Relation between bond order and delocalization index of QTAIM

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Relation between bond order and delocalization index of QTAIM

Caio L. Firme

*

_{, O.A.C. Antunes, Pierre M. Esteves}

*

Universidade Federal do Rio de Janeiro, Instituto de Química, Av. Athos da Silveira Ramos, 149, CT Bloco A, Sala 622, Cidade Universitária, Ilha do Fundão, Rio de Janeiro, BR 21941-909, Brazil

a r t i c l e

i n f o

Article history:

Received 26 August 2008 In ﬁnal form 1 December 2008 Available online 7 December 2008

a b s t r a c t

The formal bond order is half the difference between the number of bonding and anti-bonding electrons. The relation between delocalization index (DI) from QTAIM and formal bond order is linear for CC, NN, GeGe, C–Si, C–Ge and CN bonds. The electronegativity is closely related with the correspondence between the DI and the formal bond order for the CC, NN and GeGe bonds. By using the electronegativity of C, N and Ge atoms, it was obtained a general relation between the formal bond order and the DI for CC, NN and GeGe atomic pairs.

1. Introduction

In 2006 we celebrated the 90th anniversary of the Lewis bond-ing model, where he introduced the electron as a main player in chemistry. This model inspired the concept of bond order, initially developed by Pauling[1,2]. According to molecular orbital theory, the formal bond order is half the difference between the number of bonding and anti-bonding electrons in a given bond[3].

Some pair of atoms can have a number of characteristic bond lengths, rationalized in terms of multiple bonds. Pauling et al. [1,2]were the ﬁrst to establish a relation between bond length of conjugated molecules and their corresponding double bond char-acters by valence bond treatment of resonance, which was later improved by Penney[4]. Afterwards Coulson[5,6] established a new relation by using molecular orbital theory. His bond order for-mula is shown in Eq.(1)

pij¼ X ðnÞ aðnÞ i a ðnÞ j ð1Þ

where pijis the bond order (named total mobile order of a bond pij)

and aðnÞ

i is the coefﬁcient of the atomic orbital of the ith atom.

Ruedenberg[7]proposed a deﬁnition of bond order based on MO theory in Eq.(2)

pPQ¼ X

n

gncPncQn ð2Þ

where gnis the number of electrons in the nth orbital, cPnand cQn

are the coefﬁcients in P and Q atoms.

Johnston[8]showed that the bond order is expressed in terms of the experimental force constants [Eq.(3)]

n ¼k ðnÞ e kð1Þe ð3Þ where kðnÞ

e is the experimental stretching force constant for a

spe-ciﬁc bond and kð1Þ_e is the force constant for a single bond.

Guggenheimer [9], Jules and Lombardi [3] showed different relations between force constant and internuclear distance.

The quantum theory of atoms in molecules [10,11] (QTAIM) also can be used to calculate the formal bond order because the charge density at the bond critical point,

q

b, increases as the bond

length decreases[12,13]. Therefore, there is a relation between

q

b

and bond order as shown Eq.(4) [12,14]

n ¼ exp½Að

q

b BÞ ð4Þ

where B =

q

bfor a single bond (n = 1) and A = dln(nb)/d

q

b. A and B

are arbitrary constants and they are useful only for a speciﬁc pair of elements involved in the bond [3]. Bader and collaborators [12,13]established a correlation between charge density of bond critical point with formal bond order.

Matta and Hernandez-Trujillo[15,16]suggested calibrating the correlation between bond order and electron density [Eq.(4)] by means of delocalization index (DI) rather than arbitrarily bond orders.

The delocalization index, obtained from the integration of the Fermi hole density, is directly related to the bond order [17,18]. The delocalization index[19–21]is a measure of number of elec-trons that are shared or exchanged between two atoms or basins [see Eq.(5)] DI ¼ 2FðX;

X

0Þ ¼ 2½FaðX;

X

0Þ þ Fb_ðX;

_X

0 Þ ¼ 2X N i;j Si;jðXÞSi;jðX0Þ ð5Þ

where Si,j(X) is the overlap integral of

a

spin orbitalsUiandUjover

the atomic basin X and FaðX;X0_{Þ is the Fermi correlation for}

_a

* Corresponding authors.

