Prediction and characterization of the single-electron sodium bond

complexes Y–C� � �Na–H [Y = H3, H3CH2, (H3C)2H and (H3C)3]

Li Zhi-Feng,* Zhu Yuan-Cheng and Li Hui-Xue

Received 6th July 2009, Accepted 16th September 2009

First published as an Advance Article on the web 13th October 2009

DOI: 10.1039/b913363a

The prediction and characterization of the single-electron sodium bond complexes Y–C� � �Na–H

[Y = H3, H3CH2, (H3C)2H and (H3C)3] have been investigated for the first time by using

MP2/6-311++G(d,p), MP2/6-311++G(2d,2p) and MP2/aug-cc-pVDZ methods. The strength of

the interactions in H3C� � �Na–H, H3CH2C� � �Na–H, (H3C)2HC� � �Na–H, and (H3C)3C� � �Na–H

complexes has been analyzed. It is shown that the (H3C)3C radical with Na–H forms the

strongest single-electron sodium bond, followed by the (H3C)2HC radical and then the H3CH2C

radical. H3C radical forms the weakest single-electron sodium bond. NBO and AIM analyses

have also been used to estimate such conclusions. Furthermore, there are few linear/nonlinear

relationships among the several parameters in system and the interaction mode

of single-electron Na bond is LP1(C) - LP1*(Na), which is different from the single-electron

H bond and single electron halogen bond. By comparisons with some related systems,

it is concluded that the strength of single-electron bond is increased in the order:

hydrogen bond o sodium bond o bromine bond o lithium bond.

In the past decades, more-and-more attention has been paid

to the theoretical and experimental investigations on hydrogen

bonds due to their very important roles in chemistry, physics

and biology.1–3 A number of unusual weak bonds4–9 have been

proposed and extensively studied.

The nonadditivity of weak bonds is one of their interesting

and important characteristics. It is believed to play a significant

role in the explanation of many physically and chemically

important properties.10,11 Although the nonadditivity between

conventional and unconventional hydrogen bonds has often

been studied,12,13 much more recent interest has been focused

on the nonadditivity among different types of weak bonds

studied with theoretical and experimental methods.13,14

Most recently, the concept of a single-electron H bond, i.e.,

the interaction between the unpaired electron in the radical

molecule and the proton donor,11–15 has been introduced to

characterize several special hydrogen bond complexes. In most

studies on the single-electron hydrogen bond,15 single-electron

lithium bond16 and single-electron halogen bond,17 methyl

radical is often taken as a proton acceptor. This is because

the methyl radical is a simple prototype for a wide class of

organic radicals, which plays a key role as an intermediate in

the field of chemistry and biochemistry.18,19

Although the literature20 reports that, similar to the

hydrogen atom, the sodium atom radius is relatively small

and it may form weak sodium bond interactions with other

molecules, the study of the sodium bond, let alone unusual

sodium bonds, is limited until now. Since sodium is a congener

of hydrogen/lithium, it is a logical isomorphic replacement to

substitute the hydrogen/lithium with sodium in a single-

electron hydrogen bond.

In this work, we constructed H3C� � �Na–H,

H3CH2C� � �Na–H, (H3C)2HC� � �Na–H and (H3C)3C� � �Na–H.

These radicals constitute usual alkyl radicals in chemical and

biological systems. Thus, we have made an analysis of the

structures and energetics of these complexes, by which the

effect of the methyl group has been studied in the formation of

the single-electron sodium bond. To our knowledge, this is the

first investigation on the sodium bond by using the novel

single-electron sodium bond complexes. Natural bond orbital

(NBO), atom in molecules (AIM) and electrostatic potential

map (EPM) analyses have also been performed. This work

should be interesting for future theoretical investigations and

experimental works, and also deepen and enrich the content of

weaker interactions theory.

1. Computational details

Complexes and monomers were optimized at the MP2/

6-311++G(d,p), MP2/6-311++G(2d,2p) and MP2/aug-cc-

pVDZ levels21,22 because the MP2 method has more and more

been used to studying the weak interaction in recent years.

Harmonic frequency analyses were performed at the same

levels to confirm that these structures were local minima on

the energy surfaces. The interaction energies were corrected

with the basis set superposition error (BSSE). The BSSE was

evaluated by using the counterpoise method of Boys and

Bernardi.23 The atoms-in-molecules (AIM) theory of Bader24a

was applied to find the bond critical points (BCP) and

to analyze them in terms of electron densities and their

Laplacians. The AIM calculations were carried out with the

AIM2000 program. The NBO analyses were carried out with

College of Life Science and Chemistry, Tianshui Normal University,Tianshui 741001, China. E-mail: zflitsnu@163.com;Fax: +86 (0)938 8367701; Tel: +86 (0)938 8367701

This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 11113–11120 | 11113

PAPER www.rsc.org/pccp | Physical Chemistry Chemical Physics

Publ

ishe

d on

13

Oct

ober

200

9. D

ownl

oade

d by

Sta

te U

nive

rsity

of

New

Yor

k at

Sto

ny B

rook

on

27/1

0/20

14 1

1:40

:31.

