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Comments on the quantum theory of valence and bonding: Choosing between alternative definitions
~‘oloii~c 110. number 4
CHEMICAL PHYSICS LETTERS
5 October
3981
COMMENT COMMENTS ON THE QUANTUM CHOOSING
THEORY
BETWEEN ALTERNATIVE
OF VALENCE
AND BONDING:
DEFINITIONS
1. MAYER
iicc&xl
13 April 1981: in ;%a1 form 23 June 1984
Comments we rrude on the recent ose of a Lii-rvdin-orthogon;ilizcd basis for calculating bond orders and vafenccs in ztb initio SC!’ rhcory. Ir is concluded, based on an nnalpsis of connrctions with the first- and second-order density matrices, that our previous definition for bond orders 2nd valences is bctrcr founded theoretically and is free of artcfacts.
I - The alte~at~e
definitions
terrus of the original
(llol~-ort~logon~)
basis by the
usual formula
Recently, Natiello and Medrano ]I ] proposed the c&uiation of bond orders (“degrees of bonding”) and atonrio valences from ab initio SCF wavefunctions by performing a Liiwdin (S-uz) orthogonalization of the basis, transforming appropriately the density matrix and then applying the formulae introduced by Armstrong et al. [2] for the case of an orthonormal basis set. We note in this oonnection that the bond orders used in refs. [I ,2] were originally defined by Wiberg [3] and that, slightly earlier than Armstrong et al., the same formula for valence was proposed by Borisova and Semenov [4]_ The latter authors have also described [S] the procedure proposed by Natiello and Medrano [I], as early as in f 976. and applied it to a nutnber of different n~oleculcs at the extended Hiickei (EHT) level *’ _(Our prcscnt problem is in the proper handling of the overlap, and in this respect there is no essential difference between the EHT and ab initio wavefunctions.) Siruiiarly to ref. [ 11,we shall restrict our considerations to closed-shell systems; this, however, has no importance from the point of view of the problems discussed here. We defuse the closed-shell P matri.. in *I Unfortmutcly,
Borisova snd Semenov published their papers [4,5] in ZIjournal which may be difficult to fiid. I am grateful to Professor 0. Gistaiio (Sofii) for recently tailing my sttcntion to refs. fl,5]_
440
(1) where ci is the column vector formed by the LCAO coefficients of the ith occupied molecular orbital. We denote by S the usual overlap matrix of the A0 basis. Then the formulae used by Natielto and Medrano [l] (as well as by Borisova and Semenov [5]) can be cxpressed directly by means of quantities (elements of the matrices P and S) capsulated in the original nonorthogonal basis as: Bond order (“degree of bonding”) between atoms A and 3:
Valence of atom A:
A year ago I also proposed (in this journal) suitable def~itions of the bond order and valence indices for sb initio (or EHT) wavefunctions [o]. (For further discussions and applications, see refs.‘[7- lo].) These de~ni~ions differ from those used by Natiello and Medrano [i] but also go over to those of Wiberg [3], 0 ~09-~614~84~S 03.00 Q Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Volume 110, number 4
Armstrong
et al. [?I and Borisova and Semenov [4] orthonormal (i.e. the matrix S the unit matrix). The formulae I proposed
if the basis becomes
becomes are 163: Bond order: BXB = pgA
,c,
(4)
.
Valence:
The primes indicate that these quantities differ from those defined in eqs. (2) and (3j. As noted in ref. [6], in the closed-shell case the relationship (3) holds also for our definition of bond orders and valences_ (Our definitions are applicable also in the open-shell case, for which eq. (3) is not appropriate_) We feel that it is rather unsatisfactory to have dif_fcrcm deJ%tiom
for rlre same quarlrirics, especially
in view of the importance which bond orders and valences have in relating quantumchemical calculations to classical chemical concepts. This ambiguity can prevent the comparison of different calculations and may even be misleading, as the two sets of defrnitions lead to close but not identical results. (The results are identical only if the matrices S17’ and P are commutative [l 11, which is certainly not the case in general.) In such a situation, it seems very desirable to find criteria permitting one to choose between the above alternative definitions. One way is to compare numerical values. For example, the valence of the carbon in the methane molecule calculated with the 4-31G basis is much more reasonable if our definition [6] is used than that obtained by the alternative method of ref. [I] : 3.85 in ref. [6] as opposed to 3.43 in ref. [l] . However, the oppositive may also happen for some other molecule in a particular basis, so is obviously better to seek arguments of a theoretical character.
