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Charge, bond order and valence in the AB initio SCF theory
Volume 97. numbrr 3
CHARGE.
CHEMICAL
BOND ORDER AND VALENCE
PHYSICS
LETTERS
20 May 1983
IN THE AB INITIO SCF THEORY
An opcr-ltor oi_&n~ic ci~rge is introduced. the expectation values of which are blullikn’s gross atomic rhc indh idu.d atom\ Suitable definitions of the bond order (multiplicit)‘) inde\ zmd of the vslence number moksule XC .&o proposed for ths SCF LCAO MO method. (The rcsulrs apply al%o in the extended Hhckel
1_ Introduction
call be written
The intclpret-ltion oi results obtkned in ab initio c~u~s~tu~~~ihcmical c.rlculations requires the use ofdiffcrenr quantities \\lticlt ma> easily be related to _eenuine cl~~nwz~l concepts. The atomic charge. the bond order (multiplicity) and the valence number of .III at0111 in the nwleculc could undoubtedly be such parameters. ilowevrr. of these three quantities only the atomic ch.trge is used in practice ‘. and usually its definition is also considered as being somewhat arbirrary. Accordingly. the aim of the present note is to aye 111.11 1x1 Ihe L<‘AO fr.unework hiulliken’s gross 3rcvnis
populstion
113s 3 privileged
importance
among
detinitions of the .uomic charge. as well JS to propose suitable detinitions for the bond orders d 1‘1~.dcnce numbers in the SCF oh initio LCAO h10 theory. possible
XI>
7. Opemtor Cansid~ring
of atomic charge
complete orthonorm~hzed sp.rtiJ orbit& Q,(r) and the cor-
~11 .irhitrdr)
set oionc-rlectrar~
responding set ol~spiu-orbitds $J~(J-. s) = zjg(r) a(s): a = Q or /3. the operator of the number ofelectro~ls
in the second quantization
populations of an atom method.)
on in 3
formalism
as
where we use Longuet-Higgins’ notations 6:’ and ‘Q- for the creation and annihilation operators, perVP mitting us to distingish between the operators and the spin-orbit& [ I] _ III practice, of course. one uses linite basis sets. In the following we shall assume that the molecular “LCAO” basis consists of 111. generally non-orthogonal. spatial orbirals xy, each of which can be assigned to one of the atoms A in the molecule; such an assignment may simply be denoted as I, EA. Now, one may assume with no loss of generality that the infinite set of orthonormalized orbitals tip can be divided into III orbitsls o( = 1, 2, ____ HZ) which lie completely in the subspace of the LCAO basis orbitals x,, and the remaining orbitals which are ort?logonal to this subspnce. Obviously, only those terms of the sum in (1) which correspond to the former type of orbitals eP;p = 1. 2. A._.III give a non-zero result when applied to an arbitrary LCAO wavefunction (i.e. to an arbitrary wavefunction built up by using the m one-electron orbitals .yy only)_ Accordingly, the operator of the number of electrons in the LCAO case reduces to
(2) 270
0 009-26 14/S3/0000-0000/S
03.00 0 1983 North-Holland
i.e. the summation is restricted to the m orbitals lying in the subspace of the LCAO basis. These m orthonormalized orbitals *fi should span the same subspace as the non-orthogonal LCAO basis orbitals xy, but otherwise can be selected arbitrarily. We may therefore use Liiwdin-orthogonalized orbit&, i.e. write
0) where the matrix S is the overlap matrix with the elements Spy =
where f denotes the adjoint. Correcting in ref. [l] we have the anticommutation 1x;*,
$)
= x$x$-
+ &-x;+
a minor error relationship
= S,J6,,.
