Local reactivity descriptors from degenerate frontier molecular orbitals

Chemical Physics Letters 478 (2009) 310–322

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Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett

Local reactivity descriptors from degenerate frontier molecular orbitals Jorge Martínez Computational Materials Science Division, Southeastern Pacific Research Institute for Advanced Technologies (SEPARI), Universidad Técnica Federico Santa María, Av. España 1680, Edificio T, Casilla 110 V, Valparaíso, Chile

a r t i c l e

i n f o

Article history: Received 25 March 2009 In final form 26 July 2009 Available online 29 July 2009

a b s t r a c t Conceptual Density Functional Theory (DFT) has proposed a set of local descriptors to measure the reactivity on specific sites of a molecule, as an example dual descriptor has been successfully used in analyzing interesting systems to understand their local reactivity, however under the frozen orbital approximation (FOA), it is defined from non-degenerate frontier molecular orbitals (FMOs). In this work, the degeneration is taken into account to propose approximated expressions to obtain the dual descriptor, nucleophilic and electrophilic Fukui functions in closed-shell systems. The proposed expressions have been tested on molecules presenting degenerate FMOs. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Reactivity in chemistry is a key concept because it is intimately associated with reaction mechanisms thus allowing to understand chemical reactions and improve synthesis procedures to obtain new materials. A branch of Density Functional Theory (DFT) [1–3] called Conceptual DFT [4–6] has been developed and used in chemistry. As a consequence, a set of global and local descriptors to measure the reactivity of molecular systems has emerged. On the one hand, in the canonical ensemble (CE) where the energy E is written as EtðrÞ ½qðrÞ  E½N; tðrÞ, the chemical potential l and the molecular hardness g have been proposed to measure the global reactivity; they are defined as the first and second derivatives [6] of the energy functional E½N; tðrÞ with respect to N, the number of electrons in the system under study; higher order derivatives can be obtained if necessary too. All of them are classified as global reactivity descriptors (GRDs). On the other hand, after establishing the functional derivatives of the energy functional E½N; tðrÞ with respect to the external potential tðrÞ, the electronic density qðrÞ is obtained at the first order [6], the Fukui function f ðrÞ is obtained at the second order [7], the dual descriptor Df ðrÞ is obtained at the third order [8] and so on. These are called local reactivity descriptors (LRDs). In addition, it is worthwhile to mention that other LRDs have been obtained in the grand canonical ensemble (GCE) [9] where X½l; tðrÞ  E  Nl. In the GCE, the reactivity descriptors are obtained by replacing derivatives with respect to N, with derivatives with respect to l. The use of such LRDs becomes necessary when a limitation is found in one of descriptors belonging to the CE; as an example, the dual descriptor is not size extensive thus meaning it is not suitable for comparing the relative reactivity of molecules E-mail addresses: [email protected], [email protected] 0009-2614/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2009.07.086

with different sizes, in such a case, the dual descriptor defined in the CE can be replaced by its equivalent descriptor defined in the GCE which is called the second order local softness, grand canonical dual descriptor or local hypersoftness DsðrÞ [10,11]. In order to obtain numerical values of GRDs and LRDs, approximated working equations have been proposed and written in terms of FMOs, so these equations should be affected by the FMO degeneration of a molecule under study as a consequence of having enough symmetry conditions to make equivalent the energy levels of the FMOs. From the revisited bibliography it is possible to realize that the working equations involving FMOs are based on a sensible approach for obtaining some values of LRDs like f ðrÞ, Df ðrÞ and DsðrÞ, but there are no expressions adapted to the degeneration of FMOs, so that this approach fails when representing a 3-D map of any approximated LRD expression written as a function of FMOs. The goal of this article is to establish sensible working equations based on expressions for the main LRDs mentioned above which will be adapted to the FMO degeneration automatically when that phenomenon emerges in a molecular system; but if that phenomenon is absent, the conventional working equations for obtaining the LRDs should be recovered. This Letter is organized as follows: In Section 2 the theoretical basis and definitions of GRDs and LRDs mentioned in Section 1 are provided in more detail. In Section 3 the computational methodology is described and explained. Mathematical expressions adapted to the degenerate FMOs are proposed and discussed in Section 4. Section 5 contains some concluding remarks. 2. Theoretical background The Qualitative Density Functional Theory of Chemical Reactivity also called Conceptual DFT [1,12] has provided a set of reactivity descriptors based on the fact that in the CE, the electronic

J. Martínez / Chemical Physics Letters 478 (2009) 310–322

energy E can be written as a function of N, the total number of electrons, and as a functional of tðrÞ, the external potential; then E  E½N; tðrÞ. Assuming differentiability of E with respect to N and tðrÞ, a series of response functions emerge: the chemical potential that characterizes the escaping tendency of electrons from the equilibrium system is defined as [1]:

l¼



@E @N

 ð1Þ

; tðrÞ

2

@ E

g¼

@N2

!

 ¼ tðrÞ

@l @N

 :

ð2Þ

tðrÞ

Both, l and g, are global properties that are involved in the reactivity of molecular systems [6]. A three-points finite difference approximation leads to the following working equations for these quantities [1,6]

1 2

l   ðI þ AÞ; 1 2

g  ðI  AÞ;

ð3Þ ð4Þ

where I and A are the first vertical ionization potential and the first electronic affinity of the neutral molecule, respectively. Further approximation using the Koopmans’ theorem [13] from the closed-shell Hartree–Fock theory [14] (I  eHOMO and A  eLUMO ) allows one to write l and g in terms of energies of the lower unoccupied molecular orbital and higher occupied molecular orbital, LUMO (eLUMO ) and HOMO (eHOMO ), respectively:

1 2 1 g  ðeLUMO  eHOMO Þ: 2

l  ðeLUMO þ eHOMO Þ;

ð5Þ ð6Þ

In the GCE, the global softness is defined as the reciprocal of the molecular hardness:

  @N ¼ S  g1 ; @ l tðrÞ

ð7Þ

and it is a measure of the ability of a system to change for any external action [15]. As this expression indicates, it can be calculated from the molecular hardness. Returning to the CE, along with the GRDs, site selectivity is characterized by LRDs. In the first place the electronic density qðrÞ is useful as a local descriptor when reactions involves ionic compounds [16]:



dE dtðrÞ



¼ qðrÞ;

ð8Þ

N

so that net atomic charges provide a good description of the way in which the electronic density distribution of a molecular system interacts with other molecules. Reactions involving non-charged species are governed by interactions due to orbital overlapping so in such a case the Fukui function [17,1,6], f ðrÞ, describes properly the local reactivity. So the site selectivity is characterized through this function which describes the local changes occurring in the electronic density qðrÞ of the system due to changes in the total number of electrons N; through a Maxwell relation, it measures how sensitive a system’s chemical potential l is to an external perturbation tðrÞ at the point r:

f ðrÞ ¼

The function f ðrÞ reflects the ability of a molecular site to accept or donate electrons. High values of f ðrÞ are related to a high reactivity at point r [1]. Since the number of electrons N is a discrete variable, right and left derivatives of qðrÞ with respect to N have emerged. By applying a finite difference approximation to Eq. (9), two definitions of Fukui functions depending on total electronic densities are obtained:

 þ @ qðrÞ ; @N tðrÞ   @ qðrÞ ; f  ðrÞ ¼ qN ðrÞ  qN1 ðrÞ ¼ @N tðrÞ f þ ðrÞ ¼ qNþ1 ðrÞ  qN ðrÞ ¼

the molecular hardness, the resistance to charge transfer, is defined as [12]:

    @ qðrÞ dl ¼ : @N tðrÞ dtðrÞ N

ð9Þ

311

ð10Þ ð11Þ

where qNþ1 ðrÞ, qN ðrÞ and qN1 ðrÞ are the electronic densities at point r for the system with N þ 1, N and N  1 electrons, respectively. The first one f þ ðrÞ has been associated to reactivity for a nucleophilic attack so that it measures the intramolecular reactivity at the site r toward a nucleophilic reagent. The second one, f  ðrÞ, has been associated to reactivity for an electrophilic attack so that this function measures the intramolecular reactivity at the site r toward an electrophilic reagent [7]. The densities of FMOs, qLUMO ðrÞ and qHOMO ðrÞ, come to the picture since it has been shown [7,18] that when the FOA is used thus leading to a direct relation between f þ= ðrÞ with the density of the appropriate FMO thus avoiding the calculations of the system with N þ 1 and N  1 electrons:

f þ ðrÞ  qLUMO ðrÞ; f  ðrÞ  qHOMO ðrÞ;

ð12Þ ð13Þ

The use of Eqs. (12) and (13) instead of (10) and (11) allows one to diminish the computational cost without loosing the qualitative picture of the local reactivity, but this approach should be always checked by comparison of these two couples of working equations. Condensation to atoms is achieved through integration within the kth-atomic domain Xk [19,20]: þ=

fk

¼

Z

f þ= ðrÞdr:

ð14Þ

Xk

fkþ= is now an atomic index that is used to characterize the electrophilic/nucleophilic power of atom k. More recently, a new dual descriptor ½f ð2Þ ðrÞ  Df ðrÞ for chemical reactivity has been proposed [16,8,11]. It is defined in terms of the derivative of f ðrÞ with respect to N; through a Maxwell relation, the same descriptor is interpreted as the variation of g with respect to tðrÞ:

Df ðrÞ ¼

    @f ðrÞ dg ¼ : @N tðrÞ dtðrÞ N

ð15Þ

According to expressions given by Eqs. (10) and (11), it is written as the difference between nucleophilic and electrophilic Fukui functions [16]:

Df ðrÞ  f þ ðrÞ  f  ðrÞ ¼ qNþ1 ðrÞ  2qN ðrÞ þ qN1 ðrÞ:

ð16Þ

The use of densities of FMOs provides an easier-to-compute working equation:

Df ðrÞ  qLUMO ðrÞ  qHOMO ðrÞ:

ð17Þ

The computational cost is decreased owing to the use of FMO densities from the system with N electrons only. The dual descriptor allows one to obtain simultaneously the nucleophilic and the electrophilic behavior of the system at point r. The dual descriptor can also be condensed through an appropriate integration within the kth-atomic domain Xk :

312

Z

J. Martínez / Chemical Physics Letters 478 (2009) 310–322

Df ðrÞdr ¼ Dfk :

ð18Þ

Xk

When Dfk > 0 the process is driven by a nucleophilic attack on atom k and then that atom acts an electrophilic species; conversely, when Dfk < 0 as a consequence the process is driven by an electrophilic attack over atom k and therefore atom k acts as a nucleophilic species. As mentioned inSection 1, from the GCE the local hypersoftness [10] sð2Þ ðrÞ  DsðrÞ is provided according to its definition given as follows:

DsðrÞ ¼

@ 2 qðrÞ @ l2

! ¼

Df ðrÞ

tðrÞ

g2



f ðrÞ

g3

(

@g @N



) :

ð19Þ

tðrÞ

It augments the regioselectivity information from the dual descriptor with the overall molecular reactivity information from the maximum hardness principle [21–23]. This is the grand canonical dual descriptor and since the second term in brackets is negligible [24], a working equation emerges as follows:

DsðrÞ 

Df ðrÞ

g2

 Df ðrÞS2 :

ð20Þ

configuration interaction calculations by taking into account single and double excitations (CISD) [34] on the DFT optimized molecules. At the local level, 3-D maps of f þ ðrÞ, f  ðrÞ, were obtained by using the Eqs. (10)–(13); for obtaining the Df ðrÞ, Eqs. (16) and (17) were used; to obtain DsðrÞ, Eq. (20) was used. The 3-D maps of functions proposed at the present work are depicted too. 4. Results and discussion Before starting the analysis of the influence of degenerate FMOs on LRDs, this phenomenon will be analyzed in the framework of GRDs first under the assumption that the Koopmans’ theorem is satisfied. As an example the particular case of a two-fold degenerate HOMO will be considered, thus implying eHOMO ¼ eHOMO1 . As will be demonstrated, GRDs like l and g are not affected by this degeneration. Since l and g depend on FMO energies as Eqs. (5) and (6) indicate, in the case of this double degeneration the first vertical ionization potential can be expressed in terms of either the HOMO or HOMO  1; each of these two FMOs are equivalent to be considered as the FMO that drops an electron, then l can be written either in terms of eHOMO :

1 2

l  ðeLUMO þ eHOMO Þ 3. Computational methods

or in terms of Ethylene, acetylene and triaminotrinitrobenzene (TATB) molecules depicted by Fig. 1 were geometrically optimized according to the Schlegel algorithm [25] at the DFT level of theory; in particular, the functionals used in the present calculations were the Becke-3 for exchange and Perdew–Wang (B3PW91) for correlation [26–30]. For ethylene and acetylene, the 6-31G(d) basis set was used whereas for TATB a 6-31G(d,p) [31,32] basis set was used. All calculations were carried out using the GAUSSIAN 03 [33] code. The use of the Koopmans’ theorem to obtain l, g and S through the use of Eqs. (5) and (6) was checked by means of simple comparison with the three-points finite difference approximation given by Eqs. (3) and (4). For obtaining I and A of the neutral molecule, the calculation for the systems with N þ 1 and N  1 electrons were performed along with the calculation for each original system (N electrons). The way on how to introduce the electronic correlation was considered through the use of single point

eHOMO1 :

1 2

l  ðeLUMO þ eHOMO1 Þ The numerical result will be exactly the same. On the other hand, the same situation is obtained if this degree of degeneration appears in the LUMO, that is to say, eLUMO ¼ eLUMOþ1 . Extrapolating into a p-fold degenerate LUMO and a q-fold degenerate HOMO, the following possible expressions are obtained:

1 2

1 2

1 2

1 2

1 2

1 2

l  ðeLUMO þ eHOMO Þ ¼ ðeLUMO þ eHOMO1 Þ ¼    ¼ ðeLUMO þ eHOMOq Þ; l  ðeLUMO þ eHOMO Þ ¼ ðeLUMOþ1 þ eHOMO Þ ¼    ¼ ðeLUMOþp þ eHOMO Þ and all the possible combinations between the FMOs energies. Furthermore, since under the Koopmans’ theorem approach g also depends on HOMO and LUMO energies, the conclusion will be the same like in the case of l thus allowing to extrapolate into a p-fold degenerate LUMO and a q-fold degenerate HOMO and:

1 2

1 2

1 2

1 2

1 2

1 2

g  ðeLUMO  eHOMO Þ ¼ ðeLUMO  eHOMO1 Þ ¼    ¼ ðeLUMO  eHOMOp Þ;

g  ðeLUMO  eHOMO Þ ¼ ðeLUMO1  eHOMO Þ ¼    ¼ ðeLUMOq  eHOMO Þ

Fig. 1. (a) Ethylene (b) acetylene and (c) triaminotrinitrobenzene (TATB) with their respective frontier molecular orbitals (FMOs) as energetic qualitative levels. FMOs degeneration appears in acetylene (p ¼ q ¼ 2) and TATB (p ¼ 1 and q ¼ 2).

and all the possible combinations between the degenerate HOMO and LUMO energies. So the degeneration phenomenon in FMOs is not relevant for these GRDs because their mathematical expressions are not affected so giving the same numerical values. However, although LRDs do not depend on energies of FMOs, they do depend on electronic density of FMOs so although the degeneration is presented, Eqs. (12) and (13) are not sufficient in order to predict a correct local reactivity; this is due to spatial shapes of degenerate FMOs surrounding the entire molecule which have different orientations in spite of their equivalent energies. In order to simplify the demonstration, the analysis will be focused on a closed-shell system presenting as an example the case of an hypothetical molecule with two-fold degenerate FMOs, then

313

J. Martínez / Chemical Physics Letters 478 (2009) 310–322

a generalized expression can be easily proposed to correctly describe the reactivity of a closed-shell system with n-fold degenerate FMOs. Derivatives with respect to N are written after a finite difference approximation is assumed. The electronic density of a molecular system having N electrons can be expressed in terms of spatial molecular orbitals, fwi ðrÞgN=2 i¼1 , where N is an even number. So a basis set of K spatial molecular orbitals runs from i ¼ 1 up to i ¼ N=2 (HOMO). Immediately, it implies that wN=2þ1 ðrÞ corresponds to the LUMO. The electronic density of this system, qðrÞ, can be expanded through these occupied molecular orbitals:

qðrÞN ¼

N=2 X

ni jwi ðrÞj2N ;

ð21Þ

reach either the LUMO or the LUMO þ 1 so that both molecular orbitals are equivalent to be considered as a part of the nucleophilic Fukui function and the incomplete expression f þ ðrÞ  jwLUMO ðrÞj2 has to be adapted to this degeneration. So the term jwN=2þ1 ðrÞj2Nþ1 in Eq. (24) can be exchanged according to jwN=2þ2 ðrÞj2Nþ1  jwLUMOþ1 ðrÞj2 thus giving as a result an alternative expression for Eq. (25):

!þ N=2 X @jwi ðrÞj2 f ðrÞ  þ2 ; @N i¼1 !þ N=2 X @jwi ðrÞj2 :  jwLUMOþ1 ðrÞj2 þ 2 @N i¼1 þ

jwN=2þ2 ðrÞj2Nþ1

ð26Þ

i¼1

where ni is the number of occupation and jwðrÞi j2N is the electronic density of the ith-molecular orbital of the system with N electrons so that K > N=2. In a closed-shell system, ni ¼ 2. The FOA will be assumed thus meaning the shape of FMOs do not change when the total number of electrons N changes into N  1 or N þ 1, so that:

Both Eqs. (25) and (26) are equivalent but not the same because jwLUMO ðrÞj2 has a different orientation in space surrounding the whole molecule in comparison with jwLUMOþ1 ðrÞj2 , so that in order to consider the influence of these two degenerate FMOs, an average is proposed involving Eq. (25) and (26) as follows:

jwN=2 ðrÞj2Nþ1 ¼ jwN=2 ðrÞj2N1 ¼ jwN=2 ðrÞj2N  jwHOMO ðrÞj2 ;

o X @jw ðrÞj2 1n i f ðrÞ  jwLUMO ðrÞj2 þ jwLUMOþ1 ðrÞj2 þ 2 2 @N i¼1 N=2

þ

jwN=2þ1 ðrÞj2Nþ1 ¼ jwN=2þ1 ðrÞj2N1 ¼ jwN=2þ1 ðrÞj2N  jwLUMO ðrÞj2 : On the other hand, the electronic density for a system of N þ 1 electrons is given by Eq. (22):

qðrÞNþ1 ¼ jwN=2þ1 ðrÞj2Nþ1 þ 2

N=2 X

jwi ðrÞj2Nþ1 ;

and Eq. (23) gives the electronic density for a system of N  1 electrons:

qðrÞN1 ¼ jwN=2 ðrÞj2N1 þ 2

N=21 X

jwi ðrÞj2N1 :

ð23Þ

f þ ðrÞ 

4.1. Nucleophilic Fukui function adapted to the FMOs degeneration Now, the analysis will be focused on f þ ðrÞ. According to Eq. (10) the difference between Eqs. (22) and (21) can be used as an approximation as follows:

( f þ ðrÞ 

jwN=2þ1 ðrÞj2Nþ1 þ 2

N=2 X i¼1

 jwN=2þ1 ðrÞj2Nþ1 þ 2

N=2  X

) jwi ðrÞj2Nþ1

2

N=2 X

jwi ðrÞj2N ;

ð27Þ

N=2 p X 1X @jwi ðrÞj2 jwN=2þi ðrÞj2 þ 2 p i¼1 @N i¼1

!þ ;

ð28Þ

where 1 6 p 6 K  N=2. Under the FOA, the Eq. (28) reduces to Eq. (29):

i¼1

The main LRDs adapted to the degenerate FMOs will be deduced in the next subsection.

:

A generalization for a system presenting a p-fold degenerate LUMO and taking into account the orbital relaxation, allows one to propose the following expression:

ð22Þ

i¼1

!þ

f þ ðrÞ 

p 1X jw ðrÞj2 : p i¼1 N=2þi

ð29Þ

This is the nucleophilic Fukui function adapted to a p-fold degenerate LUMO. The latter is an expression to be included into a computational code because it still takes into account the labels ranging from the first unoccupied molecular orbital (i ¼ 1) until the last degenerate one (i ¼ p). A more chemically easy-to-understand expression explicitly labelling the involved degenerate unoccupied FMOs is the following:

i¼1

jwi ðrÞj2Nþ1  jwi ðrÞj2N ;

f þ ðrÞ 

i¼1

( ) N=2 X jwi ðrÞj2Nþ1  jwi ðrÞj2N 2 ;  jwN=2þ1 ðrÞjNþ1 þ 2 ðN þ 1Þ  N i¼1 ! þ N=2 X @jwi ðrÞj2  jwN=2þ1 ðrÞj2Nþ1 þ 2 ; @N i¼1 !þ N=2 X @jwi ðrÞj2 :  jwLUMO ðrÞj2 þ 2 @N i¼1