2. Connections
stand which of the two matrices S11zPS*12 and PS has the required physical significance. Wiberg’s bond index IU,,
(ps),y(ps)vp
with the density
5 OcrobL-r19S4
CHEMICAL PHYSICS LE-ITERS
matrices
The difference between the two sets of definitions that in eq. (2) the elements of the matrix SuzPSy2 are present, instead of the matrix PS used in eq. (4) Accordingly, the point is to under-
131
(6)
=
proposed ori@nally for an orthonornlalized (CNDO) basis. is essentially the archetype for both definitions
(eq. (2) and eq. (3)) of bond order - both reduce to it if the matrix S is a unit matrix. Although Wiberg perhaps defined his index on the basis of intuitive considerations, eq. (6) can be related by trivial algebra to the eschange part of the diatomic energy comPonent, obtained in the CNDO energy partitioning scheme [ 121 as SXCII-
-%I,
r
- -zYAB IUAB .
(7)
-yAB being the diatomic electron repulsion integral_ Relation (7) reveals thnr Wiberg’s bond index is connected with the exchange part of the second-order density matrks. As known [ 131, for (and only for) single-determinant wavefunctions the second-order spinless density matrix can be expressed in terms of the first-order matrix: P,(rl~‘~;‘1.‘I)=P1(‘1;‘;)P1(‘Zr’I) -
- :P~(~-,:r;)P,(~,;~~) When the two-electron
@I
part
of the total energy is calculated, the first term of(S) gives rise to the Coulomb energy and the second to the exchange energy. The latter “exchange density” term is normalized (cf. ref. [13]) as
(9) N being the number of electrons. Now, the first-order spinless density matrix can be written in the LCAO case in temis of the molecular P-matrix as
is essentially
x, being the basis atomic orbitals. Substituting (10) into (9) and performing the integrations, we arrive after some algebra at the expression 441
Volu:nc
I 10. IlnIIlber
CHEMICAL PHYSICS LETTERS
1
(11) According
the integrated
to (11).
the two-ceutre
“exchange
density”
contribution
to
is
For 311 ortl~ot~ortndized basis it equals the Wibcrg indes WAR. while in the general case one gets BykU ~ i.e. the bond order according to our defiitition, cq_ (4). It is in full accord witlt these considerations that this definition was originally obtained [6], based on an cuergy partitiouing forntul~ proposed [ 141 for the ab initio case. wliicli is similar in many respects to eq. (7) valid in the CNDO case ‘a. Note that there is no sucii connection with the escbange energy for the alternative deftnition (eq. (1)) implied by the procedure used in ref. [ 1] _ One may observe that similar derivations can be performed also for the Lbwdill-ortllogonalized basis; one then obtatins eq. (11) with matrix PS being replaced by S1/‘PSu2 everywhere. (I am indebted to tlie referee for calling my attention to the fact that this point requires explicit clarification.) Comparison v.ittt eq. (2) shows that the definition of bond order according to Natiello and hledrano [l] gives the twocenter contribution to the integrated exchange density for the pairs of “atoms‘- bearing, instead of the origiual non-ortltogonai atomic orbitals. their LBwdinorthogonalized counterparts. III other words, F!atiello and Medrano’s bond orders bold not between the original atoms but between their substitutes obtaiucd by Lbwdin-orthogonalization; of course, tltis
5 October 1984
