ORI GIN AL PA PER

Electron Density Properties and Interaction: QuantumChemical Topology Investigation on AuRnn

2+(n 5 1–6)

Li Xinying • Cao Xue

Received: 2 December 2013

� Springer Science+Business Media New York 2014

Abstract Quantum chemical calculations of the structures, stabilities and inter-

actions of the AuRnn2?(n = 1–6) series at the CCSD(T) theoretical level are per-

formed. The analyses of binding energies and average binding energies indicate that

the n = 1 and 2 systems are more stable than its neighbors. Topological analysis of

the natural bond orbital, electron density deformation, integrated charge transfer,

bond critical point properties, electron localization function, reduced density gra-

dient analysis are performed to explore the nature of the interaction. The results

show that the present Rn–Au2? interactions belong the covalent type for n = 1 and

2 systems and fall into intermediate interaction type with a pronounced covalent

character for the others.

Keywords Interaction � Electron density property � Covalent � Structure

and stability

Introduction

In recent years great effort has been expended to characterize systems in which

coinage metal (M = Cu, Ag and Au) is bound to a rare gas (RG) [1–15]. Interactions of

coinage metal ions with RG are truly remarkable since it proves the existence of a

stable compound between both types of elements considered in the past to be

archetypes of chemical inertia. Contrary to the suggestion, the chemistry of RG and

coinage metal (Cu, Ag and Au) is nowadays recognized to be broad and rich. The

features are manifestations of relativistic effects that contracts and stabilize the s and

p shells, but expanding and destabilizing the d and f shells. But what is the nature of the

L. Xinying (&) � C. Xue

School of Physics and Electronics, Institute for Computational Materials Science, Henan University,

Kaifeng 475004, People’s Republic of China

e-mail: lxying@henu.edu.cn

123

J Clust Sci

DOI 10.1007/s10876-014-0694-4

bond between coinage metal and RG? Pyykko co-workers [16, 17] suggested that most

of the bonding (BD) interaction is covalent in character and strengthens along the Ar–

Kr–Xe series based on Mulliken and natural bond analysis by performing highly

correlated CCSD(T) calculations. While the interpretation was questioned by Read

and Buckingham [18] who considered higher order multipoles to describe induced

polarization effects on the Ar atom by saying that ‘‘covalency within the RGAu?

species appears to be unproven’’. To animate the dispute, a recent review of Bellert and

Breckenridge suggested [19], using an electrostatic model and parameters, that the

XeAu? system is described by a covalent bond in which the RG acts as electron donor,

while the nature of the bond in other RGAu? ions was left undecided. Recent

investigations show that there is clearly a covalent component in the BD of Xe(Kr/

Ar)–Au? [20, 21]. Belpassi et al. [20] investigated the electron density changes upon

formation of the Ng–Au bond in detail and characterize the typical covalent bond by

the pronounced charge accumulation in the middle of the Ng–Au inter-nuclear region.

In 2001, Walker et al. [22] reported the unexpected experimental and theoretical

determination of stable MArn2? clusters and it clearly shows the pronounced

interaction between Ar and doubly charged coinage metal cations M(II), M2?

(M = Cu, Ag and Au). The results show that the binding energy of the doubly charged

Ar–Au2? is greater than that of the singly charged Ar–Au? [23]. The interactions

between RG and Au? strengthen along the He–Xe series [23], we may infer that the

interaction between RG and Au(II) would also strengthen along the RG series. Radon,

element 86, is the heaviest experimentally known RG, it is expected that Rn can

directly bonded to the doubly charged Au cation, Au(II), with considerable bond

energy, which is in fact generally regarded as the element whose chemistry is most

affected by relativistic effects [24]. Given the large differences between the second

ionization energy of Au (Au??Au2?, 20.51 eV) and that of an Rn atom (11.71 eV),

considerable charge transfer might have been expected. In particular, Rn is the largest

radiation risk in many countries such as Finland [25]. Its diffusion in air, water, and

building materials is related to its size and binding energy, thus understanding its

interaction properties is of fundamental interest. However, the theoretical investiga-

tions including geometrical structures, electronic structures, especially the roles of

interactions of small AuRnn2? clusters are less reported. Understanding of their

interaction mechanism leaves much room for improvement, and detailed theoretical

investigations are desired.