,
$5
= 8U”600* 1
hold, as linear combinations m
(6) of the operators
>?E- :
In this respect the non-orthogonality of the basis mani fests in the fact that the “true” annihilation operators $are no more the adjoints of the creation operaton ;a+ *P Another formal possibility to introduce tile same
-Urnis to consider the biorthogonal set of operators qP orbitals ,,I qp =
uGlW9,,X” -
which span the same subspace as the orbitals x, (or +J and define the creation and annihilation operator: $E’ and (3,“- corresponding to the biorthogonal orbita analogously to eq. (4), but replacing x by 9 and S1i2 b S112 everywhere [2]_ This means that in the case of a non-orthogonal basis the annihilation operator Gidefined with respect to the biorthogonal spin-orbital I$ is essentially the “true” annihilation operator for the spin-orbital xE_ as it acts according to eq. (6) exactly in the manner as the usual annihilation operators do in the orthogonal case. Now. substituting in (2) the creation operators 4:’ expressed by the opemtors 2;’ and the annihilation operators expressed by means of the operators $-. trivial algebra shows that the operator of the number of electrons decomposes in the most natural way into the sum of the operators of atomic charge BA:
(5)
which is not very convenient from a practical point of view. In fact, while operators XE’, when applied to the vacuum state IO), “create” an electron in the state x; = x,(r) a(s), the operators i;do not behave as the annihilation operators do in the usual case of orthonormalized orbitals: the anticommutator (5) implies that -*l'-~z'._.~;q-J), when XE- acts on a determinant x1 xl the result is loot a determinant for N - 1 electrons (or zero if xs $ {$;i= 1,2, ._.,N)) but is a sum ofN such determinants_ However, it is easy tc check that in the non-orthogonal case one can define also “true” annihilation operators L$- for which usual fermion anticommu;ation rules t>i;+.
20 hlay 1983
CHEMICAL PHYSICS LETTERS
Volume 97, number 3
(9) with
(10) It is easy to see that a Heitler-London-type wavefunction (VB with no ionic terms) constructed by puttingIzA electrons on the orbitals belonging to fhe atom A will be an eigenfunction of the operator IVY with the eigenvalue ~2~ _
3. Expectation
value: Mulliken’s population
Now, we shall consider a usual SCF LCAO MO wavefunction I\k>built up of the doubly filled spatial molecular orbitals (MOs) ci$ = x:=1 ciep x~_ In terms of the correspondingcreation operators Q1?= Z&C~_~ the (closed-shell) many-electron determinantal xx;+, 271
vohIme
97. nunIbfr
\bxvefunction
CHEMICAL
3
j\k) forN
=
ZJZ
electrons
can
PHYSICS
be written
as I’k) = <~~*+~‘h~?@+ Applying
.._ (i-;*g+r0>
operator
_
(11)
(10) to I*). it is easy lo ~22 thst
.V
(13 wlicle iQ: +I” -+ xE> denotes 1112X-electron determinsnr~l ~aveiuncfion in which the n~olecul~r spinorbit4 (I);” is replaced by the atomic spin-orbital $_ Tht bno~n formul~2 [3] give for the overlap of two d2t2rmin~nts i\I’) Jnd 19; gl$’ - x,“> differing in on2 orhir;tl-
LJTTERS
20 hhy 1983
characteristic of the hlulliken population anaiysis. is essentially not an arbitrary choice but is just that whit is consistent with the internal structure of the LCAO hi0 formalism_ The fact that Mulliken’s populations are often strongly basis dependent does not invalidate this conclusion: one may only hope that the basis dependence of the Mulliken charges can be reduced by taking properly into account the interatomic “basis extension” effects defined recently [2]_
4_ Bond order
and valence
In studying the decomposition of the LCAO hamiltonian into terms of different physical significance ant the related energy partitioning scheme for the SCF wavefunctions. atomic
f3y ubiiig ( ! 3) lid raking into account that (*I*) = 1_wt‘ hvr for the expectarion vJlu2 of & :
we found
electrostatic
[2]
that
the leading
term (corresponding
clirtrge approsimation)
contains
inter-
to the point-
an exchange
compo-
nent
This term originates from the inequality
I c’_lilt*t~_vptvIafioilvalue C\!_I) t>j-the operator
of die
UtOJJliCpO[JlthIiotJ (1.: OJI fitcgiwJt atom . A more formal proof showing th.31 this result is not resrrictcd 10 the SCF level. hui IS \Jlid lix .my LCAO-1) pe wavefunction (c-g_ for (‘I- 01 VB-ty;-r l’.‘d\cfunctions 100) will br given elsev&xc [ 21. In ( 14) \-.r‘ 1i.1~ inrruduced Ihe usual = closed-shell P rii.3rrix i with the 212m21ifs P 5:: l 2~~pcL,,_ 11o1eht fhc present notati% differ wnw\xh~t l;om lhose in ref. [2]_
‘Jt‘JJJ;iL dlUrgc
Tl12 tlw~e
B. Now.
we
may
define
the quantity
-Q-q iS ~%iltiliA-CJJ'SgrCJSS
J1131~s1s shms
o\crrlJp popul~rioil
beIw22n
that the %ilving” 1112 two atoms
of rhe
in question.
