Embed Size (px)
Citation preview
Accepted Manuscript
Spatial Localization of Electron Pairs in Molecules Using the Fisher InformationDensity
Andrey A. Astakhov, Vladimir G. Tsirelson
PII: S0301-0104(14)00082-2DOI: http://dx.doi.org/10.1016/j.chemphys.2014.03.006Reference: CHEMPH 9069
To appear in: Chemical Physics
Received Date: 25 January 2014Accepted Date: 12 March 2014
Please cite this article as: A.A. Astakhov, V.G. Tsirelson, Spatial Localization of Electron Pairs in Molecules Usingthe Fisher Information Density, Chemical Physics (2014), doi: http://dx.doi.org/10.1016/j.chemphys.2014.03.006
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customerswe are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, andreview of the resulting proof before it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
Spatial Localization of Electron Pairs in Molecules Using the Fisher Information Density
Andrey A. Astakhov and Vladimir G. Tsirelson
Quantum Chemistry Department, Mendeleev University of Chemical Technology,
Miusskaya Sq. 9, Moscow, 125047, Russia
Andrey A. Astakhov: E-mail [email protected], tel. +7 (499) 978-95-84
Vladimir G. Tsirelson: E-mail [email protected], tel. +7 (499) 978-97-36
Corresponding author: Vladimir G. Tsirelson
Abstract Starting from the quasi-probability distribution function by electron positions r and
momenta p and applying the minimum information principle subject to the certain physically-
grounded constrains, we obtained the approximate expression for phase-space-defined Fisher
information density (PS-FID). It provides information about an electron momentum in the position
representation and reveals the electronic shell structure for atoms with Z≤20 as well as the regions
of maximal concentration of bonding and lone electron pairs in molecules. Also, this function
enables to recognize the different types of chemical bonds as polar and non-polar covalent bonds,
the charge-shift bond as well as the weak non-covalent molecular interactions. We found that the
PS-FID behavior results from the local electron momentum uncertainty that is linked with both
information about electron real-space position (which, in turn, is related with electronic steric
factor) and the Pauli principle.
Highlights
The phase-space-defined Fisher information density is derived and studied
This function reveals the regions of maximal concentration of electron pairs
The features of this function are linked with the local electron momentum uncertainty
Keywords electron pair, electron localization, density functional theory, Fisher information, local
electron momentum
1. Introduction
The Lewis concept of electron pairs [1] is one of the most important in chemistry [2,3,4].
Physical basis for electron pairing is as follows [5,6,7]. In addition to Coulomb electron-electron
2
repulsion, the presence of a given electron in some spatial region excludes from this region all other
electrons with the same spin due to Pauli principle. Because such a behavior is peculiar to each
electron with any spin, all electrons are maximally removed from a given region, except the pair of
the ‘spin-up’ and ‘spin-down’ electrons. Eventually, all electrons in the closed-shell systems are
proved to be distributed in the pairs with different degree of localization.
However, the direct-space one-electron density, the recognized source of the information
about the chemical bonding [8,9], does not show any signatures of the electron pairing: the
electrons are indistinguishable and distributed over a whole molecule or a crystal. Therefore, the
different tools have been suggested to detect the electron localization/concentration regions. Among
them are the Laplacian of electron density [5,8,10], one-electron potential [11], electron localization
function (ELF) [12,13,14], localized orbital locator [15,16], electron localizability indicator (ELI)
[17,18,19], maximum probability domains (MPD) [20,21,22] and conditional pair density [23].
They have been recently supplemented by localized electron detector (LED) [24,25,26,27], single
exponential decay detector (SEDD) [28] and information-theoretic ELF (IT-ELF) [29], steric [30]
and Pauli potentials [31]. Some of these functions are also approximately derived from
experimental electron density and its derivatives [32,33,34]. These electron localization tools play
nowadays an important role in the chemical bonding analysis despite some disadvantages. For
example, Laplacian of electron density does not display the electronic shells for many heavy atoms
[35,36] while the LED and SEDD do not show the electron lone pairs.
The concepts of Shannon information entropy [37] and Fisher information [38], which are
nowadays widely used in physics and chemistry [39,40,41,42,43,44], provide another general
approach to electronic structure of atoms, molecules and crystals [29,45,46,47]. Especially,
Nalewajski [29] has presented the Fisher information-based ELF to demonstrate the electron
localization in molecules (IT-ELF). In this work we show that the evidence about electron
localization in chemical systems can be derived from the phase-space-(PS) defined Fisher
information. We demonstrate that the PS-Fisher information density reveals the electronic shells for
atoms and detects the most probable positions of the bonding and lone electron pairs in molecules.