nificance. It is enough to recall here, for example, that with them the overlap populations are identically zero, making it difficult to reflect in any simple fashion the effect of accumulation of electronic cliarge in the bonding regions. [See ref. [S] for a correlation between the overlap populations and the bond orders according to our definition, eq. (4)_] Another, somewhat related, argument in favour of our definitions is the following. The first-order spinless density matrix (10) is, up to a factor of 2, the kernel of the operator of projection into the subspace of the occupied molecular orbitals. In the LCAO framework, any orbital should be represented by the column vector of its expansion coefficients along tlie basis applied. In terms of such column vectors, the matrix performing the projection in question is $,P if the basis is ortltonormal and fPS in the general case. In this sense the matrices P and PS are the LCAO representations of the first-order density matrix for orthogonal and non-orthogonal basis sets, respectively. Therefore, if one wishes to generalize Wiberg*s bond indes for tlte ab initio (or EHT) case in a straightforward manner,
follows also from the very nature of the solienic applied. However, Lowdin-ortlrogonalized orbitals, although often very useful, represent nothing more than some auxiliary entities of purely mathematical sig-
then the elements Ppy of rite LCAO density matrix should be replaced by their non-orthogonal counterparts (PS)Bp. [Note that the matrix PS is not symmetric, so in general (PS),, f (PS),,, . Tlris is taken into account in the above equations.] Similar considerations apply also to valences. As Wiberg [3] pointed out, tbc actual bonding power of an orbital x, in a molecule can be given as b, = 2qp - c$, q, being the electron population in the given orbital. in fact, b, has a maximum if q, = 1, and bp = 0 for both a vacant and a doubly filled orbital. One has case and Q~ = (PS),,, i.e. ‘111= Ppp for the orthonormal hlulliken’s gross orbital population, in the non-orthogonal case. Now, summing the quantities b, for all the orbitals of atom A, but subtracting the sum of tire irltra-atomic bond orders having no chemical significance, we get [6] our definitions (5). No similar considerations apply to the alternative definitions used in ref. [l].
?a It sliould be noted that the corresponding equation (15) in wf. 161 has been misprinted: the factor 2 on the right-hand
3. Global effects
side should be l/2. Oi’c USC this opportunity also to correct the followin;: minur error in ref. 161: in the fourth lint. -CI+ _ rifhr coiunm of p. 271, rhc operator 4 should be x-a+. p ,ia
the fiitli tine front bottom, tar ci should br: c;.) 441
left Coluttin of p_ 274. the vcc-
Our definitions (eqs. (4) and (5)) were obtained [6] on tire basis of a thorough analysis of the non-orthogonal LCAO problem, connected with the introduc-
Volun~e 1 IO, number
4
CHEMICAL
PHYSICS
tion of a so-called “chemical” Hamiltonian [ 14]_ Natiello and Medrano [l] used a rather formal adaptation of formulae proposed originally for orthonormal basis sets. No doubt, the LBwdin-orthogonalization applied in ref. [I] is a straightforward approach in many respects; however, as with any ortliogonalization procedure. it is necessarily subjected to some global effects of the basis. As a consequence, such local quantities as bond orders and valences will be influenced by f3r-lying regions of the molecule, in addition to the factors included in the variational optimization of the wavefunction. The global effects involved in the procedure used in [l] can even lead to some spurious effects or artefacts. We shall illustrate this by considering 3 very simple model_ It does nor correspond to any reslistic practical problem, but is specially constructed in order to give some more immediate insight into the difference between the alternative definitions than the discussion of the formulae alone could provide. Let us consider the simples: LCAO MO wavefunction for the Hz molecule, by taking one 1s atomic orbital on each hydrogen. The valence of both atoms and the bond order between them are equal to unity according to both sets of definitions. If we formally introduce a third atom by adding a third hydrogcnic orbital to the basis, but kce@rg ir ctttpp, the wavefunction is not altered. (Whether or not 3 proton is also added at