Multiwfn -- A Multifunctional Wavefunction Analyzer
Programmed by Tian Lu (School of Chemical and Biological Engineering, University of Science and Technology Beijing, China)
Bug reporting, any question or recommend please contact: Sobereva@sina.com
Download link of Version 2.4 (2012-May-14)
Software manual (including tutorials in Chapter 4): Manual_2.4.pdf
Note: For beginners, it is strongly suggest to use Windows version. A few functions of Linux version are limited, and users may need to install some dynamical libraries by theirself in order to run Linux version.
Source code for Windows (Including all files needed by compiling under Intel Visual Fortran 12.0.0 or CVF6.5)
Multiwfn2.4_src_win.rarSource code for Linux (Including all files needed by compiling under Intel Fortran compiler 12.1.0)
Multiwfn2.4_src_linux.zipTo download older version, click "Downloads" tab and select corresponding version at righthand side.
An introductory paper on Multiwfn 2.1.2, see J. Comp. Chem., 33 ,580 http://onlinelibrary.wiley.com/doi/10.1002/jcc.22885/abstract
Recent update history
Version 2.4 (First release: 2012-May-6, latest revision time: 2012-May-14)
*Adaptive natural density partitioning (AdNDP) analysis is supported as main function 14, this function is quite useful for analyzing multi-center orbitals. See Section 3.17 and 4.14 of the manual.
*Isosurfaces can be drawn using transparent style (Linux version doesn't support this feature).
*Source function defined by Bader is supported.
*DMol3 .grd file is supported.
*Fixed some small bugs.
*2012-May-14: Fixed a fatal bug in plotting ESP on a plane when basis-sets containing diffuse functions are used.
Version 2.3.3 (2012-Apr-9)
*Orbital occupancy-perturbed Mayer bond order (namely decomposing Mayer bond-order to orbital contributions) is supported as subfunction 6 in main function 9. Please consult Section 3.11.5 and 4.9.1 of the manual.
*Transition density matrix outputted by Gaussian can be plotted by function 12 in main function 100, this is useful for analyzing spatial span of electron excitation. Please consult Section 3.100.12 and Section 4.100.5 of the manual.
*Two orbital isosurfaces can be shown simultaneously by main function 0, contour map of orbital wavefunctions of two orbitals can be drawn simultaneously by main function 4. Please consult Section 4.0.2 and 4.4.5 of the manual. This update is useful for visually analyzing orbital overlapping of NBOs.
*Isosurfaces can be plotted by more styles now, currently supported styles include: solid face, mesh, points and solid face+mesh.
Version 2.3.2 (2012-Mar-27)
*Orbital spin-type recorded in "$MOSPIN $END" field at the end of the wfn file outputted by Molden2aim 2.0.5 is supported
*Fully compatible with the g GTFs in the wfn file outputted by Molden2AIM
*Some bugs in quantitative molecular surface analysis module are fixed, internal charge separation calculated by previous version is found to be incorrect
*Accuracy of calculation of electrostatic potential is improved
*A contour line corresponding to vdW surface (electron density=0.001) can be plotted on contour/gradient line map or vector field map by choosing option 15 in post-process stage
*Fragments can be defined in bond order analysis module by option -1. If two fragments have been defined, when performing Mayer/Wiberg, Mulliken bond order analysis, the total bond order between atoms in fragment 1 and 2 will be printed together
Version 2.3.1 (2012-Mar-6)
*Analysis of orbital composition by natural atomic orbital(NAO) approach is supported as subfunction 8 of main funcition 8.
*Critical points and paths can be exported as pdb file, so that they can be conveniently viewed by external visualization softwares such as VMD
*Fixed a fatal problem in the ELF/LOL calculation on post-HF wavefunctions
*Orbital spin-type can be set customly by option 27 in in main function 6
*Many trivial improvements
*Many trivial bugs are fixed
Introduction
Multiwfn is a free, open-source, user-friendly, powerful and flexible program, aims for general wavefunction analysis, current version is running on Windows (32/64bit Windows XP/Vista/7) and 64bit Linux platform. The latest version can be downloaded at Multiwfn website http://multiwfn.codeplex.com without registering. Multiwfn accepts several kinds of files as wavefunction input: .wfn/.wfx (Conventional / Extended PROAIM wavefunction file), .31~.40 (NBO plot file), .fch (Gaussian formatted check file). Moreover, .pdb (Protein data bank format) and .chg (private format) are supported for the functions which only require atom coordinates.