View Article Online / Journal Homepage / Table of Contents for this issue

the NBO5.0 package. All other calculations were performed

with the Gaussian 03 program.24b

2. Results and discussion

2.1 Structural characteristics

The optimized structures of monomers and complexes are

shown in Fig. 1, and some selected structural parameters

calculated at the MP2/6-311++G(2d,2p) (top parameters)

and MP2/6-311++G(d,p) (bottom parameters) are also

demoted. When the hydrogen atoms of the methyl radical in

complex I are substituted with one, two and three methyl

groups respectively, complexes H3CH2C� � �Na–H (II),

(H3C)2HC� � �Na–H (III) and (H3C)3C� � �Na–H (IV) are thus

obtained. The complexes I and IV have C3v symmetry and

the other two complexes II and IV have Cs symmetry. As

Fig. 1 shows, the parameters at the MP2/6-311++G(d,p) level

are comparable to those predicted by MP2/6-311++G(2d,2p).

In order to further verify the reliability of those

structures obtained at the MP2/6-311++G(d,p) and

MP2/6-311++G(2d,2p) levels, the structures are also

calculated using the MP2/aug-cc-pVDZ method, these results

indicate that the parameters in MP2/aug-cc-pVDZ are

also close to those obtained by MP2/6-311++G(d,p) and

MP2/6-311++G(2d,2p). Thus, the following discussion is

based on MP2/6-311++G(d,p) geometries, because the MP2

method is reliable and it has been applied successfully to study

weak interactions.22

The distances of C� � �Na in complexes H3C� � �Na–H,

H3CH2C� � �Na–H, (H3C)2HC� � �Na–H and (H3C)3C� � �Na–H

are respectively 0.2885 nm, 0.2834, 0.2806 nm and 0.2780 nm,

which are all less than the sum of the van der Waals radii for

one C atom and one Na atom (0.397 nm). It is shown that

stable complexes I–IV are formed. Interesting, one can find

that the binding distances dC� � �Na are decreased in the order of

H3C� � �Na–H - H3CH2C� � �Na–H - (H3C)2HC� � �Na–H -

(H3C)3C� � �Na–H, which suggesting that the dC� � �Na in the

H3C� � �Na-H and (H3C)3C� � �Na–H clusters is the biggest

and the smallest respectively. These binding distances demonstrate

that the single-electron sodium bond is the weakest in the

H3C� � �Na–H cluster and the strongest in the (H3C)3C� � �Na–H

complex. It is observed that the C� � �Na decreases with the

increasing of the methyl group in the radical. It suggests that

the methyl group in the electron donor imposes a positive

effect on the single-electron sodium bond and enhances the

strength of the single-electron sodium bond. In the four

complexes I–IV, the Na–H bond in the electron accepter,

which is similar to hydrogen bond and single-electron halogen

bond,15,17 is also lengthened. The length of the Na–H bond in

the complex I is 0.1918 nm. As the methyl group in the radical

increases, the elongation of the Na–H bond also increases.

That shows that the strength of the Na–H bond in complexes

is decreased.

Additionally, upon the formation of the single-electron

sodium bonds, the H3C, H3CH2C, (H3C)2HC and (H3C)3C

radicals are very well stabilized. Fig. 2 shows the energies and

shapes of the main MOs of the free radicals H3C, H3CH2C,

(H3C)2HC, (H3C)3C, monomer NaH, and the corresponding

complexes I–IV. For the four free radicals H3C, H3CH2C,

(H3C)2HC and (H3C)3C, the single p electron occupies the

highest occupied molecular orbital (HOMO) with �0.383 eV,

�0.350 eV, �0.329 eV and�0.316 eV orbital energy. From the

NBO analyses, only ca. 0.0144, 0.0156, 0.0149 and 0.0120 of

an electron transfers from the H3C, H3CH2C, (H3C)2HC and

(H3C)3C to the Na–H molecule during the single-electron

sodium bond formation. Therefore, the HOMO of the four-

radical moiety remains singly occupied in four complexes I–IV

(Fig. 2). Besides, from the single occupied molecular orbitals

(SOMOs) of four complexes I–IV, it is noted that the electron

cloud of H3C, H3CH2C, (H3C)2HC and (H3C)3C radicals

approaches the Na atom slightly and the Na atom plays the

role of electron acceptor.

It is interesting that in complexes I–IV, all the double-

occupied MOs HOMO lie higher in energy than the single

occupied p orbital HOMO-1 from H3C, H3CH2C, (H3C)2HC

and (H3C)3C moieties. In some H/Li bonding systems,16,25

while the H/Li bond formation and the molecular orbital

energies of the H/Li acceptor decrease, those of the H/Li

donor increase. Similarly, since the H3C, H3CH2C,

(H3C)2HC and (H3C)3C radicals act as the Na acceptor in

complexes I–IV, the SOMO energies from the H3C, H3CH2C,

(H3C)2HC and (H3C)3C moieties decrease by 0.071 eV,

0.062 eV, 0.058 eV and 0.054 eV compared with those of the

Fig. 1 Optimized structures of monomers and complexes at MP2/6-311++G(d,p) level (bond length: � 10�1 nm, bond angle: 1).

11114 | Phys. Chem. Chem. Phys., 2009, 11, 11113–11120 This journal is �c the Owner Societies 2009

Publ

ishe

d on

13

Oct

ober

200

9. D

ownl

oade

d by

Sta

te U

nive

rsity

of

New

Yor

k at

Sto

ny B

rook

on

27/1

0/20

14 1

1:40

:31.

View Article Online

HOMO of the free H3C, H3CH2C, (H3C)2HC and (H3C)3C

radicals respectively. It shows that the four radicals are very

well stabilized upon the single-electron Na bonds’ formation.