(16) as the bond order index
between
can prove
B__1Bhas integer
to those
that
predicted
the specific treated
the index
by the classical
case of some
at the minimal
in general,
for those
basis
systems
orbitals orbitals,
respectively. order
dependence; for similar
bonds
localized
inner&hell case,
more
bonding
of course,
or less deviate
values as well as exhibit however,
one-
or lone-pail
covalent
zmpolarizeci
son of the values obtained
in
the wavefunc-
strictly
In the general
one expects,
picture
diatomics
for which
index B_dB may
the integer
On2
equal
level (as H,. N,, F2) and,
atomic
and two-center
the bond from
orbit&:
values
valence
homonuclear
tion is built up of orthonormal and two-center
A and B.
atoms
that
some
basis
the compari-
by using the same basis set
in a series of related
molecules
may
CHEMICAL
Volume 97, number 3
20 hfay 1983
PHYSICS LETTERS
alization
be instructive and give some insight into the details of the bonding situation in the systems studied- The deviation from the integer values may be caused by the non-orthogonality and delocaIization effects (like “tails” of the localized orbit&) and, especially, by the partial ionic character of the bonds formed between atoms of different electronegativities. One may observe that the bond order index BAs proposed here can be considered as an ab initio gener-
of Wiberg’s bond index
[4] which has proved
very useful in semi-empirical methods using different IeveIs of the ND0 (neglect of differential werlap) approximations * : our definition (16) of the BAB bond ’ One mw observe &at kl the CNDO method there is also B most intimate connection between W&erg’s bond index I$‘~B I4 j and the eschange part of the diatomic energy corn ponent IS]: EAB,exch_ = -~7__&‘&& -GAB being the inter atomic electron-electron repulsion inte_ml.
Table 1 Illustrative values of bond order and valence indices for some small molecules calcuhtcd approxim&e mokcular geometries Molecule
Bond orders
basis sets nnc
Valence numbers
bond
ST-O-3G
+31G
atom
Hz
H-H
1.0
1.0
H
1.0
methane
C-H
0.99
0.96
C H
3.65 0.9-I
C-N
0.98 1.01
0.96 O-94
c H
3.97 1.00
3.78 0.93
0.98 2.01
0.96 1.96
C li
3.97 1.00
3.55 0.94
C-H
c-c
0.98 3-00
0.86 3.27
C ti
3.9s 0.99
4.16 0.89
C-H
0.98
0.95
c tow C free H total H free
3.96
1.02 1.00 0.005
3.87 1.01 0.95 o.oO.4
n
1.91 0.97
1.61 b) 0.80 b)
ethane
C-C etllyIene
C-H
c-c accthylenc methyl radical
water
O-H
3)
0.80 b,
0.95
0
STO-3G
l-31G
Ci!
c-c
3.39
334
c
3.39
3.34
N2
N-N
3.0
2.67
N
3.0
2.67
co
c-o
2.52
2.14
C 0
2.52 2-52
2.14 2.14
H-C C-N
0.97 2.99
0.86 2.91
H C N
0.98 3.96 3.00
0.87 3.77 2.92
CN- ion
C-N
2.88
2.75
C N
2.88 2.88
7.75 2.75
CN radical
C-N
2.80
d
C totd C free N total N free
3.81 1.00 3.09 0.29
Cl
HCN
‘)
by using STO-3G and ?-31G
STO-6G resuits.
b, 6-31G results.
‘1 Does not converge.