Reported here are the results of the calculations undertaken on the small

AuRnn2?(n = 1–6) series using the coupled cluster method with single and double

excitation and a non-iterative correction for triple excitations (CCSD(T)) [26, 27],

with the aim of not only understanding the behavior of the systems, but also to give

an insight into the nature of the interaction mechanism between Rn atom(s) and

coinage metal cations, Au2?.

Computational Details

Interactions of the RG containing compounds often require high angular momentum

functions for accurate description [28]. For the present clusters, the relativistic

L. Xinying, C. Xue

123

effects are very important, such as gold, has a very rich and unusual chemistry with

varying physicochemical properties due to the strong relativistic effects, which

considerably decrease the size of the valence 6s orbital and lead to the expansion of

the 5d orbital resulting in this kind of unusual behavior. Our previous results

indicate that both electron correlation effects and relativistic effects play an

important role for the coinage metal-RG clusters [29], and accounted for using the

CCSD(T) method and the relativistic pseudo-potentials and the corresponding basis

sets. The 19-valence electron relativistic pseudo-potential and its matching basis

sets (37s33p22d2f1g)/[5s5p4d2f1g] are employed for the Au atom [30]. The

26-valence electron basis sets (24s18p11d2f1g)/[6s6p4d2f1g] are employed for Rn

atom(s) [31].

The calculations were performed with the Gaussian 03 W program [32]. The

basis set superposition error (BSSE) is corrected by using the counterpoise

procedure of Boys and Bernardi [33]. To explore the nature of the interactions, the

natural bond orbital (NBO) analysis is used [34, 35]. The NBO analysis is

performed with the NBO program as implemented in the Gaussian 03W.

Results and Discussion

The equilibrium structural parameters, binding energies and average binding

energies calculated at the CCSD(T) theoretical level are given in Table 1. The

nature of the optimized structures with lowest total energy as potential energy

minima have been established in all cases by verifying that all the corresponding

frequencies are positive. The binding energy is referenced to the separated atom

limit consisting of the ground state Rn atoms and Au(II) s0d9 ground states. The

global minimum energy structures of AuRnn2?(n = 1–6) are shown in Fig. 1.

Structures and Stabilities

For n = 1 system, the CCSD(T) method obtained the equilibrium Rn–Au2?

distance of 269.4 pm, and binding energies of 5.2379 eV, respectively. The similar

bond distance and greatly enhanced stability compared to the singly charged Rn–Au?

calculated at the same theoretical level with the same basis sets (265.4 pm; 1.3945 eV)

were obtained [36]. For n = 2 system, any attempt starting from bent triangular

geometries with C2v symmetry after the geometry optimization ends up as the linear

configuration with D?h symmetry. We note that the Rn–Au2? distance, 248.1 pm, is

shortened considerably compared to that of the present AuRn2? system (269.4 pm)

and the previous reported value of the singly charged AuRn2? system (264.5 pm) [36].

For n = 3 system, all pyramidal, three-dimensional initial geometries, after the

geometry optimization, end up planar structures with C2v symmetry. While the

symmetry of the singly charged AuRn3? is D3h [36], and the structural difference may

be resulted from the electronic structure of the present doubly charged Au2? (6s05d9)

and the previous singly charged Au? (6s05d10). The present structures accords with

Walker et al. [22] results of MArn2? of D?h(Linear), C2v(Planar), D4h(Planar),

Electron Density Properties and Interaction

123

C4v(Pentahedron) and D4h(Octahedron) for n = 2–6, respectively, calculated at MP4/

LANL2DZ theoretical level, while we obtained the greatly enhanced stabilities.

One can see from Fig. 1 that the AuRnn2? system can be formed by adding one Rn

atom to the stable AuRnn-12? structures without obvious changes of its structural

parameters. For n = 2, 4, and 6 systems, their D?h(Linear), D4h(Planar), and

D4h(Octahedron) symmetries arranged Au and Rn atoms in linear coordination.

While for the n = 3 and n = 5 systems, they can be formed by adding one Rn atom

to the stable n = 2 and n = 4 structures, respectively; without obvious changes of

its structural parameters thus the Rn atoms and Au are also approximately arranged

in linear coordination. It is well known that coinage metal in the valence state ?I,

M?, very much prefer linear coordination [37]. The present results indicate that the

Au(II) valence state also prefer linear coordination.