p 1X jw ðrÞj2 ; p k¼1 LUMOk

p 1X q ðrÞ; f ðrÞ  p k¼1 LUMOk

ð30Þ

þ

ð24Þ

ð25Þ

The expression given by Eq. (25) is suitable to be used on a nondegenerate FMO system taking into account the orbital relaxation. After considering the FOA and since jwLUMO ðrÞj2  qLUMO ðrÞ, Eq. (25) reduces to its well-known expression: f þ ðrÞ  qLUMO ðrÞ corresponding to Eq. (12). Now in the particular case of a two-fold degenerate LUMO system, thus meaning that LUMO and LUMO þ 1 are degenerated, by means of the Koopmans’ theorem, the incoming electron can

where the summation runs over k from 1 to p, the number of degenerate LUMOs. This re-labelled summation was possible to be established because under the FOA, the core orbitals are not relevant. Eq. (30) is intuitively easier to use than Eq. (29), but technically, they are exactly the same expressions. When p ¼ 1 implies a non-degeneration on LUMOs and as a consequence the Eq. (12) is recovered because k ¼ 1 is the upper limit of this summation so that k can be omitted as a subscript. 4.2. Electrophilic Fukui function adapted to the FMOs degeneration Similarly, an approach to obtain f  ðrÞ as Eq. (11) indicates is given by the difference between Eqs. (21) and (23):

314

J. Martínez / Chemical Physics Letters 478 (2009) 310–322



f ðrÞ  2

N=2 X

( jwi ðrÞj2N



jwN=2 ðrÞj2N1

i¼1

(

 2 jwN=2 ðrÞj2N þ

jwN=2 ðrÞj2N1 þ 2

)

Note this is equivalent to modify Eq. (23) so that the coefficient of the term jwN=21 ðrÞj2N1 inside the summation has been changed from 2 into 1 and the coefficient of the term jwN=2 ðrÞj2N1 has been changed from 1 to 2. Now after a substraction between Eqs. (33) and (34) a first possible complete expression for f  ðrÞ is obtained:

jwi ðrÞj2N1

i¼1

) jwi ðrÞj2N

i¼1

( 

N=21 X

þ2

N=21 X

N=21 X

)

(

jwi ðrÞj2N1

f ðrÞ 

i¼1

n o  2jwN=2 ðrÞj2N  jwN=2 ðrÞj2N1 (N=21 ) N=21 X X jwi ðrÞj2N  jwi ðrÞj2N1 þ2 i¼1

2jwN=2 ðrÞj2N



þ



i¼1

jwN=2 ðrÞj2N1 þ 2jwN=21 ðrÞj2N1 þ 2

After applying the FOA in Eq. (32) and since jwHOMO ðrÞj2  qHOMO ðrÞ, the well-known expression for the electrophilic Fukui function given by Eq. (13) is obtained: f  ðrÞ  qHOMO ðrÞ. Now supposing again that the molecule under study presents a two-fold degenerate HOMO, the expression f  ðrÞ  jwHOMO ðrÞj2 is incomplete because according to the Koopmans’ theorem, the leaving electron can depart either from the HOMO or the HOMO  1 and the latter is absent in this expression. Unlike the f þ ðrÞ, the f  ðrÞ implies that the occupied molecular orbitals have to be taken into account, so that the expression to compute the electronic density for a N electrons system given by Eq. (21) has to be written in a clearer way in order to better understand how to make the adaptation of f  ðrÞ to the FMOs degeneration. Re-writing the Eq. (21) by dropping the very last two terms from the summation, the following expression is obtained:

jwi ðrÞj2N

ð33Þ

i¼1

and in the case of qðrÞN1 , there are two possibilities to be considered: The first one in which the leaving electron departs from the HOMO and since jwN=2 ðrÞj2N1  jwHOMO ðrÞj2 , the following expression is obtained: N=22 X

jwi ðrÞj2N1 :

ð34Þ

i¼1

Note this corresponds to Eq. (23) as commonly written. The second possibility is that one where the leaving electron departs from the HOMO  1, then since jwN=21 ðrÞj2N1  jwHOMO1 ðrÞj2 the next expression is obtained:

qðrÞN1 ¼ 2jwN=2 ðrÞj2N1 þ jwN=21 ðrÞj2N1 þ 2

N=22 X

) jwi ðrÞj2N1 ;

and remembering that jwN=2 ðrÞj2N ¼ jwN=2 ðrÞj2N1 and jwN=21 ðrÞj2N ¼ jwN=21 ðrÞj2N1 , as a consequence Eq. (36) is obtained:

i¼1

( ) N=21 n o X jw ðrÞj2  jw ðrÞj2 2 2 i i N N1  2jwN=2 ðrÞjN  jwN=2 ðrÞjN1 þ 2 N  ðN  1Þ i¼1 ! N=21 n o X @jw ðrÞj2 2 2 i  2jwN=2 ðrÞjN  jwN=2 ðrÞjN1 þ 2 @N i¼1 ! N=21 n o X @jw ðrÞj2 i  2jwN=2 ðrÞj2N  jwN=2 ðrÞj2N þ 2 @N i¼1 ! N=21 2  X @jwi ðrÞj  jwN=2 ðrÞj2N þ 2 ; ð31Þ @N i¼1 ! N=21 X @jw ðrÞj2 2 i : ð32Þ  jwHOMO ðrÞj þ 2 @N i¼1

qðrÞN1 ¼ jwN=2 ðrÞj2N1 þ 2jwN=21 ðrÞj2N1 þ 2

) jwi ðrÞj2N

i¼1

N=21 n o X  jwi ðrÞj2N  jwi ðrÞj2N1  2jwN=2 ðrÞj2N  jwN=2 ðrÞj2N1 þ 2

N=22 X

þ2

N=22 X

(

i¼1

qðrÞN ¼ 2jwN=2 ðrÞj2N þ 2jwN=21 ðrÞj2N þ 2

2jwN=21 ðrÞj2N

N=22 X i¼1

jwi ðrÞj2N1 :

ð35Þ

f  ðrÞ  jwN=2 ðrÞj2N þ 2

N=22 X

jwi ðrÞj2N  jwi ðrÞj2N1

i¼1

( ) jwi ðrÞj2N  jwi ðrÞj2N1  þ2 N  ðN  1Þ i¼1 ! N=22 X @jw ðrÞj2 2 i  jwN=2 ðrÞjN þ 2 @N i¼1 ! N=22 X @jw ðrÞj2 i  jwHOMO ðrÞj2 þ 2 : @N i¼1 jwN=2 ðrÞj2N

N=22 X

ð36Þ

Eq. (36) is very similar to the conventional expression for f  ðrÞ given by Eq. (32), but it is not the same because this is one of two possibilities to generate f  ðrÞ. Furthermore the reader can note that jwHOMO1 ðrÞj2 is absent in Eq. (36). The second alternative expression for f  ðrÞ is given by the substraction between Eq. (33) and (35):