- Special points of Multiwfn
- (1) Very user-friendly. Multiwfn is designed as an interactive program, prompts shown in each step clearly instructs users what need to do next, Multiwfn also never print obscure messages, hence there is no any barrier even for beginners. Besides, there are more than twenty practical examples in the manual, which would be very helpful for new users.
- (2) No third-part plotting softwares are required. A high-level graphical library DISLIN is invoked internally and automatically by Multiwfn for visualizing results, most of plotting parameters are adjustable by users in an interactive interface. Thus the procedure of wavefunction analysis is remarkably simplified, especially for studying distribution of real space function.
- (3) High efficiency. The code of Multiwfn is substantially optimized. Most parts are parallelized by OpenMP technology. For time-consuming tasks, the efficiency of Multiwfn exceeds analogous programs significantly. Meanwhile, the memory requirement is very low.
- Main functions of Multiwfn
- 1) Showing molecular structure and previewing orbitals (MO, NBO, natural orbital, etc.).
- 2) Outputting all supported properties at a point.
- 3) Outputting properties in a line and plotting curve map.
- 4) Outputting properties in a plane and plotting graph. Supported graph types including filled-color map, contour map, relief map (with/without projection), gradient map, vector field map.
- 5) Outputting properties in a spatial scope, data can be exported to Gaussian-type grid file (.cub) and can be viewed in terms of isosurface. Setting up grid is very flexible and convenient.
- 6) For property calculation in one, two and three dimension, the operations between data generated from multiple files can be defined.
- 7) Promolecule and deformation properties of all built-in functions are supported.
- 8) User can decide the wavefunction informations belong to which atoms and which orbitals will be used for subsequent analysis, uninteresting parts can be discarded.
- 9) Topology analysis for electron density (AIM analysis), ELF/LOL, laplacian etc. Critical points and gradient paths can be searched and visualized in terms of 3D or plane graph. Interbasin surfaces can be drawn.
- 10) Checking and modifying wavefunction. Including assigning, exchanging, translating & duplicating, etc.
- 11) Population analysis. Hirshfeld, VDD (Voronoi deformation density), Mulliken, Löwdin, Modified MPA (SCPA, Stout & Politzer, Bickelhaupt), Becke, ADCH (Atomic dipole moment corrected Hirshfeld), CHELPG and Merz-Kollmann methods are supported.
- 12) Orbital composition analysis. Mulliken, Stout & Politzer, Ros & Schuit (SCPA), Hirshfeld and natural atomic orbital (NAO) methods are supported to partition orbital composition.
- 13) Bond order analysis. Mayer bond order, multi-center bond order (up to 6-centers), Wiberg bond order in Löwdin orthogonalized basis and Mulliken bond order are supported. Mayer and Mulliken bond order can be decomposed to orbital contributions.
- 14) Plotting Total/Partial/Overlap population density-of-states (DOS).
- 15) Plotting IR/Raman/UV-Vis spectrum, parameters (broadening function, FWHM, etc.) can be controlled by user.
- 16) Quantitative analysis of molecular surface. Surface properties such as surface area, average value and std. of mapped functions can be computed, local minima and maxima can be located.
- 17) Processing Gaussian-type cube files. User can perform mathematical operations on grid data, set value in certain range, extract data in specified plane, etc.
- 18) Adaptive natural density partitioning (AdNDP) analysis.
- 19) Many utilities involved in quantum chemistry analyses: Plotting scatter map for two functions in specific spatial scope, integrating a real space function in whole space by Becke method, evaluating overlap integral between alpha and beta orbital, monitoring SCF convergence process, generating Gaussian input file with initial guess from converged wavefunction or multiple fragment wavefunctions, calculate van der Waals volume, analyzing charge-transfer in electron transition, etc.
