Linear Response and Measures of Electron Delocalization ... - CRM2

Current Organic Chemistry, 2011, 15, 3609-3618 3609 

Linear Response and Measures of Electron Delocalization in Molecules 

János G. Ángyán* 

Équipe Modélisation Quantique et Cristallographique, CRM2, UMR 7036, Institut Jean Barriol, Nancy-University, CNRS, B.P. 239, 

F-54506 Vandoeuvre-lès-Nancy, France 

Abstract: The concept of localization and delocalization in molecules is discussed in terms of the response of the electronic system to an 

external perturbation. It is argued that both the spatial organization of electrons in pairs and the spatial distribution of the response 

intensity, reflect main features of the correlated motion of electrons, ultimately described by the pair distribution function of electrons. 

Various measures, derived from the linear charge density response function, are able to characterize localization in a rigorous way, in 

close analogy to the approach followed in solid state physics. 

Keywords: Electron localization, linear response, Resta localization index, bond order, localized orbitals, exchange hole, distributed 

polarizabilities. 

1. INTRODUCTION 

Electron localization/delocalization has often been a 

controversial subject in chemistry and physics, perhaps because this 

concept is sometimes used only in a relatively loose, qualitative 

sense in the chemical literature. However, theoretical chemists have 

always attributed a great importance to a better understanding of 

this notion and from the early days of electronic structure theory a 

considerable effort has been spent to extract precise pieces of 

information about the localization of electrons from the wave 

function. This subject has been reviewed abundantly in 2005, in a 

series of papers of a special issue of the Chemical Reviews, see e.g. 

Refs [1, 2]. Other interesting review papers [3, 4] and critical 

accounts [5, 6] were published in the same year. One should also 

mention the Special issue "90 years of chemical bonding" of the 

Journal of Computational Chemistry, with several contributions 

closely related to the subject of the present article, e.g. [7-10]. The 

field continues to develop rapidly and a considerable number of 

new ideas about the characterization of electron localization has 

appeared in the past few years. 

The purpose of the present contribution is not to provide an 

exhaustive review of these different developments, rather to give a 

somewhat specific view of the problem of electron localization, 

which is probably less familiar to the chemical community. It will 

be argued that there is a straight relationship between the localized 

or delocalized nature of an electronic system and its response 

properties, in particular its charge density response properties. 

Although this viewpoint seems to be a widely accepted one in the 

solid state physics community, following mainly the pioneering 

work of Resta and others [11-13], according to the author's opinion 

it has still not received the deserved audience among chemists. 

Resta's overview paper [14], addressed specifically to the chemists' 

community, written in a clear and accessible style, seems to have 

remained practically without any echo. It can be hoped that this 

situation will change in the future and the usefulness of this 

viewpoint, stressing the links between the theory of electric 

polarization and electron localization/delocalization, will be 

recognized in the world of finite molecular systems. 

*Address correspondence to this author at the Équipe Modélisation Quantique et 

Cristallographique, CRM2, UMR 7036, Institut Jean Barriol, Nancy-University, 

CNRS, B.P. 239, F-54506 Vandoeuvre-lès-Nancy, France; Tel: +33 3 83 68 48 74; 

Fax: +33 3 83 68 43 00; E-mail: Janos.Angyan@crm2.uhp-nancy.fr 

A measure of central importance, introduced by Resta, is the 

electron localization tensor, defined as the fluctuation of the 

electron position operator or, in more mathematical terms, the 

second cumulant moment of the electron distribution. This quantity 

is related to the imaginary part of the frequency-dependent dipole 

polarizability tensor by a well-known sum rule, the zerotemperature 

form of the fluctuation-dissipation theorem. The 

underlying physics is quite simple: an electronic system which 

undergoes strong spontaneous quantum fluctuations is, par 

excellence, delocalized. In the mean time, such a system is ready to 

respond to external perturbations, in other words, it is strongly 

polarizable. Moreover, as we shall see, not only the intensity of 

response and the delocalization of electrons can be put in parallel, 

but the spatial distribution of the response and the organization of 

electrons in opposite-spin pairs are also expected to follow similar 

trends. A further remarkable point is that the electronic 

polarizability is, in principle, an experimentally accessible physical 

property, therefore it might be possible to characterize the 

localizability of electrons on experimental grounds. 

The localization tensor is only one of the possible polarizationbased 

measures of localizability. As it will be shown, other 

quantities, like indices characterizing delocalization between 

domains as well as some variants of pointwise electron localization 

functions can be related to the linear response function of the 

electronic system. The present discussion will be focused on these 

relatively disparate subjects linked together by their relationship to 

the linear response properties of the electronic system. 

It is important to stress that the linear response viewpoint is not 

contradictory to the common interpretation of electron localization 

in terms of the Pauli exclusion principle and of the properties of the 

Fermi-hole. Since the linear response is related to the paircorrelation 

of electrons, which is strongly dominated by the Fermicorrelation, 

consequence of the Pauli principle, it is natural that 

organization of electrons in pairs be also reflected, in a certain way, 

by the response properties. 

The paper is organized as follows. In Section 2 the linear charge 

density response function and the fluctuation-dissipation theorem is 

introduced, while in Section 3 Resta's localization tensor is derived 

therefrom. In Section 4, it is shown how delocalization/localization 

indices are related to the charge-flow polarizabilities between 

appropriately selected regions. The relationship of some orbital 

localization criteria to the polarizability is briefly discussed in 

1385-2728/11 $58.00+.00 © 2011 Bentham Science Publishers

3610 Current Organic Chemistry, 2011, Vol. 15, No. 20 János G. Ángyán 

Section 5. In Section 6, the local measures of the electron 

localization are overviewed from the polarizability viewpoint, and 

finally we conclude by summarizing the main points of the present 

contribution. 