Meanwhile, the molecular orbital energies of the Na–H molecule

increase at least 0.010 eV after the single-electron Na bond

formed and its increasing degree is well in agreement with

the elongation of Na–H bonds in the sequence of complexes

I- II- III- IV, which is also consistent with the weakened

Na–H bonds’ order. It is worth pointing out here that those

changes of molecular orbital energies result in abnormal

phenomena, such as the double-occupied MOs (HOMO)

orbital from the Na–H subunit lying higher in energy than

the SOMO (HOMO-1) from the H3C, H3CH2C, (H3C)2HC

and (H3C)3C moieties. Thus, the unpaired p electrons from the

H3C, H3CH2C, (H3C)2HC and (H3C)3C radicals only occupy

the HOMO-1 orbitals of the H3C� � �Na–H, H3CH2C� � �Na–H,

(H3C)2HC� � �Na–H and (H3C)3C� � �Na–H complexes, while

the HOMOs of the H3C� � �Na–H, H3CH2C� � �Na–H,

(H3C)2HC� � �Na–H and (H3C)3C� � �Na–H complexes mainly

originate from the HOMOs of the corresponding Na–H

molecules, meanwhile the HOMO-1 of the H3C� � �Na–H,

H3CH2C� � �Na–H, (H3C)2HC� � �Na–H and (H3C)3C� � �Na–H

complexes are almost duplicates of the HOMO of the H3C,

H3CH2C, (H3C)2HC and (H3C)3C, respectively (Fig. 2).

For the typical red-shifted H-bonds, the elongation of the

proton-donating bond is often treated as strong evidence for

hydrogen interaction.26 In the four complexes I–IV, the

elongations of the Na–H bonds are by 0.0010 nm, 0.0013 nm,

0.0015 nm and 0.0016 nm (Fig. 1), which may indicate that

complexes I–IV are red-shifted single-electron Na-bonds and

that the red-shifted strength increases gradually. Upon

frequency analysis of the complexes and the monomer

Na–H, the vibrational frequencies of the Na–H bond in

complexes are decreased gradually by 14.5 cm�1, 20.2 cm�1,

23.4 cm�1 and 24.5 cm�1 for complexes I, II, III and IV, and

furthermore the red-shifted single-electron Na-bonds were

formed. This indicates that the elongation of the proton

donating bond also can be considered as an important

characterestic of the red-shifted single-electron Na-bonds.

Hermansson27 pointed out that the increasing of vibrational

intensity of X–H is another characteristic of red-shifted

H-bonds. As shown in Table 1, in single-electron sodium

complexes I, II, III and IV the vibrational intensity of Na–H

increases gradually compared to the monomer Na–H, which

supports the red-shifted single-electron Na-bond characteristics

of complexes I–IV.

Based on the above analyses and discussion, the single-

electron sodium could be termed as a form of Y� � �Na–X,

where Y is single-electron radical and the X is the electro-

negative atom/group. The redshifting single-electron bond is

characterized by an elongation of the Na–X bond, a red-

shifting of the respective Na–X stretching frequency and the

vibrational intensity increasing of Na–X, which is similar to

those of conventional hydrogen bond.

2.2 Interaction energies

Interaction energy is a powerful method of estimating the

strength of an interaction. The single-electron sodium bond

interaction energies of the H3C� � �Na–H, H3CH2C� � �Na–H,

(H3C)2HC� � �Na–H and (H3C)3C� � �Na–H complexes at

MP2/6-311++G(d,p) level are listed in Table 1. In our study

on interaction energies, the correction of BSSE is used to

calculate because the correction of EBSSE is a necessary step for

depiction the energy of weaker interactions, besides single-

electron bonds.16 As shown in Table 1, The estimated BSSE of

the Y–C� � �Na–H interaction, ranging from 2.6 to 6.0 kJ mol�1,

is rather large relative to the raw binding energy, and it

accounts for about 21–29% of the absolute value of the raw

binding energy in the four complexes. It is also seen that the

BSSE value increases when the number of methyl groups in

the radical grows.

The binding energies without (DE) and with BSSE

corrections (EBSSE) at different levels are given in Table 2.

The corrected interaction energy in single-electron sodium

complex I is calculated to be �12.4 kJ mol�1 at the MP2/

6-311++G(d,p)//MP2/6-311++G(d,p) level, which is close to

that calculated at CCSD(T)/aug-cc-pVDZ///MP2/aug-cc-

pVDZ and the other four calculated methods, indicating that

the MP2/6-311++G(d,p) method is proper to depict such an

interaction in these single-electron sodium complexes.

With these comparisons, together with the consideration of

computational cost, only the MP2/6-311++G(d,p)//MP2//

6-311++G(d,p) method was used to investigate the interaction

Fig. 2 The main molecular orbitals and corresponding energies of complexes I–IV at MP2/6-311++G(d,p) level (energy: eV).

This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 11113–11120 | 11115

Publ

ishe

d on

13

Oct

ober

200

9. D

ownl

oade

d by

Sta

te U

nive

rsity

of

New

Yor

k at

Sto

ny B

rook

on

27/1

0/20

14 1

1:40

:31.

