AIM2000—a Program to Analyze and Visualize

Software News and UpdatesAIM2000A Program to Analyze and VisualizeAtoms in Molecules

Introduction

A bout 20 years ago, Bader et al. formulatedthe theory of atoms in molecules.1 It hassince proven to be a valuable tool for the qualita-tive and quantitative analysis of molecular structureand properties. Soon after, in 19791982, a programpackage (AIMPAC2, 4) was developed that utilizedthe rigorous mathematics of the theory. AIMPACconsists of 11 different programs, which were de-veloped in Fortran. Different tasks performed bythese programs are: (a) calculation of critical pointsof the charge density and other density functions;(b) calculation of gradient paths, especially bondpaths; (c) plotting of the charge density and otherfunctions: gradient path plots, relief plots, and con-tour maps; and (d) calculation of properties ofatoms in molecules. This procedure integrates afunction f over the basin of a given atom in amolecule:

F() DZ

f (x) dx

The last procedure is very time consuming, andsometimes supplies unsatisfactory results. There-fore, alternative numerical methods have been de-veloped that overcome the computational and nu-merical difficulties.3 This led to a reimplementationand extension of AIMPAC called AIM2000.5

This article describes the objectives and featuresof AIM2000. A set of examples is given that demon-strates the components of AIM2000 and their ca-pabilities: (a) analysis of the topological structureof density functions; (b) calculation of properties

Correspondence to: F. BieglerKnig; e-mail: [email protected]

of atoms and interatomic surfaces in molecules;and (c) two- and three-dimensional visualizations ofdensity functions.

AIM2000Overview

The main intentions in the development ofAIM2000 were:

1. All components run under a single windows-and graphics-oriented user interface.

2. Most parts of the program can be operated in-tuitively. An on-line help component suppliesnecessary explanations.

3. The topological structure of a molecule isdisplayed in three dimensions as soon as itis calculated. The user immediately gets anoverview of the molecular structure.

4. Two-dimensional plots can be generated andmanipulated onscreen, and viewed on thescreen before printing out a hardcopy (wysi-wyg).

5. The algorithms to calculate properties ofatoms in molecules by integrating over atomicbasins have been improved and numerical dif-ficulties removed.

6. A component to integrate functions over inter-atomic surfaces in a molecule has been added.

7. As input, a standard wave function file(extension wfn) is needed.

AIM2000 has been implemented under WindowsNT 4.0 using Visual CCC version 6.0 and MicrosoftFoundation Classes. AIM2000 runs under the oper-ating systems Windows 95/98 and NT.

Journal of Computational Chemistry, Vol. 22, No. 5, 545559 (2001)c 2001 John Wiley & Sons, Inc.

BIEGLERKNIG, SCHNBOHM, AND BAYLES

FIGURE 1.

In addition to the above-mentioned functionality,the usual standard properties of Windows programsare supplied:

1. Saving and loading data of a session (serial-ization).

2. Output of a window content to a printer.

3. Print preview.

4. Menubar with pull-down menus. Each func-tion of AIM2000 can be triggered via a menuentry.

5. Toolbar with icons for the most frequenttasks.

6. Statusbar to indicate important information.

The working area of AIM2000 consists of fourparts (views, see Fig. 1):

7. Control View (top left), where the user choosesthe objects to work with, such as densityfunctions, atoms, interatomic surfaces, andplots.

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8. Record View (bottom right). A record of allcalculations can be displayed in this view.Control View and Record Menu define theoutput into the Record View. Examples: (a) forthe density function chosen in Control Viewa list of critical points with their proper-ties can be listed; (b) for an atom chosenin Control View a list of all atomic prop-erties calculated so far can be displayed;(c) the content of Record View is formattedfor printout. An on-line preview can be gen-erated.

9. 3D-View (bottom left). To get an impressionof the spatial structure of the molecule, itis displayed in 3D-View, including all criti-cal points and gradient paths calculated sofar. The look of 3D-View can be manipulatedvia a context menu. The view can be ro-tated, translated, and zoomed (see help com-ponent).

10. 2D-View (top right). This view can displaytwo-dimensional plots of the chosen densityfunction. These plots can be defined and mod-ified via Plot Menu and screen dialogs. Theycan also be translated, rotated, and zoomed in-teractively. There are implemented: (a) molec-ular graph plot which displays the moleculewith its atoms, critical points, bond paths,and other gradient paths connecting criticalpoints. The appearance of the different itemscan be chosen in the definition dialog. (b) Re-lief map. This is a three-dimensional plot.The first two dimensions define the planeof a cross-section of the molecule. The thirddimension displays the values of a densityfunction in this plane. A shaded display isused for this mountain. The plane of cross-section, colors, and perspective can be definedin a screen dialog. (c) Contour map. Simi-lar to the relief map, the plane of a cross-section of the molecule is defined, the val-ues of a density function are displayed usingcontour lines. The user can choose interac-tively value, style, and color of each contourline. (d) Gradient path map. This map canbe combined with a contour map into oneplot. It displays a number of gradient pathsoriginating or terminating at a chosen criti-cal point. Gradient path sources as well asline style, line color, and number of gradientpaths can be adjusted in the plot definition di-alog.

