Algorithms for Managing and Optimizing Maximum Probability Domains

The optimization algorithm of the Domains in accordance with MPDs approach has been described in references [26,38]. An atomic or ELF basin, defined on a grid, or another domain chosen by the user (a sphere, an ellipsoid, a cube, or a previously obtained MPD) can be a first guess for the MPDs.

2.1.1 Calculation of Surface and Integration

First a regular cubic grid of pointsG¼ {gi} is established with a given increment dx. The classical value of dx is 0.05 bohrs. A volume dV¼ dx³is associated to the pointsgi.

Definition of Domains

A subsetD¼ {di} ofG is defined As constituted by grid points belonging to the domain. The number of the di points is nv. The volume of the domain is:

V1[D]¼ nvdV.

Definition ofD for AIM and ELF Basin

The scalar fields(x) (where s is the electronic density for AIM or ELF) is calculated for all the points {gi}¼ G. Afterwards, the set of attractors (maxima) is calculated, first through the use of the numerical gradient ofs(x) on the grid G, and next the positions of the attractors are refined using the analytical gradient ofs. The gipoints are attributed to each attractor using the numerical gradient. V1[D] is the most accurate method of integration of the volume for ELF and AIM basins.

2.1.2 Definition of D for MPD A first guessD can be defined by:

– Using aD set of points taken from AIM or ELF analysis.

– Searching for a set of grid points interior to a guess surface (sphere, ellipse, cube, or a previous MPD surface).

The setD of points diis maintained and redefined during the surface optimiza-tion for keeping the informaoptimiza-tion about the interior–exterior space of the surface.

2.1.3 Construction of the Surface

Given the set of domain of grid pointsD, a subset S¼ {si} ofD is defined as constituting points that have less than six neighbors belonging toD. This is the surface set of grid points. The setS is then made independent of G and D, because the position ofsimay be changed.

2.1.4 Elaboration of the Surface S

The setS is then multiply screened and smoothed (the positions of si are a little modified to set triangularization).

2.1.5 Triangularization

The set of pointsS is triangularized. A set of triangles T¼ {ti} having cornerssiis defined. The process is a non-Delaunay one to allow for concavities. The triangles are defining walking in the set of nearest neighborssifor obtaining locally the most regular triangles. The setB¼ {bi} of barycenter of the trianglestiis defined. The setN¼ {ni} of normal vectors to eachtiis defined, pointing toward the external space of the domain: the set of domain grid pointsD is used for defining the versor of ni. The information about the domain is then represented byD, B, and N.

2.1.6 Variation and Optimization of the Surface

The set B is changed during optimization, bi are moved along ni. The shape derivative is calculated in the points bi and helps defining the step and sign of variation of bialong ni.

For optimization, the barycenter of the triangles (of the surface S) is moved along the normals. The displacements are proportional to the shape derivatives computed at the barycenter [40]. They are larger at the start of the optimization, smaller towards the end. During the optimization process, certain regions of domain can collapse to a surface, or even points. These low-coordinated grid points are then eliminated.

2.1.7 Redefinition of the Surface

After a controlled number of variations ofB, the set B is no more coherent with D.

So the domain and the surface are redefined: a new set of grid pointsD¼ {di} is defined internally to the surface represented byB and N. Then, the sets S, T, B, and N are redefined on the base of the newly defined D.

2.1.8 Integration Method for Probability Evaluation

The overlap matrix S D½  ¼ Sf mng between AO m and n is calculated; smn is the overlap limited to the domainD, whereas Smnis the integral extended to all the space. A Becke atomic partition method is adopted for defining the quadrature set of pointsX¼ {xi} and the weightswi:

Smn¼ wmð Þwx_i nð Þwx_i i

smn¼ wmð Þwx_i nð Þwxi if xð Þ:i

f(xi) is the domain weight function for limiting the overlap toD,wmis them-th atomic orbital of the molecule model or of the cluster chosen as a representation of the periodic system in terms of a set of local Wannier functions. The volume of the domainD can be calculated as:

V2½  ¼D X

iwif xð Þ:i

The average number of electrons in the domainh i DN ½  can be calculated in two ways:

1.

h i DN ½  ¼X^ne

n¼0

n pn

wherepnare the probabilities of finding exactly n electrons in the domainD and n_e is the number of electrons in the molecule or cluster. The set of probabilitiesP¼ pf g is a functional of the domain overlap matrix S Dn ½ .

2.

h i DN ½  ¼X

i

wirðxiÞ

whererðxⁱÞ is the electronic density calculated in grid point xi.This method is used only for checking the accuracy ofpi.

There are two methods for calculating the domain functionf(xi)

1. Given the quadrature point xi, the closest point of the grid g_iis found:f(xi) is equal to 1 ifgibelongs to the set of domain pointsD; otherwise f(xi) is 0.

2. Given the quadrature point xi, the closest barycenterbiof the surface triangles is found. The scalar productps is calculated: ps¼ (xi bi)∙ni. A “Fermi type” step function is calculated:f xð Þ ¼i ₁Exp sf ps¹½  wheresf has been optimized to a value of 50 atomic units as a compromise between a sharp step and the accuracy of the integration accuracy.

2.1.9 The Adopted Integration Methods pnandhNi[D] for MP Domains

The probabilitiesP are calculated on the basis of S[D] evaluated by the method described in Sect. 2.1.8. The f(xi) domain function is calculated by method 2 described above. hNi[D] is calculated using P. The volume V of MPD is calculated using the methodV2[D], described in Sect.2.1.8.

AIM end ELF Analysis

The probabilitiesP are evaluated on the base of S[D] matrix, calculated using the method reported in Sect.2.1.8; in this case, thef(xi) domain function is calculated using method 1 described in Sect.2.1.8.hNi[D] is calculated using P. The volume V of AIM and ELF basins are computed using the methodV1[D] as reported in section

“Definition of Domains.”