Technical Note Calculations of Orbital Overlap Range Function EDR( r ; d) and Overlap Distance D(r )using Multiwfn

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1 Technical Note Calculations of Orbital Overlap Range Function EDR( r ; d) and Overlap Distance D(r )using Multiwfn Abstract The orbital overlap range function EDR( r; d) (J. Chem. Phys. 2014, 141, ) quantifies the extent to which an electron at point r in a calculated wave function overlaps over distance d. This technical note provides a brief introduction and illustrates how to calculate and interpret EDR( r; d) and Overlap Distance D(r) using Multiwfn program. The detailed description of each interpretive tool and their important applications, utilizations and interpretation are available in the cited references at the end of this document. In this technical note, it is assumed that the user has some experience of using programs like Gaussian, GAMESS-US/UK, Firefly, Molden, Molpro, Molcas, ORCA, Q-Chem, CFour, Turbomole, PSI, MRCC and NWChem etc which produce Multiwfn Input file either.fch/.fchk,.wfn, molden,.gms and.31~.40 etc. It is strongly advised to consult Section 2 of Multiwfn Manual. Contents 1. Introduction 2. Calculations of EDR( r; d) 3. Calculations of Surface Overlap Distance D(r) 4. Quantitative analysis of molecular surface of to find D(r) extrema 5. Visualization of Surface Overlap Distance D(r) using VMD 6. References 1. Introduction The orbital overlap range function EDR( r; d) quantifies the extent to which electrons at point r in a wave function occupy orbital lobes of size d. EDR( r; d) is built from the nonlocal one-particle reduced density matrix (1-RDM) γ(r, r ) = i ψ i (r)ψ i (r ) as EDR(r; d) = ʃd 3 r g d (r, r )γ(r, r ) g d (r, r ) = ρ 1 2(r) ( 2 πd 2) 3/4 exp ( r r 2 d 2 ) 1

2 Here ρ(r) is the electron density at point r. The prefactor ensures that the EDR is between -1 and +1. EDR( r; d) quantifies the characteristic overlap length of electrons in atomic and molecular system and provides a chemically intuitive real-space picture of electron distribution. Its applications to quantify the overlap of atoms and molecules, [1] solvated electrons, [2] electrides, [3] and stretched bonds [4] can be found in the cited references. The Multiwfn implementation evaluates the EDR on a grid, for a single global input value of distance d. Section 2 of this document illustrates an example. The overlap distance D(r) is the distance d that maximizes EDR( r; d) at each point. Compact, chemically "hard" regions of small D(r) are distinguished from diffuse, chemically "soft" regions of large D(r). [5] Atomic averages of valence-electron D(r) complement the information obtailed from atomic partial charges. Plots of D(r) on density isosurfaces, and quantitative analysis of such surfaces, complements molecular electrostatic potentials. The implementation evaluates EDR( r; d i ) on a grid of distances d i, then uses a three-point numerical fit to find the maximum. Sections 3 and 4 of this document illustrate example calculations. 2. Calculations of EDR(r ; d) This example shows how to calculate EDR(r; d) at user-defined length scale d (bohr) for anionic water cluster (H2O)2 - [2]. For other systems, user should produce any of the Multiwfn Input file mentioned in abstract. For details, please consult of Multiwfn Manual. Here we are going to use a sample file available in examples directory of Multiwfn package. Boot up Multiwfn and input following commands: examples\solvatedelectron.fchk // Anionic water dimer optimized at b3lyp/ g(2d,2p) 5 // Output and plot specific property within a spatial region 20 // Orbital overlap range function EDR(r; d) // Input length scale d (Bohr). Here we consider the relatively delocalized solvated electron at d=11.22 Bohr. Further details are given in Ref. [1]. 2 // Medium quality grid, covering whole system, about points in total. Please consult Section 3.6 of Multiwfn Manual for more information about grid setting. 2

3 At this stage Multiwfn should start calculating the grid data. If the computer have multiple CPU cores (SMP parallel architecture), it is desirable to enable the parallel mode, i.e. set nthreads parameter in settings.ini to the number of CPU cores. -1 // Show isosurface graph Upon completion, a GUI window will pop up. Input isovalue of 0.74 in the Isosurface value box and press enter. The following Isosurface will appear. Fig. 1 Isosurface CCSD/ G(2d,2p) EDR( r; 11.22bohr) = 0.74 of (H2O)2 - This figure shows that at length scale of Bohr the solvated electron is between the H2O molecules. 3. Calculations of Surface Overlap Distance D(r ) This example will show the calculation of D(r) of thioformic acid and its plot on molecular electron density surface. For other systems, user should produce any of the Multiwfn Input file mentioned in abstract. For details, please consult of Multiwfn Manual. Here we are going to use a sample file available in examples directory of Multiwfn package. Boot up Multiwfn and input following commands: examples\thioformicacid.wfn // Thioformic acid optimized at b3lyp/ g(2d,2p) 5 // Output and plot specific property within a spatial region 21 // Orbital overlap length D(r) which maximizes EDR(r; d) 1 // Input total number, start and increment in EDR exponents αi=1/di 2. The overlap distance is fit using an even-tempered grid of exponents. The start value is the largest exponent α1, 3

