Download Polarization, reactivity and quantum molecular capacitance: From

Indian Journal of Chemistry
Vol. 53A, Aug-Sept 2014, pp. 1052-1057
Polarization, reactivity and quantum molecular capacitance: From electrostatics to
density functional theory
Patrice Delarue & Patrick Senet*
Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS-Université de Bourgogne, Dijon, France
Email: [email protected]
Received 31 March 2014; revised and accepted 6 May 2014
The charge distribution induced by an inhomogeneous electric potential applied to a molecule is in fact the sum of two
terms: polarization (localized) and chemical (delocalized) charge distributions. The chemical induced charge distribution is
proportional to the inhomogeneous response of the molecule to an electron transfer (Fukui function). Analogy with the
electrostatic Thomson theorem for the perfect conductors permits to define the quantum molecular capacitance.
Keywords: Theoretical chemistry, Density functional calculations, Polarization, Chemical reactivity, Electrostatics,
Quantum molecular capacitance, Fukui function
In the Born-Oppenheimer approximation, a chemical
reaction between two molecules can be described as
the variation of the electronic density of the ensemble
of the two interacting molecules due to the motions of
their nuclei. This electronic density variation is an
internal charge reorganization (polarization) of the
whole system of reagents which can be formally and
arbitrarily decomposed in responses (reactivity
indexes) of the individual molecules. Indeed, consider
for example two isolated molecules A and B with
ground-state densities ρA(r) and ρB(r), respectively.
When the molecules are at finite distance, the density
of the whole system A+B reaches the value ρAB(r).
The difference between the density ρAB(r) and the
density of each of the isolated reagents, i.e.,
∆ρ(r) = ρAB(r)-ρA(r)-ρB(r), contains information on
the respective reactivity of the individual moieties.
For example, the charge transfer on A due to B can be
calculated as ∆ ≡ Ω ρ
, where Ω is a (arbitrary) partition function1,2. For nonzero
charge transfer, one may define a linear reactivity
index ≡ Ω ∆ρ(r)/∆ .
In the density-functional theory (DFT) of the
chemical reactivity3-9, one aims to evaluate the
variation of the electronic density of a molecule
induced by its reaction with another molecule from
the ground-state properties of the individual reagents.
The perturbation induced by one molecule on another
can be either described by an external potential
(physical perturbation or polarization) or by a charge
transfer (chemical perturbation)3. In the example
described above, we aim to evaluate an approximation
of which does not depend on actual position of
B and on the choice of Ω . The so-called Fukui
function is this quantity3,10. It represents the
linear variation of the electronic density when A is
“charged” with ∆ in absence of B. However, there
is no unique definition of .
The Fukui function can be defined as a derivative of
the density relative to the number of electrons (Eq. (1)).
= …(1)
For an actual charge transfer to an isolated
molecule, the variation of the electron number, ∆ ,
is a positive or a negative integer. Therefore, the
Fukui function defined by Eq. (1) must be computed
by the finite difference between the molecular
electronic densities of the neutral and positively or
negatively charged molecules or approximated by the
HOMO or the LUMO orbitals4,5,7. Two Fukui
functions3,10 are thus defined: which describes
the molecular electrophilic reactivity (uptake of
electrons, ∆ > 0) and which describes the
molecular nucleophilic reactivity (transfer of
electrons, ∆ < 0). There is a vast interest in the
DELARUE & SENET: POLARIZATION, REACTIVITY & QUANTUM MOLECULAR CAPACITANCE
1053
applications of these electrophilic and nucleophilic
Fukui functions to understand reactivity,4,5,11 ion
solvation12, biochemical properties13-15 and to probe
reaction paths16.
If ∆ results from an internal charge transfer as
described above, there is no reason to approximate the
Fukui function by the response to an actual
charge transfer of an integer number of electrons. In
this case, is related to response of the chemical
potential of the molecule to a potential, we named the
polarization Fukui function, defined in density
functional theory of reactivity3,17,18 by Eq. (2).
= …(2)
The polarization Fukui function (Eq. (2)) is related
to the linear electronic response by Dyson
equations8,19,20. For an energy functional !