2. Method
Ghosh, Berkowitz and Parr (GBP) [48] have reformulated the density functional theory (DFT)
[49] considering the electron cloud in molecular systems as an electron gas in the effective external
potential. The N-electron quasi-probability distribution function by positions r and momenta p is
defined in a 6N-dimensional phase space [50,51] as
3
( )6
1 1
*
( ,..., , ,..., ) (2 ) ,
, , .2 2
i i i i i
Ni iN
N N i i
i
Ni i
i i i i i i i
i
F g e e
d d d
s p h r ur r p p h s
s sr r h s u
(1)
Here ( , )i i r is a wave function in the coordinate representation, ir and i stand for the
position and spin of i-th electron, correspondingly; ih , is and iu are real-space vectors and
, i ig h s is any function satisfying the condition ,0 0, 1i ig g h s . When
, 1i ig h s , function 1 1( ,..., , ,..., )N NF r r p p is the so-called Wigner function [50]. However, by
virtue of indistinguishability of particles, the electron interactions in a system are described by the
only two-particle phase-space distribution function, 1 2 1 2( , , , )F r r p p . Moreover, within the
framework of DFT, 1 1( ,..., , ,..., )N NF r r p p can be reduced to the distribution function ( , )f r p
depending on a position and momentum of only one electron [14]:
1 1
1 1
( , ) ( ) ( ) ( ,..., , ,..., )NN
i i N N i i
i i
f F d d
r p r r p p r r p p r p . (2)
The non-interacting electronic properties are extracted from ( , )f r p by integration over the
dynamical variables r and p. Especially, electron density ( ) r is expressed in terms of ( , )f r p as
( ) ( , )d f r p r p (3)
and Kohn-Sham electron kinetic energy density is
2
( ) ( , )2
st d f p
r p r p . (4)
Unlike the function (1), which is normalized to unity, ( , )f r p in (2) is normalized to the number of
electrons N. There are infinite numbers of functions ( , )f r p which satisfy eq. (3) and (4), and not
all of them are non-negative everywhere [51,52,53,54]. GBP [48] have found the most probable
form of ( , )f r p by applying the maximum entropy principle [55]. They define the local phase-space
information entropy as
( ) ( , ) ln ( , )fs d f f r p r p r p , (5)
and its global form is
( )f fS d s r r . (6)
Quantity fS is recognized as the Shannon information entropy [37], which characterizes the
information content of the function ( , )f r p . Maximum information entropy principle subject to
constrains (3) and (4) leads to the Maxwell-Boltzmann distribution function for an ideal electron
gas [48]:
4
2
3 2( , ) ( ) 2 ( ) exp
2 ( )f kT
kT
pr p r r
r. (7)
Here ( )T r is the local information electron gas temperature [48,14]. Within the non-interacting
particle approximation, GBP have associated ( )T r with electronic kinetic energy density using the
relation
3( ) ( ) ( )
2st kTr r r , (8)
which is analogue to the main equation of the molecular kinetic theory [56]. The presence of the
Boltzmann constant k just emphasizes the formal analogy between the information and
thermodynamic temperature. As opposed to the thermodynamic entropy, fS does not vanish for
quantum-mechanical electronic ground state. It is also not linked with the energy of the system.
Note that in any system at equilibrium the thermodynamic temperature is the same everywhere in
the r-space, while information temperature ( )T r varies with the position r.
Fisher information minimum principle [39] is another expression of the general principle of
physical information extreme. Now we extend the GBP approach to minimize the Fisher
information [38], which quantitatively characterizes spatial structuredness of some distribution
function [39] and plays important role in the information theory. We consider the phase-space-
defined Fisher information density (PS-FID)
( , ) ( , )( )
( , )f
f fi d
f
p pr p r pr p
r p. (9)
Here the subscript p stands for the gradient, which is taken by only momentum variables. The
function ( )fi r contains information about structuredness (or sharpness) of electronic momentum
distribution in the position representation. The global form of the PS-Fisher information
( )f fI d i p r , (10)
characterizes the total structuredness of ( , )f r p relative to the electron momentum.
Let us express fI as a functional of quasi-probability amplitude ( , ) ( , )f r p r p :
4 ( , ) ( , )fI d d p pr p r p r p . (11)
Minimization of fI (11) subject to the constrains (3) and (4) leads to the second-order differential
equation
22 ( , ) [ ( ) ( ) ] ( , ) 0
2 p
pr p r r r p (12)
in which ( ) r and ( ) r are r-depending Lagrange multipliers. Differentiation in (12) is only in the
momentum variables, therefore the position r can be regarded as a parameter. Eq. (12) is formally
5
similar to the Schrödinger equation for a harmonic oscillator, ( ) r being the ‘information energy’
and 2
( )2
p
r the ‘information harmonic potential’ [57]. Solution of (12) leads to the set of the
functions including Hermite polynomials and depending on values of three integer numbers: 1n , 2n
and 3n . Minimum information is provided by the solution with 1 2 3 0n n n ; it yields
2( , ) ( )exp ( ) f A r p r r p , (13)
where ( ) ( ) r r and ( )A r is r-depending normalization factor. Substituting (13) into (3) and
(4) and taking into account Eq. (8), we get function ( , )f r p in the form (7) with
3 2
( ) ( ) 2 ( )A kT
r r r and 1
( )2 ( )kT
rr
. Thus, minimization of fI (11) subject to the
constrains (3) and (4) yields the same distribution function as it has been obtained by GBP [48].
It is well-known that the exact Wigner distribution function can attain the negative values
[50], i.e. it is not probability distribution function (in this sense we employ the term ‘quasi-
probability’), while the function ( , )f r p (13) is non-negative everywhere since the multiplier ( )A r
is not defined if ( )T r <0. Thus ( , )f r p (13) is just Wigner-like function which correctly yields the
electron density and electronic kinetic energy density in accordance with (3) and (4), however it
does not necessary provide any other local properties. Nevertheless, calculations of electron
exchange energy and electron momentum density using ( , )f r p (13) have yielded the good results
as compare with Hartree-Fock method [58,59], and hence consideration of the function ( , )f r p in
the form (13) is physically reasonable.
A substitution of ( , )f r p (13) into (9) and integration leads to the following expression for the
phase-space-defined Fisher information density:
23 ( ) 9 ( )( )
( ) 2 ( )f
s
ikT t
r rr
r r. (14)
This is the main result of this work. While the integral PS-Fisher information fI characterizes the
global sharpness of electron momentum distribution and related to the momentum uncertainty for a
system in a given electronic quantum state, function ( )fi r does the same locally. The higher the
PS-FID value, the smaller the local uncertainty of electron momentum and vice versa. In the other
words, the phase-space defined Fisher information density ( )fi r quantitatively characterizes the
available information about momentum of electron at each point of a position space. The explicit
link between the exact PS-FID and the electron momentum uncertainty is given in Appendix.
