Bond Order and Valence Indices: A Personal Account
I. MAYER
Chemical Research Center, Hungarian Academy of Sciences, H-1525 Budapest,
PO Box 17, Hungary
Received 10 April 2006; Accepted 8 May 2006
DOI 10.1002/jcc.20494
Published online 26 October 2006 in Wiley InterScience (www.interscience.wiley.com).
Abstract: The paper accounts for the author’s activity in developing bond order and valence indices since the early 80s.
These indices represent an important conceptual link between the physical description of molecules as systems of electrons
and nuclei and the chemical picture of molecules consisting of atoms kept together by bonds. They are also useful for a
systematization and interpretation of the results obtained in the quantum chemical calculations, by permitting to extract
from the wave function different pieces of information that may be assigned chemical significance. In some cases they
can have some predictive power, too. Historically, the prototypes of such indices were introduced in the semiempirical
quantum chemistry; the most important developments were Coulson’s charge–bond order matrix in the simple Hückel
theory and the Wiberg index in the CNDO framework. (Valence indices were also introduced in the semiempirical theory.)
The definition of the ab initio bond order index emerged from the asymptotic term of the exchange energy component of
the partitioning performed in the framework of the author’s so-called “chemical Hamiltonian approach” using a “mixed”
second quantization formalism for overlapping basis sets. They can also be introduced by studying the exchange part of
the two-particle density (or of the second-order density matrix). Some properties of the bond order indices are discussed
and the author’s (until now unpublished) proof is also presented, showing the sufficient conditions under which the bond
order index of a homonuclear diatomics is equal to the “chemist’s bond order,” i.e., the half of the difference between
the number of electrons occupying bonding and antibonding orbitals. The ab initio valence indices are also introduced
and discussed, and it is stressed that for correlated wave function the same “exchange only” definition of the bond
order and valence indices should be used, which was introduced for the SCF case. The recent concept of the “atomic
decomposition of identity” is also discussed and it is utilized for introducing bond orders and valences in the framework of
the “3D analysis,” when atoms are defined not by their basis orbitals but as regions of the three-dimensional (3D) physical
space. Two versions of the 3D analysis are considered—the AIM (atoms in molecules)-type decomposing the space into
disjunct atomic domains and the “fuzzy atoms” scheme in which there are no sharp boundaries between the atoms but
they exhibit a continuous transition from one to another.
© 2006 Wiley Periodicals, Inc.
J Comput Chem 28: 204–221, 2007
Key words: bond order indices; valence indices; fuzzy atoms; “chemist’s bond order”; non-orthogonal second
quantization; chemical Hamiltonian approach
Introduction
Chemists consider molecules as consisting of atoms; physicists treat
them as systems of electrons and nuclei. Undoubtedly, the ingenious
idea of Lewis, identifying chemical bonds with electron pairs shared
between the bonded atoms, still represents the fundamental link
between these completely different descriptions. It provides one a
well-established way of describing molecules, which is relatively
easy and straightforward, as far as a qualitative picture is concerned.
However, when turning to the quantitative theory, it appeared by far
not trivial to connect the results of ab initio calculations with the
genuine chemical concepts of atoms connected by single, double,
etc., bonds, and with the electron pairs forming these bonds.
The multiplicity of a chemical bond, called also “bond order,”
is a quantity of fundamental importance in practical chemistry.
Obviously, if one wishes to discuss molecules on both chemical
and quantum mechanical (quantum chemical) levels, then one has
to find a quantum chemical counterpart of this fundamental chemical
concept.
When one started to use the concept of molecular orbitals (MOs),
it became obvious that there are “too much” valence electrons to
assume that each pair of electrons occupying a two-center MO in
Correspondence to: I. Mayer; e-mail: [email protected]
Contract/grant sponsor: Hungarian Scientific Research Fund; contract/grant
number: OTKA T43558
Contract/grant sponsor: The Spanish–Hungarian intergovernmental joint
project; contract/grant number: HH2004-0010—Magyar-Spanyol TÉT
E18/2004
© 2006 Wiley Periodicals, Inc.
Bond Order and Valence Indices: A Personal Account
a diatomics corresponds to a chemical bond—although obviously
each such MO bears a pair of electrons delocalized between the two
atoms. It became clear that one has to distinguish between bonding and antibonding orbitals, and one arrived to the definition of
“chemist’s bond order”
B=
Nbond − Nantibond
2
(1)
where Nbond and Nantibond are the number of electrons occupying
bonding and antibonding orbitals respectively. For diatomics one
could study the character of the orbitals obtained in a calculation and
count the electrons on bonding and antibonding ones. Nonetheless,
definition (1) cannot be considered a quantum chemical quantity in a
narrow sense, as it is not directly calculated from the wave function
as would, say, an expectation value of an operator.
The first quantity called bond order in quantum chemistry was
the off-diagonal matrix element of Coulson’s “charge—bond order
matrix”,1 which was identified with the π-component of the bond
order between two atoms of a conjugated (i.e., π-electron) system.
This was an extremely useful quantity, characterizing very well
indeed the degree of π-bonding between the centers involved. It
could be considered as the π-electron bond order, because it reaches
its possible maximum value equal to one for ethylene, and gives
larger values for atoms with larger delocalization of π-electrons
between them. (Excellent correlations can be observed between
Coulson’s bond orders calculated at the simplest Hückel level of
the theory and the experimental C C distances.) In accord with
the formula defining Coulson’s bond orders in terms of the orbital
coefficients Cµi ,
Dµν = 2
occ.
∗
Cµi Cνi
,
(2)
i
it can indeed be related to the degree to which the different πorbitals have significant simultaneous (and “in phase”) contributions
from the basis orbitals of both atoms in question; thus they may
be related to Lewis’s shared electron concept. This is the case
despite the fact that the number of π-electrons was considered a
sum of purely atomic contributions equal to the diagonal elements
of the “charge—bond order matrix”; thus there is no π-electron
charge shared between the atoms. (The basis orbitals were considered orthogonal in most simple π-electron models.) The use of the
off-diagonal matrix elements Dµν as bond orders is, however, limited to models with one basis orbital per atom. Note, however, that
in the early literature the name “bond order” had been often applied
for the off-diagonal Dµν matrix elements in other cases, too.
Contrary to Coulson’s bond order, Mulliken’s overlap population2 assigns a part of the electronic charge directly to the pair of
atoms considered. It characterizes the accumulation of the electrons
in the region between the chemically bonded atoms, and is a very
useful quantity often characterizing well the bond strength. However, it cannot be called bond order, because it does not represent
numbers that are close to one, two, and three for systems with single, double, and triple bonds respectively. An important property
of Mulliken’s overlap population is that it possesses the correct
rotational-hybridizational invariance that one should require for
205
any quantity assigned a physical significance. (The same holds for
Mulliken’s net and gross atomic populations; for an explicit proof,
see ref. 3.)
Wiberg4 had observed that neither Coulson’s bond order nor
Mulliken’s overlap population could be applied for Pople’s CNDOtype all-valence-electrons semiempirical theory, which was in general use at that time. The reason was that neither the individual
elements of the “density matrix” (the name of matrix D usually
applied for theories other than the π-electron ones) nor any of their
simple combinations have the correct invariance properties when the
molecule is rotated as a whole, while Mulliken’s overlap population
simply vanishes because the basis orbitals are assumed orthonormalized. One sometimes considers the orthogonal basis orbitals of
the semiempirical theories as Löwdin-orthogonalized counterparts
of some “original” ones; then one may perform a “deorthogonalization,” too, and get nonzero values of Mulliken’s overlap population.
Fortunately enough, that was not yet used in Wiberg’s time, so he
had to look for a new parameter. For that reason, he introduced a
new bond index (now bearing his name) which is quadratic in the
density-matrix elements and has the proper invariance:
WAB =
|Dµν |2 .
(3)
µ∈A ν∈B
Moreover, it appears that for singlet states most of first row homonuclear diatomics (but not for C2 ) (K. Jug, personal communication,
1985) the CNDO Wiberg index is equal to the ideal integer value
one usually assigns to the bond order in the given molecule.
This point had been investigated in detail by Borisova and
Semenov.5 They gave a strict proof, according to which one can
derive the equality (1) for the CNDO wave functions by considering the hybrid atomic orbitals (AOs) of which the individual MO are
built up. Owing to the symmetry of the homonuclear diatomics, each
MO is the (normalized) sum or difference of the hybrids of the two
atoms, which are symmetry pairs of each other. Usually there are
pairs of bonding and antibonding MOs formed of the same hybrids,
and either both of them are occupied in the SCF wave function, or
only the bonding combination is occupied and the antibonding one is
empty. Borisova and Semenov proved that in the first case these two
MOs (and the respective hybrids) do not contribute to the resulting
Wiberg index (Borisova and Semenov called it “bond multiplicity”
and defined in terms of spin-orbitals), while in the second case one
gets a contribution equal to unity—and there are no contributions
originating from cross terms between different such pairs of MOs.
The first result can simply be explained by utilizing the invariance of
determinant wave functions with respect to unitary transformations
of the occupied orbitals. In every case when the bonding and the antibonding combinations of some atomic hybrids are both occupied,
one has to perform the unitary transformation leading to the sum
and difference of these MOs. In this manner one obtains a pair of
localized orbitals, each of which is fully concentrated on one atom.
These strictly atomic lone pair orbitals (hybrids) do not, of course,
contribute to the resulting Wiberg index, and the later becomes equal
to the number of doubly occupied bonding orbitals the antibonding
counterparts of which are empty.5 This result is in full accord with
the original Lewis electron pair picture—electrons fully localized
at one of the atoms need not be counted.
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Valence is another important classical chemical concept—it
measures the ability of the atom to form chemical bonds in its
actual state, and gives the number of bonds (counted with their
multiplicities) formed by the atom in a closed-shell system. The
CNDO definition of valence index had been introduced independently by Borisova and Semenov5 and Armstrong et al.6 with a few
months priority of the former authors. Unfortunately, Borisova and
Semenov published their very important papers in a journal with
a limited circulation, and so I became aware of them only thanks
to a favor by Dr. Obis Castaño—who was their former graduate
student—after I had already published my first papers in the field.