" which
is a continuous function of , the charge transfer
Fukui function (Eq. (1)) and the polarization Fukui
function (Eq. (2)) are identical because they are
related to each other by a Maxwell relation (Eq. (3)).
= #
= $
#
%
…(3)
The relation (3) is broken once approximations
are made to compute the derivative for a
non-differentiable model of !
".
We demonstrate here that a Fukui function for the
polarization can be also defined in classical physics
by using the Thomson theorem for perfect conductors.
The analogy with the electrostatic Fukui function
permits to define a quantum molecular capacitance
and to demonstrate that the variation of the molecular
electronic density induced by an external potential can
be decomposed in a localized and delocalized (Fukui)
density variations associated to a fictitious internal
charge transfer (perturbation at constant N). This
clarifies the physical meaning of in Eq. (2) and
its relation with the molecular capacitance and the
so-called chemical hardness34.
The Thomson Theorem and the Electrostatic
Fukui Function and Hardness
In order to apply electrostatics, one represents an
isolated molecule by a perfect conductor in vacuum.
What is the electric charge distribution at the
equilibrium, −' "( , when the molecule is charged
exactly with N electrons ( = "( )? The
solution can be found by applying the variational
Thomson theorem: the energy of the actual
electrostatic field of charged conductors at
equilibrium is a minimum relative to the energies of
the fields which could be produced by any other
charge distribution on or in the conductors21.
According to this theorem, the charge distribution at
equilibrium of an isolated conductor charged with
N electrons is the one which minimizes the Coulomb
energy (EC) (Eq. (4)),
!) =
*+
. |0|
, Ω
′
…(4)
under the constraint that = "
and where Ω
is the volume of the conductor. The solution is
obtained by introducing a Lagrange multiplier λ,
2
!) − 3
−' = 0
3 = −' Ω
4 5. 6
′
|. |
…(5)
…(6)
where Eq. (6) is readily obtained by the functional
differentiation of the Eq. (4) relative to the charge
distribution −'"
. The Eq. (6) states that the
electrostatic potential generated by the charge added
(i.e., −' "( ) is constant at every point r on and
inside the conductor. In other words, there is no
electric field inside the conductor, as it must be for a
perfect conductor (which would create otherwise the
instability of the charge density added). The Eq. (6)
can be written as follows (Eq. (7)),
7* = ℎ* , 0 * 0 ′
…(7)
where we have defined a normalized function * and an energy 7* ,
* =
4 …(8)
:*
7* = − …(9)
and where the kernel ℎ* , 0 is
*+
ℎ* , 0 = |0|
…(10)
By definition, * is the electron density at
equilibrium corresponding to a charge of exactly one
electron on the conductor. The quantity 7* is twice the
INDIAN J CHEM, SEC A, AUG-SEPT 2014
energy needed to charge the cluster with one electron.
Indeed, using Eq. (6) in Eq. (4), one finds
;
!) "( = , 7* ,
…(11)
According to the definition of electrostatic
capacitance C of an isolated conductor, one
deduces21,22
7* =
*+
)
…(12)
It is worth noting that the Eqs (11) and (12) were
first derived by Max Born in its seminal work on the
volume and hydration of ions23.
We name * and 7* the electrostatic Fukui
function and hardness, respectively. The analogy
between the chemical hardness in DFT of reactivity
and the electrostatic capacitance (Eq. (12)) was
pointed by Pearson24. For a spherical perfect
conductor, the capacitance < is the radius of the
sphere21. Based on this argument, Eq. (12) leads to a
practical formula to compute the hardness of atoms as
shown numerically by Islam and Ghosh25. As shown
in electrostatics, for a perfect conductor, * is
nonzero only at the surface of the conductor21.
Therefore, using Eq. (7) and assuming that an atom is
represented by a conducting sphere with a radius
equal to its (covalent) radius (=> ), one obtains the
following equation (Eq. (13)).