6
3. Results and discussion
Let us now consider the behavior of the phase-space defined Fisher information density, eq.
(14), in some typical cases. The PS-FID is computed by the locally modified program Multiwfn 3.2
[60] from the wave functions at the Hartree-Fock/6-311G level for all cited atoms (excepting Ca,
for which the basis set 6-31G* is taken) and in the Kohn-Sham/B3LYP/aug-cc-PVQZ
approximation for molecules derived by using the Firefly v.8.0 G code [61]. The non-negative form
of the kinetic energy density
1( ) ( ) ( )
2s i i
i
t r r r , (15)
where ( )i r are the one-electron orbitals, has been used in (14). For the finite systems, at the large
distances from the nuclei the electron density square decreases more rapidly than ( )st r ; as a result
( )fi r goes to zero at r . Atomic units (a.u.) are used throughout the paper.
a) The PS-Fisher information density and atomic shells. Positions of the PS-FID extremes for
all atoms from Li to Ar are given in Table 1 together with those for ELF [12] and negative
Laplacian of electron density [8], 2 ( ) r , which are the popular descriptors of atomic electronic
shells. Fig. 1 demonstrates that PS-FID plots for atoms exhibit a series of alternative local maxima
and minima. The ELF and Laplacian show a similar behavior, functions ( )fi r and ELF are
characterized by the same numbers of local maxima and minima. The locations of ( )fi r minima are
quite close to the positions of minima in ELF, which allow us to identify these regions with the
boundaries between the atomic electronic shells. For example, for the Li atom, the local minimum
between two ( )fi r peaks is situated at a distance of 1.542 a.u. from the nucleus, as compare with
the ELF minimum at 1.528 a.u. Analogously, the ( )fi r and ELF minima are located at 0.448 and
0.470 a.u. from the nucleus for the N atom and at 0.282 and 0.297 a.u. for the Ne atom.
We checked atoms with Z20; in all cases the PS-FID function has revealed the atomic
electronic shells. For instance, in calcium atom (Fig. 1) PS-FID shows minima at 0.123; 0.602 and
2.500 a.u. from the nucleus, which can be associated with the boundaries between K, L, M and N
atomic shells, as compared with the ELF yielding 0.128; 0.618 and 2.552 a.u. for the same features.
In addition we observed that positions of ( )fi r maxima for atoms with Z20 are close to those in
2 ( ) r (see Fig. 1 and Table 1), and 2 ( ) r peaks are somewhat closer to the nucleus than
corresponding PS-FID peaks. For Ca atom, however, the functions ( )fi r and ELF exhibit four
maxima, while the Laplacian displays only three ones. Indeed, as it has been reported in [35,36], in
this case Laplacian is unable to distinguish the outermost valence electron shells.
7
Li
N
Ne
Ca
Fig. 1. The plots of the PS-Fisher information density (red), ELF (blue) and negative Laplacian of electron density
(green) for Li, N, Ne and Ca atoms. The x axis shows the distance from the nucleus. The PS-FID and ELF common
scale is on the left hand and the Laplacian scale is on the right hand. For the convenient comparison, the ELF values for
Ca atom are doubled. Atomic units (a.u.) are used throughout the paper.
The positions of ( )fi r maxima are placed between the ELF and 2 ( ) r peaks (Table 1).
Besides, for atoms with only s valence shell electrons (as Li and Ca), ( )fi r exhibits the outermost
maximum, while the ELF is failed to do that [12,62]. It is some advantage of ( )fi r as a descriptor
of the electronic valence shells.
The integration of electron density over atomic electronic shells yields the occupancies of
corresponding shells. Associating ( )fi r minima with the boundaries between shells, we obtained
the electron occupancy values for all atoms under consideration (Table 1). The occupancy of the K
shell for the second- and third-row elements are 2.02.1 e, while it is 7.88.0 e for the L shell of the
third-row elements. Kohout and Savin [62] found for the third-row atoms the ELF occupancies of
2.02.2 e for the K shells and 7.9 e for L shell.
8
Table 1. Positions of maxima and minima in ( )fi r (the first row), ELF (the second row) and 2
( ) r (the
third row) as well as the K and L shell electron occupancies derived from ( )fi r (the first row) and ELF (the second
row) for atoms from Li to Ar.
Atom rmin, a.u. rmax, a.u. Electron occupancy, e
K shell L shell K shell L shell K shell L shell
Li
1.542
1.528
0.505
2.645
-
2.463
2.0
2.0
Be
0.968
1.022
0.370
1.728
-
1.578
2.0
2.0
B
0.715
0.751
0.290
1.304
1.791
1.199
2.0
2.0
C
0.557
0.584
0.237
1.046
1.491
0.905
2.0
2.1
N
0.448
0.470
0.202
0.863
1.270
0.785
2.0
2.1
O
0.379
0.397
0.174
0.738
1.081
0.674
2.0
2.1
F
0.325
0.340
0.153
0.642
0.949
0.588
2.1
2.2
Ne
0.282
0.297
0.138
0.567
0.848
0.517
2.1
2.2
Na
0.253
0.265
0.128
2.213
2.132
0.873
0.507
0.752
0.440
3.442
-
3.545
2.1
2.2
8.0
7.9
Mg
0.227
0.236
0.116
1.625
1.678
0.768
0.450
0.653
0.392
2.615
-
2.582
2.0
2.2
7.9
7.9
Al
0.207
0.214
0.107
1.298
1.398
0.680
0.404
0.583
0.353
2.126
2.808
2.119
2.1
2.2
7.9
7.9
Si
0.188
0.195
0.100
1.156
1.152
0.612
0.367
0.515
0.322
1.819
2.318
1.790
2.1
2.2
7.8
7.9
P
0.172
0.180
0.093
0.998
1.023
0.562
0.335
0.470
0.295
1.603
2.003
1.522
2.0
2.2
7.9
7.9
S
0.159
0.166
0.087
0.884
0.908
0.511
0.309
0.428
0.274
1.425
1.768
1.349
2.1
2.2
7.8
7.9
Cl
0.148
0.154
0.082
0.793
0.815
0.472
0.286
0.391
0.256
1.283
1.581
1.208
2.0
2.2
7.9
7.9
Ar
0.138
0.143
0.077
0.715
0.737
0.440
0.267
0.360
0.238
1.168
1.462
1.088
2.1
2.2
7.8
7.9
9
We conclude that the Fisher information density obtained from the phase-space quasi-
probability distribution function (14) reveals the atomic electron shells and yields their reasonable
electron occupancy values. Thus, ( )fi r serves as a quantitative descriptor of electron localization.
