Degree of atomicity in the chemical bonding Why to return to the H 2molecule Maciej Hendzeland J ozef Spa leky Institute of Theoretical Physics Jagiellonian University
2025-04-24
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Degree of atomicity in the chemical bonding: Why to return to the H2molecule?
Maciej Hendzel∗and J´ozef Spa lek†
Institute of Theoretical Physics, Jagiellonian University,
ul. Lojasiewicza 11, PL-30-348 Krak´ow, Poland
(Dated: November 3, 2022)
We analyze two-particle binding factors for the case of H2molecule with the help of our origi-
nal Exact Diagonalization Ab Intio (EDABI) approach. Explicitly, we redefine the many-particle
covalency and ionicity factors as a function of interatomic distance. Insufficiency of those basic
characteristics is stressed and the concept of atomicity is introduced and corresponds to the Mott
and Hubbard criteria concerning the localization in many-particle systems. This additional char-
acteristic introduces atomic ingredient into the essentially molecular states and thus eliminates a
spurious behavior of the standard covalency factor with the increasing interatomic distance, as well
as provides a physical reinterpretation of the chemical bond’s nature.
I. INTRODUCTION AND MOTIVATION
The concept of chemical bond as the fundamental
quantum–mechanical characteristic of molecules such as
H2, was firmly established by Heitler–London [1] in 1927.
This pioneering quantitative paper was based, by today’s
standards, on the Hartree–Fock approximation for the
two–particle wave function of the two electrons in H2
molecule. Later, this function has been expressed by the
corresponding atomic 1shydrogen wave functions in the
form of symmetrized product with antisymmetrized spin
part, the latter reflecting the spin–singlet ground state.
Such a selection of the component atomic wave func-
tions, represented a rather drastic approximation and
has been corrected subsequently by selecting their su-
perposition of those atomic wave functions into molecu-
lar single–particle wave functions centered on individual
atoms, which have been subsequently put into a proper
two–particle form [2]. This whole procedure established
a canonical viewpoint of the covalent bond, with a de-
gree of ionicity (double occupancy of individual atoms)
introduced ad hoc later to it (Valence Bond Theory) [2].
Theory of the bonding reached its mature form with an
excellent series of papers by Ko los nad Wolniewicz [3, 4]
who have included higher (virtually) excited states, sup-
plemented with the nuclear vibrations [5] to a fully quan-
titative form, which has been subsequently tested exper-
imentally, since the bonding in H2molecule represents
one of the tests of quantum–mechanical–theory verifica-
tion in quantum chemistry [6].
In this brief paper we address, first of all, the question
why we must realize that there is a need to return to the
problem origins of the bonding nature in the H2molecule.
Namely, we have observed recently that the two–electron
wave function, representing the single bond, composed of
originally 1selectrons of hydrogen atoms contains an in-
herent inconsistency when we interpret covalency in the
standard manner [7, 8]. Explicitly, when starting from
∗maciej.hendzel@doctoral.uj.edu.pl
†jozef.spalek@uj.edu.pl (corresponding author)
an exact solution of the Heitler–London problem (with
proper molecular single–particle wave functions included
at the start), we have detected that the covalency in-
creases with the increasing distance between the nuclei,
a clearly unphysical feature. As a subsidiary observation
we have noted that the Heitler–London (Hartree–Fock)
two–electron wave function leads to nonzero (actually,
maximal) value of covalency in the limit of entirely sep-
arated atoms. Such an inconsistency has brought to our
attention the old concept of Mott [9], concerning the elec-
tron localization in condensed matter physics (see also
[10, 11]). In effect, we have decided to introduce the con-
cept of atomicity in the context of the correlated molec-
ular electronic states [7]. This concept represents a novel
nontrivial feature of the chemical bond, since it is intro-
duced as an external factor into an essentially molecular
(collective) language of the covalent bonding, including
also the ionicity. Hence, in this paper we summarize and
mainly interpret our recent results [7, 8] which, in our
view, provide a connection between (correlated) states of
small molecules and condensed matter physics, as well as
delineate the essential difference between the two.
The structure of this paper is as follows. In the next
Section we briefly summarize our method and in Sec. III,
regarded as the main part, we discuss our results and
their meaning. This is followed by a brief Outlook. In
general, the aim of the paper is to supplement previous
papers [7, 8] with detailed discussion and interpretation
of the results. Such a discussion may be of importance
when the concept of atomicity is analyzed for more com-
plicated bonds such as C–C in the hydrocarbons. The
connecting link between the condensed matter localiza-
tion and molecular atomicity may be then applied also
to other nano–systems [11].
II. METHOD
Our approach is based on Exact Diagonalization Ab
Initio (EDABI) method which has been proposed and de-
veloped in our group [12, 13]. Here we use this method to
provide complementary bonding characteristics on exam-
ple of H2molecule. The starting Hamiltonian, containing
arXiv:2210.06524v4 [physics.atm-clus] 2 Nov 2022
2
all Coulomb interactions, formulated in the second quan-
tization language, is of the form
ˆ
H=aX
i
ˆniσ +X
ijσ
0tij ˆa†
iσ ˆajσ +UX
i
ˆni↑ˆni↓
+1
2X
ij
0Kij ˆniˆnj−1
2X
ij
0JH
ij ˆ
Si·ˆ
Sj−1
4ˆniˆnj
+1
2X
ij
0J0
ij (ˆa†
i↑ˆa†
i↓ˆaj↓ˆaj↑+ H.c.)