7* =
* + ? . ′
|@ . |AB |0|
C
…(13)
By replacing * 0 by the atomic frontiers orbitals
in the Eq. (13), one recovers an earlier result derived
by Chattaraj et al.26 Surprisingly, the electrostatic
potential generated by the Fukui function27 (Eqs (1)
and (2)) evaluated at the nucleus of an atom is
also a measure of its chemical hardness28. It is
interesting to point out that the electrostatic Fukui
function is also the so-called shape function29-31
associated to the electronic density "( added to the
molecule and solution of Eq. (6), and is also the
first term in the density gradient expansion for the
Fukui function26. The Thomson theorem states
that the electrostatic Fukui function (the density
induced by charging a conductor) is solution of a
variational principle (Eq. (7)) which is closely related
to the variational principle for the Fukui function32
derived in DFT.
1054
The electrostatic Fukui function * is related to
the polarization. Indeed, the equilibrium density of
electrons 2"( induced within a conductor
(representing the molecule) by an external applied
electric potential Φ*EF (constant N) can be
deduced from the Thomson theorem as follows. One
considers the molecule in contact with an infinite
reservoir of electrons, i.e., one considers a grounded
conductor. According to the Thomson theorem, the
charge distribution −'2"( minimizes the total
electrostatic energy !,
! =
*+
,
-Ω
. 0
|. |
− ' Ω 2"
Φ*EF …(14)
that is,
Ω
*4 5. 6
0
|. |
= Φ*EF [HIJK']
…(15)
In contrast, the functional differentiation of the
Eq. (14) relative to −'2"
for an isolated conductor
under the constraints that the conductor remains
neutral (Eq. (16)),
2 M! + O' Ω 2"
P = 0
…(16)
gives Eq. (17),
−' Ω
4 5. 6
0
|. |
+ Φ*EF = O
…(17)
where O is a Lagrange multiplier associated to the
charge conservation. The particular solution of
Eq. (17) for O = 0, we name 2"QA( 0 , corresponds
to the polarization density if the conductor is
grounded, i.e. if the molecule is connected to an
infinite reservoir of electrons (Eq. (15)). From
Eq. (17), one realizes that the electron density 2"( induced by the external potential can be formulated in
function of 2"QA( 0 and of the electrostatic Fukui
function (Eq. (18)),
2"( = 2"QA( − * Δ
…(18)
where Δ ≝ Ω 2"QA( . For a grounded
conductor, Δ is the amount of charge which is
transferred from the ground (O = 0) to the conductor
due to the polarization of the molecule by the external
potential. The validity of Eq. (18) is established
DELARUE & SENET: POLARIZATION, REACTIVITY & QUANTUM MOLECULAR CAPACITANCE
1055
by inserting (18) in (17), which gives two equations,
Eqs (19) and (20),
−' Ω
TU4 5. 6
|. |
0 + Φ*EF = 0
Δ' Ω
? 0
0
|. |
V
Δ
*
W*
)
= O
…(19)
…(20)
By definition, 2"QA( is solution of the Eq. (19)
(equivalent to Eq. (15)). Using Eqs (7) and (12), one
finds the value of the Lagrange multiplier.
=
= O
…(21)
The Eq. (17) can be rewritten in an illuminating form:
Ω
*4 5. 6
0
|. |
= Φ*EF −
W*
)
(22)
Equations (18) and (22) provide a clear physical
meaning of the electrostatic Fukui function and of its
role in the electric polarization. Indeed, the density
induced by the external potential if the molecule is
connected to an infinite reservoir of electrons
(conductor grounded) is 2"QA( . The spatial
variation of this density depends on the external
potential. For example, if the external potential is the
electrostatic potential generated by a point charge, the
density induced (2"QA( will be localized in the
vicinity of the point charge and it will decrease with
the distance from this point perturbation. The
integration of this density, Δ, represents the number
of electrons transferred from the reservoir to the
molecule to build Δ = Ω 2"QA( . Because of
the conservation of the total number of electrons, the
electron number of the reservoir changes by −Δ.