Valence can probably best introduced started from the observation made by Wiberg4 in a footnote, according to which the quantity
that we shall denote bµ
2
bµ = 2Dµµ − Dµµ
(4)
measures well the covalent bonding capacity of the basis AO χµ
for the given wave function (density matrix D), because it reaches
its maximum, equal to one, for a unit electron population on the
given orbital (Dµµ = 1), and falls to zero for both an empty orbital
(Dµµ = 0) and a nonbonded pair (Dµµ = 2). At the same time, it
follows from the idempotency property D2 = 2D of the density
matrix that for closed-shell single determinant wave functions
bµ =
ν
(ν =µ)
2
Dµν
(5)
i.e., bµ is equal to the sum of all partial Wiberg indices between
orbital χµ and all the other orbitals in the molecule. Summing up
the parameters bµ for all the orbitals of the atom, but extracting the
intraatomic partial Wiberg indices lacking a chemical significance,6
we arrive to the valence of atom A:
VA = 2
Dµµ −
µ∈A
2
Dµν
.
(6)
µ,ν∈A
Alternatively, as it was done by Borisova and Semenov,5 one can
turn to the “natural hybrids” for which the intraatomic block of the
matrix D is diagonal, sum up the quantities bµ in that basis, and then
return to the original one, and obtain the same result.
Borisova and Semenov5 stressed the importance of the fact that
the valence index VA is determined by matrix elements referring
to the given atom only, i.e., by the actual valence state of the atom
in the given molecule.
It follows from the idempotency of the density matrix that for
closed-shell determinant wave functions the valence of an atom
equals to the sum of its Wiberg indices:5, 6
VA =
WAB .
(7)
A
(A=B)
As the Wiberg index measures bond multiplicity, this relationship
indeed corresponds very well to the chemical notion of valence.
While Armstrong et al.6 considered closed-shell systems only
(They only made an important remark that in the open-shell case
the equality (7) does not hold and the difference of the two sides
“should be a measure of the reactivity of the atom.” This was the
line along which I have later introduced the “free-valence index”
FA ), Borisova and Semenov5, 7 used spin-orbitals, permitting them
to treat open-shell (UHF) systems, too. Their definitions would give
a correct value of 21 for the bond order in H+
2 . However, in the two
papers5, 7 they gave different expressions for the valence index of
open-shell systems in terms of the spatial orbitals. It is my opinion
that the first is the correct one, because the use of the formula given
in the second paper 7 would give a value very close to 3 for the
valence of carbon atom in the methyl radical, which I do not think
is chemical: it is more correct to consider this carbon to be fourvalent, with one of its valences being actually free. I had a similar
objection to the definition used by Gopinathan and Jug;8 also see
ref. 9. (Chemists usually indicate this free valence by putting a dot
to the radical center on the structural formula.)
The “Chemical Hamiltonian Approach” and the Definition of the An Initio Bond Order Index
In the late 1970s I got interested in the conceptual relationships between the strict ab initio quantum chemical theory and
the genuine chemical concepts. It appeared to me that one of the
main difficulties is due to the fact that 10 “the Born–Oppenheimer
Hamiltonian does not reflect the pronounced pairwise character
of interatomic interactions in the molecular system.” In order to
discuss this problem, a special treatment has been introduced10 in
which the main elements were a “mixed” second quantization formalism for treating the overlap problem and a special projection
technique permitting to get rid (at least in some sense) of the threeand four-center integrals. This formalism got the name “Chemical
Hamiltonian Approach” (CHA). It is a typical case of the “Hilbert
space analysis” according to Hall’s terminology,11 in which the atom
is identified with the nucleus and the basis orbitals centered on it. The
alternative is the “3D analysis” in which the atom is identified with
the nucleus and a part of the three-dimensional (3D) physical space
around the nucleus; we shall consider it under the Section Atomic
Resolution of Identity and 3D Analyses. (As atoms are not true
quantum mechanical observables, one has to chose how to define
them in the quantum mechanical framework.)
The “Mixed” Second Quantization Formalism
In practice we use atom-centered basis sets, and then the quantum
mechanical problem of describing the molecular electronic structure is solely defined by the one- and two-electron integrals over the
basis AOs. The overlap of the basis orbitals centered on different
atoms reflects chemically very important interactions, but makes
difficult the calculation of the matrix elements of different operators. (That caused ab initio VB schemes to be not competitive to the
HFR approach.) The mixed second quantization formalism permits
to work in a relatively simple way with wave functions built up of
nonorthogonal orbitals, by using essentially the same techniques
that one applies in the case of an orthonormalized basis set, with the
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Bond Order and Valence Indices: A Personal Account
expense that the annihilation operators are defined in a special manner, and do not coincide with the adjoints of the respective creation
operators.
In what follows we shall use superscripts “+” for the creation
operators and superscripts “−” for the annihilation ones. However, if
the overlap matrix of the spin-orbital basis {χµ } is not a unit matrix,
i.e.,
χµ |χν = Sµν = δµν
then the operator χ̂ν−
(8)
and the respective creation and annihilation operators ψ̂µ+ , ψ̂ν− ,
which can be presented as
ψ̂µ+ =
1
(S− 2 )νµ χ̂ν+ ;
{χ̂µ+ , χ̂ν− } = χ̂µ+ χ̂ν− + χ̂ν− χ̂µ+ = Sµν .
ψ̂µ− =
ν
N̂ =
ν
−1
Sνµ
χν
N̂ =
ϕ̂ν− = (ϕ̂ν+ )† =
ν
−1 −
Sµν
χ̂ν .
(11)
Then, as it is easy to see,
{χ̂µ+ , ϕ̂ν− }
= δµν
The Operator of Atomic Population
In order to express different operators in terms of operators χ̂µ+
and ϕ̂ν− , one introduces an auxiliary Löwdin-orthogonalized set of
spin-orbitals
ψµ =
1
1
(S− 2 )νµ χν =
(S 2 )νµ ϕν
ν
ν
N̂ =
(13)
(16)
N̂A
(17)
χ̂µ+ ϕ̂µ−
(18)
A
where
N̂A =
µ∈A
is the operator of atomic population for atom A. In my previous
papers10, 12 I had given two independent proofs (one for the single determinant wave functions, another—see Appendix A—for the
general case) for the expectation value of the operator string χ̂µ+ ϕ̂ν−
χ̂µ+ ϕ̂ν− = (PS)νµ =
(12)
which means that the annihilation operator ϕ̂ν− defined with respect
to the biorthogonal basis of spin-orbitals is that operator which
acts in the nonorthogonal case exactly in the same manner as the
usual annihilation operators do in the orthogonal case. Note that we
invoke the biorthogonal basis {ϕν } only to define the true annihilation operators ϕ̂ν− corresponding to the creation operators χ̂µ+ in
the nonorthogonal case, and no other reference to the biorthogonal
basis is necessary.
χ̂µ+ ϕ̂µ− .
Grouping here the terms according to the atoms on which the spinorbitals χµ are centered, we may write
(10)
−1
where Sνµ
is a short-hand notation for the element (S−1 )νµ of
the inverse overlap matrix S−1 . As orbitals and creation operators
transform analogously, one can write
(15)
By substituting here the relationships (14) we get by a single algebra
µ
ϕµ =
(14)
ν
ψ̂µ+ ψ̂µ− .
µ
(9)
The “true” (or “effective”) annihilation operators ϕ̂ν− , for which
fermion anticommutation rules apply, can be constructed by using
the biorthogonal set of spin-orbitals
1
(S 2 )µν ϕ̂ν− .
The operator of the number of electrons N̂ has the usual form in
the orthonormalized spin-orbital basis {ψµ }:
= (χ̂ν+ )†
is only formally an annihilation operator, because its anticommutator with the creation operator χ̂µ+ is
not δµν as it were the case for an orthonormalized basis, but
207
Pµτ Sτ ν
(19)
τ
where P is the density matrix in terms of spin-orbitals. That means
N̂A =
(PS)µµ =
Pµτ Sτ µ = QA
µ∈A
(20)
µ∈A τ
i.e., the expectation value of the operator of atomic population is
Mulliken’s gross atomic population QA on the atom in question.
This result means that—irrespective of its large basis dependence
or other possible disadvantages—Mulliken’s gross atomic population has a privileged importance: it is that atomic population which
is consistent with the internal mathematical structure of the theory
using atom-centered basis orbitals. One could group the terms by
the atoms in the expression (15), too, and get the “Löwdin populations” corresponding to the individual atoms. However, the Löwdin
populations refer to the Löwdin orbitals ψµ , which are not strictly
atomic entities and vary in dependence of the chemical composition
and geometry of the system studied; in addition, it has recently been
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shown13, 14 that the Löwdin populations are not necessarily invariant with respect to the rotation of the molecule as a whole, and
equivalent atoms may be assigned different Löwdin populations.
thus the one-electron part of eq. (23) can be rewritten as
Ĥ(1) =
A
Decomposition of the Hamiltonian
ZA ZB +
ψµ |ĥ|ψν ψ̂µ+ ψ̂ν−
RAB
µ,ν
A<B
+
1 [ψλ ψµ |ψν ψ ]ψ̂λ+ ψ̂µ+ ψ̂− ψ̂ν−
2 λ,µ,ν,
(21)
where ĥ is the one-electron part of the Hamiltonian
ZA
1
ĥ = − −
,
2
rA
Sσ−1λ χλ |ĥA |χε χ̂σ+ ϕ̂ε−
−
In terms of the auxiliary basis {ψµ } one has the known expression
of the Hamiltonian
Ĥ =
ε∈A λ,σ
A
ε∈A λ,σ
Sσ−1λ χλ |
B
(B =A)
ZB
χε χ̂σ+ ϕ̂ε− .
rB
(26)
As ε ∈ A, the function ĥA |χε entering the integral in the first term
is of intraatomic character: the atomic Hamiltonian ĥA acts on a
basis orbital centered on the same atom. If the basis on A were
complete (or if |χε were an exact eigenvector of ĥA as is the case
for Schrödinger’s orbitals for a hydrogen atom), then the function
ĥA |χε could be exactly expanded in the atomic basis. This is not
the case, in general, and the function ĥA |χε has components both
in the subspace of the basis orbitals assigned to atom A and in the
orthogonal complement to that subspace. That fact may be expressed
by introducing a resolution of identity as
(22)
Î = P̂A + (1 − P̂A )
A
(27)
where
all the integrals refer to spin-orbitals (i.e., the integrations include
summations over the spins) and the convention [12|12] is used for
the two-electron integrals.