the third centre is immaterial for our problem.) After the b3sis is enlarged, the original basis function is no longer optimal in a variational sense. However. the matrix P can be calculated for non-variational functions as well, and the s3me holds for the bond orders and valences defined in its terms. One feels that a “correct” definition of the bond orders and valences must be such th3t these parameters should be uniquely determined by the actual wavefunction, irrespective of whether or not it is variational. Conceming our model, this means that, for the definitions to be adequate, the bond order between the original atoms and +eir valences must remain unchanged when the empty orbital is added to the basis. Similarly, it seems evident that one must attribute a zero valence and zero bond orders to the atom bearing only a fully empty orbital. Our definitions (eqs. (4) and (5)) fulfii these requirements. At the same time. the results obtained by using eqs. (2) and (3), corresponding to the procedure proposed by Natiello and Medrano [ I], are subject to 3 spurious influence of the empty orbital
LETTERS
5 October
19S4
as is illustmted in fig. 1 : when the ccIltre witit the empty orbital is moved, the bond orders and valences undergo significant changes u*irlw~r ~II~J* citatige iti rltc uctttai wm.efticttctiott : similarly. according to the procedure of ref. [I), a considerable valence is attributed to rhe ‘-atom” bearing the empty orbital only. NO doubt. the above model is artifici31; honever. it can illustrate the dangers connected with an uncontrolled use of LBwdin-orthogonalization. Furthermore_ this example may throw some li&t on tile origin of the troubles which Natiello and Medrano [1] encountered using double-zeta-plus-polarization (DZ + l’) type basis sets: in this case, there are a number of almost empty orbitals in the basis and the meaningless valences (e.g. 1.499 for H in H20) Natiello and Medrano obtained by using DZ + P bases c3n probably be attributed to rhe spurious influence of the basis orbitals having low population but considerable overlaps with the valence orbitals at the different centers. Such global effects are quite analogous to those discussed in our simple model are unavoidable consequences of the scheme applied in ref. [I]. References [l ]
MA Naticllo and J.A. hledrano.
Chcm. 105 (1984) 180. 131 D.R. Armstrong, P.G. Perkins snd J.J.P. Chcm. Sot. Dalton Trans. (1973) 838.
Phys.
Lsttcrs
Stcwltrd,
J.
443
Vohunr
J J 0. number 4
CHEMICAL PHYSICS LETTERS
J3f K.A. Wibcg, Tetrabcdron 24 (1966) 1083. [-I] N.P. BorBova and S.G. Semcno~, Vestn. Leningad Univ. (1973) No. 16, 119. [ 51 N.P. Borisova and S.G. S~I~ICIIOV, Vcstn. Leningrad Univ. (1976) h’o. 16, 98. [6] I. Mayer. Chem. Phys. Lctfcrs 97 (1983) 270. 171 I. RIsyer and XI. R&&z, Inorg. Cbim. Acta 77 (1983) L30.5. [8] 1. Mayer, Jntcrn. J. Quantum Chem. 16 f 1984) 151.
5 October 1984
(91 I. Mayer and P.R. Surj&r, Acta Chim. Hung., to be published. [lo] I. Mayer, to bc published. [ 1 l] P.R. Surjrfn and I. Mayer, unpublished results. [ 12j H. Fischer and Ji. Koflmar, Theorct. Chim. Acta 16 (1970) 163. [ 131 R. BfcWeeny and J3.T. Sutcliffe, bfetbods of InolecuJar quantum mechanics (Academic Press, New York, J969). [ 141 I. Mayer, Intcm. J. Quantum Chcm. 23 (1983) 341.
CHEMICAL PHYSICS LETTERS
5 October
3981
COMMENT COMMENTS ON THE QUANTUM CHOOSING
THEORY
BETWEEN ALTERNATIVE
OF VALENCE
AND BONDING:
DEFINITIONS
1. MAYER
iicc&xl
13 April 1981: in ;%a1 form 23 June 1984
Comments we rrude on the recent ose of a Lii-rvdin-orthogon;ilizcd basis for calculating bond orders and vafenccs in ztb initio SC!’ rhcory. Ir is concluded, based on an nnalpsis of connrctions with the first- and second-order density matrices, that our previous definition for bond orders 2nd valences is bctrcr founded theoretically and is free of artcfacts.