For an isolated molecule, the reservoir is the
molecule itself and the Δ electrons arises from all the
regions of the molecule according to −* Δ
(Eq. (18)). The number of electrons −Δ can be
interpreted as a virtual charge transfer of −Δ
electrons from the molecule to itself in order to build
the polarization charge, 2"QA( . Because there is no
actual charge transfer, Δ is a continuous variable and
can be fractional. Because the molecule is (virtually)
charged with – Δ electrons, the external electric
W*
potential is shifted by − ) (Eq. (22)). Consequently,
Eq. (18) tells us that an external electric potential has
two effects: it induced a polarization density 2"QA( and a chemical density −* Δ if Δ ≠ 0. For the
particular case of the electrostatic potential generated
by a point charge, 2"QA( will be localized in the
vicinity of the perturbation point charge whereas
−* Δ is delocalized and does not depend on the
position of the point charge (except through the value
Δ). It means that a point charge will induce longrange modifications of the molecular electronic density
depending on the values of the electrostatic Fukui
function * . As shown in electrostatics, for a perfect
conductor, * is nonzero only at the surface of the
conductor (at the surface of the molecule in the present
conducting simple model). Of course, a perfect
conductor is a priori a very bad model of the dielectric
properties of a molecule. However, as we show vide
infra, the physics of the Eqs (18) and (22) is preserved
in more accurate DFT models as discussed next.
The Quantum Molecular Capacitance and the
Fukui Function
One applies to a molecule (in its ground-state) an
electric external potential ΔZ*EF = −'Φ*EF .
According to the first-order DFT perturbation theory,
the electronic density induced by ΔZ*EF is the
solution of Eq. (23)3,
ℎ
, 0 2"( 0 0 = eΦ*EF + 2\
…(23)
where 2\ is the shift of the electron chemical
potential of the molecule due to the perturbation. The
hardness kernel ℎ
, 0 is defined by the second
derivative of the Hohenberg-Kohn functional (F)
relative to the electronic density at constant external
potential33, i.e.,
+]
ℎ
, 0 = 0
…(24)
Using the Eq. (2) for the polarization Fukui
function, one has Eq. (25).
2\ = −' 0 Φ*EF 0 ′
…(25)
By definition, the electron density which would
be induced by the external potential if the
molecule would be connected to an infinite
reservoir of electrons is solution of Eq. (23) with
2\ = 0, i.e.,
ℎ
, 0 2"
0
0
A( = eΦ*EF …(26)
INDIAN J CHEM, SEC A, AUG-SEPT 2014
1056
The Fukui function obeys the following equation
(Eq. (27)),
can be written as the response to an effective
self-consistent “electric” potential Φ_`a @,
ℎ
, 0 0 0 = 7
Ω
…(27)
where η is the so-called chemical hardness of the
molecule34 which plays a fundamental role in the DFT
theory of reactivity22,35-41 and in the equalization
principle of electronegativity42,43.
Therefore, using Eqs (25) to (27), one gets Eq. (28),
2\ = − 2"
0
A( ′
= − 7Δ
…(28)
in which Δ represents the number of electrons
induced on the molecule by the electric potential to
build 2" A( . Because the molecule is its own
reservoir of electrons (isolated molecule), the Δ
electrons arises from all the regions of the molecule
according to −
Δ. The number of electrons
−Δ can be interpreted as a virtual charge transfer
of −Δ electrons from the molecule to itself in
order to build the polarization charge 2" A(
as shown by Eq. (28). Because there is no actual
charge transfer, Δ is a continuous variable and can
be fractional.
Using Eq. (28), the first-order DFT equation
reads as
ℎ
, 0 2"( 0 0 = eΦ*EF − 7Δ
…(29)
or
;
*
ℎ
, 0 2"( 0 0 = Φ*EF −
*W
)
…(30)
where we define the quantum molecular capacitance
< = ' , /7 by analogy with Eq. (12). It is readily
shown that Eq. (31),
2"( = 2"
A( − Δ
*4 5. 6
0
|. |
W*
)
…(32)
where Ω is the entire space and
1
ℎ
, 0 E> 2"( 0 0
Φb>? = Φ*EF − d
'
;
− * ℎ
, 0 e 2"( 0 0
…(33)
by defining ℎ
, 0 E> and ℎ
, 0 e as the second
functional derivatives of the exchange-correlation
and kinetic energy functionals, respectively. Equation
(32) shows that the molecule can be viewed as a
perfect conductor with a capacitance < responding to
an effective “electric” potential Φb>? . Eq. (32)
permits to formulate the condition for a metal
(molecule) to behave as a perfect classical conductor.
