reasons why ND0 calculations are less convincing when applied to organometallic compounds 1201 and to the dative bond in - BH3NH3 [l i] _ It follows from the foregoing discussion that the suitability of an ND0 method for a molecule or a complex does not depend on the absence of errors but on the effectiveness of the compensation mechanism. Though a computational scheme may be reliable for the calculation of the total energies of many molecules, the incorrect partitioning of the energy is bound to lead to inconsistencies. The overestimation of the repulsion between second neighbours and its compensation by the one-centre and nearest-neighbour terms suggest a defiition of “those molecules for which the method was designed”: ND0 methods can be confident-
15 March 1978
CHEMICAL PHYSICS LETTERS
Volume 54, number 3
in the degenerate set of antibonding orbitals (or - fi), we find
ly applied to molecules with a uniform charge distribution and a certain equilibrium between the numbers of onecentre and medium-range interactions. An excess of the latter interactions is found in globular compact compounds; as can be expected from our analysis MIND0/3 underestimates their stability (fig. 3 of ref.
E(l) = (1 - S)%
PI I-
The corresponding equation obtained from ND0 methods based on (1) is: E&r
4. Equivalent interactions In clusters of atoms One special case remains to be investigated: how appropriate is the ND0 formalism in a configuration where first and second neighbours can no longer be distinguished, e.g. in an equilateral triangle? Once more we limit the investigation to a few simple problems whose results are open to interpretation. We consider a set of n identical atoms (n = 2,3,4), arranged in such a pattern that all pair interactions are equivalent: line, equilateral triangle, regular tetrahedron. Owing to the highly symmetrical structure of S we can evaluate S-L’2explicitly. With S,,P = 1 and Scly = S the eigenvalues of S are given by El=l+(n-l)s,
(12a)
e2=1
(12b)
-s,
the latter eigenvalue being (n - 1)-fold degenerate_ It is readily verified that the elements of S-1’2 are given by (S_l/?l)
PP
(s-1/2)py
=,-I
{ET”2
+ (0 - l)Q -112)
= n-1
{Q/2
_ gq
The elements of h are h,, S--L/2hS-112 are 01 f
;
_
(134 (13b)
-S)_’
[
h,,
-A%,,+
1 +(;Y
I)J;
(14a)
pdir fi = ‘hlzu =(I
-s)-’
+
= n(h pp + PE> -
(1%
(16)
From these two equations it becomes evident why (1) fads: no matter how we choose the parameters in (16) every attempt to fit (16) to reproduce (15) is doomed to fail beLause the second factor of (16) is nindependent. The best we can hope for is that the intersection of (15) and a calibrated (16) takes place near an integral value of II, so that at least one configuration of physical interest is correctly descrrbed by (16). Eq. (15) can be expressed in temrs which are entirely outside the scope of conventional ND0 methods: pair interactions in clusters decrease as the number of atoms in the cluster increases. The quantitative effects represented by (14) should not be underestimated: even for S as small as 0.1 it implies that 0 decreases by some 17% as we go from n = 2 to n = 4. It is not surprising that MIND0/3 underestimates the strain in small rings (fig. 2 of ref. [5] )_ Similarly. energies of reaction complexes may well be calculated at much too low values [l I] owing to the occurrence of atom clusters without a central atom, i.e. a large number of first-neighbour interactions_
5. Conclusions
and hflr,, and those of
'h,,
=(l
- Sh,,
lJv
It@-11)s
-
Wb)
Evaluating the one-electron energy E(l) for a set of n electrons in a configuration with two electrons in the bonding orbital {a + (n - 1)/3) and (n - 2) electrons
Before we sketch some possible quantitative developments of the ideas presented in this paper we summarize the qualitative conclusions that can be drawn from our analysis. The most pessimistic interpretation of eq. (9) and its consequence that j3r3 is considerably smaller than it is usually assumed to be implies that all semi-empirical calculations performed since 1965 can be discarded, except those of n-electron theoreticians, who have maintained for nearly 50 years that PI3 1s negligrble. In our opinion such a conclusion is an exaggeration_ Many reliable results have been obtained for ground state energies, and the accompanying SCF wavefunctions 403
Volume 54, number 3
CHEWCAL PHYSICS LETTERS
should be very close to those which are to be expected from a revision according to our suggestions: the proposed modificecions of the one-electron terms in the Hartree-Fock matrix elements can be regarded as rather weak perturbations. Probably in many cases the most interesting results will be obtained from revised calculations of some of the more delicate features, such as the potential field around a molecule, where calculations according to present-day ND0 methods are not invarrably successful 125,261 and the consequences of the proposed revision of the theory are not easily predicted. More interesting results can be expected for those molecules and complexes where the best available ND0 methods show unusually large deviations. The fact that the signs of several of these deviations can be explained by eqs. (9)-( 11) suggests that a revised computatron method, based throughout on a full recognition of the consequences of complete orthogonalisation, is definitely promising. The most intriguing aspect of (9) is, however, that is may well be the key to entirely new developments and applications of ND0 methods: eq. (9) is the first recognition in ND0 language of the fact that some interactions are weak, even at a distance of only a few A. Moreover, it states that such weak interactions are stereospecific: the interaction between atoms 1 and 3 is sensitive to the nature and the exact location of atom 2. A related conclusion can be drawn from section 4: the total interaction in a cluster cannot be written as a sum of unperturbed pair interactions. Now the interrelation of the concepts of weak interactions, stereospecificity and many-body interactions has been emphasized for many years among others by Jansen et al., in their investigations, in terms of indirect exchange, of such problems as crystal stability, magnetic ordering and rotational barriers [27] _ Hydrogen bonding is another example of a weak interaction, where manycentre terms and stereospecificity play an important role [28]_ The possibility to study such interactions by means of convenient and cheap ND0 methods can become valuable.