Calculation of Molecular Structure andAtomic Properties

Both Calculation Menu and toolbar contain en-tries to trigger the calculations of AIM2000.

CALCULATION OF CRITICAL POINTS

To calculate critical points of the chosen densityfunction, Newtons method is used to find zeros ofits gradient. Taking the charge density (x) as thedensity function, the following equation describesone step of the iteration

xiC1 D xi H((xi1 (xiwhere H((xi)) is the Hessian matrix of second par-tial derivatives of , and the Newton stepsize.With D 1, the classical Newtons method is em-ployed. This stepsize together with desired accuracyand maximal number of iterations can be set in theoptions dialog for critical points.

In general, Newtons method converges rapidly,and supplies accurate results. However, success ishighly dependent on the choice of starting values.The calculation dialog for critical points presentsseveral choices of starting values:

1. Starting values at nuclear positions. This isrecommended for the charge density to findthe density maxima near the nuclear positions.

2. Starting values at mean values of density max-ima pairs. When applied to the charge densitythis usually computes the (3,1)-bond criticalpoints. Sometimes it is necessary to reduce theNewton stepsize to 0.5 to prevent the itera-tion from leaving the considered area.

3. Starting values at mean values of three densitymaxima. When applied to the charge densitythis procedure looks for ring- and cage-criticalpoints.

4. Single starting values: the user can choose ar-bitrary starting values or compose a set ofstarting values from nuclear positions and al-ready calculated critical points.

5. Grid search for critical points: a cube can bedefined and a grid of starting values insidethe cube to search for critical points system-atically. This option is useful for other densityfunctions like L D 14r2, which can have agreat number of critical points.

JOURNAL OF COMPUTATIONAL CHEMISTRY 547


CALCULATION OF MOLECULAR GRAPH

The molecular graph of a molecule consists of allgradient paths of the charge density that connectcritical points uniquely:

1. A bond path connects a (3,1)-critical point xbwith a charge density maximum xm. Thesepaths can be calculated going uphill from xb inthe direction of the eigenvector of the positiveeigenvalue of the Hessian matrix of xb.

2. A unique path connects a (3,C1)-criticalpoint xr with a (3,1)-critical point xb, whichlies in the ring plane. Unfortunately, no uniquedirection defines this path. There is a two-dimensional set of paths going uphill from xras well as a two-dimensional set of paths go-ing downhill from xb. The common path has tobe calculated iteratively. A bisection iterationto find the uphill direction from xr is em-ployed. This method can be time consuming,especially in cases where the gradient pathsstrongly bend.

3. A unique path connects a charge density min-imum xc with a (3,C1)-critical point xr, whichbelongs to the cage. These paths can be calcu-lated going downhill from xr in the directionof the eigenvector of the negative eigenvalueof the Hessian matrix of xr.

To calculate points along a gradient path the fol-lowing initial value problem is solved:

g(xm) D

x(l) 2 R3 dx(l)dl D r(x(l), X)kr(x(l), X)k2 ;

x(0) D xm

where l is the arclength of the gradient path.This representation ensures an equal distribution ofpoints along a gradient path. Following the path up-hill means l < 0, downhill means l > 0.

CALCULATION OF PROPERTIES OF ATOMS ANDINTERATOMIC SURFACES

After choosing an atom in Control View andclicking Integration over atomic basin in the Calcula-tion Menu, a dialog appears that controls the calcu-lation of atomic properties. The integration is donein a polar atomic coordinate system, which has itsorigin at the nuclear position, and which is rotatedsuch that molecular symmetry can be used. This co-ordinate system is displayed in 3D-View. A list of

the coordinates of all nuclei and critical points in theatomic coordinate system is displayed. The user canset the borders of the -integration, which usuallyruns from 0 to 360 degrees. If the integration bordersare equal, rotational symmetry is assumed. Accord-ing to the integration intervall in , a symmetryfactor is displayed. All integration results are mul-tiplied by this factor. In case of rotational symmetry,this factor would be 2 . The user can overwrite thisfactor.

The user can choose between two integrationmethods. The first algorithm uses radial coordinatesto integrate inside a sphere with radius and nat-ural coordinates ouside. It is described in ref. 3.This algorithm does not involve calculation of in-teratomic surfaces, and is usually the faster one.The other algorithm integrates the whole basin inradial coordinates and computes the intersectionsof a ray defined by the angles and and theinteratomic surface using a bisection method. A ba-sic form of this algorithm has already been part ofAIMPAC.

Accuracy requirements and default values canbe set in an option dialog. Computer time con-sumption is high for the integration procedures,and very sensitive to changes in accuracy require-ments.