273
\i~luxnc
9 i. nwuhcr
3
CHEMICAL
PHYSICS
inde\ goes wer IO thdr of thecorresponding Wibeg rf the ovrrlrlp matri.. S becomes the unit matrix. AI the s.nnc rime our bond index B,,B cm also be used
Index
lrl rhr exrsrldrd Htickl method. for inll~ces .we not apphcahle_ U?- pIerAihg Wibcrg‘s idea (set p. 10~0 in rei_ [-I], \xc may consider = 2(PS) a< (PS$, _ .I . measure of
which Wiberg’s rhe foornore
on
the quantity 6, the bonding ptwcr ~s-t’,n orbir31 in :! given 11w!ax11~: b, is equal to one try iiie t>rblldl \\irh n ,\lullIkcn population of unity .md f.11:~IV _wro ior lx~111 ;I vx.mt and ~1doubly filled ~H?M~_I!. 11 IS CJS) to see 111~1for .I closed-shell system
LETTERS
6-A = VA -
70 May 1983
c
BAB _
B (Be_-1)
of the total actual valence of the atom. eq. (17). and the sum of the bond orders formed by it. We note hen that the valence numbers VA and F_4 defined above may be considered as ab initio generalizations of the quantities discussed in ref. [6] for the CNDO level. Like the bond orders. both V, and FA assume in-
teger values in the specific cases discussed above. In the general case they are expected to have values close to those predicted by rhe classical valence concepts and may be used to link the ab initio SCF (or the extended Hiickel) results with rhe latter. Table 1 contains the results of illustrative calculations for some small molecules. We note :hat the somewhat
I ,=2
c
f.’ -1
(PS),,
c
fi: z-1
w3,_.(W,,~
_
(17)
(19)
smaller
numerical values obtained for the 43 I G basis may immediately be related with the known “overpolarization” of the bonds characteristic of this basis set. The question how lo generalize the bond order and valence indices lo the case of refined (as CL etc.) wave funcrions requires further investigation. Nevertheless. it seems that even at the SCF lcve! useful information mdy be gained by calculating these parameters: an application for discussing the behaviour of the “hypertalent-‘ sulphur atom in some sulphonyl chloride molecules will be described elsewhere.
References (1s) II C. Lonp~et-l&ins. in: Qumtuin throry of ;IIoms. molcculcs and ~hc solid stdtc. cd. P.-O. Lixldin (Academic l’rcs. Xes York. 1966) p_ 105. 1. XI> cr. f‘ourth 1nternation.d Congrc\s in Qu.mtum Chsmistr>. Upps~l~.June 1981; Intern. J. OUCII~~UI~ Cheln 10 be publkhed. P.-O. LG\rdin. Phys. Kev. 97 (1955) 1971. K;.A. U’iberg. TetrAedron 2-l ( 1966) 1083. 11. i‘ischer .md 11. Kiollrnsr,Theorer. Chin]. ActA 16 (1970 163. D.R. Armstrong. P.C. Perkins snd J_J.IDs Srewxt. J. Chem. SW. D.dIon ( 1973) 838.
CHARGE.
CHEMICAL
BOND ORDER AND VALENCE
PHYSICS
LETTERS
20 May 1983
IN THE AB INITIO SCF THEORY
An opcr-ltor oi_&n~ic ci~rge is introduced. the expectation values of which are blullikn’s gross atomic rhc indh idu.d atom\ Suitable definitions of the bond order (multiplicit)‘) inde\ zmd of the vslence number moksule XC .&o proposed for ths SCF LCAO MO method. (The rcsulrs apply al%o in the extended Hhckel
1_ Introduction
call be written
The intclpret-ltion oi results obtkned in ab initio c~u~s~tu~~~ihcmical c.rlculations requires the use ofdiffcrenr quantities \\lticlt ma> easily be related to _eenuine cl~~nwz~l concepts. The atomic charge. the bond order (multiplicity) and the valence number of .III at0111 in the nwleculc could undoubtedly be such parameters. ilowevrr. of these three quantities only the atomic ch.trge is used in practice ‘. and usually its definition is also considered as being somewhat arbirrary. Accordingly. the aim of the present note is to aye 111.11 1x1 Ihe L<‘AO fr.unework hiulliken’s gross 3rcvnis
populstion
113s 3 privileged
importance
among
detinitions of the .uomic charge. as well JS to propose suitable detinitions for the bond orders d 1‘1~.dcnce numbers in the SCF oh initio LCAO h10 theory. possible
XI>
7. Opemtor Cansid~ring
of atomic charge
complete orthonorm~hzed sp.rtiJ orbit& Q,(r) and the cor-
~11 .irhitrdr)
set oionc-rlectrar~
responding set ol~spiu-orbitds $J~(J-. s) = zjg(r) a(s): a = Q or /3. the operator of the number ofelectro~ls
in the second quantization
populations of an atom method.)