2. FROM THE POLARIZABILITY TO THE EXCHANGE- 

CORRELATION HOLE 

The response of an electronic system to an external, possible 

time-dependent, electric perturbation is characterized by a linear 

response function, (r, r ;) , which can be regarded as a 

generalization of its polarizability: 

(r;)= dr(r, r ;)V( r ;), . (1) 

where V( r ;) is a perturbing potential at point r oscillating by a 

frequency , (r;) is the induced charge density at r and the 

linear charge density response function is defined as 

(r,r ' ;)= lim 1 

0 + n0 

 

0| ˆ(r)|nn | ˆ(r' )|0 

 

+ i 0n 

0| ˆ(r' )|nn | ˆ(r)|0 

i + 0n 

 

 

. (2) 

The zero-temperature fluctuation-dissipation theorem 

establishes a relationship between the imaginary part of the charge 

density response function and the charge density fluctuation 

autocorrelation function: 

 

 

d m(r, r ;)=S(r, r ). (3) 

0 

The charge density autocorrelation function, sometimes called 

static form factor, is a ground state expectation value, 

S(r, r )=0| ˆ(r) r )|0, (4) 

where ˆ(r)= ˆ(r)(r) is the charge density fluctuation operator 

taken with respect to the mean ground state density, (r) . The 

function S(r, r ) is closely related to the exchange-correlation hole, 

h xc 

(r, r )= 2 

(r, r )/(r)( r ) : 

S(r, r )=(r)h xc 

(r, r )+ (r)(r r ), (5) 

as it can be easily proven by considering the definition of the pair 

density operator, ˆ 2 

(r, r )= ˆ(r) ˆ( r )ˆ(r)(r r ) , and inserting it in 

the definition of the form factor. 

An approximate relationship can be established between the xchole 

density and the static charge density response, 

(r, r )=(r, r ;0). Let us define a position dependent parameter 

(r, r ) such that 

(r, r )= 2 

S(r, r ) 

(r, r ) . (6) 

(r, r ) plays the role of an average energy denominator for a pair 

of coordinates and allows us to express the frequency dependent 

polarizability in a single-pole form, c.f. Eq. (7) of Ref. [15]. If one 

neglects the spatial dependence of (r, r ) by replacing it by a 

space average, (r, r ) , an approximate proportionality can be 

established between the static polarizability and the xc-hole density: 

(r, r ) (r)h xc 

(r, r )+ (r)(r r ). (7) 

The qualitative meaning of the above relationship is that in a 

point r the amplitude of the electronic charge density response to 

an external perturbation at r can be predicted from the knowledge 

of the exchange-correlation hole belonging to the reference point 

r . The expected response will be different from zero only in those 

points of the space where the xc-hole is non-vanishing. In other 

words the polarization of the electrons remains confined within the 

space occupied by the xc-hole. If the xc-hole extends over the 

whole molecular framework, the electron is strongly delocalized. 

This would be a finite, molecular analog of the conducting state in 

an infinite solid. On the contrary, if the xc-hole occupies only a 

limited portion of the molecular space, one should consider the 

electron as localized. Therefore, as far as one understands the 

spatial distribution of the xc-hole, one has a key to predict the 

charge density response and vice versa. Both of these quantities, 

response function and Fermi-hole, provide appropriate bases for our 

understanding of electron localization. 

A related example for using a mapping between response 

functions and the Fermi-hole density has been given by the 

interpretation of the of Fermi-contact contributions to the nuclear 

spin-spin coupling constants [16-19]. Visualization of the response 

spin densities and of the Fermi-hole belonging to a reference 

electron placed at an atomic position has convincingly 

demonstrated the relationship between these two quantities. 

Furthermore, a proportionality has been found between the spinspin 

coupling constants and the delocalization index between the 

atom pair, measuring the sharing of the Fermi-hole between the 

atomic volumes [16]. 

The charge density polarizability function as well as the xc-hole 

are complicated two-variable functions, which are not really 

appropriate to obtain a quick insight and understanding. Further 

analysis of these functions is needed either in terms of integrated 

quantities or by using a kind of coarse-grain representation, e.g. by 

space domains or other kind of partitions of the molecule. In the 

following sections it will be shown how these functions are related 

to various simpler measures of electron localizability. 

3. RESTA LOCALIZATION TENSOR 

A useful global measure of the electron localizability has been 

defined by Resta [12-14], via the second cumulant moment of the 

electron distribution. It can be easily shown that the following 

expectation value, describing the electron position fluctuation per 

electron, 

N 

ˆr ˆr 

c 

= 1 N {0| ˆr 

(i)ˆr 

( j)|00| ˆr 

(i)|00| ˆr 

( j)|0}, (8) 

i, j=1 

i=1 

j=1 

is strictly equivalent to the second moment of the static form factor 

normalized to the number of electrons, N : 

N 

ˆr ˆr 

c 

= 1 ' 

drdrr 

r 

S(r, r ). (9) 

N 

In the above equations ˆr 

(i) the = x, y,z component of the 

i -th electronic position vector. We can see that Eq. (9) establishes 

a direct link between fundamental two-variable functions discussed 

in the preceding section. The intensive quantity, ˆr ˆr 

c 

, is called 

also second cumulant moment per electron, or localization tensor 

[14]. 

An explicitly origin-independent equivalent form of the 

localization tensor can be obtained by using the chargeconservation 

sum rule of the form factor: 

ˆr ˆr 

c 

= 1 dr dr (r r ) 

(r r') 

S(r, r ), (10) 

2N 

N

Linear Response and Measures of Electron Delocalization Current Organic Chemistry, 2011, Vol. 15, No. 20 3611 

Table 1. Selected "Experimental" Resta Localization Indices, r 2 exp c 

= 3 S DODS 2 N 1 

, from DOSD Data. The Systems are Listed in the Order of Increasing 

Localization Index. The Degree of Delocalization, as Measured by this Quantity, is Supposed to Increase from the the Top to the Bottom of 

the Table 

Molecule 

r 2 c 

exp 

Ref. Molecule r 2 c 

exp 

Ref. 

