The AIM utility allows to analyse charge densities produced by the ABINIT code. The AIM analysis (Atom-In-Molecule) has been proposed by Bader. Thanks to topological properties of the charge density, the space is partitioned in non-overlapping regions, each containing a nucleus. The charge density of each region is attributed to the corresponding nucleus, hence the concept of Atom-In-Molecule.

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- 1. Introduction
- 2. Input and output files
- 3. Typical use of input variables
- 4. List of input variables

The Bader technique allows to partition the space in attraction regions. Each of these regions is centered on one atom. The only input for this technique is the total charge density : the density gradient line starting from one point in space leads to one unique attracting atom. (References to the relevant litterature are to be provided).

Around each atom, the basin of attraction forms a irregular, curved polyhedron. Different polyhedra might have faces, vertices of apices in common. Altogether, these polyhedra span the whole space.

The points where the density gradient vanishes are called "critical points" (CP). They all belong to the surface of some Bader polyhedra. According to the number of positive eigenvalues of the Hessian of the density, one will distinguish :

- "Bonding CPs" (a minimum along one eigenvector of the Hessian, but a maximum along the two other eigenvectors of the Hessian), that belong to a face of the Bader volume (one BCP per face)
- "Ring CPs" (a minimum along two eigenvectors of the Hessian, but a maximum along the remaining eigenvector of the Hessian), that belong to a vertex of the Bader volume (one RCP per vertex)
- "Cage CPs" (a global minimum, defining an apex of the Bader volume.

In the present implementation, the user is required to specify one atom for which he/she wants to compute the Bader volume, surfaces or critical points. Different runs are needed for different atoms.

In case of the search for critical points, one start from the middle of the segment with a neighbouring atom (all neighbouring atoms are examined in turn), and evolves towards a nearby point with zero gradient. Then, in case crit equals 2, one checks that the CP that has been found belongs to the attraction region of the desired atom. This last step is by no means trivial. In the present implementation, the check is done by means of the straight line (radius) connecting the point with the atom. In case the Bader volume is not convex, it might be that a correctly identified CP of the Bader volume defines a radius that goes through a region that does not belong to the Bader volume : the CP is "hidden" from the atom defining the attraction region. In this case, the CP is considered as NOT being part of the Bader volume, unfortunately. The reader is adviced to look at the automatic test of the MgO molecule to see such a pathology : no cage critical point is found for the Mg atom. By chance, this problem is not a severe one, when the user is interested by other aspects of the Bader analysis, as described below.

In case of the search for the Bader surface, or the integral of the charge within the Bader surface, the user should define a series of radii of integration, equally spread over the theta and phi angular variables. Symmetries can be used to decrease the angular region to be sampled. Along each of these radii, the algorithm will determine at which distance the radius crosses the Bader surface. The integral of the charge will be done up to this distance. For this search of the Bader surface, the information needed from the critical points analysis is rather restricted : only an estimation of the minimal and maximal radii of the Bader surface. This is why the fact that not all CP have been determined is rather unimportant. On the other hand, the fact that some part of the Bader surface might not be "seen" by the defining atom must be known by the user. There might be a small amount of "hidden" charge as well. Numerical tests show that the amount of hidden charge is quite small, likely less than 0.01 electron.

The determination of density, gradient of density and hessian of the density is made thanks to an interpolation scheme, within each (small) parallelipiped of the FFT grid (there are n1*n2*n3 such parallelipiped). Numerical subtleties associated with such a finite element scheme are more delicate than for the usual treatment of the density within the main ABINIT code ! There are many more parameters to be adjusted, defining the criteria for stopping the search for a CP, or the distance beyond which CPs are considered different. The user is strongly advised to experiment with the different parameters, in order to have some feeling about the robutness of his/her calculations against variation of these. Examples from the automatic tests should help him/her, as well as the associated comments in the corresponding README files.

Note that the AIM program can also determine the Bader distance for one given angular direction, or determine the density and laplacian at several, given, points in space, according to the user will.

To run the program one needs to prepare two files:

- [files-file] file which contains the name of the input file, the root for the names of the different output files, and the name of other data files.
- [input-file] file which gives the values of the input variables

Except these files you need the valence density in real space (*_DEN file, output of ABINIT) and the core density files (*.fc file, output of the FHI pseudopotential generator code, actually available from the ABINIT Web page)

The files file (called for example aim.files) could look like:

aim.in # input-file abo_DEN # valence density (output of ABINIT) aim # the root of the different output files at1.fc # core density files (in the same order as at2.fc # in the ABINIT files-file ) ...

