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Reply to “comment on direct consequences of the bond index statistical interpretation”
Volume 141, number 5
CHEMICAL PHYSICS LETTERS
20 November 1987
COMMENT REPLY TO “COMMENT ON DIRECT CONSEQUENCES OF THE BOND INDEX STATISTICAL INTERPRETATION”
Myriam S. DE GIAMBIAGI, Mario GIAMBIAGI Centro Brasileiro de Pesquisas Fisicas, Run Xavier Sigaud 150, 22290 Rio de Janeiro, RJ, Brasil
and Paulo PITANGA Institute de Fisica. UniversidadeFederal do Rio de Janeiro, Ilha do FundEo, 21910 Rio de Janeiro, RJ, Brasil Received 2 October 1987
In this reply to the Comment by Jorge and Batista (Chem. Phys. Letters I38 (1987) 115) we show that the softness of an atom in a molecule is actually proportional to the atom’s MO self-charge, as stated before.
In a previous paper [ 11, Jorge and Batista claim that the atomic charge fluctuation - which in a former work signed Giambiagi, Giambiagi and Jorge [ 21 was related to the self-charge in the atom - is actually related to the atom’s valence. Ref. [ 31, which is the subject of their comment, merely links the mentioned fluctuation to atomic softness. When dealing with closed shells in ref. [ 21, we work with orbitals, not with spin orbitals, and this becomes explicit through the factor 2 in the first-order density matrix 2i7,
(1) If spin orbitals are used no factor 2 appears. Due to the idempotency of n [ 41, Mulliken’s charge qA (the mean value of the electronic charge operator d) may be written as [ 4,5]
(2)
where 4IAAis the self-charge of A and, ZABthe bond index between atoms A and B [4]. 466
Let us remind some basic formulae. If Vi is a MO for the ith eigenfunction and 9, (or 4”) the basis set functions, we may write either [ 61 ~j=~x’O#O (I
or
Y/l=~&,&.
(3)
b
Longuet-Higgins [7] reports these and the related formulae with covariant basis, but this is easily extended to the bi-orthogonal basis. Therefore
where i runs over all states. To these basis functions and molecular functions the corresponding creation-annihilation operators are associated. We have ~2~71
and for the diagonal part of the reduced second-order density matrix in the one-determinant approximation [ 7,8] (~~~~~-“~-b)=4(n,bn-~~~~).
(6)
The errorin formula (8) of ref. [ 1] was not to per-
CHEMICAL PHYSICS LETTERS
Volume 141. number 5
ceive that in the anticommutation relations [ 71 {V,
VA= 1 IA, .
{q&~-~}=~X,,Xjb{!P:, Ii
!P;>=s:
(7)
due to the completeness relation, the sum is carried over all MOs, and not only over the occupied ones [ 4,9]. It is entirely irrelevant that a and b belong or not to the same atom. Adding only over the occupied MOsdoesnotleadto6f:buttoIT~ (orIZ:ifa=b). Let us work out in detail how the factorization of the second-order density matrix in the onedeterminant approximation appears in the present formulation (~h’~~~-“~-b>=-(~b’~I:~-b9-“>
(8)
Introducing (5) and (6) in (8) we have =4l7;l?:-2l7:J7;.
(9)
Mayer mentioned that there was a factor 2 missing [ lo] in some of our formulae of ref. [ 21. Adding over
aeA, beB the left-hand member of (9) we had (IO)
which in fact, by (9), should read (d~~s--(b>(~~B=--fIAB.
’
(11)
We mentioned in ref. [ 31 that core electrons and lone pairs must be subtracted from the self-charge in (11) for, being additive constants in (2), they do not contribute to the fluctuation. The fluctuation in (11) is related to the softness S, of atom A in the molecule [ 31 s.4=P((831>-(&a2),
/f=llkT.
(12)
In ref. [ 11, due to the mentioned error, it is concluded that
where [5,11,12]
v
(15)
However, in this case one can always write (16)
which, following (15), is verified only for (Y= 1, i.e. qA= qB, If, instead, by (11) (17)
(qA/%)‘=IAAII~~=sAIsB
US)
confirming the appropriateness of our softness defynition in ref. [ 31. As a consequence of their association between valence and softness, the authors of ref. [ l] state that they “obtain satisfactory results for the hardness of any first-row element in a molecule”, increasing symmetrically from carbon to lithium and to fluorine. This disagrees with the known hardness parameters for atoms [ 14,151.
(10’)
As, once again, no restriction appears in (8) or (9) to a and b belonging or not to the same atom, we also have (431>-(6>‘=-U/l,4
($~>-(~AA>2=(Bs>-{4B>2-
it is easily seen that for any a, [ 131
=(~b’4-b~n’~-a)-nf:<~,+c-a>.