- The real space functions supported by Multiwfn
- 1 Electron density
- 2 Gradient norm of electron density
- 3 Laplacian of electron density
- 4 Value of orbital wavefunction
- 5 Electron Spin density
- 6 Hamiltonian kinetic K(r)
- 7 Lagrangian kinetic G(r)
- 8 Electrostatic potential from nuclear / atomic charges
- 9 Electron Localization Function (ELF) defined by Becke and the one defined by Tsirelson
- 10 Localized orbital locator (LOL) defined by Becke and the one defined by Tsirelson
- 11 Local information entropy
- 12 Total electrostatic potential (ESP)
- 13 Reduced density gradient (RDG)
- 14 Reduced density gradient with promolecular approximation
- 15 Sign(lambda2)*rho (The product of the sign of the second largest eigenvalue of electron density Hessian matrix and electron density)
- 16 Sign(lambda2)*rho with promolecular approximation
- 17 Fermi hole / correlation factor function (Only available for single determinant wavefunction)
- 18 Average local ionization energy
- 19 Source function
- 20 Other functions, such as potential energy density and electron energy density
Multiwfn also reserves a custom function, the code can be easily filled by users to extend the function of Multiwfn.
Citing Multiwfn
If Multiwfn is involved in your paper, please cite as:
Tian Lu, Feiwu Chen, J. Comp. Chem. 33, 580 (2012)
I sincerely hope users can send me their accepted paper by E-mail (sobereva@sina.com)
Discussion zone
http://emuch.net/bbs/forumdisplay.php?fid=290&type=997Note that this forum needs register (at
http://emuch.net/bbs/register.php). Non-chinese speaking users are welcome to discuss in English.
Other resources
Multiwfn_poster.jpg 715KB, presented at the 28th CCS congress (2012, Apr, 13-16)
A very brief introduction to Multiwfn 2.2.ppt 3.83MB, uploaded at 2011-NOV-22.
A brief introduction to Multiwfn 1.6.ppt 2.53MB, uploaded at 2010-DEC-12. Notice that Version 1.6 was not formally released.
"Study multi-center bonds by AdNDP approach as well as ELF/LOL and multi-center bond order" (in Chinese) by Tian Lu detailedly introduced the usage of AdNDP module in Multiwfn by four practical example, meanwhile similarities and differences between AdNDP, ELF/LOL and multi-center bond order methods are compared.
http://hi.baidu.com/sobereva/blog/item/80082a604736dfff8db10d9c.html"On the calculation methods of orbital composition" (in Chinese) by Tian Lu deeply discussed pros and cons of various calculation methods of orbital composition, the usage of orbital composition analysis module of Multiwfn are described in detail.
http://hi.baidu.com/sobereva/blog/item/09c390cd99aec44e0fb345e4.html"Using Multiwfn to plot NBO and related orbitals" (in Chinese) by Tian Lu
http://hi.baidu.com/sobereva/blog/item/1aae2feeafc1afe4b3fb9562.html"Using Multiwfn to plot difference map for electron density" (in Chinese) by Tian Lu
http://hi.baidu.com/sobereva/blog/item/1926e3ca721ec998c817684a.html"Using Multiwfn to perform topology analysis and calculate angle of lone pairs" (in Chinese) by Tian Lu
http://hi.baidu.com/sobereva/blog/item/d9e49111a38cd9dba7ef3fde.html"Visual research of weak interaction by Multiwfn" (in Chinese) by Tian Lu detailed the analysis method of weak interaction by using reduced density gradient (RDG) and sign(lambda2)*rho function, a lot of instances were given.
The latest version of the text, figures and related files can be downloaded at:
Multiwfn_NCI(2011-FEB-14).rar"Visual research of electron localization" (in Chinese) by Tian Lu graphically introduced ELF, LOL and laplacian function by using Multiwfn.
http://hi.baidu.com/sobereva/blog/item/f3c56817b2e99b0c4a90a7d6.html"Making anime to analyze electron structure characteristic" (in Chinese) by Tian Lu introduced how to create anime by using Multiwfn and shell script.
http://hi.baidu.com/sobereva/blog/item/7a1a7ff73cc37134720eec86.html"Reply some questions about Multiwfn and the development plan of Multiwfn" (in Chinese) is written by Tian Lu at 2011-NOV-13
http://hi.baidu.com/sobereva/blog/item/f0861a816bdb01c5bd3e1e4c.htmlBy using
molden2aim program written by W. Zou, Molden input files can be converted to .wfn format, so that Multiwfn can support more quantum chemistry packages such as Molpro, Molcas, ORCA, Dalton etc. For detail please visit
http://people.smu.edu/wzou/program/index.html and consult Section 5.1 of Multiwfn manual.