Substituting the expressions (13) and (14) into (21), we obtain
the Born–Oppenheimer Hamiltonian in the mixed second quantized
form containing only integrals over the “original” spin-orbitals and
the creation and true annihilation operators corresponding them:
Ĥ =
P̂A =
µ,ν∈A
−1
|χµ S(A)µν
χν |
(28)
−1
is
is the projection operator on the atomic basis. (In eq. (27) S(A)µν
a short-hand for a matrix element of the inverse intraatomic overlap
matrix.) One may write
ZA ZB
+
Sσ−1λ χλ |ĥ|χε χ̂σ+ ϕ̂ε−
RAB
λ,ε,σ
χλ |ĥA |χε = χλ |P̂A ĥA |χε + χλ |(1 − P̂A )ĥA |χε .
(29)
A<B
+
1
2
γ ,ϑ,κ,,η,ε
−1 −1
Sηγ
Sεϑ [χγ χϑ |χκ χ ]χ̂η+ χ̂ε+ ϕ̂− ϕ̂κ− .
(23)
The elements of the inverse overlap matrix present in the Hamiltonian (23) reflect global effects in the molecule, and one needs
a further effort to separate out the true intraatomic and diatomic
interactions. For that reason a special projection technique had
been introduced.10 As the simplest example, let us consider the
intraatomic part of the one-electron Hamiltonian corresponding to
atom A:
ZA
1
ĥA = − −
.
2
rA
(24)
Obviously, P̂A ĥA |χε is that component of the function ĥA |χε which
enters the problem of the free atom A and which is transferable to
any chemical environment in which atom A occurs. Therefore, it is
of meaning to introduce the approximation
χλ |ĥA |χε ∼
= χλ |P̂A ĥA |χε =
µ,ν∈A
A ε,λ,σ ∈A
ZB
rB
B
(B =A)
(25)
−1
Sλµ S(A)µν
χν |ĥ|χε (30)
and to consider χλ |(1 − P̂A )ĥA |χε as a finite basis correction term.
(It is quite analogous to the terms causing the so-called “basis set
superposition error” in the theory of intermolecular interactions.15 )
Note that eq. (30) becomes a strict equality if both λ, ε ∈ A.
Substituting the approximation (30), the first sum on the righthand side of expression (26) becomes
For every atom A we may write
ĥ = ĥA −
−1
A
+ −
S(A)σ
λ χλ |ĥ |χε χ̂σ ϕ̂ε
(31)
—an expression that contains only one-center quantities, and represents a sum of effective atomic one-electron Hamiltonians. (They
are “effective,” because expression (31) is written down in the
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Bond Order and Valence Indices: A Personal Account
global many-atomic basis, and contains therefore the “effective”
annihilation operators ϕ̂ε− .)
Similarly, we may consider the function (ZB /rB )|χε in the second term of the expression (26) as a diatomic entity (ε ∈ A; A = B),
which may be approximated
by projecting it on the union of atomic
subspaces AB = A B. Furthermore, one may perform analogous
manipulations with the two-electron function (1/r12 )χκ (1)χ (2)
entering the two-electron integrals in the Hamiltonian (23), by
introducing a projector for each electron. (One has to introduce
the projectors P̂A (1)P̂A (2) or P̂AB (1)P̂AB (2) depending on whether
orbitals χκ and χ are centered on the same atom A or on different
atoms A and B.)
In this manner one gets the expression of the Hamiltonian in the
following form
Ĥ =
ĤA +
A
A ε,λ,σ ∈A
+
1
2
κ,λ,µ,ν,,τ ∈A
The first term of the diatomic Hamiltonian (34) describes the
internuclear repulsion; it need not be discussed. In the terms describing electron-nuclear attraction and interelectron repulsion, it is of
meaning to separate out those contributions that correspond to the
electrostatic (and exchange) interactions and those that are due to
“differential overlap densities” χµ∗ (r )χν (r ) of the orbitals centered
on different atoms. If there were no differential overlap, then the
sums on the right-hand side of eq. (34) would not run on the whole
diatomic basis AB but only on one of the atoms A or B and the
intraatomic blocks of the inverse overlap matrix S−1
AB would coin−1
cide with the respective atomic inverse overlap matrix S−1
A or SB .
One may write, therefore
el stat
= ĤAB
+ ĤAB
ĤAB
(32)
A<B
(35)
where
−1
A
+ −
S(A)σ
λ χλ |ĥ |χε χ̂σ ϕ̂ε
Hierarchy of Diatomic Interactions
overlap
ĤAB
+ Ĥfin bas
where ĤA and ĤAB
are effective atomic and diatomic operators
(These are “effective” operators because they contain the “effective”
annihilation operators ϕ̂ε− . As a consequence, they are not Hermitian) and Ĥfin bas collects the finite basis corrections of either atomic
or diatomic nature, which are connected with the remainders of the
projective integrals approximations. Obviously, they are expected
to be really negligible only if very large basis sets are used.
The effective atomic and diatomic Hamiltonians contain only
terms related to the given atom and diatomic fragment respectively
(The remainders of the projective approximations leading to the
effective atomic Hamiltonians ĤA have components that can be
expanded in the basis sets of diatomic fragments. One could regroup
these terms from Ĥfin bas to the effective diatomic operators ĤAB
, and
get an expansion in which the sum of atomic and diatomic Hamiltonians recovers the exact one for diatomic molecules. Terms of such
type were considered in the energy decompositions,16, 17 but are of
no relevance as far as our present subject is concerned):
ĤA =
209
−1
−1
S(A)νµ
S(A)λτ
[χµ χτ |χκ χ ]χ̂ν+ χ̂λ+ ϕ̂− ϕ̂κ−
ε,µ,ν∈A
−
+
ZB
|χε χ̂µ+ ϕ̂ε−
rB
−1
S(B)µν
χν |
1 2
κ,λ,µ∈A ν,,τ ∈B
ZA
|χε χ̂µ+ ϕ̂ε−
rA
−1
−1
S(A)µλ
S(B)ντ
[χλ χτ |χκ χ ]χ̂µ+ χ̂ν+ ϕ̂− ϕ̂κ−
1 −1
−1
+
S(A)µλ S(B)ντ
[χλ χτ |χκ χ ]χ̂µ+ χ̂ν+ ϕ̂− ϕ̂κ−
2 κ,λ,µ∈B
ν,,τ ∈A
+
ZA Z B
RAB
(36)
el stat
is the difference between ĤAB
and ĤAB
. (We do not
and ĤAB
need its explicit expression10 here.)
Now, we consider the asymptotic behaviour of the integrals entering the electrostatic component (36) of the Hamiltonian for large
interatomic distances; it is easy to see that they behave asymptotically as the interactions of point charge(s) proportional to the
respective overlap integral(s):
overlap
(33)
Z A ZB
ZB
−1
−
S(AB)µν
χν | |χε χ̂µ+ ϕ̂ε−
RAB
rB
µ,ν∈AB
ε∈A
ZA
1 −1
−1
+
χν | |χε χ̂µ+ ϕ̂ε− +
S(AB)µλ
S(AB)ντ
r
2
A
ε∈B
λ,µ,ν,τ ∈AB

×
[χλ χτ |χκ χ ]χ̂µ+ χ̂ν+ ϕ̂− ϕ̂κ−
=
ĤAB
ZB
Sνε ZB
|χε ∼
;
rB
RAB
[χλ χτ |χκ χ ] ∼
Sλκ Sτ .
RAB
(37)
By substituting the asymptotic expressions (37) into (36) and
performing the summations leading to Kronecker deltas, we get, by
using the definition of the operator of atomic population (18), the
Hamiltonian of electrostatic interactions in the point-charge approximation, reflecting the overall electrostatic balance of the molecule
(One gets the operator string as χ̂µ+ χ̂ν+ ϕ̂ν− ϕ̂µ− , which obviously
equals χ̂µ+ ϕ̂µ− χ̂ν+ ϕ̂ν− because χµ and χν are centered on different
atoms; therefore µ = ν):
κ∈A ∈B
κ∈B ∈A
−1
S(A)µν
χν |
ε,µ,ν∈B
χν |

+ + − −
+
[χλ χτ |χκ χ ]χ̂µ χ̂ν ϕ̂ ϕ̂κ
el stat
=−
ĤAB
(34)
point
ĤAB
and we refer to ref. 10 for an explicit expression of Ĥfin bas .
Journal of Computational Chemistry
=
1
(−N̂A ZA − N̂B ZA + N̂A N̂B + ZA ZB )
RAB
1
=
(N̂A − ZA )(N̂B − ZB ).