(~~A~)-(~~A)(4o)=-IAB,
Let us show, through a very simple example, that (13) leads to a fragrant contradiction. In a diatomic molecule AB where A and B contribute one orbital each, VA=I’,. If (13) was true, then
SA=-:BLl,
=-[-(~b’~-b~~~-“>+n~:~,‘~-a>l
(0:>-<&4A*==t*
(14)
B#A
V>=J,,,
(@;$-b@;@-“)
20 November 1987
(13)
References [ I ] F.E. Jorge and A.B. Batista, Chem. Phys. Letters 138 (1987) 115.
[ 21 M.S. de Giambiagi, M. Giambiagi and F.E. Jorge, Theoret. Chim. Acta 68 (1985) 337.
[ 3 ] P. Pitanga, M. Giambiagi and M.S. de Giambiagi, Chem. Phys. Letters 128 (1986)411. [ 41 M. Giambiagi, MS. de Giambiagi, D.R. Grempel and CD. Heymann, J. Chim. Phys. 72 (1975) 15. [5] I. Mayer, Chem. Phys. Letters 97(1983) 270. [ 6) MS. de Giambiagi, M. Giambiagiand F.E. Jorge, Z. Naturforsch. 39a (1984) 1259. [ 71 H.C. Longuet-Higgins, in: Quantum theory of atoms, molecules and the solid state, ed. P.-O. Liiwdin (Academic Press, New York, 1966) p. 105. [8] I. Mayer, Intern. J. Quantum Chem. 23 (1983) 341. [ 91 M.S. de Giambiagi and M. Giambiagi, Chem. Phys. Letters 1 (1968) 563. [lo] I. Mayer, Intern. J. Quantum Chem. 29 (1986) 73. [ 1I ] D.R. Armstrong, P.G. Perkins and J.J.P. Stewart, J. Chem. Sot. Dalton (1973) 838,2273.
467
Volume 14I +number 5
CHEMICAL PHYSICS LETTERS
[ 121N.P. Borisova and S.G. Semenov, Vestn. Leningr. Univ. No. 16 (1973) 119. [ 131 P. Pitanga, M.S. de Giambiagi and M. Ciambiagi, Quim. Nova, to be published.
468
[ 141 R.G. ParrandR.G.Pearson,
20 November 1987
J.Am. Chem. Sot. 105 (1983) 7512. [ 151 J.L. Gkquez, A. Vela and M. Gal&, in: Electronegativity, ed. K.D. Sen (Springer, Berlin, 1987).
CHEMICAL PHYSICS LETTERS
20 November 1987
COMMENT REPLY TO “COMMENT ON DIRECT CONSEQUENCES OF THE BOND INDEX STATISTICAL INTERPRETATION”
Myriam S. DE GIAMBIAGI, Mario GIAMBIAGI Centro Brasileiro de Pesquisas Fisicas, Run Xavier Sigaud 150, 22290 Rio de Janeiro, RJ, Brasil
and Paulo PITANGA Institute de Fisica. UniversidadeFederal do Rio de Janeiro, Ilha do FundEo, 21910 Rio de Janeiro, RJ, Brasil Received 2 October 1987
In this reply to the Comment by Jorge and Batista (Chem. Phys. Letters I38 (1987) 115) we show that the softness of an atom in a molecule is actually proportional to the atom’s MO self-charge, as stated before.
In a previous paper [ 11, Jorge and Batista claim that the atomic charge fluctuation - which in a former work signed Giambiagi, Giambiagi and Jorge [ 21 was related to the self-charge in the atom - is actually related to the atom’s valence. Ref. [ 31, which is the subject of their comment, merely links the mentioned fluctuation to atomic softness. When dealing with closed shells in ref. [ 21, we work with orbitals, not with spin orbitals, and this becomes explicit through the factor 2 in the first-order density matrix 2i7,
(1) If spin orbitals are used no factor 2 appears. Due to the idempotency of n [ 41, Mulliken’s charge qA (the mean value of the electronic charge operator d) may be written as [ 4,5]
(2)
where 4IAAis the self-charge of A and, ZABthe bond index between atoms A and B [4]. 466
Let us remind some basic formulae. If Vi is a MO for the ith eigenfunction and 9, (or 4”) the basis set functions, we may write either [ 61 ~j=~x’O#O (I
or
Y/l=~&,&.
(3)
b
Longuet-Higgins [7] reports these and the related formulae with covariant basis, but this is easily extended to the bi-orthogonal basis. Therefore
where i runs over all states. To these basis functions and molecular functions the corresponding creation-annihilation operators are associated. We have ~2~71
and for the diagonal part of the reduced second-order density matrix in the one-determinant approximation [ 7,8] (~~~~~-“~-b)=4(n,bn-~~~~).