E-mail addresses: cﬁrme@iq.ufrj.br (C.L. Firme), pesteves@iq.ufrj.br (P.M. Esteves).

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electrons of atomic basinXdelocalized into another atomic basin X0_[Eq._(6)] FaðX;

X

0Þ ¼ X Na i;j NSi;jðXÞSi;jðX0Þ ð6Þ

Cioslowski and Mixon[22] used overlap integral Sii(X) from

diatomic contributions to obtain the bond order which is different from the formula of delocalization index.

Angyan et al.[23]used the formula of DI to measure the bond order directly. Twice sharing index of Fulton and Mixon [24], which is similar to DI formula, also was used to measure bond or-der directly. Topological bond oror-der was used by Malcolm et al. [25]to corroborate the bonding picture of REER and R2EER2

sys-tems, where R = H, Me and E = Ge, Si, and Sn.

In this work we show that the delocalization index can also be used to estimate formal bond order in QTAIM, since a linear corre-lation between DI and formal bond order was found.

2. Computational details

The geometries of the studied species were optimized by using standard techniques [26]. Vibrational analysis on the optimized geometries of selected points on the potential energy surface was carried to determine whether the resulting geometries are true minima or transition states, by checking the existence of imaginary frequencies. The calculations were performed at B3LYP/6-311++G**

and MP2/6-311++G**_levels_[27–33]_{by using G}_AUSSIAN_{03 package}

[34]. The electronic density was derived from the Kohn-Sham orbi-tals obtained at B3LYP/6-311++G**_{and MP2/6-311++G}**_{levels and}

further used for QTAIM calculations by means of AIM2000 software

[35]. The geometry optimization and electronic density calcula-tions also were performed at CCSD(T)/6-311++G**_{for some of the}

studied species.

3. Results and discussion

3.1. Can the delocalization index measure the bond order?

The calculation of the Fermi hole density shows that there is not a pair of electrons in a chemical bond, except for the electron pairs in the core of the atoms[21,36,37]. Although the concept of DI is different from Lewis concept of bond order, the delocalization index can be used to estimate the bond order of a covalent bond in an alternative way. Matta and Hernandez-Trujillo[15]related the DI and the

q

b in an equation that enables calibration of the

experimental charge density in the bond critical point by means

of calculated delocalization indexes[17]. Angyan et al.[23] and Fulton and Mixon[24]used the formula of DI to measure the bond order directly.

Fig. 1 shows the relation between Ruedenberg’s or Coulson’s bond order and the delocalization index. There is a good correspon-dence between the delocalization index and Coulson’s bond order for C–C bonds[38].

3.2. Characteristics of the delocalization index

By using the wavefunction obtained from coupled cluster the-ory with single, double and triple excitations (CCSD(T)), the DI

val-0 1 0 1 2 Ruedenberg BO Ethane Hexatriene (C-C) Anthracene (C1–C2) Ethylene Anthracene (C2–C3) Hexatriene (C=C) 1 2 1 2 Coulson BO Benzene Hexatriene (C-C) Hexatriene (C=C) Ethylene

Bond order and delocalization index

of CC bonds

Bond order and delocalization index of CC bonds 0.5 1.5 2.5 1.2 1.4 1.6 1.8 2.2 1.2 1.4 1.6 1.8 2.2 0.2 0.4 0.6 0.8 1.2 DI DI

Fig. 1. Bond order and delocalization index of CC bonds in selected hydrocarbons. Table 1

Values of delocalization indexes of CC bond of ethane, ethylene and acetylene and of NN bond of hydrazine (H2N–NH2), diazene (H–N@N–H) and N2from B3LYP and CCSD(T) theories.