6. Towards
a parametrization
Now that the consequences of a duly performed orthogonalisation have been described we have to in404
15 March1978
dicate how the revised ND0 method can be implemented so as to produce numerical results. In the parametrisation process we can distinguish two steps: first, we have to find suitable expressions for the empirical A0 integrals in terms of appropriate variables, and next numerical values for the parameters in these expressions must be established. For the latter step the numerical fitting procedure described by Dewar [S] will be adequate; the first step, however, should be designed so as to accommodate the results of the foregoing analysis. It is evident that any ND0 method must admit a considerable empirical element in order to make up for its many deficiencies, such as the neglect of orbitalother core overlap, the approximation of the wavefuntion by a onedeterminantal function expressed in terms of a minimal basis, and the inaccuracy of the ND0 approximation itself. An attempt to formulate the consequences of eqs. (9)-( 11) in terms of a generally applicable empirical formalism should further satisfy the requirements stated before: it must be unbiased and simple. Just as in the earlier stages of the analysis we have no objections against the treatment of the two-electron ter_ms in earlier ND0 methods; a minimum set of rotation invariant TAR should be sufficient. In the derivation of the one-electron part of the Hartree-Fock matrix elements, however, we propose considerable deviations from traditional procedures. The MO’s are described as linear combinations of Ldwdin orbit&, so that the one-electron part of F must be an empirical version of integrals over the same orbitals. However, because these orbitals depend on the size and the geometry of the whole molecule, the corresponding integrals cannot be determined directly: they must be derived by the transformation b = R-‘/+$-l/”
(17)
from parametric matrix elements valid for the Slater orbitals, which can be regarded as invariant from one molecule to another. This treatment deviates strongly from the conventional one, where the MO’s are expressed in Liiwdin orbitals, and the one-electron terms in Fccc( and refer to pure orbitals partly
Volume 54.
CHEMICAL PHYSICS LETTCRS
number 3
transformation with S-L’2 is sufficient to obtain useful diagonal elements “hpp for the Lowdin basis: A
-
c BPA
(1%
ZBrAB*
Up, = - fj (Ifi + API -
(Z.4 -
-
(1%
The off-diagonal elements h,, require some care. First unlike jifiv, they cannot be regarded as local terms; Slater-type charge distributions (~1 interact strongly with the whole core. Further, if h,, were approximated by Mulliken’s formula h,, =$Sflv(hplr + h,,), then after orthogonalisation all p’s, even those between neighbours, would be quite small. On the other hand, we conclude from Mulliken’s correct treatment of the diatomic case that h,, is proportional to S,, because the same proportionality is valid for the two other terms in (6)_ We may therefore tentatively write h PV = i&,(h~fl
+ h,,)
f $,j-(cr,
n, R) t
f can be written as a constant Cam for a given pair of orbitals. The one-electron part of the Hartree-Fock matrix is now defined by (17)-(20). A final remark concerns the types of expression which can be used for f&u,v, R) in (20). in the simplest version of the comparable expression for fl in existent ND0 theories f is replaced by a constant for a given pair of atoms, without regard to the orbitals involved (CNDO: f = f (& + 0:)). The most detailed parametrisation, untried and undesirable, could attempt to establish one functron for each pair of atomic orbitals. MIND0/3 [S] steers a well-defined middle course between these two extremes by maintaining three elements in f: a constant factor, characteristic of the atom pair; a factor $ (I, + 1,) which distinguishes the individual orbital pairs, and, at least in principle, the option of a further R-dependence, though the exploration of the latter degree of freedom proved to be fruitless. We regard the introduction of experimental orbital characteristics as a useful addition to the flexibrlity of
15 hiarch 1978