Ne 0.5702 [22] H 2CO 1.1766 [21] 

SF 6 0.6739 [23] SiH 4 1.3233 [24] 

SiF 4 0.6951 [25] CH 3OCH 3 1.3321 [26] 

Ar 0.7307 [22] C 2H 5OC 2H 5 1.3900 [26] 

Cl 2 0.8523 [27] CH 3OC 3H 7 1.3975 [26] 

N 2 1.0161 [28] NH 3 1.3978 [28] 

CO 1.0422 [29] C 2H 2 1.4357 [30] 

H 2S 1.0942 [25] C 2H 4 1.5290 [31] 

H 2O 1.0974 [29] C 2H6 1.5625 [31] 

He 1.1168 [21] H 2 2.3250 [28] 

and Eq. (10) can be further transformed to a form, which is easy to 

interpret as the density-weighted second moment of the xc-hole 

function: 

ˆr ˆr 

c 

= 1 dr(r)dr (r r ) 

(r r') 

h xc 

(r, r ), (11) 

2N 

establishing a relationship between the xc-hole (dominated by the 

exchange- or Fermi-hole) and the Resta localization tensor. Clearly, 

and xc-hole with large second moment components can be 

considered as an indication for the presence of strongly delocalized 

electrons. 

The expression of the second cumulant moment in terms of the 

frequency-dependent dipole polarizability tensor, 

, is 

essentially the same as that of the S 1 

, the –1 order dipole oscillator 

strength sum rule (see e.g. Ref. [20]). Comparing 

 

dm 

 

0 

()=N r 

r 

c 

(12) 

with the usual form of the sum rule, one finds a simple 

proportionality between dipole oscillator strength sum rule S 1 

and 

the trace of the localization tensor, which can be called Resta 

localization index, r 2 c 

: r 2 c 

= 3 

2N S 1 

. Thus it can be expected 

that the spherical average of the localization tensor is an 

information accessible from experimental data, e.g. via the dipole 

oscillator strength distributions (DOSD) reported in the literature. 

The existence of such a relationship is quite remarkable, because it 

makes possible an experimental characterization of a fundamentally 

conceptual quantity, the electron localizability. A set of illustrative 

examples is presented in Table 1, partly relying on a compilation of 

experimental data by Olney [21] and using also some more recent 

DOSD determinations. 

The S 1 

oscillator strength sum rule, and therefore the trace of 

the localization tensor, is connected to the second moment of the 

intracule and extracule densities by rigorous relationships, which 

has been derived for the case of many-electron atoms [32]. It can be 

shown also that S 1 

(and the trace of the localization tensor) is 

proportional to the the negative of the mean squared distance 

between electrons. 

The Resta localization index, in a somewhat paradoxical 

manner, increases with the degree of delocalization. In view of this 

fact, it may seem legitimate to call this quantity rather 

"delocalization index". Alternatively one might choose e.g. the 

inverse of r 2 c 

, which would increase by increasing degree of 

localization of the electron. However, such a change of 

nomenclature would lead to confusion with earlier works, therefore 

the original terminology has been retained in this work. 

The discussion up to now has been about the in principle exact 

xc-hole, which supposes that both the ground state wave function in 

the expectation values and the linear response function appearing in 

the fluctuation-dissipation theorem are exact. Usually one has only 

some approximations available for these quantities, e.g. in the form 

of an independent particle method, like Hartree-Fock (HF) or 

Kohn-Sham (KS). In these cases, the single determinant wave 

function provides the exchange (Fermi) hole, which can be 

calculated from the first order density matrix. In principle, the same 

quantity can be obtained from the fluctuation-dissipation formula, 

using the the non-interacting (uncoupled Hartree-Fock, UCHF) 

response function. To reproduce expectation value results in this 

way, extremely large basis sets would be needed due to the slow 

convergence of the resolution of identity in Gaussian basis sets. A 

further possibility is to use the interacting, TDCHF (time-dependent 

coupled Hartree-Fock) polarizabilities in the fluctuation-dissipation 

formula, giving access to a low-order approximation of the 

correlation hole, but the problem of the basis set incompleteness 

remains still a crucial one. As far as direct correlation effects on the 

expectation value of the second cumulant moment are concerned, a 

few recent full CI calculations have been reported for linear Li n and 

Be n chains in order to monitor the appearance of a metal-insulator 

type phase transitions as a function of the interatomic spacings 

[33, 34]. 

The localization tensor itself can be related to the dynamic 

dipole polarizability tensor and it is approximately proportional to 

the static polarizability tensor of the system. The tensorial character 

of the localization function reflects the fact that the localizability 

can be different in different directions. For instance, one expects 

higher mobility and larger tensor elements in an aromatic plane of a 

molecule than in the perpendicular direction. A few examples of 

theoretically calculated localization tensor are listed in Table 2, 

obtained at the HF/6-311G** level, using a development version of 

the MOLPRO program package [35]. It is interesting that aromatic


Table 2. The trace of the Resta Localization Tensor r 2 theor 

c 

Calculation were done at the HF/6-311G** Level 

and the Corresponding Anisotropies, Defined as 2 = 1 2 [3Tr (r r 2 ) Tr(r 

c 

 

r 

c 

) 2 ]. 