Molecule Atomic pair Delocalization index

B3LYP CCSD(T) H3C–CH3 CC 1.00 0.99 H2C@CH2 CC 1.99 1.85 HC–CH CC 2.71 2.31 H2N–NH2 NN 1.20 1.19 H–N@N–H NN 2.17 2.16 N„N NN 3.03 2.99 N N N H N H H H H H O H C O

-

+ 1 2 3 4 H H H H O 5 6 7 8 9 10 11 12

DI= 1.00 DI= 1.99 DI= 2.71

DI= 1.02

DI= 1.76

DI= 2.42

DI= 0.93 DI= 1.58 DI= 1.82

DI= 1.39

DI= 1.25

DI= 0.97 (DI= 0.99) (DI= 1.86) (DI= 2.71)

(DI= 1.37) (DI= 0.92) (DI= 1.64) (DI= 2.30) (DI= 1.12) (DI= 0.84) (DI= 1.42) (DI= 1.59) (DI= 0.88) H H H H H H H H N H H H H H O H H H N H H

Fig. 2. Delocalization indexes of the CC, CN and CO atomic pairs of the molecules 1– 12 from B3LYP. Delocalization indexes between parentheses are from MP2.

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ues of CC bond of ethane and ethylene and the DI values for NN bond of hydrazine (H2N–NH2), diazene (H–N@N–H) and N2were

very similar to the corresponding DI values obtained from B3LYP (Table 1). The DI values of CC bond of acetylene from B3LYP and CCSD(T) are reasonably similar.

Careful analysis of DI values obtained by HF and B3LYP from Sannigrahi and coworkers[39]indicates that most DI values from B3LYP are slightly higher than those from HF,1_{but there are also}

considerable number of DI values in which the difference between B3LYP and HF is irrelevant.2

In this study, we also show that the difference between DI val-ues obtained from MP2 and B3LYP is small. In average, difference of DI values from MP2 and B3LYP is 0.07 (Figs. 2 and 3), which is in the same order of magnitude of a very weak CC interaction. In case of SiSi bonds, the delocalization index difference is much higher (0.32). For CN, CO and SiSi bonds, the delocalization index from B3LYP is, in average, 0.13 higher than that from MP2. For NN and CC bonds, the delocalization index from B3LYP is, in aver-age, 0.02 higher than that from MP2. For C–Si and C–Ge bonds, the DI from B3LYP is 0.09 and 0.07, respectively, higher than those from MP2 with exception of C–Ge with triple bond where DI from B3LYP is 0.04 smaller than that from MP2. Conversely, for GeGe bonds, the delocalization index from B3LYP is 0.02, in average, smaller than that from MP2. Thus, the delocalization indexes from B3LYP are slightly higher than those from MP2 for the studied systems.

The value of the delocalization index is speciﬁc for each atomic pair. From the general formula HnX–XHn(where n is the number of

hydrogen atoms and X is B, C, N, O, Al, Si, P, or S atom) one can see

that the DI of the X–X bond increases from X@B to X@O and from X@Al to X@S (Table 2). Thus, there is an increasing trend of the DI and

q

bwith the electronegativity.Table 2also shows the values

of the localization index (LI) of C, N, O, Al, Si, P, or S atoms. The localization index indicates the number of electrons localized within an atomic basin [Eq.(8)]

LI ¼ FðX;

XÞ ¼

X N

i;j

S2i;jðXÞ ð8Þ

3.3. The linear relation between delocalization index and formal bond order

In order to investigate the relation between DI and formal bond order, we chose the molecular systems whose atomic pairs have formal bond orders varying from 1 to 3. Group 14 atoms, nitrogen and oxygen atoms have well-known molecular systems in which

13 2.305 2.223 2.444 2.172 2.105 2.356 1.095 1.238 1.480 (2.343) (2.164) (2.101) (2.450) _(2.289) _(2.221) (1.473) _(1.259) (1.120) 14 15 16 17 18 19 20 21 25 26 27 22 23 24 1.966 _1.778 1.720 1.708 1.649 1.885 (1.877) (1.711) _(1.632) (1.962) _(1.791) (1.713)

Fig. 3. Optimized geometries of the species 13–27 with corresponding Si–Si, Ge–Ge and N–N bond lengths (in Angstroms) from B3LYP and MP2 (between parentheses).