MIND0/3, but we doubt whether the ionisation potentials I alone are sufficient. Mulliken’s argument [21] in favour of a dependence on 1, referring to a VB description of Hz in terms of non-orthogonal orbitals, is rathe. vague: fi is described in terms of a perturbation which is taken to be dependent on the energy of the electron in the unperturbed atom. In our opinion the finite probability of finding two electrons near any of the nuclei suggests that the perturbation should also be connected with an electron affinity. A similar conclusion can be drawn from the general form of the hamiltonian matrix elements which occur in VB tneory bdsed on orthogonalised orbitals [29]. In this approach &, appears exclusrvely in off-dragonal matrix elements HP4 where the configurationsp and 4 differ by one spin AO. It is readily derived that in a triatomic molecule ABC a matrix element flAB will be obtained from the following pairs of configurations: ABC and A+B-C, ABC and A-B%, ABiCand A+BC-, and A-BCt and AB-C+. It is seen that in this approach configurations with positive and negative A and B ions occur in exactly the same way, so that a description of flAB as a symmetrrcal functron of ionisation potentrals and eIectron affinities seems more likely to be correct than one in terms of I alone. The argument is quite close to the one used in CNDO/2 [ 131 for the approximation of U,, as a function of Ifi + A, we wish to account satrsfactorrly for the tendency cf an atomic orbital both to acquire and to lose electrons. A final choice between the various possible implementations of (20) can only be made on the basis of a comparison of the optimal numerical results to be obtained from each alternative. From our general analysis it appears that our modification of the ND0 methods is more consistent than its predecessors, and that it explains several of their specific shortcomings. The details of its actual elaboration and application will no doubt reflect once more the conflict between the two traditional enemies, convenience and accuracy, though we do not exclude the possibdity of their reconciliation_
References [ 11 J-A. Pople, D.P. Sentry and G-A. Segal, J. Chem. Phys. 43 (1965) S129. [2] J-A. Pople, D.L. Beveridge and P.A. Dobosh, J. Chem. Phys. 47 (1967) 2026. 405
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PHYSICS
R. Sustmann, J.E. Williams, M.J.S. Dewar, L.C. Allen and P. von R. Schleyer, J. Am. Chem. Sot. 91 (1969) 5350. [4] NC. Baud and M.J.S. Dewar, 3. Chem. Phys. 50 (1969) 1262. [S] R.C. Bingham. hf J.S. Dewar and D.H. Lo, J. Am. Chem. Sot_ 97 (1975) 1285. [6] XfJ.S. Dewar, J. Am. Chem. Sot. 97 (1975) 6591. [7] J.A. Pople, J. Am. Chem. Sot. 97 (1975) 5306. [S] WJ. Hehre, J. Am. Chem. Sot. 97 (1975) 5308. [9] H. Dits, N.Lf.M. Nrbbering and J-W. Verhoeven, Chem. Phys. Letters 51 (1977) 95. [IO] M-IS. Dewar and D. Landmann, J. Am. Chem. Sot. 99 (1977) 4633. [ 11 J S. Bantle and R. Ahlrichs, Chem. Phys. Letters 53 (1978) 148. [ 121 A.R. Gregory and M.N. Paddon-Row, J. Am. Chem. Sot. 98 (1976) 7521. [ 131 J.A. Pople and D.L. Beveridge, Approximate molecular orbital theory (McGraw-Hill, New York, 1970). [ 141 P.-Q. Loudin, J. Chem. Phys. 18 (1950) 365.
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[ 151 R.S. hfulltken, J. Chem. Phys. 46 (1949) 497. [ 161 I. Fischer-Hjalmars. J. Chem. Phys. 42 (1965) 1962. [ 171 M-IS. Deuar, The molecular orbital theory of organic chemistry (McGraw-Hf, New York, 1969). [18] R.C. Parr, J. Chem. Phys. 20 (1952) 1499. [ 191 S. de Bruijn, Theoret- Chim. Acta 17 (1970) 293. [20] P. Ros, Jerusalem Symposium on Quantum Chemistry and Biochemistry (1974) p. 207. [21] R.S. Mulhken, J. Phys. Chem. 56 (1952) 295. [22] D-P. Chong, hfol. Phys. 10 (!966) 67. 1231 S. de Bruijn, unpublished calculations. [24] J. Linderbeg, Chem. Phys. Letters 1 (1967) 39. [25] C. Giessner-Prettre and A. Pullman, Theoret. Chim. Acta 37 (1975) 335. 1261 E-N. Svendsen and T. Stroyer-Hansen. Theoret. Chim. Acta 45 (1977) 53. [27] L. Jansen and R. Block, Angew. Chem. Intern. Ed. 16 (1977) 294, and references therein. [ 281 A. Witkowsky, Mol. Phys. 29 (1975) 1441. 1291 R. McWeeny, Proc. Roy. Sot. A223 (1954) 63,306.
RESONANCE
CHEMICALPHYSICSLElTERS
INTEGRALS
IN SEMI-EMPIRICAL
15 March 1978
MO THEORIES
S. DE BRUIJN Institute of Theoretical chemistr_v. University of Amsterdam Amsterdam. l%e Netherlands
.
Received 18 November 1977 Revised manuscript received 13 January 1978
Several shortcomings of semi-empirical methods based on the neglect of differential overlap (NDO) hpproxrmatron can be ascrrbcd to inconsistencies in the basic formulae of the methods. _4 rewed treatment results, without any addtttondl .ISsumptions, in e formalism which describes strong bonds as well es weak, stereospecrfic mterxtions m molecules and molecular complexes.