Table 1 Structures and stabilities performed at CCSD(T) theoretical level

n R1 (pm) R2 (pm) A (�) Eb (eV) Eb-ave (eV)

1 269.4 5.2379 2.6190

2 248.1 A312 = 180.0 7.6466 2.5489

3 289.2 263.2 A314 = 144.7 8.2329 2.0582

4 276.4 A213 = 90.0 11.4294 2.2859

5 306.5 278.2 A214 = 168.4 11.9555 1.9926

6 323.4 278.5 A214 = 180.0 12.3970 1.7710

Fig. 1 CCSD(T) structures for AuRnn2? clusters. Details of structures are given in Table 1

L. Xinying, C. Xue

123

In cluster physics, the binding energy (Eb) and average binding energy (Eb-ave)

are sensitive quantities that reflect the relative stability of clusters. Here they are

defined as:

EbðnÞ ¼ EðAu2þÞ þ nEðRnÞ � EðAuRn2þn Þ ð1Þ

Eb�aveðnÞ ¼ ½EðAu2þÞ þ nEðRnÞ � EðAuRn2þn Þ�=nþ 1 ð2Þ

where E(….) is the total energy of the corresponding system.

The results collected in Table 1 clearly show that the binding energies increase

monotonically as the size of n increase, which means that the clusters can

continuously gain energy during the growth process. One can see from Fig. 1 that

the Au atom was found to be located between the Rn atoms, thus the number of Au–Rn

interaction increases monotonically as the size of n increase while the Rn–Rn

interaction does not have the same behavior; thus it results in the monotonically

increase of binding energies and irregular variable trend of the average binding

energies. The results collected in Table 1 clearly shows the enhanced stabilities of

the n = 1 and 2 systems.

NBO Analysis

To understand the interaction mechanism, we performed the NBO analysis of the

stable species, AuRn2? and AuRn22?. The NBO’s are a set of localized orbitals that

fulfill the requirements of orthonormality and maximum occupancy on the grounds

of the calculated MO’s after diagonalizing the corresponding electron density

matrices. The transformation to NBO produces both highly occupied and nearly

empty localized orbital. The former can be classified as lone pairs (LP) or BD pairs.

For AuRn2? system, the NBO results show that there are one alpha r-bond and

one beta r-bond. Each BD (rAu–Rn) can be written in terms of two directed valence

hybrids, hAu and hRn, on the bonded centers Au and Rn, respectively.

The alpha BD orbital can be expressed as

rAu�Rn ¼ 0:9329hRn þ 0:3602hAu ð3aÞThe hRn and hAu can be described as linear combination of the natural atomic

orbital on its center as follows:

hAu ¼ �0:9249ð6sÞ � 0:3586ð6pzÞ ð3bÞhRn ¼ �0:2241ð6sÞ þ 0:9706ð6pzÞ ð3cÞ

For the beta BD:

rAu�Rn ¼ 0:7164hRn þ 0:6977hAu ð4aÞhAu ¼ �0:4412ð6sÞ � 0:1663ð6pzÞ � 0:8767ð5dz2Þ ð4bÞ

hRn ¼ 0:9932ð6pzÞ ð4cÞFor alpha orbital, the Au–Rn r-BD is resulted from the overlap of a 6s6pz

(mainly 6s) hybrid on Au and 6s6pz (mainly 6pz) hybrid on Rn atom; it can be seen

from the natural atomic orbital occupancies, 6s0.1226pz0.018 for Au and 6s0.9916pz

0.870

Electron Density Properties and Interaction

123

for Rn. For beta orbital, the 5d orbital plays an important role. The interaction

comes from the overlap of the 6s6p5d (mainly 5dz2) hybrid on Au and the 6pz orbital

of Rn atom. The corresponding orbital occupancies are 6s0.0956pz0.0155dz2

0.375 for Au

and 6pz0.506 for Rn.