Choosing an interatomic surface in Control Viewand clicking Integration over interatomic surface in theCalculation Menu brings up a dialog that controlscalculation of properties of interatomic surfaces.Similar to the calculation of atomic properties, sym-metry can be used to accelerate the integration overinteratomic surfaces. The method of integration alsouses a natural coordinate system, and is described inref. 3.

The functions that are integrated in the integra-tion procedures can be selected using two optiondialogs: Functions for Basin Integration and Functionsfor Surface Integrations. By default, all functions areselected.

Help Component

The help component of AIM2000 is implementedin HTML. It is available stand alone via the Internet5

and as on-line help during a program run. Typ-ing F1 displays the main page, while Shift-F1 isa context-sensitive help component and displaysinformation about the current program part or di-alog.

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TABLE I.Definitions of Atomic Properties for Basin Integration.

Name of Atomic Property Definition of Property for Atom A with Basin

Number of electrons N(A) D R dx (x) D R dx(N R d 0 )Laplacian of atom L(A) D R dx L(x) D R dx( 14r2)Lagrangian kinetic energy G(A) D R dx G(x) D R dx ((Nh2/2m)N R d 0 r r )Hamiltonian kinetic energy K(A) D R dx K(x) D R dx ((Nh2/2m)N R d 0 r2 C r2 )Virial field V(A) D R dx (x r (x)Cr (x (x)))Energy of atom E(A) D R dx (K(x) (1 R))Missing information I(A) D R dx ((/N) ln(/N))Atomic average of 1/r R1(A) D

R dx (/rA)

Atomic average of r R1(A) DR dx ( rA)

Atomic average of r2 R2(A) DR dx ( r2A)

Atomic average of r4 R4(A) DR dx ( r4A)

Atomic average of r x/rA GR1(A) DR dx (r x/rA)

Atomic average of r x GR0(A) DR dx (r x)

Atomic average of r x rA GR1(A) DR dx (r x rA)

Atomic average of r x r2A GR2(A) DR dx (r x r2A)

Electric dipole (x)

Electric dipole (y)

0@Mx(A)My(A)Mz(A)

1A D R dx ( dA)Electric dipole (z)

Attraction of density A by nucleus A V0ne(A) DR dx (ZA(x)/rA)

Attraction of density A by nucleus A (corr.) VC0ne(A) D 2(1R)R V0ne(A)Attraction of density A by all nuclei Vne(A) D

R dx (

PB ZB(x)/rB)

Attraction of density A by all nuclei (corr.) VCne(A) D 2(1R)R Vne(A)Electronelectron repulsion contribution

to energy of atom A Vee(A) DR dx Vee(x)

Electronelectron repulsion contributionto energy of atom A (corr.) VCee(A) D 2(1R)R Vee(A)

Potential energy of repulsion (corr.) Vrep(A) D 2E(A) VCne(A)Total potential energy of atom Vtot(A) D VCne(A)C Vrep(A)Atomic quadrupole (xx)

Atomic quadrupole (xy)

Atomic quadrupole (xz)

0B@Qxx(A) Qxy(A) Qxz(A)Qxy(A) Qyy(A) Qyz(A)Qxz(A) Qyz(A) Qzz(A)

1CAD R dx (3 dA dTA C dTA dA E)Atomic quadrupole (yy)Atomic quadrupole (yz)

Atomic quadrupole (zz)

Force exerted on nucleus A by density of A (x)Force exerted on nucleus A by density of A (y)

0@FAx(A)FAy(A)FAz(A)

1A D R dx ( ZA dA/r3A)Force exerted on nucleus A by density of A (z)

Force exerted on all other nuclei by density of A (x)

Force exerted on all other nuclei by density of A (y)

0@FBx(A)FBy(A)FBz(A)

1A D R dx (PB ZB dB/r3B)Force exerted on all other nuclei by density of A (z)



TABLE I.(Continued)

Name of Atomic Property Definition of Property for Atom A with Basin

Total integrated volume (0.001 au isosurface) VOL1(A) DR dx (1, if (x) 0.001, else 0)

Total integrated volume (0.002 au isosurface) VOL2(A) DR dx (1, if (x) 0.002, else 0)

Electron density over integrated volume (0.001 au isosurface) NVOL1(A) DR dx ((x), if (x) 0.001, else 0)

Electron density over integrated volume (0.002 au isosurface) NVOL2(A) DR dx ((x), if (x) 0.002, else 0)

Basin virial VBAS(A) DR dx (x r (x))

Surface virial VSURF(A) D 2K(A) VBAS(A)Ehrenfest force (x)

Ehrenfest force (y)

0B@EFx(A)EFy(A)EFz(A)

1CA D R dx (r (x))Ehrenfest force (z)

TABLE II.Definition of Properties for Surface Integration.