To clarify such behavior of the phase-space-defined Fisher information density we note that
total non-interacting kinetic electronic energy density ( )st r is proportional to the electron
momentum variance density (see Appendix), i.e. ( )st r yields a local quantitative estimation of the
momentum uncertainty. It can be decomposed into the two parts [63]:
( ) ( ) ( )s W Pt t t r r r , (16)
where Weizsäcker energy density [64],
1 ( ) ( )( )
8 ( )Wt
r rr
r, (17)
is exact kinetic energy for non-interacting bosonic system, and the Pauli energy density, ( )Pt r ,
arising from the fact that electrons obey the Pauli exclusion principle [65]. Note that Weizsäcker
energy density per electron, ( ) ( )Wt r r , is proportional to the square of imaginary part of the local
electron momentum [66,67,68,69], ( ) ( ) 2 ( ) p r r r , which describes the electron ‘quantum
fluctuations’ or ‘quantum spread’ [68] and recently has been used to investigation of electron
localization in atomic [25] and molecular [26,27] systems. On the other hand, ( )Wt r up to a
multiplier represents the Fisher information density of electron distribution
( ) ( )( )
( )i
r rr
r, (18)
which can be regarded as a local measure of sharpness of electron density ( ) r [70]. Thus, ( )i r
and therefore ( )Wt r carry information about the single electron position regardless both of its spin
and positions of another particles. If ( )Wt r is large at the point r, it means that probability to obtain
an electron in some neighboring point is quite different, than that in r. At the same time by virtue of
additivity ( )Pt r also has the meaning of information; it is the information about electron position
provided that another electron with the same spin is localized at the point r. The better electron pair
is localized around point r, the smaller ( )Pt r value i.e. the smaller information about another
electron pair.
Thus, the electron momentum uncertainty hidden in ( )st r consists of two contributions. The
first one (as it requires the uncertainty principle) results from the predominant location of electrons
in atoms and molecules ‘somewhere near the nuclei’ due to electrostatic attraction. According to
[71,30], it can be associated with a definite size of atomic electronic shells, i.e. with an electronic
10
steric factor. This causes nonzero quantum spread term (which vanishes everywhere only for the
total electron delocalization) and, hence, defines the behavior of function ( )Wt r . In hypothetical
bosonic-like system our knowledge of particle position in the real space given by ( ) r is the only
source of the momentum uncertainty. In the actual many-electron system, the exchange correlation
between the same spin electrons increases the local electron kinetic energy by the amount of ( )Pt r
and contributes to total uncertainty of electron momentum. Between atomic electronic shells, both
( )Wt r and ( )Pt r are relatively large and, correspondingly, ( )fi r is small. At the same time ( )Pt r is
small within the shells, correspondingly ( )fi r peaks in these regions. We note that is the function
( )Pt r that defines the electron pairs localization in ELF [12,72]. The quantum spread term in PS-
FID, as described by ( )Wt r , which is related with electrostatic attraction of electrons to the nuclei,
shifts the ( )fi r maxima towards the nucleus as compared with ELF (see Table 1). Thus, additional
account for the quantum spread term in the PS-FID makes ( )fi r preferable descriptor of the spatial
localization of electron pairs in atoms and molecules.
b) The PS-Fisher information density of covalent bonds in some simple molecules. The PS-
FID images for BH3, BF3, NF3 and PF3 molecules have depicted in Fig. 2 (a–d) in comparison with
corresponding ELF maps (Fig. 2 (e–h)). In BH3, the large ( )fi r peaks (Fig. 2a) are situated between
B and H nuclei pointing out the areas where bonding electron pairs are mainly localized. The same
picture is inherent to the ELF distribution (Fig. 2e). Positions and shapes of all maxima are the same
for the both functions; however, the ELF maxima are wider. The spherical maximum around the B
nucleus position corresponds to the 1s electron shell of this atom.
The PS-FID and ELF in polar B–F bonds of BF3 (Fig. 2b,f) shows that bonding electron pairs
are shifted toward more electronegative F atoms reflecting some ionic component of these bonds.
The local maxima on the B–F lines are somewhat closer to F atoms in the ( )fi r function. The both
functions exhibit the core-related maxima around all the nuclei positions and also show the two
electron lone-pair peaks behind each F atom. This also obviously demonstrates that information
about electron momentum is larger in the more electronegative atom.
Similar agreement between ( )fi r and ELF is observed in NF3 molecule as well (Fig. 2c,g).
The local bonding maxima on N–F bonds are only slightly shifted to the N nucleus unlike BF3,
because the electronegativity difference between N and F atoms is less than between B and F ones.
In PF3 molecule (Fig. 2d and 2h), however, we observe the ( )fi r displacement towards the F atoms
similar to BF3. Thus, ( )fi r as ELF reveals a polarity of the single P–F bond. The K and L electron
shell maxima in P atom are also visible in PS-FID figure.