For AuRn22? system, NBO results show that there are one alpha r-bond, one

beta r-bond and one beta p-bond between Au and each Rn atom. Both r-bonds are

also resulted from the overlap of 6s6p (alpha)/6s6p5d (beta) hybrid on Au and 6s6pz

hybrid on Rn atom. While for the p-bond, it can be expressed as:

pAu�Rn ¼ 0:9132hRn þ 0:4076hAu ð5aÞhAu ¼ 0:3241ð6pxÞ þ 0:6121ð6pyÞ þ 0:3285ð5dxzÞ þ 0:6204ð5dyzÞ ð5bÞ

hRn ¼ 0:4659ð6pxÞ þ 0:8797ð6pyÞ ð5cÞThe present p-bond is resulted from the overlap of a 6px6py5dxz5dyz hybrid on Au

and 6px6py hybrid on Rn atom; it can be seen from the natural atomic orbital

occupancies, 6px0.0286py

0.0405dxz0.8535dyz

0.481 for Au and 6px0.9456py

0.840 for Rn. For a

p-bond, the orbital share a nodal plane which passes through both of the involved

nuclei (Au and Rn). Since the overlap of the orbital to form a p-bond is not as great

as the overlap obtained from r-bond (which is directed along the bond axis.

Figure 2), p-bond in general is weaker than the r-bond. A p-bond, along with an

r-bond forms a ‘‘double’’ bond. We note that the p-bond could not be found for

other systems (n = 1, 3–6). The existence of the p-bonds greatly shortens the

Rn–Au distance, therefore the n = 2 system has the smallest Rn–Au distance

among the present n = 1–6 systems (Table 1).

As mentioned above, considerable charge transfer from Rn atom(s) to Au might

have been expected due to the great difference of ionization energies between Au

and Rn. The natural population analysis (NPA) and occupancies of valence orbital

of Au and Rn clearly show the charge transfer mechanism. From Fig. 3 we note

that, for Rn atom, the occupancy changes of the 6s orbital, 2.0-Occ6s, are \0.1 for

all the systems while considerable changes can be found for the 6p orbital. It

indicates that the electron transferred mainly from the 6p orbital of Rn atom to the

Au atom. For Au atom, the NPA values (the 2.0-NPA value gives the amount of

electrons transferred from Rn atom(s) to Au atom) decreased monotonically with

cluster size n and it indicates that more and more electrons transferred to the Au

atom. Note that for n = 5 and 6 systems, the NPAs of Au are negative; it indicates

that more than 2.0 electrons transferred to Au atom. The occupancies of the 6s (Au)

increase rapidly from n = 1–2 and then reach a plateau for n = 2–6 systems. The

occupancies of 5d orbital decrease from n = 1–3 and then increase to a plateau for

n = 4–6 systems. While for 6p orbital, the occupancy increase monotonically with

the cluster size n and it indicates that more and more charge transferred to the 6p

orbital and thus they play an important role in the interaction. It should be pointed

out that the small values of occupancy changes of the 5d orbital of Au for n = 2 and

3 systems do not indicate that the 5d orbital play the trivial role. For example for

n = 2 system, the occupancy changes of the 5d orbital, 0.24, is very small, while the

NBO analyses show that the 5dz2 orbital is involved in the r-bond and the 5dxz5dyz

L. Xinying, C. Xue

123

orbital is involved in the p-bond, and they play very important role in the

interaction, as can be seen from the occupancy, 5d1:999xy 5d1:851

xz 5d1:479yz 5d1:999

x2�y2 5d1:918z2 .

Electron Density Properties

Topological analysis of the electron density deformation, integrated charge transfer,

bond critical point (BCP) properties, electron localization function (ELF) and

reduced density gradient (RDG) are performed on AuRn2? and AuRn22? to explore

the nature of interaction. The results were calculated by Multiwfn and plotted by

VMD program [38, 39].

Electron Density Deformation Analysis

We look at a graphical representation of the electron density deformation upon the

formation of the interactions between the fragments (Au2? and Rn) to take a

qualitative and insightful approach to understand the nature of the Rn–Au2?

interaction. The electron density deformation function, Dq(r), is defined as the

difference between the total electron density of the system and the promolecule

Fig. 2 The r-bond and p-bondof AuRn2

2?. The isosurfacevalue is 0.05

Fig. 3 Occupancies of valence orbital and NPA of Au and Rn. For Rn atom, the value of occupancychanges of 6s and 6p are, 2.0-Occ6s and 6.0-Occ6p. The occupancy change of 5d of Au: Occ5d-9.0. Forn = 3 system, three Rn atoms are arranged in different position and labeled 2, 3 and 4 (Fig. 1), they havedifferent NPAs and orbital occupancies

Electron Density Properties and Interaction

123

density of the non-interacting fragments (in the same positions); this function is

regarded as a deformation density associated with the redistribution of the electron

charge when the system forms from the constituent fragments [40]. In Fig. 4, the

contours of the electron density difference between the complexes and the non-

interacting fragments (in the same positions) for AuRn2? and AuRn22? are plotted,

the blue contours refer to positive differences (density accumulation) and the red

contours are negative ones (density depletion). Obvious changes of the electron

density distribution can be seen in Fig. 4, there is electron accumulation in the

Rn–Au2? interaction region, and it enhances the stability of the species.