Definition of Property for Surface S betweenName of Surface Property Atoms A and B

Surface integral of electron density N(S) D RS dx(x)Surface integral of Laplacian density L(S) D RS dx L(x)Lagrangian kinetic energy G(S) D RS dx G(x)Hamiltonian kinetic energy K(S) D RS dx K(x)Surface integral of r n(x) GRN(S) D RS dx (r n(x))Hypervirial gradient (n D 1), atom A VR1,A(S) D

RS dx (d

TA n(x)/rA)

Hypervirial gradient (n D 1), atom B VR1,B(S) DR

S dx (dTB n(x)/rB)Hypervirial gradient (n D 0), atom A VR0,A(S) D

RS dx (d

TA n(x))


S dx (dTB n(x))Hypervirial gradient (n D 1), atom A VR1,A(S) D

RS dx (d

TA n(x)rA)


S dx (dTB n(x)rB)Hypervirial gradient (n D 2), atom A VR2,A(S) D

RS dx (d

TA n(x)r2A)


S dx (dTB n(x)r2B)Hypervirial gradient (n D 1), total VR1 D VR1,A C VR1,BHypervirial gradient (n D 0), total VR0 D VR0,A C VR0,BHypervirial gradient (n D 1), total VR1 D VR1,A C VR1,BHypervirial gradient (n D 2), total VR2 D VR2,A C VR2,BVirial of force exerted on surface of A VA(S) D

RS dx (dTA n(x))

Virial of force exerted on surface of B VB(S) DR

S dx (dTB n(x))

Total virial of force exerted on surface V(S) D VA(S)C VB(S)Total force exerted on electrons of atom A (x)

Total force exerted on electrons of atom A (y)

0B@SGNx(S)SGNy(S)SGNz(S)

1CA D RS dx ( n(x))Total force exerted on electrons of atom A (z)

Gradient of force exerted on electrons of atom A SGN(S) D RS dx(div( ) n(x))Total integrated area (0.001 au isosurface) AREA(S) D RS dx (1, if 0.001, else 0)

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Example Sessions with AIM2000

The results of example sessions for three mole-cules are presented: benzene, diborane, and tryp-tophan. The wave functions have been calculatedusing the program package Gaussian.6

The definitions of calculated properties (refer toTables I and II) are based on the following notations:XA is the nuclear position of atom A, dA D x XA,rA D kdAk2; ZA is the nuclear charge of atom A;R D V/T is the virial ratio of the molecule. Fora molecule in an equilibrium geometry, R shouldequal 2 if the virial theorem is satisfied; is thewave function of the molecule;

Rd 0 is the summa-

tion over all spin and integration over all electroniccoordinates but one; Vee(x) is the electronelectronrepulsion energy density; (x) is the quantum stresstensor; n(x) is the normal to an interatomic surfacebetween atoms A and B at a point x pointing out-ward from atom A; and E is the 3 3 unit matrix.

BENZENE

We used a 6-311CCg(2d,2p) basis set for theGaussian6 calculation of the wave function. To ex-amine the benzene molecule with AIM2000, thefollowing steps have been carried out:

1. Load the wave function using the entry in thefile menu.

FIGURE 2.

2. Call the dialog for the calculation of criticalpoints from the Calculation Menu. All criticalpoints are determined successfully with de-fault settings by using nuclear positions andmean values of density maxima pairs as start-ing values.

3. Call the dialog for the calculation of gradientpaths of the molecular graph. With default set-tings there is no problem calculating bond andring paths. View the results in 3D-View.

TABLE III.Integrated Properties of Atoms in Benzene (Total Calculated Energy: 230.769094545203,Virial R D V/T D 2.00075285).Property Results for Atom C1 Results for Atom H7

N 5.9830404197eC000 1.0156066265eC000L 2.5183347861e004 4.8882029469e005G 3.7798492684eC001 6.3320930006e001K 3.7798744518eC001 6.3325818209e001V 7.5597237202eC001 1.2664674821eC000E 3.7827201303eC001 6.3373493051e001I 6.4307280415e001 1.6114365003e001

R1 1.4949653752eC001 1.3035968140eC000R1 6.1544559570eC000 1.1257716923eC000R2 9.6959117609eC000 1.6540791938eC000R4 4.6821363599eC001 7.1840929549eC000

GR1 2.7077505869eC001 1.8719611975eC000GR0 1.3099327563eC001 2.1793844089eC000GR1 1.5771465438eC001 3.3235768794eC000GR2 3.1050068914eC001 6.3444175448eC000



TABLE III.(Continued)

Property Results for Atom C1 Results for Atom H7

Mx 1.9729632176e006 2.9425537273e007My 5.2223158481e002 1.1988216649e001Mz 3.4612873009e005 5.7207975436e006V0ne 8.9697922510eC001 1.3035968140eC000