Molecule r 2 theor 

c 

Molecule r 2 theor 

c 

 

Ne 0.5856 0.0000 CHCH 1.8202 0.3610 

SF 6 0.8250 0.0000 CH 3-CH 3 1.8400 0.0056 

F 2 0.8470 0.1932 C 8H + 9 (homotropylium) 1.8424 0.1907 

Ar 0.9105 0.0000 CH 2=CH-CCH 1.8446 0.2991 

He 1.1765 0.0000 CH 2=CH 2 1.8510 0.1888 

H 2O 1.1791 0.0971 C 6H 6 1.8599 0.2171 

CO 1.2609 0.1489 CH 2-CH=CH-CH 2 1.8660 0.2613 

ClNO 1.2983 0.2473 C 10H 8 (naphtalene) 1.8711 0.2594 

N 2 1.3389 0.2720 CH 4 1.8960 0.0000 

H 2C=O 1.3840 0.1627 C 16H 10 (pyrene) 1.9028 0.2904 

CS 2 1.5537 0.4480 C 24H 12 (coronene) 1.9281 0.3224 

NH 3 1.5657 0.0602 CH 2=C=C=CH 2 1.9412 0.4889 

H 2C=NH 1.6315 0.1913 B 2H 6 2.1485 0.0822 

CH=C=NH 2 1.6761 0.2874 H 2 2.5961 0.3855 

systems, which are conventionally considered as strongly 

delocalized, have a mean localization tensor around 1.86-1.90 and 

show a relatively moderate anisotropy. The methane molecule is 

characterized by a value of 1.896, slightly larger than that of the 

benzene and naphtalene. The presence of heteroatoms, i.e. a polar 

character of the electronic structure decreases dramatically the 

mean value of the localization tensor, indicating a stronger 

localization of electrons in these systems. 

4. Delocalization Indices and Distributed Polarizabilities 

While in the previous section we have seen that a global 

characterization of the localizability of electrons can be achieved 

via the Resta localization tensor, which is closely related to the total 

molecular dipole polarizability, in this section we continue our 

analysis in terms of a coarse grain, domain-based representation of 

the full nonlocal charge density response function. 

Consider a partition of the full molecular space in disjoint, nonoverlapping 

domains, { a 

} . Although at this point the explicit 

nature of this partition is irrelevant, in order to fix the ideas, we can 

think about Bader's atoms in molecule (AIM) topological 

partitioning, as an example. Following Stone's concept of 

distributed polarizability [36], the total charge density response can 

be partitioned to multipolar contributions assigned to pairs of 

domains. The appropriate choice of the domains is quite important 

in order to have physically meaningful polarizability values. For 

instance, the Bader's AIM scheme seems to be a convenient choice, 

because of the excellent transferability properties of the atomic 

domains obtained in this framework [37-39]. The lowest-order 

multipolar contribution to the distributed polarizabilities is due to 

the variation of the domain net charges under the effect of an 

external potential or field: electric charges flow from one domain 

(atom) to another one, driven by the electrostatic field (potential 

difference) acting between them. The charge-flow polarizability is 

given by 

ab 

()= dr 

a 

dr(r, r ;), (13) 

b 

and it can be calculated at any level of response theory, provided 

one has transition matrix elements of the electron population 

operator over the various domains [37, 38]. Since the net electric 

charge of the molecule should be invariant under the effect of an 

external field, in addition to the symmetry property, ab 

= ba 

, the 

following charge-conservation sum rule holds: 

ab 

()=0. (14) 

b 

The charge conservation rule implies that the sum of the 

negative two-center charge-flow polarizabilities compensates 

exactly the on-site charge-charge polarizability, aa 

= 

ba ab 

. 

The fluctuation-dissipation theorem can be applied to the 

charge-flow polarizabilities ab 

() leading to the interdomain 

correlation of fluctuations (covariances) of the number of electrons, 

described by the domain (atomic) population operators, ˆNa 

 

 

d m ()= ˆN 

0 ab a 

ˆNb c 

. (15) 

ˆN a 

ˆNb c 

can be regarded as a domain decomposed form-factor, 

ˆN a 

ˆNb c 

= dr drS(r, r )= ab 

N a 

I ab 

. (16) 

a 

b 

The delocalization index, I ab 

, between different domains is the 

covariance of the populations (see e.g. Ref. [2]) 

I ab 

= dr(r) 

a 

drh xc 

(r, r ), (17) 

and reflects the number of electrons shared by the two domains. In 

the a=b case I aa 

is related to the variance of the electron 

b


F 

Cl 

O 

O 

1.3489 

1.7571 1.4697 1.2269 

1.1630 N 

1.1738 

N 

P 

N 

O 

133.25 

o 

O 

O 

129.77 

o 

O 

O 

O 

O 

O 

O 

O 

S 

O 

1.1569 

1.1601 

1.7375 

1.2857 

1.2916 

C 

1.7336 

C 

C 

C 

F 

o 

F 

Cl 

108.27 113.20 

o 

Cl 

S 

S 

O 

O 

F 

Cl 

O 

O 

1.3010 1.7453 1.4050 1.4134 

B 

B 

S 

B 

F 

F 

Cl 

Cl 

O 

O 

O 

O 

Fig. (1). Schematic illustration of the family of Y-conjugated systems. The numbers are the bond lengths. After Ref. [43]. 

ΑAA 

2.5 

2.0 

1.5 

1.0 

0.5 

0.0 0.5 1.0 1.5 2.0 2.5 

N 2 

A c 

ΑAB 

0.0 

0.2 

0.4 

0.6 

0.8 

1.0 0.8 0.6 0.4 0.2 0.0 

N A N B c 

Fig. (2). Correlation of Bader atomic population fluctuations (a) and atom-atom fluctuations (b) with the corresponding charge-flow polarizabilities for Y- 

conjugated systems. On (a) green points (dashed line) are central atoms, red points (solid lines) are the oxygen ligands, and other type of ligand atoms are 

marked by blue points (dotted line). On (b) only the bonds with the central atom are considered. The linear regression lines were calculated by excluding a few 

outlier points, indicated by different colors. After Ref. [43]. 

population and it can be considered as a domain localization index, 

giving the number of electrons localized in a 

[2]. A similar 

analysis can be performed [40] from the viewpoint of the domain 

averaged Fermi hole (DAFH) concept [8, 41, 42]. 

