Electronegativity is one of the basic concepts in Chemistry as well as the theory of a chemical bond. After Linus Pauling proposed a scale of electronegativity of ...
V. K. Kochnev
ENSEMBLE N-REPRESENTABILITY AND ELECTRONEGATIVITY ABSOLUTE ELECTRONEGATIVITY IN GAS c 2017
г.
1. Variational formulas. V. K. Kochnev 2. Consequences and remarks. V. K. Kochnev, A. D. Izotov, corresponding member of RAS
1. Electronegativity is one of the basic concepts in Chemistry as well as the theory of a chemical bond. After Linus Pauling proposed a scale of electronegativity of Elements based on the chemical bonds formation heat in diatomic molecules many other electronegativity scales occured, sometimes without an immediate thermodynamic basis. Among them, there is an exact scale of absolute electronegativity based on the chemical potentials of atoms and molecules. For a Fermi system the number of particles
N
is usually given, and the chemical potential
a function of the number to consider the
µ
µ
is considered as
N . In practical calculations it is more convenient
as an independent variable. Because of commutation
between operator of the number of particles with the Hamiltonian of the system the quantities
N
and
µ
are dual. In the present paper this
property manifests itself in the variational calculation of the electronic energy of an ideal gas, by a simple relation to the energy of the system
E = µN , where N
is the number of electrons. From the formula, the exact
algebraic relationship between absolute electronegativity and the hardness of molecules follows, which has long been discussed in the literature.
Let us take one mole of an ideal gas consisting of several components
Bi
(molecules of arbitrary compounds in gaseous phase) with corresponding mole fractions gas
ρtot (1)
xi
. We will consider the electron density of the whole
as a sum of electron densities of all molecules, assuming that
molecules are located far away from each other in the different points in
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V. K. Kochnev
space and are not interacting (i.e. have frozen electron densities). Individual molecules in the ground states have electronic chemical potentials aligned between atoms (absolute electronegativity of molecules) [1, 2]. The total system is not a gigantic "supermolecule" , which prevents the (very) old arguments for chemical potential equalization from working. Also, in the limit of noninteracting systems it is known that electronegativity is equalized only in rather complicated mathematical sense, owing to the inherent nondifferentiability of the density functional in that limit [3]. As the molecules in gas are not infinitely separated (finite volume) , the "paradox" that arises from nondifferentiability is avoided, as long as the Radial parts of the electron density functions of molecules extend to infinity. Ideality of the gas means that the energies of interactions between gas molecules, which are of the smallness order of
∼ r−5 − r−6
[4], are infinitesimal of a higher
order on the average than the sum of the gas molecules energies. The electron density for one molecule of the component expressed [5] in terms of its natural orbitals numbers
ϕk i
Bi
may be
and their occupation
nki , when the ground state of gas (multi-electron system) as well
as the states of its molecules could be represented on the basis of the classification of individual electrons’ states (ensemble N-representability):
ρi (1) =
X
nki |ϕki |2 , 0 ≤ nki ≤ 2.
(1)
ki The representing of all electrons in the whole gas is
N=
X
xi NA
i
NA
nki =
Bi
(2)
}
N (i)
, the functional
N (i) ,
i
{z
is the Avogadro constant,
of the component
X
ki
| where
X
xi NA
is the number of molecules
N (i) = N (i) [xi , nki ]
of all the electrons in the component
2
Bi ,
gives the number
and the internal sum
V. K. Kochnev
X
nki = Ni ,
(3)
ki is the number of electrons in one molecule of the component The functions
Z where
ϕk i
Bi
.
satisfy orthonormalization constraints,
ϕ∗pm (1) ϕqn (1) dτ1 = hϕpm |ϕqn i = δpq δmn
(4)
δpq relates to the orthogonality of the orbitals within one molecule
due to the shape of these functions, and
δmn
relates to the orthogonality
of the orbitals between different molecules due to the location of these functions in different points in space at a sufficiently large average distance from each other (ideal gas), when the exact value of
δmn
is obtained after
discarding infinitesimals of a higher order of smallness.