VC0ne 8.9731674346eC001 1.3040873358eC000Vne 1.4724279847eC002 1.0425437205eC001

VCne 1.4729820349eC002 1.0429360124eC001Vee 4.2124297686eC001 4.5647708861eC000

VCee 4.2140148358eC001 4.5664885334eC000Vrep 7.1632703325eC001 9.1601617402eC000Vtot 7.5665500163eC001 1.2691983838eC000Qxx 1.8548650917eC000 1.4133164458e001Qxy 1.8412957890e004 1.1097481745e005Qxz 1.7379950836e004 2.5267563228e005Qyy 1.6479168406eC000 5.0143892420e001Qyz 9.3339043898e005 1.1647766754e005Qzz 3.5027819324eC000 3.6010727962e001FAx 4.6528251227e004 1.0116564073e005FAy 2.7002749490e001 2.2076064572e001FAz 5.1963854693e004 1.9822745194e005FBx 5.2461905170e005 1.2239485993e006FBy 8.1784552399eC000 2.2415281255eC000FBz 2.7795674655e005 1.2187364828e005

VOL1 8.3230752472eC001 4.9379654052eC001VOL2 7.0782453725eC001 3.7121284413eC001

NVOL1 5.9632763188eC000 9.9237355348e001NVOL2 5.9453895243eC000 9.7483921719e001VBAS 7.5331435647eC001 1.1620346392eC000VSURF 2.6580155560e001 1.0443284290e001

EFx 3.4210324811e004 9.2911642225e006EFy 2.4512599750e002 6.5749013942e002EFz 5.9522957178e004 1.8411686352e005

4. In Plot Menu choose a new plot and moleculargraph to display the molecular graph of ben-zene in 2D-View (see Fig. 2).

5. After analyzing the molecular structure, prop-erties of atoms in molecules can be calculated.After choosing an atom in Control View, callthe dialog for integration over atomic basins.To get better results, relative and absolute ac-curacy are both set to 105 in the options dia-log. Integration in natural coordinaates needs3228167 wave function evaluations to inte-grate atom C1 and 474888 wave function eval-uations to integrate atom H7. For results, see

Table III. To get correct dipole and quadrupolemoments, no symmetry was used. Results forother atoms follow by symmetry.

6. After choosing an interatomic surface in Con-trol View, call the dialog for integrationover interatomic surfaces from the calculationmenu. Using defaul settings, integration is car-ried out for the surfaces between C1 and C2,and between C1 and H7. For results see Ta-ble IV. Results for other surfaces follow bysymmetry.

7. After calculating these values, integrationaccuracy can be examined using Gauss

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TABLE IV.Integrated Properties of Interatomic Surfacesin Benzene.

Results for Surface Results for Surfacebetween between

Property A D C1 and B D C2 A D C1 and B D H7

N 1.435488 0.996293L 0.343893 0.108499G 0.934137 0.544816K 0.590245 0.436317

GRN 0.000000 0.000000VR1,A 1.081951 0.657707VR1,B 1.081951 0.735505VR0,A 1.882754 1.084239VR0,B 1.882754 0.867596VR1,A 3.490877 1.862423VR1,B 3.490878 1.178076VR2,A 7.029438 3.344751VR2,B 7.029439 1.913162VR1 2.163901 1.393212VR0 3.765508 1.951835VR1 6.981755 3.040500VR2 14.058877 5.257912VA 0.118583 0.028916VB 0.118583 0.104887V0 0.237166 0.133803

SGNx 0.078299 0.000002SGNy 0.045206 0.065824SGNz 0.000000 0.000011SGN 0.000000 0.208372AREA 39.208610 32.480316

theorem.3 With exact calculation, we have forvolume integrals:Z

L(x) dx D 0.

For surface integrals we have:ZSr(x)T n(x) dx D 0.

Gauss theorem connects both integral types:

g(A) 3 ZA

(x) dxCZA

r(x)T x dx

XSA

ZSA

(x)(dA(x)T n(x)

dx D 0

FIGURE 3.

where the sum runs over all interatomic sur-faces SA of A. With values from Tables IIIand IV we get:

g(C1) D 0.0000466 andg(H7) D 0.0001614.

Also, the total number of electrons in the mole-cule should be 42:

6 ZC

(x) dxC 6 ZH

(x) dx D 41.991882.

8. Produce various plots in 2D-View: Contourmap (Fig. 3), gradient path map (Fig. 4), re-lief map (Fig. 5), and relief map of Lagrangianfunction L(x) (Fig. 6).

9. Save work as an aim-file by clickingSave As: : : in the file menu.

DIBORANE

Similar calculations have been carried outfor the diborane molecule. However, because ofthe structure of the molecule, some difficultiesarise:

1. With the default settings, the program cannotdetermine correct ring paths, because these are



FIGURE 4.

highly bent. Increasing the maximal numberof bisection iterations in differential equationoptions to 60, and minimal distance to criticalpoints to 0.1 produces accurate results.

FIGURE 6.

2. The distribution of gradient paths in thesurface between B1 and H5 (a hydrogenbonded to both borans) is very uneven.The program cannot satisfy the default in-

FIGURE 5.