View Article Online

energies of other complexes in this letter. It can be seen from

Table 1 that the absolute interaction energies of single-electron

sodium increased in the order: H3C� � �Na–H(12.4 kJ mol�1)oH3CH2C� � �Na–H(16.8 kJ mol�1) o (H3C)2HC� � �Na–H

(19.1 kJ mol�1) o (H3C)3C� � �Na–H (20.7 kJ mol�1), which

is consistent with the sequence of the increasing number of

methyl groups in complexes I–IV. That is, the greater the

number of methyl groups, the bigger the absolute interaction

energy and the higher the strength of single-electron bond.

This result demonstrates that the methyl group in the electron

donor plays a positive contribution to the single-electron

sodium bond, which is in agreement with that in halogen17

and hydrogen bonds.28 When the methyl hydrogen atom in the

H3C� � �Na–H complex is replaced with one methyl group, the

corrected interaction energy increases by 4.4 kJ mol�1 at the

MP2/6-311++G(d,p) level. However, the contribution of

the methyl group in O� � �H–O and H3C� � �Br–O increases by

2.5 kJ mol�1 and 2.6 kJ mol�1 in absolute value,17 which

shows that the contribution of the methyl group is bigger in

the single-electron sodium bond complex. When two methyl

hydrogen atoms in the complex H3C� � �Na–H are replaced

with two methyl groups, the calculated interaction energy

is �16.8 kJ mol�1. Its absolute value increases by 6.7 kJ mol�1

relative to H3C� � �Na–H. When all methyl hydrogen atoms in

H3C� � �Na–H are replaced with methyl groups, the increase of

the corrected interaction energy is the greatest (8.3 kJ mol�1).

The result shows that the presence of a positive methyl group

enhances the strength of single-electron sodium bonds.

In order to have a strength comparison of single-electron

lithium bond, single-electron sodium bond, single-electron

hydrogen bond and single-electron bromine bond, we also

calculated the interaction energies of single-electron bond

H3C� � �X–F (X = H, Li, Na, Br) at MP2/6-311++G(d,p)

level. When the Na atom in H3C� � �Na–F complex is replaced

with H, Li, Br atoms, the corrected interaction energies

are �8.7 kJ mol�1, �21.2 kJ mol�1 and �14.9 kJ mol�1,

respectively, suggesting the single-electron sodium bond is

weaker than that of single-electron lithium bond and it is

stronger than that of single-electron hydrogen bond. Thus, the

strength of single-electron bond is increased in the order:

hydrogen bondo sodium bondo bromine bondo lithium bond

because the interaction energy of H3C� � �Na-F is �14.5 kJ mol�1.

Moreover, plots of interaction energy (EBSSE) against

the dC� � �Na are good linear with correlation coefficients of

0.99786 with the equation of y = 3.04069 + 0.01243x

(Fig. 3). Typically, the relationship between EBSSE and dNa–H

is in good agreement with the equation of y = 1.99887 �7.32394 � 10�4 x with r = 0.99872, which suggests that with

increasing methyl group number, the absolute interaction

energy of the complex is greater, the elongation of the Na–H

bond is bigger and the Na–H bond strength is much weaker.

Fig. 4 shows that the dipole moments of the complexes

correlate well with interaction energies EBSSE and the linear

equation is y= 82.41� 11.91x with r= 0.985, suggesting that

the bigger the dipole moment (m), the more negative the

interaction energies, and the stronger the interaction.

2.3 NBO and AIM analysis

For a better understanding of the contribution to the

cooperativity of the single-electron sodium bond to orbital

interactions, NBO analysis has been carried out for the

complexes at the MP2/6-311++G(d,p) level. The results are

given in Table 1.

Let us first repeat that the formation of a hydrogen-bonded

complex, either a conventional hydrogen bond or a blueshifting

hydrogen bond, involves charge-transfer (CT) from the proton

acceptor to the proton donor. This results in the increase of

electron density in the X–H antibonding orbitals of the proton

donor. Since the charge-transfer accompanies the formation of

Table 1 Interaction energies (EBSSE), vibrational frequencies (v), stabilization energies (E(2)ij ), hybridization of C(spn), methyl charge change (Dq),

fractional number of electrons transferred (DN) and the ionic character of C� � �Na bond

I (H3C� � �Na–H) II (H3CH2C� � �Na–H) III ((H3C)2HC� � �Na–H) IV ((H3C)3C� � �Na–H)

BSSE/kJ mol�1 2.6 3.6 4.7 6.0EBSSE/kJ mol�1 �12.4 �16.8 �19.1 �20.7vNa–H/cm

�1 a1182.4 (304) 1167.9 (366) 1162.2 (383) 1159.0 (401) 1157.9 (410)DvNa–H/cm

�1 �14.5 �20.2 �23.4 �24.5E(2)ij [LP1(C) - LP*1(Na)]/kJ mol�1 20.6 20.1 17.3 13.3

C(spn) 2.69 2.82 2.95 3.08Dq(e)(Na–H) �0.0144 �0.0156 �0.0149 �0.0120DN 0.0403 0.0175 0.0030 �0.0060Ionic character of C� � �Na bond (%) 96.8 97.1 97.6 98.2

a The vNa–H of Na–H bond in monomer and its vibrational intensity are in brackets.