The analog of the charge-conservation sum rule for the 

delocalization indices is a simple consequence of the normalization 

of the exchange hole: ˆN 2 a 

c 

= 

ˆN ba a 

ˆNb c 

and we can 

anticipate the following proportionality: 

ˆN a 2 c 

aa 

and ˆN a 

ˆNb c 

ab 

(a b) (18) 

The proportionality factor is in principle system-dependent and 

it behaves like a kind of mean excitation energy, characteristic to 

the atom pair. The expected proportionality between (static) charge 

flow polarizabilities and delocalization indices (and on-site chargecharge 

polarizabilities) has been tested for a family of related Y- 

conjugated compounds [43], shown on Fig. (1). 

It is clear from Fig. (2) that both the one- and two-center 

indices are in fact proportional to the corresponding charge-flow 

polarizabilities. In the one-center case the atoms have been 

classified in different categories: central B, C, N, S, P atoms (with 

the exclusion of the carbon atoms of CS 3 and COCl 2 ), oxygen 

ligand atom and halogen (F,Cl) and S ligand atoms. For the twocenter 

case only formally bonded atom pairs are shown, since the 

too weak bonding/polarizability does not lead to interpretable 

correlations. Disregarded of the above mentioned outliers, very 

good linear regressions could be found, indicating that for these 

special bonds there is a unique mean excitation energy. In some 

cases the delocalization indices as well as the single-center 

localization indices are quite low, below 0.5. One can consider such 

values as indicators for the presence of strongly ionic bonding. 

The benzene molecule is an example where the Fermi-hole 

extends farther than the nearest-neighbour atoms and the charge 

density response remains large even in the para (1,4) position. The 

-electron system of the benzene is usually considered as a


delocalized system, because the Fermi-hole of the -electrons is not 

confined to atoms and pairs of atoms, and it extends over the whole 

molecular skeleton. 

We can compare the charge flow polarizabilities and the 

delocalization indices in the case of the benzene molecule. The 

well-known relationships in organic chemistry between orthometa- 

and para-positions in benzene are quite well reflected by both 

the charge-flow polarizabilities and the delocalization indices. 

Distributed polarizability calculations [38] with the time dependent 

coupled Hartree-Fock (TDCHF) method provide the following 

charge-flow polarizabilities: -0.800 (ortho), +0.103 (meta) and - 

0.316 (para), while the delocalization indices [44] are 1.386 (ortho), 

0.072 (meta) and 0.098 (para). It is remarkable that the charge-flow 

polarizability for the meta atom is positive, which means that (at 

this level of approximation) the electron flow between atoms in 

meta position is in the direction of the sign of electric potential 

difference, although for negatively charge electrons it is expected to 

be the opposite. Applying the proportionality rule between the 

charge density response and the xc-hole one concludes that in the 

region of the meta carbon atoms the xc-hole function changes its 

sign. We remind that TDCHF includes some approximate 

Coulomb-correlation effects, while the UCHF polarizabilities which 

are related to the Fermi-hole function, should be negative 

everywhere in the space. Therefore it seems that the Coulombcorrelation 

effect enhances considerably the difference in the 

behavior of the meta and para carbon atoms. A comparison of these 

sites show considerably less difference in terms of delocalization 

indices calculated from the single determinant. 

The question arises, whether it can be defined in general terms 

the criteria of having well-localized domains in a molecule In the 

light of the above discussion, such a criterion can be defined by the 

equality, I aa 

N a 

, which implies that the xc-hole must be, at least 

approximately, normalized over the domain a 

, i.e. 

drh xc 

(r, r ) 1 . Under such circumstances the local sum rule 

a 

dr drS(r, r ) 0 (19) 

a 

a 

holds true, providing a condition of the localizability over the 

domain a 

. 

Depending on the chemical nature of the atom in question, 

Bader's AIM domains can be either well-localized or delocalized 

character. For instance, domains corresponding to a mono- or 

polyatomic ion in a strongly ionic structure will be naturally 

localized, just like subsets of atoms that constitute a closed shell 

molecule in a weakly bound intermolecular complex. On the other 

hand, atoms connected by a covalent bond are not well-localized 

because their domains comprise only a part of the xc-hole, 

extending over two or more atomic basins. Genuinely welllocalized 

domains must be constructed according to appropriate 

criteria. The main purpose of numerous local functions 

characterizing localization is precisely to identify regions of highly 

localizable and less localizable character. For instance, ELF 

(electron localization function) attractor domains [45] are probably 

good candidates for naturally well-localized regions of space and it 

can be expected that the inter-region fluctuations are relatively low. 

5. MOLECULAR ORBITAL LOCALIZATION CRITERIA 

In the following it will be shown that some of the most popular 

localization criteria can be, in fact, traced back to the requirement 

of minimum deformability (polarizability) of the one-electron 

orbitals building up the Slater-determinant of an independent 

particle wave function. 

In his 1960 paper, entitled “Construction of some molecular 

orbitals to be approximately invariant for changes from one 

molecule to another” Boys proposed to “impose a mathematical 

constraint of maximum insensitivity to alterations in the distant 

nuclear charges”. Such a criterion was supposed “to lead orbitals 

which are localized around the chemical valency links and the 

atomic lone pairs” [46]. This condition means that the shape of 

these one-electron functions (orbitals) ensures their maximal 

stiffness and minimal polarizability and, by consequence their 

insensitivity to the chemical environment. In other, more familiar 

terms, such orbitals are supposed to be maximally transferable. 

The final form of the Boys-Foster localization criterion [47], 

established only a few years after the above-cited work, requires 

that , the spread 

= i 

| r 2 | i 

i 

| r | i 

2 (20) 

i 

be minimal after a series of unitary transformations of the orbitals. 