Indeed, comparing Eqs (32) and (33) with Eq. (22),
one deduces that a system will behave as a perfect
conductor if
[ℎ
, 0 E> + ℎ
, 0 e ]2"( 0 0 = 0
…(34)
Because of the charge conservation, Eq. (34) is
equivalent to Eq. (35),
ℎ
, 0 E> + ℎ
, 0 e = f
…(35)
where K is an arbitrary constant. The Eq. (35) is a
nontrivial condition. For example, for a metal
described by the Thomas-Fermi-Dirac (TFD) model8,
the condition (35) is not fulfilled. In TFD, one has
(Eq. (36)),
ℎ g]h , 0 E + ℎ g]h , 0 e
,j.l(m
=i
…(31)
is the solution of Eq. (30). Equation (31) clearly
shows the physical meaning of the polarization Fukui
function (defined by Eq. (2)); it controls the internal
electron flows needed to build the polarization charge
density 2" A( induced by an external potential.
The quantity −
Δ is a delocalized chemical
charge distribution induced by an external potential.
Then Eq. (30) resembles the electrostatic Eq. (22) and
= Φb>? −
n
o . −
j.p,p
+
q 2
− ′
o . …(36)
where the units of the kernel are eV Å2. The condition
(34) reads as
,j.l(m
i
n
o . −
j.p,p
+
q 2"( = 0
o . …(37)
The term in brackets should be zero for a perfect
conductor which is not the case in the TFD model.
1057
DELARUE & SENET: POLARIZATION, REACTIVITY & QUANTUM MOLECULAR CAPACITANCE
This means that the electric potential inside a
conductor described by the TFD model is not
constant.
Conclusions
We demonstrate that the electron density induced
in a molecule by an external potential can be
decomposed in the sum of two contributions
(Eq. (31)); 2" A( and −
Δ and involved a
(generally nonzero) virtual charge transfer, – Δ. By
analogy with electrostatics, this charge transfer can be
interpreted as the amount of charge which would be
transferred from an infinite reservoir of electrons in
order to build the polarization charge, 2" A( .
Because an isolated molecule is its own reservoir, this
charge transfer arises from the molecule itself from all
regions of the molecule proportionally to the value of
the (polarization, Eq. (2)) Fukui function, . A high
value of within a molecule means that this region
is a local reservoir of electrons for any perturbation.
Consequently, a localized perturbation may have an
effect far away from its vicinity, in the regions of the
local reservoirs of electrons, if – Δ ≠ 0. The local
reservoirs of electrons should play a direct role in the
chemical reactivity of the molecules.
The analogy with electrostatics allows us to define
a quantum molecular capacitance C. As in
electrostatics (Eq. (7)), the potential associated to the
charge distribution −'
Δ is a constant equal to
W*
− ) . For any external potential, the linear response
of the molecule can be formulated as the response to
an effective electric potential (Φb>? ) as shown by
Eq. (32). Therefore, one may hope that appropriate
models inspired from electrostatics can be used to
model the reactivity of (large) molecules.
Implementation of such models and of the Eqs (31)
and (32) are currently being undertaken.
Acknowledgement
This work was funded by the Conseil Regional de
Bourgogne under the program PARI-NANO2BIO.
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
References
1
2
3
4
Bultinck P, Fias S, Van Alsenoy C, Ayers P W &
Carbó-Dorca R, J Chem Phys, 127 (2007) 034102.
Fuentealba P, Florez E & Tiznado W, J Chem Theory
Comput, 6 (2010) 1470.
Parr R G & Yang W, Density-Functional Theory of Atoms
and Molecules (Oxford University Press, New York) 1989.