The density of whole gas
ρtot (1)
determines all properties of the united
system [6], and implies the existence of energy functional for whole gas
Eυtot
. According to the assumption about the existence of an expression
of the density
ρtot (1)
by the natural orbitals of molecules it implies the
existence of a natural orbital energy functional which is a sum of the functionals components
Bi
Eυtot = EυN O, tot [xi , nki , ϕki ],
EυN O, i [nki , ϕki ] of all molecules of all the
of the ideal gas,
EυN O, tot [xi , nki , ϕki ]
=
X
xi NA EυN O, i [nki , ϕki ]
(5)
i where
x i NA
is still the number of molecules of the component
and the existence of the functionals
EυN O, i [nki , ϕki ]
Bi
,
is implied by Eq. (1),
because again the density determines all the properties of a ground state. The functionals
EυN O, i [nki , ϕki ]
EυN O, i [nki , ϕki ]
=
X
usually are written as
Z nki
|ϕki |2 υi (1)dτ1 + FiN O [nki , ϕki ]
ki 3
(6)
V. K. Kochnev
where
υi (1)
is external potential, and
FiN O [nki , ϕki ]
is the sum of the
kinetic energy functional and the electron-electron interaction energy functional.
Then for any set
{x0i , n0ki , ϕ0ki }
satisfying the constraints of Eqs. (1)-
(4) under the condition that the gas density
ρtot (1)
is expressable by
the natural orbitals of the molecules (ensemble-N-representation), there is a valid wavefunction with a density
ρtot (1)
(a given density might even
have more than one natural orbital resolution [5]), thus the functional
EυN O, tot [x0i , n0ki , ϕ0ki ],
defined by Eq. (5) obeys a variational inequality
EυN O, tot [x0i , n0ki , ϕ0ki ] ≥ EυN O, tot [xi , nki , ϕki ]
(7)
Or, incorporating the constraints with Lagrange multipliers, one has the stationary principle
( EυN O, tot [x0i , n0ki , ϕ0ki ]
δ
− µ1 N
(1)
[x01 , n0k1 ]
+ . . . µi N
(i)
[x0i , n0ki ]
+ ...
) −
X
λpqmn hϕ0pm |ϕ0qn i
=0
(8)
pqmn The Lagrange multipliers
EυN O, tot [xi , nki , ϕki ]
µi
are the sensitivities of the energy
with respect to the constraints
derivatives of the energy minimum
E
µi =
N (i)
, and are the
with respect to the values of
∂E ∂N (i)
N (i)
(9)
υ
The definitions of chemical potentials proceeding from macroscopic and
N (i)
.
The expansion of Eq. (8) gives the stationary condition
4
:
E
V. K. Kochnev
0=
X
δx0i
"
i
∂EυN O, tot ∂x0i
+
XX i
− µi x0j , n0 , ϕ0
∂EυN O, tot ∂n0ki
δn0ki
ki
(i)
∂N ∂x0i
#
x0j , n0 , ϕ0
!
(i)
− µi x0 , n0j , ϕ0
∂N ∂n0ki
!
x0 , n0j , ϕ0
!
+
XX i
δϕ0ki
ki
∂EυN O, tot ∂ϕ0ki x0 , n0 , ϕ0j )! ( X λpqmn hϕ0pm |ϕ0qn i +δ −
(10)
pqmn The last term of Eq. (10) relates to the constraints of Eqs. (4). To get the stationary conditions from Eq. (10) we now compute derivatives from Eqs. (2) and (5). Differentiation of
x0i
gives
∂EυN O, tot = NA EυN O, i [nki , ϕki ] 0 ∂xi x0j , n0 , ϕ0 X ∂N (i) = N n0ki , A 0 ∂xi x0 , n0 , ϕ0 ki
j
while differentiation of
n0ki
∂EυN O, tot ∂n0ki
!
(i)
!
∂N ∂n0ki
(11)
gives
= x0i NA x0 , n0j , ϕ0
∂EυN O, i ∂n0ki
! x0 , n0j , ϕ0
(12)
= x0i NA . x0 , n0j , ϕ0
So, from the second term of Eq. (10) and Eq. (12) we infer that for the true ground state
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V. K. Kochnev
∂EυN O, tot ∂n0ki
0=
!