Of course, such transformations, preserving the orthonormality of 

the orbitals do not not change the occupied manifold and leave the 

single-determinant wave function invariant. The condition of 

minimizing the spread has been introduced in solid state physics 

literature many years after the formulation of Boys localization 

critera [11] leading to maximally localized Wannier functions in 

solids. 

The spread is the sum of the dipole moment fluctuations 

associated with the individual orbitals and therefore it can be 

considered as a measure of the stiffness of the orbitals. At the 

minimum of this functional the sum of the one-electron 

polarizabilities associated to the orbitals is supposed to be the 

smallest possible. Resta has shown that the trace of the localization 

function is a lower bound of the spread [14]. In fact, in case of a 

single determinant wave function, the trace of the localization 

tensor can be written as 

N = | i 

| r | j 

| 2 . (21) 

i 

Note that has been defined as the second cumulant moment 

per electron, therefore one has to multiply the lhs by N . Since is 

positive and invariant with respect to the unitary transformation of 

the orbitals, the minimum of will correspond to the minimum of 

squared sum of the off-diagonal dipole moment matrix elements, or 

to the maximum of the squared sum of the diagonal dipole moment 

matrix elements. The scheme of the Boys localization can be thus 

summarized as, 

ji 

i 

min{ i 

| r 2 | i 

i 

| r | i 

2 }. (22) 

i 

 

invariant 

i 

 

maximal 

Although these auxiliary criteria are more often practical in 

computational implementations, it is important to keep in mind that 

the underlying physical criterion is the maximum transferability or 

minimum polarizability of the orbitals, expressed by the condition 

of minimal dipole fluctuation. 

One can proceed in an analogous way if the localizability is 

measured by the delocalization and localization indices over 

domains. Supposing a given partition of the molecular space, the


sum of localization indices (cf. Eqs (16)-(18)) for all the regions 

can be written as 

i 

| ˆNa | i 

| i 

| ˆNa | j 

| 2 . (23) 

a 

i 

a 

While this quantity is invariant to unitary transformations of the 

orbitals, it is not so for the sum of single-orbital localization 

indices: 

D = i 

| ˆNa | i 

| i 

| ˆNa | i 

| 2 , (24) 

a 

i 

which should be minimized in order to get optimally localized 

orbitals. Following a similar reasoning as previously, the 

minimization of the fluctuation can be replaced by maximization of 

the sum of square of the orbital populations. Summarizing these 

localization schemes based on the minimization of atomic 

populations over the orbitals: 

min{ 

 

i 

| ˆNa | i 

 

i 

a 

invariant 

 

ij 

a 

i 

i 

| ˆNa | i 

2 }. (25) 

i 

a 

maximal 

This is the Pipek-Mezey localization criterion, which has been 

generalized to multideterminant wave functions and Bader atomic 

domain populations (instead of the originally used Mulliken 

populations) by Cioslowski [48]. 

Popular in quantum chemistry before the appearance of more 

fundamental approaches in the years '90 and widely used to 

characterize bonding in molecules, localized orbitals are often 

considered as suspect and theoretically unfounded tools for the 

analysis of chemical bonding. Paradoxically, in solid state 

electronic structure studies, the equivalent maximally localized 

Wannier functions enjoy a certain popularity and play a crucial role 

in the interpretation of electric polarization phenomena in solids. 

The so-called "Wannier centers" [11], i.e. the centroids of localized 

orbitals, are routinely used to interpret the evolution of the 

electronic structure during ab initio molecular dynamics 

trajectories. Is there a deeper reason for the success of localized 

orbitals 

The answer is affirmative and it is based on a strong connection 

between localized orbitals and the Fermi-hole. This relationship can 

be understood by taking a simple model for a well-localized system 

described by a single-determinant wave function, expressed in 

terms of a set of localized spin-orbitals, { i 

(r)} [49]. If this 

system is well-localized, the orbitals are strictly localized, i.e. the 

product (differential overlap) i 

(r) j 

(r) of two different 

localized orbitals ( i j ) of -spin electrons is (vanishingly) 

small. In such a case the Fermi-hole, can be approximately 

represented in a “diagonal” form: 

h x 

(r, r ) 1 

occ 

| i 

(r)| 2 | i 

( r )| 2 . (26) 

 

(r) i 

The regions occupied by the strictly localized orbitals partition 

the molecular space is to approximately disjoint regions, i 

, 

characterized by the condition that | i 

(r)| 2 / 

(r) 1 . With the 

help of the function i 

(r) , defined as unity if r i 

and 0 

otherwise, the Fermi-hole function is approximately 

occ 

h x 

(r,r') i 

(r)| i 

(r') | 2 . (27) 

i 

The above equation expresses the rule observed by Luken [50- 

52], who remarked that the shape of the Fermi-hole remains stable 

within a domain of a given electron pair. Moreover, in these cases 

the shape of the Fermi-hole strongly resembles to the square of the 

localized orbital. In this way one can establish a qualitative or semiquantitative 

relationship between the localization from a "physical" 

linear response viewpoint, and the localized orbital picture, 

sometimes considered in the modern quantum chemical literature as 

an obsolete one. The relationship between localized orbitals and 

domain averaged Fermi hole functions has been raised by Matito 

and Salvador on the one hand and Mayer on the other in the 

Faraday Discussion on Chemical Concepts from Quantum 

Mechanics [53]. 

6. LOCALIZABILITY CHARACTERIZED BY SINGLE- 

VARIABLE FUNCTIONS OVER THE MOLECULAR SPACE 

In addition to a global characterization of localization, like 

Resta localization tensor or index, or coarse grain localization 

measures like the intra- and intra-domain (e.g. atomic and diatomic) 

localization and delocalization indices (sometimes referred to as 

covalent bond orders and valences), there is a definite need for 

pointwise measures of electron localization as well. The ultimate 

expectation towards such functions is that they provide a pictorial 

representation about the organization of electron pairs in the space 

and visualize the Lewis dot structure of molecules. 