Examples
The graphs below are generated by Multiwfn directly, any other external programs are not required, only the file carrying wavefunction information is needed as input.
The 0.08 isosurface of two natural bond orbitals (NBO) of NH2COH, the first one is lone pair of nitrogen, the second one is anti-bonding orbital between carbon and oxygen. The secondary perturbation energy due to their interaction reached about 60kcal/mol.
Contour map of the two NBOs shown above, the drawing plane is perpendicular to molecular plane and passed through both carbon and nitrogen atoms.
Critical points and bond paths of electron density of imidazole - magnesium porphyrin complex. Some of interbasin surfaces are shown by yellow surfaces.
(3, -3) and (3,-1) critical points and corresponding topology path of ELF of pyrazine. The purple spheres beside nitrogen atoms reveal the position of lone pairs, while the purple spheres between each two atoms shows that electrons are highly localized in the covalent bond regions.
Spin density in the line defined by carbon and oxygen nuclei of triplet state methanamide.
Color-filled map of gradient norm of electron density of benzene, the black points at the center of chemical bonds are bond critical points (3,-1) in AIM theory.
Localized orbital locator (LOL) map of a small part of graphene, isovalue of the contour line is 0.5. The wavefunction of graphene primitive cell is calculated by PBC function of Gaussian, then Multiwfn is used to extend the wavefunction to periodic plane.
Contour map of electrostatic potential of ClF3 in molecular plane, crimson and black lines correspond to positive and negative part respectively. The bold blue line shows the van der Waals surface (electron density=0.001, which is defined by Bader)
Gradient vector field with contour lines of electron density of uracil in molecular plane
Filled color relief map with projection map of ELF (Electron localization function) of Li6 cluster
The 0.5 isosurface of reduced density gradient (RDG) of phenol dimer. The regions enclosed by the surfaces shows the region where the weak interactions are present. The spatial scope of data is shown by blue framework.
Gradient map of electron density with contour lines of magnesium porphyrin. Brown, blue, and orange circles denote (3,-3), (3,-1) and (3,+1) critical points respectively, deep brown lines depict bond paths, deep blue lines reveals interbasin path.
Deformation electron density map of magnesium porphyrin, the solid lines represent the region in which electron density increased during chemical bond formation, the dash lines represent the region that density decreased.
Total / Partial / Overlap density-of-state (DOS) map of ferrocene. For clarity, isosurfaces of corresponding molecular orbitals were appended on the graph by external tools.
Minima (blue spheres) and maxima (red spheres) of average local ionization energy on van der Waals surface of phenol. The location of minima above and below the conjugated ring perfectly explained the effect of hydroxyl as a ortho-para directing group. Minimum 8 (at back) and 9 correspond to the easily polarized lone pair of oxygen.
Deformation density map during pushing two hydrogens with like-spin electron together. To draw the anime, generate wavefunction files of each step first, then write a script to invoke Multiwfn to process them and output corresponding graphs, finally use ImageMagick to combine graphs to gif anime file.
Two of three 5-center orbitals of B13+ cluster produced by adaptive natural density partitioning (AdNDP) approach.
Development plan
Support basin integral (Version 2.5)
Support charge decomposition analysis (may be in Version 2.6)
Support plotting orbital interaction diagram (may be in Version 2.6)
Support 2-Particle density matrix generated by post-HF wavefunctions
Support the quantum chemistry softwares using plane-wave
Acknowledgement
The author thanks to following users (in no particular order), who provided valuable suggestions or reported bugs, users' feedbacks are very important for the development of Multiwfn.