Chermette H, J Comput Chem, 20 (1999) 129.
40
41
42
43
Ayers P W & Levy M, Theor Chem Acc, 103 (2000) 353.
Geerlings P, De Proft F & Langenaeker W, Chem Rev, 103
(2003) 1793.
Chattaraj P K, Sakar U & Roy D R, Chem Rev, 106 (2006)
2065.
Ayers P W, Yang W T & Bartolotti L J, Fukui Function
in Chemical Reactivity Theory: A Density Functional View,
edited by P K Chattaraj (CRC Press, Boca Raton) 2009, 255.
Senet P, J Chem Phys, 105 (1996) 6471.
Parr R G & Yang W, J Am Chem Soc, 106 (1984) 4049.
Komorowski L, Lipinski J, Szarek P & Ordon P, J Chem
Phys, 135 (2011) 014109.
Vuilleumier R & Sprik M, J Chem Phys, 115 (2001) 3454.
Baeten A, De Proft F & Geerlings P, Int J Quant Chem, 60
(1996) 931.
Senet P & Aparicio F, J Chem Phys, 126 (2007) 145105.
Krishtal A, Senet P & Van Alsenoy C, J Chem Phys, 131
(2009) 044312.
Vuilleumier R & Sprik M, Chem Phys Lett, 365 (2002) 305.
Ayers P W, De Proft F, Borgoo A & Geerlings P, J Chem
Phys, 126 (2007) 224107.
Sablon N, De Proft F, Ayers P W & Geerlings P, J Chem
Phys, 126 (2007) 224108.
Senet P, J Chem Phys, 107 (1997) 2516.
Cohen M H, Ganduglia-Pirovano M V & Kudrnovsky J,
J Chem Phys, 101 (1994) 8988.
Landau L D, Lifshitz E M & Pitaevskii L P, Electrodynamics
of Continuous Media, 2nd Edn, (Butterworth-Heineman,
Oxford) 2000.
Senet P, Chem Phys Lett, 275 (1997) 527.
Born M, Z Phys A, 1 (1920) 45.
Pearson R G, Inorg Chem, 27 (1988) 734.
Islam N & Ghosh D C, Mol Phys, 109 (2011) 1533.
Chattaraj P K, Cedillo A & Parr R G, J Chem Phys, 103
(1995) 10621.
Berkowitz M, J Am Chem Soc, 109 (1987) 4823.
Cardenas C, Chem Phys Lett, 513 (2011) 127.
Parr R G & Bartolotti L, J Phys Chem, 87 (1983) 2810.
De Proft F, Ayers P W, Sen K D & Geerlings P, J Chem
Phys, 120 (2004) 9969.
Ayers P W, Proc Natl Acad Sci USA, 97 (2000) 1959.
Chattaraj P K, Cedillo A & Parr R G, J Chem Phys, 103
(1995) 7645.
Berkowitz M & Parr R G, J Chem Phys, 88 (1988) 2554.
Parr R G & Pearson R G, J Am Chem Soc, 105 (1983) 7512.
Yang W T, Lee C & Ghosh S K, J Phys Chem, 89 (1985)
5412.
Ghosh S K, Chem Phys Lett, 172 (1990) 77.
Pearson R G, Chemical Hardness, (Wiley-VCH, Weinheim,
Germany) 1997.
Ayers P W, Faraday Disc, 135 (2007) 161.
Pearson R G, The Hardness of Closed Systems, in Chemical
Reactivity Theory: A Density Functional View, edited by
P K Chattaraj (CRC Press, Boca Raton) 2009, 155.
Torrent-Sucarrat M, De Proft F, Ayers P W & Geerlings P,
Phys Chem Chem Phys, 12 (2010) 1072.
Ghosh D C & Islam N, Int J Quant Chem, 111 (2011) 1931.
Mortier W J, Ghosh S K & Shankar S, J Am Chem Soc, 108
(1986) 4315.
Bultinck P, Langenaeker W, Lahorte P, De Proft F, Geerlings P,
Waroquier M & Tollenaere J P, J Phys Chem A, 106 (2002) 7887.