∂N (i) ∂n0ki
− µi x0 , n0j , ϕ0
= x0i NA
∂EυN O, i ∂n0ki
! = x0 , n0j , ϕ0
(13)
! − µi x0i NA x0 , n0j , ϕ0
i.e.,
µi = for all indices of orbitals
ki
∂EυN O, i ∂nki
(14)
x, nj , ϕ
and for all mole fractions of components of
the gas. And from the first term of Eq. (10) and Eq. (11) we infer that for the true ground state
0=
∂EυN O, tot ∂x0i =
− µi
x0j , n0 , ϕ0
∂N (i) ∂x0i
NA EυN O, i [nki , ϕki ]
= x0j , n0 , ϕ0
− µi NA
X
n0ki
(15)
ki i.e.,
Eυi = µi Ni for all components
Bi
(16)
in the equilibrium state.
Equations (14) show that all natural orbitals of particular component
Bi have the same electronic chemical potential (absolute Electronegativity) not only throughout the orbitals of one molecule, but also throughout all orbitals of all molecules of
Bi
of an ideal gas. Although the case of an
ideal gas is itself essential, an equally good argument could have been made by taking the "interacting molecules" case where the old arguments work [1, 2], and slowly turning off the interactions until the present case arises in the limit. However, in contrast to the direct calculation of the
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V. K. Kochnev
energy variation it would be necessary, in addition to previous arguments, to “prohibit” the equalization of electronegativities of different components
Bi
. We can imagine for example a mixture of water and benzene vapors
that coexist indefinitely long and do not undergo any chemical changes, i.e. retain the original values of
µH2 O
and
µC6 H6 .
Comparing the Eqs. (9) and (14) one gets
µi =
∂E ∂N (i)
=
υ
∂EυN O, i ∂nki
(17)
x, nj , ϕ
The left of Eq. (17) is a macroscopical definition of chemical potential for whole gas of molecules, and on the right is the value for the individual orbital of the individual molecule. And this is not a meaningless tautology, but a reflection of the fact that for degenerate (indistinguishable) electrons that property does not depend on the scale.
Eq. (16) shows that the approximation formula
E ≈ µN ,
which is
the direct consequence of the chemical potential definition as a derivative
µ = (∂E/∂N )υ , turns to exact formula (E = µN ) for the equilibrium state of gas; contributions of components of particles
N
Bi
Eυi = µi Ni
are nothing else than the partial energies
in an ideal gas. It is especially notable that the number
and the dual value of the chemical potential
µ are perfectly
symmetric in one expression for energy (in more detail about the relation between
N
and
µ
see in [7, с. 93] ). And the formulas (16) do not mean
that the total energy depends linearly on the number of electrons, because of the electronic chemical potentials themselves depend on the number of electrons (i.e. values
η=
E(N ) = µ(N ) · N ).
Moreover, the corresponding hardness
2
1∂ E 2 ∂N 2 [8] can be found by definition, using Eq. (16),
1 ∂ 2E 1 ∂µ 1 ∂ = η= = 2 2 ∂N 2 ∂N 2 ∂N 7
E(N ) N
E 1µ − N = 2 N
(18)
V. K. Kochnev
i.e.,
ηN =
1 µ−E 2
(19)
The Eq. (19) provides the relationship between absolute electronegativity and hardness, giving an analytical expression of the well known assumption [9] that electronegativity and hardness are two different descriptors of the same fundamental property of the system (the factor
1 2 was introduced by
Parr [8] just for obtaining a formula for hardness similar to the Mulliken formula for electronegativiy, and is not important). Lets examine the Eq. (19) in more detail.
2. Free gas of electrons. For an absolutely free ideal electronic gas of noninteracting particles the relation
E = µN
is the obvious property of the
extensiveness of the energy of the system with a constant
η=
1 ∂µ 2 ∂N
=0
µ. The hardness
as a consequence of the independence of electronic chemical
potential on the number of particles. The formula (19) then indicates that the average energy of one electron leads to the Eq.
E = µN ,
E =
E N is exactly the
µ,
which again
that is obvious in the situation.