Table 2 Binding energies (EBSSE, kJ mol�1) of H3C� � �Na–H complex calculated at different levels

DE BSSE EBSSE

MP2/6-311++G(d,p)//MP2/6-311++G(d,p) �15.0 2.6 �12.4CCSD(T)/6-311++G(d,p)//MP2/6-311++G(d,p) �15.4 2.9 �12.5MP2/6-311++G(2d,2p)//MP2/6-311++G(2d,2p) �14.3 1.6 �12.8CCSD(T)/6-311++G(2d,2p)//MP2/6-311++G(2d,2p) �14.8 1.7 �13.1MP2/aug-cc-pVDZ//MP2/aug-cc-pVDZ �15.1 1.9 �13.2CCSD(T)/aug-cc-pVDZ//MP2/aug-cc-pVDZ �15.5 1.9 �13.6

11116 | Phys. Chem. Chem. Phys., 2009, 11, 11113–11120 This journal is �c the Owner Societies 2009

Publ

ishe

d on

13

Oct

ober

200

9. D

ownl

oade

d by

Sta

te U

nive

rsity

of

New

Yor

k at

Sto

ny B

rook

on

27/1

0/20

14 1

1:40

:31.

View Article Online

hydrogen bonds and plays a major role in it, the E(2)ij can be

taken as an index to judge the strength of hydrogen bonds. For

the single-electron sodium bond, the case is substantially

different. Although the CT also occurrs between the two

fragments in complexes I–IV, the charge-transfer from the

single-electron of electron donor to the Na atom acceptor is

mainly directed to the LP*1(Na) rather than s*(Na–H).

Addition, the main orbital interactions LP1(C) - LP*1(Na)

in interactions I–IV are also different from the electron

transferring of single-electron halogen bond17 and single-electron

hydrogen bond26 complexes. The change tendency of

donor–acceptor interaction stabilization energy E(2)ij [LP1(C) -

LP*1(Na)] in Table 1 for complexes from I to IV is opposite to

the order of their intermolecular interaction energies, suggesting

that the role of CT is not the most important one compared

with hydrogen bonds. Moreover, it can be seen from Table 1

that the E(2)ij [LP1(C) - LP*1(Na)] is decreased in complexes

with the increasing number of methyl groups in the radical.

In exploring the single-electron sodium bonds of

H3C� � �Na–H, H3CH2C� � �Na–H, (H3C)2HC� � �Na–H and

(H3C)3C� � �Na–H, the effect of hybridization of the electron

donor atom (C) on the strength of the C� � �Na bond was

studied (Table 1). As compared with the complex

H3C� � �Na–H, the interaction energy of the H3CH2C� � �Na–H

increases by 35.5% in absolute value. The C� � �Na binding

distance in the H3CH2C� � �Na–H complex (0.2834 nm)

decreases relatively to that in the H3C� � �Na–H complex

(0.2885 nm). A larger shortening of about 0.0051 nm is found

for the C� � �Na binding distance as the carbon hybridization

of electron donor C is from sp2.69 (methyl radical) to sp2.82

(ethyl radical). The results demonstrate that the strength of

the C� � �Na single-electron sodium bond is increased in the

H3CH2C� � �Na–H complex. As the number of methyl groups

in the radical grows (I - II - III - IV), we also find that

the p character of hybridization of C in C� � �Na single-electron

sodium bond increases in the following order: I(72.9%) oII(73.8%) o III(74.7%) o IV(75.5%), which agrees with

the increasing sequence of the single-electron sodium bond

absolute interaction energies.

If one concentrates on the relationship between EBSSE and

E(2)ij [LP1(C) - LP*1(Na)], then the nonlinear relationship

between EBSSE and E(2)ij [LP1(C) - LP*1(Na)] is found, that

is, they are in well agreement with the nonlinear equation of

y = 20.85 � 0.0002 � 0.6x because of the r2 is 0.996. We also

find that the relationship of dC� � �Na and E(2)ij [LP1(C) -

LP*1(Na)] is nonlinear and it is satisfied with the exponential

asymptotic function of y = 21.08 + 8.26 � 1038 � 2.1�14x

(r2 = 0.990).

It is interesting to note from Table 1 that the 7EBSSE7increases with the increasing number of the methyl group.

However, the methyl charge change (Dq) is irregular. It is

shown that from complex I to II, II to III, and III to IV, the

effect of direct hyperconjugation on EBSSE and Dq are

less-and-less. In a single-electron H-bond system, as the

proton acceptor adjoins with an electropositive group the

strength of the single-electron H-bond is increased. The methyl

group in the electron donor is electron-donating, making a

positive contribution to the formation of single-electron

sodium bond and results in the interaction energies of the

complexes increasing as the number of methyl group grows.

If any two systems A and B are brought together, a single

system will be formed with a constant value of m. In this case,

there is a transfer of electrons from the less electronegative

system and the fractional number of electrons transferred

DN is given by ref. 29

DN ¼ wA � wB2ðZA þ ZBÞ

ð1Þ

where w = �m where w is called the absolute electronegativity.

Chemical potential (m) and chemical hardness (Z) are two

important quantities, which are used to characterize any

chemical system. They are defined as m = (I + A)/2 and

Z = (I�A)/2, where I and A are the ionization energy and

electron affinity of the system. A large value of DN represents a

strong and favorable interaction between A and B. It is noted

that the DN values for complexes I–IV are decreased, which

indicates that the ability of electron transference from I to II,

II to III and III to IV is decreased, which also consistent with

the E(2)ij [LP1(C) - LP*1(Na)] decreasing gradually. However,

the absolute interaction energy of complexes from I to IV is

increased, which is not consistent with increasing sequence of

DN value in four complexes. That is, the value of DN cannot

be used to predict the strength of single-electron Na-bond.