Table 2

Charge density of the bond critical point (qb), the delocalization index (DI) of the X–X bond, and the localization index (LI) of the X atom from the general formula HnX–XHn. General formula HnX–XHn qb(au.)a(X–X bond) DIa(X–X bond) LI (X atom)a

H2B–BH2 0.18 0.82 2.59 H3C–CH3 0.23 1.00 3.96 H2N–NH2 0.27 1.20 6.13 HO–OH 0.27 1.26 7.58 H2Al–AlH2 0.06 0.26 10.27 H3Si–SiH3 0.09 0.70 10.61 H2P–PH2 0.10 0.95 12.51 HS–SH 0.12 1.22 14.76 a From B3LYP/6-311++G** . 1

Average difference of delocalization indexes is in 102

order of magnitude. 2

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formal bond orders range from 1 to 3. However, in the case of mol-ecules with Si and Ge atoms, we opted for the simplest systems.

Fig. 2shows the DI’s of a bond between two carbon atoms or be-tween a carbon atom and a heteroatom (oxygen or nitrogen atom) in the molecular systems 1–12. Hereafter, the deﬁnition of formal

bond order[3]will be applied to compare with the delocalization index of the studied compounds. The C–C, C–O and C–N bonds in the systems 1–12 have different formal bond orders. The values of the DI’s of CN bonds are nearly the same as the values of the DI’s of CC bonds. Conversely, the values of the DI’s of C–O bonds are reasonably different from those of CC bonds.

The optimized geometries of the species 13–27 are shown in Fig. 3. The bond length differences between B3LYP and MP2 values of these species are in average 0.011 Å, except for GeGe bond lengths where average difference is 0.042 Å. The SiSi bond lengths ranges from 2.140 to 2.160 ÅA

0

for stable disilenes[40,41]and nearly 2.062ÅA

0

for stable disilynes [42]. The GeGe bond lengths ranges from 2.213 to 2.285 ÅA

0

for stable digermenes [43,44] and from 2.060 to 2.285 ÅA0 for stable digermynes[45–48]. The C–Ge single bond is 1.98 ÅA

0

and C–Ge double bond is 1.80 ÅA 0

[49]for stable germ-anes and germenes, respectively. The C–Si bond of methylsilane [50]is 1.867 ÅA0. All of these experimental bond lengths are very close to our obtained values (Fig. 3), except for 0.1ÅA

0

difference be-tween experimental and theoretical values of C–Si double bond [50].

Fig. 4shows some silicon, germanium and nitrogen compounds with formal bond order ranging from 1 to 3. There is a direct rela-tion between the DI and the formal bond order of inorganic atomic pairs, Si–Si bond and Ge–Ge bonds, Si–C, Ge–C and N–N bonds (species 13–27 inFig. 4). The DI values for SiSi bonds are higher than those for GeGe bonds. In N–N bonds, the values of DI are very close to their bond order values (Fig. 4).

The direct use of DI to measure bond order[23,24]would be inconvenient for Si–Si, C–Si and Ge–Ge bonds since their DI’s for single, double and triple bonds are very different from those of C–C atomic pair. Thus, there would not be standard values of bond order for different atomic pairs. In this work, the correlation be-tween DI and formal bond order standardize the calculation of for-mal bond order for different atomic pairs. Likewise, Bader and coworkers established a correlation between formal bond order and charge density of the bond critical point (See Supplementary material). 25 26 27 22 23 24 N N 13 14 15 16 17 18 19 20 21

DI= 0.70 DI= 0.91 DI= 1.81

DI= 0.81 _{DI= 1.39} DI= 2.02

DI= 1.20 DI= 2.17 DI= 3.03

(DI= 0.18) (DI= 0.63) (DI= 1.64)

(DI= 0.82) _{(DI= 1.41)} _{(DI= 2.00)}

(DI= 1.19) (DI= 2.15) (DI= 3.01)

N N H H H H N N H H Si Si H H H H H H Si Si H H H H Si Si H H Ge Ge H H H H H H Ge Ge H H H H Ge Ge H H Si H H H H H H Ge H H H H H H Ge H H H H Si H H H H _Si H H Ge H H

DI= 0.53 DI= 1.19 DI= 1.93

DI= 2.64 DI= 1.66

DI= 0.83

(DI= 0.76) (DI= 1.58) (DI= 2.30)

(DI= 1.92) (DI= 1.02)

(DI= 0.44)