11
a)
e)
b)
f)
c)
g)
d)
h)
Fig. 2. The PS-FID (left) and ELF (right) distributions in BH3, BF3, NF3 and PF3 molecules. The PS-FID lines
are drawn with the step of 0.1 in the interval 01.0 and with the step 0.2 in the interval 1.03.0. The ELF lines are
depicted with the step 0.05 in the interval 0 to 0.50 and with step 0.10 in the interval 0.50 to 1.00. Some lines were
added for clarity.
12
Note that volumes occupied by the bonded and lone
electron pairs in the PS-FID representation (see Fig 2-4
below) are compressed and located somewhat closer to the
nuclei as compare with the ELF due to account for the
quantum spread.
It is instructive to consider PS-FID distributions in the
ethane, ethylene and acetylene (Fig. 3). It demonstrates that
PS-FID distinguishes the single, double and triple carbon-
carbon bonds. The compact high bonding maximum on the
C–C bond in the ethane is spread in the ethylene and
acetylene. One can say that the increase in the bond
multiplicity locally extends the electron momentum
distribution in the interatomic CC regions decreasing,
respectively, the information about electron momentum due
to repulsion of the bonding electron pairs. This feature,
however, is not common. For example, in the N2O5
molecule (which is not shown here), the ( )fi r peaks are
higher in the single N–O bonds rather than on the one-and-
a-half N–O bonds. Thus, the electron lone pairs of O atoms,
which are absent in C atoms, effect the bonding electrons in
N2O5 decreasing the electron momentum fluctuations.
c) The PS-Fisher Information Density in the Charge-
Shift and Non-Covalent Interactions. Fig. 4 depicts the PS-
FID distributions in the gas-phase molecule N2O4 and van
der Waals dimer (Cl2)2. In N2O4, function ( )fi r explicitly
demonstrates the difference between covalent N–O bonds
and weak N–N bond [73] (Fig. 4a). Indeed, there is only one
local ( )fi r maximum on each N–O bond, while two the PS-
FID maxima become apparent in N–N bond, indicating the
charge separation along interatomic line. It agrees with the
conclusion that the N–N bond is formed due to electron pair
fluctuation between two nitrogen atoms and that molecular stabilization is achieved due to the
resonance energy [74]. It also allows to consider this bond as intermediate between covalent C–C
a)
b)
c)
Fig. 3. The PS-FID distributions in
a) ethane, b) ethylene and c) acetylene. The
lines are drawn in the interval 03.0 with
the step of 0.2 a.u.
13
bond in ethane and so-called charge-shift bond [75,76], which is characterized by the electron shift
from the middle of the bond distance to the atomic basins.
a)
b)
Fig. 4. PS Fisher information density in N2O4 and (Cl2)2. The lines are in the interval 01.0 with the step 01 and the
interval 1.0to 3.0 with the step 0.2.
PS-FID distributions in the van der Waals dimer (Cl2)2 (Fig. 4b) shows the charge-shift
intramolecular bonds ClCl in each of Cl2 molecules [75] and the weak intermolecular Cl2···Cl2
interaction. PS-FID shows maxima at the middle of intramolecular ClCl bonds and K and L core
electronic shells around Cl atoms. It is clearly seems from Pic. 4b that non-covalent intermolecular
Cl2···Cl2 interaction happens in such a way that the chlorine lone electron pair in the one molecule
faces depleted area in at ( )fi r of another molecule. This picture reflects a key-and-lock mechanism
of non-covalent bonding in Cl2…Cl2 dimer already reported in [77,32] in terms of the Laplacian of
electron density and ELF. PS-FID confirms that the one Cl2 molecule in a dimer (Cl2)2 acts as a
Lewis base while the second one serves a Lewis acid. Thus PS-FID produces chemically corrected
results for weak atomic and molecular interactions as well.
d) The ( )fi r Basin Analysis. The gradient field of scalar function ( )fi r allows analyzing the
topology of the PS-FID function similar to that for ELF [78]. We have separated the basins in ( )fi r
which can be attributed to core electron shells, valence electrons and electron lone pairs. Using
terminology from the ELF analysis [78], we can distinguish in the PS-FID distribution the
monosynaptic core basins, monosynaptic electron lone pair basins and disynaptic valence basins
corresponding to the location of the bonding electron pairs.
14
In BH3, ( )fi r shows one monosynaptic basin, corresponding to the core shell of B atom and
three protonated disynaptic basins, which include the H nuclei. The electron density integration
over these mentioned basins showed that each of them contains 2.0 electrons i.e. one electron pair.
Therefore, exactly one electron pair is the share on each single B–H bond. In the BF3 we observe
the monosynaptic core basins in each atom and three disynaptic basins located on the B–F bonds
and shifted towards F. The B and F core basins contain 2.0 and 2.1 electrons, correspondingly. At
the same time, each of the valence disynaptic basin includes only 1.6 electrons, in contrast to the
Lewis model. It results from significant polarity of the B–F bonds and electron density
displacement to the F atoms: two equal monosynaptic basins of each F atom, corresponding to the
electron lone pairs, are occupied by 3.2 e each. Similar situation takes place in NF3 molecule: we
have found four monosynaptic core basins, three disynaptic bonding basins and valence
monosynaptic basins with lone pairs around F and N atoms. However, in PF3 there are no
disynaptic bonding basins in PS-FID; corresponding density is merged into the electron lone pair
regions.
In the C2H6 molecule, the valence disynaptic basins corresponding to C–C bond and C–H
bonds are occupied with 2.0 e, as well as in the C atom core shell. This is entirely consistent with
the Lewis model. In the ethylene and acetylene, however, this model is violated. We have found
only 3.2 e on the C–C bond in C2H4 and only 5.4 e in C2H2 molecules instead of 2 and 3 electron
pairs respectively. The other electrons are equally distributed over the C–H bonds.