on in 3
formalism
as
where we use Longuet-Higgins’ notations 6:’ and ‘Q- for the creation and annihilation operators, perVP mitting us to distingish between the operators and the spin-orbit& [ I] _ III practice, of course. one uses linite basis sets. In the following we shall assume that the molecular “LCAO” basis consists of 111. generally non-orthogonal. spatial orbirals xy, each of which can be assigned to one of the atoms A in the molecule; such an assignment may simply be denoted as I, EA. Now, one may assume with no loss of generality that the infinite set of orthonormalized orbitals tip can be divided into III orbitsls o( = 1, 2, ____ HZ) which lie completely in the subspace of the LCAO basis orbitals x,, and the remaining orbitals which are ort?logonal to this subspnce. Obviously, only those terms of the sum in (1) which correspond to the former type of orbitals eP;p = 1. 2. A._.III give a non-zero result when applied to an arbitrary LCAO wavefunction (i.e. to an arbitrary wavefunction built up by using the m one-electron orbitals .yy only)_ Accordingly, the operator of the number of electrons in the LCAO case reduces to
(2) 270
0 009-26 14/S3/0000-0000/S
03.00 0 1983 North-Holland
i.e. the summation is restricted to the m orbitals lying in the subspace of the LCAO basis. These m orthonormalized orbitals *fi should span the same subspace as the non-orthogonal LCAO basis orbitals xy, but otherwise can be selected arbitrarily. We may therefore use Liiwdin-orthogonalized orbit&, i.e. write
0) where the matrix S is the overlap matrix with the elements Spy =
where f denotes the adjoint. Correcting in ref. [l] we have the anticommutation 1x;*,
$)
= x$x$-
+ &-x;+
a minor error relationship
= S,J6,,.
,
$5
= 8U”600* 1
hold, as linear combinations m
(6) of the operators
>?E- :
In this respect the non-orthogonality of the basis mani fests in the fact that the “true” annihilation operators $are no more the adjoints of the creation operaton ;a+ *P Another formal possibility to introduce tile same
-Urnis to consider the biorthogonal set of operators qP orbitals ,,I qp =
uGlW9,,X” -
which span the same subspace as the orbitals x, (or +J and define the creation and annihilation operator: $E’ and (3,“- corresponding to the biorthogonal orbita analogously to eq. (4), but replacing x by 9 and S1i2 b S112 everywhere [2]_ This means that in the case of a non-orthogonal basis the annihilation operator Gidefined with respect to the biorthogonal spin-orbital I$ is essentially the “true” annihilation operator for the spin-orbital xE_ as it acts according to eq. (6) exactly in the manner as the usual annihilation operators do in the orthogonal case. Now. substituting in (2) the creation operators 4:’ expressed by the opemtors 2;’ and the annihilation operators expressed by means of the operators $-. trivial algebra shows that the operator of the number of electrons decomposes in the most natural way into the sum of the operators of atomic charge BA:
(5)
which is not very convenient from a practical point of view. In fact, while operators XE’, when applied to the vacuum state IO), “create” an electron in the state x; = x,(r) a(s), the operators i;do not behave as the annihilation operators do in the usual case of orthonormalized orbitals: the anticommutator (5) implies that -*l'-~z'._.~;q-J), when XE- acts on a determinant x1 xl the result is loot a determinant for N - 1 electrons (or zero if xs $ {$;i= 1,2, ._.,N)) but is a sum ofN such determinants_ However, it is easy tc check that in the non-orthogonal case one can define also “true” annihilation operators L$- for which usual fermion anticommu;ation rules t>i;+.