1. Introduction Since the publication of the CNDO/l method [ 1] in 1965 a considerable number of semi-empirical MO methods has been developed_ Attempts to improve the original version of the theory have exploited the possibilities inherent in modifications of the crude ND0 approximation (INDO [2] , NDDO [3] , MIND0 [4] )_ A further improvement of the accuracy of ND0 results is due to calibration procedures in which a wide variety of experimental data is used, as in the case of MlND0/3 [5]. Though the latter method can be regarded as virtually the best that can be obtained within the framework of conventional ND0 theories, the necessity of an
awareness of its limitations was stated by Dewar [5,6], confirmed by Pople [7] and Hehre 181, and documented by further calculations [9-l I] _ Some cases where MIND0/3 results are inaccurate were formulated by Dewar: the method underestimates the strain in small rings; on the other hand, it also underestimates stabilities, e.g. those of compact globular compounds [5]. In one of the problems studied by Bantle and Ahlrichs [ 1 l] the difficulties are ascribed to the presence of a three-centre four-electron bond; and though an analysis [ 121 of the inability of CNDO/2 and INDO to cope with non-bonded, i.e. weak, interactions does not directly refer to MIND0/3, it will prove to be equally significant as the work reported in ref. [ll] -
Now, if within the mathematical framework of MIND0/3 no further significant progress is possible [5], then the formulae themselves should be reconsidered (cf_ ref. [7] )_ In the present paper we focus the attention on one assumption whose validity is taken for granted in all ND0 methods proposed until now, where the one-electron two-centre integral fl is assumed to be a function of two atomic orbitals @A, VB) and the distance between the corresponding nuclei, i e. PpA,Vu =f@A*"ByRAB)-
(1)
We shall show that this expression is inappropnate in the important case of polyatomic molecules, because it ignores the dependence of fl on several highly relevant variables. Our proposed revision of the definition of /3 in terms of A0 integrals ~111 lead to a qualitative explanation of some systematic inaccuracres of MIND013 results. However desuable a revision of the ND0 formalism may be, it will be necessary to keep in mind two of the criteria for a decent semi-empirical method as formulated by Pople and Beveridge [ 131: is it possible to modrfy (1) and yet to keep a theory that is both unbiased and simple? It will become clear that our proposed modifications result in energy expressions which are as unbiased as those which occur in present-day ND0 theories. A part of their simplicity, however, has to be sacrificed; as a compensation for the additional labour 399
Volume 54, number 3
CHEMICAL
we get energy expressions which show clearly that interactions in polyatomic molecules are more complicated and more varied than one might conclude from the formulae obtained from earlier ND0 theories.
2. Justifications of the ND0 approximation Our analysis will be based throughout on some consequences of the orthogonalisation procedure introduced by Ldwdm [ 141 and appiied by Pople [I], i.e. A.= *s-l/’
,
(2)
where K represents the non-orthogonal minimal basis set of Slater orbitals. It is well known that in the h basis all overlap integrals vanish, and that the neglect of all two-electron integrals except “(wlvv) is justitied, at least to first order in S, by Mulliken’s approximation [ 151 (PI “$S,“{(IyII
+bvl3 -
(3)
The most elaborate discussion of (2) and (3) is to be found in Fischer-Hjalmars’ analysis of the ZDO approximation [ 16]_ Before we proceed to apply (2) we consider a few alternative arguments for the elimination of all terms in (pvl from the energy expression. In the first place we mention the procedure preferred by Dewar [5,17], which is quite similar to the one applied in the very first version of the ZDO approxlmation [ 181. Without any refererlce to an orthogonalized basis set Dewar argues that in the LCAO MO version of the expression for the electronic energy E= (*IHl*)l(*liP),
(4)
the numerator is built up from atomic orbital multicentre integrals, whereas the overlap integrals appear in the denominator. In view of (3) we may assume that (4) remains approximately the same if we neglect both the overlap and the multicentre integrals. An argued exception is made for the one-electron integrals (ylh 1~); the most efficient explanation of this exception is that Mulliken’s approximation (3) does not apply to (JL1V2 Iv) because of the absence of a charge distribution (~.lvI. All surviving A0 integrals are now to be evaluated by means of purely empirical methods. In our opinion Dewar’s justification of the ND0 approximation is as good as Pople’s, except for one feature: it offers no guidance with regard to the next steps to be 400
15 March J978
PHYSICS LETTERS
taken; immediately
after the simplification of (4) we seas of sheer param-
are set adrift on the unmapped
etrisation. Now it may be argued that this separate discussion of Dewar’s treatment of (4) is superfluous, because it could be replaced by the simplification based on (2) without any change to his MIND0 method. Sure, but that is only part of the truth. From a semi-empirical point of view Dewar’s approach is so flexible that he might have effectively obtained the same conclusions as in the present paper, but for one incorrect statement in ref. [S] : contrary to Dewar’s expressed opinion it will appear in the following section that p can be made to depend on bond angles without a violation of the requirement of rotational invariance. In the second place we give some a_ttention to a possible refinemenr of (2). It is well known that the Mulliken approximation when applied to o-orbitals is less accurate than in the case of sr-orbitals. We may try to improve our justification of the neglect of differential overlap by replacing Sell2 in (2) by S-1/3U, the unitary matrix U being chosen in such a way that the equivalence of atomic orbit& coupled by a molecular symmetry operation is maintained. From the allowed matrices we can choose U such as to minimalise the inaccuracy of (3), a conceivable criterion being the minimalisation of the sum of the absolute values of the transformed (y lpo) with p # v and/or p # u. We have two objections against such an improved orthogonalisation. First, the unknown matrix U precludes the general analysis we wish to offer; second, the refinement is superfluous: many of the multicentre integrals may emerge from a standard orthogonalisation treatment with values of several tenths of an eV without seriously affecting the accuracy of an ND0 method. As an example we discuss the integrals Xblw) which occur in the energy formula E=CP,,
AhPa
with a coefficient P,,P,,_
If for any A0 ZJ,Pu, can be
assumed to be approximately constant for a large set of molecules in different states where the same JL is found on the neighbouring atom, then the energy term &~~~~(“I~JII) is effectively dependent on Pw done, vy hOly(vv) can be incorporated in the empiri-
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CHEMICAL
15
PHYSICS LETTERS
cal value of flflv. For further examples we refer to a similar analysis of energy terms in s-electron theory [ 193 _We mention in passing that the assumption of a constant Pvv in the discussion of such minor terms seems to be quite reasonabie for organic molecules. It should not be used when one compares different states
March
1978
one s-orbital only. In the given configuration S13 is considerably smaller than S17 and Sz3.
of metallo-organic systems, where extensive opposite migrations of n- and u-electrons occur frequently 1201.