(

2jwN=2 ðrÞj2N þ 2jwN=21 ðrÞj2N þ 2

f  ðrÞ 

N=22 X i¼1

( 

)

jwi ðrÞj2N

2jwN=2 ðrÞj2N1

þ

jwN=21 ðrÞj2N1

þ2

N=22 X

) jwi ðrÞj2N1

;

i¼1

and since jwN=2 ðrÞj2N ¼ jwN=2 ðrÞj2N1 and jwN=21 ðrÞj2N ¼ jwN=21 ðrÞj2N1 , this substraction is reduced to:

f  ðrÞ  jwN=21 ðrÞj2N þ 2

N=22 X i¼1 N=22 X

jwi ðrÞj2N  jwi ðrÞj2N1 (

jwi ðrÞj2N  jwi ðrÞj2N1 N  ðN  1Þ i¼1 ! N=22 X @jw ðrÞj2 2 i  jwN=21 ðrÞjN þ 2 @N i¼1 ! N=22 X @jw ðrÞj2 i  jwHOMO1 ðrÞj2 þ 2 : @N i¼1  jwN=21 ðrÞj2N þ 2

)

ð37Þ

Note that in this case, the jwHOMO ðrÞj2 term is not included. Both Eqs. (36) and (37) are equivalent expressions but not the same because, as mentioned before, jwHOMO ðrÞj2 usually has a different orientation is space around the molecule when comparing with jwHOMO1 ðrÞj2 , so in order to consider the influence of these two degenerate FMOs, an average is proposed from Eqs. (36) and (37) as follows:

f  ðrÞ 

N=22 o X @jw ðrÞj2 1n i jwHOMO ðrÞj2 þ jwHOMO1 ðrÞj2 þ 2 2 @N i¼1

! : ð38Þ

By simple inspection, a generalized expression for a system presenting a q-fold degenerate HOMO, along with the orbital relaxation, is proposed as follows:

J. Martínez / Chemical Physics Letters 478 (2009) 310–322

f  ðrÞ 

N=2q q X @jw ðrÞj2 1X i jwN=2þ1i ðrÞj2 þ 2 q i¼1 @N i¼1

! ð39Þ

;

where 1 6 q 6 N=2  1. In the most extreme case, when all occupied molecular orbitals are degenerated (q ¼ N=2), the relaxation  2  i ðrÞj terms does not appear from given by the summation of @jw@N this mathematical analysis, only the first summation will remain in the Eq. (39) and then q ¼ N=2 will be satisfied, in any other case the restriction 1 6 q 6 N=2  1 must be satisfied. Under the FOA, Eq. (39) is reduced to:

f  ðrÞ 

q 1X jw ðrÞj2 ; q i¼1 N=2þ1i

ð40Þ

where 1 6 q 6 N=2. Eq. (40) is the electrophilic Fukui function adapted to a q-fold degenerate HOMO. This expression is suitable to be inserted into a computational code, nevertheless, by the same exposed reason for the nucleophilic Fukui function, a more useful expression keeping the same numerical result can be used: q 1X f  ðrÞ  jw ðrÞj2 ; q k¼1 HOMOk

f  ðrÞ 

ð41Þ

q 1X q ðrÞ; q k¼1 HOMOk

Df ðrÞ 

p q 1X 1X jwN=2þi ðrÞj2  jw ðrÞj2 : p i¼1 q i¼1 N=2þ1i

ð43Þ

Eq. (43) is the dual descriptor adapted to p-fold LUMO and q-fold HOMO degenerations; when p ¼ q ¼ 1, Eq. (17) is recovered. Like nucleophilic and electrophilic Fukui functions adapted to degeneration of FMOs, Eq. (43) is easier to be understood when it is written in terms of labels involving the degenerate FMOs only:

Df ðrÞ 

p q 1X 1X jwLUMOk ðrÞj2  jw ðrÞj2 ; p k¼1 q k¼1 HOMOk

Df ðrÞ 

p q 1X 1X qLUMOk ðrÞ  q ðrÞ: p k¼1 q k¼1 HOMOk

ð44Þ

In addition, as Eq. (20) indicates, the local hypersoftness descriptor adapted to p-fold LUMO and q-fold HOMO degenerations and taking into account the orbital relaxation, it will be given as follows:

( p q 1X 1X jwN=2þi ðrÞj2  jw ðrÞj2 p i¼1 q i¼1 N=2þ1i !) N=2 X @jwi ðrÞj2 : þ2 @N i¼N=2qþ1

2

Once the expressions to obtain Fukui functions have been adapted to the FMOs degeneration, it is easier to propose general expressions for Df ðrÞ and DsðrÞ written as functions of degenerate FMOs and the respective relaxation terms. Then according to the definition of Df ðrÞ, when it is adapted to the FMOs degeneration, it is defined as a difference between Eqs. (28) and (39):

!þ ) ( N=2 p X 1X @jwi ðrÞj2 jwN=2þi ðrÞj2 þ 2 p i¼1 @N i¼1 ! ) ( N=2q q X X 1 @jwi ðrÞj2 jwN=2þ1i ðrÞj2 þ 2  q i¼1 @N i¼1

þ

And to better understand the physical meaning, an expression chemically more suitable is shown as follows:

DsðrÞ  S2 

( ) p q 1X 1X qLUMOk ðrÞ  qHOMOk ðrÞ : p k¼1 q k¼1

ð47Þ

4.4. Physical support to the proposed expressions

ð42Þ @jwi ðrÞj2 @N

ð46Þ

Note that Eqs. (46) and (47) are technically the same expressions.

p q 1X 1X  jwN=2þi ðrÞj2  jw ðrÞj2 p i¼1 q i¼1 N=2þ1i ! N=2 X @jwi ðrÞj2 ; þ2 @N i¼N=2qþ1



ð45Þ

Under the FOA, Eq. (45) reduces to Eq. (46) ready to be implemented into a computational code:

( ) p q 1X 1X 2 2 DsðrÞ  S  jw ðrÞj  jw ðrÞj : p i¼1 N=2þi q i¼1 N=2þ1i

4.3. Dual and local hypersoftness descriptors adapted to the FMOs degeneration

where in Eq. (42) it has been assumed that  2 i ðrÞj  @jw@N .

this LRD the orbital relaxation becomes more important as the degeneration increases. By taking into account the FOA, Eq. (42) reduces to:

DsðrÞ  S2 

where the summation runs over k from 1 to q, the number of degenerate HOMOs. When q ¼ 1 implies a non-degeneration on HOMOs and the subscript k ¼ 1 can be omitted thus recovering the Eq. (13). Note that Eqs. (40) and (41) are technically the same expressions.