Can Xu; GuangYao Zhou; TianWaiLaiKe; jsbach; Beefly; Emilio Jose Juarez-Perez; YangChunBaiXue; XinYing Li; Yang Yang; Andy Kerridge; junjian; jinyun Wang;Zhuo Yang
The papers cited Multiwfn
1 Yang Yang, Hexacoordinate Bonding and Aromaticity in Silicon Phthalocyanine. J. Phys. Chem. A, 114, 13257 (2010)
http://pubs.acs.org/doi/abs/10.1021/jp109278v2 Lin Xu et al., Comparative insight into the halogen bonding of 4-chloropyridine and its metal [CuI, ZnII] coordinations with halide ions: a theoretical study on M-C-X···X’. Chem. Phys., 379, 66 (2011)
http://dx.doi.org/10.1016/j.chemphys.2010.11.0073 Li Xinying and Cao Xue, Interaction and Electron Density Properties of MKr42+ (M = Cu, Ag and Au): ab initio Calculation. Aust. J. Chem., 64, 339 (2011)
http://www.publish.csiro.au/paper/CH103734 Lin Xu et al., Evaluation of nucleotide C–BrO–P contacts from ONIOM calculations: Theoretical insight into halogen bonding in nucleic acids. Chem. Phys. Lett., 509, 175 (2011)
http://www.sciencedirect.com/science/article/pii/S000926141100546X5 Fang Wang, HongChen Du, JianYing Zhang, XueDong Gong, DFT studies on the structures and stabilities of N5+-containing salts, Struct. Chem., 22, 1067 (2011)
http://www.springerlink.com/content/741275kr45322x656 Xinying Li, Ab initio study of MKr n2+ (M = Cu, Ag, and Au, n = 1-6) clusters, J. Mol. Model. (2011)
http://www.springerlink.com/content/m7u68kh73n7381567 X. Li and X. Cao, Nature of M-Ng interaction in the MNg 42+ (M = Cu, Ag and Au; Ng = He and Ne) molecules: ab initio calculations, Eur. Phys. J. D, 64, 221 (2011)
http://www.springerlink.com/content/5451543mp62552208 Patrik Neuhaus, Michael Winkler, Wolfram Sander, EPR spectroscopic and computational characterization of the 2-dehydro-m-xylylene and 4-dehydro-m-xylylene triradicals, J. Phys. Organ. Chem., 24, 976 (2011)
http://onlinelibrary.wiley.com/doi/10.1002/poc.1911/abstract9 HuiYing Xu, Wei Wang, Interaction between Mg-porphyrin and Nitrogen, Oxygen Heterocyclic Compounds, Acta Phys. -Chim. Sin., 27, 2565 (2011)
http://www.whxb.pku.edu.cn/CN/abstract/abstract27755.shtml10 Ran Li, Qingzhong Li, Jianbo Cheng and Wenzuo Li, The structure, properties, and nature of unconventional π halogen bond in the complexes of Al 42- and halohydrocarbons, J. Mol. Model. (2011)
http://www.springerlink.com/content/e484222j75567p83/11 Dongdong Qi, Jianzhuang Jiang, Toward Panchromatic Organic Functional Molecules: Density Functional Theory Study on the Electronic Absorption Spectra of Substituted Tetraanthracenylporphyrins, J. Phys. Chem. A, 115, 13811 (2011)
http://pubs.acs.org/doi/abs/10.1021/jp208963a12 Tian Lu, Feiwu Chen, Meaning and Functional Form of Electron Localization Function, Acta Phys. -Chim. Sin., 27, 2786 (2011)
http://www.whxb.pku.edu.cn/EN/abstract/abstract27788.shtml13 Tian Lu, Feiwu Chen, Comparison of Computational Methods for Atomic Charges, Acta Phys. -Chim. Sin., 28, 1 (2012)
http://www.whxb.pku.edu.cn/EN/abstract/abstract27818.shtml14 Ran Lia, Qingzhong Lia, Zhenbo Liua, et. al., Enhancing the function, non-additivity, and substitution position effect of the Li atom in the cation–π interaction and its mechanism: an ab initio study of Li+ ··· Li-substituted benzene complexes, Mol. Phys., 110, 65 (2012)
http://www.tandfonline.com/doi/abs/10.1080/00268976.2011.63310715 Tian Lu, Feiwu Chen, Calculation of Molecular Orbital Composition, Acta Chim. Sin., 69, 2393 (2011)
http://sioc-journal.cn/Jwk_hxxb/CN/abstract/abstract340458.shtml16 Yang Yang, Metal-Ligand Coordination in Subphthalocyanines and Phthalocyanines: DFT, AIM and ELF Analyses, Polyhedron, 33, 310 (2012)
http://www.sciencedirect.com/science/article/pii/S027753871100721217 Yunsheng Xue, Lin An, Youguang Zheng, et. al., Structure and electronic spectral property of coumarin-chalcone hybrids: a comparative study using conventional and long-range corrected hybrid functionals, Comp. Theo. Chem., 981, 90 (2012)
http://www.sciencedirect.com/science/article/pii/S2210271X1100634718 Tian Lu, Feiwu Chen, Multiwfn: A multifunctional wavefunction analyzer, J. Comp. Chem., 33, 580 (2012)
http://onlinelibrary.wiley.com/doi/10.1002/jcc.22885/abstract19 Wenming Sun, Yuxiang Bu, Yixuan Wang, Interaction and protection mechanism between li@C60 and nucleic acid bases (NABs): Performance of PM6-DH2 on noncovalent interaction of NABs-Li@C60, J. Comp. Chem., 33, 490 (2012)
http://onlinelibrary.wiley.com/doi/10.1002/jcc.22881/abstract20 Cheng Cheng, Li Sheng & Zesheng Li, Is it possible to synthesize organic Ar compound? a theoretical study, Mol. Phys.
