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

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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

all Coulomb interactions, formulated in the second quan-

tization language, is of the form

H=aX

ˆniσ +X

ijσ

0tij ˆa†

iσ ˆajσ +UX

ˆni↑ˆni↓

0Kij ˆniˆnj−1

0JH

ij ˆ

Si·ˆ

Sj−1

4ˆniˆnj

0J0

ij (ˆa†

i↑ˆa†

i↓ˆaj↓ˆaj↑+ H.c.)

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.

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Degreeofatomicityinthechemicalbonding:WhytoreturntotheH2molecule?MaciejHendzelandJozefSpalekyInstituteofTheoreticalPhysics,JagiellonianUniversity,ul.Lojasiewicza11,PL-30-348Krakow,Poland(Dated:November3,2022)Weanalyzetwo-particlebindingfactorsforthecaseofH2moleculewiththehelpofourorigi-nalExactDi...

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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

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