One of the widely used function to characterize electron 

localization is the negative of the Laplacian of the electron density, 

L(r)= 2 (r) . The Laplacian reflects the shell structure of atoms 

and, as it has been first pointed out on empirical grounds by Bader 

and coworkers [54], there is a good correspondence between the 

topology of the negative Laplacian and the domains where the 

electron pairings are the strongest, providing thus a link between 

L(r) and Lewis or VSEPR models of chemical bonding [7]. The 

regions of maxima of L(r) correspond to local charge 

concentrations, while negative regions of L(r) can be associated 

with local charge depletion. This behavior of the 3-dimensional 

Laplacian function could be put into correspondence with the 

properties of the 6-dimensional Fermi-hole distribution. In 

particular there is a homeomorphism between the Laplacian of the 

charge density and the Laplacian of the conditional pair distribution 

function. 

As we were able to relate the xc-hole to the change density 

response function by a sum rule, the fluctuation-dissipation 

theorem, analogously, one can relate the Laplacian of the charge 

density to another sum rule, the generalized f-sum rule, which is 

reads as [55] 

+ 

c(r, r )= 1 

dm(r, r ;). (28) 

 

The real function c(r, r ) can be expressed as a ground state 

expectation value of the double commutator of the Hamiltonian 

with the charge density operator at two different positions, 

c(r 1 

,r )=0|[[Ĥ, ˆ(r 2 1)], ˆ(r 2 

)] | 0 = 1 

1 ( (r 1 

r 2 

)(r 1 

)). (29) 

After having taken the derivatives on the rhs, one of the terms is 

equal to the negative of Laplacian multiplied by the Dirac delta 

function, (r 1 

r 2 

) 2 1 

(r 1 

) . The usual dipole oscillator strength 

Thomas-Reiche-Kuhn sum rule can be obtained by double 

integrating the second moment of c(r 1 

,r 2 

) . In particular, the integral


of the second moment of 

L(r)= 2 (r) turns out to be 

proportional to the number of electrons. 

Another family of localization functions is based on various 

forms of the local covariance of the electron pair distribution [6]. A 

first example of this category of pointwise functions has been the 

ELF proposed by Becke and Edgecombe [56] and later interpreted 

and reformulated by others, see e.g. [45, 57, 58]. The ELF is based 

on a second order expansion of the Fermi-hole density and use an 

appropriate transformation function, which has the role to 

renormalize its possible values between 0 and 1 and accentuate the 

graphical expression of the shell structure. Among the numerous 

possible formulations of electron localization functions one can cite 

e.g. the localized orbital locator (LOL) of Schmider and Becke [59], 

the electron localizability indicator (ELI) of Kohout [60]. An 

interesting generalization of this type of approach consists in 

extending the radius of spherical interation from an infinitesimal 

domain to a larger finite one, leading to the radial exchange density 

function of Geier [61]. 

Still another possibility to construct local functions can be 

based on integrated or global properties of the xc-hole distribution, 

h xc 

(r, r ) belonging to the reference point, r . One of the 

characteristic features of the hole function normalized to -1 is the 

position of its centroid or its center of charge: 

' 

D 

(r)= drr 

h xc 

(r, r ), (30) 

which is a three-dimensional vector ( = x, y,z ) in the laboratory 

reference frame. The xc-hole distribution belonging to the reference 

point r , is centered in D(r) , which may serve as a natural 

expansion center for non-local hole models. Furthermore, according 

to the charge density reconstitution sum rule [62], the total 

electronic dipole moment μ 

can be obtained by the density 

weighted average of the xc-hole centroids: 

μ 

= dr(r)D 

(r). (31) 

This equation appears as the fuzzy generalization of the formula 

to obtain the polarization vector in terms of the Wannier centers, as 

it is used in solid state theory. Notice that the x-hole centroid 

function, D(r) has appeared as an auxiliary quantity in the Becke- 

Johnson model for dispersion forces [63], where these authors 

introduced the notion of "exchange-hole dipole moment" for the 

case of single-determinant wave functions. Associating a simple, 

classical picture to the negative unit point charge at r , representing 

the electron, and a positive unit charge in D(r) , representing the 

hole, the exchange-hole dipole moment function becomes: 

d(r) = D(r) - r, (32) 

which will be useful in the forthcoming discussions. 

As a tentative local scalar characterization of localization, one 

can start from the explicitly origin-independent form of the second 

cumulant moment expression (11) and define the following 

function, 

(r)= 1 2 (r) dr (r r )(r 

r )h 

(r, r ) 

(33) 

xc 

measuring the trace of the second moment of the xc-hole, i.e. its 

spread in the space, weighted by the electron density at the 

reference point. This quantity has the advantage, that by its 

integration over the whole space one obtains the global Resta 

localization index, . 

Taking into account the fact that the hole-centroid, D(r) 

appears as a natural expansion centre for multipolar analysis of the 

hole, an "intrinsic" second moment distribution can be defined as 

(r)= 1 2 (r) dr (D(r) r )(D(r) r )h xc 

(r, r ). (34) 

Unfortunately this quantity does not integrate to the second 

cumulant moment of the total electronic system. 

Finally we can consider the difference of (r) and (r) and 

construct the following function [64], 

(r)=2( (r)(r) )= (r)d(r) d(r), (35) 

which integrates to the density weighted squared exchange-hole 

dipole moment 

d 2 = dr(r)= dr(r)d 2 (r). (36) 

The quantity d 2 can be regarded as an approximation to the 

correlated second cumulant moment of the electronic system, 

calculated, somewhat paradoxically, from a single determinant 

wave function, i.e. from the exchange hole [15, 63, 65]. 