20 hlay 1983
CHEMICAL PHYSICS LETTERS
Volume 97, number 3
(9) with
(10) It is easy to see that a Heitler-London-type wavefunction (VB with no ionic terms) constructed by puttingIzA electrons on the orbitals belonging to fhe atom A will be an eigenfunction of the operator IVY with the eigenvalue ~2~ _
3. Expectation
value: Mulliken’s population
Now, we shall consider a usual SCF LCAO MO wavefunction I\k>built up of the doubly filled spatial molecular orbitals (MOs) ci$ = x:=1 ciep x~_ In terms of the correspondingcreation operators Q1?= Z&C~_~ the (closed-shell) many-electron determinantal xx;+, 271
vohIme
97. nunIbfr
\bxvefunction
CHEMICAL
3
j\k) forN
=
ZJZ
electrons
can
PHYSICS
be written
as I’k) = <~~*+~‘h~?@+ Applying
.._ (i-;*g+r0>
operator
_
(11)
(10) to I*). it is easy lo ~22 thst
.V
(13 wlicle iQ: +I” -+ xE> denotes 1112X-electron determinsnr~l ~aveiuncfion in which the n~olecul~r spinorbit4 (I);” is replaced by the atomic spin-orbital $_ Tht bno~n formul~2 [3] give for the overlap of two d2t2rmin~nts i\I’) Jnd 19; gl$’ - x,“> differing in on2 orhir;tl-
LJTTERS
20 hhy 1983
characteristic of the hlulliken population anaiysis. is essentially not an arbitrary choice but is just that whit is consistent with the internal structure of the LCAO hi0 formalism_ The fact that Mulliken’s populations are often strongly basis dependent does not invalidate this conclusion: one may only hope that the basis dependence of the Mulliken charges can be reduced by taking properly into account the interatomic “basis extension” effects defined recently [2]_
4_ Bond order
and valence
In studying the decomposition of the LCAO hamiltonian into terms of different physical significance ant the related energy partitioning scheme for the SCF wavefunctions. atomic
f3y ubiiig ( ! 3) lid raking into account that (*I*) = 1_wt‘ hvr for the expectarion vJlu2 of & :
we found
electrostatic
[2]
that
the leading
term (corresponding
clirtrge approsimation)
contains
inter-
to the point-
an exchange
compo-
nent
This term originates from the inequality
I c’_lilt*t~_vptvIafioilvalue C\!_I) t>j-the operator
of die
UtOJJliCpO[JlthIiotJ (1.: OJI fitcgiwJt atom . A more formal proof showing th.31 this result is not resrrictcd 10 the SCF level. hui IS \Jlid lix .my LCAO-1) pe wavefunction (c-g_ for (‘I- 01 VB-ty;-r l’.‘d\cfunctions 100) will br given elsev&xc [ 21. In ( 14) \-.r‘ 1i.1~ inrruduced Ihe usual = closed-shell P rii.3rrix i with the 212m21ifs P 5:: l 2~~pcL,,_ 11o1eht fhc present notati% differ wnw\xh~t l;om lhose in ref. [2]_
‘Jt‘JJJ;iL dlUrgc
Tl12 tlw~e
B. Now.
we
may
define
the quantity
-Q-q iS ~%iltiliA-CJJ'SgrCJSS
J1131~s1s shms
o\crrlJp popul~rioil
beIw22n
that the %ilving” 1112 two atoms
of rhe
in question.
(16) as the bond order index
between
can prove
B__1Bhas integer
to those
that
predicted
the specific treated
the index
by the classical
case of some
at the minimal
in general,
for those
basis
systems
orbitals orbitals,
respectively. order
dependence; for similar
bonds
localized
inner&hell case,
more
bonding
of course,
or less deviate
values as well as exhibit however,
one-
or lone-pail
covalent
zmpolarizeci
son of the values obtained
in
the wavefunc-
strictly
In the general
one expects,
picture
diatomics
for which
index B_dB may
the integer
On2
equal
level (as H,. N,, F2) and,
atomic
and two-center
the bond from
orbit&:
values
valence
homonuclear
tion is built up of orthonormal and two-center
A and B.
atoms
that
some
basis
the compari-
by using the same basis set
in a series of related
molecules
may
CHEMICAL
Volume 97, number 3
20 hfay 1983
PHYSICS LETTERS
alization
be instructive and give some insight into the details of the bonding situation in the systems studied- The deviation from the integer values may be caused by the non-orthogonality and delocaIization effects (like “tails” of the localized orbit&) and, especially, by the partial ionic character of the bonds formed between atoms of different electronegativities. One may observe that the bond order index BAs proposed here can be considered as an ab initio gener-
of Wiberg’s bond index
[4] which has proved
very useful in semi-empirical methods using different IeveIs of the ND0 (neglect of differential werlap) approximations * : our definition (16) of the BAB bond ’ One mw observe &at kl the CNDO method there is also B most intimate connection between W&erg’s bond index I$‘~B I4 j and the eschange part of the diatomic energy corn ponent IS]: EAB,exch_ = -~7__&‘&& -GAB being the inter atomic electron-electron repulsion inte_ml.