Df ðrÞ 

315





@jwi ðrÞj2 @N



Note that the degeneration affects the influence of relaxation terms in the dual descriptor. Whereas f þ ðrÞ includes N=2 relaxation terms and f  ðrÞ includes N=2  q relaxation terms, the dual descriptor only takes into account q relaxation terms, thus meaning that a system with no degeneration on HOMO (q ¼ 1) will ðrÞj2 Þ have only one relaxation term corresponding to the ð@jwHOMO @N derivative. If HOMO degeneration increases (q > 1), then the num2 i ðrÞj Þ from Eq. (42) inber of terms inside the summation of ð@jw@N creases too so that for Df ðrÞ and any other function depending on

Although the proposed expressions for f þ ðrÞ, f  ðrÞ, Df ðrÞ and DsðrÞ look sensible, a physical support should be established in order to take these expressions into account for using in future calculations. The deduction of the Fukui functions given by Eqs. (30) and (41) can be understood under a fundamental property of the degenerate orbitals establishing that any linear combination is also a solution of the Schrödinger equation. As an example, in the case of a two-fold degenerate HOMO, wHOMO1 and wHOMO2 , a set of orthogonal orbitals can be defined through the following linear combinations:

1 wHOMO ðrÞa ¼ pffiffiffi fwHOMO1 ðrÞ þ wHOMO2 ðrÞg; 2 1 wHOMO ðrÞb ¼ pffiffiffi fwHOMO1 ðrÞ  wHOMO2 ðrÞg; 2 as a consequence, the absolute squares of these expressions are given by:

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J. Martínez / Chemical Physics Letters 478 (2009) 310–322

 2 1 1 w  ¼ jw ðrÞj2 þ wHOMO1 ðrÞ  wHOMO2 ðrÞ HOMO ðrÞa 2 HOMO1 2 1 1 þ wHOMO2 ðrÞ  wHOMO1 ðrÞ þ jwHOMO2 ðrÞj2 ; 2 2  2 1 1  2 w  ¼ jwHOMO1 ðrÞj  wHOMO1 ðrÞ  wHOMO2 ðrÞ HOMO ðrÞb 2 2 1  1  wHOMO2 ðrÞ  wHOMO1 ðrÞ þ jwHOMO2 ðrÞj2 : 2 2

ð48Þ

ð49Þ

By taking into account these two expressions and considering the FMO approximation, there would be two plausible electrophilic Fukui functions: f  ðrÞa  jwHOMO ðrÞa j2 and f  ðrÞb  jwHOMO ðrÞb j2 , both of them are valid expressions to be used as an approach to obtain the electrophilic Fukui function. Since the involved orbitals belong to an orthogonal set, after integration within an atomic domain Xk , the

Table 1 Global reactivity descriptors: chemical potential l, molecular hardness g and global softness S from the three-points finite difference (TPFD) approximation by using I and A and from the Koopmans’ theorem through the use of eHOMO and eLUMO . All values are given in eV with the exception of S which is given in eV1 . Level of theory

I

A

l

g

S

B3PW91 CISD

10.39176 9.89914

3.24449 3.97813

3.57364 2.96051

6.81812 6.93863

0.14667 0.14412

HC„CH

B3PW91 CISD

11.14270 10.66946

4.30823 5.20200

3.41723 2.73373

7.72547 7.93573

0.12944 0.12601

TATB

B3PW91

9.02884

1.14963

5.08923

3.93960

0.25383

l

g

S

Molecule TPFD approximation H2C@CH2

eHOMO Koopmans’ theorem H2C@CH2

eLUMO

B3PW91 CISD

0.27043 0.37173

0.01668 0.18056

3.45247 2.60102

3.90636 7.51435

0.25599 0.13308

HC„CH

B3PW91 CISD

0.28604 0.39967

0.05008 0.21625

3.21043 2.49558

4.57318 8.38008

0.21867 0.11933

TATB

B3PW91 CISD

0.26944 0.37471

0.10465 0.03557

5.08979 4.61427

2.24210 5.58219

0.44601 0.17914

Fig. 2. From Table 1: Comparisons between the use of the Koopmans’ theorem and the TPFD approximation at the B3PW91 level of theory to obtain (a) the electronic chemical potential and (b) the molecular hardness. Comparisons between the electronic correlation given by the B3PW91 and CISD level of theories are shown by (c) the electronic chemical potential and (d) the molecular hardness.

J. Martínez / Chemical Physics Letters 478 (2009) 310–322

integral of the products wHOMO1 ðrÞ  wHOMO2 ðrÞ and wHOMO2 ðrÞ wHOMO1 ðrÞ will be practically equal to zero. On the other hand, as the Fukui function has to satisfy the normalization condition, the linear combination 12 jwHOMO ðrÞa j2 þ 1 jwHOMO ðrÞb j2 is suitable and complete because it includes the 2 two possible expressions for the electrophilic Fukui functions satisfying the normalization condition too. So when taking into account the average between Eqs. (48) and (49), the following is obtained:

2  2 o 1 1 n 1 wHOMO ðrÞa  þ wHOMO ðrÞb  ¼ jwHOMO1 ðrÞj2 þ jwHOMO2 ðrÞj2 ; 2 2 2 1 1 f  ðrÞ  jwHOMO1 ðrÞj2 þ jwHOMO2 ðrÞj2 ; 2 2 1  fqHOMO1 ðrÞ þ qHOMO2 ðrÞg; 2 where it has been assumed that f  ðrÞ  12 fjwHOMO ðrÞa j2 þ jwHOMO ðrÞb j2 g. An analog expression can be deduced to obtain f þ ðrÞ in case of a two-fold degenerate LUMO system:

317

2  2 o 1 1 n 1 wLUMO ðrÞa  þ wLUMO ðrÞb  ¼ jwLUMO1 ðrÞj2 þ jwLUMO2 ðrÞj2 ; 2 2 2 1 1 f þ ðrÞ  jwLUMO1 ðrÞj2 þ jwLUMO2 ðrÞj2 ; 2 2 1  fqLUMO1 ðrÞ þ qLUMO2 ðrÞg: 2 In the case of a three-fold degenerate HOMO there will be also a proper set of linear combinations of FMOs as follows:

1 wHOMO ðrÞa ¼ pffiffiffi fwHOMO1 ðrÞ þ wHOMO2 ðrÞ þ wHOMO3 ðrÞg; 3 1 wHOMO ðrÞb ¼ pffiffiffi fwHOMO1 ðrÞ  wHOMO2 ðrÞ þ wHOMO3 ðrÞg; 3 1 wHOMO ðrÞc ¼ pffiffiffi fwHOMO1 ðrÞ þ wHOMO2 ðrÞ  wHOMO3 ðrÞg; 3 1 wHOMO ðrÞd ¼ pffiffiffi fwHOMO1 ðrÞ  wHOMO2 ðrÞ  wHOMO3 ðrÞg; 3 which will allow one to write expressions in terms of absolute squares of FMOs only without including cross products between

Fig. 3. Ethylene: From the right to the left, all isosurfaces of the nucleophilic, electrophilic and dual descriptors 3-D maps are obtained at the B3PW91 and CISD levels of theory and depicted at 1  103 a:u: (a) by means of finite differences and (b) by means of FMO approximation with the conventional expressions. Nucleophilic attacks will be oriented toward red lobes; electrophilic attacks, toward yellow lobes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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J. Martínez / Chemical Physics Letters 478 (2009) 310–322

the FMOs; so that by taking into account the selected linear combinations, it is possible to easily check that jwHOMO ðrÞa j2 þ jwHOMO ðrÞb j2 þ jwHOMO ðrÞc j2 þ jwHOMO ðrÞd j2 is equal to:

o 4n 2 2 2 jwHOMO1 ðrÞj þ jwHOMO2 ðrÞj þ jwHOMO3 ðrÞj ; 3 then the average is obtained by multiplying this expression by 1=4 h thus giving f  ðrÞ  14 jwHOMO ðrÞa j2 þ jwHOMO ðrÞb j2 þ jwHOMO ðrÞc j2 þ jwHOMO ðrÞd j2  and providing the electrophilic Fukui function adapted to this three-fold degenerate HOMO system:

o 1n 2 2 2 jwHOMO1 ðrÞj þ jwHOMO2 ðrÞj þ jwHOMO3 ðrÞj ; 3 1  fqHOMO1 ðrÞ þ qHOMO2 ðrÞ þ qHOMO3 ðrÞg: 3

f  ðrÞ 

And a similar expression is obtained for the nucleophilic Fukui function adapted to a three-fold degenerate LUMO system:

o 1n 2 2 2 jwLUMO1 ðrÞj þ jwLUMO2 ðrÞj þ jwLUMO3 ðrÞj ; 3 1  fqLUMO1 ðrÞ þ qLUMO2 ðrÞ þ qLUMO3 ðrÞg: 3

f þ ðrÞ 

In the case of a q-fold degenerate HOMO and p-fold degenerate LUMO system, there will be always a proper set of linear combinations of FMOs to generate the approximated Fukui functions adapted to the FMOs degeneration given by Eqs. (30) and (41). From this point of view, any approximation to obtain nucleophilic/electrophilic Fukui functions adapted to the degenerate FMOs will have a physical support. Eqs. (30), (41), (44) and (47) have a probabilistic interpretation so that coefficients 1p and 1q can be understood as the probability of each event thus leading to the constraint that the sum of the probabilities is equal to 1 for each generalized expression corresponding to the normalization condition of Fukui function [17,19,20].

Fig. 4. Acetylene: From the right to the left, all isosurfaces of the nucleophilic, electrophilic and dual descriptors 3-D maps are obtained at the B3PW91 and CISD levels of theory and depicted at 1  103 a:u: (a) by means of finite differences and (b) by means of FMO approximation with the conventional expressions. Nucleophilic attacks will be oriented toward red lobes; electrophilic attacks, toward yellow lobes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

J. Martínez / Chemical Physics Letters 478 (2009) 310–322

4.5. Applications Three molecules have been chosen: ethylene, acetylene and triaminotrinitrobenzene (TATB) given by Fig. 1 along with their respective qualitative energetic levels of FMOs and degree of degeneration, p and q. Their main GRDs, l and g, have been calculated at the B3PW91 and CISD levels of theory through the use of the three-points finite difference (TPFD) approximation and the Koopmans’ theorem (Table 1). It is possible to realize by watching the linear relationships depicted by Fig. 2a and b that errors due to the use of Koopmans’ theorem can systematically be corrected; on the other hand the introduction of the electronic correlation does not dramatically depend on the level of theory as shown by Fig. 2c and d. Therefore the use of Koopmans’ theorem is enough to qualitatively assess the proposed mathematical expressions. Ethylene is a very simple molecule [Fig. 1a] belonging to the D2h group of symmetry. After a geometrical optimization given by a quantum mechanical calculation as explained in Section 3, the qualitative energetic levels of the FMOs and their respective closest neighbor molecular orbitals, HOMO  1 and LUMO þ 1, are sketched in Fig. 1a. The nucleophilic and electrophilic Fukui functions along with the dual descriptor have been obtained as 3-D maps in terms of total densities thus meaning through the use of Eqs. (10), (11) and (16), respectively as Fig. 3a shows. Additionally, the lack of degeneration ðp ¼ q ¼ 1Þ allows one to use the conventional equations for obtaining the approximated Fukui functions and dual descriptor 3-D maps, that means in terms of the FMO densities given by Eqs. (12), (13) and (17), respectively as Fig. 3b shows. As depicted by Fig. 3, these 3-D maps have been obtained at the DFT and CISD levels of theory. There is no a noticeable difference when introducing the electronic correlation by means of CISD or DFT calculations, but some differences can be appreciated when comparing the use of Eqs. (10), (11) and (16) depending on total respective electronic densities with Eqs. (12), (13) and (17) depending on FMO densities; it is possible to visualize that nucle-

319

ophilic and electrophilic Fukui functions strongly depend on relaxation terms because the 3-D maps of these functions are not exactly the same; however the dual descriptor obtained by Eq. (16) or by Eq. (17) makes evident to be a more robust function since there is not an exaggerated difference between the 3-D maps obtained by means of these two last equations. Anyway the qualitative local reactivity information is coherently conserved. Independently of the way of calculation and the level of theory, each depicted function belongs to the totally symmetric irreducible representation of the D2h group of symmetry, Ag , and since this molecule does not exhibit any degeneration on its FMOs, the symmetry criterion will be suitable to evaluate whether a LRD has been well represented or not on any other molecule. When the acetylene [Fig. 1b] molecule is considered to be analyzed by the same set of equations, since this molecule exhibits a two-fold degeneration on HOMOs and LUMOs ðp ¼ q ¼ 2Þ, some differences are expected in its LRDs in comparison to those belonging to ethylene. However, the depicted nucleophilic, electrophilic and dual descriptor functions given by Fig. 4 are suspiciously very similar to the LRDs given by Fig. 3; this does not depend on how the functions are obtained either by using the total densities [Eqs. (10), (11) and (16)] or the FMO densities [Eqs. (12), (13) and (17)]; neither does it depend on how the electronic correlation has been introduced, thus meaning either by means of a CISD or a DFT level of theory. Ethylene is a planar molecule that does not have the same symmetry that acetylene does; the latter belongs to the D1h group of symmetry and then the respective LRDs should belong to the respective totally symmetric irreducible representation, Rþ g. In spite of the predicted local reactivity for acetylene by Fig. 4 is qualitatively correct because both carbon atoms show the same local reactivity, these 3-D maps do not belong to the irreducible representation Rþ g , thus indicating that expressions given by the Eqs. (12), (13) and (17) are incomplete from the symmetry point of view. After using the Eqs. (30), (41) and (44), correct shapes to represent these LRDs as 3-D maps are depicted by Fig. 6a because they do belong to Rþ g.

Fig. 5. TATB: From the right to the left, all isosurfaces of the nucleophilic, electrophilic and dual descriptors 3-D maps are obtained at the B3PW91 level of theory only and depicted at 1  103 a:u: (a) by means of finite differences and (b) by means of FMO approximation with the conventional expressions. Nucleophilic attacks will be oriented toward red lobes; electrophilic attacks, toward yellow lobes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