4 subsequent exponents are incremented by αi+1/αi = 1/αinc. The defaults suffice for common systems. After selecting the manual or default option, a list of exponents will be appear which will be used in evaluation of D(r). 2 // Medium quality grid, covering whole system, about points in total (cf. Section 3.6 of Multiwfn Manual). At this stage Multiwfn will start calculations. This step is slow. It is recommended to enable parallel mode, i.e. set nthreads parameter in settings.ini to the number of CPU cores. Wait until calculations are finished. Option 2 and 3 on return menu, can be used to export the grid data to Gaussian cube file or to plain text file respectively in current directory. The next step is to generate molecular (spin)density isosurface. 0 // Return to main menu 5 // Output and plot specific property within a spatial region 1 // Electron density Or 5 // Electron spin density 2 // Medium quality grid, covering whole system, about points in total. The grid should be same as for D(r) calculations. Export data in current folder by selecting option 2 or 3. The D(r) grid data can be plotted on molecular (spin)density isosurface by any program like VMD, GassView or MoproViewer etc. Fig. 2 surface overlap length D(r) of Thioformic acid from 2.8 bohr (red) to 3.5 bohr (blue) plotted over electrons/bohr 3 electron density surface. 4

5 D(r) plotted between 2.9 bohr (red) to 3.4 bohr (blue) distinguishes the chemically hard oxygen lone pair (red) from the softer sulfur lone pair (blue). 4. Quantitative analysis of molecular surface to find D(r ) extrema The example illustrates the quantitative analysis of surface overlap distance D(r) on molecular (spin)density isosurface of thioformic acid. Boot up Multiwfn and input following commands: examples\thioformicacid.wfn // Thioformic acid optimized at b3lyp/ g(2d,2p) 12 // Quantitative analysis of molecular surface 2 // Select mapped function 6 // Orbital overlap length D(r) which maximizes EDR(r; d) 1 // Manually input total number, start and increment in EDR exponents. Please consult Section 3 for more information. Or 2 // Use default values i.e // Start analysis now! Now the analysis will start. This step is slow step and will take some time. Once calculations are finished following results will be printed on screen along with other information: Global surface minimum: a.u. at Ang Global surface maximum: a.u. at Ang The number of surface minima: 4 # Value X/Y/Z coordinate(angstrom) * The number of surface maxima: 10 # Value X/Y/Z coordinate(angstrom) *

6 Now select: 0 // View molecular structure, surface minima and maxima This will show molecular structure and surface extrema (red and blue spheres correspond to maxima and minima, respectively). The GUI window will show that surface minima is present on oxygen atom due to compact lone pair and surface maxima is located on sulfur atom due to its more diffuse and weakly bound lone pair as shown in Fig. 3. Fig. 3 Molecular structure and surface extrema of thioformic acid. Red and blue spheres correspond to surface maxima and minima, respectively. 5. Visualization of Surface Overlap Distance D(r ) using VMD This example will show the visualization of D(r) over electron density surface using an opensource molecular visualization program VMD. We shall use electron density and D(r) cube files 6

7 density.cub and EDRDmax.cub respectively of thioformic acid generated in section 3 of this document. Start VMD File > New Molecule > Browse Choose density.cub >Load On the same window of Load files for: i.e. 0:density.cube, Browse > EDRDmax.cub >Load NOTE: While loading EDRDmax.cub DO NOT select New Molecule in Load files for: option. It will load EDRDmax.cub as new process. On VMD Main windows Graphics > Representations > Draw style (1 st tab) > Drawing Method: Isosurface > IsoValue: >Vol: density.cub > Draw: Solid Surface > Show: Isosurface Coloring Method: Volume > EDRDmax.cub Trajectory (3 rd tab): Color Scale Data Range: On VMD Main windows Graphics > Colors > Color Scale > Method: RGB Categories: Display > Names: Background > Colors: 8 white On VMD Main windows Display > Axes > Off The above procedure will results the following plot of D(r) molecular surface. Fig. 4 surface overlap length D(r) of Thioformic acid from 2.8 bohr (red) to 3.5 bohr (blue) plotted over electrons/bohr 3 electron density surface using VMD program. 7

8 References [1] a) B. G. Janesko, G. Scalmani, M. J. Frisch. How far do electrons delocalize? J. Chem. Phys. 2014, 141, ; b) A. Mehmood, B. G. Janesko. An Orbital-Overlap Complement to Atomic Partial Charge. Angew. Chem. Int. Ed. 2017, 56, [2] B. G. Janesko, G. Scalmani, M. J. Frisch. Quantifying solvated electrons' delocalization. PCCP 2015, 17, [3] B. G. Janesko, G. Scalmani, M. J. Frisch. Quantifying Electron Delocalization in Electrides. J. Chem. Theory Comput. 2016, 12, [4] A. Mehmood, B. G. Janesko. The electron delocalization range in stretched bonds. International Journal of Quantum Chemistry 2016, [5] B. G. Janesko, K. B. Wiberg, G. Scalmani, M. J. Frisch. Electron Delocalization Range in Atoms and on Molecular Surfaces. J. Chem. Theory Comput. 2016, 12,