I - The alte~at~e
definitions
terrus of the original
(llol~-ort~logon~)
basis by the
usual formula
Recently, Natiello and Medrano ]I ] proposed the c&uiation of bond orders (“degrees of bonding”) and atonrio valences from ab initio SCF wavefunctions by performing a Liiwdin (S-uz) orthogonalization of the basis, transforming appropriately the density matrix and then applying the formulae introduced by Armstrong et al. [2] for the case of an orthonormal basis set. We note in this oonnection that the bond orders used in refs. [I ,2] were originally defined by Wiberg [3] and that, slightly earlier than Armstrong et al., the same formula for valence was proposed by Borisova and Semenov [4]_ The latter authors have also described [S] the procedure proposed by Natiello and Medrano [I], as early as in f 976. and applied it to a nutnber of different n~oleculcs at the extended Hiickei (EHT) level *’ _(Our prcscnt problem is in the proper handling of the overlap, and in this respect there is no essential difference between the EHT and ab initio wavefunctions.) Siruiiarly to ref. [ 11,we shall restrict our considerations to closed-shell systems; this, however, has no importance from the point of view of the problems discussed here. We defuse the closed-shell P matri.. in *I Unfortmutcly,
Borisova snd Semenov published their papers [4,5] in ZIjournal which may be difficult to fiid. I am grateful to Professor 0. Gistaiio (Sofii) for recently tailing my sttcntion to refs. fl,5]_
440
(1) where ci is the column vector formed by the LCAO coefficients of the ith occupied molecular orbital. We denote by S the usual overlap matrix of the A0 basis. Then the formulae used by Natielto and Medrano [l] (as well as by Borisova and Semenov [5]) can be cxpressed directly by means of quantities (elements of the matrices P and S) capsulated in the original nonorthogonal basis as: Bond order (“degree of bonding”) between atoms A and 3:
Valence of atom A:
A year ago I also proposed (in this journal) suitable def~itions of the bond order and valence indices for sb initio (or EHT) wavefunctions [o]. (For further discussions and applications, see refs.‘[7- lo].) These de~ni~ions differ from those used by Natiello and Medrano [i] but also go over to those of Wiberg [3], 0 ~09-~614~84~S 03.00 Q Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Volume 110, number 4
Armstrong
et al. [?I and Borisova and Semenov [4] orthonormal (i.e. the matrix S the unit matrix). The formulae I proposed
if the basis becomes
becomes are 163: Bond order: BXB = pgA
,c,
(4)
.
Valence:
The primes indicate that these quantities differ from those defined in eqs. (2) and (3j. As noted in ref. [6], in the closed-shell case the relationship (3) holds also for our definition of bond orders and valences_ (Our definitions are applicable also in the open-shell case, for which eq. (3) is not appropriate_) We feel that it is rather unsatisfactory to have dif_fcrcm deJ%tiom
for rlre same quarlrirics, especially
in view of the importance which bond orders and valences have in relating quantumchemical calculations to classical chemical concepts. This ambiguity can prevent the comparison of different calculations and may even be misleading, as the two sets of defrnitions lead to close but not identical results. (The results are identical only if the matrices S17’ and P are commutative [l 11, which is certainly not the case in general.) In such a situation, it seems very desirable to find criteria permitting one to choose between the above alternative definitions. One way is to compare numerical values. For example, the valence of the carbon in the methane molecule calculated with the 4-31G basis is much more reasonable if our definition [6] is used than that obtained by the alternative method of ref. [I] : 3.85 in ref. [6] as opposed to 3.43 in ref. [l] . However, the oppositive may also happen for some other molecule in a particular basis, so is obviously better to seek arguments of a theoretical character.
2. Connections
stand which of the two matrices S11zPS*12 and PS has the required physical significance. Wiberg’s bond index IU,,
(ps),y(ps)vp
with the density
5 OcrobL-r19S4
CHEMICAL PHYSICS LE-ITERS
matrices
The difference between the two sets of definitions that in eq. (2) the elements of the matrix SuzPSy2 are present, instead of the matrix PS used in eq. (4) Accordingly, the point is to under-
131
(6)
=
proposed ori@nally for an orthonornlalized (CNDO) basis. is essentially the archetype for both definitions
(eq. (2) and eq. (3)) of bond order - both reduce to it if the matrix S is a unit matrix. Although Wiberg perhaps defined his index on the basis of intuitive considerations, eq. (6) can be related by trivial algebra to the eschange part of the diatomic energy comPonent, obtained in the CNDO energy partitioning scheme [ 121 as SXCII-
-%I,
r
- -zYAB IUAB .