320

J. Martínez / Chemical Physics Letters 478 (2009) 310–322

In order to find a more noticeable failure on the conventional expressions to obtain the Fukui functions and dual descriptor under the FMO approximation [Eqs. (12), (13) and (17)], the TATB molecule (p ¼ 1 and q ¼ 2) was analyzed from the same point of view. It belongs to the D3h group of symmetry. Since a CISD calculation provides the same local reactivity information that a DFT calculation does, Fig. 5 exhibits only 3-D maps of LRDs obtained through the use of a DFT-B3PW91 level of theory calculation. By symmetry reasons, it is expected that on the one hand the carbon atoms linked to –NO2 groups have the same reactivity among them because each has identical environment and on the other hand, the carbon atoms linked to –NH2 groups should have the same reactivity among them due to the same reasons, so that there would be two kinds of carbon atoms. The nucleophilic Fukui function f þ ðrÞ belonging to the totally symmetric irreducible representation of the D3h group of symmetry, A01 , correctly represents the local reactivity on atoms of this molecule so that it is possible to distinguish two kinds of carbon atoms in this molecule, however those functions depending on the degenerate HOMO, that is to say, the electrophilic Fukui

function f  ðrÞ and the dual descriptor Df ðrÞ do not represent the local reactivity properly. They do not belong to A01 . This can be easily observed because of one of the carbon atoms linked to a –NO2 group shows different reactivity in comparison with the remaining carbon atoms linked to their respective –NO2 groups. This cannot be possible in this system in equilibrium. Note that the f  ðrÞ 3-D maps were obtained by means of the Eqs. (11) and (13) as Fig. 5 shows. And in the case of the Df ðrÞ, the Eqs. (16) and (17) were used to produce the respective 3-D maps depicted by Fig. 5. It is very clear that those equations depending on densities of HOMOs [Eqs. (13) and (17)] should be corrected in order to explicitly take into account the degeneration. So after using the expressions for the electrophilic Fukui function [Eq. (41)] and the dual descriptor [Eq. (44)], proper 3-D maps of the respective LRDs are depicted by Fig. 6b. Now, all of these 3-D maps belong to the totally symmetric irreducible representation, A01 . To sum up, Eqs. (30), (41), and (44) have been used to the molecules mentioned above according to their p-fold LUMO and q-fold HOMO degenerations:

Fig. 6. (a) Acetylene and (b) TATB. From the right to the left, nucleophilic, electrophilic and dual descriptors isosurfaces 3-D maps are depicted at 1  103 a:u: They were obtained by means of FMO approximation with the suggested expressions adapted to the FMOs degeneration. Nucleophilic attacks will be oriented toward red lobes; eledctrophilic attacks, toward yellow lobes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

J. Martínez / Chemical Physics Letters 478 (2009) 310–322

321

Ethylene:

f þ ðrÞ 

1 X

qLUMOk ðrÞ

k¼1

 qLUMO1 ðrÞ  qLUMO ðrÞ; f  ðrÞ 

1 X

qHOMOk ðrÞ

k¼1

 qHOMO1 ðrÞ  qHOMO ðrÞ;

Df ðrÞ 

1 X

qLUMOk ðrÞ 

k¼1

1 X

qHOMOk ðrÞ

k¼1

 qLUMO1 ðrÞ  qHOMO1 ðrÞ  qLUMO ðrÞ  qHOMO ðrÞ; where qHOMO1 ðrÞ and qLUMO1 ðrÞ are the respective ordinary FMOs, so that they can be simply called qHOMO ðrÞ and qLUMO ðrÞ, respectively. Ethylene shows that the conventional Eqs. (12), (13) and (17) are correctly enough to be used as local reactivity predictive tools. Acetylene:

f þ ðrÞ 

2 1X q ðrÞ 2 k¼1 LUMOk

1 ðrÞ þ qLUMO2 ðrÞg; fq 2 LUMO1 2 1X f  ðrÞ  q ðrÞ 2 k¼1 HOMOk 

1 ðrÞ þ qHOMO2 ðrÞg; fq 2 HOMO1 2 2 1X 1X Df ðrÞ  qLUMOk ðrÞ  q ðrÞ 2 k¼1 2 k¼1 HOMOk 



1 1 ðrÞ þ qLUMO2 ðrÞg  fqHOMO1 ðrÞ þ qHOMO2 ðrÞg: fq 2 LUMO1 2

TATB:

f þ ðrÞ 

1 X

qLUMOk ðrÞ

k¼1

 qLUMO1 ðrÞ  qLUMO ðrÞ; f  ðrÞ 

2 1X q ðrÞ 2 k¼1 HOMOk

1 fq ðrÞ þ qHOMO2 ðrÞg; 2 HOMO1 1 2 X 1X Df ðrÞ  qLUMOk ðrÞ  q ðrÞ 2 k¼1 HOMOk k¼1 

1  qLUMO1 ðrÞ  fqHOMO1 ðrÞ þ qHOMO2 ðrÞg 2 1  qLUMO ðrÞ  fqHOMO1 ðrÞ þ qHOMO2 ðrÞg: 2 These two last molecules have been useful to state that Eqs. (30), (41) and (44) are able to better predict the local reactivity than Eqs. (12), (13) and (17) because these three last equations are incomplete; this is very important before carrying out a condensation of FMO Fukui functions [17,35,19] and any function that depends of FMOs. To finish, it is worthwhile to mention that DsðrÞ given by Eq. (47) shows the correct behavior along with the characteristic capability to describe the relative reactivity of molecules with different sizes. As can be observed in Fig. 7, now the local reactivity is properly described because molecules as ethylene and acetylene possess p-electrons which are less polarizable than the aromatic

Fig. 7. (a) Ethylene (b) acetylene and (c) TATB. 3-D maps of local hypersoftness isosurfaces DsðrÞ given by Eq. (47) are depicted at 1  104 a:u: The values of S were extracted from Table 1 at the B3PW91 level of theory. Nucleophilic attacks will be oriented toward red lobes; electrophilic attacks, toward yellow lobes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

p-electrons [36] of TATB, so that in these non-aromatic molecules the difference between nucleophilic regions (yellow lobes) and electrophilic regions (red lobes) are relatively bigger than the difference exhibited by the nucleophilic and electrophilic regions in TATB; the latter does not show a noticeable difference when comparing DsðrÞ given by Fig. 7c with Df ðrÞ depicted by Fig. 6b, whereas in the case of acetylene it is possible to distinguish an evident difference when comparing DsðrÞ given by Fig. 7b with Df ðrÞ depicted by Fig. 6a; like acetylene, in the case of ethylene this distinction can also be observed when comparing DsðrÞ given by Fig. 7a with Df ðrÞ depicted by Fig. 3b. A similar procedure can be used in generating a 3-D map of the multiphilic descriptor [37]. 5. Concluding remarks Eqs. (30), (41), (44) and (47) for predicting qualitatively (and quantitatively if desired by means of a proper integration) local reactivities on closed-shell molecular systems that present degeneration in their frontier molecular orbitals have been proposed so that p and q are the degree of degeneration of LUMO and HOMO, respectively. Conventional expressions for the Fukui, dual and local hypersoftness descriptors are special cases that can be deduced from the proposed expressions at the present work. The set of expressions for obtaining LRDs have been tested on three molecules: two of them exhibit the degeneration phenomenon on their FMOs and the remaining one does not show any degeneration phenomenon on its FMOs which was used as a control molecule to gauge the suggested expressions in this article. It has been demonstrated that the degeneration considerations of FMOs on LRDs were very sensible since the local reactivity is predicted as expected in well-known molecular systems according to the respective group of symmetry. These very simple expressions adapted to symmetry can help to predict local reactivities on highly symmetric molecules thus allowing to condensate, through a suitable integration procedure, the values on lobes that are not displayed when using the conventional expressions non-adapted to the degenerate FMOs. Acknowledgement The author wishes to thank the financial support from Program B: HPC & Materials SEPARI Centro TIC, FONDEF project CTICV002. References [1] R.G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989. [2] P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864.

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