Fig. 4. Delocalization indexes of the Si–Si, Ge–Ge, N–N, SiC and GeC atomic pairs of the molecules 13–27 from B3LYP. Delocalization indexes between parentheses are from MP2. 0 1 2 3 4 0 1 2 3 4 CC bond NN bond GeGe bond 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 C-C bond Si-Si bond N-N bond Ge-Ge bond C-N bond Si-C bond Ge-C bond C-C bond Si-Si bond N-N bond Ge-Ge bond C-N bond Si-C bond Ge-C bond 0.5 1.5 2.5 3.5

Formal bond order Formal bond order

DI DI DI DI 0.5 1.5 2.5 3.5 0 1 2 3 0 1 2 3 CC bond NN bond GeGe bond 0.5 1.5 2.5 3.5 0.5 1.5 2.5 3.5 3.5 2.5 1.5 0.5

MP2

B3LYP

0.5 1.5 2.5

Fig. 5. (A and C) Formal bond order versus DI plot for CC, SiSi, NN, GeGe, CN, SiC and GeC bonds from B3LYP and MP2, respectively; (B and D) formal bond order versus DI plot for CC, NN and GeGe derived from the relation with their corresponding electronegativities from B3LYP and MP2, respectively.

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Fig. 5A shows the relation between the DI and the formal bond order for CC, NN, GeGe, SiSi, SiC, GeC and CN atomic pairs. There is a direct relation between the formal bond orders and their corre-sponding DI’s. This relation is speciﬁc for each sort of atomic pair because different atomic pairs, with the same formal bond order, have different delocalization indexes (Table 2). Each plot was de-rived from the studied compounds (1–3, 5–7, 13–27) with formal bond orders ranging from 1 to 3. These plots are similar to that ob-tained from the relation between the charge density of bond criti-cal point and the formal bond order for CC bonds[12,14]. However, the relation between the formal bond order and delocalization in-dex is linear (Fig. 5A and C). In addition, the delocalization inin-dexes of CC, NN, GeGe, SiC, GeC and CN bonds have a very good correla-tion with the formal bond order. In the case of SiSi bonds, there is a moderate correlation between their DI’s and the formal bond or-ders from MP2 calculation (Fig. 5C).

We found a relation between the electronegativity of C, N and Ge atoms and the angular coefficients of the CC, NN and GeGe curves ofFig. 5A and C, whose linear and angular coefficients were used to build the plots of theFig. 5B and D (See Supplementary material). The angular coefficient of the curves ofFig. 5A and C rep-resents the increasing tendency of DI with formal bond order. The DI value is unique for a specific atomic pair whose atoms have a specific electronegativity. We found the link among electronegativ-ity, DI and formal bond order by building the plots ofFig. 5B and D. Then, the electronegativity of C, N and Ge atoms could be used to give a single curve of linear relation between the formal bond order and the delocalization index encompassing the CC, NN and GeGe bonds. Hence, the electronegativity is closely related with the correspondence between the delocalization index and the for-mal bond order for the CC, NN and GeGe bonds. The plots ofFig. 5B and D shows that it is possible to obtain a relatively general rela-tion between the formal bond order and the delocalizarela-tion index with different atomic pairs, as a consequence of the direct relation between the delocalization index and the electronegativity of the atom of the atomic pair (Table 2).

Thus, the DI can be used to estimate the formal bond order of a chemical bond in an alternative way from QTAIM. The bond order of any CC, GeGe, CN, SiC, GeC and NN bond can be obtained from the corresponding linear relation inFig. 5A and C. The electroneg-ativity is closely related with the correspondence between the delocalization index and the bond order for the CC, NN and GeGe bonds. Finally, a relatively general relation between the formal bond order and the delocalization index with different atomic pairs can be obtained.

4. Conclusions

The delocalization index from QTAIM can be used to estimate the formal bond order of a chemical bond in an alternative way. The relation between DI and formal bond order is linear for the CC, NN, GeGe, SiC, GeC and CN bonds. From the electronegativity of carbon, nitrogen and germanium atoms, it is obtained a single relation between the delocalization index and the formal bond or-der encompassing the CC, NN and GeGe bonds.

Acknowledgments

Authors thank FAPERJ and CNPq for ﬁnancial support.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, in the online version, atdoi:10.1016/j.cplett.2008.12.004.

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