RAB
DOI 10.1002/jcc
(38)
210
I. Mayer
point
Expectation Value of ĤAB
•
Vol. 28, No. 1
•
Journal of Computational Chemistry
In terms of the spatial orbitals, it can be rewritten as
and the Bond Order Index
point
Now, we calculate the expectation value of operator ĤAB by using
the equality (20) for the expectation value of the atomic population operator N̂A . For calculating the expectation value of the
product N̂A N̂B we need the expectation value of the operator string
χ̂µ+ ϕ̂µ− χ̂ν+ ϕ̂ν− = χ̂µ+ χ̂ν+ ϕ̂ν− ϕ̂µ− . This can be obtained as a special case
of the general relationship
χ̂µ+ χ̂ν+ ϕ̂− ϕ̂κ− = (PS)κµ (PS)ν − (PS)κν (PS)µ
BAB = 2
(Pα S)µν (Pα S)νµ + (Pβ S)µν (Pβ S)νµ . (44)
µ∈A ν∈B
Here Pα and Pβ are the density matrices for the orbitals occupied
with spins α and β, respectively. They have the usual expression in
terms of the orbital coefficients of the occupied orbitals:
(39)
α
Pµν
10
valid for single determinant wave functions. (Here matrices P and
S refer to the basis of spin-orbitals.) The proof of this relationship—
it has not been published previously—is outlined in Appendix B.
Using this result, we get
N̂A N̂B =
α α∗
Cµi
Cνi ;
i=1


1 
=
(PS)µν (PS)νµ 
q A qB −
RAB
ν∈B
(41)
µ∈A
—it differs from the electrostatic interaction of the resulting atomic
charges qA = ZA − QA by a term, proportional to the exchange
component on the right-hand side of eq. (40)
(PS)µν (PS)νµ .
(42)
BAB = 2
BAB =
(PS)µν (PS)νµ .
µ∈A ν∈B
(43)
β∗
(45)
(DS)µν (DS)νµ
(46)
where D = Pα + Pβ .
After publication of my paper,12 I got a letter from Ms. Giambiagi (Rio de Janeiro) with a copy of their paper 19 in which essentially
the same definition (46) was proposed for the “all valence electron”
semiempirical theories with overlap (practically extended Hückel)
as a formal generalization of the Wiberg index. Their paper—
probably because of the use of unusual and rather cumbersome
notations and too lapidary a presentation—did not receive the proper
attention in the literature.
In my first paper about the subject,12 I have suggested to use
definition (46) also to the closed-shell case, and only somewhat later
(roughly when I had read the papers of Borisova and Semenov,5, 7
and probably not without being influenced by them) I have realized,
that the general definition (43) expressed in terms of spin orbitals in
the open-shell case should be rewritten to spatial orbitals either as
eq. (44) or, introducing the spin-density matrix Ps = Pα − Pβ , as
µ∈A ν∈B
Returning home by train from the International Congress of
Quantum Chemistry held in June 1982 in Uppsala (I will be forever grateful for the invitation to this congress I got from Professor
Per-Olov Löwdin; it had a decisive impact on my scientific carrier),
suddenly I realized that this quantity, if multiplied with a constant,
will give a value 1, 2, and 3 for the simplest systems with single, double, and triple bonds. Then I also found that the new parameter has an
obvious analogy with the Wiberg index of the CNDO-type theories:
in the orthonormalized basis it simply reduces to the Wiberg index.
(There is a full analogy also in the sense that the CNDO energy
partitioning mentioned in the work of Fischer and Kollmar 18 also
exhibits an exchange energy component proportional to the Wiberg
index.12 ) Thus I had arrived at the definition of the ab initio bond
order index in terms of spin-orbitals:
β
Cµi Cνi .
µ∈A ν∈B
(40)
point
ĤAB =
nβ
i=1
BAB =
µ∈A ν∈B
As could be expected, the right-hand side contains a direct, or
“Coulombic,” term and a term of exchange type; note that the matrix
element (PS)µν can differ from zero only if the spin-orbitals χµ and
χν are of the same spin.
point
Thus the expectation value of the operator ĤAB is10
β
Pµν
In the closed-shell case Pα = Pβ and the definition (44) reduces
simply to12
χ̂µ+ χ̂ν+ ϕ̂ν− ϕ̂µ− = QA QB −
(PS)µν (PS)νµ .
µ∈A ν∈B
=
nα
(DS)µν (DS)νµ + (Ps S)µν (Ps S)νµ
(47)
µ∈A ν∈B
and not as eq. (46), which is valid in the closed-shell case only (The
corrections (44) and (47) had been published20 as an “addendum”
to my paper.12 )
Some Properties of the Ab Initio Bond Order Index
A very important property of the bond order index defined earlier
is that it is invariant with respect to the most general rotationalhybridizational transformations mixing the basis orbitals on the
individual atoms.3 In fact, one may consider Mulliken’s overlap
population and the bond order index as the simplest (if not the only)
invariant quantities, representing linear and quadratic combinations
of the interatomic density matrix elements, respectively.
It follows from the equality (40) that the emergence of the bond
order index BAB in the case of single determinant wave functions
Journal of Computational Chemistry
DOI 10.1002/jcc
Bond Order and Valence Indices: A Personal Account
may directly be connected with the fact that the expectation value of
the operator product N̂A N̂B differs from the product of the respective
expectation values: N̂A N̂B = N̂A N̂B = QA QB . In fact, one has
according to the equalities (40) and (43)
BAB = −2 N̂A N̂B − N̂A N̂B .
(48)
Soon after publication of my paper,12 de Giambiagi et al.21
rewrote the relationship (48) to the equivalent form
BAB = −2
N̂A − N̂A N̂B − N̂B (49)
which permits to give a “statistical” interpretation to the bond order
index: it measures the degree in which the fluctuations of the electron
populations on the two atoms—i.e., their deviations from the mean
(expectation) values—are correlated with each other: if the atoms
are connected with a covalent bond, then the decrease of the electron
density on the one atom involves the increase of it on the partner
atom, and vice versa.
It is to be noted that some authors use instead of the bond order
index as defined in eqs. (44) and (47) the Wiberg index calculated in a
Löwdin-orthogonalized counterpart of the actual basis set. This possibility was already stressed in the fundamental papers of Borisova
and Semenov,5, 7 and later it was independently proposed by Natiello
and Medrano.22 It has the disadvantage that the different quantities
(including atomic populations) calculated in a Löwdin basis are
not necessarily rotationally invariant—in particular, they are not if
the popular 6-31G** basis set is used.13, 14 The invariance problem
can be excluded13, 14 if one uses Davidson’s version of Löwdinorthogonalization,23 in which the orbitals on the individual atoms
are preorthogonalized, and one uses the Löwdin-orthogonalization
only for treating interatomic overlap. However, this scheme results
in completely different numbers than the conventional variant of
Löwdin-orthogonalization.
Another argument is connected with the delocalized and global
character of the Löwdin-orthogonalized basis functions, which may
be a source of unphysical effects. Thus, one can artificially change
the bond order values by changing the position of a basis function
that is completely unoccupied in the given wave function,24 but
overlaps with some of basis functions having an actual importance.
This indicates that well-pronounced local effects may be distorted
by the presence of some (nearly) empty basis orbitals centered in
other parts of the system.
Nonetheless, the use of a Löwdin-orthogonalized basis (preferably in the Davidson’s version) may be the only possibility of a
Hilbert-space analysis in the cases when the basis contains diffuse
functions lacking any pronounced atomic character and therefore
Mulliken’s atomic populations and the related bond order indices
defined earlier become ill-behaved. However, for basis sets of reasonably atomic nature the use of definitions eqs. (44) or (47) is
clearly preferable.
We may note here that there are cases in which one uses a plane
wave basis set in the calculations, and then performs a Hilbert-space
analysis by projecting the MO obtained on an auxiliary AO basis
set. Conceptually one can imagine a similar approach also in the
211
cases when the basis contains some off-centered orbitals, e.g., bond
functions.
Exchange Density and Bond Order
In the 1980s I had got 9, 25 a possibly deeper understanding of the
bond order indices by considering the normalization integral of the
exchange part of the second-order density matrix ρ2 (1, 2; 1 , 2 ).
Later I have understood3 that exactly the same considerations may
be accomplished by using a mathematically much simpler entity,
the exchange density. Exchange density is the diagonal part of the
exchange component of the second-order density matrix, which one
obtains if the primed and unprimed quantities are set equal (or Fermi
hole).
The electron density ρ(r ) gives the probability density of finding
an electron around the point r ; it can be calculated as the expectation value of the operator ρ̂(r ) = Ni=1 δ(ri − r ). Analogously, the
two-particle density ρ2 (r1 , r2 ) gives the probability density of finding one electron around the point r1 and, simultaneously, another
electron around the point r2 . It represents the expectation value of
the operator 3
ρ̂2 (r1 , r2 ) =
δ(ri − r1 )δ(rj − r2 )
(50)
i,j
(i =j)
ρ2 (r1 , r2 ) differs from the product of ρ(r1 )ρ(r2 ) because of the antisymmetry of the wave function, and of electron correlation, if the
latter is also taken into account. For single determinant wave function only the antisymmetry (“exchange”) plays a role and we may
define the exchange density ρ2x (r1 , r2 ) through the relationship
ρ2 (r1 , r2 ) = ρ(r1 )ρ(r2 ) − ρ2x (r1 , r2 ).
(51)
Assuming that we are using a single determinant wave function
built up of nα orbitals ai (r ) filled with spin α and nβ orbitals bi (r )
filled with spin β, then one can easily calculate the expectation
value of operator ρ̂2 (r1 , r2 ) by using the general formulae for matrix
elements of two-electron operators, and obtain after some simple
manipulations3 the expression for ρ2x (r1 , r2 ) as
2x (r1 , r2 ) =
nα
ai∗ (r1 )aj (r1 )aj∗ (r2 )ai (r2 )
i,j=1
+
nβ
bi∗ (r1 )bj (r1 )bj∗ (r2 )bi (r2 ).
(52)
i,j=1
Integrating 2x (r1 , r2 ) over both variables, we have
Journal of Computational Chemistry
2x (r1 , r2 ) dv1 dv2 =
nα
ai |aj aj |ai +
i,j=1
=
nα
i,j=1
DOI 10.1002/jcc
nβ
bi |bj bj |bi i,j=1
δij +
nβ
i,j=1
δij = nα + nβ = N.
(53)
212
I. Mayer
•
Vol. 28, No. 1
•
Journal of Computational Chemistry
By substituting here the LCAO expansion of the orbitals and
performing trivial manipulations, we get 3
m
[(Pα S)τ ν (Pα S)ντ + (Pβ S)τ ν (Pβ S)ντ ] = N.