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TABLE V.Integrated Properties of Atoms in Diborane (Total Calculated Energy: 52.832614782625,Virial R D V/T D 2.00039683).Property Results for Atom B1 Results for Atom H3 Results for Atom H7

N 2.9278818803eC000 1.6838933109eC000 1.7038301374eC000L 2.3158810578e004 8.9828148536e005 2.5455891963e005G 2.3750825903eC001 8.6347456503e001 9.2780636365e001K 2.3751057491eC001 8.6356439318e001 9.2783181954e001V 4.7501883394eC001 1.7270389582eC000 1.8556381832eC000E 2.3760482623eC001 8.6390708143e001 9.2820001104e001I 4.8369114547e001 6.1520224463e001 5.9573643714e001

R1 1.0398673618eC001 1.7084237794eC000 1.7438946534eC000R1 1.7513728182eC000 2.2805288740eC000 2.2005758313eC000R2 1.8687159211eC000 3.8486116219eC000 3.3914955843eC000R4 5.8324779443eC000 1.8642590252eC001 1.2273504710eC001

GR1 1.9203811134eC001 2.6546852562eC000 2.2536324882eC000GR0 6.7229485181eC000 3.6323676181eC000 2.7935394446eC000GR1 4.1228579830eC000 6.3214127095eC000 4.3385580066eC000GR2 4.8644234632eC000 1.3335419643eC001 8.0833146915eC000Mx 6.1620751365e002 1.6476958740e001 1.3002932021e005My 6.1381114382e006 3.2274079204e001 4.2556984940e006Mz 4.7441074179e006 4.6927788898e006 4.4244732441e001V0ne 5.1993368089eC001 1.7084237794eC000 1.7438946534eC000

VC0ne 5.2003682307eC001 1.7087626891eC000 1.7442405996eC000Vne 6.2407196219eC001 9.3529298333eC000 1.1321241737eC001

VCne 6.2419576286eC001 9.3547852267eC000 1.1323487596eC001Vee 1.9328069364eC001 4.2446827201eC000 5.1967049588eC000

VCee 1.9331903582eC001 4.2455247617eC000 5.1977358585eC000Vrep 1.4899490322eC001 7.6263004003eC000 9.4662408062eC000Vtot 4.7520085964eC001 1.7284848264eC000 1.8572467895eC000Qxx 9.3587297497e002 9.7362995500e002 4.8869539931e001Qxy 3.2014158666e005 3.6233530777e001 1.7980376780e004Qxz 2.1933317412e005 1.1369439812e005 1.5385552973e005Qyy 1.4665331410e001 3.9323131045e001 1.4822974116e001Qyz 9.8624812239e005 4.8288526452e006 2.1484320813e005Qzz 2.4024061160e001 4.9059430595e001 3.4046565815e001FAx 1.3379192369e001 1.7278364288e001 9.4276240870e006FAy 2.1787346245e004 3.0175930628e001 1.0564320247e005FAz 1.0102027929e003 1.6050374112e005 3.4513495926e001FBx 1.5685202260eC000 1.4567554141eC000 1.0235664602e005FBy 3.4471363223e006 1.9271921175eC000 4.9078022456e006FBz 3.2593080746e008 2.3093719873e005 2.3437419389eC000

VOL1 1.9769486072eC001 8.2954371177eC001 5.9325909444eC001VOL2 1.8366020652eC001 6.3861218827eC001 4.8931048694eC001

NVOL1 2.9258317567eC000 1.6465346820eC000 1.6854130256eC000NVOL2 2.9238034222eC000 1.6192981632eC000 1.6705121628eC000VBAS 4.6863546154eC001 1.1850678196eC000 1.0077801617eC000VSURF 6.3833724041e001 5.4197113865e001 8.4785802151e001

EFx 7.4539344428e002 1.7570988470e001 2.5327984850e005EFy 1.0585631229e004 2.9569830622e001 8.4662530720e006EFz 1.0350520882e003 2.1388574763e005 3.0656573473e001



TABLE VI.Integrated Properties of Interatomic Surfaces inDiborane.


Property A D B1 and B D H3 A D B1 and B D H5

N 0.938160 0.706251L 0.204176 0.271561G 0.653846 0.553541K 0.449670 0.281980

GRN 0.000000 0.000000VR1,A 0.469683 0.328220VR1,B 0.762191 0.607242VR0,A 0.572611 0.459372VR0,B 1.419255 1.139803VR1,A 0.713886 0.730035VR1,B 2.800173 2.194189VR2,A 0.860460 1.383931VR2,B 5.904543 4.362299VR1 1.231875 0.935462VR0 1.991866 1.599175VR1 3.514060 2.924225VR2 6.765003 5.746230VA 0.227334 0.091851VB 0.541805 0.415265V 0.769139 0.507116


tegration accuracy of 105 when integrat-ing over this surface. Relaxing the absoluteaccuracy to 104 produces satisfactory re-sults.