(6)
The errorin formula (8) of ref. [ 1] was not to per-
CHEMICAL PHYSICS LETTERS
Volume 141. number 5
ceive that in the anticommutation relations [ 71 {V,
VA= 1 IA, .
{q&~-~}=~X,,Xjb{!P:, Ii
!P;>=s:
(7)
due to the completeness relation, the sum is carried over all MOs, and not only over the occupied ones [ 4,9]. It is entirely irrelevant that a and b belong or not to the same atom. Adding only over the occupied MOsdoesnotleadto6f:buttoIT~ (orIZ:ifa=b). Let us work out in detail how the factorization of the second-order density matrix in the onedeterminant approximation appears in the present formulation (~h’~~~-“~-b>=-(~b’~I:~-b9-“>
(8)
Introducing (5) and (6) in (8) we have =4l7;l?:-2l7:J7;.
(9)
Mayer mentioned that there was a factor 2 missing [ lo] in some of our formulae of ref. [ 21. Adding over
aeA, beB the left-hand member of (9) we had (IO)
which in fact, by (9), should read (d~~s--(b>(~~B=--fIAB.
’
(11)
We mentioned in ref. [ 31 that core electrons and lone pairs must be subtracted from the self-charge in (11) for, being additive constants in (2), they do not contribute to the fluctuation. The fluctuation in (11) is related to the softness S, of atom A in the molecule [ 31 s.4=P((831>-(&a2),
/f=llkT.
(12)
In ref. [ 11, due to the mentioned error, it is concluded that
where [5,11,12]
v
(15)
However, in this case one can always write (16)
which, following (15), is verified only for (Y= 1, i.e. qA= qB, If, instead, by (11) (17)
(qA/%)‘=IAAII~~=sAIsB
US)
confirming the appropriateness of our softness defynition in ref. [ 31. As a consequence of their association between valence and softness, the authors of ref. [ l] state that they “obtain satisfactory results for the hardness of any first-row element in a molecule”, increasing symmetrically from carbon to lithium and to fluorine. This disagrees with the known hardness parameters for atoms [ 14,151.
(10’)
As, once again, no restriction appears in (8) or (9) to a and b belonging or not to the same atom, we also have (431>-(6>‘=-U/l,4
($~>-(~AA>2=(Bs>-{4B>2-
it is easily seen that for any a, [ 131
=(~b’4-b~n’~-a)-nf:<~,+c-a>.
(~~A~)-(~~A)(4o)=-IAB,
Let us show, through a very simple example, that (13) leads to a fragrant contradiction. In a diatomic molecule AB where A and B contribute one orbital each, VA=I’,. If (13) was true, then
SA=-:BLl,
=-[-(~b’~-b~~~-“>+n~:~,‘~-a>l
(0:>-<&4A*==t*
(14)
B#A
V>=J,,,
(@;$-b@;@-“)
20 November 1987
(13)
References [ I ] F.E. Jorge and A.B. Batista, Chem. Phys. Letters 138 (1987) 115.
[ 21 M.S. de Giambiagi, M. Giambiagi and F.E. Jorge, Theoret. Chim. Acta 68 (1985) 337.
[ 3 ] P. Pitanga, M. Giambiagi and M.S. de Giambiagi, Chem. Phys. Letters 128 (1986)411. [ 41 M. Giambiagi, MS. de Giambiagi, D.R. Grempel and CD. Heymann, J. Chim. Phys. 72 (1975) 15. [5] I. Mayer, Chem. Phys. Letters 97(1983) 270. [ 6) MS. de Giambiagi, M. Giambiagiand F.E. Jorge, Z. Naturforsch. 39a (1984) 1259. [ 71 H.C. Longuet-Higgins, in: Quantum theory of atoms, molecules and the solid state, ed. P.-O. Liiwdin (Academic Press, New York, 1966) p. 105. [8] I. Mayer, Intern. J. Quantum Chem. 23 (1983) 341. [ 91 M.S. de Giambiagi and M. Giambiagi, Chem. Phys. Letters 1 (1968) 563. [lo] I. Mayer, Intern. J. Quantum Chem. 29 (1986) 73. [ 1I ] D.R. Armstrong, P.G. Perkins and J.J.P. Stewart, J. Chem. Sot. Dalton (1973) 838,2273.
467
Volume 14I +number 5
CHEMICAL PHYSICS LETTERS
[ 121N.P. Borisova and S.G. Semenov, Vestn. Leningr. Univ. No. 16 (1973) 119. [ 131 P. Pitanga, M.S. de Giambiagi and M. Ciambiagi, Quim. Nova, to be published.
468
[ 141 R.G. ParrandR.G.Pearson,
20 November 1987
J.Am. Chem. Sot. 105 (1983) 7512. [ 151 J.L. Gkquez, A. Vela and M. Gal&, in: Electronegativity, ed. K.D. Sen (Springer, Berlin, 1987).