About the _DEN file:

Usually, the grid in the real space for the valence density should be
finer than the one proposed by ABINIT.
(For example for the lattice parameter 7-8~a.u. , ngfft at least 64
gives the precision of the Bader charge estimated to be better
than 0.003~electrons).

About the core density files:

LDA core density files have been generated for the
whole periodic table, and are available on the ABINIT web site.
Since the densities are weakly dependent on the choice of the
XC functional, and moreover, the charge density analysis
is mostly a qualitative tool, these files can be used
for other functionals. Still, if really accurate
values for the Bader charge analysis are needed, one should
generate core density files with the same XC functional as
for the valence density.

The main executable file is called aim. Supposing that the "files" file is called aim.files, and that the executable is placed in your working directory, aim is run interactively (in Unix) with the command

*aim < aim.files >& log*

or, in the background, with the command

*aim < aim.files >& log &*

where standard out and standard error are piped
to the log file called "log"
(piping the standard error, thanks to the '&' sign placed after '>'
is **really important**
for the analysis of eventual failures, when not due
to AIM, but to other sources, like disk full problem ...).
The user can specify
any names he/she wishes for any of these files.
Variations of the above commands
could be needed,
depending on the flavor of UNIX that is used on the platform
that is considered for running the code.

The syntax of the input file is quite similar to the syntax of the main abinit input files : the file is parsed, keywords are identified, comments are also identified. However, the multidataset mode is not available.

Note that you have to specify what you want to calculate (default = nothing). An example of the simple input file for Oxygen in bulk MgO is given in ~ABINIT/Test_v3/t57.in . There are also corresponding output files in this directory.

Before giving the description of the input variables for the aim input file, we give some explanation concerning the output file.

The atomic units and cartesian coordinates are used for all output parameters. The names of the output files are of the form root+suffix. There are different output files:

- [*.out] - the central output file - there are many informations
which are clearly described.
Here is some additional information on the integration - there is separated
integrations of the core and the valence density.
In both cases the radial integration
is performed using cubic splines and the angular ones by Gauss
quadrature. However the principal part of the core density
**of the considered atom**is integrated in the sphere of minimal Bader radius using spherical symmetry. The rest of the core density (of this atom, out of this sphere) together with all the core contributions of the neighbors are added to the valence density integration. In the ouput file, there is the result of the**complete**integration of the core density of the atom, then the two contributions (spherical integration vs. the others) and then the total Bader charge. - [*.crit] - the file with the critical points (CPs) -
they are listed in the order Bond CPs (BCPs), Ring CPs (RCPs)
and Cage CPs (CCPs).
The line of the output contains these informations (in latex, sorry for this):
$$ position \quad \frac{eigen\ values}{of\ Hessian} \quad \frac{index\ of\ the}{bonded\ atom}\footnote{BCPs only} \quad \Delta \rho_c \quad \rho_c, $$

where position is done with respect to the considered atom. - [*.surf] - the file with the Bader surface - there is a head
of the form (again in latex):
\begin{tabular}{ccc} index of the atom & \multicolumn{2}{c}{position} \\ ntheta & thetamin & thetamax \\ nphi & phimin & phimax \\ \end{tabular}

and the list of the Bader surface radii:$$ \theta \quad \phi \quad r(\theta,\phi) \quad W(\theta,\phi)\footnote{the weight for the Gauss quadrature}. $$

The minimal and maximal radii are given at the last line. - [*.log] - the log file - a lot of informations but it is not very nice actually.
- [*.gp] - gnuplot script showing the calculated part of the Bader surface with lines.

load 'file'(quotes are necessary). Note, that this isn't considered as the visualization (it is only for working purpose)!

There are two groups of input variables : those that define the behaviour of the driver and those related to the numerics (all the others). Here is the list of the driver variables :

A standard determination of the electronic charge around one atom will use : Then, the user will have to specify the atom, thanks to atom, and the angular integration, typically with phimax, thetamax, nphi, ntheta.Finally, the following input variables might often need some tuning : lgrad2, lstep2, dpclim, maxatd, maxcpd.

In most cases, the other input variables will not have to be modified.

Atomic units and cartesian coordinates are used for all input parameters.

** Alphabetical list of AIM input variables.**

A.
atom
atrad

B.

C.
coff1
coff2
crit

D.
denout
dltyp
dpclim

E.