Table 1 Illustrative values of bond order and valence indices for some small molecules calcuhtcd approxim&e mokcular geometries Molecule
Bond orders
basis sets nnc
Valence numbers
bond
ST-O-3G
+31G
atom
Hz
H-H
1.0
1.0
H
1.0
methane
C-H
0.99
0.96
C H
3.65 0.9-I
C-N
0.98 1.01
0.96 O-94
c H
3.97 1.00
3.78 0.93
0.98 2.01
0.96 1.96
C li
3.97 1.00
3.55 0.94
C-H
c-c
0.98 3-00
0.86 3.27
C ti
3.9s 0.99
4.16 0.89
C-H
0.98
0.95
c tow C free H total H free
3.96
1.02 1.00 0.005
3.87 1.01 0.95 o.oO.4
n
1.91 0.97
1.61 b) 0.80 b)
ethane
C-C etllyIene
C-H
c-c accthylenc methyl radical
water
O-H
3)
0.80 b,
0.95
0
STO-3G
l-31G
Ci!
c-c
3.39
334
c
3.39
3.34
N2
N-N
3.0
2.67
N
3.0
2.67
co
c-o
2.52
2.14
C 0
2.52 2-52
2.14 2.14
H-C C-N
0.97 2.99
0.86 2.91
H C N
0.98 3.96 3.00
0.87 3.77 2.92
CN- ion
C-N
2.88
2.75
C N
2.88 2.88
7.75 2.75
CN radical
C-N
2.80
d
C totd C free N total N free
3.81 1.00 3.09 0.29
Cl
HCN
‘)
by using STO-3G and ?-31G
STO-6G resuits.
b, 6-31G results.
‘1 Does not converge.
273
\i~luxnc
9 i. nwuhcr
3
CHEMICAL
PHYSICS
inde\ goes wer IO thdr of thecorresponding Wibeg rf the ovrrlrlp matri.. S becomes the unit matrix. AI the s.nnc rime our bond index B,,B cm also be used
Index
lrl rhr exrsrldrd Htickl method. for inll~ces .we not apphcahle_ U?- pIerAihg Wibcrg‘s idea (set p. 10~0 in rei_ [-I], \xc may consider = 2(PS) a< (PS$, _ .I . measure of
which Wiberg’s rhe foornore
on
the quantity 6, the bonding ptwcr ~s-t’,n orbir31 in :! given 11w!ax11~: b, is equal to one try iiie t>rblldl \\irh n ,\lullIkcn population of unity .md f.11:~IV _wro ior lx~111 ;I vx.mt and ~1doubly filled ~H?M~_I!. 11 IS CJS) to see 111~1for .I closed-shell system
LETTERS
6-A = VA -
70 May 1983
c
BAB _
B (Be_-1)
of the total actual valence of the atom. eq. (17). and the sum of the bond orders formed by it. We note hen that the valence numbers VA and F_4 defined above may be considered as ab initio generalizations of the quantities discussed in ref. [6] for the CNDO level. Like the bond orders. both V, and FA assume in-
teger values in the specific cases discussed above. In the general case they are expected to have values close to those predicted by rhe classical valence concepts and may be used to link the ab initio SCF (or the extended Hiickel) results with rhe latter. Table 1 contains the results of illustrative calculations for some small molecules. We note :hat the somewhat
I ,=2
c
f.’ -1
(PS),,
c
fi: z-1
w3,_.(W,,~
_
(17)
(19)
smaller
numerical values obtained for the 43 I G basis may immediately be related with the known “overpolarization” of the bonds characteristic of this basis set. The question how lo generalize the bond order and valence indices lo the case of refined (as CL etc.) wave funcrions requires further investigation. Nevertheless. it seems that even at the SCF lcve! useful information mdy be gained by calculating these parameters: an application for discussing the behaviour of the “hypertalent-‘ sulphur atom in some sulphonyl chloride molecules will be described elsewhere.
References (1s) II C. Lonp~et-l&ins. in: Qumtuin throry of ;IIoms. molcculcs and ~hc solid stdtc. cd. P.-O. Lixldin (Academic l’rcs. Xes York. 1966) p_ 105. 1. XI> cr. f‘ourth 1nternation.d Congrc\s in Qu.mtum Chsmistr>. Upps~l~.June 1981; Intern. J. OUCII~~UI~ Cheln 10 be publkhed. P.-O. LG\rdin. Phys. Kev. 97 (1955) 1971. K;.A. U’iberg. TetrAedron 2-l ( 1966) 1083. 11. i‘ischer .md 11. Kiollrnsr,Theorer. Chin]. ActA 16 (1970 163. D.R. Armstrong. P.C. Perkins snd J_J.IDs Srewxt. J. Chem. SW. D.dIon ( 1973) 838.