3.
Long distance interactions in polyatomic molecules There is a perfectly sound principle underlying all
well-considered
parametrisation
methods
in semi-em-
pirical theories: the empirical parameter is expressed in terms of A0 integrals, the expression obtained is used in order to describe the parameter as a plausible function of the relevant variables, and the rest is a problem of calibration, i.e. fitting to a given, appropriate type of curve. This principle was correctly applied to a minority of chemical problems, viz. to diatomic molecules exclusively. If we can take it as established that &, is the empirical counterpart of “h 12, then it is easily found that, up to first order in S, Ddir = hhdir = W h,U lur
- t S,“($,
+ h”,),
In all semi-empirical methods the mutual orthogcnality of the orbitals X, and 5 is duly taken into account when PI3 1s computed on the basis of (6) and (7); however, the fact that both of them should be regarded as crthcgcnalised with respect to X7 IS not considered. In order to calculate %t3 we expand S-1/3 = (1 + A)-*/* up to second order in A ; maintaining all those terms in which no more than two off-diagonal elements of S and h occur we find that
+ it,,
-
$,,h,3 _
(7) We can also accept the working hypothesis that for cases other than those considered in ref. 1211 fldir is roughly proportional to SP,. What we have to show next is that in apolyatomic molecule the identification of fl and pdlr IS _ impossible; that it is, indeed, the oversimplification which is responsible for most of the failures of ND0 theories. Let us consider a triatomic molecule with a bondangle somewhere between 100” and 180” (smaller bondangles are more conveniently approached from the extreme case to be discussed in section 4)_ In order to keep both the derivation and the interpretation of the formulae as simple as possible we restrict the discussion to an elementaxy case: on each centre we consider
03)
using (6) we can rearrange this expression to yield x/l 13 =&
-$s,,Pg
+ ~Si3S~3(~l~~
where our superscript “dir” emphasizes that the equation applies to the direct interaction between two atoms. Eq. (6) was introduced and, for a few simple cases, evaluated by Mulliken [21] ; it is obvious that
_ ;
-$s&~ -
u12z +/l33)-
(9)
We see that Ah 13 is the sum of three positive and three negative terms, all of the same order of magnitude. It is impossible to predict off-hand even the sign of their sum; it is equally impossible to replace the expression by its first term, which is the only one that does not depend on atom 2. Any /&formula based on eq. (1) is invalid for the Interaction between non-neighbcur atoms. We conclude from (9) that the p which is to describe the interaction between second neighbcurs depends critically on aI1 elements that make up the geometry of the relevant triangle. Consequently, Dewar [S] would have been correct if he had introduced the bondangle 0 into his p-function. The idea that an angular dependence must violate the principle of rotational invariance is wrong: the interactions are not made to depend on the choice of the coordinate axes, but on the direction of the bonds. It is a consequence of these directed physical perturbations that the directions in an atom in a molecule are no longer equivalent;even s-crbitals lose their spherical symmetry on crthcgcnalisaticn. 401
Volume 54, number
CHEMICALPHYSICSLETTERS
3
It is another question whether the explicit introduction of 0 is expedient. We do not think it is, for two reasons. First, the angIes are redundant parameters: the geometry of a molecule is determ?red by the set of interatomic distances, which we need anyway for the calculation of the overIap integrals. Second, 8 will be a highly inconvenient parameter in polyatomic molecules where the Iine connecting atoms 1 and 3 is an eIement of more than one triangle_ There is some computational evidence in favour of the validity of (9).
In n-electron
theory we have a situ-
ation which is comparable with the one described above: we consider one orbital per atcm, and all these orbitals are equivalent. Chong [22J found that in benzene %h,3 = +0.07 eV, a calculation [23] of PI3 based on finderberg’s relation [24] /3 =R-‘ds/$R gave the same result for benzene, and values between +0.03 and to.07 eV for the various second-neighbour interactions in naphthalene. Needless to say, positive p’s are at variance with the prescriptions of ali established ND0 methods: they can give onIy negative vaIues, the order
of magnitude being -1 eV for second neighbours. Now if eq. (1) is fimdamentahy wrong, how can it possibly have served as the starting point for an impressive series of successful calculations? In order to answer this question we need not immediately invoke the freedom inherent to a parametrisation. A consideration of the formulae
erful compensation
will reveal the presence
or a pow-
mechanism.