Basin analysis in N2O4 shows disynaptic basins with electron populations of 2.6 between N
and O atoms, while the weak N–N bond [74,75] exhibits two monosynaptic basins populated by 1.4
e. It means that two N atoms share approximately three electrons, two of which are capable to
pairing. Thus a chemical bond between N atoms can be considered as formed due to the resonance
fluctuations between single electron in one N atom and electron pair in another. In addition, each O
atom shows two monosynaptic electron lone pair basins with occupancies of 2.7 and 2.3 e, the more
occupied basins faced to each other within the molecule.
In each Cl2 molecule, the charge-shift bonding mechanism of intra-molecule interaction
agrees well with occupancy of disynaptic valence basins of 1.2 e. This basin does not split into two
monosynaptic ones because Cl–Cl interaction is stronger than N–N interaction in N2O4. This small
occupancy demonstrates in agreement with [75] that intramolecular Cl–Cl bond does not formed by
the electron pair, but rather by electron fluctuation between two Cl atoms.
4. Summary
In this work, we combined the information-theoretical description of electron density with
density functional theory and derived the expression for the Fisher information density, ( )fi r , eq.
15
(14), by applying the minimum information principle to the phase-space quasi-probability
distribution function. Analysis of this PS-FID function revealed the electron shell structure in atoms
with Z≤20 and allowed to locate the core, valence and lone pair regions of electron in some
instructive molecules. Moreover, PS-FID is capable to distinguish the bonds with different
multiplicity in ethane, ethylene, and acetylene and to reveal the features of the charge-shift bond
and non-covalent molecular interactions. The link between our approach and local electron
momentum theory has been established. The regions of maximally probable electron pair location
as defined by PS-FID reflect the local electron momentum uncertainty related with both nonzero
quantum spread, which can be associated with an electronic steric factor, and the exchange
correlation between the same spin electrons as defined by the Pauli energy density, ( )Pt r . An
additional account for the quantum spread term in ( )fi r makes it preferable tool of the spatial
localization of electron pairs in atoms and molecules as compared with the other descriptors.
Acknowledgement
This work was supported by the Russian Foundation for Basic Research, grant 13-03-00767a.
Appendix
It can be proved that the PS Fisher information density for any exact non-negative quasi-
probability electron position-momentum distribution function ( , )F r p (not necessary of the form
(7)),
( , ) ( , )( )
( , )F
F Fi d
F
p pr p r pr p
r p, (A1)
satisfies the following inequality
29 ( )( )
( ( ))Fi
Var
rr
p r, (A2)
where ( ( ))Var p r is the electronic momentum variance density [79]
2( ( )) [ ( ) ( ) ] ( , )Var d F p r p p r r p r p , (A3)
and ( )p r is the electronic momentum average density [66,68]
( ) ( , )d F p r pp r p . (A4)
The expression (A2) is nothing else as the Cramer-Rao bound [80] in a local form. In atomic and
molecular systems in a stationary electronic state at the equilibrium ( )p r equals to 0. It implies that
variance density for these systems is proportional to the kinetic energy density ( ( )) 2 ( )sVar tp r r
(in the orbital approximation). Hence
16
29 ( )( ) ( )
2 ( )F f
s
i it
rr r
r. (A5)
Thus, ( )fi r in eq. (14) is a lower bound for the exact PS-Fisher information density (A1); the
equality in (A5) holds only for Gaussian distribution (i.e. only for this distribution the Fisher
information density ( )Fi r is inversely related to the variance density, in agreement with eq. (14)).
Note that expression (A5) can be treated as a local form of Stam’s uncertainty relation for the Fisher
information and the mean square momentum [81,41].
References
[1] G.N. Lewis, The atom and the molecule, J. Am. Chem. Soc. 38, 762 (1916).
[2] R.G. Gillespie, I. Hargittai, The VSEPR model of molecular geometry. Boston: Allyn and Bacon, 1991.
[3] R.J. Gillespie, P.L.A. Popelier, Molecular geometry and chemical bonding: From Lewis to electron densities, New
York, Oxford University Press, 2001.
[4] A.A. Simões, In between worlds: G.N. Lewis, the shared pair bond and its multifarious contexts, J. Comput. Chem.
28, 62 (2007).
[5] R.F.W. Bader, The physical basis of the Lewis electron pair model, Molecules in Physics, Chemistry and Biology,
3, 73 (1989).
[6] R.F.W. Bader, S. Johnson, T.-H. Tang, P.L.A. Popelier, The Lewis model and beyond, J. Phys. Chem. 100, 15398
(1996).
[7] R.F.W. Bader, J.Hernández-Trujillo, F. Cortés-Guzmán, Chemical bonding: From Lewis to atoms in molecules, J
Comput. Chem. 28, 4 (2007).
[8] R.F.W. Bader, Atoms in molecules: A quantum theory, Oxford University Press, Oxford, 1994, p. 438.
[9] V.G. Tsirelson, R.P. Ozerov, Electron density and bonding in crystals: Principles, theory and X-ray diffraction
experiments in solid state physics and chemistry, Bristol and Philadelphia: IOP, 1996.
[10] R.F.W. Bader, P.J. MacDougall, C.D.H. Lau, Bonded and nonbonded charge concentration and their relation to
molecular geometry and reactivity, J. Am. Chem. Soc. 106, 1594 (1984).
[11] G. Hunter, Atoms in molecules from the exact one-electron wave function, Can. J. Chem. 74, 1008 (1996).
[12] A.D. Becke, K.E. Edgecombe, A simple measure of electron localization in atomic and molecular systems, J.
Chem. Phys. 92, 5397 (1990).