The rigorous AIM theory30 has been successfully applied in

characterizing hydrogen bonds, halogen bonds and lithium

bonds in a wide variety of molecular complexes. Popelier31

proposed a set of criteria for the existence of H bonding within

the AIM formalism. The most prominent evidence of hydrogen

Fig. 3 The relationships among EBSSE, d�Na–H and dC���Na

(’, K, m and . denote complexes I, II, III and IV respectively).

Fig. 4 The relationships between dipole moments m and EBSSE

(’, K, m and . denote complexes I, II, III and IV respectively).

This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 11113–11120 | 11117

Publ

ishe

d on

13

Oct

ober

200

9. D

ownl

oade

d by

Sta

te U

nive

rsity

of

New

Yor

k at

Sto

ny B

rook

on

27/1

0/20

14 1

1:40

:31.

View Article Online

bonding is the existence of a bond path between the donor

hydrogen nucleus and the acceptor, containing a interatomic

surface (IAS) and a bond critical point (BCP) at which the

electron density (r) ranges form 0.002 to 0.035 a.u. and the

Laplacian of the electron density (r2r) ranges from 0.024 to

0.139 a.u. Topological parameters are helpful for understanding

the effect of the methyl group on the single-electron sodium

bond. In the present study, the first three AIM criteria of

hydrogen bonds will be systematically applied to the single-

electron sodium bonds in order to gain a deeper insight into

this special type of interaction. According to the AIM theory

of Bader, the molecular graph is intuitionistic for the topological

property of electron density, which can also display the

structure of the bond system.

Fig. 5 detects the four molecular graphs of complexes I–IV,

which indicates that there are bond critical points [BCP]

existing in corresponding complexes, suggesting a bond action

between corresponding atoms. The expected bond paths

associated with the single-electron bond BCPs can also be

visualized in Fig. 5.

In the sets of three topological parameters given in Fig. 5,

the top value refers to electron densities (r), the middle one to

Laplacians (r2r) and the bottom one to ellipticity (e).Popelier31 proposed that for covalent bonds the value of the

r2r is negative. For ionic bonds, hydrogen bonds, and van der

Waals interactions, values of the r2r are positive. Interestingly,

for four single-electron sodium complexes, the r and r2rvalues of the C� � �Na bonds are 0.0068–0.0094 a.u. and

0.0285–0.0408 a.u., which are just within the range of the rand r2r values of hydrogen bonds, besides they are closed to

the hydrogen bond lower limit of Popelier. It may suggest that

the strength of the single-electron Na bond is comparable to

that of the general hydrogen bond and its strength is weaker

compared with the hydrogen bond. In Fig. 5, the r and r2rvalues of the C� � �Na bonds increase in the order H3C� � �Na–H

(r: 0.0068 a.u.;r2r: 0.0285)oH3CH2C� � �Na–H (r: 0.0079 a.u.;r2r: 0.0336 a.u.) o (H3C)2HC� � �Na–H (r: 0.0086 a.u.; r2r:0.0372 a.u.)o (H3C)3C� � �Na–H (r: 0.0094 a.u.;r2r: 0.0408 a.u.)while those of the Na–H bonds decrease gradually. Again it

has been shown that r and r2r are related to the bond order

and thus to the bond strength. As a result, the values for r and

r2r in complexes I are smallest and in IV are largest for

C� � �Na bonds compared to those in other three complexes

respectively, which is also consistent with the interaction

analysis in section 2.2.

The ellipticity e is defined as l1/l2� 1, of which the l1 and l2are the two eigenvalues of the Hessian matrix of electron

density. The ellipticity provides a measure for not only the

p character of a bond but also its structural stability. Substantial

bond ellipticities reflect structural instability, that is, the bond can

easily be ruptured. However, as shown in Fig. 5, the results

derived from ellipticity criteria are different from that of electron

densities (r), Laplacians (r2r) and interaction energies.

We also found that the increase in the absolute interaction

energies of the complexes leads to an increase of r and r2r(Fig. 6), suggesting that the strength of interaction gets

stronger-and-stronger with increasing r and r2r. Typically,the relationship between EBSSE and r is linear (y = 0.00296 �3.02305 � 10�4x, r = 0.989), and it is the same as the

relationship between EBSSE and dNa–H with the same correlation

equation of y = 1.99887 � 7.32394 � 10�4 x with r = 0.999,

indicating that the effect of interaction energies on dNa–H and

r are at the same level (Fig. 3 and 6).

From the above discussion, we can see that the first three

criteria for the hydrogen bond are all echoed in the redshifting

single-electron sodium bond, excepting that of ellipticity.

This indicates the different nature of the two types of inter-

molecular interactions.

In addition, the ionic character is predominant because the

value of r2r on C� � �Na is positive, which is also consistent

with the result of NRT (natural resonance theory) analysis

(the ionic character of C� � �Na bond) as shown in Table 1.

In recent years, many researchers have used graphic models,

especially electrostatic potential maps (EPM), as a tool in

conformational analysis because they have been used primarily

for predicting sites and relative reactivities toward electro-

philic attack, and in studies of biological recognition and

hydrogen bonding interactions.32–34 The electrostatic potential

Fig. 5 Molecular graphs of I–IV complexes (the parameters from top to bottom are values of r, r2r and e).