On investigating a number of available CNDO, INDO and MIND0 results we found that the sum of the sec-
ond-neighbour terms PMy&,, was non-negligible and positive, as a consequence of the predominance cf small negative bond orders. For a medium-sized molecule the sum of the computed second-neighbour interactions can be as Iarge as 5-10 eV. If the few p-values calculated from duly orthogonalised orbitah are iepresentative, then the whole computed second-neighbour interaction energy is spurious, and in a theory which aims at an accuracy of a few kcaI/moIe this constitutes a considerabie error. How can it be corrected? To begin with, we can identify two errors with the opposite sign. The first-neighbour interactions and the one-centre integrals are given by (10) and (11) respectively, i.e. %l 13 =@
- 3sz,@
f&3523(hll 402
+h,,
- $,,@I? -a,,);
(10)
15 March 1978
(11) In Ah12 there is a large leading term which is defined in agreement with (1); there are five small correction dir , and three of them terms, each with a factor S,, or f313 are positive. Therefore, PI2 values which are on the whole slightly too negative can be conveniently formulated in agreement with (1). Similarly, if %, r is identified with a free-atom parameter we neglect at least the two following terms, which are positive. The overall situation can be summarized as follows: in the usual evaluation of the second-neighbour terms we introduce a spurious positive energy, due to a large error in &3 and to small negative bond orders. In the terms in “h12 and hhll we neglect small positive terms in integrals which are to be multiplied by large positive bond orders. Even if these errors do not cancel it is quite conceivable that their resultant can be roughly represented
as a function
of the number of bonds in the
molecule. If that is the case, then a simple adaptation of the parameters is sufficient to obtain highly accurate total energies. The two Iast terms in (11) deserve some attention. If in the triatomic molecule the charge distribution is fairly uniform (P,, = l), it is readily shown that these terms do not contribute
to the energy. Therefore,
in
many organic molecules their neglect is of no consequence. If, however, there are appreciable differences between the charge densities on the atoms, then the fina! terms in (11) become important_ The neglect of these terms may be one of the reasons why ND0 calculations are less convincing when applied to organometallic compounds 1201 and to the dative bond in - BH3NH3 [l i] _ It follows from the foregoing discussion that the suitability of an ND0 method for a molecule or a complex does not depend on the absence of errors but on the effectiveness of the compensation mechanism. Though a computational scheme may be reliable for the calculation of the total energies of many molecules, the incorrect partitioning of the energy is bound to lead to inconsistencies. The overestimation of the repulsion between second neighbours and its compensation by the one-centre and nearest-neighbour terms suggest a defiition of “those molecules for which the method was designed”: ND0 methods can be confident-
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CHEMICAL PHYSICS LETTERS
Volume 54, number 3
in the degenerate set of antibonding orbitals (or - fi), we find
ly applied to molecules with a uniform charge distribution and a certain equilibrium between the numbers of onecentre and medium-range interactions. An excess of the latter interactions is found in globular compact compounds; as can be expected from our analysis MIND0/3 underestimates their stability (fig. 3 of ref.
E(l) = (1 - S)%
PI I-
The corresponding equation obtained from ND0 methods based on (1) is: E&r
4. Equivalent interactions In clusters of atoms One special case remains to be investigated: how appropriate is the ND0 formalism in a configuration where first and second neighbours can no longer be distinguished, e.g. in an equilateral triangle? Once more we limit the investigation to a few simple problems whose results are open to interpretation. We consider a set of n identical atoms (n = 2,3,4), arranged in such a pattern that all pair interactions are equivalent: line, equilateral triangle, regular tetrahedron. Owing to the highly symmetrical structure of S we can evaluate S-L’2explicitly. With S,,P = 1 and Scly = S the eigenvalues of S are given by El=l+(n-l)s,
(12a)
e2=1
(12b)
-s,
the latter eigenvalue being (n - 1)-fold degenerate_ It is readily verified that the elements of S-1’2 are given by (S_l/?l)
PP
(s-1/2)py
=,-I
{ET”2
+ (0 - l)Q -112)
= n-1
{Q/2
_ gq
The elements of h are h,, S--L/2hS-112 are 01 f
;
_
(134 (13b)
-S)_’
[
h,,
-A%,,+
1 +(;Y
I)J;
(14a)
pdir fi = ‘hlzu =(I
-s)-’
+
= n(h pp + PE> -
(1%
(16)
From these two equations it becomes evident why (1) fads: no matter how we choose the parameters in (16) every attempt to fit (16) to reproduce (15) is doomed to fail beLause the second factor of (16) is nindependent. The best we can hope for is that the intersection of (15) and a calibrated (16) takes place near an integral value of II, so that at least one configuration of physical interest is correctly descrrbed by (16). Eq. (15) can be expressed in temrs which are entirely outside the scope of conventional ND0 methods: pair interactions in clusters decrease as the number of atoms in the cluster increases. The quantitative effects represented by (14) should not be underestimated: even for S as small as 0.1 it implies that 0 decreases by some 17% as we go from n = 2 to n = 4. It is not surprising that MIND0/3 underestimates the strain in small rings (fig. 2 of ref. [5] )_ Similarly. energies of reaction complexes may well be calculated at much too low values [l I] owing to the occurrence of atom clusters without a central atom, i.e. a large number of first-neighbour interactions_