To make the picture of the BD which emerges from the density difference plots

more quantitative, we plot the curve representing the partial integral of the electron

density difference Dq along the inter-nuclear (z) axis, integrated charge transfer

Q(z), and it can measures the actual electronic charge fluctuation with respect to the

isolated fragments, as it moves along the inter-nuclear axis (from -? to z) and is

defined as follows:

QðzÞ ¼Z�1

�1

dx

Z�1

�1

dy

Z�1

�1

Dqðx; y; zÞdz ð6Þ

Imagining a point, z = z0, on the inter-nuclear z axis to identify a perpendicular

plane passing through that point, the corresponding value of Q(z0) measures the

amount of charge has moved from the right side to the left side of the plane, with

respect to the situation in the non-interacting fragments. Therefore the negative

Fig. 4 Integrated charge transfers as a function of the inter-nuclear distance and contour plots of theelectron density difference of AuRn2? and AuRn2

2?. The Q values are calculated with the followinglower and upper limit: x = [-6, 6], y = [-6, 6], z = [-6, 6]. AuRn2?: Au (0, 0, 0), Rn (0, 0, -2.694);AuRn2

2?: Au (0, 0, 0), Rn (0, 0, -2.481), Rn (0, 0, 2.481). Contour levels are 2m 9 10n a.u., with m = 1,2, 3 for every n of set n = -3, -2, -1, 0

L. Xinying, C. Xue

123

value indicates charge from left to right. The difference between two Q values,

Q(z1) and Q(z2), shows the net electron influx into the region delimited by the two

planes (z = z1 and z = z2). The regions of the Q(z) curve with a negative slope

correspond clearly to zones of charge depletion (red contours), while charge

accumulates where Q picks up (blue contours). The first thing we note that the Q is

negative everywhere in the left molecular region (z \ 0), indicating that, at each

point, there has been a net shift of charge toward the Au atom. The plots make clear

the pronounced electron depletion around the Rn site and charge transfer to the

Rn–Au inter-nuclear region.

For AuRn2? (Black curve in Fig. 4), from z = -6 to -2.69 A, left side of the Rn

atom, the Q curve has the negative slope and it indicates the net electron charge loss.

The loss takes place until 1.64 A, z = -1.64, from the Au site, and there is an

inversion and charge starts to re-accumulate rapidly until about 0.51 A, z = -0.51,

when part of the lost charge is recovered, it (from z = -1.64 to -0.51) corresponds

to the blue contour regions, the interaction region of AuRn2?. We can estimate the

amount of charge accumulate to the interaction inter-nuclear region, Q(-0.51) to

Q(-1.64) = 0.297 a.u., and it enhances the stability and appears to be an indicator

of the interaction strength. In addition, we note that around the Au nuclear a narrow

zone (about z = -0.25 to 0.25) where Q picks up rapidly and becomes zero,

implying a corresponding considerable net electron accumulation on this zone. It

indicates that the charge is transferred not only to the interaction inter-nuclear

region but also to the region surrounding the Au atom itself, which may be

interpreted as the partial filling of the empty valence 6s6p orbital and accords

qualitatively with the occupancies analysis results (see ‘‘NBO Analysis’’ section).

For AuRn22? (Green curve in Fig. 4), we can estimate that about 0.301 a.u.

[Q(-0.524) to Q(-1.679) = 0.301] charges accumulate to the interaction inter-

nuclear region. We note that, around the Au nuclear, the Q picks up more rapidly

than that of n = 1 system, it indicate that the occupancies of 6s6p for n = 2 system

are greater than those of n = 1 system, as can be seen from Fig. 3.

There are two r-bonds (one alpha bond and one beta bond) in Au–Rn interaction

region for n = 1 system, while there are three bonds (one alpha r-bond, one beta

r-bond and one p-bond) in the region for n = 2 system. About 0.30 electrons

transferred to the interaction region for both systems. One can infer that the amount

of electrons, engaged in the r-bonds, in n = 2 system are smaller than those of

n = 1 system; namely, the r-bonds in n = 2 system are weaker than those of n = 1

system. In general, the p-bond is weaker than the r-bond. Therefore both systems

have similar stability according to the average binding energies analysis.