(54)
ν,τ =1
We can group the terms on the left-hand side according to the atoms
on which the basis orbitals are centered:
[(Pα S)τ ν (Pα S)ντ + (Pβ S)τ ν (Pβ S)ντ ] = N.
Comparing this expression with the definition in eq. (44), we see that
the bond order BAB between atoms A and B is the diatomic contribution to the integral of the exchange density ρ2x (r1 , r2 ). (The factor 2
present in the definition in eq. (44) is also recovered, because the sum
for A and B in expansion (55) runs over all the atoms independently.)
We may note that eq. (54) could formally be obtained also as
a trivial consequence of the idempotency of the density matrices
(Pσ S)2 = Pσ S valid for the single determinant wave functions and
of the property Tr(Pσ S) = nσ . Based on this, it was also possible to introduce the product of three (and even more) density
matrices and define26 some genuinely three-center (or many-center)
bond order indices that may be used to identify true three-center
(many-center) chemical bonds—like the two-electron three-center
bonds in diborane molecule. (Note that the existence of a threecenter bond manifests also by the appearance of some conventional
“two-center” bond order—and the respective attractive exchange
interaction—between the two external atoms, even if they are too
far apart to have any direct interactions with each other.27 This effect
may be important for the stability of diborane molecule and similar
systems.)
Bond Orders in Homonuclear Diatomics
The derivation of the relationship (1) for homonuclear diatomics,
given by Borisova and Semenov for an orthogonal basis, could
not be generalized for the overlapping case, although the practice
indicates that this relationship does hold to a good accuracy provided that a minimal basis set is applied in the ab initio calculations
(c.f. Table 1). Therefore I have looked for another treatment.28
Table 1. Bond Orders of Singlet Homonuclear Diatomics Calculated by
Molecule
H2
Li2
Be2
B2
C2
N2
O2
F2
Bond order
1.0000
0.9980
1.9987
2.9994
3.3328
3.0000
2.0000
1.0000
χaµ
|χbν
= sµ δµν .
(55)
A,B ν∈A τ ∈B
STO-6G Basis Set at the Equilibrium Bond Distances.
We should utilize the invariance of the bond order indices with
respect to the rotational–hybridizational transformations of the basis
orbitals and Löwdin’s “pairing theorem”29 originally proved by
Amos and Hall30 (also see ref. 3). This means that we replace the
original bases of spatial orbitals {χaµ (r )} and {χbµ (r )} on the two
atoms by orthonormalized ones {χaµ
(r )} and {χbµ
(r )}, where subscripts a and b indicate that the orbitals in question are localized
on atoms A and B respectively, and subject the latter to unitary
transformation, providing the new basis orbitals to be also “paired”
(56)
These transformations do not change the subspace of the orbitals
assigned to the individual atoms, and so should leave invariant every
physically meaningful quantity. (In what follows we shall omit the
primes for the sake of simplicity.)
Relationship (56) means that a given basis orbital has a nonzero
overlap at most with one orbital of the other atom. (In this construct the pairs of orbitals that have nonzero overlap are often
called “corresponding orbitals”.) It follows from the symmetry of
the homonuclear diatomics that the pairs of corresponding orbitals
transform into each other under the interchange of the atoms (or
may be selected so if there are degenerate sµ values). We construct
the normalized sums and differences of the pairs of corresponding
orbitals as
φµb = [2(1 + sµ )]− 2 (χaµ + χbµ )
1
(57)
φµa = [2(1 − sµ )]− 2 (χaµ − χbµ ).
1
Here φµb and φµa are the µ-th bonding and antibonding orbitals, constructed of the pair of corresponding orbitals χaµ and χbµ . (It is
supposed that the pairing of the orbitals is performed by an algorithm3 in which the phases of the orbitals are selected so as to provide
the overlap integrals sµ to be nonnegative real numbers.)
Now we consider single determinant wave functions built up
of these orbitals filled with the respective spins σ = α or β. We
assume that the order of the basis orbitals is selected as χa1 , χb1 , χa2 ,
χb2 , . . . ; then both the overlap matrix S and the density matrices P σ
are block-diagonal with 2 by 2 nonzero blocks on the main diagonal.
As a consequence, the bond order index (44) will represent a sum
of the contributions originating from the individual pairs of basis
orbitals χaµ , χbµ . We may study them independently of each other,
for which it is sufficient to consider the respective 2 by 2 blocks of
the matrices S and P σ . We shall denote these blocks as Sµ and Pµσ ,
respectively.
Now, let us first consider the case when the bonding orbital φµb
is occupied with spin σ in the wave function, but its antibonding
counterpart is not. In that case, as it is easy to see,
Pµσ =
1
1
1
1
2(1 + sµ ) 1
(58)
and the matrix product Pµσ Sµ is
Pµσ Sµ =
Journal of Computational Chemistry
1
1
2(1 + sµ ) 1
DOI 10.1002/jcc
1
1
1
sµ
sµ
1
=
1 1
2 1
1
.
1
(59)
Bond Order and Valence Indices: A Personal Account
It follows from this result that (Pµσ Sµ )12 (Pµσ Sµ )21 = 41 , and
by substituting into the definition (44) we see that the orbital φµb
contributes 21 to the bond order BAB if it is occupied once. (There
is a common factor of 2 in the equation.) If this bonding orbital is
doubly occupied, its contribution to the bond order index is unity:
we get a contribution of 21 for both spins σ = α and β.
One gets exactly the same result if the antibonding orbital φµa
is singly or doubly occupied (but φµb is empty). Although for the
ground state of a homonuclear diatomics that can hardly be the
case, this fact indicates that the bond order indices (unlike the energy
components16, 17, 31 ) not always are able to distinguish between the
bonding and antibonding situations.
If both orbitals φµb and φµa are occupied with the spin σ , then we
get for matrix Pµσ
Pµσ =
1
1 − sµ2
1
−sµ
−sµ
1
(60)
and for Pµσ Sµ
Pµσ Sµ =
1
0
0
.
1
(61)
Therefore, if both the bonding and antibonding combinations of a
pair of corresponding orbitals are occupied with a given spin, then
there is a complete cancellation and the given pair of corresponding
orbitals does not contribute to the bond order index BAB .
These results indicate that in the homonuclear diatomics the
sufficient condition of getting a half-integer or integer value for
the bond order index is that the occupied orbitals coincide with the
bonding and antibonding combinations of corresponding orbitals
obtained in the pairing procedure. One may formally reduce to this
case also all wave functions in which there is only a single occupied
orbital in every symmetry species. In that case we may omit from
the basis all the hybrid orbitals forming the virtual MOs; as they are
empty, their omission changes neither the wave function nor any
parameter computed from it. Then we may perform the pairing of
the remaining basis orbitals and ensure the required block-diagonal
character of the matrices.
The conditions of the derivation were exactly fulfilled—even if
a minimal basis is used—only if one could neglect completely the
interaction (overlap) of the core orbitals—actually the 1s orbitals
(K shell) for the first row atoms—with all the orbitals of the partner atom. In that case the 1s AO would not be mixed with any other
orbitals in the pairing procedure, and their sum and difference would
coincide with the 1σg and 1σu canonic MOs. After the core orbitals
are separated out, the remaining part of the minimal basis is so small
that the sufficient conditions discussed earlier are usually satisfied
because of the symmetry considerations. (This is not the case, however, for the singlet C2 molecule (K. Jug, personal communication,
1985)). As the atomic number increases in the series from Li2 to F2 ,
the 1s orbitals become relatively more compact and more separated
from the valence shells, and thus the bond orders deviate less from
the ideal integer values.
For larger basis sets the conditions of the derivation are not fulfilled and the bond order indices differ from the respective classical
213
values—but not too much. Usually there is a significant interaction only between strongly overlapping basis orbitals, and for the
homonuclear diatomics this will lead to the consequence that the
hybrids building up the SCF orbitals do not significantly deviate
from those that one would get in the pairing procedure. (The most
important factor is the similarity of the hybrids making up the pairs
of bonding and antibonding orbitals corresponding to each other.
This requirement is not fulfilled for the C2 molecule even in a minimal basis set, and its bond order is far from an integer.) These
qualitative considerations hold also for heteronuclear diatomics and
for molecules containing more than two atoms. This means that the
bond orders will not deviate too much from their classical values—
although for such systems one does not get strict integers even in
minimal basis sets, as the bond orders are influenced by different
bond polarity and delocalization effects, too.
It may be of interest to note that for the homonuclear diatomics
treated at the minimal basis level—assuming that the core orbitals
may be considered fully separated—one obtains exactly the same
bond order values also by turning to the Löwdin-orthogonalized
basis and calculating the Wiberg indices for it. (This is not strictly
true in any other case.) The explanation is as follows. We start from
an orthonormalized and paired basis on the two atoms; then Löwdin
orthogonalization will mix only pairs of the corresponding orbitals,
owing to the block-diagonal character of the overlap matrix. Then
it is enough to observe that due to symmetry reasons the bonding
and antibonding orbitals coincide with the bonding and antibonding
combinations of the Löwdin-orthogonalized AOs, and the Wiberg
indices are also equal to the ideal bond multiplicity values, as it
was proved in the work of Borisova and Semenov5 discussed in
the introduction. It may be of interest to observe the following.
If the given occupied orbital and its antibonding counterpart are
both occupied, then one may replace the orbitals φµb and φµa by their
√
normalized sum and difference (1/ 2)(φµb ±φµa ), without changing
the determinant wave function. Now, owing to the difference of the
normalization factors in eq. (57), this sum and difference do not
recover the individual AO-s χaµ and χbµ , but—as it is discussed in
ref. 3—, give their Löwdin-orthogonalized counterparts. Thus, the
wave function in which both φµb and φµb are occupied is equivalent
to a wave function in which there are two strictly local orbitals in
terms of the Löwdin basis, which do not contribute to the Wiberg
index, of course.