Electrons in a metal. Althrough electrons in a metal can not be
considered noninteracting the same relationships are obtained for them. While an unlimited increase of the number of electrons part of Eq. (19) obviously does not depend on of Eq. (19) must not depend on
N,
N,
N →∞
the right
and hence the left part
which is possible only if the hardness
of the electronic system of the metal is approaching to zero. As there’s an uncertainty of the type
0×∞
in the left part of Eq. (19) some additional
information is required for the resolution of it. If one puts the left part of Eq. (19) equal to zero, then it exactly follows the equality
E = µN ,
which is often used in the physics of solids. Moreover, after renaming for the situation the average energy of one electron to
8
E = and rewriting the
V. K. Kochnev
right part of Eq. (19) as
−( − µ), thus, nothing more than the numerator
of the Fermi-Dirac exponent
−µ
e− kT
for the probability distribution of one
electron in a metal.
Electrons in molecules. According with Eq. (19) for a system of
interacting electrons it turns out it turns out that the average value of the energy of one electron
E =
E N is not the same as electronic chemical
potential, and the difference is the greater the higher the hardness
η
of the
electronic system, i.e. stronger interactions between electrons. Comparing the ratio Eq. (16), written as
E==
µ=
E N and the average energy of one electron
E N one can conclude, that they can differ because of the fact that
the averaging in the first and second cases is performed on different sets. For the first case, in the course of (1)-(16) here is the energy of the entire system, which is a quantum average value calculated on the set of possible quantum states of the system. For the second case there is an average value of the energy of one electron in the set of electrons in the molecule in the ground state, and, as long as the electrons are indistinguishable, this reduces to one division by
N . To find the difference between these two
values and not to lose generality, just by definition, differentiation (18) is required.
CONCLUSION. The variational formula is obtained for the relationship between total electronic energy and absolute electronegativity. This allows to formalize the assumption that the electronegativity and hardness of a molecular system are two different descriptors of the same property of the system. The property is the difference between the average electron energy when calculating this quantity as the quantum average over the states of the whole system (i.e. the chemical potential value
µ)
on the one hand and
as the average "specific" energy of one electron in the ground state of the
9
V. K. Kochnev
system (i.e.
N
E=
E N ), on the other hand. The expression
Eυ (N ) = µ(N ) ×
is a requirement of an exact theory, although it is not satisfied for
the most modern methods of calculating electronic energy, except for the systems consisting exclusively of hydrogen atoms. In the latter case, the relationship between the total electronic energy potential
µH2 (r) and interatomic distance rH2
EH2 (r), electronic chemical
was considered in one of the
previous papers [10]. Experimental data on the estimation of electronic chemical potentials of some molecules, including hydrogen, are given in [11].
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СПИСОК ЛИТЕРАТУРЫ
V. K. Kochnev
СПИСОК ЛИТЕРАТУРЫ
Список литературы [1] R. G. Parr, R. A. Donnelly, M. Levy, and W. E. Palke, J. Chem. Phys. 1978, 68(8), 3801-3807; [2] R. P. Itzkowski, and J. L. Margrave, J. Am. Chem. Soc. 1961, 83, 3547-3551; [3] P. W. Ayers, Theor. Chem. Acc. 2007, 118, 371-381; [4] Ландау Л.Д., Лифшиц Е.М. Теоретическая физика: Учеб. Пособ.: Для вузов. В 10 т. Т.III. Квантовая механика (нерелятивистская теория). – 5-е изд., стереот. – М.:ФИЗМАТЛИТ, 2002, 808 с. – ISBN 5-9221-0057-2 (Т.III); [5] D. W. Smith, Phys. Rev. 1966, 147, 896-898; [6] P. Hohenberg and W. Kohn, Phys. Rev. B. 1964, 136, 864-871; [7] Абрикосов А.А., Горьков Л.П., Дзялошинский И.Е. Методы квантовой теории поля в статистической физике. М. Физматгиз, 1962 г., 444 с. [8] R. G. Pearson, Coord. Chem. Rev. 1990, 100, 403-425; [9] D.C. Ghosh, N. Islam, International Journal of Quantum Chemistry. 2011, 111, 40-51; [10] V. K. Kochnev and N. T. Kuznetsov, Russ. J. Inorg. Chem. 2015, 60(7), 875-878; [11] R.G. Pearson, J. Org. Chem. 1989, 54, 1423-1430.
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