F.
foldep
follow
folstp

G.
gpsurf

H.

I.
inpt
irho
ivol

J.

K.

L.
lapout
lgrad
lgrad2
lstep
lstep2

M.
maxatd
maxcpd

N.
ngrid
nphi
nsa
nsb
nsc
ntheta

O.

P.
phimax
phimin

Q.

R.
radstp
ratmin
rsurf
rsurdir

S.
scal
surf

T.
thetamax
thetamin

U.

V.
vpts

W.

X.

Y.

Z.

atom

Characteristic: numerics

Variable type: integer

Default: 1

Index of the investigated atom.

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atrad

Characteristic: numerics

Variable type: real

Default: 1.0

A first estimation of the Bader radius (not too important - it is used only two times)

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coff1

Characteristic: numerics

Variable type: real

Default: 0.98

See the input variable ratmin.

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coff2

Characteristic: numerics

Variable type: real

Default: 0.95

See the input variable ratmin.

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crit

Characteristic: driver

Variable type: integer

Default: 0 (no computation of critical points)

Drives the computation of critical points.

- [0] not
- [-1] reading from the file ``root''.crit
- [1] calculated (simplified version)
- [2] calculated (standard version - recommended)
- [3] calculated (the original version)

The original version searches all critical points (CPs) starting from the center betwen two and three atoms (atom - neighbor(s)) by Newton-Raphson algorithm - without tests (not recommended) - don't use together with surface analysis !

The simplified and standard versions search CP(3,-1) starting
from the center of the pairs~atom-neighbor; then CP(3,1) from the
center between two CP(3,-1) and finally CP(3,3) from the center
between two CP(3,1). The robust Popeliers's algorithm is used.
The difference between the two is based in the fact
that the standard version makes the test if the CP is really
on the Bader surface of the calculated atom for each CP, while
the simplified version does this only for CP(3,-1). When CP analysis is
rather fast (with respect to surface determination), 2 is recommended.
In all cases the number of neighbors considered is limited by
distance cutoff (variable maxatd)

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** | ** Go to the AIM input variable list

denout

Characteristic: driver

Variable type: integer

Default: 0

Output of the electronic density. The specification of the line (plane) in the real space must be given in the input variable vpts and grid in ngrid. It is also possible to get only the valence density or the core density (see dltyp).

- 0, no output
- 1, 1D distribution
- 2, 2D distribution

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dltyp

Characteristic: driver

Variable type: integer

Default: 0

Specification of the contribution of the electronic density corresponding to the density and/or laplacian output (see denout and lapout)

- 0, total electronic density
- 1, only the valence density
- 2, only the core density

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dpclim

Characteristic: numerics

Variable type: real

Default: 1.d-2

If two "numerically different" critical points are separated by less than

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foldep

Characteristic: numerics

Variable type: real foldep(3)

Default: 0.0, 0.0, 0.0

Needed in the case follow=1 only. Defines the starting point.

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follow

Characteristic: driver

Variable type: integer

Default: 0

Follow the gradient path to the corresponding atom starting from the position specified in the input variable foldep.

- 0, not calculated
- 1, calculated

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folstp

Characteristic: numerics

Variable type: real

Default: 0.5

The first step for following the gradient path.

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gpsurf

Characteristic: driver

Variable type: integer

Default: 0

Drives the graphic output (gnuplot script) of the irreducible part of the calculated Bader surface.

- 0, not output
- 1, output

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inpt

Characteristic: numerics

Variable type: integer

Default: 100

Number of radial points used for integration of the Bader charge (not too sensitive).

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irho

Characteristic: driver

Variable type: integer

Default: 0

Drives the integration of the charge of the Bader atom.

- 0, not calculated
- 1, calculated (usual mode)

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ivol

Characteristic: driver

Variable type: integer

Default: 0

Drives the integration of the volume of the Bader atom.

- 0, not calculated
- 1, calculated

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lapout

Characteristic: driver

Variable type: integer

Default: 0

Output of the laplacian of electronic density. The specification of the line (plane) in the real space must be given in the input variable vpts and grid in ngrid. It is also possible to get only the valence density or the core density (see dltyp).

- 0, no output
- 1, 1D distribution
- 2, 2D distribution

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lgrad

Characteristic: numerics

Variable type: real

Default: 1.d-12

The search for one particular CP is decided to be succesfull when either the norm of the gradient of the electron density is smaller than

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lgrad2

Characteristic: numerics

Variable type: real

Default: 1.d-5

Determines the criterion for deciding that a CP has been found. See lgrad for more details.