The Ab Initio Valence Indices
We have discussed in the introduction the quantity bµ defined in
the spirit of Wiberg’s footnote.4 In the ab initio case we have to
take into account overlap and should define it through Mulliken’s
gross orbital populations (DS)µµ instead of simply Dµµ used in
eq. (4). A disadvantage of Mulliken’s gross orbital populations is
that they are not strictly limited to the interval between 0 and 2. As
a consequence, it is possible, at least in principle, that one gets a
negative bµ value for some basis orbitals. Thus, we have
bµ = 2(DS)µµ − (DS)2µµ .
(62)
In order to get valence, we shall sum the quantities bµ for the given
atom and extract from the sum the partial bond orders between the
Journal of Computational Chemistry
DOI 10.1002/jcc
214
I. Mayer
•
Vol. 28, No. 1
•
Journal of Computational Chemistry
orbitals within the given atom. In the closed-shell case this means
that we have to subtract
(DS)µν (DS)νµ
(63)
µ,ν∈A
(µ=ν)
and arrive to the definition
VA = 2
(DS)µµ −
(DS)µν DS)νµ .
µ∈A
(64)
µ,ν∈A
Alternatively, one may turn to the (generalized) natural hybrids32 in
which the intraatomic block of the matrix DS is diagonal, sum up
the quantities bµ in that basis and then return to the original one,
and obtain the same result.
It follows from the idempotency properties of the matrix DS that
in the closed-shell SCF case we have the relationship
VA =
BAB
(65)
B
(B =A)
quite similar to that which was valid in the semiempirical case.
Following the scheme of Armstrong et al. used in the CNDO
case,6 we accept the definition (64) also for the open-shell case. Then
equality (65) does not hold and we may define the free-valence index
FA as the difference (The definition (66) gives the free valence in
terms of the actual wave function and does not refer to any external
quantity. Therefore, it conceptually differs from the definitions of
the quantity also called “free valence” either in the old Hückel theory
or by Gopinathan and Jug8 in a semiempirical all-valence electrons
theory. In those cases free valence measures the deviation of the
actual sum of bond orders from some theoretical (maximal or ideal)
value)
FA = V A −
BAB .
(66)
B
(B =A)
In the RHF case the free-valence index of all atoms vanishes, FA =
0, while in the UHF one it can be expressed via the spin-density
matrix Ps as
FA =
(Ps S)µν Ps S)νµ
(67)
µ,ν∈A
(For an explicit derivation, see ref. 3.).
In light of this expansion, the square root of the free-valence
index may be in some sense considered as the “number of spins”
on the given atom. (Note that the sum of the free valences for all
the atoms is equal to 1 only in the simplest cases—e.g., one electron
delocalized along a regular polygon—but usually exceeds 1 because
the spin polarization phenomenon.)
Correlated Wave Functions
All the above-mentioned considerations were related to the SCF
wave functions. In the DFT case one can apply them to the single
determinant built up of the Kohn–Sham orbitals (see e.g., ref. 33)—
although, strictly speaking, the latter are not attributed any definite
physical meaning. It is an important question, how one should generalize the bond order and valence indices for the case in which
electron correlation is taken into account explicitly.
There are two conceptually different approaches to the definition
of bond order indices in the correlated case. They are based on two
different expressions, which, however, give coinciding results in the
single determinant case. The first starts from the expression (49) of
the correlation between the fluctuations of the atomic populations,
and uses it as a general definition of the bond order. That is equivalent
of using the whole difference
−ρ2xc (r1 , r2 ) = ρ2 (r1 , r2 ) − ρ(r1 )ρ(r2 )
(68)
between the actual pair density ρ2 (r1 , r2 ) and the product ρ(r1 )ρ(r2 )
of one-electron densities, and decomposing its integral according to
the different atoms. The notation ρ2xc in eq. (68) indicates that it
reflects both exchange (antisymmetry) and the correlation effects—
in other words it includes both the “Fermi-hole,” which is due to the
antisymmetry requirement, and the “Coulomb-hole” due to electron
correlation. This definition is seemingly very attractive, and no doubt
the quantity calculated in this manner may be of some interest.
However, this quantity has serious drawbacks if used for defining
the bond order, and I cannot recommend its use.
In my opinion, the most important argument against using a
bond order definition based on the relationships (49) or (68) is the
fact 25 that such a definition gives the value 0.39 for Weinbaum’s
classical wave function for H2 . This wave function represents the
solution of the full CI problem for the minimal basis of Slatertype orbitals with optimized exponents. It accounts for ∼84% of
the whole binding energy of the H2 molecule, which is pretty fair,
especially if one takes into account that the free hydrogen atoms are
described exactly in this basis. One simply must not call bond order
a quantity that is only 0.39 for the prototype single chemical bond in
the H2 molecule, described with Weinbaum’s prototype correlated
wave function. That is simply not chemical. I think that this is the
case, even if other correlated calculations of H2 give results34 in
which the deviation from unity is much less dramatic. The same
conclusion can be drawn from the results presented in ref. 35 in the
3D AIM (atoms in molecules) framework: bond indices exceeding
5 have been obtained for N2 and F2 by using the definition including
the Coulomb hole, while the “exchange only” definition discussed
below produced chemically reasonable numbers. It appears that the
authors of ref. 35 misunderstood the message of my paper,25 and
assumed that I am proposing the use of the relationship (49) and
have overlooked that I had proved there its inadequacy. Essentially
the “exchange only” scheme has been rediscovered also in ref. 36.
In one of my papers25 I had proposed to connect bond order
index with only the exchange part of the second-order density matrix
which is formally constructed from the first-order density matrix
exactly in the same manner as in the single determinant case. The
same can be done for the simpler entity, the exchange density, used in
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Bond Order and Valence Indices: A Personal Account
215
Figure 1. C C and C H bond orders, and total and free valences of carbon for the dissociation of the
ethylene molecule into two triplet methylenes, treated at the (4,6) CAS level of theory by using 6-31G**
basis set.
the present paper. One has simply to observe that the expansion (52)
is nothing else than a particular case of a more general expression
2x (r1 , r2 ) =
niα njα ai∗ (r1 )aj (r1 )aj∗ (r2 )ai (r2 )
i,j=1
+
β β
ni nj bi∗ (r1 )bj (r1 )bj∗ (r2 )bi (r2 )
(69)
i,j=1
where ai and bi are the natural spin-orbitals of spins α and β respectively. (One obtains eq. (52) from eq. (69) by observing that in the
single determinant case the occupation numbers niσ (σ = α or β)
are equal to either 1 or 0.)
The use of the exchange density eq. (69) means that one has to
use exactly the same eqs. (44) or (47) and (64), (66) for defining
bond orders, and total and free valences in the correlated case, too.
Thus there is no need to work with the second-order density matrix,
as were the case if we used the fluctuation type definition, and so
the calculation of bond orders and valences may be accomplished
in the correlated case exactly in the same manner in terms of the
first-order matrix alone, as one does for single determinant wave
functions. The only difference is that the equality (65) does not hold
any more even for closed-shell molecules, and thus FA = 0 in the
correlated case.
It is my opinion that this way of defining the bond order and
valence indices for the correlated case is very chemical, and is
applicable not only near the equilibrium distances, but is able to
describe the whole process of bond formation/dissociation. This is
illustrated well in Figure 1, which displays some results for the ethylene molecule dissociating into two triplet methylenes, as calculated
with a (4,6) CAS wave function by using 6-31G** basis set. One
may see that the C C bond order that is nearly 2 at the equilibrium
distance gradually decreases and tends to 0 at the large distances—
as it should. Simultaneously with this, there appears a free valence
on the carbon, tending to a limit close to 2 at the large distances, in
agreement with the fact that there are two unpaired electrons in the
triplet methylene. The sum of the C C bond order and of the carbon free valence is almost constant, thus the carbon atom remains
practically four-valent during the whole dissociation. (Note that the
ground state of methylene is the triplet; contrary to the CAS scheme,
the RHF method is only able to describe dissociation of ethylene
into two singlet methylenes and results in divalent carbons with no
free valences.) (The C H bond order stays nearly constant at a value
close to 1.)
Atomic Resolution of Identity and 3D Analyses
Atomic Resolution of Identity
As already noted, the concept of atoms within a molecule is not a
well cut one. Strictly speaking, the atoms do not directly appear in
the quantum mechanical description of molecules: the Schrödinger
equation is written down for the individual particles (electrons
and nuclei). Thus one has to introduce some definition of the
atom “from outside,” which necessarily means some arbitrariness.
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The Hilbert space analysis considered earlier is based on the fact
that practical quantum chemistry mostly uses atom-centered basis
sets. However, neither the use of basis atom-centered basis sets nor
the Hilbert-space approach to the analysis is obligatory. A widely
used alternative is the 3D analysis in which the physical space is
decomposed into regions attributed to the individual atoms.
Most recently36 we have proposed the “atomic decomposition
of identity,” permitting to treat different definitions of the AIM on
equal footing, i.e., in a framework of a common formalism. It is
applicable for analyzing different physical quantities in terms of
contributions coming from the individual atoms or pairs of atoms,
including population analysis, bond orders, and energy partitioning
schemes. In this approach one presents the identity operator Î as
a sum of some operators ρ̂A corresponding to the individual atoms
according to the scheme of analysis selected:
Î =
we get the expansion (72) with the definition of the gross atomic
populations
QA =
A
(DSA )µµ =
Dµν Sνµ
.
It is trivial to check that the definitions (73) and (76) lead to identical
results.
By introducing two atomic resolutions of identity into each integral in the expansion (71), one to the “bra” and one to the ket, we
can get the decomposition of the total electron charge into the sum
of net atomic populations qAA and overlap populations qAB
N=
ρ̂A .
(76)
µ∈A ν
µ∈A
qAA +
A
(70)
qAB
(77)
A,B
(A =B)
A
Note that the individual operators ρ̂A are not necessarily Hermitian.