FIGURE 8.

FIGURE 7.

Integration results are presented in Table V forvolume integration and Table VI for surface integra-tion. Check of integration accuracy:

g(B1) D 0.003269, g(H3) D 0.000056, andg(H5) D 0.038345.

Number of electrons:

2 ZB1

(x) dxC 4 ZH3

(x) dx

C 2 ZH5

(x) dx D 15.998996.

Pictures and plots: Screen shot of 3D-View(Fig. 7), gradient path map in ring plane (Fig. 8),gradient path map of interatomic surface betweenB1 and H5 (Fig. 9).

TRYPTOPHAN

Another less simple example is the trypto-phan molecule (wave function basis set: 6-311g).Critical points and bond paths can be calculatedwithout difficulty. Again, there are problems cal-culating ring paths. Because of a very uneven dis-tribution of gradient paths in the ring plane nearthe hydrogen bridge, the program is not able tocompute the connection between the ring criticalpoint and the bond critical point between H12and O25.

556 VOL. 22, NO. 5

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TABLE VII.Integrated Properties of Atoms in Tryptophan (Total Calculated Energy: 682.358536381611,Virial R D V/T D 2.00054192).Property Results for Atom H12 Results for Atom N14 Results for Atom H15 Results for Atom O25

N 9.6524198408e001 8.5051521102eC000 5.5834781751e001 9.3033617925eC000L 3.3442096066e004 1.7482094135e003 9.0547017536e005 2.2525970707e004G 6.1141740326e001 5.5297745721eC001 4.4201928515e001 7.5596480574eC001K 6.1108298230e001 5.5295997512eC001 4.4210983217e001 7.5596255315eC001V 1.2225003856eC000 1.1059374323eC002 8.8412911731e001 1.5119273589eC002E 6.1141414039e001 5.5325963519eC001 4.4234942033e001 7.5637222437eC001I 6.7634351391e002 4.2902229802e001 3.9162812206e002 4.4797887010e001

R1 1.2613689643eC000 1.9418072622eC001 8.7845673077e001 2.3195599377eC001R1 1.0435294613eC000 9.4412505568eC000 5.2747944592e001 9.5481328403eC000R2 1.4751412657eC000 1.4923710772eC001 6.8886306186e001 1.4263233105eC001R4 5.6217611070eC000 6.2924301263eC001 2.4976782060eC000 5.6332688156eC001

GR1 1.7678197665eC000 3.5917573169eC001 1.1730273130eC000 4.4764882602eC001GR0 1.9768650437eC000 1.9671442469eC001 1.1788728115eC000 2.4597789057eC001GR1 2.8451612239eC000 2.5680079331eC001 1.5915660651eC000 3.1149415442eC001GR2 5.0437816481eC000 4.8602909136eC001 2.7386852466eC000 5.5510943873eC001Mx 7.5291310276e002 3.9971869863e002 3.9190013718e002 8.9715493833e004My 7.5574658923e002 1.6647319098e001 1.5879025299e001 2.0475332298e001Mz 4.0547142861e002 1.5015984315e002 1.1408297637e002 2.2640593141e001V0ne 1.2613689643eC000 1.3592650835eC002 8.7845673077e001 1.8556479502eC002

VC0ne 1.2617106522eC000 1.3596332902eC002 8.7869469293e001 1.8561506203eC002Vne 1.8765656397eC001 2.8768597402eC002 9.9403522198eC000 3.2820993253eC002

VGne 1.8770739762eC001 2.8776390430eC002 9.9430449280eC000 3.2829884020eC002Vee 8.7480490102eC000 9.7994250382eC001 4.5264832826eC000 1.0251962038eC002

VCee 8.7504187395eC000 9.8020795711eC001 4.5277094462eC000 1.0254739157eC002Vrep 1.7545529003eC001 1.7709055832eC002 9.0571191310eC000 1.7700422597eC002Vtot 1.2252107583eC000 1.1067334598eC002 8.8592579704e001 1.5129461423eC002Qxx 2.7011956681e001 2.5975745188e001 3.2425268661e002 7.2478685212e001Qxy 2.8884610650e001 4.9015655706e001 2.6143644359e002 2.6557651505e001Qxz 1.0286837480e001 4.6424609103e001 1.8476134825e003 3.2129037902e001Qyy 1.4280121392e002 1.8156393524eC000 6.9320455247e002 3.4704683714e001Qyz 1.0296182712e001 4.6886443139e001 9.8119010497e003 3.3098154009e001Qzz 2.8439968820e001 2.0753968043eC000 3.6895186586e002 3.7774001498e001FAx 1.5284311282e001 1.7202065439e001 1.3093819580e002 4.0547629231e002FAy 1.1873655969e001 7.0310995952e001 5.1661645804e002 1.4477397823eC000FAz 4.8286162798e002 1.1182135931e001 4.3186102673e003 1.5176518629eC000FBx 1.1993236423eC000 3.9670325298eC000 3.7667869883e001 8.8140612748eC000FBy 1.9770732800eC000 1.7556344492eC001 1.5998988219eC000 1.8226432238eC001FBz 9.6808885500e001 3.1801737796eC000 1.9031857373e001 7.6055170502e001