[13] A. Savin, A.D. Becke, J. Flad, R. Nesper, H. Preuss, H. von Schnering, A new look et electron localization,
Angew. Chem. Int. Ed. Engl. 30, 409 (1991).
[14] P.W. Ayers, R.G. Parr, A. Nagy, Local kinetic energy and local temperature in the density-functional theory of
electronic structure, Int. J. Quant. Chem. 90, 309 (2002).
[15] H.L. Schmider, A.D. Becke, Chemical content of the kinetic energy density, J. Mol. Struct. (Theochem) 527, 51
(2000).
[16] H.L. Schmider, A.D. Becke, Two functions of density matrix and their relation to the chemical bond, J. Chem.
Phys. 116, 3184 (2002).
[17] M. Kohout, A measure of electron localizability, Int. J. Quant. Chem. 97, 651 (2004).
[18] M. Kohout, K. Pernal, F.R. Wagner, Y. Grin, Electron localizability indicator for correlated wavefunctions. I.
Parallel-spin pairs, J. Theor. Chem. Acc. 112, 453 (2004).
[19] M. Kohout, K. Pernal, F.R. Wagner, Y. Grin, Electron localizability indicator for correlated wavefunctions. II.
Antiparallel-spin pairs, J. Theor. Chem. Acc. 113, 287 (2005).
[20] A. Savin, K.D. Sen (ed) In: Reviews of modern quantum chemistry. A celebration of the contributions of Robert G
Parr. World Scientific, Singapore, 2002.
[21] E. Cancès, R. Keriven, F. Lodier, A. Savin, How electrons guard the space: Shape optimization with probability
distribution criteria, Theor. Chem. Acc. 111, 373 (2004).
[22] M. Causà, A. Savin, Maximum probability domains in crystals: The rock-salt structure, J. Phys. Chem. A 115,
13139 (2011).
[23] R.F.W. Bader, G.L. Heard, The mapping of the conditional pair density onto electron density, J. Chem. Phys. 111,
8789 (1999).
[24] M. Kohout, A. Savin, H. Preuss, Contribution to the electron distribution analysis. I. Shell structure of atoms, J.
Chem. Phys. 95, 1928 (1991).
17
[25 . . oh rque , R.J. Boyd, On the local representation of the electronic momentum operator in atomic systems, J.
Chem. Phys. 129, 024110 (2008).
[26 . . oh rque , R.J. Boyd, A localized electron detector for atomic and molecular systems, J. Theor. Chem. Acc.
127, 393 (2010).
[27 . . oh rque , C. Matta, R.J. Boyd, The localized electron detector as an ab initio representation of molecular
structures, Int. J. Quant. Chem. 110, 2418 (2010).
[28] P. de Silva, J. Korchowiec, T.A. Wesolowski, Revealing the bonding pattern from the molecular electron density
using single exponential decay detector: An orbital-free alternative to the electron localization function,
ChemPhysChem. 13, 3462 (2012).
[29] R.F. Nalewajski, A.M. Köster, S. Escalante, Electron localization function as information measure, J. Phys. Chem.
109, 10038 (2005).
[30] V.G. Tsirelson, A.I. Stash, S. Liu, Quantifying steric effect with experimental electron density, J. Chem. Phys. 133,
114110 (2010).
[31] V.G. Tsirelson, A.I. Stash, V.V. Karasiev, S. Liu, Pauli potential and Pauli charge from experimental electron
density, J. Comp. Theor. Chem. 1006, 92 (2013).
[32] V.G. Tsirelson, A.I. Stash, Determination of the electron localization function from electron density, Chem. Phys.
Lett. 351, 142 (2002).
[33] V.G. Tsirelson, A.I. Stash, Analyzing experimental electron density with the localized-orbital locator, Acta Cryst.
B58, 780 (2002).
[34] V. Tsirelson, A. Stash, On functions and quantities derived from the experimental electron density, Acta Cryst.
A60, 418 (2004).
[35] R.P. Sagar, A.C.T. Ku, V.H. Smith, A.M. Simas, The Laplacian of the charge density and its relationship to the
shell structure of atoms and ions, J. Chem. Phys. 88, 4367 (1988).
[36] H. Schmider, R.P. Sagar, V.H. Smith, On the determination of atomic shell boundaries, J. Chem. Phys. 94, 8627
(1991).
[37] C.E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27, 379 (1948).
[38] R.A. Fisher, Theory of statistical estimation, Proc. Cambridge Philos. Soc. 22, 700 (1925).
[39] R.B. Frieden, Science from Fisher information: An unification, Cambridge University Press, Cambridge, 2004, p.
490.
[40] R.F. Nalewajki, Information origins of the chemical bond, Nova Science Publishers, New York, 2010, p. 196.
[41] E. Romera, J.C. Angulo, J.S. Dehesa, Fisher entropy and uncertaintylike relationships in many-body systems, Phys.
Rev. A 59, 4064 (1999).
[42] I. Bialynicki-Birula, J. Mycielski, Uncertainty relations for information entropy in wave mechanics, Commun.
Math. Phys. 44, 129 (1975).
[43] S. López-Rosa, R.O. Esquivel, J.C. Angulo, J. Antolín, J.S. Dehesa, N. Flores-Gallegos, Fisher information study
in position and momentum spaces for elementary chemical reactions, J. Chem. Theory Comput. 6, 145 (2010).
[44] A. Mohajeri, M. Alipour, Shannon information entropy of fractional occupation probability as an electron
correlation measure in atoms and molecules, Chem. Phys. 360, 132 (2009).
[45] A. Borgoo, F. De Proft, P. Geerlings, K.D.Sen, Complexity of Dirac–Fock atom increases with atomic number,
Chem. Phys. Lett. 444, 186 (2007).