5. Conclusions
and hflr,, and those of
'h,,
=(l
- Sh,,
lJv
It@-11)s
-
Wb)
Evaluating the one-electron energy E(l) for a set of n electrons in a configuration with two electrons in the bonding orbital {a + (n - 1)/3) and (n - 2) electrons
Before we sketch some possible quantitative developments of the ideas presented in this paper we summarize the qualitative conclusions that can be drawn from our analysis. The most pessimistic interpretation of eq. (9) and its consequence that j3r3 is considerably smaller than it is usually assumed to be implies that all semi-empirical calculations performed since 1965 can be discarded, except those of n-electron theoreticians, who have maintained for nearly 50 years that PI3 1s negligrble. In our opinion such a conclusion is an exaggeration_ Many reliable results have been obtained for ground state energies, and the accompanying SCF wavefunctions 403
Volume 54, number 3
CHEWCAL PHYSICS LETTERS
should be very close to those which are to be expected from a revision according to our suggestions: the proposed modificecions of the one-electron terms in the Hartree-Fock matrix elements can be regarded as rather weak perturbations. Probably in many cases the most interesting results will be obtained from revised calculations of some of the more delicate features, such as the potential field around a molecule, where calculations according to present-day ND0 methods are not invarrably successful 125,261 and the consequences of the proposed revision of the theory are not easily predicted. More interesting results can be expected for those molecules and complexes where the best available ND0 methods show unusually large deviations. The fact that the signs of several of these deviations can be explained by eqs. (9)-( 11) suggests that a revised computatron method, based throughout on a full recognition of the consequences of complete orthogonalisation, is definitely promising. The most intriguing aspect of (9) is, however, that is may well be the key to entirely new developments and applications of ND0 methods: eq. (9) is the first recognition in ND0 language of the fact that some interactions are weak, even at a distance of only a few A. Moreover, it states that such weak interactions are stereospecific: the interaction between atoms 1 and 3 is sensitive to the nature and the exact location of atom 2. A related conclusion can be drawn from section 4: the total interaction in a cluster cannot be written as a sum of unperturbed pair interactions. Now the interrelation of the concepts of weak interactions, stereospecificity and many-body interactions has been emphasized for many years among others by Jansen et al., in their investigations, in terms of indirect exchange, of such problems as crystal stability, magnetic ordering and rotational barriers [27] _ Hydrogen bonding is another example of a weak interaction, where manycentre terms and stereospecificity play an important role [28]_ The possibility to study such interactions by means of convenient and cheap ND0 methods can become valuable.
6. Towards
a parametrization
Now that the consequences of a duly performed orthogonalisation have been described we have to in404
15 March1978
dicate how the revised ND0 method can be implemented so as to produce numerical results. In the parametrisation process we can distinguish two steps: first, we have to find suitable expressions for the empirical A0 integrals in terms of appropriate variables, and next numerical values for the parameters in these expressions must be established. For the latter step the numerical fitting procedure described by Dewar [S] will be adequate; the first step, however, should be designed so as to accommodate the results of the foregoing analysis. It is evident that any ND0 method must admit a considerable empirical element in order to make up for its many deficiencies, such as the neglect of orbitalother core overlap, the approximation of the wavefuntion by a onedeterminantal function expressed in terms of a minimal basis, and the inaccuracy of the ND0 approximation itself. An attempt to formulate the consequences of eqs. (9)-( 11) in terms of a generally applicable empirical formalism should further satisfy the requirements stated before: it must be unbiased and simple. Just as in the earlier stages of the analysis we have no objections against the treatment of the two-electron ter_ms in earlier ND0 methods; a minimum set of rotation invariant TAR should be sufficient. In the derivation of the one-electron part of the Hartree-Fock matrix elements, however, we propose considerable deviations from traditional procedures. The MO’s are described as linear combinations of Ldwdin orbit&, so that the one-electron part of F must be an empirical version of integrals over the same orbitals. However, because these orbitals depend on the size and the geometry of the whole molecule, the corresponding integrals cannot be determined directly: they must be derived by the transformation b = R-‘/+$-l/”
(17)
from parametric matrix elements valid for the Slater orbitals, which can be regarded as invariant from one molecule to another. This treatment deviates strongly from the conventional one, where the MO’s are expressed in Liiwdin orbitals, and the one-electron terms in Fccc( and refer to pure orbitals partly
Volume 54.