VOL1 4.5474971854eC001 1.0571214988eC002 2.7581636481eC001 1.1986309256eC002VOL2 3.6126972456eC001 9.0600175894eC001 2.0427856041eC001 1.0118691585eC002

NVOL1 9.5026880815e001 8.4828784174eC000 5.4561370511e001 9.2740220904eC000NVOL2 9.3684586898e001 8.4612042260eC000 5.3542110278e001 9.2470831909eC000VBAS 1.1141822683eC000 1.0866928242eC002 8.4820372664e001 1.4991981062eC002VSURF 1.0831811726e001 1.9244608087eC000 3.5925390669e002 1.2729252685eC000

EFx 4.9455387350e002 1.3353976453e001 1.1580957843e002 8.8568219128e002EFy 4.2283435049e002 5.2708976547e001 4.7916213426e002 4.7137753226e001EFz 1.8726939634e002 5.6504605985e002 3.2732164758e003 1.4251216015e001



FIGURE 9.

Integration works without problems. Table VIIshows results of the volume integration for H12,N14, H15, and O25. Table VIII contains results of in-tegration over the interatomic surfaces between C11

FIGURE 10.

and H12, and between H12 and O25. Picture: Screenshot of 3D-View (Fig. 10).

Check of accuracy whith Gauss theorem:g(H12) D 0.002065.

TABLE VIII.Integrated Properties of Interatomic Surfaces in Tryptophan.


Property A D C11 and B D H12 A D H12 and B D O25

N 1.009142 0.041466L 0.098455 0.043299G 0.539633 0.033937K 0.441178 0.009362

GRN 0.000000 0.000000VR1,A 0.656200 0.033392VR1,B 0.723824 0.037881VR0,A 1.090584 0.087647VR0,B 0.833279 0.125613VR1,A 1.877027 0.236285VR1,B 1.094899 0.421105VR2,A 3.335375 0.659732VR2,B 1.670486 1.430675VR1 1.380023 0.071273VR0 1.923863 0.213260VR1 2.971925 0.657390VR2 5.005861 2.090607VA 0.046191 0.012611VB 0.097668 0.019397V 0.143859 0.032008


558 VOL. 22, NO. 5

AIM2000

References

1. Bader, R. F. W. Atoms in Molecules: A Quantum Theory;Clarendon Press: Oxford, 1990.

2. BieglerKnig, F.; Bader, R. F. W.; Tang, T. H. J ComputChem 1982, 13, 317.

3. BieglerKnig, F. J Comput Chem 2000, 21, 1040.

4. AIMPAC software to download from: http://www.chemistry.mcmaster.ca/aimpac/.

5. AIM2000 homepage and software to download: http://gauss.fh-bielefeld.de/aim2000.

6. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria,G. E.; Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.;Montgomery, J. A.; Stratmann, R. E.; Burant, J. C.;Dapprich, S.; Millam, J. M.; Daniels, A. D.; Kudin, K. N.;Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi,M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.;Clifford, S.; Ochterski, J.; Petersson, G. A.; Ayala, P. Y.;Cui, Q.; Morokuma, K.; Malick, D. K.; Rabuck, A. D.;Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz,J. V.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Ko-

maromi, I.; Gomperts, R.; Martin, L.; Fox, D. J.; Keith, T.;Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Gonza-lez, C.; Challacombe, M.; Gill, P. M. W.; Johnson, B. G.; Chen,W.; Wong, M. W.; Andres, J. L.; HeadGordon, M.; Replogle,E. S.; Pople, J. A. Gaussian 98; Gaussian, Inc.: Pittsburgh, PA,1998.

FRIEDRICH BIEGLERKNIGJENS SCHNBOHMFachbereich Mathematik

und TechnikFachhochschule BielefeldAm Stadtholz 2433609 Bielefeld, Germany

DAVID BAYLESDepartment of ChemistryMcMaster University1280 Main St. W, HamiltonOntario, L8S4M1, Canada


IntroductionAIM2000---OverviewFIGURE 1.

Calculation of Molecular Structure and Atomic PropertiesHelp ComponentTABLE I.TABLE I. (CONTINUED)TABLE II.

Example Sessions with AIM2000FIGURE 2.TABLE III.TABLE III. (CONTINUED)TABLE IV.FIGURE 3.FIGURE 4.FIGURE 6.FIGURE 5.TABLE V.TABLE VI.FIGURE 8.FIGURE 7.TABLE VII.FIGURE 9.FIGURE 10.TABLE VIII.

References

AIM2000—a Program to Analyze and Visualize

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