It has been argued [64], that since (r) measures the squared 

distance of the electron to its exchange-hole centroid, it has a small 

value for the regions where the electron is close to the center of its 

own exchange-hole, therefore it indicates a high probability of 

having strongly paired electrons. An electron localization function, 

which has been called Fermi-hole locality indicator (FHLI), can be 

constructed [64] by the help of a transformation function, as 

FHLI(r)= ( 1+(r) ) 1 . (37) 

The FHLI function is quite different from the ELF-like 

functions, since instead of using strictly local features of the 

exchange hole, it is constructed on the basis of a strongly non-local 

property, the distance of the electron to the centroid of its hole. 

(a) acetylene (b) ethylene (c) ethane 

Fig. (3). Ethylene, ethane and acetylene FHLI functions. (After Ref. [64]). 

A few illustrative examples of the FHLI function have been 

calculated by a development version of the MOLPRO program 

package [35]. The 6-311G** basis set has been used and the FHLI 

functions were represented on a regular grid in the Gaussian cube 

format and visualized by the VMD molecular graphics package 

[66]. 

The function FHLI provides a quite reasonable picture of the 

electron localization, which resembles, at least in some respects, to 

conventional ELF images. In the series of the prototypes for 

carbon-carbon triple (Fig. 3) double and single bonds one can 

clearly distinguish the signature of different bonding types. At the 

chosen isodensity contour the triple and double bonds are more 

voluminous than the carbon-carbon -bond. However, the 

representation of the bonds between non-hydrogen atoms seems to 

differ markedly from the ELF pictures. While the X-H bonds


(a) butatriene (CH 2=C=C=CH 2 ) (b) vinylacetylene (HC C CH=CH 2) 

Fig. (4). FHLI functions of butatriene and vinylacetylene. 

appear as a single domain, the electrons that contribute to a given 

X-Y bond leave their signatures in the form of a more-or-less 

spacious FHLI domains, located near the atomic centers which 

participate in the bonding. The -bond of ethane can be recognized 

from the two small domains localized on the C-C axis near the 

atoms. In the case of ethylene the four electrons of the double bond 

appear as symmetrically arranged lobes above and below the 

molecular plane, corresponding to a “banana-type” representation. 

The triple bond in acetylene can be recognized from the cylindric 

shapes located near the carbon atoms. 

In the case of isomeric C 4 H 4 structures of butatriene and 

vinylacetylene, one can distinguish the triple and double bonds in 

the latter, while in the butatriane one clearly sees the allen-like 

arrangement of three double bonds. It is interesting to observe in 

butatriene that the two central atoms contribute by significantly less 

spacious in-plane lobes than the terminal atoms. In the case of 

vinylacetylene, the deformation of the disks representing the triple 

bonds is obviously due to the perturbative effect of the molecular 

skeleton (Fig. 4). 

In the case of aminoacetylene (Fig. 5a) the most significant 

structure is the double disk of the triple bond. Around the NH 2 

group, the H atoms and the lone pair form an almost regular 3-fold 

symmetric arrangement. The methylenimine (Fig. 5b), which is isoelectronic 

with ethylene one can observe a kind of fusion of the 

nitrogen lone pair and the p z -like lobes of the double bond located 

on the same nitrogen. 

localized electrons are less, delocalized electron are more 

polarizable - can be put in more precise terms and establish a clear 

relationship with the more conventional descriptors of electron 

localization, based on the xc-hole or more specifically on the 

Fermi-hole and its properties. The view advocated in this account is 

inspired by recent developments in solid state electronic structure 

theory, based on a parallel between the modern theory of 

polarization and the notion of electron localization. It has been 

shown that the Resta localization index, used until now almost 

exclusively as a global measure to characterize the itinerant or nonitinerant 

character of electrons in solids, can be of some utility for 

molecular systems as well. Moreover, using experimental dipole 

oscillator strength distributions, the localization index seems to be 

accessible on experimental grounds as well. 

The fluctuation or covariance of the domain populations in a 

molecule, widely used as coarse-grain measures to describe 

localization and delocalization between atoms (or other domains) 

has been related to the charge-flow polarizability, which can be 

derived directly from the charge density response function. The 

approximate proportionality between charge-flow polarizabilities 

and delocalization indices has been illustrated on a few selected 

examples. 

Orbital localization criteria can also be reformulated as a 

requirement of constructing the most transferable and therefore the 

less polarizable set of one-electron functions. The Boys localization 

criterion is based on the dipole fluctuation, while the Pipek-Mezey 

and related criteria can be derived from the requirement of minimal 

fluctuation of atomic (or domain) populations. 

Finally, it has been shown that the Fermi hole locality indicator, 

which is able to characterize the electron localization by measuring 

some non-local features of the Fermi-hole, integrates to an 

approximate value of the second cumulant moment, thereby related 

again to the linear response function. 

Hopefully, the above cited examples could illustrate the 

intimate link between the basic physical mechanisms that govern 

the response properties of an electronic system and the specific 

organization of fermions, which leads to the important chemical 

concept of electron localization. 

8. ACKNOWLEDGEMENTS 

The author is grateful to Dr. A. Savin (Paris) for reading the 

manuscript and for his comments. The financial support of the 

Agence Nationale de la Recherche (ANR) in the framework of the 

project WADEMECOM (ANR-07-BLAN-0272) is gratefully 

acknowledged. 

REFERENCES 

(a) aminoacetylene (HC C NH ) 

2 

(b) methylenimine (CH2 =NH) 

Fig. (5). FHLI functions of methylenimine and aminoacetylene. 

Further examples have been presented in Ref. [64], which 

demonstrate that the FHLI is able to provide an useful pictorial 

representation of the electron pair structure of molecules. 

7. CONCLUSIONS 

The electron localization/delocalization problem has been 

presented from the viewpoint of the linear response properties of 

the molecular charge density to an external electric field 

perturbation. It has been shown that a seemingly loose statement - 

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Received: 03 March, 2010 Revised: 08 May, 2010 Accepted: 08 May, 2010

Linear Response and Measures of Electron Delocalization ... - CRM2

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