The use of this approach is very simple. For instance, the total
number of electrons may be expressed in terms of the natural spinβ
orbitals ai and bi having occupation numbers niα and ni introduced
earlier as
N=
ρ(r ) dv =
niα ai |ai +
i
with
qAA =
niα ai |ρ̂A† ρ̂A |ai +
i
(71)
i
qAB =
By inserting an “atomic resolution of identity” to each “ket,” one can
present the number of electrons as a sum over the atomic electron
populations
niα ai |ρ̂A† ρ̂B |ai +
QA
niα ai |ρ̂A |ai +
i
(79)
respectively. It is easy to see that the equality
QA = qAA +
(72)
qAB
(80)
B
(A =B)
where
QA =
β
ni bi |ρ̂A† ρ̂B |bi i
A
(78)
and
β
ni bi |bi .
i
N=
β
ni bi |ρ̂A† ρ̂A |bi i
β
ni bi |ρ̂A |bi .
(73)
holds.
By integrating the two sides of the expression (69) inserting one
atomic decomposition of identity to each ket we may write
i
2x (r1 , r2 ) dv1 dv2
Alternatively, by using the LCAO expansion of the first-order
density matrix, one can write
N=
µ
(DS)µµ =
=
Dµν Sνµ
nα
niα njα ai |aj aj |ai +
i,j=1
(74)
µ,ν
=
A,B


SAνµ = χν |ρ̂A |χµ (75)
β β
ni nj bi |bj bj |bi i,j=1
nα
niα njα ai |ρ̂A |aj aj |ρ̂B |ai i,j=1
point
where, as in Section Expectation Value of ĤAB and the Bond Order
Matrix, D = Pα + Pβ . By substituting the atomic resolution of
identity (70) to the ket in each Sνµ = χν |χµ , and introducing the
“atomic overlap matrices” SA with the elements
nβ
+
nβ

β β
ni nj bi |ρ̂A |bj bj |ρ̂B |bi  .
(81)
i,j=1
Thus, by picking up the contribution to this sum, corresponding
to a given pair of atoms A and B, we get the most general expression
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217
of the bond order as
Atomic Operators: 3D Analysis

In the 3D analysis one introduces a weight function wA (r ) for each
atom and every point r of the 3D space, which should satisfy the
conditions
BAB = 2 
nα
niα njα ai |ρ̂A |aj aj |ρ̂B |ai 
i,j=1
+
nβ
β β
ni nj bi |ρ̂A |bj bj |ρ̂B |bi 
wA (r ) ≥ 0;
(82)
wA (r ) = 1
(87)
A
i,j=1
everywhere. Then the atomic operator ρ̂A may be defined as
where the factor 2 is introduced because in eq. (81) the sum over
atoms A and B runs independently and all terms are symmetric with
respect to the interchange of A and B. Substituting here the LCAO
expansion of the natural orbitals, this can simply be rewritten as
BAB = 2
Pα SA µν Pα SB )νµ + (Pβ SA )µν (Pβ SB )νµ . (83)
µ,ν
Analogously, the general definition of the total valence can be
given as
VA = 2
DSA )µµ −
(DSA )µν (DSA )νµ
µ
(84)
µ,ν
ρ̂A = wA (r )|r =r .
Here the notation r = r indicates that one should replace r by
r after the action of all the operators on the wave functions (as
functions of r ) had been evaluated, but before the integration over r
is carried out. Thus, quantum mechanical operators act only on the
electronic wave functions but not on the weight function wA (r ).
The AIM Case
Two conceptually different types of the weight functions are possible. In the first case the 3D space is decomposed into disjunct atomic
domains A , which means that in each domain one weight function
is equal to 1 and all the others vanish:
wA (r ) =
and eq. (66) applies for the free valence.
Atomic Operators: Hilbert-Space Analysis
The Hilbert-space analysis discussed in the previous sections may
be recovered if one defines the atomic operators as
ρ̂A =
|χµ ϕµ |
(85)
A
Sµν
=
A
= χν |ρ̂A |χµ =
Sνµ
Sνµ
0
if µ ∈ A
.
otherwise
(86)
This result means that the atomic overlap matrix consist of the
“intraatomic” columns of the original overlap matrix, while all the
other columns are zeroed. (Thus the atomic overlap matrices trivially
sum to the original one, as they should.) It is easy to see that by using
the atomic overlap matrix given by the equality (86) we recover
immediately Mulliken’s gross, net, and overlap populations from
the general expressions (76), (78), and (79), respectively, and the
expressions (83) and (84) of the bond order and valence also become
identical with the definitions in eqs. (44) and (64) respectively.
1
0
if r ∈ A
.
otherwise
(89)
The most widely used scheme of this type is Bader’s Atoms in
Molecules (AIM) Theory,37 in which the dividing surfaces between
atomic domains are determined based on the topological properties of the electron density. Owing to the disjunct character of the
domains, the atomic overlap integrals are given simply as
µ∈A
where |ϕµ is the biorthogonal counterpart of the basis orbital |χµ ,
introduced in eq. (10). Then, according to the definition (75) the
elements of the atomic overlap matrix become
(88)
wA (r )χµ∗ (r )χν (r ) dv =
A
χµ∗ (r )χν (r ) dv
(90)
i.e., the integration is restricted to the given atomic domain A .
Bond orders, and total and free valences are then given by eqs. (83),
(84), and (66) mentioned earlier.
Besides bond order and valence calculations, the AIM scheme
has a great number of other useful applications, which are out of our
present scope. The fact that the atomic domains are defined on the
basis of the electron density calculated from actual wave function
represents a very important advantage of the AIM method, especially because it reduces the arbitrariness in defining the individual
atoms. It is also the source of some minor drawbacks. Thus, the
atomic domains have complex (and physically not always appealing) forms, which make costly their determination and the numerical
integrations according to eq. (90). (But that is a more serious problem if one wishes to do energy partitioning17, 38 than for bond order
calculations.) Also, in some cases there appear so-called nonnuclear
attractors i.e., domains containing no nuclei—e.g., in the middle of
the acetylene triple bond. They reflect some real physical peculiarities of the electron density and are not “artifact” in that sense, but
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it is rather difficult to find any chemical interpretation for them. For
some systems, e.g., for boron compounds, the total atomic charges
are also difficult to interpret.
In the AIM scheme all the overlap populations vanish (qAB = 0
if A = B) because of the disjunct character of the domains, and
therefore the net and gross atomic populations coincide:
Qa = qAA =
w(r )ρ(r ) dv =
A
ρ(r ) dv
(91)
where ρ(r ) is the electron density
ρ(r ) =
µ,ν
Dµν χµ∗ (r )χν (r ).
(92)
Of course, one can also apply eq. (76).
The bond order index has been introduced in the AIM theory by
Ángyán et al.39 Somewhat later, Fradera et al.40 introduced a “delocalization index” which in the single determinant case is identical
with the index of Ángyán et al.,39 and most authors who are using
it for the the Hartree-Fock or DFT cases are not aware of the original paper.39 The delocalization index of Fradera et al.40 is defined
in such a way that for the explicitly correlated case it involves the
use of the total ρ xc (r1 , r2 ) in eq. (68) mentioned earlier, which,
as already noted, I cannot recommend. Bader and Stephens41 have
introduced a parameter similar to bond order as early as in 1975, but
not for atoms but for the so-called two-electron “loges,” i.e., spatial
domains housing an electron pair—a concept, which was introduced
by Daudel in the 60s–70s and has became completely obsolete nowadays. As already mentioned, the use of the definition in ref. 40 leads
to chemically meaningless bond indices exceeding 5 for the N2 and
F2 molecules,35 while no such problem occurs if one uses the definition advocated here (and also by Ángyán et al.39 ). Thus the use of
definition (83) based on the “exchange only” part (69) of the secondorder density matrix is more chemical. (As already noted, it is much
more economical, as well, as it requires the use of the first-order
density matrix only, and not that of the second-order one.)
The “Fuzzy Atoms” Case
An alternative to using disjunct atomic domains is to use so-called
fuzzy atoms, i.e., such divisions of the 3D space into atomic regions
in which the regions assigned to the individual atoms have no sharp
boundaries but exhibit a continuous transition from one to another.
Fuzzy atoms were first introduced by Hirshfeld,42 who defined the
hypothetical “promolecule” consisting of unperturbed and noninteracting free atoms placed at the atoms’ actual positions in the
molecule, and then the weight function w(r ) is calculated by using
the atomic electron densities of these free atoms as
0 (r )
wA (r ) = A
.
K0 (r )
(93)
K
This is often called the “stockholder’s principle.” Obviously definition (93) satisfies conditions (87). This is a transparent and
easy-to-apply definition; it has, however, serious drawbacks, too.17
The hydrogen atom has no core electrons and, as a consequence, the
wA (r ) function of the heavy atom to which the hydrogen is attached
usually has a significant value at the proton’s position. That means
the electron density near the proton is partly assigned to the adjacent
heavy atom, and thus the individual atoms are not “well cut.”
Recently we have proposed to use Becke’s weight function43
for calculating bond order and valence indices,44 as well as to use
it in energy partitioning.45 Becke originally proposed that function
for performing effectively the numerical integrations necessary in
the DFT framework. This is a relatively simple algebraic function,
which is calculated in an iterative manner. We shall not repeat here
the algorithm, which is almost easier to program46 than to describe
(see the original paper 43 and the appendix in our paper 44 ), but only
mention that it satisfies requirements (87) and has the value strictly
equal to 1 at the position of “own” nucleus (therefore the weight
functions corresponding to all the other atoms vanish there). The
nature of the atoms is introduced by the use of empirical atomic
radii; actually only the ratio of the radii of atoms that are close
to each other is of importance. There is also a stiffness parameter determining the speed of transition from one atomic region to
another. It may be also mentioned that one may locate the stationary point of charge density along the interatomic axis connecting
bonded atoms and use its position for determining the ratio of
the atomic radii. In this manner one gets a scheme that combines
fuzzy atoms with some advantages (and disadvantages) of the AIM
method.