(7)
-yAB being the diatomic electron repulsion integral_ Relation (7) reveals thnr Wiberg’s bond index is connected with the exchange part of the second-order density matrks. As known [ 131, for (and only for) single-determinant wavefunctions the second-order spinless density matrix can be expressed in terms of the first-order matrix: P,(rl~‘~;‘1.‘I)=P1(‘1;‘;)P1(‘Zr’I) -
- :P~(~-,:r;)P,(~,;~~) When the two-electron
@I
part
of the total energy is calculated, the first term of(S) gives rise to the Coulomb energy and the second to the exchange energy. The latter “exchange density” term is normalized (cf. ref. [13]) as
(9) N being the number of electrons. Now, the first-order spinless density matrix can be written in the LCAO case in temis of the molecular P-matrix as
is essentially
x, being the basis atomic orbitals. Substituting (10) into (9) and performing the integrations, we arrive after some algebra at the expression 441
Volu:nc
I 10. IlnIIlber
CHEMICAL PHYSICS LETTERS
1
(11) According
the integrated
to (11).
the two-ceutre
“exchange
density”
contribution
to
is
For 311 ortl~ot~ortndized basis it equals the Wibcrg indes WAR. while in the general case one gets BykU ~ i.e. the bond order according to our defiitition, cq_ (4). It is in full accord witlt these considerations that this definition was originally obtained [6], based on an cuergy partitiouing forntul~ proposed [ 141 for the ab initio case. wliicli is similar in many respects to eq. (7) valid in the CNDO case ‘a. Note that there is no sucii connection with the escbange energy for the alternative deftnition (eq. (1)) implied by the procedure used in ref. [ 1] _ One may observe that similar derivations can be performed also for the Lbwdill-ortllogonalized basis; one then obtatins eq. (11) with matrix PS being replaced by S1/‘PSu2 everywhere. (I am indebted to tlie referee for calling my attention to the fact that this point requires explicit clarification.) Comparison v.ittt eq. (2) shows that the definition of bond order according to Natiello and hledrano [l] gives the twocenter contribution to the integrated exchange density for the pairs of “atoms‘- bearing, instead of the origiual non-ortltogonai atomic orbitals. their LBwdinorthogonalized counterparts. III other words, F!atiello and Medrano’s bond orders bold not between the original atoms but between their substitutes obtaiucd by Lbwdin-orthogonalization; of course, tltis
5 October 1984
nificance. It is enough to recall here, for example, that with them the overlap populations are identically zero, making it difficult to reflect in any simple fashion the effect of accumulation of electronic cliarge in the bonding regions. [See ref. [S] for a correlation between the overlap populations and the bond orders according to our definition, eq. (4)_] Another, somewhat related, argument in favour of our definitions is the following. The first-order spinless density matrix (10) is, up to a factor of 2, the kernel of the operator of projection into the subspace of the occupied molecular orbitals. In the LCAO framework, any orbital should be represented by the column vector of its expansion coefficients along tlie basis applied. In terms of such column vectors, the matrix performing the projection in question is $,P if the basis is ortltonormal and fPS in the general case. In this sense the matrices P and PS are the LCAO representations of the first-order density matrix for orthogonal and non-orthogonal basis sets, respectively. Therefore, if one wishes to generalize Wiberg*s bond indes for tlte ab initio (or EHT) case in a straightforward manner,
follows also from the very nature of the solienic applied. However, Lowdin-ortlrogonalized orbitals, although often very useful, represent nothing more than some auxiliary entities of purely mathematical sig-
then the elements Ppy of rite LCAO density matrix should be replaced by their non-orthogonal counterparts (PS)Bp. [Note that the matrix PS is not symmetric, so in general (PS),, f (PS),,, . Tlris is taken into account in the above equations.] Similar considerations apply also to valences. As Wiberg [3] pointed out, tbc actual bonding power of an orbital x, in a molecule can be given as b, = 2qp - c$, q, being the electron population in the given orbital. in fact, b, has a maximum if q, = 1, and bp = 0 for both a vacant and a doubly filled orbital. One has case and Q~ = (PS),,, i.e. ‘111= Ppp for the orthonormal hlulliken’s gross orbital population, in the non-orthogonal case. Now, summing the quantities b, for all the orbitals of atom A, but subtracting the sum of tire irltra-atomic bond orders having no chemical significance, we get [6] our definitions (5). No similar considerations apply to the alternative definitions used in ref. [l].