+1
2X
ij
0Vij (ˆniσ + ˆnjσ )(ˆa†
i¯σˆaj¯σ+ H.c.) + Hion-ion ,(1)
where H.c. denotes the Hermitian conjugation, ˆaiσ (ˆa†
iσ)
are fermionic annihilation (creation) operators for state i
and spin σ, ˆniσ ≡ˆa†
iσ ˆaiσ, and ˆni≡ˆni↑+ ˆni↓≡ˆniσ + ˆni¯σ.
The spin operators are defined as ˆ
Si≡1
2Pαβ ˆa†
iασαβ
iˆaiβ
with σirepresenting Pauli matrices. The primed sum-
mations mean that i6=j. The Hamiltonian contains
the atomic and hopping parts (∝aand tij , respec-
tively), the so-called Hubbard term ∝U; representing
the intra-atomic interaction between the particles on the
same atomic site iwith opposite spins, the direct intersite
Coulomb interaction ∝Kij , Heisenberg exchange ∝JH
ij ,
and the two-particle and the correlated hopping terms
(∝J0
ij and Vij , respectively). The last term describes
the ion-ion Coulomb interaction which is adopted here in
its classical form.
By way of diagonalization of Hamiltonian (Eq. (1))
one can write ground state energy with the ground–
state two–particle wave function, obtained in the form
ψG(r1,r2) = ψcov(r1,r2) + ψion(r1,r2), where ionic and
covalent parts are
ψcov(r1,r2) = 2(t+V)
p2D(D−U+K)[w1(r1)w2(r2) (2)
+w1(r2)w2(r1)][χ↑(1)χ↓(2) −χ↓(1)χ↑(2)],
ψion(r1,r2) = −1
2sD−U+K
√2D[w1(r1)w1(r2) (3)
+w2(r2)w2(r1)][χ↑(1)χ↓(2) −χ↓(1)χ↑(2)],
with
D≡p(U−K)2+ 16(t+V)2,(4)
and
wiσ(r) = β[φiσ (r)−γφjσ (r)],(5)
with i= 1, j= 2 or i= 2, or j= 1, in this case. The
two functions are molecular functions and come out nat-
urally within our method, in which the two neighboring
atomic functions φi(r) are mixed, with βand γas mixing
parameters. These atomic functions can be in the form
of Slater or Gaussian form (Slater or Gaussian type or-
bitals, STO or GTO). Furthermore, Eqs. (2) and (3) can
be rewritten, with use of Eq. (5), in the following way
[7]
ψcov(r1,r2) = Cβ2(1 + γ2)−2γIβ2[φ1(r1)φ2(r2)
(6)
+φ2(r1)φ1(r2)][χ↑(1)χ↓(2) −χ↓(1)χ↑(2)],
and
ψion(r1,r2) = Iβ2(1 −γ2)−2γCβ2[φ1(r1)φ1(r2)
(7)
+φ2(r1)φ2(r2)][χ↑(1)χ↓(2) −χ↓(1)χ↑(2)],
where Cand Iare coefficients from (2) and (3), respec-
tively.
Parenthetically, for the sake of comparison one can
write postulated VB two–particle wave function
ψV B
cov (r1,r2) = 1
p2(1 + S2)[φ1(r1)φ2(r2) + φ2(r1)φ1(r2)]
(8)
×1
√2[χ↑(1)χ↓(2) −χ↓(1)χ↑(2)],
and
ψV B
ion (r1,r2)=[φ1(r1)φ1(r2) + φ2(r1)φ2(r2)] (9)
×1
√2[χ↑(1)χ↓(2) −χ↓(1)χ↑(2)],
where Sis the overlap between the neighboring atomic
wave functions. However, the total wave function, con-
sisting of sum of the (8) and (9) has not been obtained
directly as a solution of the respective Schr¨odinger equa-
tion, whereas in our approach its form comes out ex-
plicitly from our exact solution and represents the exact
treatment of the Heitler–London problem.
Based on these functions we redefine the ionicity and
covalency [8] and define atomicity [7], the last is the com-
plementary characteristic to the two former.
A remark is in place at this point. As said above, he
two–electron component wave functions (6) and (7) have
formally the same form as their VB correspondents (8)
and (9), albeit with the two principal differences. First,
the coefficients before the covalent and ionic parts, ψcov
and ψion, are different as they contain all Coulomb–
interaction terms between the particles composing the
bond. Second, the orbital size (α−1) of the original
atomic wave functions, composing those functions are ad-
justed in the resultant two–particle ground state. These
two factors, in addition to the exact expression for the
two–particle wave function, are the qualitative differences
with the original Heitler–London theory.
摘要：
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Degreeofatomicityinthechemicalbonding:WhytoreturntotheH2molecule?MaciejHendzelandJozefSpalekyInstituteofTheoreticalPhysics,JagiellonianUniversity,ul.Lojasiewicza11,PL-30-348Krakow,Poland(Dated:November3,2022)Weanalyzetwo-particlebindingfactorsforthecaseofH2moleculewiththehelpofourorigi-nalExactDi...
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