CHEMICAL PHYSICS LETTCRS
number 3
transformation with S-L’2 is sufficient to obtain useful diagonal elements “hpp for the Lowdin basis: A
-
c BPA
(1%
ZBrAB*
Up, = - fj (Ifi + API -
(Z.4 -
-
(1%
The off-diagonal elements h,, require some care. First unlike jifiv, they cannot be regarded as local terms; Slater-type charge distributions (~1 interact strongly with the whole core. Further, if h,, were approximated by Mulliken’s formula h,, =$Sflv(hplr + h,,), then after orthogonalisation all p’s, even those between neighbours, would be quite small. On the other hand, we conclude from Mulliken’s correct treatment of the diatomic case that h,, is proportional to S,, because the same proportionality is valid for the two other terms in (6)_ We may therefore tentatively write h PV = i&,(h~fl
+ h,,)
f $,j-(cr,
n, R) t
f can be written as a constant Cam for a given pair of orbitals. The one-electron part of the Hartree-Fock matrix is now defined by (17)-(20). A final remark concerns the types of expression which can be used for f&u,v, R) in (20). in the simplest version of the comparable expression for fl in existent ND0 theories f is replaced by a constant for a given pair of atoms, without regard to the orbitals involved (CNDO: f = f (& + 0:)). The most detailed parametrisation, untried and undesirable, could attempt to establish one functron for each pair of atomic orbitals. MIND0/3 [S] steers a well-defined middle course between these two extremes by maintaining three elements in f: a constant factor, characteristic of the atom pair; a factor $ (I, + 1,) which distinguishes the individual orbital pairs, and, at least in principle, the option of a further R-dependence, though the exploration of the latter degree of freedom proved to be fruitless. We regard the introduction of experimental orbital characteristics as a useful addition to the flexibrlity of
15 hiarch 1978
MIND0/3, but we doubt whether the ionisation potentials I alone are sufficient. Mulliken’s argument [21] in favour of a dependence on 1, referring to a VB description of Hz in terms of non-orthogonal orbitals, is rathe. vague: fi is described in terms of a perturbation which is taken to be dependent on the energy of the electron in the unperturbed atom. In our opinion the finite probability of finding two electrons near any of the nuclei suggests that the perturbation should also be connected with an electron affinity. A similar conclusion can be drawn from the general form of the hamiltonian matrix elements which occur in VB tneory bdsed on orthogonalised orbitals [29]. In this approach &, appears exclusrvely in off-dragonal matrix elements HP4 where the configurationsp and 4 differ by one spin AO. It is readily derived that in a triatomic molecule ABC a matrix element flAB will be obtained from the following pairs of configurations: ABC and A+B-C, ABC and A-B%, ABiCand A+BC-, and A-BCt and AB-C+. It is seen that in this approach configurations with positive and negative A and B ions occur in exactly the same way, so that a description of flAB as a symmetrrcal functron of ionisation potentrals and eIectron affinities seems more likely to be correct than one in terms of I alone. The argument is quite close to the one used in CNDO/2 [ 131 for the approximation of U,, as a function of Ifi + A, we wish to account satrsfactorrly for the tendency cf an atomic orbital both to acquire and to lose electrons. A final choice between the various possible implementations of (20) can only be made on the basis of a comparison of the optimal numerical results to be obtained from each alternative. From our general analysis it appears that our modification of the ND0 methods is more consistent than its predecessors, and that it explains several of their specific shortcomings. The details of its actual elaboration and application will no doubt reflect once more the conflict between the two traditional enemies, convenience and accuracy, though we do not exclude the possibdity of their reconciliation_
References [ 11 J-A. Pople, D.P. Sentry and G-A. Segal, J. Chem. Phys. 43 (1965) S129. [2] J-A. Pople, D.L. Beveridge and P.A. Dobosh, J. Chem. Phys. 47 (1967) 2026. 405
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PHYSICS
R. Sustmann, J.E. Williams, M.J.S. Dewar, L.C. Allen and P. von R. Schleyer, J. Am. Chem. Sot. 91 (1969) 5350. [4] NC. Baud and M.J.S. Dewar, 3. Chem. Phys. 50 (1969) 1262. [S] R.C. Bingham. hf J.S. Dewar and D.H. Lo, J. Am. Chem. Sot_ 97 (1975) 1285. [6] XfJ.S. Dewar, J. Am. Chem. Sot. 97 (1975) 6591. [7] J.A. Pople, J. Am. Chem. Sot. 97 (1975) 5306. [S] WJ. Hehre, J. Am. Chem. Sot. 97 (1975) 5308. [9] H. Dits, N.Lf.M. Nrbbering and J-W. Verhoeven, Chem. Phys. Letters 51 (1977) 95. [IO] M-IS. Dewar and D. Landmann, J. Am. Chem. Sot. 99 (1977) 4633. [ 11 J S. Bantle and R. Ahlrichs, Chem. Phys. Letters 53 (1978) 148. [ 121 A.R. Gregory and M.N. Paddon-Row, J. Am. Chem. Sot. 98 (1976) 7521. [ 131 J.A. Pople and D.L. Beveridge, Approximate molecular orbital theory (McGraw-Hill, New York, 1970). [ 141 P.-Q. Loudin, J. Chem. Phys. 18 (1950) 365.
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[ 151 R.S. hfulltken, J. Chem. Phys. 46 (1949) 497. [ 161 I. Fischer-Hjalmars. J. Chem. Phys. 42 (1965) 1962. [ 171 M-IS. Deuar, The molecular orbital theory of organic chemistry (McGraw-Hf, New York, 1969). [18] R.C. Parr, J. Chem. Phys. 20 (1952) 1499. [ 191 S. de Bruijn, Theoret- Chim. Acta 17 (1970) 293. [20] P. Ros, Jerusalem Symposium on Quantum Chemistry and Biochemistry (1974) p. 207. [21] R.S. Mulhken, J. Phys. Chem. 56 (1952) 295. [22] D-P. Chong, hfol. Phys. 10 (!966) 67. 1231 S. de Bruijn, unpublished calculations. [24] J. Linderbeg, Chem. Phys. Letters 1 (1967) 39. [25] C. Giessner-Prettre and A. Pullman, Theoret. Chim. Acta 37 (1975) 335. 1261 E-N. Svendsen and T. Stroyer-Hansen. Theoret. Chim. Acta 45 (1977) 53. [27] L. Jansen and R. Block, Angew. Chem. Intern. Ed. 16 (1977) 294, and references therein. [ 281 A. Witkowsky, Mol. Phys. 29 (1975) 1441. 1291 R. McWeeny, Proc. Roy. Sot. A223 (1954) 63,306.