The main feature of the interaction is a pronounced charge accumulation in the

middle of the region between the Rn and Au nuclei, this feature is a clear indication

of the formation of a covalent bond.

Atoms in Molecules (AIM) Analysis

According to the AIM theory of Bader [41], the chemical BD can be characterized

by the existence of a (3, -1) type of BCP and the corresponding bond path. Its

nature is revealed by descriptors at the BCP such as the electron density q(r) and

Electron Density Properties and Interaction

123

Laplacian r2q, which is the sum of three curvatures of the electron density Hessian

matrix, k1, k2, and k3. Chemical BDs can be divided into two groups according to

the Laplacians: shared interactions and closed shell interactions, which correspond

to the negative and positive Laplacian. It is also suggested that local energy density

E(r) at the BCP, the sum of the kinetic energy density and the potential energy

density, to distinguish between covalency and ionicity [42]. Chemical BD

characterized by positive Laplacian r2q(r) and negative E(r) is referred as

‘‘intermediate type’’ [43]. Note that, for n = 1 and 2 systems, the Laplacian values,

0.046 and 0.054, are very small; moreover, the values of energy densities and k2 are

more negative than others, taking the binding energies into consideration, the

interaction can be classfied in the covalent type. While interactions in n = 3–6

systems (negative E(r) and positive Laplacian value, Table 2) all fall into the

intermediate type.

ELF Analysis

The ELF introduced by Becke and Edgecombe has been widely used as a

convenient descriptor of the chemical bond in a great variety of systems [44]. It is

conveniently mapped on the interval (0, 1) and provides a rigorous basis for the

analysis of the interaction. A region of the space with a high ELF value is

interpreted as a region where is more probable to find an electron or a pair

of localized electrons. ELF = 0 correspond to a delocalized system, 1 to a

completely localized situation, and 0.5 is the value one should obtain for the

homogenous electron gas. The concept of basin (domain) was first introduced by

Bader in AIM theory [41], and it was transplanted to the analysis of ELF by Savin

and Silvi [45] and Savin et al. [46]. Topological analysis of the ELF provides a

partition of the molecular space into basins of attractors which have a clear chemical

signification: (i) core basins are located around nuclei, and (ii) valence basins are

characterized by their synaptic orders: monosynaptic basins represent the LP,

whereas disynaptic basins belong to the covalent interaction.

The reduction of the localization domains obtained by the ELF approach (Fig. 5)

yields important information about electron localization processes. It was previously

Table 2 BCP properties at CCSD(T) level

n BCP (A–B) q (10-2) k2 (10-2) Lap(10-1) E (10-2)

1 (Au–Rn2) 7.8220 -6.7815 0.4584 -2.7014

2 (Au–Rn2, 3) 10.1294 -6.7503 0.5406 -4.3102

3 (Au–Rn2) 4.9809 -3.4451 0.9605 -1.0418

(Au–Rn3, 4) 6.9468 -6.2500 1.4062 -2.7075

4 (Au–Rn2, 3, 4, 5) 6.3717 -4.8657 0.9344 -1.7669

5 (Au–Rn6) 3.5660 -2.3126 0.7979 -0.4779

(Au–Rn2, 3, 4, 5) 6.1454 -4.6719 0.9385 -1.6326

6 (Au–Rn6, 7) 2.6666 -1.6288 0.6379 -0.2172

(Au–Rn2, 3, 4, 5) 6.1062 -4.6386 0.9366 -1.6127

L. Xinying, C. Xue

123

adopted as a tool for the understanding the chemical BD [47]. For the ELF value

equal to 0.5—corresponding to a totally delocalized electron density—there are

only distinguishable valence basins of the Rn atom(s) and one pseudo-valence

domain of gold V(Au1). For AuRn2?, a further bifurcation occurs at ELF = 0.58,

giving rise to BD point attractors disynaptic domain V(Rn2, Au1). The ELF = 0.80

map clearly shows the non-BD ring attractor domain V(Rn2) and core basin C(Rn2).