Fig. 6 The relationships between rC� � �Na andr2rC� � �Na with EBSSE at

the MP2/6-311++G(d,p) level (’, K, m and . denote complexes I,

II, III and IV respectively).

11118 | Phys. Chem. Chem. Phys., 2009, 11, 11113–11120 This journal is �c the Owner Societies 2009

Publ

ishe

d on

13

Oct

ober

200

9. D

ownl

oade

d by

Sta

te U

nive

rsity

of

New

Yor

k at

Sto

ny B

rook

on

27/1

0/20

14 1

1:40

:31.

View Article Online

ep is defined as being the energy of interaction of a positive

point charge with the nuclei and the electrons of a molecule.35

ep ¼XnucleusA

ZA

RAp�Xbasem

Xfunctionv

Pmv

ZfmðrÞfvðrÞ

rPdr ð2Þ

The first summation is that of nucleus A. The Z terms are the

atomic numbers and RAp are the distances between the nuclei

and the pints charge. The second part of the summation is the

basis functions f. P is the density matrix, and the integrals

reflect Coulombic interactions between the electrons and the

point charge, where rP is the distance between them. Positive

potential values reflect nucleus predominance, while negative

values represent rearrangements of electronic charges and lone

pairs of electrons. The fundamental application of this study is

the analysis of non-covalent interactions. Fig. 7 plotted the

EPM for the complexes I–IV.

The maps show the negative potential sites that include

electronegative atoms and their neighborhoods as well as the

positive potential sites located around the hydrogen atoms.

The main negative center includes the radical C atom, which

should be responsible for the interaction with the active

electron–donor. It is also clear in maps that the Na atom

could be act as the electron-acceptors because its electrostatic

potential is positive (about 0.180 a.u.). Based on above

reasons, the single-electron sodium bond complexes

Y–C� � �Na–H [Y = H3, H3CH2, (H3C)2H and (H3C)3] are

formed with the radical C atom as electron–donor and the Na

atom in Na–H as electron–acceptor, which suggests that the

interactions are mainly electrostatic between fragments Y–C

[Y = H3, H3CH2, (H3C)2H and (H3C)3] and Na–H and also is

in good agreement with the results obtained by r2r in Fig. 5

and the ionic character of C� � �Na bond in Table 1.

3. Conclusions

In the present work, the prediction and characterisation of

novel single-electron sodium bond complexes has been

performed for the first time. The complexes H3C� � �Na–H,

H3CH2C� � �Na–H, (H3C)2HC� � �Na–H, and (H3C)3C� � �Na–H

have been studied at the MP2/6-311++G(d,p), MP2/

6-311++G(2d,2p) and MP2/aug-cc-pVDZ levels. From this

study, the following conclusions can be obtained:

(1) Four single-electron sodium bond complexes exhibit

redshifting sodium characters. An increase of 0.0010–0.0016 nm

of the Na–H bond length upon dimer formation is observed,

and the corresponding Na–H stretching frequencies are lower

by 14.5–25.4 cm�1.

(2) In the four complexes I–IV, the main orbital interactions

are all LP1(C)-LP*1(Na), which is not only different from the

single-electron H-bonds and single-electron halogen bonds,

but also the other typical weak interaction bonds, such as

hydrogen bonds, halogen bonds and sodium bonds.

(3) By comparison with related systems, it is concluded that

the strength of a single-electron bond increases in the order:

hydrogen bond o sodium bond o bromine bond olithium bond.

(4) Investigations show that the non-additivity of the methyl

group has a significant effect on the single-electron sodium

bond of the H3C� � �Na–H complex. With the increasing

number of the methyl group in the radical, the stabilization

energy E(2)ij [LP1(C) - LP*1(Na)] and fractional number of

electrons transferred (DN) decreases while the Na–H length,

BSSE energy, absolute interaction energy EBSSE, the ionic

characters of C� � �Na bond, the proportion of the p orbital

in carbon atom hybridized orbital of the radical and the

topological parameters (r and r2r) increase.(5) By application of the first three hydrogen-bonding

criteria within the AIM formalism to the redshifting

single-electron sodium bond, the analysis discloses that the

single-electron sodium bond complex and hydrogen bond

complex have no essential difference.

(6) In the single-electron sodium bond complex, the

relationships between EBSSE and E(2)ij [LP1(C) - LP*1(Na)],

dC� � �Na and E(2)ij [LP1(C) - LP*1(Na)] are all nonlinear with

the exponential asymptotic function of y = a + bcx.

Acknowledgements

This work was supported by Foundation of Education

Committee of Gansu Province (Grant No. 0708-11) and

‘QingLan’ Talent Engineering Funds of Tianshui Normal

University.

References

1 S. Scheiner, Hydrogen bonding, Oxford University Press,New York, 1997.

2 G. J. Zhao and K. L. Han, J. Phys. Chem. A, 2007, 111, 2469.3 Y. Jiang, A. A. Alcaraz, J. M. Chen, H. Kobayashi, Y. J. Lu andJ. P. Snyder, J. Med. Chem., 2006, 49, 1891.

4 H. Cybulski, E. Tyminska and J. Sadlej, ChemPhysChem, 2006, 7,629.

5 P. Hobza and Z. Havlas, Chem. Rev., 2000, 100, 4253.6 G. J. Zhao and K. L. Han, ChemPhysChem, 2008, 9, 1842.7 Z. R. Li, D. Wu, Z. S. Li, X. R. Huang, F. M. Tao and C. C. Sun,J. Phys. Chem. A, 2001, 105, 1163.