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lstep

Characteristic: numerics

Variable type: real

Default: 1.d-10

Determines the criterion for deciding a CP has been found. See lgrad for more details.

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lstep2

Characteristic: numerics

Variable type: real

Default: 1.d-5

Determines the criterion for deciding that a CP has been found. See lgrad for more details.

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maxatd

Characteristic: numerics

Variable type: real

Default: 10.0

Atoms within this maximal distance are considered in order to start the search of a CP.

Note that the supercell,
determined by
nsa,
nsb, and
nsc might
be too small to actually lead to the consideration of all the
desired atoms.

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** | ** Go to the AIM input variable list

maxcpd

Characteristic: numerics

Variable type: real

Default: 5.0

The CPs are searched for within this maximal distance.

Note that the supercell,
determined by
nsa,
nsb, and
nsc might
be too small to actually lead to the consideration of all the
critical points.

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** | ** Go to the AIM input variable list

ngrid

Characteristic: numerics

Variable type: integer ngrid(2)

Default: 30,30

Defines the grid in real space, for the density and laplacian outputs, governed by denout and lapout.

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nphi

Characteristic: numerics

Variable type: integer

Default: 48

With ntheta, this variable defines the angular grid for the integration within the Bader volume, in particular, the number of phi angles, to be used between phimin and phimax. When the difference between these two variables is 2 pi, the recommended value of

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nsa, nsb, nsc

Characteristic: numerics

Variable type: integer

Default: 3

These variables define a "supercell", from the primitive cell repeated along each primitive direction. This supercell is build as follows :

do isa=-nsa,nsa do isb=-nsb,nsb do isc=-nsc,nsc -> here, the cell is translated by the vector -> (isa,isb,isc) in crystallographic coordinates -> and accumulated, to give the supercell enddo enddo enddo

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ntheta

Characteristic: numerics

Variable type: integer

Default: 32

With nphi, this variable defines the angular grid for the integration within the Bader volume, in particular, the number of theta angles, to be used between thetamin and thetamax. When the difference between these two variables is pi, the recommended value of

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phimin, phimax

Characteristic: numerics

Variable type: real

Default: 0.0 for phimin, 2 pi for phimax

Angular limits of integration of the Bader volume for the phi variables. The number of integration points is given by nphi. The range of integration can be decreased if there are symmetry reasons for doing this.

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radstp

Characteristic: numerics

Variable type: real

Default: 0.05

The length of the first step in the search of the exact Bader radius.

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ratmin

Characteristic: numerics

Variable type: real

Default: 1.0

The first estimation of the smallest radius of the bassin of the atom (the distance at which the procedure that follows the gradient path annonces that the gradient path finishes in the corresponding atom) This parameter is very important for the speed of the calculation, but this first estimation is not usually used because the program makes a new one based on the knowledge of CPs. In fact after the CP analysis, the new estimation is done by the product of the adhoc parameter coff1 (default 0.98) by the distance of the nearest bonding CP. If there is a problem later, coff2 (default 0.95) is used instead.

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rsurdir

Characteristic: numerics

Variable type: real rsurdir(2)

Default: 0.0, 0.0

In the case rsurf=1, gives the direction (angular coordinates theta,phi) along which the radius of the Bader surface is to be determined.

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rsurf

Characteristic: driver

Variable type: integer

Default: 0

Drive the computation of the radius of the Bader surface for the angles specified in the input variable rsurdir

- 0, not calculated
- 1, calculated

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scal

Characteristic: numerics

Variable type: real scal(3)

Default: 1.0, 1.0, 1.0

The scaling of the cartesian coordinates for the computation of the distances (x'[i]=x[i]/scal[i]). Not really useful.

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surf

Characteristic: driver

Variable type: integer

Default: 0

Drive the computation of the full Bader surface.

- 0, not calculated
- 1, calculated

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thetamin, thetamax

Characteristic: numerics

Variable type: real

Default: 0.0 for thetamin, pi for thetamax

Angular limits of integration of the Bader volume for the theta variables. The number of integration points is given by ntheta. The range of integration can be decreased if there are symmetry reasons for doing this.

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vpts

Characteristic: numerics

Variable type: real vpts(6) or vpts(9)

Default: all zero

Basic vectors of the line or rectangle in real space, defining the points for which the density or laplacian will be computed, thanks to denout or lapout

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