While for AuRn22?, four attractors around the Rn2 atom (ELF1 = ELF2 = 0.717,

ELF3 = ELF4 = 0.823) were found, therefore there are four valence domains,

Vi(Rn2), i = 1–4, as shown in Fig. 5, and volumes of the V1(Rn2) and V2(Rn2) are

very small. The Rn3 atom has the same trend. According to the NBO analysis, the

disynaptic basins V(Rn2, Au1) clearly corresponds to the r-bond of n = 1 system,

while the V1(Rn2) and V2(Rn2) correspond to the p-bond for AuRn22? system.

After the basins were generated, some analyses based on the basins can be

conducted to extract information of chemical interest. One can integrate electron

density in the basins to acquire electron population numbers in the basins

�NXi¼Z

Xi

qðrÞdv ð7Þ

and investigate the localization index (LI) in the basins and delocalization index

(DI) between basins. LI measures how many electrons are localized in a basin in

average, while DI is a quantitative measure of the number of electrons delocalized

(or say shared) between two basins. The results are given in Table 3. The near-grid

method [48] is utilized in calculation by Multiwfn program [38].

For n = 1 system, AuRn2?, the C(Rn2) and V(Au1) domains contain 17.466 and

17.408 electrons, which is in line with the fact that the Rn atom has 18 (5s25p65d10)

electrons in its pseudo-core and Au atom has 17.636 (the NPA of Au is 1.364)

electrons. Also, the average population number in V(Au1, Rn2) is 0.708,

approximately reflecting that in average there is 0.708 electron shared between

Au and Rn and thus it enhanced the stability. Between C(Rn2) and V(Au1), namely

DI(C(Rn2), V(Au1)), 0.076, the value is trivial, reflecting that the electron

delocalization of atomic core of Rn atom is rather difficult, which is in accord with

the NBO analysis that the interaction come mainly from the 6s6p hybrid of Rn atom.

DI(C(Rn2), V(Rn2)) and DI(C(Rn2), V(Au1, Rn2)) are 3.936 and 0.215,

representing that the electrons in C(Rn2) have more pronounced probability to

exchange with the ones in V(Rn2) compared to V(Au1, Rn2) since the C(Rn2) was

surrounded by V(Rn2). The DI(V(Rn2), V(Au1)) is 0.628, it quantitative indicate

that there are 0.628 electrons delocalized (shared) between the two basins and

accords with the average population in V(Au1, Rn2). LI value of V(Rn2) and

V(Rn2, Au1), 4.880 and 0.135, is less than their average electron population

numbers distinctly, 7.416 and 0.708, revealing that the electrons in these basins do

not express very strong localization character.

For n = 2 system, AuRn22?, between Rn and Au region, one can not find any

disynaptic basins. While the small domain V1,2(Rn2) give evidence of the p-bond

and its average population number, 0.490, is less than that of V(Au1, Rn2) in n = 1

system. We note that the average populations of the V(Au1), 17.408 and 17.456 for

Electron Density Properties and Interaction

123

n = 1 and n = 2 systems, are similar; while the LI of the latter, 16.276, is smaller

than that of the former, 16.840. It indicates that the valence electrons of Au of n = 2

system express stronger localization character. NBO analysis shows that for n = 2

Fig. 5 The reduction of the localization domains in AuRn2? and AuRn22?

Table 3 The DI and LI of AuRn2? and AuRn22?

DI V(Au1) V(Rn2) V(Au1,Rn2) C(Rn2) LI

AuRn2? V(Au1) 1.128 0.628 0.424 0.076 16.840 (17.408)

V(Rn2) 0.628 5.071 0.507 3.936 4.880 (7.416)

V(Au1, Rn2) 0.424 0.507 1.146 0.215 0.135 (0.708)

C(Rn2) 0.076 3.936 0.215 4.228 15.352 (17.466)aAuRn2

2? DI V(Au1) V3(Rn2) V1(Rn2) C(Rn2) LI

V(Au1) 2.359 0.374 0.179 0.077 16.276 (17.456)

V3(Rn2) 0.374 3.934 0.209 1.902 1.707 (3.674)

V1(Rn2) 0.179 0.209 0.836 0.185 0.072 (0.490)

C(Rn2) 0.077 1.902 0.185 4.261 15.308 (17.439)

The electron population numbers are given in parenthesesa For AuRn2

2?, there are 11 basins, V(Au1), four valence basins and one core basin for each Rn atom.