http://www.tandfonline.com/doi/abs/10.1080/00268976.2011.64434221 Xiaoyan Li, Jie Sun, Lingpeng Meng, Yanli Zeng and Shijun Zheng, Comparative study on the Al–Al multiple bond in Na2[Arx′AlAlArx′] and H2[Arx′AlAlArx′] (Arx′ = C6H3-2, 6-(C6H5)2), Theor. Chem. Acc., 131, 1116 (2012)
http://www.springerlink.com/content/33326p843h756t0522 Qingzhong Li, Ran Li, Xiaofeng Liu, Wenzuo Li, Jianbo Cheng, Pnicogen-Hydride Interaction between FH2X (X = P and As) and HM (M = ZnH, BeH, MgH, Li, and Na), J. Phys. Chem. A, (2012)
http://pubs.acs.org/doi/abs/10.1021/jp211435b23 Yifang Ouyang, Peng Wang, Peng Xiang, Hongmei Chen, Yong Du, Density-functional theory study of Aln and Aln−1Mg (n = 2–17) clusters, Comp. Theo. Chem. (2012)
http://www.sciencedirect.com/science/article/pii/S2210271X1200026624 Dongdong Qi, Lijuan Zhang, Luyang Zhao, Xue Cai, Jianzhuang Jiang, A Special Conjugated Model around sp3 Carbon Atoms: Density Functional Theory Study on the Homoaromatic Electron Delocalization and Their Applications of Benzofused Tetra(triptycene)porphyrins, ChemPhysChem (2012)
http://onlinelibrary.wiley.com/doi/10.1002/cphc.201200076/abstract25 Xinying Li, Xiuying Cheng, Xue Cao, Quantum chemical topology investigation on structure, electronic properties and interaction of CuNg n + (n = 1–3, Ng = He, Ne), Struct. Chem. (2012)
http://www.springerlink.com/content/04456n3065547254/26 Likai Yan, Shizheng Wen, Jianping Wang, Zhongmin Su, Influence of the central atom on the electronic properties of Lindqvist polyoxometalates, Comp. Theo. Chem. (2012)
http://www.sciencedirect.com/science/article/pii/S2210271X1200066727 Meiyuan Guo, Rongxing He, Yulan Dai, Wei Shen, Ming Li, Electron-Deficient Pyrimidine Adopted in Porphyrin Sensitizers: A Theoretical Interpretation of π-Spacers Leading to Highly Efficient Photo-to-Electric Conversion Performances in Dye-Sensitized Solar Cells, J. Phys. Chem. C (2012)
http://pubs.acs.org/doi/full/10.1021/jp210982928 Huiying Xu, Wei Wang, Xiangyang Xu, Molecular Recognition of Pyromellitic Imide-azacyclophane to Organic Pollutant, Chinese J. Struct. Chem., 4, 591 (2012)
http://159.226.150.102/qikan/Cpaper/zhaiyao.asp?bsid=1524029 Yunfang Yang, Guijuan Cheng, Jun Zhu, Xinhao Zhang, Shigeyoshi Inoue, Yundong Wu, Silicon-Containing Formal 4π-Electron Four-Membered Ring Systems: Antiaromatic, Aromatic, or Nonaromatic?, Chem.-Eur. J., (2012)
http://onlinelibrary.wiley.com/doi/10.1002/chem.201103443/abstract30 Xinying Li, Xue Cao, Yusheng Wang, Structure, Electronic Properties and Interaction of MRn n+ (n = 1–3, M = Cu, Ag and Au) Clusters: Ab Initio Calculations, J. Clust. Sci., (2012)
http://www.springerlink.com/content/163787616k191011/