[46] D.K. Seo, C.J. Weng, Orbital interpretation of kinetic energy density and a direct space comparison of chemical
bonding in tetrahedral network solids, J. Phys. Chem. A 112, 7705 (2008).
[47] V.G. Tsirelson, A. Nagy, Binding entropy and its application to solids, J. Phys. Chem. A 113, 9022 (2009).
[48] K. Ghosh, M. Berkowitz, R.G. Parr, Transcription of ground-state density functional theory into a local
thermodynamics, Proc. Natl. Acad. Sci. USA 81, 8028 (1984).
[49] R.G. Parr, W. Yang, Density functional theory of atoms and molecules, Oxford University Press, Oxford, 1989.
[50] E.P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749 (1932).
[51] L. Cohen, Generalized phase-space distribution functions, J. Math. Phys. 7, 781 (1966).
[52] L. Cohen, Local kinetic energy in quantum mechanics, J. Chem. Phys. 70, 788 (1979).
[53] L. Cohen, Representable local kinetic energy, J. Chem. Phys. 80, 4277 (1984).
[54] L. Cohen, Y.I. Zaparovanny, Positive quantum joint distributions, J. Math. Phys. 21, 794 (1980).
[55] E.T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106, 620 (1957).
[56] J.O. Hirschfeld, C. Curttis, R. Bird, Molecular theory of gases and liquids, Wiley, 1964.
[57] See, for example, S.P. Flego, A. Plastino, A.R. Plastino, Special features of the relation between Fisher information
and Schrödinger eigenvalue equation, J. Math. Phys. 52, 082103 (2011) and related links.
[58] R.G. Parr, K. Rupnik, S.K. Ghosh, Phase-space approach to the density-functional calculation of Compton profiles
of atoms and molecules, Phys. Rev. Lett. 56, 1555 (1986).
[59] C. Lee, R.G. Parr, Gaussian and other approximations to the first-order density matrix of electronic systems, and
the derivation of various local-density-functional theories, Phys. Rev. 35, 2377 (1987).
[60] T. Lu, F. Chen, Multiwfn: A multifunctional wavefunction analyser, J. Comp. Chem. 33, 580 (2012).
[61] A.A. Granovsky, Firefly version 8.0, www http://classic.chem.msu.su/gran/firefly/index.html, the last accessed
16.01.2014.
18
[62] M. Kohout, A. Savin, Atomic shell structure and electron numbers, Int. J. Quant. Chem. 60, 875 (1996).
[63] A. Holas, N.H. March, Construction of the Pauli potential, Pauli energy, and effective potential from the electron
density, Phys. Rev. A 44, 5521 (1991).
[64] C.F. von Weizsäcker, Zur Theorie der Kernmassen, Z. Phys. 96, 431 (1935).
[65] M. Levy and H. Ou-Yang, Exact properties of the Pauli potential for the square root of the electron density and the
kinetic energy functional, Phys. Rev. A 38, 625 (1988).
[66] L. Cohen, Local values in quantum mechanics, Phys. Lett. A 212, 315 (1996).
[67] M.J.W. Hall, Exact uncertainty relations, Phys. Rev. A 64, 052103 (2001).
[68] S. Luo, Statistics of local values in quantum mechanics, Int. J. Theor. Phys. 41, 1713 (2002).
[69] I.P. Hamilton, R.A. Mosna, L. Delle Site, Classical kinetic energy, quantum fluctuation terms and kinetic-energy
functionals. Theor. Chem. Acc. 118, 407 (2007).
[70] S.B. Sears, R.G. Parr, U. Dinur, On the quantum-mechanical kinetic energy as a measure of the information in a
distribution, Israel J. Chem. 19, 165 (1980).
[71] S.B. Liu, Steric effect: a quantitative description from density functional theory, J. Chem. Phys. 126, 244103
(2007).
[72] P.W. Ayers, Electron localization functions and local measures of the covariance, J. Chem. Sci. 117, 441 (2005).
[73] V.G. Tsirelson, A.V. Shishkina, A.I. Stash, S. Parsons, The experimental and theoretical QTAIMC study of the
atomic and molecular interactions in dinitrogen tetraoxide, Acta Cryst. B65, 647 (2009).
[74] T.M. Cardozo, M.A.C. Nascimento, Chemical Bonding in N2 molecule and the role of quantum mechanical
interference effect, J. Phys. Chem. A 113, 12541 (2009).
[75] S. Shaik, D. Danovich, B. Silvi, D.L. Lauvergnat, P.C. Hiberty, Charge-shift bonding – a class of electron-pair
bonds that emerges from valence bond theory and is supported by the electron localization function approach, Chem.
Eur. J. 11, 6358 (2005).
[76] S. Shaik, D. Danovich, W. Wu, P.C. Hiberty, Charge-shift bonding and its manifestation in chemistry, Nat. Chem.
1, 443 (2009).
[77] V.G. Tsirelson, P.F. Zou, T.-H. Tang, R.F.W. Bader, Topological definition of crystal structure: Determination of
bonded interactions in solid molecular chlorine, Acta Cryst. A 51, 143 (1995).
[78] B. Silvi, A. Savin, Classification of chemical bonds based on topological analysis of electron localization
functions, Nature 371, 683 (1994).
[79] J.G. Muga, J.P. Palao, R. Sala, Average local values and local variances in quantum mechanics, Phys. Lett. A 238,
90 (1998).
[80] H. Cramer, Mathematical Methods of Statistics. Princeton Univ. Press, Princeton, 1946.
[81] A.J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Inf. Control 2, 101
(1959).
Highlights
The phase-space-defined Fisher information density is derived and studied
This function reveals the regions of maximal concentration of electron pairs
The features of this function are linked with the local electron momentum uncertainty