The DI should be an 11 9 11 matrix. Here the DI of the V(Au1), V3(Rn2), V1(Rn2) and C(Rn2) are

given. The diagonal terms are the sums of the elements in corresponding row/column of the 11 9 11

matrix

L. Xinying, C. Xue

123

system the 5dz2 and 5dxz5dyz orbitals are involved in the r-bonds and p-bonds,

respectively; while for n = 1 system only the 5dz2 orbital engages in the interaction.

RDG Analysis

Yang and co-workers [49] developed an approach to detect the weak interactions in

real space based on the electron density and its derivatives. The RDG is a

fundamental dimensionless quantity coming from the density and its first derivative

RDG ¼ 1=ð2ð3p2Þ1=3Þ rqj j=q4=3 ð8ÞThe weak interactions can be isolated as regions with low electron density and

low RDG value. To understand the interaction in more detail, the sign of k2 (the

second eigenvalue of the electron density Hessian matrix) is utilized to distinguish

the bonded (k2 \ 0) from nonbonded (k2 [ 0) interactions. The plot of the RDG

versus the electron density q multiplied by the sign of k2 can analyze and visualize a

wide range of interactions types. As can be seen in Fig. 6, spikes are found in the

low-density, low-gradient region, indicative of weak interactions in the system and

the electron density value at the RDG versus sign(k2)q peaks (electron density of

BCP) itself provides the information about the strength of interaction. Large,

negative values of sign(k2)q are indicative of stronger attractive interactions (spikes

in the left part in Fig. 6), while if it is large and positive, the interaction is repulsion

(spikes in the right part) [49]. RDG analysis can discriminate between different

types of interactions: (1) very low density values (i.e., q\ 0.005 a.u.) generally

map to weaker dispersion interactions, (2) slightly higher density values (i.e.,

0.005 \ q\ 0.05 a.u.) map to stronger noncovalent interactions.

Fig. 6 Plots of the RDG versus sign(k2)q and RDG isosurface (RDG = 0.3 and 0.5) for AuRn2? andAuRn2

2?

Electron Density Properties and Interaction

123

From Fig. 6 we find that the RDG = 0.3 line crosses only the attractive

interaction spikes while the RDG = 0.5 line crosses not only the attractive but also

the repulsive spikes. In the first case (RDG = 0.3 isosurface), the low-density, low-

gradient region corresponds to the region between Au and Rn atom(s), indicative the

interaction region, the blue region clear shows the stronger attractive interaction

between Au and Rn. The RDG = 0.5 isosurface clearly shows the steric repulsion

(Red annulus).

For n = 1 and 2 systems, the integrated charge transfer and considerable qBCP

show clear evidence of covalent character, the Laplacian values are trivial, and the

interactions belong to the covalent type. In particular, the existence of the disynaptic

basin V(Rn2, Au1) enhances the covalent character. For n = 3–6 systems, the

positive Laplacian values of BCP do not reveal any strong covalent interactions, and

it falls into the intermediate type with the negative energy density. While electron

density deformation Dq(r), integrated charge transfer Q and NPA analysis clearly

show that there are considerable electron transferred from Rn atom(s) to the

interaction inter-nuclear region and around Au atom, taking the binding energy and

RDG analysis into account; it suggests that the interactions between the Au and Rn

of n = 3–6 systems fall into intermediate interaction type with a pronounced

covalent character.

Conclusion

Investigations of the AuRnn2? (n = 1–6) series at the CCSD(T) theoretical level

with extended basis sets provide reliable structures and stabilities as well as insights

into its nature of interaction mechanism. The Au2? valence states prefer linear

coordination and the stable structures are of D?h, C2v, D4h, C4v and D4h for

n = 2–6, respectively. The average binding energies clearly show that the n = 1

and 2 systems are more stable than its neighbors.

The NBO analysis shows the r-bond in all the systems, and the p-bonds were

found in the n = 2 system. Topological analysis on electron density properties

(Laplacian values, energy density electron density deformation analysis, integrated

charge transfer Q, BCP properties, ELF, RDG) clearly show that the interactions of

n = 1 and 2 systems belong to the covalent type and those of n = 3–6 systems

prefer the intermediate type with pronounced covalent character.

Acknowledgments Supports from the Program for Innovative Research Team (in Science and

Technology) in University of Henan Province (No. 13IRTSTHN017) and Projects for Youth Key Teacher

by Henan Province (No. 2011GGJS-029) are gratefully acknowledged.

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