In the fuzzy atoms scheme one has nonzero values of the overlap densities, which may be calculated either by using eq. (79), or
simply as
qAB =
wA (r )wB (r )ρ(r ) dv.
(94)
This “fuzzy atoms overlap population” is a 3D analogue of
Mulliken’s overlap populations used in the Hilbert-space analysis,
and similarly, it reflects the degree to which the electronic charge
is shared between the atoms A and B—but it cannot be called bond
order either. The fuzzy atoms’ bond orders and valences may be
calculated by using eqs. (83), (84), and (66) with the values of the
atomic integrals calculated as
A
Sµν
=
wA (r )χµ∗ (r )χν (r ) dv.
(95)
The “fuzzy atoms” scheme permits the numerical integrations to
be performed very easily, and has also the important advantage that
the results exhibit quite moderate basis set dependence and, unlike
the results of Hilbert-space analysis, do have a well-defined basis
set limit, and thus may be used also for large basis sets, or those
including diffuse orbitals. The AIM results do have a basis set limit,
too.
Our program FUZZY46 —it may be downloaded freely and is
applicable after a “Gaussian” run—performs calculations of both
Hilbert space and fuzzy atoms bond orders and valences and fuzzy
atoms overlap populations, for the case of both SCF and correlated
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Bond Order and Valence Indices: A Personal Account
wave functions. Programs BORDER and BO-VIR, applicable also
for large systems, are presently designed for the SCF (RHF. UHF,
may be DFT) case only, but would require only minor changes—
input of the density matrices and not orbital coefficients—for the
correlated calculations.
Discussion
Bond order and valence indices may have two different roles. First,
they help to link the quantum mechanical description of molecules
as systems of particles with their classical chemical picture of atoms
connected by bonds, and to find quantum mechanical counterparts
of such genuine chemical parameters as the multiplicity of a bond
and the actual valence of an atom in a molecule. It is my opinion
that this possibility has a great conceptual importance.
The bond order and valence indices have also different practical applications. First of all, they may be used for interpretation
and systematization of the results, obtained in quantum chemical
calculations. In this respect, bond orders and valences represent theoretical tools permitting to extract from the wave function different
pieces of information that may be assigned chemical significance.
(As the wave functions usually represent very large sets of numbers for a human, this is also useful to get a comprehensible picture
of the molecule, and may be considered a sort of data compression.) Probably this is the manner in which the indices have been
applied in most cases (for e.g., in ref. 47). One may identify chemical bonds, intramolecular hydrogen bonds, or hydrogen-bond like
interaction, get an idea about the intensity of nonbonded interactions, etc. (As noted earlier, bond orders cannot always distinguish
between bonding and antibonding character of the secondary interactions; energy components are more sensitive (and expensive)
parameters.16, 17, 31, 45 ), decide whether the atoms in question may be
considered bonded at all… (I myself am not a chemist and probably
could not survey properly the applications of this type.)
As there are strong correlations between the strengths of the different bonds and their bond orders calculated by a given basis set of
sufficiently atomic character (e.g., the classical 6-31G** basis), in
some cases one may use bond orders not only for interpretations but
also for predictions: one could, for example, predict which bonds
will undergo rupture in a pyrolysis experiment. (That was possible
even by using Wiberg indices at the CNDO/2 semiempirical level
of theory.48 ) Of course, weaker bonds are usually longer and vice
versa, and so in some cases it may be difficult to decide whether the
bond is weak because some external (e.g., steric) effects force it to
be longer or it is long because it is weak due to some features of
the electronic structure. Our method for predicting the primary bond
cleavages in the electron impact mass spectroscopic experiment 49, 50
is truly predictive because it is based on effects of this latter type.
In that method we consider a Koopmans-type vertical ionization
of the molecule, deleting one electron from a canonic HF orbital,
and comparing the bond orders (and often also energy components)
calculated for the ion and the parent neutral molecule. If there is a
bond the bond order index of which is dropping dramatically as a
consequence of ionization, then one may expect that it will be ruptured in the MS experiment. There are systems for which ionization
from the HOMO does not lead to any bond cleavage—they usually
exhibit a big peak corresponding to the original molecular ion—and
219
fragmentation occurs only if an electron is removed from a lower
lying orbital. This is the case e.g., for molecules in which the HOMO
is a π-orbital of an aromatic ring, ionization from which does not
lead to fragmentation. In such cases characteristic fragments may
be due to ionization from, e.g., a lone pair of a heteroatom in a side
chain. However, it if often difficult—or even impossible—to get
converged SCF solutions for such “hole states”; our experience indicates that it is not necessary either: in order to see what will happen
in the MS apparatus, it is sufficient to consider the Koopmans-type
vertical ionization of the neutral molecule without considering any
orbital relaxation. (Moreover, that also usually works at the semiempirical MNDO level, too—in particular, if combined with energy
decomposition.51, 52 )
Although the predictions discussed earlier are related to the possible outcome of some dynamic events, they are based on static
parameters, and must be considered with due care, as they are giving some information only about the starting point of the process
in question. It is, however, also possible to follow some dynamic
processes in some detail, by investigating different points of the
reaction path on the potential surface. Thus, for instance, in one
of our works53 we could detect (by computing fuzzy atoms bond
orders) how the presence of a Ni atom assists in a process of forming a new C C bond by first developing a “transient” Ni C bond
along the reaction path, which is then replaced by the newly formed
C C one. Very instructive are the studies of Lendvay,54, 55 who
studied the changes of bond orders along the reaction paths of different reactions. He demonstrated that by calculating bond orders
one may follow in detail how one bond is breaking and another is
forming along the reaction path, check the “bond order conservation rule,” study whether the electronic structure of the transition
state is more similar to the reactants or to the products, reveal
whether the reaction is synchronous or asynchronous, and so on,
i.e., the use of bond order and valence indices permits to answer the
different qualitative questions a chemist may put forward about a
reaction.
Acknowledgments
I am deeply indebted to my coworkers—coauthors of several papers
on the subject—Dr. Andrea Hamza (Budapest) and Dr. Pedro
Salvador (Girona), who in the recent years participated in this
research. I am very grateful to Dr. G. Lendvay for many useful
discussions.
Appendix A: Proof of the Relationship Eq. (19)
For an arbitrary wave function (it need not be a single determinant)
which can be expanded by using the AO basis {χµ } of spin-orbitals,
the spatial distribution of the electron charge and spin density is
given by the expansion
ρ(r , s) =
µ,ν
Pνµ χµ∗ (r , s)χν (r , s).
As ρ(r , s) is real, the “density matrix” P is Hermitian.
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ρ(r , s) is a quantum mechanical observable, it is equal to the
expectation value of the respective operator
ρ̂(r , s) =
N
δ(r − ri )δssi .
According to the anticommutation relationship (12) we have for the
anticommutator
{φ̂i+ , ϕ̂ν− } =
(A2)
µ
Cµi {χµ+ , ϕ̂ν− } =
Cµi δµν = Cνi .
(B3)
Cκi Cj |(φi → χµ ; φj → χν )
(B4)
µ
i=1
As any symmetric sum of the one-electron operators, ρ̂(r , s) has
also a second quantized representation in terms of the auxiliary
Löwdin-orthonormalized basis of spin-orbitals {ψµ }:
ρ̂(r , s) =
µ,ν
µ,ν,ε
χ̂µ+ χ̂ν+ ϕ̂− ϕ̂κ− | =
ψµ (1)|δ(r − r1 )δss1 |ψν (1)ψ̂µ+ ψ̂ν− .
(A3)
where |(φi → χµ ; φj → χν ) is the determinant wave function,
which can be obtained from | by replacing spin-orbitals φi and φj
with the basis spin-orbitals χµ and χν , respectively. It follows from
the general expression4 for the overlap of two determinant wave
functions built up of nonorthogonal orbitals, that
φi |χµ φi |χν |(φi → χµ ; φj → χν ) = .
φj |χµ φj |χν −1
Sεµ
χµ (1)|δ(r − r1 )δss1 |χν 1χ̂ε+ ϕ̂ν− . (A4)
Owing to the presence of the delta-function and of the Kronecker symbol, one can perform explicitly the integration and the
summation over the spins:
ρ̂(r , s) =
µ,ν,ε
−1 ∗
Sεµ
χµ (r , s)χν (r , s)χ̂ε+ ϕ̂ν− .
µ,ν,ε
ε
σ
(A6)
Now, by comparing with the general formula (A1) we have
φk |χλ =
Cσ∗ k Sσ λ
(B6)
and using the expansion (B4) we get
−1
Sεµ
χ̂ε+ ϕ̂ν− χµ∗ (r , s)χν (r , s).
Pµν =
(B5)
Expanding the 2 by 2 determinant in (B5) by substituting the
integrals as
(A5)
By taking the expectation vale we have
ρ(r , s) =
i,j
Substituting here the expansion (13) of the functions ψ and expressing operators ψ̂µ+ and ψ̂ν− through the operators χ̂µ+ and ϕ̂ν− by using
the relationships (14), we get
ρ̂(r , s) =
As a consequence,
−1
Sεµ
χ̂ε+ ϕ̂ν− |χ̂µ+ χ̂ν+ ϕ̂− ϕ̂κ− |
∗
∗
∗
∗
Cκi Cj
Cσ i Sσ µ
Cλj Sλν −
C σ i Sσ ν
Cλj Sλµ
=
σ
i,j
λ
σ
λ
= (PS)κµ (PS)ν − (PS)κν (PS)µ .
(B7)
(A7)
References
or, after trivial manipulations:
χ̂ε+ ϕ̂ν− = (PS)νµ =
Pν Sµ .
(A8)
Appendix B: Proof of the Relationship Eq. (39)
Let
| = φ̂1+ φ̂2+ · · · φ̂N+ |vac
(B1)
be an N-electron single determinant wave function built up of the
orthonormalized spin-orbitals
φi =
µ
Cµi χµ .
(B2)
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