?a It sliould be noted that the corresponding equation (15) in wf. 161 has been misprinted: the factor 2 on the right-hand
3. Global effects
side should be l/2. Oi’c USC this opportunity also to correct the followin;: minur error in ref. 161: in the fourth lint. -CI+ _ rifhr coiunm of p. 271, rhc operator 4 should be x-a+. p ,ia
the fiitli tine front bottom, tar ci should br: c;.) 441
left Coluttin of p_ 274. the vcc-
Our definitions (eqs. (4) and (5)) were obtained [6] on tire basis of a thorough analysis of the non-orthogonal LCAO problem, connected with the introduc-
Volun~e 1 IO, number
4
CHEMICAL
PHYSICS
tion of a so-called “chemical” Hamiltonian [ 14]_ Natiello and Medrano [l] used a rather formal adaptation of formulae proposed originally for orthonormal basis sets. No doubt, the LBwdin-orthogonalization applied in ref. [I] is a straightforward approach in many respects; however, as with any ortliogonalization procedure. it is necessarily subjected to some global effects of the basis. As a consequence, such local quantities as bond orders and valences will be influenced by f3r-lying regions of the molecule, in addition to the factors included in the variational optimization of the wavefunction. The global effects involved in the procedure used in [l] can even lead to some spurious effects or artefacts. We shall illustrate this by considering 3 very simple model_ It does nor correspond to any reslistic practical problem, but is specially constructed in order to give some more immediate insight into the difference between the alternative definitions than the discussion of the formulae alone could provide. Let us consider the simples: LCAO MO wavefunction for the Hz molecule, by taking one 1s atomic orbital on each hydrogen. The valence of both atoms and the bond order between them are equal to unity according to both sets of definitions. If we formally introduce a third atom by adding a third hydrogcnic orbital to the basis, but kce@rg ir ctttpp, the wavefunction is not altered. (Whether or not 3 proton is also added at the third centre is immaterial for our problem.) After the b3sis is enlarged, the original basis function is no longer optimal in a variational sense. However. the matrix P can be calculated for non-variational functions as well, and the s3me holds for the bond orders and valences defined in its terms. One feels that a “correct” definition of the bond orders and valences must be such th3t these parameters should be uniquely determined by the actual wavefunction, irrespective of whether or not it is variational. Conceming our model, this means that, for the definitions to be adequate, the bond order between the original atoms and +eir valences must remain unchanged when the empty orbital is added to the basis. Similarly, it seems evident that one must attribute a zero valence and zero bond orders to the atom bearing only a fully empty orbital. Our definitions (eqs. (4) and (5)) fulfii these requirements. At the same time. the results obtained by using eqs. (2) and (3), corresponding to the procedure proposed by Natiello and Medrano [ I], are subject to 3 spurious influence of the empty orbital
LETTERS
5 October
19S4
as is illustmted in fig. 1 : when the ccIltre witit the empty orbital is moved, the bond orders and valences undergo significant changes u*irlw~r ~II~J* citatige iti rltc uctttai wm.efticttctiott : similarly. according to the procedure of ref. [I), a considerable valence is attributed to rhe ‘-atom” bearing the empty orbital only. NO doubt. the above model is artifici31; honever. it can illustrate the dangers connected with an uncontrolled use of LBwdin-orthogonalization. Furthermore_ this example may throw some li&t on tile origin of the troubles which Natiello and Medrano [1] encountered using double-zeta-plus-polarization (DZ + l’) type basis sets: in this case, there are a number of almost empty orbitals in the basis and the meaningless valences (e.g. 1.499 for H in H20) Natiello and Medrano obtained by using DZ + P bases c3n probably be attributed to rhe spurious influence of the basis orbitals having low population but considerable overlaps with the valence orbitals at the different centers. Such global effects are quite analogous to those discussed in our simple model are unavoidable consequences of the scheme applied in ref. [I]. References [l ]
MA Naticllo and J.A. hledrano.
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