8 X. Y. Hao, Z. R. Li, D. Wu, Y. Wang, Z. S. Li and C. C. Sun,J. Chem. Phys., 2003, 118, 83.

Fig. 7 Calculated electrostatic potentials on the molecular surface for complexes I–IV at MP2/6-311++G(d,p) level.

This journal is �c the Owner Societies 2009 Phys. Chem. Chem. Phys., 2009, 11, 11113–11120 | 11119

Publ

ishe

d on

13

Oct

ober

200

9. D

ownl

oade

d by

Sta

te U

nive

rsity

of

New

Yor

k at

Sto

ny B

rook

on

27/1

0/20

14 1

1:40

:31.

View Article Online

9 Z. F. Li, X. N. Shi, Y. Z. Liu, H. A. Tang and J. Y. Zhang, Chin. J.Chem. Phys., 2009, 22, 303.

10 M. Nsangou, Z. Dhaouadi, N. Jaıdane and Z. B. Lakhdar, J. Mol.Struct. (THEOCHEM), 2007, 819, 142.

11 B. Skwara, W. Bartkowiak, A. Zawada and R. W. Gora, Chem.Phys. Lett., 2007, 436, 116.

12 M. A. Blatchford, P. Raveendran and S. L.Wallen, J. Phys. Chem. A,2003, 107, 10311.

13 T. Kar and S. Scheiner, Int. J. Quantum Chem., 2006, 106, 843.14 Q. Z. Li, X. L. An, F. Luan, W. Z. Li, B. A. Gong, J. B. Cheng and

J. Z. Sun, Int. J. Quantum Chem., 2008, 108, 558.15 M. Igarashia, T. Ishibashia and H. Tachikawa, J. Mol. Struct.

(THEOCHEM), 2002, 594, 61.16 Y. Li, D. Wu, Z. R. Li, W. Chen and C. C. Sun, J. Chem. Phys.,

2006, 125, 84317.17 Q. Z. Li, X. L. An, B. A. Gong and J. B. Cheng, J. Mol. Struct.

(THEOCHEM), 2008, 866, 11.18 A. Romieu, S. Bellon, D. Gaparutto and J. Cadet,Org. Lett., 2000,

2, 1085.19 S. Hix, M. B. Kadiis, R. P. Mason and O. Augusto, Chem. Res.

Toxicol., 2000, 13, 1056.20 B. S. Ault, J. Am. Chem. Soc., 1978, 100, 2426.21 M. D. Pluth, R. G. Bergman and K. N. Raymond, J. Am. Chem.

Soc., 2008, 130, 6362.22 P. I. Nagy and P. W. Erhardt, J. Phys. Chem. A, 2008, 112, 4342.23 S. F. Boys and F. Bernardi, Mol. Phys., 1970, 19, 553.24 (a) R. F. W. Bader, Atom in Molecules—A Quantum Theory,

Oxford University Press, Oxford, 1990; (b) M. J. Frisch,G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb,J. R. Cheeseman, J. A. Montgomery, Jr., T. Vreven,K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar,J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani,N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara,K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima,

Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox,H. P. Hratchian, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo,R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin,R. Cammi, C. Pomelli, J. Ochterski, P. Y. Ayala, K. Morokuma,G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski,S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas,D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman,J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski,B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi,R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng,A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. G. Johnson,W. Chen, M. W. Wong, C. Gonzalez and J. A. Pople, GAUSSIAN03 (Revision E.1), Gaussian, Inc., Wallingford, CT, 2004.

25 X. S. Li, L. Liu and H. B. Schlegel, J. Am. Chem. Soc., 2002, 124,9639.

26 G. R. Desiraju and T. Steiner, The Weak Hydrogen Bond inStructural Chemistry and Biology, Oxford University Press Inc.,New York, 1999.

27 K. Hermansson, J. Phys. Chem. A, 2002, 106, 4695.28 Q. Z. Li, X. L. An, F. Luan, W. Z. Li, B. A. Gong and J. B. Cheng,

J. Phys. Chem. A, 2008, 112, 3985.29 R. G. Parr and R. G. Pearson, J. Am. Chem. Soc., 1983, 105, 7512.30 R. F. W. Bader, Chem. Rev., 1991, 91, 893.31 P. L. A. Popelier, J. Phys. Chem. A, 1998, 102, 1873.32 Rafel Cabot and A. Christopher. Hunter, Chem. Commun., 2009,

2005.33 C. Munoz-Caro, A. Nino, M. L. Sement, J. M. Leal and S. Ibeas,

J. Org. Chem., 2000, 65, 405.34 Chemical Applications of Atomic and Molecular Electrostatic

Potentials, ed. P. Politzer and D. G. Truhlar, Plenum Press,New York, 1981.

35 J. Hehre, A Guide to Molecular Mechanics and QuantumChemical Calculations, Wavefunction, Inc., Irvine, CA, USA,2003, ch. 4 and 19.

11120 | Phys. Chem. Chem. Phys., 2009, 11, 11113–11120 This journal is �c the Owner Societies 2009

Publ

ishe

d on

13

Oct

ober

200

9. D

ownl

oade

d by

Sta

te U

nive

rsity

of

New

Yor

k at

Sto

ny B

rook

on

27/1

0/20

14 1

1:40

:31.

View Article Online